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Research ArticleModeling and Chaotic Dynamics of the Laminated CompositePiezoelectric Rectangular Plate
Minghui Yao Wei Zhang and D M Wang
College of Mechanical Engineering Beijing University of Technology Beijing 100124 China
Correspondence should be addressed to Minghui Yao ymhbjuteducn
Received 12 October 2013 Accepted 27 December 2013 Published 2 March 2014
Academic Editor Rongni Yang
Copyright copy 2014 Minghui Yao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper investigates the multipulse heteroclinic bifurcations and chaotic dynamics of a laminated composite piezoelectricrectangular plate by using an extended Melnikov method in the resonant case According to the von Karman type equationsReddyrsquos third-order shear deformation plate theory andHamiltonrsquos principle the equations of motion are derived for the laminatedcomposite piezoelectric rectangular plate with combined parametric excitations and transverse excitation The method of multiplescales and Galerkinrsquos approach are applied to the partial differential governing equation Then the four-dimensional averagedequation is obtained for the case of 1 3 internal resonance and primary parametric resonance The extended Melnikov method isused to study the Shilnikov typemultipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectricrectangular plate The necessary conditions of the existence for the Shilnikov type multipulse chaotic dynamics are analyticallyobtained From the investigation the geometric structure of the multipulse orbits is described in the four-dimensional phasespace Numerical simulations show that the Shilnikov type multipulse chaotic motions can occur To sum up both theoreticaland numerical studies suggest that chaos for the Smale horseshoe sense in motion exists for the laminated composite piezoelectricrectangular plate
1 Introduction
The need for high-speed light-weight and energy-savingstructures in the aerospace and aviation industry has ledto the composite materials instead of traditional materi-als Additional requirements for multifunctionality activevibration shape control vibration suppression and acousticcontrol have made the development of smart and intelligentstructures A piezoelectric composite laminate is composedof piezoelectric layers which are embedded in laminatedcomposite structures or are boned on the surface of struc-turesThe direct and converse piezoelectric effects are used tosuppress the transient vibration and to control the deforma-tion shape and buckling of the structures Such lightweightflexible structures generate large deformations geometricalnonlinearity and structural instability when piezoelectriccomposite laminates are subjected to the coupling betweenthe mechanical and electrical loads Therefore it is neces-sary to study geometrically nonlinear effects on dynamiccharacteristics of structures in order to accurately design
and effectively control vibrations of piezoelectric compositelaminate structures It is very important to investigate thelarge amplitude nonlinear vibrations of smart structures withpiezoelectric materials in order to achieve and predict thedesired performance of the systems
Recently the studies on dynamics of composite structureswith piezoelectric materials have made some progress Tzouet al [1] used spatially distributed orthogonal piezoelectricactuators to perform the distributed structural control ofelastic shell They utilized a gain factor and a spatially dis-tributed mode actuator function to describe modal feedbackfunctions Purekar et al [2] presented phased array filterswith piezoelectric sensors to detect damage in isotropicplates and adopted wave propagation to describe platedynamics Ishihara and Noda [3] took into account theeffect of transverse shear to analyze the dynamic behaviorof the laminate composed of fiber-reinforced laminae andpiezoelectric layers constituting a symmetric cross-ply lam-inate rectangular plate with simply supported edges Oh [4]considered snap-through thermopiezo-elastic behaviors to
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 345072 19 pageshttpdxdoiorg1011552014345072
2 Mathematical Problems in Engineering
examine the buckling bifurcation and sling-shot buckling ofactive piezo-laminated plates Lee et al [5] employed third-order shear deformation theory and nonlinear finite elementto canvass deflection suppression characteristics of laminatedcomposite shell structures with smart material laminaePanda andRay [6] exploited the first-order shear deformationtheory and the three-dimensional finite element method todelve into the open-loop and closed-loop nonlinear dynamicsof functionally graded plates with the piezoelectric fiber-reinforced composite material under the thermal environ-ment Dumir et al [7] used the extendedHamiltonrsquos principleto derive the coupled nonlinear equations of motion and theboundary conditions for buckling and vibration of symmet-rically laminated hybrid angle-ply piezoelectric panels underin-plane electrothermomechanical loading Yao and Zhang[8] employed the third-order shear deformation plate theoryto explore the bifurcations and chaotic dynamics of the four-edge simply supported laminated composite piezoelectricrectangular plate in the case of the 1 2 internal resonances
The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefrontof nonlinear dynamics for the past two decades There aretwo ways of solutions on Shilnikov type chaotic dynamics ofhigh-dimensional nonlinear systems One is Shilnikov typesingle-pulse chaotic dynamics and the other is Shilnikovtype multipulse chaotic dynamics Most researchers focusedon Shilnikov type single-pulse chaotic dynamics of high-dimensional nonlinear systems Much research in this fieldhas concentrated on Shilnikov type single-pulse chaoticdynamics of thin plate structures Feng and Sethna [9]utilized the global perturbation method to study the globalbifurcations and chaotic dynamics of the thin plate underparametric excitation and obtained the conditions in whichthe Shilnikov type homoclinic orbits and chaos can occurTien et al [10] applied the Melnikov method to investigatethe global bifurcation and chaos for the Smale horseshoesense of a two-degree-of-freedom shallow arch subjectedto simple harmonic excitation for the case of 1 2 internalresonance Malhotra and Sri Namachchivaya [11] employedthe averaging method and Melnikov technique to canvassthe local global bifurcations and chaotic motions of a two-degree-of-freedom shallow arch subjected to simple har-monic excitation for the case of 1 1 internal resonance Theglobal bifurcations and chaotic dynamics were investigatedby Zhang [12] for the simply supported rectangular thinplates subjected to the parametrical-external excitation andthe parametrical excitation Yeo and Lee [13] made use ofthe global perturbation technique to examine the globaldynamics of an imperfect circular plate for the case of1 1 internal resonance and obtained the criteria for chaoticmotions of homoclinic orbits and heteroclinic orbits Yu andChen [14] adopted the global perturbationmethod to explorethe global bifurcations of a simply supported rectangularmetallic plate subjected to a transverse harmonic excitationfor the case of 1 1 internal resonance
While most of studies are on the Shilnikov typesingle-pulse global bifurcations and chaotic dynamics ofhigh-dimensional nonlinear systems there are researchersinvestigating the Shilnikov type multipulse homoclinic and
heteroclinic bifurcations and chaotic dynamics So far thereare two theories of the Shilnikov type multipulse chaoticdynamics One is the extended Melnikov method and theother theory is the energy phase method Much achievementis made in the former theory of high-dimensional nonlinearsystems In 1996 Kovacic and Wettergren [15] used a modi-fiedMelnikovmethod to investigate the existence of the mul-tipulse jumping of homoclinic orbits and chaotic dynamicsin resonantly forced coupled pendula Furthermore Kaperand Kovacic [16] studied the existence of several classes ofthe multibump orbits homoclinic to resonance bands forcompletely integral Hamiltonian systems subjected to smallamplitude Hamiltonian and damped perturbations Camassaet al [17] presented a new Melnikov method which is calledthe extended Melnikov method to explore the multipulsejumping of homoclinic and heteroclinic orbits in a class ofperturbed Hamiltonian systems Until recently Zhang andYao [18] introduced the extended Melnikov method to theengineering field They came up with a simplification of theextended Melnikov method in the resonant case and utilizedit to analyze the Shilnikov type multipulse homoclinic bifur-cations and chaotic dynamics for the nonlinear nonplanaroscillations of the cantilever beam
The study on the second theory of the Shilnikov typemul-tipulse chaotic dynamics was stated by Haller and Wiggins[19] They presented the energy phase method to investigatethe existence of the multipulse jumping homoclinic andheteroclinic orbits in perturbed Hamiltonian systems Upto now few researchers have made use of the energy phasemethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsMalhotra et al [20] used the energy-phase method to inves-tigate multipulse homoclinic orbits and chaotic dynamics forthe motion of flexible spinning discs Yu and Chen [21] madeuse of the energy-phase method to examine the Shilnikovtype multipulse homoclinic orbits of a harmonically excitedcircular plate
This paper focuses on the Shilnikov typemultipulse orbitsand chaotic dynamics for a simply supported laminatedcomposite piezoelectric rectangular plate under combinedparametric excitations and transverse load Based on thevon Karman type equations and Reddyrsquos third-order sheardeformation plate theory Hamiltonrsquos principle is employedto obtain the governing nonlinear equations of the lami-nated composite piezoelectric rectangular plate with com-bined parametric excitation and transverse load We applyGalerkinrsquos approach and the method of multiple scales to thepartial differential governing equations to obtain the four-dimensional averaged equation for the case of 1 3 internalresonance and primary parametric resonance From theaveraged equation the theory of normal form is used tofind the explicit formulas of normal form We study theheteroclinic bifurcations of the unperturbed system and thecharacteristic of the hyperbolic dynamics of the dissipativesystem respectively Finally we employ the extended Mel-nikov method to analyze the Shilnikov type multipulse orbitsand chaotic dynamics in the laminated composite piezoelec-tric plate In this paper the extended Melnikov function
Mathematical Problems in Engineering 3
x
a
y
b
qcosΩ3t
qxcosΩ1t
qycosΩ2t
Figure 1 The model of a laminated composite piezoelectric rectangular plate is given
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric plate under the case of 1 3 internal resonancesThe analysis indicates that there exist the Shilnikov typemultipulse jumping orbits in the perturbed phase space forthe averaged equations We present the geometric structureof the multipulse orbits in the four-dimensional phase spaceThe results from numerical simulation also show that thechaotic motion can occur in the motion of the laminatedcomposite piezoelectric plate which verifies the analyticalprediction The Shilnikov type multipulse orbits are discov-ered from the results of numerical simulation In summaryboth theoretical and numerical studies demonstrate thatchaos for the Smale horseshoe sense in themotion existsThispaper demonstrates how to employ the extended Melnikovmethod to analyze the Shilnikov type multipulse heteroclinicbifurcations and chaotic dynamics of high-dimensional non-linear systems in engineering applications
The laminated composite piezoelectric rectangular platesare widely applied in space stations satellite solar panelssensors and actuators for the active control of structuresand so on In this paper we have investigated the multipulseglobal bifurcations and chaotic dynamics of a laminated com-posite piezoelectric rectangular plate by using an extendedMelnikov method and numerical simulations in detail Wehave understood nonlinear vibration characteristics of alaminated composite piezoelectric rectangular plate Ourtheoretical results can be used to solve some engineeringproblems Since these smart structures are generally lightweight and relatively large structural flexibility laminatedcomposite piezoelectric rectangular plates can induce largevibration deformation during the rapid deployment In orderto eliminate or suppress large vibration and chaotic motiontheoretical results can help optimize the design of thestructural parameters of laminated composite piezoelectricrectangular plates Therefore the theoretical studies on themultipulse global bifurcations and chaotic dynamics of lam-inated composite piezoelectric rectangular plates play a very
important role in applications in aerospace and mechanicalengineering
2 Equations of Motion andPerturbation Analysis
We consider a four-edge simply supported laminated com-posite piezoelectric rectangular plate where the length thewidth and the thickness are denoted by 119886 119887 and ℎ respec-tively The laminated composite piezoelectric rectangularplate is subjected to in-plane excitation transverse excita-tion and piezoelectric excitation as shown in Figure 1 Weconsider the laminated composite piezoelectric rectangularplate as regular symmetric cross-ply laminates with 119899 layerswith respect to principal material coordinates alternativelyoriented at 0∘ and 90∘ to the laminated coordinate axes Someof layers are made of the piezoelectric materials as actuatorsand the other layers are made of fiber-reinforced compositematerials It is assumed that different layers of the symmetriccross-ply composite laminated piezoelectric rectangular plateare perfectly clung to each other and piezoelectric actuatorlayers are embedded in the plate The fiber direction ofodd-numbered layers is the 119909-direction of the laminate Thefiber direction of even-numbered layers is the 119910-directionof the laminate Simply supported plate with immovableedges satisfies the symmetry requirement that eliminatesthe coupling between bending and extension However thedisplacement of 119909 is free to move at the edge of 119910 = 0 andthe displacement of 119910 is free to move at the edge of 119909 = 0Therefore the membrane stress is smaller and there exists thecoupling between bending and extension A Cartesian coor-dinate system 119874119909119910119911 is located in the middle surface of thecomposite laminated piezoelectric rectangular plate Assumethat (119908 V 119906) and (119908
0 V0 1199060) describe the displacements of
an arbitrary point and a point in the middle surface of thecomposite laminated piezoelectric rectangular plate in the119909 119910 and 119911 directions respectively It is also assumed thatin-plane excitations of the composite laminated piezoelectricrectangular plate are loaded along the 119910-direction at 119909 = 0
4 Mathematical Problems in Engineering
and the 119909-direction at 119910 = 0 with the form of 1199020+ 119902119909cosΩ1119905
and 1199021+119902119910cosΩ2119905 respectively Transverse excitation loaded
to the composite laminated piezoelectric rectangular plate isexpressed as 119902 = 119902
3cosΩ3119905The dynamic electrical loading is
represented by 119864119911= 119864119911cosΩ4119905
Considering Reddyrsquos third-order shear deformationdescription of the displacement field we have
119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601
119909(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119909+
1205971199080
120597119909
)
(1a)
V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601
119910(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119910+
1205971199080
120597119910
)
(1b)
119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)
where (1199060 V0 1199080) are the deflection of a point on the middle
surface (119906 V 119908) are the displacement components along the(119909 119910 119911) coordinate directions and 120601
119909and 120601
119910represent the
rotation components of normal to the middle surface aboutthe 119910 and 119909 axes respectively
The nonlinear strain-displacement relations are assumedto have the following form
120576119909119909
=
120597119906
120597119909
+
1
2
(
120597119908
120597119909
)
2
120576119909119911
=
1
2
(
120597119906
120597119911
+
120597119908
120597119909
)
120576119909119910
=
1
2
(
120597119906
120597119910
+
120597V120597119909
+
120597119908
120597119909
120597119908
120597119910
)
120576119910119910
=
120597V120597119910
+
1
2
(
120597119908
120597119910
)
2
120576119910119911
=
1
2
(
120597V120597119911
+
120597119908
120597119910
)
120576119911119911
=
120597119908
120597119911
(2)
Stress constitutive relations are presented as follows
120590119894119895= 120590119904
119894119895119896119897
120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)
where120590119894119895and 120576119896119897denote themechanical stresses and strains in
extended vector notation 120590119904119894119895119896119897
represents the elastic stiffnesstensor 119864
119896stands for the electric field vector and 119890
119894119895is the
piezoelectric tensorAccording to Hamiltonrsquos principle the nonlinear gov-
erning equations of motion in terms of generalized dis-placements (119906
0 V0 1199080 120601119909 120601119910) for the composite laminated
piezoelectric rectangular plate are given in the previousstudies as follows [8]
11986011
1205972
1199060
1205971199092
+ 11986066
1205972
1199060
1205971199102
+ (11986012
+ 11986066)
1205972V0
120597119909120597119910
+ 11986011
1205971199080
120597119909
1205972
1199080
1205971199092
+ 11986066
1205971199080
120597119909
1205972
1199080
1205971199102
+ (11986012
+ 11986066)
1205971199080
120597119910
1205972
1199080
120597119909120597119910
= 11986800+ 1198691
120601119909minus 11988811198683
1205970
120597119909
(4a)
11986066
1205972V0
1205971199092
+ 11986022
1205972V0
1205971199102
+ (11986021
+ 11986066)
1205972
1199060
120597119909120597119910
+ 11986066
1205971199080
120597119910
1205972
1199080
1205971199092
+ 11986022
1205971199080
120597119910
1205972
1199080
1205971199102
+ (11986021
+ 11986066)
1205971199080
120597119909
1205972
1199080
120597119909120597119910
= 1198680V0+ 1198691
120601119910minus 11988811198683
1205970
120597119910
(4b)
11986066
1205971199080
120597119909
1205972
1199060
1205971199102
minus 119867221198882
1
1205974
1199080
1205971199104
+ 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
1205973
120601119910
1205971199101205971199092
+ 1198881(11986522
minus 119867221198881)
1205973
120601119910
1205971199103
minus 119867111198882
1
1205974
1199080
1205971199094
+ 11986011
1205971199080
120597119909
1205972
1199060
1205971199092
+ (119865441198882
2
minus 2119863441198882+ 11986044)
120597120601119910
120597119910
+ 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
120601119909
1205971199102
120597119909
minus 1198882
1
(11986721
+ 411986766
+ 11986712)
1205974
1199080
1205971199102
1205971199092
+ (11986044
minus 119873119875
119910
cos (Ω4119905) + 119865
441198882
2
minus 2119863441198882)
1205972
1199080
1205971199102
minus
120597119873119875
119910
120597119910
cos (Ω2119905)
1205971199080
120597119910
+ (11986021
+ 411986066
+ 11986012)
times
1205971199080
120597119909
1205971199080
120597119910
1205972
1199080
120597119910120597119909
+ 1198881(11986511
minus 119867111198881)
1205973
120601119909
1205971199093
+ (11986021
+ 11986066)
1205971199080
120597119910
1205972
1199060
120597119910120597119909
+ 11986021
1205971199060
120597119909
1205972
1199080
1205971199102
times 11986066
1205971199080
120597119910
1205972V0
1205971199092
+ 11986022
1205971199080
120597119910
1205972V0
1205971199102
Mathematical Problems in Engineering 5
+
1
2
(11986012
+ 211986066) (
1205971199080
120597119910
)
2
1205972
1199080
1205971199092
+ 11986022
1205972
1199080
1205971199102
120597V0
120597119910
+ (11986012
+ 11986066)
1205971199080
120597119909
1205972V0
120597119910120597119909
+
1
2
(11986021
+ 211986066)
1205972
1199080
1205971199102
times (
1205971199080
120597119909
)
2
+
3
2
11986011(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092
+ 11986011
1205972
1199080
1205971199092
1205971199060
120597119909
+ 11986012
1205972
1199080
1205971199092
120597V0
120597119910
+ 211986066
1205972
1199080
120597119910120597119909
120597V0
120597119909
+ 211986066
1205972
1199080
120597119910120597119909
1205971199060
120597119910
+
3
2
11986022(
1205971199080
120597119910
)
2
1205972
1199080
1205971199102
+ (11986055
+ 119902119909cos (Ω
1119905) minus 119873
119875
119909
cos (Ω4119905)
+ 119865551198882
2
minus 2119863551198882)
1205972
1199080
1205971199092
minus
120597119873119875
119909
120597119909
cos (Ω3119905)
120597119908
120597119909
+ (119865551198882
2
minus 2119863551198882+ 11986055)
120597120601119909
120597119909
minus 119902 cos (Ω3119905) + 119896119891
1205971199080
120597119905
= 11986800minus 1198882
1
1198686(
1205972
0
1205971199092
+
1205972
0
1205971199102
)
+ 11988811198683(
1205970
120597119909
+
120597V0
120597119910
) + 11988811198694(
120597
120601119909
120597119909
+
120597
120601119910
120597119910
)
(4c)
(11986311
minus 2119865111198881+ 119867111198882
1
)
1205972
120601119909
1205971199092
+ (11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119909
1205971199102
minus 1198881(11986511
minus 119867111198881)
1205973
1199080
1205971199093
minus (119865551198882
2
minus 2119863551198882+ 11986055)
1205971199080
120597119909
+ (11986312
+ 11986366
+ 119867661198882
1
minus 2119865661198881+ 119867121198882
1
minus 2119865121198881)
times
1205972
120601119910
120597119910120597119909
minus 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
times
1205973
1199080
1205971199102
120597119909
+ (2119863551198882minus 11986055
minus 119865551198882
2
) 120601119909
= 11986910+ 1198702
120601119909minus 11988811198694
1205970
120597119909
(4d)
(11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119910
1205971199092
minus 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
1199080
1205971199101205971199092
+ (119867211198882
1
+ 11986366
+ 11986321
minus 2119865211198881+ 119867661198882
1
minus 2119865661198881)
times
1205972
120601119909
120597119910120597119909
+ (119867221198882
1
+ 11986322
minus 2119865221198881)
1205972
120601119910
1205971199102
minus 1198881(11986522
minus 119867221198881)
1205973
1199080
1205971199103
minus (119865441198882
2
minus 2119863441198882+ 11986044)
times
1205971199080
120597119910
+ (2119863441198882minus 119865441198882
2
minus 11986044) 120601119910
= 1198691V0+ 1198702
120601119910minus 11988811198694
1205970
120597119910
(4e)
The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]
119909 = 0 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5a)
119909 = 119886 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5b)
119910 = 0 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5c)
119910 = 119887 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5d)
int
ℎ
0
119873119909119909
1003816100381610038161003816119909=0
119889119911 = minusint
ℎ
0
(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)
int
ℎ
0
119873119910119910
10038161003816100381610038161003816119910=0
119889119911 = minusint
ℎ
0
(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)
The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906
0 V0 1199080 120601119909 and 120601
119910 It is desirable
that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906
0 V0 1199080 120601119909 and
120601119910in the following forms
1199060= 11990601(119905) cos 120587119909
2119886
cos120587119910
2119887
+ 11990602(119905) cos 3120587119909
2119886
cos120587119910
2119887
(6a)
V0= V1(119905) cos
120587119910
2119887
cos 1205871199092119886
+ V2(119905) cos
120587119910
2119887
cos 31205871199092119886
(6b)
1199080= 1199081(119905) sin 120587119909
119886
sin120587119910
119887
+ 1199082(119905) sin 3120587119909
119886
sin120587119910
119887
(6c)
120601119909= 1206011(119905) cos 120587119909
119886
sin120587119910
119887
+ 1206012(119905) cos 3120587119909
119886
sin120587119910
119887
(6d)
120601119910= 1206013(119905) cos
120587119910
119887
sin 120587119909
119886
+ 1206014(119905) cos
120587119910
119887
sin 3120587119909
119886
(6e)
Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
examine the buckling bifurcation and sling-shot buckling ofactive piezo-laminated plates Lee et al [5] employed third-order shear deformation theory and nonlinear finite elementto canvass deflection suppression characteristics of laminatedcomposite shell structures with smart material laminaePanda andRay [6] exploited the first-order shear deformationtheory and the three-dimensional finite element method todelve into the open-loop and closed-loop nonlinear dynamicsof functionally graded plates with the piezoelectric fiber-reinforced composite material under the thermal environ-ment Dumir et al [7] used the extendedHamiltonrsquos principleto derive the coupled nonlinear equations of motion and theboundary conditions for buckling and vibration of symmet-rically laminated hybrid angle-ply piezoelectric panels underin-plane electrothermomechanical loading Yao and Zhang[8] employed the third-order shear deformation plate theoryto explore the bifurcations and chaotic dynamics of the four-edge simply supported laminated composite piezoelectricrectangular plate in the case of the 1 2 internal resonances
The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefrontof nonlinear dynamics for the past two decades There aretwo ways of solutions on Shilnikov type chaotic dynamics ofhigh-dimensional nonlinear systems One is Shilnikov typesingle-pulse chaotic dynamics and the other is Shilnikovtype multipulse chaotic dynamics Most researchers focusedon Shilnikov type single-pulse chaotic dynamics of high-dimensional nonlinear systems Much research in this fieldhas concentrated on Shilnikov type single-pulse chaoticdynamics of thin plate structures Feng and Sethna [9]utilized the global perturbation method to study the globalbifurcations and chaotic dynamics of the thin plate underparametric excitation and obtained the conditions in whichthe Shilnikov type homoclinic orbits and chaos can occurTien et al [10] applied the Melnikov method to investigatethe global bifurcation and chaos for the Smale horseshoesense of a two-degree-of-freedom shallow arch subjectedto simple harmonic excitation for the case of 1 2 internalresonance Malhotra and Sri Namachchivaya [11] employedthe averaging method and Melnikov technique to canvassthe local global bifurcations and chaotic motions of a two-degree-of-freedom shallow arch subjected to simple har-monic excitation for the case of 1 1 internal resonance Theglobal bifurcations and chaotic dynamics were investigatedby Zhang [12] for the simply supported rectangular thinplates subjected to the parametrical-external excitation andthe parametrical excitation Yeo and Lee [13] made use ofthe global perturbation technique to examine the globaldynamics of an imperfect circular plate for the case of1 1 internal resonance and obtained the criteria for chaoticmotions of homoclinic orbits and heteroclinic orbits Yu andChen [14] adopted the global perturbationmethod to explorethe global bifurcations of a simply supported rectangularmetallic plate subjected to a transverse harmonic excitationfor the case of 1 1 internal resonance
While most of studies are on the Shilnikov typesingle-pulse global bifurcations and chaotic dynamics ofhigh-dimensional nonlinear systems there are researchersinvestigating the Shilnikov type multipulse homoclinic and
heteroclinic bifurcations and chaotic dynamics So far thereare two theories of the Shilnikov type multipulse chaoticdynamics One is the extended Melnikov method and theother theory is the energy phase method Much achievementis made in the former theory of high-dimensional nonlinearsystems In 1996 Kovacic and Wettergren [15] used a modi-fiedMelnikovmethod to investigate the existence of the mul-tipulse jumping of homoclinic orbits and chaotic dynamicsin resonantly forced coupled pendula Furthermore Kaperand Kovacic [16] studied the existence of several classes ofthe multibump orbits homoclinic to resonance bands forcompletely integral Hamiltonian systems subjected to smallamplitude Hamiltonian and damped perturbations Camassaet al [17] presented a new Melnikov method which is calledthe extended Melnikov method to explore the multipulsejumping of homoclinic and heteroclinic orbits in a class ofperturbed Hamiltonian systems Until recently Zhang andYao [18] introduced the extended Melnikov method to theengineering field They came up with a simplification of theextended Melnikov method in the resonant case and utilizedit to analyze the Shilnikov type multipulse homoclinic bifur-cations and chaotic dynamics for the nonlinear nonplanaroscillations of the cantilever beam
The study on the second theory of the Shilnikov typemul-tipulse chaotic dynamics was stated by Haller and Wiggins[19] They presented the energy phase method to investigatethe existence of the multipulse jumping homoclinic andheteroclinic orbits in perturbed Hamiltonian systems Upto now few researchers have made use of the energy phasemethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsMalhotra et al [20] used the energy-phase method to inves-tigate multipulse homoclinic orbits and chaotic dynamics forthe motion of flexible spinning discs Yu and Chen [21] madeuse of the energy-phase method to examine the Shilnikovtype multipulse homoclinic orbits of a harmonically excitedcircular plate
This paper focuses on the Shilnikov typemultipulse orbitsand chaotic dynamics for a simply supported laminatedcomposite piezoelectric rectangular plate under combinedparametric excitations and transverse load Based on thevon Karman type equations and Reddyrsquos third-order sheardeformation plate theory Hamiltonrsquos principle is employedto obtain the governing nonlinear equations of the lami-nated composite piezoelectric rectangular plate with com-bined parametric excitation and transverse load We applyGalerkinrsquos approach and the method of multiple scales to thepartial differential governing equations to obtain the four-dimensional averaged equation for the case of 1 3 internalresonance and primary parametric resonance From theaveraged equation the theory of normal form is used tofind the explicit formulas of normal form We study theheteroclinic bifurcations of the unperturbed system and thecharacteristic of the hyperbolic dynamics of the dissipativesystem respectively Finally we employ the extended Mel-nikov method to analyze the Shilnikov type multipulse orbitsand chaotic dynamics in the laminated composite piezoelec-tric plate In this paper the extended Melnikov function
Mathematical Problems in Engineering 3
x
a
y
b
qcosΩ3t
qxcosΩ1t
qycosΩ2t
Figure 1 The model of a laminated composite piezoelectric rectangular plate is given
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric plate under the case of 1 3 internal resonancesThe analysis indicates that there exist the Shilnikov typemultipulse jumping orbits in the perturbed phase space forthe averaged equations We present the geometric structureof the multipulse orbits in the four-dimensional phase spaceThe results from numerical simulation also show that thechaotic motion can occur in the motion of the laminatedcomposite piezoelectric plate which verifies the analyticalprediction The Shilnikov type multipulse orbits are discov-ered from the results of numerical simulation In summaryboth theoretical and numerical studies demonstrate thatchaos for the Smale horseshoe sense in themotion existsThispaper demonstrates how to employ the extended Melnikovmethod to analyze the Shilnikov type multipulse heteroclinicbifurcations and chaotic dynamics of high-dimensional non-linear systems in engineering applications
The laminated composite piezoelectric rectangular platesare widely applied in space stations satellite solar panelssensors and actuators for the active control of structuresand so on In this paper we have investigated the multipulseglobal bifurcations and chaotic dynamics of a laminated com-posite piezoelectric rectangular plate by using an extendedMelnikov method and numerical simulations in detail Wehave understood nonlinear vibration characteristics of alaminated composite piezoelectric rectangular plate Ourtheoretical results can be used to solve some engineeringproblems Since these smart structures are generally lightweight and relatively large structural flexibility laminatedcomposite piezoelectric rectangular plates can induce largevibration deformation during the rapid deployment In orderto eliminate or suppress large vibration and chaotic motiontheoretical results can help optimize the design of thestructural parameters of laminated composite piezoelectricrectangular plates Therefore the theoretical studies on themultipulse global bifurcations and chaotic dynamics of lam-inated composite piezoelectric rectangular plates play a very
important role in applications in aerospace and mechanicalengineering
2 Equations of Motion andPerturbation Analysis
We consider a four-edge simply supported laminated com-posite piezoelectric rectangular plate where the length thewidth and the thickness are denoted by 119886 119887 and ℎ respec-tively The laminated composite piezoelectric rectangularplate is subjected to in-plane excitation transverse excita-tion and piezoelectric excitation as shown in Figure 1 Weconsider the laminated composite piezoelectric rectangularplate as regular symmetric cross-ply laminates with 119899 layerswith respect to principal material coordinates alternativelyoriented at 0∘ and 90∘ to the laminated coordinate axes Someof layers are made of the piezoelectric materials as actuatorsand the other layers are made of fiber-reinforced compositematerials It is assumed that different layers of the symmetriccross-ply composite laminated piezoelectric rectangular plateare perfectly clung to each other and piezoelectric actuatorlayers are embedded in the plate The fiber direction ofodd-numbered layers is the 119909-direction of the laminate Thefiber direction of even-numbered layers is the 119910-directionof the laminate Simply supported plate with immovableedges satisfies the symmetry requirement that eliminatesthe coupling between bending and extension However thedisplacement of 119909 is free to move at the edge of 119910 = 0 andthe displacement of 119910 is free to move at the edge of 119909 = 0Therefore the membrane stress is smaller and there exists thecoupling between bending and extension A Cartesian coor-dinate system 119874119909119910119911 is located in the middle surface of thecomposite laminated piezoelectric rectangular plate Assumethat (119908 V 119906) and (119908
0 V0 1199060) describe the displacements of
an arbitrary point and a point in the middle surface of thecomposite laminated piezoelectric rectangular plate in the119909 119910 and 119911 directions respectively It is also assumed thatin-plane excitations of the composite laminated piezoelectricrectangular plate are loaded along the 119910-direction at 119909 = 0
4 Mathematical Problems in Engineering
and the 119909-direction at 119910 = 0 with the form of 1199020+ 119902119909cosΩ1119905
and 1199021+119902119910cosΩ2119905 respectively Transverse excitation loaded
to the composite laminated piezoelectric rectangular plate isexpressed as 119902 = 119902
3cosΩ3119905The dynamic electrical loading is
represented by 119864119911= 119864119911cosΩ4119905
Considering Reddyrsquos third-order shear deformationdescription of the displacement field we have
119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601
119909(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119909+
1205971199080
120597119909
)
(1a)
V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601
119910(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119910+
1205971199080
120597119910
)
(1b)
119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)
where (1199060 V0 1199080) are the deflection of a point on the middle
surface (119906 V 119908) are the displacement components along the(119909 119910 119911) coordinate directions and 120601
119909and 120601
119910represent the
rotation components of normal to the middle surface aboutthe 119910 and 119909 axes respectively
The nonlinear strain-displacement relations are assumedto have the following form
120576119909119909
=
120597119906
120597119909
+
1
2
(
120597119908
120597119909
)
2
120576119909119911
=
1
2
(
120597119906
120597119911
+
120597119908
120597119909
)
120576119909119910
=
1
2
(
120597119906
120597119910
+
120597V120597119909
+
120597119908
120597119909
120597119908
120597119910
)
120576119910119910
=
120597V120597119910
+
1
2
(
120597119908
120597119910
)
2
120576119910119911
=
1
2
(
120597V120597119911
+
120597119908
120597119910
)
120576119911119911
=
120597119908
120597119911
(2)
Stress constitutive relations are presented as follows
120590119894119895= 120590119904
119894119895119896119897
120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)
where120590119894119895and 120576119896119897denote themechanical stresses and strains in
extended vector notation 120590119904119894119895119896119897
represents the elastic stiffnesstensor 119864
119896stands for the electric field vector and 119890
119894119895is the
piezoelectric tensorAccording to Hamiltonrsquos principle the nonlinear gov-
erning equations of motion in terms of generalized dis-placements (119906
0 V0 1199080 120601119909 120601119910) for the composite laminated
piezoelectric rectangular plate are given in the previousstudies as follows [8]
11986011
1205972
1199060
1205971199092
+ 11986066
1205972
1199060
1205971199102
+ (11986012
+ 11986066)
1205972V0
120597119909120597119910
+ 11986011
1205971199080
120597119909
1205972
1199080
1205971199092
+ 11986066
1205971199080
120597119909
1205972
1199080
1205971199102
+ (11986012
+ 11986066)
1205971199080
120597119910
1205972
1199080
120597119909120597119910
= 11986800+ 1198691
120601119909minus 11988811198683
1205970
120597119909
(4a)
11986066
1205972V0
1205971199092
+ 11986022
1205972V0
1205971199102
+ (11986021
+ 11986066)
1205972
1199060
120597119909120597119910
+ 11986066
1205971199080
120597119910
1205972
1199080
1205971199092
+ 11986022
1205971199080
120597119910
1205972
1199080
1205971199102
+ (11986021
+ 11986066)
1205971199080
120597119909
1205972
1199080
120597119909120597119910
= 1198680V0+ 1198691
120601119910minus 11988811198683
1205970
120597119910
(4b)
11986066
1205971199080
120597119909
1205972
1199060
1205971199102
minus 119867221198882
1
1205974
1199080
1205971199104
+ 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
1205973
120601119910
1205971199101205971199092
+ 1198881(11986522
minus 119867221198881)
1205973
120601119910
1205971199103
minus 119867111198882
1
1205974
1199080
1205971199094
+ 11986011
1205971199080
120597119909
1205972
1199060
1205971199092
+ (119865441198882
2
minus 2119863441198882+ 11986044)
120597120601119910
120597119910
+ 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
120601119909
1205971199102
120597119909
minus 1198882
1
(11986721
+ 411986766
+ 11986712)
1205974
1199080
1205971199102
1205971199092
+ (11986044
minus 119873119875
119910
cos (Ω4119905) + 119865
441198882
2
minus 2119863441198882)
1205972
1199080
1205971199102
minus
120597119873119875
119910
120597119910
cos (Ω2119905)
1205971199080
120597119910
+ (11986021
+ 411986066
+ 11986012)
times
1205971199080
120597119909
1205971199080
120597119910
1205972
1199080
120597119910120597119909
+ 1198881(11986511
minus 119867111198881)
1205973
120601119909
1205971199093
+ (11986021
+ 11986066)
1205971199080
120597119910
1205972
1199060
120597119910120597119909
+ 11986021
1205971199060
120597119909
1205972
1199080
1205971199102
times 11986066
1205971199080
120597119910
1205972V0
1205971199092
+ 11986022
1205971199080
120597119910
1205972V0
1205971199102
Mathematical Problems in Engineering 5
+
1
2
(11986012
+ 211986066) (
1205971199080
120597119910
)
2
1205972
1199080
1205971199092
+ 11986022
1205972
1199080
1205971199102
120597V0
120597119910
+ (11986012
+ 11986066)
1205971199080
120597119909
1205972V0
120597119910120597119909
+
1
2
(11986021
+ 211986066)
1205972
1199080
1205971199102
times (
1205971199080
120597119909
)
2
+
3
2
11986011(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092
+ 11986011
1205972
1199080
1205971199092
1205971199060
120597119909
+ 11986012
1205972
1199080
1205971199092
120597V0
120597119910
+ 211986066
1205972
1199080
120597119910120597119909
120597V0
120597119909
+ 211986066
1205972
1199080
120597119910120597119909
1205971199060
120597119910
+
3
2
11986022(
1205971199080
120597119910
)
2
1205972
1199080
1205971199102
+ (11986055
+ 119902119909cos (Ω
1119905) minus 119873
119875
119909
cos (Ω4119905)
+ 119865551198882
2
minus 2119863551198882)
1205972
1199080
1205971199092
minus
120597119873119875
119909
120597119909
cos (Ω3119905)
120597119908
120597119909
+ (119865551198882
2
minus 2119863551198882+ 11986055)
120597120601119909
120597119909
minus 119902 cos (Ω3119905) + 119896119891
1205971199080
120597119905
= 11986800minus 1198882
1
1198686(
1205972
0
1205971199092
+
1205972
0
1205971199102
)
+ 11988811198683(
1205970
120597119909
+
120597V0
120597119910
) + 11988811198694(
120597
120601119909
120597119909
+
120597
120601119910
120597119910
)
(4c)
(11986311
minus 2119865111198881+ 119867111198882
1
)
1205972
120601119909
1205971199092
+ (11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119909
1205971199102
minus 1198881(11986511
minus 119867111198881)
1205973
1199080
1205971199093
minus (119865551198882
2
minus 2119863551198882+ 11986055)
1205971199080
120597119909
+ (11986312
+ 11986366
+ 119867661198882
1
minus 2119865661198881+ 119867121198882
1
minus 2119865121198881)
times
1205972
120601119910
120597119910120597119909
minus 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
times
1205973
1199080
1205971199102
120597119909
+ (2119863551198882minus 11986055
minus 119865551198882
2
) 120601119909
= 11986910+ 1198702
120601119909minus 11988811198694
1205970
120597119909
(4d)
(11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119910
1205971199092
minus 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
1199080
1205971199101205971199092
+ (119867211198882
1
+ 11986366
+ 11986321
minus 2119865211198881+ 119867661198882
1
minus 2119865661198881)
times
1205972
120601119909
120597119910120597119909
+ (119867221198882
1
+ 11986322
minus 2119865221198881)
1205972
120601119910
1205971199102
minus 1198881(11986522
minus 119867221198881)
1205973
1199080
1205971199103
minus (119865441198882
2
minus 2119863441198882+ 11986044)
times
1205971199080
120597119910
+ (2119863441198882minus 119865441198882
2
minus 11986044) 120601119910
= 1198691V0+ 1198702
120601119910minus 11988811198694
1205970
120597119910
(4e)
The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]
119909 = 0 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5a)
119909 = 119886 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5b)
119910 = 0 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5c)
119910 = 119887 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5d)
int
ℎ
0
119873119909119909
1003816100381610038161003816119909=0
119889119911 = minusint
ℎ
0
(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)
int
ℎ
0
119873119910119910
10038161003816100381610038161003816119910=0
119889119911 = minusint
ℎ
0
(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)
The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906
0 V0 1199080 120601119909 and 120601
119910 It is desirable
that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906
0 V0 1199080 120601119909 and
120601119910in the following forms
1199060= 11990601(119905) cos 120587119909
2119886
cos120587119910
2119887
+ 11990602(119905) cos 3120587119909
2119886
cos120587119910
2119887
(6a)
V0= V1(119905) cos
120587119910
2119887
cos 1205871199092119886
+ V2(119905) cos
120587119910
2119887
cos 31205871199092119886
(6b)
1199080= 1199081(119905) sin 120587119909
119886
sin120587119910
119887
+ 1199082(119905) sin 3120587119909
119886
sin120587119910
119887
(6c)
120601119909= 1206011(119905) cos 120587119909
119886
sin120587119910
119887
+ 1206012(119905) cos 3120587119909
119886
sin120587119910
119887
(6d)
120601119910= 1206013(119905) cos
120587119910
119887
sin 120587119909
119886
+ 1206014(119905) cos
120587119910
119887
sin 3120587119909
119886
(6e)
Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
x
a
y
b
qcosΩ3t
qxcosΩ1t
qycosΩ2t
Figure 1 The model of a laminated composite piezoelectric rectangular plate is given
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric plate under the case of 1 3 internal resonancesThe analysis indicates that there exist the Shilnikov typemultipulse jumping orbits in the perturbed phase space forthe averaged equations We present the geometric structureof the multipulse orbits in the four-dimensional phase spaceThe results from numerical simulation also show that thechaotic motion can occur in the motion of the laminatedcomposite piezoelectric plate which verifies the analyticalprediction The Shilnikov type multipulse orbits are discov-ered from the results of numerical simulation In summaryboth theoretical and numerical studies demonstrate thatchaos for the Smale horseshoe sense in themotion existsThispaper demonstrates how to employ the extended Melnikovmethod to analyze the Shilnikov type multipulse heteroclinicbifurcations and chaotic dynamics of high-dimensional non-linear systems in engineering applications
The laminated composite piezoelectric rectangular platesare widely applied in space stations satellite solar panelssensors and actuators for the active control of structuresand so on In this paper we have investigated the multipulseglobal bifurcations and chaotic dynamics of a laminated com-posite piezoelectric rectangular plate by using an extendedMelnikov method and numerical simulations in detail Wehave understood nonlinear vibration characteristics of alaminated composite piezoelectric rectangular plate Ourtheoretical results can be used to solve some engineeringproblems Since these smart structures are generally lightweight and relatively large structural flexibility laminatedcomposite piezoelectric rectangular plates can induce largevibration deformation during the rapid deployment In orderto eliminate or suppress large vibration and chaotic motiontheoretical results can help optimize the design of thestructural parameters of laminated composite piezoelectricrectangular plates Therefore the theoretical studies on themultipulse global bifurcations and chaotic dynamics of lam-inated composite piezoelectric rectangular plates play a very
important role in applications in aerospace and mechanicalengineering
2 Equations of Motion andPerturbation Analysis
We consider a four-edge simply supported laminated com-posite piezoelectric rectangular plate where the length thewidth and the thickness are denoted by 119886 119887 and ℎ respec-tively The laminated composite piezoelectric rectangularplate is subjected to in-plane excitation transverse excita-tion and piezoelectric excitation as shown in Figure 1 Weconsider the laminated composite piezoelectric rectangularplate as regular symmetric cross-ply laminates with 119899 layerswith respect to principal material coordinates alternativelyoriented at 0∘ and 90∘ to the laminated coordinate axes Someof layers are made of the piezoelectric materials as actuatorsand the other layers are made of fiber-reinforced compositematerials It is assumed that different layers of the symmetriccross-ply composite laminated piezoelectric rectangular plateare perfectly clung to each other and piezoelectric actuatorlayers are embedded in the plate The fiber direction ofodd-numbered layers is the 119909-direction of the laminate Thefiber direction of even-numbered layers is the 119910-directionof the laminate Simply supported plate with immovableedges satisfies the symmetry requirement that eliminatesthe coupling between bending and extension However thedisplacement of 119909 is free to move at the edge of 119910 = 0 andthe displacement of 119910 is free to move at the edge of 119909 = 0Therefore the membrane stress is smaller and there exists thecoupling between bending and extension A Cartesian coor-dinate system 119874119909119910119911 is located in the middle surface of thecomposite laminated piezoelectric rectangular plate Assumethat (119908 V 119906) and (119908
0 V0 1199060) describe the displacements of
an arbitrary point and a point in the middle surface of thecomposite laminated piezoelectric rectangular plate in the119909 119910 and 119911 directions respectively It is also assumed thatin-plane excitations of the composite laminated piezoelectricrectangular plate are loaded along the 119910-direction at 119909 = 0
4 Mathematical Problems in Engineering
and the 119909-direction at 119910 = 0 with the form of 1199020+ 119902119909cosΩ1119905
and 1199021+119902119910cosΩ2119905 respectively Transverse excitation loaded
to the composite laminated piezoelectric rectangular plate isexpressed as 119902 = 119902
3cosΩ3119905The dynamic electrical loading is
represented by 119864119911= 119864119911cosΩ4119905
Considering Reddyrsquos third-order shear deformationdescription of the displacement field we have
119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601
119909(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119909+
1205971199080
120597119909
)
(1a)
V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601
119910(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119910+
1205971199080
120597119910
)
(1b)
119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)
where (1199060 V0 1199080) are the deflection of a point on the middle
surface (119906 V 119908) are the displacement components along the(119909 119910 119911) coordinate directions and 120601
119909and 120601
119910represent the
rotation components of normal to the middle surface aboutthe 119910 and 119909 axes respectively
The nonlinear strain-displacement relations are assumedto have the following form
120576119909119909
=
120597119906
120597119909
+
1
2
(
120597119908
120597119909
)
2
120576119909119911
=
1
2
(
120597119906
120597119911
+
120597119908
120597119909
)
120576119909119910
=
1
2
(
120597119906
120597119910
+
120597V120597119909
+
120597119908
120597119909
120597119908
120597119910
)
120576119910119910
=
120597V120597119910
+
1
2
(
120597119908
120597119910
)
2
120576119910119911
=
1
2
(
120597V120597119911
+
120597119908
120597119910
)
120576119911119911
=
120597119908
120597119911
(2)
Stress constitutive relations are presented as follows
120590119894119895= 120590119904
119894119895119896119897
120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)
where120590119894119895and 120576119896119897denote themechanical stresses and strains in
extended vector notation 120590119904119894119895119896119897
represents the elastic stiffnesstensor 119864
119896stands for the electric field vector and 119890
119894119895is the
piezoelectric tensorAccording to Hamiltonrsquos principle the nonlinear gov-
erning equations of motion in terms of generalized dis-placements (119906
0 V0 1199080 120601119909 120601119910) for the composite laminated
piezoelectric rectangular plate are given in the previousstudies as follows [8]
11986011
1205972
1199060
1205971199092
+ 11986066
1205972
1199060
1205971199102
+ (11986012
+ 11986066)
1205972V0
120597119909120597119910
+ 11986011
1205971199080
120597119909
1205972
1199080
1205971199092
+ 11986066
1205971199080
120597119909
1205972
1199080
1205971199102
+ (11986012
+ 11986066)
1205971199080
120597119910
1205972
1199080
120597119909120597119910
= 11986800+ 1198691
120601119909minus 11988811198683
1205970
120597119909
(4a)
11986066
1205972V0
1205971199092
+ 11986022
1205972V0
1205971199102
+ (11986021
+ 11986066)
1205972
1199060
120597119909120597119910
+ 11986066
1205971199080
120597119910
1205972
1199080
1205971199092
+ 11986022
1205971199080
120597119910
1205972
1199080
1205971199102
+ (11986021
+ 11986066)
1205971199080
120597119909
1205972
1199080
120597119909120597119910
= 1198680V0+ 1198691
120601119910minus 11988811198683
1205970
120597119910
(4b)
11986066
1205971199080
120597119909
1205972
1199060
1205971199102
minus 119867221198882
1
1205974
1199080
1205971199104
+ 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
1205973
120601119910
1205971199101205971199092
+ 1198881(11986522
minus 119867221198881)
1205973
120601119910
1205971199103
minus 119867111198882
1
1205974
1199080
1205971199094
+ 11986011
1205971199080
120597119909
1205972
1199060
1205971199092
+ (119865441198882
2
minus 2119863441198882+ 11986044)
120597120601119910
120597119910
+ 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
120601119909
1205971199102
120597119909
minus 1198882
1
(11986721
+ 411986766
+ 11986712)
1205974
1199080
1205971199102
1205971199092
+ (11986044
minus 119873119875
119910
cos (Ω4119905) + 119865
441198882
2
minus 2119863441198882)
1205972
1199080
1205971199102
minus
120597119873119875
119910
120597119910
cos (Ω2119905)
1205971199080
120597119910
+ (11986021
+ 411986066
+ 11986012)
times
1205971199080
120597119909
1205971199080
120597119910
1205972
1199080
120597119910120597119909
+ 1198881(11986511
minus 119867111198881)
1205973
120601119909
1205971199093
+ (11986021
+ 11986066)
1205971199080
120597119910
1205972
1199060
120597119910120597119909
+ 11986021
1205971199060
120597119909
1205972
1199080
1205971199102
times 11986066
1205971199080
120597119910
1205972V0
1205971199092
+ 11986022
1205971199080
120597119910
1205972V0
1205971199102
Mathematical Problems in Engineering 5
+
1
2
(11986012
+ 211986066) (
1205971199080
120597119910
)
2
1205972
1199080
1205971199092
+ 11986022
1205972
1199080
1205971199102
120597V0
120597119910
+ (11986012
+ 11986066)
1205971199080
120597119909
1205972V0
120597119910120597119909
+
1
2
(11986021
+ 211986066)
1205972
1199080
1205971199102
times (
1205971199080
120597119909
)
2
+
3
2
11986011(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092
+ 11986011
1205972
1199080
1205971199092
1205971199060
120597119909
+ 11986012
1205972
1199080
1205971199092
120597V0
120597119910
+ 211986066
1205972
1199080
120597119910120597119909
120597V0
120597119909
+ 211986066
1205972
1199080
120597119910120597119909
1205971199060
120597119910
+
3
2
11986022(
1205971199080
120597119910
)
2
1205972
1199080
1205971199102
+ (11986055
+ 119902119909cos (Ω
1119905) minus 119873
119875
119909
cos (Ω4119905)
+ 119865551198882
2
minus 2119863551198882)
1205972
1199080
1205971199092
minus
120597119873119875
119909
120597119909
cos (Ω3119905)
120597119908
120597119909
+ (119865551198882
2
minus 2119863551198882+ 11986055)
120597120601119909
120597119909
minus 119902 cos (Ω3119905) + 119896119891
1205971199080
120597119905
= 11986800minus 1198882
1
1198686(
1205972
0
1205971199092
+
1205972
0
1205971199102
)
+ 11988811198683(
1205970
120597119909
+
120597V0
120597119910
) + 11988811198694(
120597
120601119909
120597119909
+
120597
120601119910
120597119910
)
(4c)
(11986311
minus 2119865111198881+ 119867111198882
1
)
1205972
120601119909
1205971199092
+ (11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119909
1205971199102
minus 1198881(11986511
minus 119867111198881)
1205973
1199080
1205971199093
minus (119865551198882
2
minus 2119863551198882+ 11986055)
1205971199080
120597119909
+ (11986312
+ 11986366
+ 119867661198882
1
minus 2119865661198881+ 119867121198882
1
minus 2119865121198881)
times
1205972
120601119910
120597119910120597119909
minus 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
times
1205973
1199080
1205971199102
120597119909
+ (2119863551198882minus 11986055
minus 119865551198882
2
) 120601119909
= 11986910+ 1198702
120601119909minus 11988811198694
1205970
120597119909
(4d)
(11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119910
1205971199092
minus 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
1199080
1205971199101205971199092
+ (119867211198882
1
+ 11986366
+ 11986321
minus 2119865211198881+ 119867661198882
1
minus 2119865661198881)
times
1205972
120601119909
120597119910120597119909
+ (119867221198882
1
+ 11986322
minus 2119865221198881)
1205972
120601119910
1205971199102
minus 1198881(11986522
minus 119867221198881)
1205973
1199080
1205971199103
minus (119865441198882
2
minus 2119863441198882+ 11986044)
times
1205971199080
120597119910
+ (2119863441198882minus 119865441198882
2
minus 11986044) 120601119910
= 1198691V0+ 1198702
120601119910minus 11988811198694
1205970
120597119910
(4e)
The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]
119909 = 0 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5a)
119909 = 119886 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5b)
119910 = 0 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5c)
119910 = 119887 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5d)
int
ℎ
0
119873119909119909
1003816100381610038161003816119909=0
119889119911 = minusint
ℎ
0
(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)
int
ℎ
0
119873119910119910
10038161003816100381610038161003816119910=0
119889119911 = minusint
ℎ
0
(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)
The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906
0 V0 1199080 120601119909 and 120601
119910 It is desirable
that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906
0 V0 1199080 120601119909 and
120601119910in the following forms
1199060= 11990601(119905) cos 120587119909
2119886
cos120587119910
2119887
+ 11990602(119905) cos 3120587119909
2119886
cos120587119910
2119887
(6a)
V0= V1(119905) cos
120587119910
2119887
cos 1205871199092119886
+ V2(119905) cos
120587119910
2119887
cos 31205871199092119886
(6b)
1199080= 1199081(119905) sin 120587119909
119886
sin120587119910
119887
+ 1199082(119905) sin 3120587119909
119886
sin120587119910
119887
(6c)
120601119909= 1206011(119905) cos 120587119909
119886
sin120587119910
119887
+ 1206012(119905) cos 3120587119909
119886
sin120587119910
119887
(6d)
120601119910= 1206013(119905) cos
120587119910
119887
sin 120587119909
119886
+ 1206014(119905) cos
120587119910
119887
sin 3120587119909
119886
(6e)
Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
and the 119909-direction at 119910 = 0 with the form of 1199020+ 119902119909cosΩ1119905
and 1199021+119902119910cosΩ2119905 respectively Transverse excitation loaded
to the composite laminated piezoelectric rectangular plate isexpressed as 119902 = 119902
3cosΩ3119905The dynamic electrical loading is
represented by 119864119911= 119864119911cosΩ4119905
Considering Reddyrsquos third-order shear deformationdescription of the displacement field we have
119906 (119909 119910 119911 119905) = 1199060(119909 119910 119905) + 119911120601
119909(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119909+
1205971199080
120597119909
)
(1a)
V (119909 119910 119911 119905) = V0(119909 119910 119905) + 119911120601
119910(119909 119910 119905)
minus 1199113
4
3ℎ2
(120601119910+
1205971199080
120597119910
)
(1b)
119908 (119909 119910 119911 119905) = 1199080(119909 119910 119905) (1c)
where (1199060 V0 1199080) are the deflection of a point on the middle
surface (119906 V 119908) are the displacement components along the(119909 119910 119911) coordinate directions and 120601
119909and 120601
119910represent the
rotation components of normal to the middle surface aboutthe 119910 and 119909 axes respectively
The nonlinear strain-displacement relations are assumedto have the following form
120576119909119909
=
120597119906
120597119909
+
1
2
(
120597119908
120597119909
)
2
120576119909119911
=
1
2
(
120597119906
120597119911
+
120597119908
120597119909
)
120576119909119910
=
1
2
(
120597119906
120597119910
+
120597V120597119909
+
120597119908
120597119909
120597119908
120597119910
)
120576119910119910
=
120597V120597119910
+
1
2
(
120597119908
120597119910
)
2
120576119910119911
=
1
2
(
120597V120597119911
+
120597119908
120597119910
)
120576119911119911
=
120597119908
120597119911
(2)
Stress constitutive relations are presented as follows
120590119894119895= 120590119904
119894119895119896119897
120576119896119897minus 119890119894119895119896119864119896 (119894 119895 119896 119897 = 119909 119910 119911) (3)
where120590119894119895and 120576119896119897denote themechanical stresses and strains in
extended vector notation 120590119904119894119895119896119897
represents the elastic stiffnesstensor 119864
119896stands for the electric field vector and 119890
119894119895is the
piezoelectric tensorAccording to Hamiltonrsquos principle the nonlinear gov-
erning equations of motion in terms of generalized dis-placements (119906
0 V0 1199080 120601119909 120601119910) for the composite laminated
piezoelectric rectangular plate are given in the previousstudies as follows [8]
11986011
1205972
1199060
1205971199092
+ 11986066
1205972
1199060
1205971199102
+ (11986012
+ 11986066)
1205972V0
120597119909120597119910
+ 11986011
1205971199080
120597119909
1205972
1199080
1205971199092
+ 11986066
1205971199080
120597119909
1205972
1199080
1205971199102
+ (11986012
+ 11986066)
1205971199080
120597119910
1205972
1199080
120597119909120597119910
= 11986800+ 1198691
120601119909minus 11988811198683
1205970
120597119909
(4a)
11986066
1205972V0
1205971199092
+ 11986022
1205972V0
1205971199102
+ (11986021
+ 11986066)
1205972
1199060
120597119909120597119910
+ 11986066
1205971199080
120597119910
1205972
1199080
1205971199092
+ 11986022
1205971199080
120597119910
1205972
1199080
1205971199102
+ (11986021
+ 11986066)
1205971199080
120597119909
1205972
1199080
120597119909120597119910
= 1198680V0+ 1198691
120601119910minus 11988811198683
1205970
120597119910
(4b)
11986066
1205971199080
120597119909
1205972
1199060
1205971199102
minus 119867221198882
1
1205974
1199080
1205971199104
+ 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
1205973
120601119910
1205971199101205971199092
+ 1198881(11986522
minus 119867221198881)
1205973
120601119910
1205971199103
minus 119867111198882
1
1205974
1199080
1205971199094
+ 11986011
1205971199080
120597119909
1205972
1199060
1205971199092
+ (119865441198882
2
minus 2119863441198882+ 11986044)
120597120601119910
120597119910
+ 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
120601119909
1205971199102
120597119909
minus 1198882
1
(11986721
+ 411986766
+ 11986712)
1205974
1199080
1205971199102
1205971199092
+ (11986044
minus 119873119875
119910
cos (Ω4119905) + 119865
441198882
2
minus 2119863441198882)
1205972
1199080
1205971199102
minus
120597119873119875
119910
120597119910
cos (Ω2119905)
1205971199080
120597119910
+ (11986021
+ 411986066
+ 11986012)
times
1205971199080
120597119909
1205971199080
120597119910
1205972
1199080
120597119910120597119909
+ 1198881(11986511
minus 119867111198881)
1205973
120601119909
1205971199093
+ (11986021
+ 11986066)
1205971199080
120597119910
1205972
1199060
120597119910120597119909
+ 11986021
1205971199060
120597119909
1205972
1199080
1205971199102
times 11986066
1205971199080
120597119910
1205972V0
1205971199092
+ 11986022
1205971199080
120597119910
1205972V0
1205971199102
Mathematical Problems in Engineering 5
+
1
2
(11986012
+ 211986066) (
1205971199080
120597119910
)
2
1205972
1199080
1205971199092
+ 11986022
1205972
1199080
1205971199102
120597V0
120597119910
+ (11986012
+ 11986066)
1205971199080
120597119909
1205972V0
120597119910120597119909
+
1
2
(11986021
+ 211986066)
1205972
1199080
1205971199102
times (
1205971199080
120597119909
)
2
+
3
2
11986011(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092
+ 11986011
1205972
1199080
1205971199092
1205971199060
120597119909
+ 11986012
1205972
1199080
1205971199092
120597V0
120597119910
+ 211986066
1205972
1199080
120597119910120597119909
120597V0
120597119909
+ 211986066
1205972
1199080
120597119910120597119909
1205971199060
120597119910
+
3
2
11986022(
1205971199080
120597119910
)
2
1205972
1199080
1205971199102
+ (11986055
+ 119902119909cos (Ω
1119905) minus 119873
119875
119909
cos (Ω4119905)
+ 119865551198882
2
minus 2119863551198882)
1205972
1199080
1205971199092
minus
120597119873119875
119909
120597119909
cos (Ω3119905)
120597119908
120597119909
+ (119865551198882
2
minus 2119863551198882+ 11986055)
120597120601119909
120597119909
minus 119902 cos (Ω3119905) + 119896119891
1205971199080
120597119905
= 11986800minus 1198882
1
1198686(
1205972
0
1205971199092
+
1205972
0
1205971199102
)
+ 11988811198683(
1205970
120597119909
+
120597V0
120597119910
) + 11988811198694(
120597
120601119909
120597119909
+
120597
120601119910
120597119910
)
(4c)
(11986311
minus 2119865111198881+ 119867111198882
1
)
1205972
120601119909
1205971199092
+ (11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119909
1205971199102
minus 1198881(11986511
minus 119867111198881)
1205973
1199080
1205971199093
minus (119865551198882
2
minus 2119863551198882+ 11986055)
1205971199080
120597119909
+ (11986312
+ 11986366
+ 119867661198882
1
minus 2119865661198881+ 119867121198882
1
minus 2119865121198881)
times
1205972
120601119910
120597119910120597119909
minus 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
times
1205973
1199080
1205971199102
120597119909
+ (2119863551198882minus 11986055
minus 119865551198882
2
) 120601119909
= 11986910+ 1198702
120601119909minus 11988811198694
1205970
120597119909
(4d)
(11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119910
1205971199092
minus 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
1199080
1205971199101205971199092
+ (119867211198882
1
+ 11986366
+ 11986321
minus 2119865211198881+ 119867661198882
1
minus 2119865661198881)
times
1205972
120601119909
120597119910120597119909
+ (119867221198882
1
+ 11986322
minus 2119865221198881)
1205972
120601119910
1205971199102
minus 1198881(11986522
minus 119867221198881)
1205973
1199080
1205971199103
minus (119865441198882
2
minus 2119863441198882+ 11986044)
times
1205971199080
120597119910
+ (2119863441198882minus 119865441198882
2
minus 11986044) 120601119910
= 1198691V0+ 1198702
120601119910minus 11988811198694
1205970
120597119910
(4e)
The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]
119909 = 0 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5a)
119909 = 119886 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5b)
119910 = 0 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5c)
119910 = 119887 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5d)
int
ℎ
0
119873119909119909
1003816100381610038161003816119909=0
119889119911 = minusint
ℎ
0
(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)
int
ℎ
0
119873119910119910
10038161003816100381610038161003816119910=0
119889119911 = minusint
ℎ
0
(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)
The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906
0 V0 1199080 120601119909 and 120601
119910 It is desirable
that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906
0 V0 1199080 120601119909 and
120601119910in the following forms
1199060= 11990601(119905) cos 120587119909
2119886
cos120587119910
2119887
+ 11990602(119905) cos 3120587119909
2119886
cos120587119910
2119887
(6a)
V0= V1(119905) cos
120587119910
2119887
cos 1205871199092119886
+ V2(119905) cos
120587119910
2119887
cos 31205871199092119886
(6b)
1199080= 1199081(119905) sin 120587119909
119886
sin120587119910
119887
+ 1199082(119905) sin 3120587119909
119886
sin120587119910
119887
(6c)
120601119909= 1206011(119905) cos 120587119909
119886
sin120587119910
119887
+ 1206012(119905) cos 3120587119909
119886
sin120587119910
119887
(6d)
120601119910= 1206013(119905) cos
120587119910
119887
sin 120587119909
119886
+ 1206014(119905) cos
120587119910
119887
sin 3120587119909
119886
(6e)
Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
+
1
2
(11986012
+ 211986066) (
1205971199080
120597119910
)
2
1205972
1199080
1205971199092
+ 11986022
1205972
1199080
1205971199102
120597V0
120597119910
+ (11986012
+ 11986066)
1205971199080
120597119909
1205972V0
120597119910120597119909
+
1
2
(11986021
+ 211986066)
1205972
1199080
1205971199102
times (
1205971199080
120597119909
)
2
+
3
2
11986011(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092
+ 11986011
1205972
1199080
1205971199092
1205971199060
120597119909
+ 11986012
1205972
1199080
1205971199092
120597V0
120597119910
+ 211986066
1205972
1199080
120597119910120597119909
120597V0
120597119909
+ 211986066
1205972
1199080
120597119910120597119909
1205971199060
120597119910
+
3
2
11986022(
1205971199080
120597119910
)
2
1205972
1199080
1205971199102
+ (11986055
+ 119902119909cos (Ω
1119905) minus 119873
119875
119909
cos (Ω4119905)
+ 119865551198882
2
minus 2119863551198882)
1205972
1199080
1205971199092
minus
120597119873119875
119909
120597119909
cos (Ω3119905)
120597119908
120597119909
+ (119865551198882
2
minus 2119863551198882+ 11986055)
120597120601119909
120597119909
minus 119902 cos (Ω3119905) + 119896119891
1205971199080
120597119905
= 11986800minus 1198882
1
1198686(
1205972
0
1205971199092
+
1205972
0
1205971199102
)
+ 11988811198683(
1205970
120597119909
+
120597V0
120597119910
) + 11988811198694(
120597
120601119909
120597119909
+
120597
120601119910
120597119910
)
(4c)
(11986311
minus 2119865111198881+ 119867111198882
1
)
1205972
120601119909
1205971199092
+ (11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119909
1205971199102
minus 1198881(11986511
minus 119867111198881)
1205973
1199080
1205971199093
minus (119865551198882
2
minus 2119863551198882+ 11986055)
1205971199080
120597119909
+ (11986312
+ 11986366
+ 119867661198882
1
minus 2119865661198881+ 119867121198882
1
minus 2119865121198881)
times
1205972
120601119910
120597119910120597119909
minus 1198881(211986566
+ 11986512
minus 2119867661198881minus 119867121198881)
times
1205973
1199080
1205971199102
120597119909
+ (2119863551198882minus 11986055
minus 119865551198882
2
) 120601119909
= 11986910+ 1198702
120601119909minus 11988811198694
1205970
120597119909
(4d)
(11986366
minus 2119865661198881+ 119867661198882
1
)
1205972
120601119910
1205971199092
minus 1198881(11986521
+ 211986566
minus 119867211198881minus 2119867661198881)
1205973
1199080
1205971199101205971199092
+ (119867211198882
1
+ 11986366
+ 11986321
minus 2119865211198881+ 119867661198882
1
minus 2119865661198881)
times
1205972
120601119909
120597119910120597119909
+ (119867221198882
1
+ 11986322
minus 2119865221198881)
1205972
120601119910
1205971199102
minus 1198881(11986522
minus 119867221198881)
1205973
1199080
1205971199103
minus (119865441198882
2
minus 2119863441198882+ 11986044)
times
1205971199080
120597119910
+ (2119863441198882minus 119865441198882
2
minus 11986044) 120601119910
= 1198691V0+ 1198702
120601119910minus 11988811198694
1205970
120597119910
(4e)
The simply supported boundary conditions of the com-posite laminated piezoelectric rectangular plate can be repre-sented as follows [8 22]
119909 = 0 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5a)
119909 = 119886 119908 = 0 120601119910= 0 119873
119909119910= 0 119872
119909119909= 0
(5b)
119910 = 0 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5c)
119910 = 119887 119908 = 0 120601119909= 0 119873
119909119910= 0 119872
119910119910= 0
(5d)
int
ℎ
0
119873119909119909
1003816100381610038161003816119909=0
119889119911 = minusint
ℎ
0
(1199020+ 119902119909cosΩ1119905) 119889119911 (5e)
int
ℎ
0
119873119910119910
10038161003816100381610038161003816119910=0
119889119911 = minusint
ℎ
0
(1199021+ 119902119910cosΩ2119905) 119889119911 (5f)
The boundary condition (5f) includes the influence of thein-plane load We consider complicated nonlinear dynamicsof the composite laminated piezoelectric rectangular plate inthe first two modes of 119906
0 V0 1199080 120601119909 and 120601
119910 It is desirable
that we select an appropriate mode function to satisfy theboundary conditionThus we can rewrite 119906
0 V0 1199080 120601119909 and
120601119910in the following forms
1199060= 11990601(119905) cos 120587119909
2119886
cos120587119910
2119887
+ 11990602(119905) cos 3120587119909
2119886
cos120587119910
2119887
(6a)
V0= V1(119905) cos
120587119910
2119887
cos 1205871199092119886
+ V2(119905) cos
120587119910
2119887
cos 31205871199092119886
(6b)
1199080= 1199081(119905) sin 120587119909
119886
sin120587119910
119887
+ 1199082(119905) sin 3120587119909
119886
sin120587119910
119887
(6c)
120601119909= 1206011(119905) cos 120587119909
119886
sin120587119910
119887
+ 1206012(119905) cos 3120587119909
119886
sin120587119910
119887
(6d)
120601119910= 1206013(119905) cos
120587119910
119887
sin 120587119909
119886
+ 1206014(119905) cos
120587119910
119887
sin 3120587119909
119886
(6e)
Bymeans of the Galerkin method substituting (6a) (6b)(6c) (6d) (6e) into (4a) (4b) (4c) (4d) (4e) integrating
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
and neglecting all inertia terms in (4a) (4b) (4d) and (4e)we obtain the expressions of 119906
01 11990602 V1 V2 1206011 1206012 1206013 and
1206014via 1199081and 119908
2as follows
11990601
= 11989611199082
1
+ 11989621199082
2
+ 119896311990811199082 (7a)
11990602
= 11989641199082
1
+ 11989651199082
2
+ 119896611990811199082 (7b)
V1= 11989671199082
1
+ 11989681199082
2
+ 119896911990811199082 (7c)
V2= 119896101199082
1
+ 119896111199082
2
+ 1198961211990811199082 (7d)
1206011= 119896191199081 120601
2= 119896201199082 (7e)
1206013= 119896211199081 120601
4= 119896221199082 (7f)
where the coefficients presented in (7a) (7b) (7c) (7d) (7e)(7f) can be found in the previous studies [8]
In order to obtain the dimensionless governing equationsof motion we introduce the transformations of the variablesand parameters
119906 =
1199060
119886
V =
V0
119887
119908 =
1199080
ℎ
120601119909
= 120601119909 120601
119910
= 120601119910 119909 =
119909
119886
119910 =
119910
119887
119902 =
1198872
119864ℎ3
119902 119902119909
=
1198872
119864ℎ3
119902119909 119902
119910
=
1198872
119864ℎ3
119902119910
119905 = 1205872
(
119864
119886119887120588
)
12
119905 Ω119894=
1
1205872
(
119886119887120588
119864
)
12
Ω119894
(119894 = 1 2)
119860119894119895=
(119886119887)12
119864ℎ2
119860119894119895 119861
119894119895=
(119886119887)12
119864ℎ3
119861119894119895
119863119894119895=
(119886119887)12
119864ℎ4
119863119894119895 119864
119894119895=
(119886119887)12
119864ℎ5
119864119894119895
119865119894119895=
(119886119887)12
119864ℎ6
119865119894119895 119867
119894119895=
(119886119887)12
119864ℎ8
119867119894119895
119868119894=
1
(119886119887)(119894+1)2
120588
119868119894
(8)
For simplicity we drop the overbar in the followinganalysis Substituting (5a) (5b) (5c) (5d) (5e) (5f)ndash(8) into(4c) and applying the Galerkin procedure we obtain thegoverning equations of motion of the composite laminatedpiezoelectric rectangular plate for the dimensionless as fol-lows
1+ 12058311+ 1205962
1
1199081
+ (1198862cosΩ1119905 + 1198863cosΩ2119905 minus 1198864cosΩ4119905) 1199081
+ 11988651199082
1
1199082+ 11988661199082
2
1199081+ 11988671199083
1
+ 11988681199083
2
= 1198911cosΩ3119905
(9a)
2+ 12058322+ 1205962
2
1199082
+ (1198872cosΩ1119905 + 1198873cosΩ2119905 + 1198874cosΩ4119905) 1199082
+ 11988751199082
2
1199081+ 11988761199082
1
1199082+ 11988771199083
2
+ 11988781199083
1
= 1198912cosΩ3119905
(9b)
where the coefficients presented in (9a) (9b) are given in theprevious studies [8]
The above equations include the cubic terms in-planeexcitation transverse excitation and piezoelectric excitationEquation (9a) (9b) can describe the nonlinear transverseoscillations of the composite laminated piezoelectric rectan-gular plate We only study the case of primary parametricresonance and 1 3 internal resonances In this resonant casethere are the following resonant relations
1205962
1
=
1205962
9
+ 1205761205901 120596
2
2
= 1205962
+ 1205761205902
Ω3= 120596 Ω
1= Ω2= Ω4=
2120596
3
1205962asymp 31205961
(10)
where 1205901and 120590
2are two detuning parameters
The method of multiple scales [23] is employed to (9a)(9b) to find the uniform solutions in the following form
1199081(119905 120576) = 119909
10(1198790 1198791) + 120576119909
11(1198790 1198791) + sdot sdot sdot (11a)
1199082(119905 120576) = 119909
20(1198790 1198791) + 120576119909
21(1198790 1198791) + sdot sdot sdot (11b)
where 1198790= 119905 1198791= 120576119905
Substituting (10) and (11a) (11b) into (9a) (9b) andbalancing the coefficients of corresponding powers of 120576 onthe left-hand and right-hand sides of equations the four-dimensional averaged equations in the Cartesian form areobtained as follows
1= minus
1
2
12058311199091minus
1
2
12059011199092+
1
4
(1198862+ 1198863minus 1198864) 1199092
minus
3
2
11988671199092(1199092
1
+ 1199092
2
) minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(12a)
2= minus
1
2
12058311199092+
1
2
12059011199091+
1
4
(1198862+ 1198863minus 1198864) 1199091
+
3
2
11988671199091(1199092
1
+ 1199092
2
) +
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(12b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(12c)
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
) minus
1
12
1198912
(12d)
3 Computation of Normal Form
In order to assist the analysis of the Shilnikov type multipulseorbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate it is necessary to reduce theaveraged equation (12a) (12b) (12c) (12d) to a simplernormal form It is found that there are 119885
2oplus 1198852and 119863
4
symmetries in the averaged equation (12a) (12b) (12c) (12d)without the parameters Therefore these symmetries are alsoheld in normal form
We take into account the excitation amplitude 1198912as a
perturbation parameter Amplitude 1198912can be considered as
an unfolding parameter when the Shilnikov type multipulseorbits are investigated Obviously when we do not considerthe perturbation parameter (12a) (12b) (12c) (12d) become
1= minus
1
2
12058311199091+ (1198910minus
1
2
1205901)1199092minus
3
2
11988671199092(1199092
1
+ 1199092
2
)
minus
1
2
11988651199094(1199092
1
minus 1199092
2
)
minus 11988661199092(1199092
3
+ 1199092
4
) + 1198865119909111990921199093
(13a)
2= minus
1
2
12058311199092+ (1198910+
1
2
1205901)1199091+
3
2
11988671199091(1199092
1
+ 1199092
2
)
+
1
2
11988651199093(1199092
1
minus 1199092
2
)
+ 11988661199091(1199092
3
+ 1199092
4
) + 1198865119909111990921199094
(13b)
3= minus
1
2
12058321199093minus
1
6
12059021199094minus
1
3
11988761199094(1199092
1
+ 1199092
2
)
minus
1
2
11988771199094(1199092
3
+ 1199092
4
) minus
1
6
11988781199092(31199092
1
minus 1199092
2
)
(13c)
4= minus
1
2
12058321199094+
1
6
12059021199093+
1
3
11988761199093(1199092
1
+ 1199092
2
)
+
1
2
11988771199093(1199092
3
+ 1199092
4
) +
1
6
11988781199091(1199092
1
minus 31199092
2
)
(13d)
where 1198910= (14)(119886
2+ 1198863minus 1198864)
Executing the Maple program given by Zhang et al [24]the nonlinear transformation used here is given as follows
1199091= 1199101minus
1
4
11988671199103
1
+
31198865(1205902minus 6)
1205902
2
1199102
1
1199103minus 119886611991011199102
3
minus 119886611991011199102
4
minus
31198865(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199102
2
1199103
+
1198865(61205902
2
minus 721205902+ 432)
1205903
2
119910111991021199104
(14a)
1199092= 1199102+
3
2
11988671199103
2
+
3
4
11988671199102
1
1199102+
31198865
1205902
1199102
1
1199104
minus
31198865(1205902
2
minus 121205902+ 72)
1205903
2
1199102
2
1199104minus
61198865(1205902minus 6)
1205902
2
119910111991021199103
(14b)
1199093= 1199103minus
1198878
1205902
1199103
1
+
31198878(1205902
2
minus 121205902+ 72)
1205903
2
11991011199102
2
minus
1
3
1198876119910111991021199104
(14c)
1199094= 1199104+
1198878(1205903
2
minus 181205902
2
+ 2161205902minus 1296)
1205904
2
1199103
2
minus
31198878(1205902minus 6)
1205902
2
1199102
1
1199102+
1
3
1198876119910111991021199103
(14d)
Substituting (14a) (14b) (14c) (14d) into (13a) (13b)(13c) (13d) yields a simpler 3rd-order normal form with theparameters for averaged equation (12a) (12b) (12c) (12d) asfollows
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (15a)
1199102= 12059011199101minus 1205831
1199102+ 11988661199101(1199102
3
+ 1199102
4
) +
3
2
11988671199103
1
(15b)
1199103= minus1205832
1199103minus 12059021199104minus
1
3
11988761199102
1
1199104minus
1
2
11988771199104(1199102
3
+ 1199102
4
) (15c)
1199104= 12059021199103minus 1205832
1199104+
1
3
11988761199102
1
1199103+
1
2
11988771199103(1199102
3
+ 1199102
4
) minus 1198912
(15d)
where the coefficients are 1205831
= (12)1205831 1205832
= (12)1205832 1205902=
(16)1205902 and 119891
2
= (112)1198912 respectively
Further let
1199103= 119868 cos 120574 119910
4= 119868 sin 120574 (16)
Substituting (16) into (15a) (15b) (15c) (15d) yields
1199101= minus1205831
1199101+ (1 minus 120590
1) 1199102 (17a)
1199102= 12059011199101minus 1205831
1199102+ 119886611991011198682
+
3
2
11988671199103
1
(17b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (17c)
119868 120574 = 1205902119868 +
1
3
11988761199102
1
119868 +
1
2
11988771198683
minus 1198912
cos 120574 (17d)
In order to get the unfolding of (17a) (17b) (17c) (17d) alinear transformation is introduced
[
1199101
1199102
] = radic3
radic10038161003816100381610038161198866
1003816100381610038161003816
radic10038161003816100381610038161198876
1003816100381610038161003816
[
1 minus 1205901
0
1205831
1] [
1199061
1199062
] (18)
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Substituting (18) into (17a) (17b) (17c) (17d) and can-celing nonlinear terms including the parameter 120590
1yield the
unfolding as follows
1= 1199062 (19a)
2= minus120583119906
1minus 12058331199062+ 12057811199063
1
+ 119886611990611198682
(19b)
119868 = minus1205832
119868 minus 1198912
sin 120574 (19c)
119868 120574 = 1205902119868 + 11988661199062
1
119868 + 12057221198683
minus 1198912
cos 120574 (19d)
where 120583 = 1205832
1
minus 1205901(1 minus 120590
1) 1205833= 21205831
1205781= 91198866119886721198876and
1205722= (12)119887
7
The scale transformations to be introduced into (19a)(19b) (19c) (19d) are
1205832
997888rarr 1205761205832
1205833997888rarr 120576120583
3 119891
2
997888rarr 1205761198912
1205781997888rarr 1205781 120572
2997888rarr 1205722 119886
6997888rarr 1198866
(20)
Then normal form (19a) (19b) (19c) (19d) can berewritten in the form with the perturbations
1=
120597119867
1205971199062
+ 1205761198921199061= 1199062 (21a)
2= minus
120597119867
1205971199061
+ 1205761198921199062= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
minus 12057612058331199062 (21b)
119868 =
120597119867
120597120574
+ 120576119892119868
= minus1205761205832
119868 minus 1205761198912
sin 120574 (21c)
119868 120574 = minus
120597119867
120597119868
+ 120576119892120574
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
minus 1205761198912
cos 120574 (21d)
where the Hamiltonian function119867 is of the form
119867(1199061 1199062 119868 120574) =
1
2
1199062
2
+
1
2
1205831199062
1
minus
1
4
12057811199064
1
minus
1
2
11988661198682
1199062
1
minus
1
2
12059021198682
minus
1
4
12057221198684
(22)
and 1198921199061 1198921199062 119892119868 and 119892
120574 are the perturbation terms inducedby the dissipative effects
1198921199061= 0 119892
1199062= minus12058331199062
119892119868
= minus1205832
119868 minus 1198912
sin 120574 119892120574
= minus1198912
cos 120574(23)
4 Heteroclinic Bifurcations ofUnperturbed System
In this section we focus on studying the nonlinear dynamicscharacteristic of the unperturbed system When 120576 = 0 itcan be known that system from (21a) (21b) (21c) (21d) isan uncoupled two-degree-of-freedom nonlinear system Thevariable 119868 appears in the subspace (119906
1 1199062) of (21a) (21b)
(21c) (21d) as a parameter since 119868 = 0 Consider the first twodecoupled equations of (21a) (21b) (21c) (21d)
1= 1199062 (24a)
2= minus120583119906
1+ 12057811199063
1
+ 119886611990611198682
(24b)
Since 1205781
gt 0 (24a) (24b) can exhibit the heteroclinicbifurcations It is obvious from (24a) (24b) that when 120583 minus
11988661198682
lt 0 the only solution to (24a) (24b) is the trivial zerosolution (119906
1 1199062) = (0 0) which is the saddle point On the
curve defined by 120583 = 11988661198682 that is
11986812
= plusmn[
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(25)
the trivial zero solution bifurcates into three solutionsthrough a pitchfork bifurcation which are given by 119902
0= (0 0)
and 119902plusmn(119868) = (119861 0) respectively where
119861 = plusmn
1
1205781
[1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
]
12
(26)
From the Jacobian matrix evaluated at the nonzerosolutions it can be found that the singular point 119902
0is the
center point and the singular points 119902plusmn(119868) are saddle points It
is observed that 119868 and 120574 actually represent the amplitude andphase of vibrations Therefore we assume that 119868 ge 0 and (25)becomes
1198681= 0 119868
2= [
1205832
1
minus 1205901(1 minus 120590
1)
1198866
]
12
(27)
such that for all 119868 isin [1198681 1198682] (24a) (24b) have two hyperbolic
saddle points 119902plusmn(119868) which are connected by a pair of
heteroclinic orbits 119906ℎplusmn
(1198791 119868) that is lim
1198791rarrplusmninfin
119906ℎ
plusmn
(1198791 119868) =
119902plusmn(119868) Thus in the full four-dimensional phase space the set
defined by
119872 = (119906 119868 120574) | 119906 = 119902plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587 (28)
is a two-dimensional invariant manifoldFrom the results obtained by Feng et al [9ndash11] it is
known that two-dimensional invariant manifold 119872 is nor-mally hyperbolic The two-dimensional normally hyperbolicinvariant manifold 119872 has the three-dimensional stable andunstable manifolds represented as 119882
119904
(119872) and 119882119906
(119872)respectively The existence of the heteroclinic orbit of (24a)(24b) to 119902
plusmn(119868) = (119861 0) indicates that 119882119904(119872) and 119882
119906
(119872)
intersect nontransversally along a three-dimensional mani-fold denoted by Γ which can be written as
Γ = (119906 119868 120574) | 119906 = 119906ℎ
plusmn
(1198791 119868) 1198681lt 119868 lt 119868
2
120574 = int
1198791
0
119863119868119867(119906ℎ
plusmn
(1198791 119868) 119868) 119889119904 + 120574
0
(29)
We analyze the dynamics of the unperturbed system of(21a) (21b) (21c) (21d) restricted to 119872 Considering theunperturbed system of (21a) (21b) (21c) (21d) restricted to119872 yields
119868 = 0 (30a)
119868 120574 = 119863119868119867(119902plusmn(119868) 119868) 119868
1lt 119868 lt 119868
2 (30b)
Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering 9
where
119863119868119867(119902plusmn(119868) 119868) = minus
120597119867 (119902plusmn(119868) 119868)
120597119868
= 1205902119868 + 12057221198683
+ 11988661198681199062
1
(31)
From the results obtained by Feng et al [9ndash11] it is knownthat if 119863
119868119867(119902plusmn(119868) 119868) = 0 119868 = constant is called a periodic
orbit and if 119863119868119867(119902plusmn 119868) = 0 119868 = constant is known as a
circle of the singular points Any value of 119868 isin [1198681 1198682] at which
119863119868119867(119902plusmn 119868) = 0 is a resonant value 119868 and these singular points
are resonant singular points We denote a resonant value by119868119903such that
119863119868119867(119902plusmn 119868) = 120590
2119868119903+ 12057221198683
119903
+ 11988661198681199031199062
1
= 0 (32)
Then we obtain
119868119903= plusmn
12059021205781+ 1198866[1205832
1
minus 1205901(1 minus 120590
1)]
1198862
6
minus 12057221205781
12
(33)
The geometric structure of the stable and unstable mani-folds of 119872 in the full four-dimensional phase space for theunperturbed system of (21a) (21b) (21c) (21d) is given inFigure 2 Since 120574 represents the phase of the oscillationswhen119868 = 119868119903 the phase shift Δ120574 of oscillations is defined by
Δ120574 = 120574 (+infin 119868119903) minus 120574 (minusinfin 119868
119903) (34)
The physical interpretation of the phase shift is the phasedifference between the two end points of the orbit In thesubspace (119906
1 1199062) there exists a pair of heteroclinic orbits
connecting to saddle points Therefore the homoclinic orbitin the subspace (119868 120574) is in fact a heteroclinic connecting inthe full four-dimensional space (119906
1 1199062 119868 120574) The phase shift
denotes the difference of the value 120574 when a trajectory leavesand returns to the basin of attraction of 119872 We will use thephase shift in subsequent analysis to obtain the condition forthe existence of the Shilnikov typemultipulse orbitThephaseshift will be calculated later in the heteroclinic orbit analysis
We consider the heteroclinic orbits of (24a) (24b) Let1205761= 120583 minus 119886
61198682 and 120583
3= 1205762 (24a) (24b) can be rewritten as
1= 1199062 (35a)
2= minus12057611199061+ 12057811199063
1
minus 12057612057621199062 (35b)
Set 120576 = 0 (35a) (35b) is a system with the Hamiltonianfunction
119867(1199061 1199062) =
1
2
1199062
2
+
1
2
12057611199062
1
minus
1
4
12057811199064
1
(36)
When 119867 = 0 there is a heteroclinic loop Γ0 which
consists of the two hyperbolic saddle points 119902plusmn(119868) and a pair
of heteroclinic orbits 119906plusmn(1198791) In order to calculate the phase
shift and the extended Melnikov function it is necessary to
obtain the equations of a pair of heteroclinic orbits which aregiven as follows
1199061(1198791) = plusmnradic
1205761
1205781
tanh(
radic21205761
2
1198791) (37a)
1199062(1198791) = plusmn
1205761
radic21205781
sech2 (radic21205761
2
1198791) (37b)
We turn our attention to the computation of the phaseshift Substituting the first equation of (37a) (37b) into thefourth equation of the unperturbed system of (21a) (21b)(21c) (21d) and integrating yield
120574 (1198791) = 1205961199031198791minus
1198866radic21205761
1205781
tanh(
radic21205761
2
1198791) + 1205740 (38)
where 120596119903= 1205902+ 12057221198682
+ 119886612057611205781
At 119868 = 119868119903 there is 120596
119903equiv 0 Therefore the phase shift may
be expressed as
Δ120574 = [minus
21198866radic21205761
1205781
]
119868=119868119903
= minus
21198866
1205781
radic2 [1205832
1
minus 1205901(1 minus 120590
1) minus 11988661198682
119903
]
(39)
5 Existence of Multipulse Orbits
After obtaining detailed information on the nonlineardynamic characteristics of the subspace (119906
1 1199062) for the
unperturbed system from (21a) (21b) (21c) (21d) the nextstep is to examine the effects of small perturbation terms (0 lt
120576 ≪ 1) on the unperturbed system from (21a) (21b) (21c)(21d) The extended Melnikov method developed by Kovacicet al [15ndash17] is utilized to discover the existence of the multi-pulse orbits and chaotic dynamics of the nonlinear vibrationfor the laminated composite piezoelectric rectangular plateWe start by studying the influence of such small perturbationson themanifold119872The objective of the research is to identifythe parameter regions where the existence of the multipulseorbits is possible in the perturbed phase space The main aimis to verify whether these parameters satisfy the transversalitycondition of multipulse chaotic dynamics It will be shownthat thesemultipulse orbits can occur in theHamilton systemwith dissipative perturbations if the parameters meet thetransversality condition The existence of such multipulseorbits provides a robust mechanism for the existence ofthe complicated dynamics in the perturbed system In thissection the emphasis is put on the application aspects of theextended Melnikov method to (21a) (21b) (21c) (21d)
51 Dissipative Perturbations of the Homoclinic Loop Weanalyze dynamics of the perturbed system and the influenceof small perturbations on 119872 Based on the analysis byKovacic et al [15ndash17] we know that 119872 along with its stableand unstable manifolds is invariant under small sufficientlydifferentiable perturbations It is noticed that 119902
plusmn(119868) in (24a)
10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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10 Mathematical Problems in Engineering
M
I
120574
times
u2
u1
(a)
0 1205742120587
I = I1
I = Ir
I = I2
(b)
Figure 2 The geometric structure of manifolds119872119882119904(119872) and119882119906
(119872) is given in the full four-dimensional phase space
(24b) maintains the characteristic of the hyperbolic singularpoint under small perturbations in particular 119872 rarr 119872
120576
Therefore we obtain
119872 = 119872120576= (119906 119868 120574) | 119906 = 119902
plusmn(119868) 1198681lt 119868 lt 119868
2 0 le 120574 lt 2120587
(40)
Considering the last two equations of (21a) (21b) (21c)(21d) yields
119868 = minus1205832
119868 minus 1198912
sin 120574 (41a)
120574 = 1205902+ 12057221198682
+ 11988661199062
1
minus
1198912
cos 120574119868
(41b)
It is known from the above analysis that the last two equationsof (21a) (21b) (21c) (21d) are of a pair of pure imaginaryeigenvalues Therefore the resonance can occur in (41a)(41b) Also introduce the scale transformations
1205832
997888rarr 1205761205832
119868 = 119868119903+ radic120576ℎ
1198912
997888rarr 1205761198912
1198791997888rarr
1198791
radic120576
(42)
Substituting the above transformations into (41a) (41b)yields
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 minus radic1205761205832
ℎ (43a)
120574 = minus
2120575
1205781
119868119903ℎ minus radic120576(
120575
1205781
ℎ2
+
1198912
119868119903
cos 120574) (43b)
where 120575 = 1198862
6
minus 12057221205781
When 120576 = 0 (43a) (43b) become
ℎ = minus120583
2
119868119903minus 1198912
sin 120574 (44a)
120574 = minus
2120575
1205781
119868119903ℎ (44b)
The unperturbed system from (44a) (44b) is a Hamiltonsystem with the function
119863(ℎ 120574) = minus120583
2
119868119903120574 + 1198912
cos 120574 +
120575
1205781
119868119903ℎ2
(45)
The singular points of (44a) (44b) are given as
1198750= (0 120574
119888) = (0 minus arcsin(
1205832
119868119903
1198912
))
1198760= (0 120574
119904) = (0 120587 + arcsin(
1205832
119868119903
1198912
))
(46)
Based on the characteristic equations evaluated at the twosingular points119875
0and119876
0 we can know the stabilities of these
singular points Therefore it is known that the singular point1198750is a center pointThe singular point119876
0is a saddle which is
connected to itself by a homoclinic orbit The phase portraitof system for (44a) (44b) is shown in Figure 3(a)
It is found that for the sufficiently small parameter 120576the singular point 119876
0remains a hyperbolic singular point
119876120576of the saddle stability type For small perturbations the
singular point 1198750becomes a hyperbolic sink 119875
120576 The phase
portrait of the perturbed system from (43a) (43b) is depictedin Figure 3(b)
Using the function (45) at ℎ = 0 and substituting 120574119904in
(46) into (45) the estimate of the basin of the attractor for120574min is obtained as
120574min minus
1198912
1205832
119868119903
cos 120574min = 120587 + arcsin1205832
119868119903
1198912
+
radic119891
2
2
minus 1205832
2
1198682
119903
1205832
119868119903
(47)
Define an annulus 119860120576near 119868 = 119868
119903as
119860120576= (1199061 1199062 119868 120574) | 119906
1= 119861 119906
2= 0
1003816100381610038161003816119868 minus 119868119903
1003816100381610038161003816lt radic120576119862 120574 isin 119879
119871
(48)
where 119862 is a constant and is sufficiently large so that theunperturbed orbit is enclosed within the annulus
It is noticed that the three-dimensional stable and unsta-ble manifolds of 119860
120576 denoted as119882119904(119860
120576) and119882
119906
(119860120576) are the
subsets of the manifolds 119882119904(119872120576) and 119882
119906
(119872120576) respectively
We will indicate that for the perturbed system the saddlefocus 119875
120576on 119860120576has the multipulse orbits which come out
of the annulus 119860120576and can return to the annulus in the full
four-dimensional spaceThese orbits which are asymptotic tosome invariant manifolds in the slow manifold119872
120576 leave and
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
h
120574
P0
120574min 120574c 120574s
Q0
(a)
h
120574Q120576
P120576
(b)
Figure 3 Dynamics on the normally hyperbolic manifold is described (a) the unperturbed case (b) the perturbed case
enter a small neighborhood of 119872120576multiple times and finally
return and approach an invariant set in119872120576asymptotically as
shown in Figure 4 In Figure 4 this is an example of the three-pulse jumping orbit which depicts the formation mechanismof the multipulse orbits
52 The 119896-Pulse Melnikov Function Most researchers fo-cused on Shilnikov type single-pulse chaotic dynamics ofthe high-dimensional nonlinear systems from the thin platestructures in the past There exist multipulse chaotic dynam-ics in the practical engineering systems The extended Mel-nikov method is a kind of theory which can be used to inves-tigate the multipulse jumping orbits in the high-dimensionalnonlinear systems Since the theory on multipulse chaoticdynamics is very esoteric and abstract it is difficult to beextended to solve the engineering problems Up to nowfew researchers have made use of the extended Melnikovmethod to study the Shilnikov type multipulse homoclinicand heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications
The extended Melnikov method was first presented byKovacic et al [15ndash17] which is an extension of the global per-turbation method developed by Feng et al [9ndash11] Camassaet al [17] gave the detailed procedure of mathematical proofon the extended Melnikov method which unifies severaldisjoint perturbation theoretical methods This method canbe also utilized to detect the Shilnikov typemultipulse homo-clinic or heteroclinic orbits to the slow manifolds of four-dimensional near-integrable Hamilton systems or higher-dimensional nonlinear systems The extended Melnikovfunction is different from the usual Melnikov function anddescribes slow dynamics of the multipulse orbits on thehyperbolic manifold
The key of the extended Melnikov method is how tocalculate the extended Melnikov methodThe extended Mel-nikov function is computed by a recursion procedure fromthe usual 1-pulseMelnikov function and depends on the smallperturbation parameter 120576 through a logarithmic functionwhich calculates the asymptotic in the particularly delicatesmall 120576 limit In this paper the extended Melnikov function
120574
|u|
h
120574c 120574c + 3Δ120574
p120576qc
Figure 4 The Shilnikov type three-pulse orbits are obtained
can be simplified in the resonant case and does not dependon the perturbation parameter We have used the extendedMelnikovmethod to investigate heteroclinic bifurcations andmultipulse chaotic dynamics of the laminated compositepiezoelectric rectangular plate
We use the extended Melnikov method described byKovacic et al [15ndash17] to find the Shilnikov type multipulseorbits for nonlinear vibration for the laminated compositepiezoelectric rectangular plate We search for the multipulseexcursions to find the nondegenerate zeroes of the extendedMelnikov function119872
119896(120576 119868 120574
0 1205832
) with the certain combina-tion of parameters 120576 119868 120574
0 and 120583
2
which we name the 119896-pulseMelnikov function
It is important to obtain the detailed expression of the 119896-pulse Melnikov function We compute the 1-pulse Melnikovfunction based on the formula obtained by Kovacic et al[15ndash17] at the resonant case 119868 = 119868
119903 The 1-pulse Melnikov
function 1198721(120576 119868119903 1205740 1205832
) coincides with the standard Mel-nikov function 119872(119868
119903 1205740 1205832
) The 1-pulse Melnikov function119872(119868119903 1205740 1205832
) on both heteroclinic manifolds 119882119904
(119872) and119882119906
(119872) is given as follows
119872(119868119903 1205740 1205832
1205781 1198866 1205761)
= int
+infin
minusinfin
⟨n (119901ℎ
(119905)) g (119901ℎ (119905) 1205832
0)⟩ 1198891198791
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
= int
+infin
minusinfin
(
120597119867
1205971199061
1198921199061+
120597119867
1205971199062
1198921199062+
120597119867
120597119868
119892119868
+
120597119867
120597120574
119892120574
)1198891198791
= minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus 1198866
radic21205761
1205781
) minus cos(1205740+ 1198866
radic21205761
1205781
)]
(49)
Based on the results given by Kovacic et al [15ndash17] it is known that the 119896-pulse Melnikov function119872119896(120576 119868119903 1205740 1205832
) (119896 = 1 2 ) is defined as
119872119896(120576 119868119903 1205740 1205832
)
=
119896minus1
sum
119895=0
119872(119868119903 119895Δ120574 (119868
119903) + Γ119895(120576 119868119903 1205740 1205832
) + 1205740 1205832
)
(50)
where
Γ119895(120576 119868119903 1205740 1205832
) =
Ω (1199090(119868119903) 119868119903)
120582 (119868119903)
119895
sum
119903=1
log100381610038161003816100381610038161003816100381610038161003816
120589 (119868119903)
120576119872119903(120576 119868119903 1205740 1205832
)
100381610038161003816100381610038161003816100381610038161003816
(51)
for 119895 = 1 119896 minus 1 and Γ0(120576 119868119903 1205740 1205832
) = 0It is noticed that the angle function Γ
119895(120576 119868119903 1205740 1205832
) isthe complex formula where 119872
119896(120576 119868119903 1205740 1205832
) appears as theargument of a logarithm When resonance occurs theperiodic orbit corresponding to the value 119868
119903degenerates
into a circle of equilibria Under this case there existsΓ119895(120576 119868119903 1205740 1205832
) = 0 (119895 = 0 1 119896 minus 1) Based on theexpression obtained by Kovacic et al [15ndash17] the 119896-pulseMelnikov function can be written as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
=
119896minus1
sum
119895=0
119872(119868119903 1205740+ 119895Δ120574 (119868
119903) 1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus
21198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
+ sdot sdot sdot
minus 1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1) 1198866
radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)]
minus
2radic21205833
31205781
12057632
1
minus 2radic211988661205832
1198682
119903
12057612
1
1205781
= minus1198912
119868119903[cos(120574
0minus
1198866radic21205761
1205781
minus 2 (119896 minus 1)
1198866radic21205761
1205781
)
minus cos(1205740+
1198866radic21205761
1205781
)]
minus
2radic21198961205833
31205781
12057632
1
minus 2radic211989611988661205832
1198682
119903
12057612
1
1205781
(52)
If we set Δ120574 = minus21198866(radic212057611205781) and 120574
119896minus1= 1205740+ (119896 minus
1)(Δ1205742) (52) can be rewritten as follows
119872119896(119868119903 1205740 1205832
1205781 1198866 1205761)
= 119872119896(119868119903 120574119896minus1
minus (119896 minus 1)
Δ120574
2
1205832
1205781 1198866 1205761)
= minus1198912
119868119903[cos(120574
119896minus1+
1
2
119896Δ120574)
minus cos(120574119896minus1
minus
1
2
119896Δ120574)]
+
11989612058331205761
31198866
Δ120574 + 1205832
1198682
119903
(119896Δ120574)
= 21198912
119868119903sin 120574119896minus1
sin(
1
2
119896Δ120574)
+
12058331205761
31198866
(119896Δ120574) + 1205832
1198682
119903
(119896Δ120574)
(53)
Based on Proposition 31 given by Kovacic et al [15ndash17]the nonfolding condition is always satisfied in the resonantcase We obtain the following two conditions
10038161003816100381610038161003816100381610038161003816100381610038161003816
(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
311988661198912
119868119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
lt 1
1
2
119896Δ120574 = 119899120587 119899 = 0 plusmn1 plusmn2
(54)
The main aim of the following analysis focuses onidentifying simple zeroes of the 119896-pulse Melnikov functionDefine a set that contains all such simple zeroes to be
119885119899
minus
= (119868119903 120574119896minus1
1205832
1205781 1198866 1205761) | 119872119896= 0119863
1205740
119872119896
= 0 (55)
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
The 119896-pulse Melnikov function has two simple zeroes in theinterval 120574
119896minus1isin [0 120587]
120574119896minus11
= minus arcsin(12) 119896Δ120574
sin ((12) 119896Δ120574)
(12058331205761+ 311988661205832
1198682
119903
)
(311988661198912
119868119903)
120574119896minus12
= 120587 minus 120574119896minus11
(56)
53 Geometric Structure of Multipulse Orbits Based on theaforementioned analysis we obtain the following conclu-sions When the parameters of 119896 120583
3 1205761 1205832
1198866 and 119891
2
satisfy condition (54) the 119896-pulse Melnikov function (53)has simple zeroes at 120574
119896minus1= 120574119896minus11
and 120574119896minus1
= 120574119896minus12
=
120587 minus 120574119896minus11
For 119894 = 1 or 119894 = 2 when the 119895-pulse Melnikovfunction 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) has no simple zeroes the
stable and unstable manifolds119882119904(119872120576) and119882
119906
(119872120576) intersect
transversely along a symmetric pair of the two-dimensional119896-pulse surfaces sum
1205832120578111988661205761
plusmn120576(120574119896minus1119894
) This signifies that thepresence of the Shilnikov type 119899-pulse orbits leads to chaoticdynamics in the sense of the Smale horseshoes for thenonlinear motion for the laminated composite piezoelectricrectangular plate In the phase space of the unperturbedsystem from (21a) (21b) (21c) (21d) this symmetric pair ofthe two-dimensional 119896-pulse surfaces breaks down smoothlyonto a pair of limiting 119896-pulse surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus1119894
)parametrized by (37a) (37b) and (38) with 119868 = 119868
119903 1205740
=
120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574 and an arbitrary ℎ The sign in(49) is determined by the sign of the corresponding 119895-pulseMelnikov function119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761)
From the discussion given by Kovacic et al [15ndash17] itis easily found that for 120574
0119894
= 120574119896minus1119894
minus (119896 minus 1)(Δ1205742) + 119895Δ120574
(119894 = 1 or 119894 = 2) the values of the 119895-pulse Melnikovfunctions 119872
119895(119868119903 1205740119894
1205832
1205781 1198866 1205761) are not zero for all 119895 =
1 119896 minus 1 and all 119895 have the same sign It is knownthat this sign is negative for 120574
01
and positive for 12057402
Therefore the 119896-pulse heteroclinic surfaces sum1205832 12057811198866 1205761
plusmn120576(120574119896minus11
)
and sum1205832120578111988661205761
plusmn120576(120574119896minus12
) indeed exist and the limiting 119896-pulsesurfaces sum
1205832120578111988661205761
plusmn0
(120574119896minus11
) and sum1205832120578111988661205761
plusmn0
(120574119896minus12
) also existwhen 120576 = 0 Since the regions enclosed by the stable andunstable heteroclinic manifolds119882119904(119872) and119882
119906
(119872) are bothconvex and the normal vector
n = ((minus1205831199061+ 12057811199063
1
+ 11988661198682
1199061) minus1199062 0 0) (57)
is known to point out of these manifolds it demonstratesthat the orbits forming each of the surfaces sum1205832 12057811198866 1205761
plusmn0
(120574119896minus11
)
are parametrized by (37a) (37b) and (38) with the alter-nating signs and the orbits forming each of the surfacessum1205832120578111988661205761
plusmn0
(120574119896minus12
) are parametrized by (37a) (37b) and (38)with the same signs
For the parameter 1205832
= 120583 there exist119873minus1 orbit segments119874119894(120583) (119894 = 2 119873) on the annulus119872 where the end points
of the segments 119874119894(120583) are 119889
119894(120583) and 119888
119894(120583) respectively The
trajectories of (44a) (44b) on the segments119874119894(120583) travel from
the end points 119889119894(120583) to 119888
119894(120583) in forward time Therefore the
end points 119889119894(120583) and 119888
119894(120583) are respectively referred to as
the departure and landing points of the heteroclinic jumpingΓ119894 In addition the line 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903) transversely
intersects the segments 119874119894(120583) at the end point 119888
119894(120583) for 119894 =
2 119873 while the line 120574 = 1205740119894
(119868119903 120583) + Δ120574
+
(119868119903) transversely
intersects the segments119874119894+1
(120583) at the end point 119889119894+1
(120583)when119894 = 1 119873 minus 1 For all 119894 = 2 119873 minus 1 the difference inthe coordinates ℎ of two end points 119888
119894(120583) and 119889
119894+1(120583) is zero
namely
ℎ (119888119894(120583)) minus ℎ (119889
119894+1(120583)) = 0 (58)
For each 119894 = 2 119873 minus 1 one of the heteroclinic orbitsrepresented by Γ
119894and contained in the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) at the value 120583 = 120583 connects two intersectionpoints 119888
119894(120583) and 119889
119894+1(120583) Therefore a heteroclinic orbit Γ
1
on the limiting surfaces sum1205832 12057811198866 12057610
(12057401
) connects the certainpoint 119888
1(120583) on the annulus 119872 to the end point 119889
2(120583) on the
segment 1198742(120583) It is also known that a heteroclinic orbit Γ
119873
on the limiting surfaces sum1205832120578111988661205761
0
(1205740119873
) connects the endpoint 119888
119873(120583) on the segments 119874
119873(120583) to the certain point
119889119873+1
(120583) on the annulus119872 According to the study of Kovacicet al [15ndash17] there exists an 119899-bump singular transitionorbit or a modified 119873-bump singular transition orbit The3-bump jumping orbit depicted in Figure 5 consists of theheteroclinic orbits Γ
119894( 119894 = 1 2 3) on the limiting surfaces
sum1205832120578111988661205761
0
(1205740119894
) (119894 = 1 2 3) at the parameter 120583 = 120583 and theorbit segments 119874
1(120583) and 119874
2(120583) of (44a) (44b) It is known
from the above analysis that the orbit segments 119874119894(120583) (119894 =
2 119873) intersect transversely with the lines 120574 = 1205740119894
(119868119903 120583) +
Δ120574+
(119868119903) and 120574 = 120574
0119894
(119868119903 120583) minus Δ120574
minus
(119868119903)
The 2-bump singular surface shown in Figure 6 is com-posed of two single-pulse singular intersection surfacessum1205832120578111988661205761
0
(120574119896minus11
) andsum1205832120578111988661205761
0
(120574119896minus12
)This surface connectsthe singular points of (44a) (44b) that lie on the line 120574 = 120574
01
minus
Δ120574minus to those of (44a) (44b) that lie on the line 120574 = 120574
01
minusΔ120574+
on the annulus119872We obtain a countable infinity of the singular heteroclinic
jumping orbits as follows Each orbit starts along one branchof the manifold 119882(119876
0) of the saddle 119876
0on the annulus 119872
Then the singular heteroclinic jumping orbit departs fromthe annulus 119872 goes along one of the singular 119896-pulse orbitsΓ119896 and lands back at a point on the separatrix that connects
the saddle1198760to itself on the annulus119872 After traveling along
the separatrix for a while the singular heteroclinic jumpingorbit takes off again along the singular 119897-pulse orbit Γ
119897and
continues such process Eventually the singular heteroclinicjumping orbit lands back on the separatrix
Therefore it is concluded that the multipulse orbits of(21a) (21b) (21c) (21d) consist of several portions of the slowtime scale on the hyperbolic manifold119872
120576andmany fast time
scale heteroclinic pulses leaving from the manifold 119872120576 and
these multipulse heteroclinic orbits form a consecutive andrecurrence process
6 Numerical Results of Chaotic Motions
Based on the above qualitative analysis for the multipulseorbits and chaotic dynamics of the laminated composite
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
120574
|u|h
Γ3
Γ2
Γ1
M
O2O3
12057403 + Δ120574+ 12057402 + Δ120574+ 12057401 + Δ120574+12057403 minus Δ120574minus 12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 5 The 3-bump orbit with the single-pulse is depicted
piezoelectric rectangular plate the conditions of the chaoticmotion under the sense of the Smale horses are obtainedTheheteroclinic bifurcations of (12a) (12b) (12c) (12d) appearwhen 120578
1gt 0 Therefore the above theoretical analysis
is focused on the situation which there exist heteroclinicbifurcations in (12a) (12b) (12c) (12d) The parameter 120578
1
is the combination of the parameters 1198866 1198867 and 119887
6 where
1205781
= (911988661198867)21198876 In this section we have only performed
numerical simulations of the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateunder heteroclinic bifurcations in order to further verify thetheoretical analysis Consequently the parameters 119886
6 1198867 and
1198876are chosen to satisfy 120578
1gt 0
We choose the averaged equation (12a) (12b) (12c) (12d)to conduct numerical simulations A numerical approachthrough the computer software Matlab is utilized to explorethe existence of the Shilnikov type multipulse chaoticmotions in the laminated composite piezoelectric rectangularplate Based on the above qualitative analysis it is found thatthe damping coefficients 120583
1 1205832and transverse excitation 119891
2
play an important role in the multipulse chaotic motionsof the laminated composite piezoelectric rectangular plateIn addition the parameters 119886
2and 119886
3are related to the in-
plane excitation in the119909-direction and the in-plane excitationin the 119910-direction respectively The parameter 119886
4is the
piezoelectric excitation which reflects the characteristics ofthe piezoelectric material Hence the parameters 120583
1 1198862 1198864
and 1198912are selected as the controlling parameters to discover
the law for complicated nonlinear dynamics of the laminatedcomposite piezoelectric rectangular plate
We begin to draw bifurcation diagrams of the parameters1198912 1205831 1198862 and 119886
4 Bifurcation diagrams describe the vibration
law of the modal displacements 1199091and 119909
3 respectively when
the parameters 1198912 1205831 1198862 and 119886
4change in a certain region
We draw bifurcation diagrams according to the rules of theRunge-Kutta algorithm and the Poincare map theory For theperiodic motions Poincare map is of several separate pointsFor a chaotic motion the Poincare map consists of a numberof points on the limited Poincare sectionTherefore it can beobserved that chaotic motion and periodic motion of nonlin-ear systemappear frombifurcation diagramsThe chaotic and
120574
|u|h
M
Γsum (12057401 12057402)
12057402 + Δ120574+ 12057401 + Δ120574+
12057402 minus Δ120574minus 12057401 minus Δ120574minus
Figure 6 The 2-pulse singular surfaces sum(12057401
12057402
) are depicted
periodic responses can be identified by several conventionalcriteria such as phase portraits and Poincare map Basedon the response law of bifurcation diagrams phase portraitsand Poincare map are utilized to further verify the existenceof the chaotic motions and the periodic motions In orderto compare the influence of these parameters 119891
2 1205831 1198862
and 1198864on nonlinear vibration in the laminated composite
piezoelectric rectangular plate we choose the same initialconditions to carry out numerical simulation
Figure 7 illustrates the bifurcation diagram of the lam-inated composite piezoelectric rectangular plate when theexcitation 119891
2varies in the interval 119891
2= 2 sim 200
Other parameters and initial conditions are chosen as 1205901=
183 1205902
= 197 1205831
= 02 1205832
= 02 1198862
= 230 1198863
=
120 1198864= 130 119886
5= minus101 119886
6= minus203 119886
7= minus205 119887
6=
407 1198877= minus308 119887
8= 109 119909
10= minus001 119909
20= minus005 119909
30=
minus001 11990940
= minus001 Figures 7(a) and 7(b) represent thebifurcation diagram on the plane (119909
1 1198912) and (119909
3 1198912) respec-
tively It is observed from Figure 7 that the excitation 1198912
is an important parameter that influences on the nonlineardynamic responses of the laminated composite piezoelectricrectangular plate Figure 7 shows that the chaotic motionof the laminated composite piezoelectric rectangular plateappears first followed by a periodic motion of that With theincrease of excitation 119891
2 Figure 7 presents the following law
chaotic motionrarrmulti-period motionWe study the impact of the damping parameter on the
nonlinear dynamic responses of the laminated compositepiezoelectric rectangular plate Figure 8 is the bifurcationdiagramof the laminated composite piezoelectric rectangularplate with the damping coefficient 120583
1 The figure demon-
strates that system is beginning to enter into the region ofthe chaoticmotion then appears the periodicmotionwindowand finally comes into the region of the chaotic motion againas the damping coefficient 120583
1varies in the interval 120583
1=
001 sim 07 Other parameters and initial conditions are thesame as those in Figure 7 when excitation is chosen as 119891
2=
827 Figures 8(a) and 8(b) describe the nonlinear motion ofthe laminated composite piezoelectric rectangular plate on
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
50 100 150minus2
minus15
minus1
minus05
0
05
1
x1
f2
(a)
20 40 60 80 100 120 140 160 180
minus3
minus25
minus2
minus15
x3
f2
(b)
Figure 7 The bifurcation diagram is obtained for the excitation 1198912
= 2sim200 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
=
minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1198912
) (b) the bifurcation diagram on the plane (1199093
1198912
)
0 01 02 03 04 05 06 07minus25
minus2
minus15
minus1
minus05
0
05
1
15
x1
1205831
(a)
0 01 02 03 04 05 06 07minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
x1
1205831
x3
1205831
x3
(b)
Figure 8 The bifurcation diagram is obtained for the damping coefficient 1205831
= 001sim07 the excitation 1198912
= 827 and initial conditions11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane (1199091
1205831
) (b) the bifurcation diagram on theplane (119909
3
1205831
)
the planes (1199091 1205831) and (119909
3 1205831) respectively as well as the
impact of the damping coefficient 1205831on the system
Figure 9 portraysthe bifurcation diagram for the lami-nated composite piezoelectric rectangular plate when the in-plane excitation 119886
2in the 119909-direction varies in the interval
1198862= 2sim65 Other parameters and initial conditions remain
the same as those in Figure 8 when the damping coefficient1205831is selected as 120583
1= 02 Figures 9(a) and 9(b) display
the bifurcation diagram on the plane (1199091 1198862) and (119909
3 1198862)
respectively Figure 9 presents that the beginning movementof the system is the periodic motion then the system appearsthe chaotic motion With the increase of the excitation1198862 Figure 9 shows the following evolution law periodic
motionrarr chaotic motion
Figure 10 indicates the bifurcation diagram for the lam-inated composite piezoelectric rectangular plate when thepiezoelectric excitation 119886
4varies from 119886
4= 2 to 119886
4= 120
Other parameters and initial conditions remain the sameas those in Figure 9 when the in-plane excitation 119886
2is
chosen as 1198862
= 23 Figures 10(a) and 10(b) demonstratethe bifurcation diagram on the planes (119909
1 1198864) and (119909
3 1198864)
respectively It is observed from Figure 10 that the piezoelec-tric excitation 119886
4has a significant influence on the compli-
cated nonlinear dynamic behaviors of the laminated com-posite piezoelectric rectangular plate As the piezoelectricexcitation 119886
4increases Figure 10 reveals the following
law chaotic motionrarrmultiperiod motionrarr one-periodmotionrarrmultiperiod motionrarr chaotic motion
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
0 10 20 30 40 50 60
minus3
minus2
minus1
0
1
x1
a2
(a)
0 10 20 30 40 50 60minus34
minus32
minus3
minus28
minus26
minus24
minus22
minus2
minus18
a2
x3
(b)
Figure 9 The bifurcation diagram is obtained for the in-plane excitation 1198862
= 2sim65 the damping coefficients 1205831
= 02 and 1205832
= 02 theexcitation 119891
2
= 827 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
= minus001 (a) the bifurcation diagram on the plane(1199091
1198862
) (b) the bifurcation diagram on the plane (1199093
1198862
)
0 20 40 60 80 100 120minus25
minus2
minus15
minus1
minus05
0
05
1
x1
a4
(a)
0 20 40 60 80 100 120
minus35
minus3
minus25
minus2
a4
x3
(b)
Figure 10 The bifurcation diagram is obtained for the piezoelectric excitation 1198864
= 2sim120 the excitation 1198912
= 827 the damping coefficients1205831
= 02 and 1205832
= 02 the in-plane excitations 1198862
= 23 and 1198863
= 120 and initial conditions 11990910
= minus001 11990920
= minus005 11990930
= minus001 11990940
=
minus001 (a) the bifurcation diagram on the plane (1199091
1198864
) (b) the bifurcation diagram on the plane (1199093
1198864
)
Based on the above bifurcation diagram the excitations1198912 1198862 1198864 and the damping coefficient 120583
1are selected as
specific values in order to find themultipulse chaoticmotionsof the laminated composite piezoelectric rectangular plateFigure 11 indicates existence of themultipulse chaotic motionof the laminated composite piezoelectric rectangular platewhen the excitation 119891
2is 827 In this case the chosen
parameters and initial conditions are the same as those inFigure 7 Figures 11(a) and 11(b) are the three-dimensionalphase portrait in the space (119909
1 1199092 1199093) and the Poincare
map on the plane (1199091 1199092) respectively Figure 11 shows that
the excitation 1198912has a noticeable effect on the existence of
the multipulse chaotic motions on the laminated compositepiezoelectric rectangular plate
Besides the excitations 1198912 1198862 1198864and the damping coef-
ficient 1205831 the multipulse chaotic motions of the laminated
composite piezoelectric rectangular plate also depend onother parameters Figure 12 is obtained when the parametersand initial conditions are chosen as 120590
1= 1437 120590
2=
1142 1205831= 02 120583
2= 02 119886
2= 300 119886
3= 750 119886
4= 450
1198865
= minus1166 1198866
= 1227 1198912
= 1227 1198867
= minus268 1198876
=
minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 0511990930
= minus001 11990940
= 916 Comparing with Figures 11and 12 it is found that there are differences in the phaseportrait and the Poincare map respectively From thethree-dimensional phase portrait in Figure 12 we can see thatthere exists obvious multipulse jumping phenomenon Thethree-dimensional phase portrait is composed of the four
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
minus1minus050051
minus05
0
05
minus25
minus2
minus15
minus1
x1
x2
x3
(a)
minus15 minus1 minus05 0 05 1 15 2
minus1
minus05
0
05
1
x1
x2
(b)
Figure 11 The multipulse chaotic motion is obtained when 1198912
= 827 and 1205831
= 02 (a) the phase portrait in the three-dimensional space(1199091
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
minus50
5
minus50
5minus6
minus4
minus2
0
2
4
6
x1x2
x3
(a)
minus10 minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 12 The multipulse chaotic motion is obtained when 1205901
= 1437 1205902
= 1142 1198862
= 300 1198863
= 750 1198864
= 450 1198865
= minus1166 1198866
=
1227 1198912
= 1227 1198867
= minus268 1198876
= minus22 1198877
= minus969 1198878
= minus2232 11990910
= minus108 11990920
= 05 11990930
= minus001 11990940
= 916 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
regionsThe different regions are connected by themultipulseorbits
In the following numerical simulations several differ-ent sets of parameters and initial conditions are givenin order to investigate the different shapes of the multi-pulse chaotic motion Figure 13 demonstrates the multipulsechaotic response in the laminated composite piezoelectricrectangular plate for 119891
2= 9238 Some parameters and
initial conditions are chosen as 1205901= 361 120590
2= 313 119886
5=
minus1501 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
=
minus005 11990940
= minus005 In this case other parameters are thesame as those in Figure 7 From Figure 13 we can see thatthere is another shape for the multipulse chaotic motion Itis found that the shapes of these two phenomena depicted inFigures 12 and 13 are completely different From the three-dimensional phase portrait in Figure 13 it is found thatmultipulse jumping phenomenon is more prominent
7 Conclusions
In this paper the nonlinear vibrations of the laminated com-posite piezoelectric rectangular plate are studied by applyingthe theories of the global bifurcations and chaotic dynamicsfor high-dimensional nonlinear systems The multipulseheteroclinic orbits and chaotic dynamics are investigatedusing the extended Melnikov method for the case where theaveraged equations have one nonsemisimple double zero anda pair of pure imaginary eigenvaluesThe extendedMelnikovmethod can be applied to study the Shilnikov typemultipulseheteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applicationsAnalysis of themultipulse heteroclinic orbits in the laminatedcomposite piezoelectric rectangular plate demonstrates thatsuch an analysis is a typical singular perturbation problemin which there are two different time scales Dynamics onthe hyperbolic manifold 119872
120576are of the slow time scale and
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
minus50
5
minus10
0
10
minus6
minus4
minus2
0
2
x1x2
x3
(a)
minus5 0 5 10
minus10
minus5
0
5
10
x1
x2
(b)
Figure 13 The multipulse chaotic motion is obtained when 1205901
= 361 1205902
= 313 1198862
= 230 1198863
= 120 1198864
= 130 1198865
= minus1501 1198866
=
minus203 1198912
= 9238 1198867
= minus205 1198876
= 407 1198877
= minus308 1198878
= 409 11990910
= minus001 11990920
= minus009 11990930
= minus005 11990940
= minus005 (a) the phase portrait inthe three-dimensional space (119909
1
1199092
1199093
) (b) Poincare map on the plane (1199091
1199092
)
the multipulse jumping orbits taking off from this manifoldare of the fast time scale It is shown that the transfer ofenergy between the two different modes occurs throughthe multipulse jumping orbits The studies have led to thefollowing conclusions
(1) There exist the Shilnikov type multipulse chaoticmotions in nonlinear vibration of the laminated compositepiezoelectric rectangular plate The geometric interpretationof the 119896-pulse Melnikov function is a signed distance mea-sured along the normal to a heteroclinic manifold whichgives the more delicate local estimates near the hyperbolicmanifold In the resonant case the 119896-pulse extended Mel-nikov function 119872
119896(120576 119868 120574
0 1205832
) does not depend on the smallperturbation parameter 0 lt 120576 ≪ 1 and the nonfolding condi-tion is automatically satisfied resulting in the angle functionΓ119895(120576 119868119903 1205740 1205832
) (119895 = 0 1 119896 minus 1) being zero Therefore thecomputing procedure of the extendedMelnikov function canbe simplified
(2) In order to verify the theoretical predictions numer-ical simulation is used to examine the bifurcations andchaotic motions of the laminated composite piezoelectricrectangular plate Several types of the bifurcation diagramsare obtained when the transverse excitation 119891
2 the in-plane
excitation 1198862 the piezoelectric excitation 119886
4 and the damping
coefficient 1205831are chosen as several different kinds of control
parameters Based on the bifurcation diagrams the nonlinearcomplicated dynamic behavior of the laminated compositepiezoelectric rectangular plate is controlled by varying theexcitations 119891
2 1198862 1198864and the damping coefficient 120583
1 respec-
tively Therefore the excitations 1198912 1198862 1198864and the damping
coefficient 1205831have important influence on the nonlinear
dynamics responses of the laminated composite piezoelectricrectangular plate
(3) There exist different shapes of the chaotic motionsin the nonlinear oscillations of the laminated compositepiezoelectric rectangular plate under different excitationsparameters and initial conditions It is found from numerical
simulations that the shapes of the chaotic motions are com-pletely different From the three-dimensional phase portraitsin Figures 12 and 13 it is found that there exist obviousmultipulse jumping phenomena Therefore parameters andinitial conditions impact the shapes of the multipulse chaoticmotions
(4)There existmultipulse chaoticmotions in the averagedequations It is well known that the multipulse chaoticmotions in the averaged equations can lead to the multi-pulse amplitude modulated chaotic vibrations in the originalsystem under certain conditions Therefore the multipulseamplitudemodulated chaoticmotions occur in the laminatedcomposite piezoelectric rectangular plate
In summary both theoretical and numerical studiessuggest that chaos for the Smale horseshoe sense in nonlinearmotion of the simply supported laminated composite piezo-electric rectangular plate exists
Conflict of Interests
The authors declare that there is no conflict of interests in thispaper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NNSFC)through Grant nos 11172009 11372015 10872010 1129015210732020 and 11072008 the National Science Foundationfor Distinguished Young Scholars of China (NSFDYSC)through Grant no 10425209 the Funding Project for Aca-demic Human Resources Development in Institutions ofHigher Learning under the Jurisdiction of Beijing Munici-pality (PHRIHLB) the Foundation of Beijing University ofTechnology through Grant no X4001015201301 the PhDPrograms Foundation of Beijing University of Technology(DPFBUT) through Grant no 52001015200701
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 19
References
[1] H S Tzou J P Zhong and J J Hollkamp ldquoSpatially distributedorthogonal piezoelectric shell actuators theory and applica-tionsrdquo Journal of Sound and Vibration vol 177 no 3 pp 363ndash378 1994
[2] A S Purekar D J Pines S Sundararaman and D E AdamsldquoDirectional piezoelectric phased array filters for detectingdamage in isotropic platesrdquo Smart Materials and Structures vol13 no 4 pp 838ndash850 2004
[3] M Ishihara and N Noda ldquoControl of mechanical deformationof a laminate by piezoelectric actuator taking into account thetransverse shearrdquo Archive of Applied Mechanics vol 74 no 1-2pp 16ndash28 2004
[4] I K Oh ldquoThermopiezoelastic nonlinear dynamics of activepiezolaminated platesrdquo Smart Materials and Structures vol 14no 4 pp 823ndash834 2005
[5] S J Lee J N Reddy and F Rostam-Abadi ldquoNonlinear finiteelement analysis of laminated composite shells with actuatinglayersrdquo Finite Elements in Analysis and Design vol 43 no 1 pp1ndash21 2006
[6] S Panda and M C Ray ldquoActive constrained layer dampingof geometrically nonlinear vibrations of functionally gradedplates using piezoelectric fiber-reinforced compositesrdquo SmartMaterials and Structures vol 17 no 2 Article ID 025012 2008
[7] P C Dumir P Kumari and S Kapuria ldquoAssessment of thirdorder smeared and zigzag theories for buckling and vibration offlat angle-ply hybrid piezoelectric panelsrdquoComposite Structuresvol 90 no 3 pp 346ndash362 2009
[8] M H Yao and W Zhang ldquoMulti-Pulse chaotic motions ofhigh-dimension nonlinear system for a laminated compositepiezoelectric rectangular platerdquoMeccanica 2013
[9] Z C Feng and P R Sethna ldquoGlobal bifurcations in the motionof parametrically excited thin platesrdquo Nonlinear Dynamics vol4 no 4 pp 389ndash408 1993
[10] W M Tien N S Namachchivaya and A K Bajaj ldquoNon-linear dynamics of a shallow arch under periodic excitationmdashI1 2 internal resonancerdquo International Journal of Non-LinearMechanics vol 29 no 3 pp 349ndash366 1994
[11] N Malhotra and N Sri Namachchivaya ldquoChaotic motion ofshallow arch structures under 1 1 internal resonancerdquo Journalof Engineering Mechanics vol 123 no 6 pp 620ndash627 1997
[12] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239 no5 pp 1013ndash1036 2001
[13] M H Yeo and W K Lee ldquoEvidences of global bifurcations ofan imperfect circular platerdquo Journal of Sound and Vibration vol293 no 1-2 pp 138ndash155 2006
[14] W Yu and F Chen ldquoGlobal bifurcations of a simply supportedrectangular metallic plate subjected to a transverse harmonicexcitationrdquo Nonlinear Dynamics vol 59 no 1-2 pp 129ndash1412010
[15] G Kovacic and T A Wettergren ldquoHomoclinic orbits in thedynamics of resonantly driven coupled pendulardquo Zeitschrift furAngewandte Mathematik und Physik vol 47 no 2 pp 221ndash2641996
[16] T J Kaper and G Kovacic ldquoMulti-bump orbits homoclinic toresonance bandsrdquo Transactions of the American MathematicalSociety vol 348 no 10 pp 3835ndash3887 1996
[17] R Camassa G Kovacic and S K Tin ldquoA Melnikov methodfor homoclinic orbits with many pulsesrdquo Archive for RationalMechanics and Analysis vol 143 no 2 pp 105ndash193 1998
[18] W Zhang and M H Yao ldquoTheories of multi-pulse globalbifurcations for high-dimensional systems and application tocantilever beamrdquo International Journal of Modern Physics B vol22 no 24 pp 4089ndash4141 2008
[19] G Haller and S Wiggins ldquoN-pulse homoclinic orbits inperturbations of resonant hamiltonian systemsrdquo Archive forRationalMechanics and Analysis vol 130 no 1 pp 25ndash101 1995
[20] N Malhotra N Sri Namachchivaya and R J McDonaldldquoMultipulse orbits in the motion of flexible spinning discsrdquoJournal of Nonlinear Science vol 12 no 1 pp 1ndash26 2002
[21] W Yu and F Chen ldquoOrbits homoclinic to resonances in aharmonically excited and undamped circular platerdquoMeccanicavol 45 no 4 pp 567ndash575 2010
[22] J N ReddyMechanics of LaminatedComposite Plates and ShellsSpringer New York NY USA 2004
[23] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[24] W Zhang F Wang and J W Zu ldquoComputation of normalforms for high dimensional non-linear systems and applicationto non-planar non-linear oscillations of a cantilever beamrdquoJournal of Sound and Vibration vol 278 no 4-5 pp 949ndash9742004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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