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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 418374, 26 pageshttp://dx.doi.org/10.1155/2013/418374
Research ArticleVibration, Stability, and Resonance of Angle-Ply CompositeLaminated Rectangular Thin Plate under Multiexcitations
M. Sayed1,2 and A. A. Mousa1,3
1 Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 888, Al-Taif, Saudi Arabia2Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt3 Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shibin El-Kom, Egypt
Correspondence should be addressed to M. Sayed; moh 6 [email protected]
Received 10 November 2012; Revised 15 April 2013; Accepted 1 May 2013
Academic Editor: Dane Quinn
Copyright ยฉ 2013 M. Sayed and A. A. Mousa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plateunder parametric and external excitations is presented.Themethod of multiple time scale perturbation is applied to solve the non-linear differential equations describing the systemup to and including the second-order approximation. All possible resonance casesare extracted at this approximation order.The case of 1 : 1 : 3 primary and internal resonance, whereฮฉ
3โ ๐1,๐2โ ๐1, and๐
3โ 3๐1,
is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influ-ences of the cubic terms onnonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied.The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integrationof themodal equations. Reliability of the obtained results is verified by comparison between the finite differencemethod (FDM) andRunge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of suchsystem. Such cases should be avoided as working conditions for the system. Variation of the parameters ๐
1, ๐2, ๐ผ7, ๐ฝ8, ๐1, ๐2, ๐1, ๐2
leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of thesystem are reported. Comparison with the available published work is reported.
1. Introduction
Composite laminated plates that are widely used in severalengineering fields such as machinery, shipbuilding, aircraft,automobiles, robot arm, watercraft-hydropower, and wingsof helicopters are made of the angle-ply composite laminatedplates. Several researchers have focused their attention onstudying the nonlinear dynamics, bifurcations, and chaos ofthe composite laminated plates.
Internal resonance has been found in many engineeringproblems in which the natural frequencies of the system arecommensurable. Ye et al. [1] investigated the local and globalnonlinear dynamics of a parametrically excited symmetriccross-ply composite laminated rectangular thin plate underparametric excitation. The study is focused on the case of 1 : 1internal resonance and primary parametric resonance. Zhang[2] dealt with the global bifurcations and chaotic dynamics of
a parametrically excited, simply supported rectangular thinplate. The method of multiple scales is used to obtain theaveraged equations. The case of 1 : 1 internal resonance andprimary parametric resonance is considered. Guo et al.[3] studied the nonlinear dynamics of a four-edge simplysupported angle-ply composite laminated rectangular thinplate excited by both the in-plane and transverse loads. Theasymptotic perturbation method is used to derive the fouraveraged equations under 1 : 1 internal resonance. Zhang et al.[4] investigated the local and global bifurcations of a para-metrically and externally excited simply supported rectan-gular thin plate under simultaneous transversal and in-planeexcitations.The studies are focused on the case of 1 : 1 internalresonance and primary parametric resonance. Tien et al. [5]applied the averaging method and Melnikov technique tostudy local, global bifurcations and chaos of a two-degrees-of-freedom shallow arch subjected to simple harmonic
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2 Mathematical Problems in Engineering
excitation for case of 1 : 2 internal resonances. Sayed andMousa [6] investigated the influence of the quadratic andcubic terms on nonlinear dynamic characteristics of theangle-ply composite laminated rectangular plate with para-metric and external excitations. Two cases of the subhar-monic resonances cases in the presence of 1 : 2 internal reso-nances are considered.Themethod ofmultiple time scale per-turbation is applied to solve the nonlinear differential equa-tions describing the system up to and including the second-order approximation. Zhang et al. [7] gave further studies onthe nonlinear oscillations and chaotic dynamics of a para-metrically excited simply supported symmetric cross-plylaminated composite rectangular thin plate with the geo-metric nonlinearity and nonlinear damping. Zhang et al. [8]dealt with the nonlinear vibrations and chaotic dynamics ofa simply supported orthotropic functionally graded mate-rial (FGM) rectangular plate subjected to the in-plane andtransverse excitations together with thermal loading in thepresence of 1 : 2 : 4 internal resonance, primary parametricresonance, and subharmonic resonance of order 1/2. Zhanget al. [9] investigated the bifurcations and chaotic dynamics ofa simply supported symmetric cross-ply composite laminatedpiezoelectric rectangular plate subject to the transverse, in-plane excitations and the excitation loaded by piezoelectriclayers. Zhang and Li [10] analyzed the resonant chaoticmotions of a simply supported rectangular thin plate withparametrically and externally excitations using exponentialdichotomies and an averaging procedure. Zhang et al. [11]analyzed the chaotic dynamics of a six-dimensional nonlinearsystem which represents the averaged equation of a compos-ite laminated piezoelectric rectangular plate subjected to thetransverse, in-plane excitations and the excitation loaded bypiezoelectric layers. The case of 1 : 2 : 4 internal resonances isconsidered. Zhang and Hao [12] studied the global bifurca-tions and multipulse chaotic dynamics of the composite lam-inated piezoelectric rectangular plate by using the improvedextended Melnikov method. The multipulse chaotic motionsof the system are found by using numerical simulation, whichfurther verifies the result of theoretical analysis. Guo andZhang [13] studied the nonlinear oscillations and chaoticdynamics for a simply supported symmetric cross-ply com-posite laminated rectangular thin plate with parametric andforcing excitations. The case of 1 : 2 : 3 internal resonance isconsidered. The method of multiple scales is employed toobtain the six-dimensional averaged equation.The numericalmethod is used to investigate the periodic and chaoticmotions of the composite laminated rectangular thin plate.Eissa and Sayed [14โ16] and Sayed [17] investigated the effectsof different active controllers on simple and spring pendulumat the primary resonance via negative velocity feedback orits square or cubic. Sayed and Kamel [18, 19] investigated theeffects of different controllers on the vibrating system and thesaturation control to reduce vibrations due to rotor blade flap-ping motion. The stability of the system is investigated usingboth phase-plane method and frequency response curves.Sayed and Hamed [20] studied the response of a two-degree-of-freedom systemwith quadratic coupling under parametricand harmonic excitations. The method of multiple scaleperturbation technique is applied to solve the nonlinear
differential equations and obtain approximate solutions up toand including the second-order approximations. Amer et al.[21] investigated the dynamical system of a twin-tail aircraft,which is described by two coupled second-order nonlin-ear differential equations having both quadratic and cubicnonlinearities under different controllers. Best active controlof the system has been achieved via negative accelera-tion feedback. The stability of the system is investigatedapplying both frequency response equations and phase-plane method. Hamed et al. [22] studied the nonlineardynamic behavior of a string-beam coupled system subjectedto external, parametric, and tuned excitations that are pre-sented.The case of 1 : 1 internal resonance between themodesof the beam and string, and the primary and combined reson-ance for the beam is considered. The method of multiplescales is applied to obtain approximate solutions up to andincluding the second-order approximations. All resonancecases are extracted and investigated. Stability of the systemis studied using frequency response equations and the phase-plane method. Awrejcewicz et al. [23โ25] studied the chaoticdynamics of continuous mechanical systems such as flexibleplates and shallow shells.The considered problems are solvedby the Bubnov-Galerkin, Ritz method with higher approxi-mations, and finite difference method. Convergence and val-idation of those methods are studied. Awrejcewicz et al. [26]investigated the chaotic vibrations of flexible nonlinear Euler-Bernoulli beams subjected to harmonic load andwith variousboundary conditions. Reliability of the obtained results isverified by the finite difference method and finite elementmethodwith the Bubnov-Galerkin approximation for variousboundary conditions and various dynamic regimes.
In this paper, the perturbation method and stabilityof the composite laminated piezoelectric rectangular plateunder simultaneous transverse and in-plane excitations areinvestigated. The method of multiple scales are applied toobtain the second-order uniform asymptotic solutions. Allpossible resonance cases are extracted at this approximationorder. The study is focused on the case of 1 : 1 : 3 internalresonance and primary resonance.The stability of the systemand the effects of different parameters on system behaviorhave been studied using frequency response curves. Stabilityis performed of figures by solid and dotted lines. Theanalytical results given by the method of multiple timescale is verified by comparison with results from numericalintegration of the modal equations. It is quite clear thatsome of the simultaneous resonance cases are undesirable inthe design of such system. Such cases should be avoided asworking conditions for the system. Some recommendationsregarding the different parameters of the system are reported.Comparison with the available published work is reported.
2. Mathematical Analysis
Consider a simply supported four edges composite laminatedpiezoelectric rectangular plate of lengths ๐, ๐ and thickness โ,as shown in Figure 1. The composite laminated piezoelectricrectangular plate is considered as regular symmetric cross-ply laminates with ๐ layers. A Cartesian coordinate system is
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Mathematical Problems in Engineering 3
y
b
q cosฮฉ3t
q0 + qx cosฮฉ1t
q1 + qy cosฮฉ2t
x
a
Figure 1: The model of the composite laminated piezoelectricrectangular plate.
located in the middle surface of the plate. Assume that ๐ข, V,and ๐ค represent the displacements of an arbitrary point ofthe composite laminated piezoelectric rectangular plate in the๐ฅ, ๐ฆ, and ๐ง directions, respectively. The in-plane excitationsare loaded along the ๐ฆdirection at ๐ฅ = 0 in the form๐0+ ๐๐ฅcosฮฉ1๐ก, and the excitations are loaded along the ๐ฅ
direction at ๐ฆ = 0 in the form ๐1+ ๐๐ฆcosฮฉ2๐ก. The trans-
verse excitation subjected to the composite laminated piezo-electric rectangular plate is represented by ๐ cosฮฉ
3๐ก. The
dynamic electrical loading is expressed as ๐ธ๐งcosฮฉ4๐ก. Based
on Reddyโs third-order shear deformation plate theory [27],the displacement field at an arbitrary point in the compositelaminated plate is expressed as [13]
๐ข (๐ฅ, ๐ฆ, ๐ก) = ๐ข0(๐ฅ, ๐ฆ, ๐ก) + ๐ง๐
๐ฅ(๐ฅ, ๐ฆ, ๐ก) โ ๐ง
3 4
3โ2(๐๐ฅ+๐๐ค0
๐๐ฅ) ,
(1a)
V (๐ฅ, ๐ฆ, ๐ก) = V0(๐ฅ, ๐ฆ, ๐ก) + ๐ง๐
๐ฆ(๐ฅ, ๐ฆ, ๐ก) โ ๐ง
3 4
3โ2(๐๐ฆ+๐๐ค0
๐๐ฆ) ,
(1b)
๐ค (๐ฅ, ๐ฆ, ๐ก) = ๐ค0(๐ฅ, ๐ฆ, ๐ก) , (1c)
where ๐ข0, V0, and ๐ค
0are the original displacement on the
midplane of the plate in the๐ฅ,๐ฆ, and ๐งdirections, respectively.Let ๐๐ฅand ๐
๐ฆrepresent the midplane rotations of transverse
normal about the ๐ฅ and ๐ฆ axes, respectively. From thevan Karman-type plate theory and Hamiltonโs principle, thenonlinear governing equations of motion of the compositelaminated piezoelectric rectangular plate are given as follows[12]:
๐๐๐ฅ๐ฅ
๐๐ฅ+
๐๐๐ฅ๐ฆ
๐๐ฆ= ๐ผ0๏ฟฝฬ๏ฟฝ0+ ๐ฝ1๏ฟฝฬ๏ฟฝ๐ฅโ4
3โ2๐ผ3
๐๏ฟฝฬ๏ฟฝ0
๐๐ฅ, (2a)
๐๐๐ฅ๐ฆ
๐๐ฅ+
๐๐๐ฆ๐ฆ
๐๐ฆ= ๐ผ0Vฬ0+ ๐ฝ1๏ฟฝฬ๏ฟฝ๐ฆโ4
3โ2๐ผ3
๐๏ฟฝฬ๏ฟฝ0
๐๐ฆ, (2b)
๐๐๐ฅ
๐๐ฅ+
๐๐๐ฆ
๐๐ฆ+๐
๐๐ฅ(๐๐ฅ๐ฅ
๐๐ค0
๐๐ฅ+ ๐๐ฅ๐ฆ
๐๐ค0
๐๐ฆ)
+๐
๐๐ฆ(๐๐ฅ๐ฆ
๐๐ค0
๐๐ฅ+ ๐๐ฆ๐ฆ
๐๐ค0
๐๐ฆ)
+4
3โ2(๐2๐๐ฅ๐ฅ
๐๐ฅ2+ 2
๐2๐๐ฅ๐ฆ
๐๐ฅ๐๐ฆ+
๐2๐๐ฆ๐ฆ
๐๐ฆ2) + ๐
= ๐ผ0๏ฟฝฬ๏ฟฝ0โ16
9โ4๐ผ6(๐2๏ฟฝฬ๏ฟฝ0
๐๐ฅ2+๐2๏ฟฝฬ๏ฟฝ0
๐๐ฆ2)
+4
3โ2[๐ผ3(๐๏ฟฝฬ๏ฟฝ0
๐๐ฅ+๐Vฬ0
๐๐ฆ) + ๐ฝ4(๐๏ฟฝฬ๏ฟฝ๐ฅ
๐๐ฅ+
๐๏ฟฝฬ๏ฟฝ๐ฆ
๐๐ฆ)] ,
(2c)
๐๐๐ฅ๐ฅ
๐๐ฅ+
๐๐๐ฅ๐ฆ
๐๐ฆโ ๐๐ฅ= ๐ฝ1๏ฟฝฬ๏ฟฝ0+ ๐2๏ฟฝฬ๏ฟฝ๐ฅโ4
3โ2๐ฝ4
๐๏ฟฝฬ๏ฟฝ0
๐๐ฅ, (2d)
๐๐๐ฅ๐ฆ
๐๐ฅ+
๐๐๐ฆ๐ฆ
๐๐ฆโ ๐๐ฆ= ๐ฝ1Vฬ0+ ๐2๏ฟฝฬ๏ฟฝ๐ฆโ4
3โ2๐ฝ4
๐๏ฟฝฬ๏ฟฝ0
๐๐ฆ. (2e)
Theboundary conditions of the simply supported rectangularcomposite laminated plate are expressed as follows:
at ๐ฅ = 0, ๐ฅ = ๐ : ๐๐ข๐๐ฅ= 0,
๐ค = ๐ข = ๐๐ฅ= ๐๐ฆ= ๐๐ฅ๐ฅ= ๐๐ฅ๐ฆ= ๐๐ฅ= 0,
(3a)
at ๐ฆ = 0, ๐ฆ = ๐ : ๐V๐๐ฆ= 0,
๐ค = V = ๐๐ฅ๐ฆ= ๐๐ฆ๐ฆ= ๐๐ฆ= 0,
(3b)
โซ
๐
0
๐๐ฅ๐ฅ
๐ฅ=0, ๐๐๐ฆ = โซ
๐
0
(๐0+ ๐๐ฅcosฮฉ1๐ก) ๐๐ฆ, (3c)
โซ
๐
0
๐๐ฆ๐ฆ
๐ฆ=0, ๐๐๐ฅ = โซ
๐
0
(๐1+ ๐๐ฆcosฮฉ2๐ก) ๐๐ฅ. (3d)
Applying the Galerkin procedure, we obtain that the dimen-sionless differential equations of motion for the simplysupported symmetric cross-ply rectangular thin plate areshown as follows [11, 28]:
๏ฟฝฬ๏ฟฝ1+ ๐1๏ฟฝฬ๏ฟฝ1+ ๐2
1๐ข1
+ (๐11cosฮฉ1๐ก + ๐12cosฮฉ2๐ก + ๐14cosฮฉ4๐ก) ๐ข1
+ ๐ผ1๐ข2
1๐ข2+ ๐ผ2๐ข2
1๐ข3+ ๐ผ3๐ข2
2๐ข1+ ๐ผ4๐ข2
2๐ข3
+ ๐ผ5๐ข2
3๐ข1+ ๐ผ6๐ข2
3๐ข2+ ๐ผ7๐ข3
1+ ๐ผ8๐ข3
2
+ ๐ผ9๐ข3
3+ ๐ผ10๐ข1๐ข2๐ข3
= ๐1cosฮฉ3๐ก,
(4a)
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4 Mathematical Problems in Engineering
๏ฟฝฬ๏ฟฝ2+ ๐2๏ฟฝฬ๏ฟฝ2+ ๐2
2๐ข2
+ (๐21cosฮฉ1๐ก + ๐22cosฮฉ2๐ก + ๐24cosฮฉ4๐ก) ๐ข2
+ ๐ฝ1๐ข2
1๐ข2+ ๐ฝ2๐ข2
1๐ข3+ ๐ฝ3๐ข2
2๐ข1+ ๐ฝ4๐ข2
2๐ข3
+ ๐ฝ5๐ข2
3๐ข1+ ๐ฝ6๐ข2
3๐ข2+ ๐ฝ7๐ข3
1+ ๐ฝ8๐ข3
2
+ ๐ฝ9๐ข3
3+ ๐ฝ10๐ข1๐ข2๐ข3
= ๐2cosฮฉ3๐ก,
(4b)
๏ฟฝฬ๏ฟฝ3+ ๐3๏ฟฝฬ๏ฟฝ3+ ๐2
3๐ข3
+ (๐31cosฮฉ1๐ก + ๐32cosฮฉ2๐ก + ๐34cosฮฉ4๐ก) ๐ข3
+ ๐พ1๐ข2
1๐ข2+ ๐พ2๐ข2
1๐ข3+ ๐พ3๐ข2
2๐ข1+ ๐พ4๐ข2
2๐ข3
+ ๐พ5๐ข2
3๐ข1+ ๐พ6๐ข2
3๐ข2+ ๐พ7๐ข3
1+ ๐พ8๐ข3
2
+ ๐พ9๐ข3
3+ ๐พ10๐ข1๐ข2๐ข3
= ๐3cosฮฉ3๐ก,
(4c)
where ๐ข1, ๐ข2, and ๐ข
3are the vibration amplitudes of the com-
posite laminated piezoelectric rectangular plate for the first-order, second-order, and the third-order modes, respectively,๐1, ๐2, and ๐
3are the linear viscous damping coefficients,
๐1, ๐2, and ๐
3are the natural frequencies of the rectangular
plate, andฮฉ1,ฮฉ2,ฮฉ3, andฮฉ
4are the excitations frequencies.
๐๐1, ๐๐2, ๐๐3, and ๐
๐(๐ = 1, 2, 3) are the amplitudes of
parametric and external excitation forces corresponding tothe three nonlinear modes, and ๐ผ
๐, ๐ฝ๐, and ๐พ
๐(๐ = 1, 2, . . . , 10)
are the nonlinear coefficients.The linear viscous damping andexciting forces are assumed to be
๐๐= ๐๐๐, ๐๐1= ๐๐๐1, ๐
๐2= ๐๐๐2,
๐๐3= ๐๐๐3, ๐๐= ๐2๐๐, ๐ = 1, 2, 3,
(5)
where ๐ is a small perturbation parameter and 0 < ๐ โช 1.The external excitation forces ๐
๐are of the order 2, and
the linear viscous damping ๐๐, parametric exciting forces๐
๐1,
๐๐2, and ๐
๐3are of the order 1.
To consider the influence of the cubic terms on non-linear dynamic characteristics of the composite laminatedpiezoelectric rectangular plate, we need to obtain the second-order approximate solution of (4a), (4b), and (4c). Methodof multiple scales [29โ31] is applied to obtain a second-order approximation for the system. For the second-orderapproximation, we introduce three time scales defined by
๐0= ๐ก, ๐
1= ๐๐ก, ๐
2= ๐2๐ก. (6)
In terms of these scales, the time derivatives become
๐
๐๐ก= ๐ท0+ ๐๐ท1+ ๐2๐ท2, (7a)
๐2
๐๐ก2= ๐ท2
0+ 2๐๐ท
0๐ท1+ ๐2(๐ท2
1+ 2๐ท0๐ท2) , (7b)
where ๐ท๐= ๐/๐๐
๐, ๐ = 0, 1, 2. We seek a uniform approxi-
mation to the solution of (4a), (4b), and (4c) in the form:
๐ข๐ (๐ก, ๐) = ๐๐ข๐1 (๐0, ๐1, ๐2) + ๐
2๐ข๐2(๐0, ๐1, ๐2)
+ ๐3๐ข๐3(๐0, ๐1, ๐2) + ๐ (๐
4) , ๐ = 1, 2, 3.
(8)
Terms of ๐(๐4) and higher orders are neglected. Substituting(5) and (7a)โ(8) into (4a), (4b), and (4c) and equating thecoefficients of like powers of ๐, we obtain the following.
Order ๐
(๐ท2
0+ ๐2
1) ๐ข11= 0, (9a)
(๐ท2
0+ ๐2
2) ๐ข21= 0, (9b)
(๐ท2
0+ ๐2
3) ๐ข31= 0. (9c)
Order ๐2
(๐ท2
0+ ๐2
1) ๐ข12= โ2๐ท
0๐ท1๐ข11โ ๐1๐ท0๐ข11
โ (๐11cosฮฉ1๐0+ ๐12cosฮฉ2๐0
+๐14cosฮฉ4๐0) ๐ข11+ ๐1cosฮฉ3๐0,
(10a)
(๐ท2
0+ ๐2
2) ๐ข22= โ2๐ท
0๐ท1๐ข21โ ๐2๐ท0๐ข21
โ (๐21cosฮฉ1๐0+ ๐22cosฮฉ2๐0
+๐24cosฮฉ4๐0) ๐ข21+ ๐2cosฮฉ3๐0,
(10b)
(๐ท2
0+ ๐2
3) ๐ข32= โ2๐ท
0๐ท1๐ข31โ ๐3๐ท0๐ข31
โ (๐31cosฮฉ1๐0+ ๐32cosฮฉ2๐0
+๐34cosฮฉ4๐0) ๐ข31+ ๐3cosฮฉ3๐0.
(10c)
Order ๐3
(๐ท2
0+ ๐2
1) ๐ข13= โ๐ท2
1๐ข11โ 2๐ท0๐ท2๐ข11โ 2๐ท0๐ท1๐ข12
โ ๐1(๐ท0๐ข12+ ๐ท1๐ข11)
โ (๐11cosฮฉ1๐0+ ๐12cosฮฉ2๐0
+ ๐14cosฮฉ4๐0) ๐ข12
โ ๐ผ1๐ข2
11๐ข21โ ๐ผ2๐ข2
11๐ข31โ ๐ผ3๐ข2
21๐ข11
โ ๐ผ4๐ข2
21๐ข31โ ๐ผ5๐ข2
31๐ข11โ ๐ผ6๐ข2
31๐ข21
โ ๐ผ7๐ข3
11โ ๐ผ8๐ข3
21โ ๐ผ9๐ข3
31โ ๐ผ10๐ข11๐ข21๐ข31,
(11a)
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Mathematical Problems in Engineering 5
(๐ท2
0+ ๐2
2) ๐ข23= โ๐ท2
1๐ข21โ 2๐ท0๐ท2๐ข21โ 2๐ท0๐ท1๐ข22
โ ๐2(๐ท0๐ข22+ ๐ท1๐ข21)
โ (๐21cosฮฉ1๐0+ ๐22cosฮฉ2๐0
+๐24cosฮฉ4๐0) ๐ข22
โ ๐ฝ1๐ข2
11๐ข21โ ๐ฝ2๐ข2
11๐ข31โ ๐ฝ3๐ข2
21๐ข11
โ ๐ฝ4๐ข2
21๐ข31โ ๐ฝ5๐ข2
31๐ข11โ ๐ฝ6๐ข2
31๐ข21
โ ๐ฝ7๐ข3
11โ ๐ฝ8๐ข3
21โ ๐ฝ9๐ข3
31โ ๐ฝ10๐ข11๐ข21๐ข31,
(11b)
(๐ท2
0+ ๐2
3) ๐ข33= โ๐ท2
1๐ข31โ 2๐ท0๐ท2๐ข31โ 2๐ท0๐ท1๐ข32
โ ๐3(๐ท0๐ข32+ ๐ท1๐ข31)
โ (๐31cosฮฉ1๐0+ ๐32cosฮฉ2๐0
+๐34cosฮฉ4๐0) ๐ข32
โ ๐พ1๐ข2
11๐ข21โ ๐พ2๐ข2
11๐ข31โ ๐พ3๐ข2
21๐ข11
โ ๐พ4๐ข2
21๐ข31โ ๐พ5๐ข2
31๐ข11โ ๐พ6๐ข2
31๐ข21
โ ๐พ7๐ข3
11โ ๐พ8๐ข3
21โ ๐พ9๐ข3
31โ ๐พ10๐ข11๐ข21๐ข31.
(11c)
The general solutions of (9a), (9b), and (9c) can be written inthe form
๐ข11= ๐ด1(๐1, ๐2) exp (๐๐
1๐0) + ๐๐, (12a)
๐ข21= ๐ด2(๐1, ๐2) exp (๐๐
2๐0) + ๐๐, (12b)
๐ข31= ๐ด3(๐1, ๐2) exp (๐๐
3๐0) + ๐๐, (12c)
where๐ด1,๐ด2, and๐ด
3are a complex function in๐
1, ๐2which
can be determined from eliminating the secular terms at thenext approximation and ๐๐ stands for the complex conjugateof the preceding terms. Substituting (12a), (12b), and (12c)into (10a), (10b), and (10c) and eliminating the secular terms,then the first-order approximations are given by
๐ข12= ๐ธ1exp (๐๐
1๐0) + ๐ธ2exp (๐ (ฮฉ
1+ ๐1) ๐0)
+ ๐ธ3exp (๐ (ฮฉ
1โ ๐1) ๐0) + ๐ธ4exp (๐ (ฮฉ
2+ ๐1) ๐0)
+ ๐ธ5exp (๐ (ฮฉ
2โ ๐1) ๐0) + ๐ธ6exp (๐ (ฮฉ
4+ ๐1) ๐0)
+ ๐ธ7exp (๐ (ฮฉ
4โ ๐1) ๐0) + ๐๐,
(13a)
๐ข22= ๐ธ8exp (๐๐
2๐0) + ๐ธ9exp (๐ (ฮฉ
1+ ๐2) ๐0)
+ ๐ธ10exp (๐ (ฮฉ
1โ ๐2) ๐0) + ๐ธ11exp (๐ (ฮฉ
2+ ๐2) ๐0)
+ ๐ธ12exp (๐ (ฮฉ
2โ ๐2) ๐0) + ๐ธ13exp (๐ (ฮฉ
4+ ๐2) ๐0)
+ ๐ธ14exp (๐ (ฮฉ
4โ ๐2) ๐0) + ๐๐,
(13b)
๐ข32= ๐ธ15exp (๐๐
3๐0) + ๐ธ16exp (๐ (ฮฉ
1+ ๐3) ๐0)
+ ๐ธ17exp (๐ (ฮฉ
1โ ๐3) ๐0) + ๐ธ18exp (๐ (ฮฉ
2+ ๐3) ๐0)
+ ๐ธ19exp (๐ (ฮฉ
2โ ๐3) ๐0) + ๐ธ20exp (๐ (ฮฉ
4+ ๐3) ๐0)
+ ๐ธ21exp (๐ (ฮฉ
4โ ๐3) ๐0) + ๐ธ22exp (๐ฮฉ
3๐0) + ๐๐,
(13c)
where ๐ธ๐(๐ = 1, 2, . . . , 22) are the complex functions in ๐
1
and ๐2. From (12a)โ(13c) into (11a), (11b), and (11c) and elim-
inating the secular terms, the second-order approximation isgiven by
๐ข13(๐0, ๐1, ๐2) = ๐ป
1exp (๐๐
2๐0) + ๐ป2exp (๐๐
3๐0)
+ ๐ป3exp (3๐๐
1๐0) + ๐ป4exp (3๐๐
2๐0)
+ ๐ป5exp (3๐๐
3๐0)
+ ๐ป6exp (๐ (๐
2ยฑ 2๐1) ๐0)
+ ๐ป7exp (๐ (๐
3ยฑ 2๐1) ๐0)
+ ๐ป8exp (๐ (2๐
2ยฑ ๐1) ๐0)
+ ๐ป9exp (๐ (2๐
3ยฑ ๐1) ๐0)
+ ๐ป10exp (๐ (๐
2ยฑ 2๐3) ๐0)
+ ๐ป11exp (๐ (๐
3ยฑ 2๐2) ๐0)
+ ๐ป12exp (๐ (๐
3ยฑ ๐2ยฑ ๐1) ๐0)
+ ๐ป13exp (๐ฮฉ
3๐0)
+ ๐ป14exp (๐ (ฮฉ
3ยฑ ฮฉ1) ๐0)
+ ๐ป15exp (๐ (ฮฉ
3ยฑ ฮฉ2) ๐0)
+ ๐ป16exp (๐ (ฮฉ
3ยฑ ฮฉ4) ๐0)
+ ๐ป17exp (๐ (ฮฉ
1ยฑ ๐1) ๐0)
+ ๐ป18exp (๐ (ฮฉ
2ยฑ ๐1) ๐0)
+ ๐ป19exp (๐ (ฮฉ
4ยฑ ๐1) ๐0)
+ ๐ป20exp (๐ (2ฮฉ
1ยฑ ๐1) ๐0)
+ ๐ป21exp (๐ (2ฮฉ
2ยฑ ๐1) ๐0)
+ ๐ป22exp (๐ (2ฮฉ
4ยฑ ๐1) ๐0)
+ ๐ป23exp (๐ (ฮฉ
2ยฑ ฮฉ1ยฑ ๐1) ๐0)
+ ๐ป24exp (๐ (ฮฉ
4ยฑ ฮฉ1ยฑ ๐1) ๐0)
+ ๐ป25exp (๐ (ฮฉ
4ยฑ ฮฉ2ยฑ ๐1) ๐0) + ๐๐,
(14a)๐ข23(๐0, ๐1, ๐2) = ๐ป
26exp (๐๐
1๐0) + ๐ป27exp (๐๐
3๐0)
+ ๐ป28exp (3๐๐
1๐0) + ๐ป29exp (3๐๐
2๐0)
-
6 Mathematical Problems in Engineering
+ ๐ป30exp (3๐๐
3๐0)
+ ๐ป31exp (๐ (๐
2ยฑ 2๐1) ๐0)
+ ๐ป32exp (๐ (2๐
2ยฑ ๐1) ๐0)
+ ๐ป33exp (๐ (๐
3ยฑ 2๐1) ๐0)
+ ๐ป34exp (๐ (2๐
3ยฑ ๐1) ๐0)
+ ๐ป35exp (๐ (๐
2ยฑ 2๐3) ๐0)
+ ๐ป36exp (๐ (2๐
2ยฑ ๐3) ๐0)
+ ๐ป37exp (๐ (๐
3ยฑ ๐2ยฑ ๐1) ๐0)
+ ๐ป38exp (๐ฮฉ
3๐0)
+ ๐ป39exp (๐ (ฮฉ
3ยฑ ฮฉ1) ๐0)
+ ๐ป40exp (๐ (ฮฉ
3ยฑ ฮฉ2) ๐0)
+ ๐ป41exp (๐ (ฮฉ
4ยฑ ฮฉ3) ๐0)
+ ๐ป42exp (๐ (ฮฉ
1ยฑ ๐2) ๐0)
+ ๐ป43exp (๐ (ฮฉ
2ยฑ ๐2) ๐0)
+ ๐ป44exp (๐ (ฮฉ
4ยฑ ๐2) ๐0)
+ ๐ป45exp (๐ (2ฮฉ
1ยฑ ๐2) ๐0)
+ ๐ป46exp (๐ (2ฮฉ
2ยฑ ๐2) ๐0)
+ ๐ป47exp (๐ (2ฮฉ
4ยฑ ๐2) ๐0)
+ ๐ป48exp (๐ (ฮฉ
2ยฑ ฮฉ1ยฑ ๐2) ๐0)
+ ๐ป49exp (๐ (ฮฉ
4ยฑ ฮฉ1ยฑ ๐2) ๐0)
+ ๐ป50exp (๐ (ฮฉ
4ยฑ ฮฉ2ยฑ ๐2) ๐0) + ๐๐,
(14b)๐ข33(๐0, ๐1, ๐2) = ๐ป
51exp (3๐๐
3๐0) + ๐ป52exp (๐๐
1๐0)
+ ๐ป53exp (3๐๐
1๐0) + ๐ป54exp (๐๐
2๐0)
+ ๐ป55exp (3๐๐
2๐0)
+ ๐ป56exp (๐ (๐
2ยฑ 2๐1) ๐0)
+ ๐ป57exp (๐ (2๐
2ยฑ ๐1) ๐0)
+ ๐ป58exp (๐ (๐
3ยฑ 2๐1) ๐0)
+ ๐ป59exp (๐ (2๐
3ยฑ ๐1) ๐0)
+ ๐ป60exp (๐ (๐
2ยฑ 2๐3) ๐0)
+ ๐ป61exp (๐ (2๐
2ยฑ ๐3) ๐0)
+ ๐ป62exp (๐ (๐
3ยฑ ๐2ยฑ ๐1) ๐0)
+ ๐ป63exp (๐ฮฉ
3๐0)
+ ๐ป64exp (๐ (ฮฉ
3ยฑ ฮฉ1) ๐0)
+ ๐ป65exp (๐ (ฮฉ
3ยฑ ฮฉ2) ๐0)
+ ๐ป66exp (๐ (ฮฉ
3ยฑ ฮฉ4) ๐0)
+ ๐ป67exp (๐ (ฮฉ
1ยฑ ๐3) ๐0)
+ ๐ป68exp (๐ (ฮฉ
2ยฑ ๐3) ๐0)
+ ๐ป69exp (๐ (ฮฉ
4ยฑ ๐3) ๐0)
+ ๐ป70exp (๐ (2ฮฉ
1ยฑ ๐3) ๐0)
+ ๐ป71exp (๐ (2ฮฉ
2ยฑ ๐3) ๐0)
+ ๐ป72exp (๐ (2ฮฉ
4ยฑ ๐3) ๐0)
+ ๐ป73exp (๐ (ฮฉ
2ยฑ ฮฉ1ยฑ ๐3) ๐0)
+ ๐ป74exp (๐ (ฮฉ
4ยฑ ฮฉ1ยฑ ๐3) ๐0)
+ ๐ป75exp (๐ (ฮฉ
4ยฑ ฮฉ2ยฑ ๐3) ๐0) + ๐๐,
(14c)
where ๐ป๐(๐ = 1, 2, . . . , 75) are the complex functions in ๐
1
and ๐2. From the above derived solutions, the reported
resonance cases are the following.
(i) Primary resonance:ฮฉ1โ ๐๐,ฮฉ2โ ๐๐,ฮฉ3โ ๐๐,ฮฉ4โ
๐๐, and ๐ = 1, 2, 3.
(ii) Subharmonic resonance: ฮฉ1โ 2๐๐, ฮฉ2โ 2๐๐, ฮฉ4โ
2๐๐, and ๐ = 1, 2, 3.
(iii) Internal or secondary resonance: ๐1โ ๐2, ๐2โ ๐3,
๐3โ ๐1, ๐1โ 3๐๐ , ๐2โ 3๐๐, ๐3โ 3๐๐, ๐ = 2, 3,
๐ = 1, 3, and๐ = 1, 2.(iv) Combined resonance: ๐
3ยฑ ๐2โ 2๐1, ๐3ยฑ ๐1โ 2๐2,
๐1ยฑ๐2โ 2๐3,๐2ยฑ2๐3โ ๐1,๐3ยฑ2๐2โ ๐1,๐3ยฑ2๐1โ
๐2, ๐1ยฑ 2๐3โ ๐2, ๐2ยฑ 2๐1โ ๐3, ๐1ยฑ 2๐2โ ๐3,
ฮฉ3ยฑ ฮฉ๐กโ ๐๐, ฮฉ4ยฑ ฮฉ๐โ 2๐๐, ฮฉ2ยฑ ฮฉ1โ 2๐๐,
๐ก = 1, 2, 4, ๐ = 1, 2, and ๐ = 1, 2, 3.(v) Simultaneous or incident resonance.
Any combination of the previous resonance cases is con-sidered as simultaneous resonance.
3. Stability Analysis
Thebehavior of such a system can be very complex, especiallywhen the natural frequencies and the forcing frequency sat-isfy certain internal and external resonance conditions. Thestudy is focused on the case of 1 : 1 : 3 primary resonance andinternal resonance, where ฮฉ
3โ ๐1, ๐2โ ๐1, and ๐
3โ 3๐1.
To describe how close the frequencies are to the resonanceconditions, we introduce detuning parameters as follows:
ฮฉ3= ๐1+ ๐1= ๐1+ ๐๏ฟฝฬ๏ฟฝ1,
๐2= ๐1+ ๐2= ๐1+ ๐๏ฟฝฬ๏ฟฝ2,
๐3= 3๐1+ ๐3= 3๐1+ ๐๏ฟฝฬ๏ฟฝ3,
(15)
-
Mathematical Problems in Engineering 7
where๐1and๐2,๐3are called the external and internal detun-
ing parameters, respectively. Eliminating the secular termsleads to solvability conditions for the first- and second-orderexpansions as follows:
2๐๐1๐ท1๐ด1= โ๐๐
1๐1๐ด1+๐1
2exp (๐๏ฟฝฬ๏ฟฝ
1๐1) , (16a)
2๐๐2๐ท1๐ด2= โ๐๐
2๐2๐ด2+๐2
2exp (๐ (๏ฟฝฬ๏ฟฝ
1โ ๏ฟฝฬ๏ฟฝ2) ๐1) , (16b)
2๐๐3๐ท1๐ด3= โ๐๐
3๐3๐ด3, (16c)
2๐๐1๐ท2๐ด1
= โ๐ท2
1๐ด1โ ๐1๐ท1๐ด1
โ {๐2
11
2 (ฮฉ2
1โ 4๐2
1)+
๐2
12
2 (ฮฉ2
2โ 4๐2
1)+
๐2
14
2 (ฮฉ2
4โ 4๐2
1)}๐ด1
โ {2๐ผ1๐ด1๐ด1+ 2๐ผ6๐ด3๐ด3+ 3๐ผ8๐ด2๐ด2}๐ด2exp (๐๏ฟฝฬ๏ฟฝ
2๐1)
โ ๐ผ1๐ด2
1๐ด2exp (โ๐๏ฟฝฬ๏ฟฝ
2๐1) โ ๐ผ2๐ด2
1๐ด3exp (๐๏ฟฝฬ๏ฟฝ
3๐1)
โ {2๐ผ3๐ด2๐ด2+ 2๐ผ5๐ด3๐ด3+ 3๐ผ7๐ด1๐ด1}๐ด1
โ ๐ผ4๐ด2
2๐ด3exp (๐ (๏ฟฝฬ๏ฟฝ
3โ 2๏ฟฝฬ๏ฟฝ2) ๐1)
โ ๐ผ10๐ด1๐ด2๐ด3exp (๐ (๏ฟฝฬ๏ฟฝ
3โ ๏ฟฝฬ๏ฟฝ2) ๐1) ,
(17a)2๐๐2๐ท2๐ด2
= โ๐ท2
1๐ด2โ ๐2๐ท1๐ด2
โ {๐2
21
2 (ฮฉ2
1โ 4๐2
2)+
๐2
22
2 (ฮฉ2
2โ 4๐2
2)+
๐2
24
2 (ฮฉ2
4โ 4๐2
2)}๐ด2
โ {2๐ฝ3๐ด2๐ด2+ 2๐ฝ5๐ด3๐ด3+ 3๐ฝ7๐ด1๐ด1}
ร ๐ด1exp (โ๐๏ฟฝฬ๏ฟฝ
2๐1)
โ ๐ฝ3๐ด2
2๐ด1exp (๐๏ฟฝฬ๏ฟฝ
2๐1) โ ๐ฝ1๐ด2
1๐ด2exp (โ2๐๏ฟฝฬ๏ฟฝ
2๐1)
โ {2๐ฝ6๐ด3๐ด3+ 2๐ฝ1๐ด1๐ด1+ 3๐ฝ8๐ด2๐ด2}๐ด2
โ ๐ฝ2๐ด2
1๐ด3exp (๐ (๏ฟฝฬ๏ฟฝ
3โ ๏ฟฝฬ๏ฟฝ2) ๐1)
โ ๐ฝ10๐ด1๐ด2๐ด3exp (๐ (๏ฟฝฬ๏ฟฝ
3โ 2๏ฟฝฬ๏ฟฝ2) ๐1)
โ ๐ฝ4๐ด2
2๐ด3exp (๐ (๏ฟฝฬ๏ฟฝ
3โ 3๏ฟฝฬ๏ฟฝ2) ๐1) ,
(17b)2๐๐3๐ท2๐ด3
= โ๐ท2
1๐ด3โ ๐3๐ท1๐ด3
โ {๐2
31
2 (ฮฉ2
1โ 4๐2
3)+
๐2
32
2 (ฮฉ2
2โ 4๐2
3)+
๐2
34
2 (ฮฉ2
4โ 4๐2
3)}๐ด3
โ ๐พ10๐ด1๐ด2๐ด3exp (โ๐๏ฟฝฬ๏ฟฝ
2๐1)
โ ๐พ10๐ด1๐ด2๐ด3exp (๐๏ฟฝฬ๏ฟฝ
2๐1)
โ ๐พ7๐ด3
1exp (โ๐๏ฟฝฬ๏ฟฝ
3๐1)
โ {2๐พ2๐ด1๐ด1+ 2๐พ4๐ด2๐ด2+ 3๐พ9๐ด3๐ด3}๐ด3
โ ๐พ1๐ด2
1๐ด2exp (๐ (๏ฟฝฬ๏ฟฝ
2โ ๏ฟฝฬ๏ฟฝ3) ๐1)
โ ๐พ3๐ด2
2๐ด1exp (๐ (2๏ฟฝฬ๏ฟฝ
2โ ๏ฟฝฬ๏ฟฝ3) ๐1)
โ ๐พ8๐ด3
2exp (๐ (3๏ฟฝฬ๏ฟฝ
2โ ๏ฟฝฬ๏ฟฝ3) ๐1) .
(17c)
From (7a), multiplying both sides by 2๐๐๐, we get
2๐๐๐
๐๐ด๐
๐๐ก= ๐2๐๐
๐๐ท1๐ด๐+ ๐22๐๐๐๐ท2๐ด๐,
๐ = 1, 2, 3.
(18)
To analyze the solutions of (16a)โ(17c), we express ๐ด๐in the
polar form as follows:
๐ด๐(๐1, ๐2) =
๐๐
2exp (๐๐
๐) ,
๐๐= ๐๐๐, (๐ = 1, 2, 3) ,
(19)
where ๐๐and ๐
๐are the steady-state amplitudes and phases
of the motion, respectively. Substituting (16a)โ(17c) and (19)into (18) and equating the real and imaginary parts, we obtainthe following equations describing the modulation of theamplitudes and phases of the response:
ฬ๐1= โ
๐1
2๐1+ {
๐1
2๐1
โ๐1๐1
4๐2
1
} sin ๐1
+๐1๐1
8๐2
1
cos ๐1โ๐ผ1
8๐1
๐2
1๐2sin ๐2
โ๐ผ2
8๐1
๐2
1๐3sin ๐3โ๐ผ4
8๐1
๐2
2๐3sin (๐3โ 2๐2)
โ๐ผ6
4๐1
๐2
3๐2sin ๐2โ3๐ผ8
8๐1
๐3
2sin ๐2
โ๐ผ10
8๐1
๐1๐2๐3sin (๐3โ ๐2) ,
๐1๏ฟฝฬ๏ฟฝ1= {โ
๐2
1
8๐1
+ฮ1
2๐1
}๐1โ {
๐1
2๐1
โ๐1๐1
4๐2
1
} cos ๐1
+๐1๐1
8๐2
1
sin ๐1+3๐ผ1
8๐1
๐2
1๐2cos ๐2+๐ผ2
8๐1
๐2
1๐3cos ๐3
+๐ผ3
4๐1
๐1๐2
2+๐ผ4
8๐1
๐2
2๐3cos (๐
3โ 2๐2)
+๐ผ5
4๐1
๐1๐2
3+๐ผ6
4๐1
๐2
3๐2cos ๐2+3๐ผ7
8๐1
๐3
1
-
8 Mathematical Problems in Engineering
+3๐ผ8
8๐1
๐3
2cos ๐2+๐ผ10
8๐1
๐1๐2๐3cos (๐
3โ ๐2) ,
ฬ๐2= โ
๐2
2๐2+ {
๐2
2๐2
โ(๐1โ ๐2) ๐2
4๐2
2
} sin (๐1โ ๐2)
+๐2๐2
8๐2
2
cos (๐1โ ๐2) +
๐ฝ1
8๐2
๐2
1๐2sin 2๐
2
โ๐ฝ2
8๐2
๐2
1๐3sin (๐3โ ๐2) +
๐ฝ3
8๐2
๐2
2๐1sin ๐2
โ๐ฝ4
8๐2
๐2
2๐3sin (๐3โ 3๐2) +
๐ฝ5
4๐2
๐2
3๐1sin ๐2
+3๐ฝ7
8๐2
๐3
1sin ๐2โ๐ฝ10
8๐2
๐1๐2๐3sin (๐3โ 2๐2) ,
๐2๏ฟฝฬ๏ฟฝ2= {โ
๐2
2
8๐2
+ฮ2
2๐2
}๐2
+ {(๐1โ ๐2) ๐2
4๐2
2
โ๐2
2๐2
} cos (๐1โ ๐2)
+๐2๐2
8๐2
2
sin (๐1โ ๐2) +
๐ฝ1
8๐2
๐2
1๐2cos 2๐
2
+๐ฝ1
4๐2
๐2
1๐2+๐ฝ2
8๐2
๐2
1๐3cos (๐
3โ ๐2)
+3๐ฝ3
8๐2
๐2
2๐1cos ๐2+๐ฝ4
8๐2
๐2
2๐3cos (๐
3โ 3๐2)
+๐ฝ5
4๐2
๐2
3๐1cos ๐2+๐ฝ6
4๐2
๐2
3๐2+3๐ฝ7
8๐2
๐3
1cos ๐2
+3๐ฝ8
8๐2
๐3
2+๐ฝ10
8๐2
๐1๐2๐3cos (๐
3โ 2๐2) ,
ฬ๐3= โ
๐3
2๐3โ๐พ1
8๐3
๐2
1๐2sin (๐2โ ๐3)
โ๐พ3
8๐3
๐2
2๐1sin (2๐
2โ ๐3) +
๐พ7
8๐3
๐3
1sin ๐3
โ๐พ8
8๐3
๐3
2sin (3๐
2โ ๐3) ,
๐3๏ฟฝฬ๏ฟฝ3= {โ
๐2
3
8๐3
+ฮ3
2๐3
}๐3+๐พ1
8๐3
๐2
1๐2cos (๐
2โ ๐3)
+๐พ2
4๐3
๐2
1๐3+๐พ3
8๐3
๐2
2๐1cos (2๐
2โ ๐3)
+๐พ4
4๐3
๐2
2๐3+๐พ7
8๐3
๐3
1cos ๐3
+๐พ8
8๐3
๐3
2cos (3๐
2โ ๐3) +
3๐พ9
8๐3
๐3
3
+๐พ10
4๐3
๐1๐2๐3cos ๐2,
(20)
where
ฮ๐= {
๐2
๐1
2 (ฮฉ2
1โ 4๐2๐)+
๐2
๐2
2 (ฮฉ2
2โ 4๐2๐)+
๐2
๐4
2 (ฮฉ2
4โ 4๐2๐)} ,
๐ = 1, 2, 3,
๐1= ๏ฟฝฬ๏ฟฝ1๐1โ ๐1,
๐2= ๏ฟฝฬ๏ฟฝ2๐1+ ๐2โ ๐1,
๐3= ๏ฟฝฬ๏ฟฝ3๐1+ ๐3โ 3๐1.
(21)
Steady-state solutions of the system correspond to the fixedpoints of (20), which in turn correspond to
๏ฟฝฬ๏ฟฝ1= ๐1,
๏ฟฝฬ๏ฟฝ2= ๐1โ ๐2,
๏ฟฝฬ๏ฟฝ3= 3๐1โ ๐3.
(22)
Hence, the fixed points of (20) are given by
โ๐1
2๐1+ {
๐1
2๐1
โ๐1๐1
4๐2
1
} sin ๐1
+๐1๐1
8๐2
1
cos ๐1โ๐ผ1
8๐1
๐2
1๐2sin ๐2
โ๐ผ2
8๐1
๐2
1๐3sin ๐3โ๐ผ4
8๐1
๐2
2๐3sin (๐3โ 2๐2)
โ๐ผ6
4๐1
๐2
3๐2sin ๐2โ3๐ผ8
8๐1
๐3
2sin ๐2
โ๐ผ10
8๐1
๐1๐2๐3sin (๐3โ ๐2) = 0,
๐1๐1+ {
๐2
1
8๐1
โฮ1
2๐1
}๐1
+ {๐1
2๐1
โ๐1๐1
4๐2
1
} cos ๐1โ๐1๐1
8๐2
1
sin ๐1
โ3๐ผ1
8๐1
๐2
1๐2cos ๐2โ๐ผ2
8๐1
๐2
1๐3cos ๐3
โ๐ผ3
4๐1
๐1๐2
2โ๐ผ4
8๐1
๐2
2๐3cos (๐
3โ 2๐2) โ
๐ผ5
4๐1
๐1๐2
3
โ๐ผ6
4๐1
๐2
3๐2cos ๐2โ3๐ผ7
8๐1
๐3
1โ3๐ผ8
8๐1
๐3
2cos ๐2
โ๐ผ10
8๐1
๐1๐2๐3cos (๐
3โ ๐2) = 0,
โ๐2
2๐2+ {
๐2
2๐2
โ(๐1โ ๐2) ๐2
4๐2
2
} sin (๐1โ ๐2)
+๐2๐2
8๐2
2
cos (๐1โ ๐2)
+๐ฝ1
8๐2
๐2
1๐2sin 2๐
2โ๐ฝ2
8๐2
๐2
1๐3sin (๐3โ ๐2)
-
Mathematical Problems in Engineering 9
+๐ฝ3
8๐2
๐2
2๐1sin ๐2โ๐ฝ4
8๐2
๐2
2๐3sin (๐3โ 3๐2)
+๐ฝ5
4๐2
๐2
3๐1sin ๐2+3๐ฝ7
8๐2
๐3
1sin ๐2
โ๐ฝ10
8๐2
๐1๐2๐3sin (๐3โ 2๐2) = 0,
๐2(๐2โ ๐1) โ {
๐2
2
8๐2
โฮ2
2๐2
}๐2
+ {(๐1โ ๐2) ๐2
4๐2
2
โ๐2
2๐2
} cos (๐1โ ๐2)
+๐2๐2
8๐2
2
sin (๐1โ ๐2) +
๐ฝ1
8๐2
๐2
1๐2cos 2๐
2
+๐ฝ1
4๐2
๐2
1๐2+๐ฝ2
8๐2
๐2
1๐3cos (๐
3โ ๐2)
+3๐ฝ3
8๐2
๐2
2๐1cos ๐2+๐ฝ4
8๐2
๐2
2๐3cos (๐
3โ 3๐2)
+๐ฝ5
4๐2
๐2
3๐1cos ๐2+๐ฝ6
4๐2
๐2
3๐2+3๐ฝ7
8๐2
๐3
1cos ๐2
+3๐ฝ8
8๐2
๐3
2+๐ฝ10
8๐2
๐1๐2๐3cos (๐
3โ 2๐2) = 0,
โ๐3
2๐3โ๐พ1
8๐3
๐2
1๐2sin (๐2โ ๐3) โ
๐พ3
8๐3
๐2
2๐1sin (2๐
2โ ๐3)
+๐พ7
8๐3
๐3
1sin ๐3โ๐พ8
8๐3
๐3
2sin (3๐
2โ ๐3) = 0,
๐3(๐3โ 3๐1) โ {
๐2
3
8๐3
โฮ3
2๐3
}๐3+๐พ1
8๐3
๐2
1๐2cos (๐
2โ ๐3)
+๐พ2
4๐3
๐2
1๐3+๐พ3
8๐3
๐2
2๐1cos (2๐
2โ ๐3)
+๐พ4
4๐3
๐2
2๐3+๐พ7
8๐3
๐3
1cos ๐3+๐พ8
8๐3
๐3
2cos (3๐
2โ ๐3)
+3๐พ9
8๐3
๐3
3+๐พ10
4๐3
๐1๐2๐3cos ๐2= 0.
(23)
There are six possibilities besides the trivial solution asfollows:
(1) ๐1ฬธ= 0, ๐2= 0, ๐
3= 0 (single mode),
(2) ๐2ฬธ= 0, ๐1= 0, ๐
3= 0 (single mode),
(3) ๐1ฬธ= 0, ๐2ฬธ= 0, ๐3= 0 (two modes),
(4) ๐1ฬธ= 0, ๐3ฬธ= 0, ๐2= 0 (two modes),
(5) ๐2ฬธ= 0, ๐3ฬธ= 0, ๐1= 0 (two modes),
(6) ๐1ฬธ= 0, ๐2ฬธ= 0, ๐3ฬธ= 0 (three modes).
Case 1. In this case, where ๐2= 0, ๐
3= 0, the frequency
response equation is given by
9๐ผ2
7
64๐2
1
๐6
1+ [๐ 3+3๐ผ7๐1
4๐1
] ๐4
1+ [๐ 2+ ๐2
1+๐2
1๐1
4๐1
โฮ1๐1
๐1
] ๐2
1
โ๐2
1๐2
1
64๐4
1
โ ๐ 2
1= 0.
(24)
Case 2. In this case, where ๐1= 0, ๐
3= 0, the frequency
response equation is given by
9๐ฝ2
8
64๐2
2
๐6
2+ [๐3+3๐ฝ8(๐2โ ๐1)
4๐2
] ๐4
2
+ [๐2+ (๐2โ ๐1)2โ๐2
2(๐2โ ๐1)
4๐2
+ฮ2(๐2โ ๐1)
๐2
] ๐2
2
โ๐2
2๐2
2
64๐4
2
โ ๐2
1= 0.
(25)
Case 3. In this case, where ๐3= 0, the frequency response
equations are given by
9๐ผ2
7
64๐2
1
๐6
1+ [๐ 3+3๐ผ7๐1
4๐1
] ๐4
1+ [๐ 2+ ๐2
1+๐2
1๐1
4๐1
โฮ1๐1
๐1
] ๐2
1
โ๐2
1๐2
1
64๐4
1
โ ๐ 2
1โ9๐ผ2
1
64๐2
1
๐4
1๐2
2โ9๐ผ1๐ผ8
32๐2
1
๐2
1๐4
2
โ9๐ผ2
8
64๐2
1
๐6
2+3๐ 1๐ผ8
4๐1
๐3
2+3๐ 1๐ผ1
4๐1
๐2
1๐2= 0,
9๐ฝ2
8
64๐2
2
๐6
2+ [๐3+3๐ฝ8(๐2โ ๐1)
4๐2
] ๐4
2
+ [๐2+ (๐2โ ๐1)2โ๐2
2(๐2โ ๐1)
4๐2
+ฮ2(๐2โ ๐1)
๐2
] ๐2
2
โ๐2
2๐2
2
64๐4
2
โ ๐2
1+3๐1๐ฝ3
4๐2
๐2
2๐1+3๐1๐ฝ7
4๐2
๐3
1+๐1๐ฝ1
4๐2
๐2
1๐2
โ9๐ฝ2
3
64๐2
2
๐4
2๐2
1โ [
๐ฝ2
1
64๐2
2
+9๐ฝ3๐ฝ7
32๐2
2
] ๐4
1๐2
2โ9๐ฝ2
7
64๐2
2
๐6
1
โ3๐ฝ1๐ฝ7
32๐2
2
๐5
1๐2โ3๐ฝ1๐ฝ3
32๐2
2
๐3
1๐3
2= 0.
(26)
Case 4. In this case, where ๐2= 0, the frequency response
equations are given by
9๐ผ2
7
64๐2
1
๐6
1+ [๐ 3+3๐ผ7๐1
4๐1
] ๐4
1+ [๐ 2+ ๐2
1+๐2
1๐1
4๐1
โฮ1๐1
๐1
] ๐2
1
โ๐2
1๐2
1
64๐4
1
โ ๐ 2
1โ๐ผ2
2
64๐2
1
๐4
1๐2
3+๐ 1๐ผ2
4๐1
๐2
1๐3= 0,
-
10 Mathematical Problems in Engineering
0 50 100 150 200โ1
โ0.5
0
0.5
1
Timeโ0.4 โ0.2 0 0.2 0.4
โ4
โ2
0
2
4
Velo
city
0 50 100 150 200โ0.4
โ0.2
0
0.2
0.4
Timeโ0.2 โ0.1 0 0.1 0.2
โ1
โ0.5
0
0.5
1
Velo
city
0 50 100 150 200โ0.4
โ0.2
0
0.2
0.4
Timeโ0.2 โ0.1 0 0.1 0.2โ1
โ0.5
0
0.5
1Ve
loci
ty
Am
plitu
deu1
Am
plitu
deu2
Am
plitu
deu3
u1
u2
u3
Figure 2: Time response and phase-plane diagrams of the system for nonresonance case.
9๐พ2
9
64๐2
3
๐6
3+ [๐พ2+3๐พ9(๐3โ 3๐1)
4๐3
] ๐4
3
+ [๐พ1+ (๐3โ 3๐1)2โ๐2
3(๐3โ 3๐1)
4๐3
+ฮ3(๐3โ 3๐1)
๐3
] ๐2
3
โ๐พ2
7
64๐2
3
๐6
1= 0.
(27)
Case 5. In this case, where ๐1= 0, the frequency response
equations are given by
9๐ฝ2
8
64๐2
2
๐6
2+ [๐3+3๐ฝ8(๐2โ ๐1)
4๐2
] ๐4
2
+ [๐2+ (๐2โ ๐1)2โ๐2
2(๐2โ ๐1)
4๐2
+ฮ2(๐2โ ๐1)
๐2
] ๐2
2
โ๐2
2๐2
2
64๐4
2
โ ๐2
1+๐1๐ฝ4
4๐2
๐2
2๐3โ๐ฝ2
4
64๐2
2
๐4
2๐2
3= 0,
9๐พ2
9
64๐2
3
๐6
3+ [๐พ2+3๐พ9(๐3โ 3๐1)
4๐3
] ๐4
3
+ [๐พ1+ (๐3โ 3๐1)2โ๐2
3(๐3โ 3๐1)
4๐3
+ฮ3(๐3โ 3๐1)
๐3
] ๐2
3
โ๐พ2
8
64๐2
3
๐6
2= 0.
(28)
Case 6. In this case, where ๐1ฬธ= 0, ๐2ฬธ= 0, and ๐
3ฬธ= 0, this is the
practical case, the frequency response equations are given by
9๐ผ2
7
64๐2
1
๐6
1+ [๐ 3+3๐ผ7๐1
4๐1
] ๐4
1+ [๐ 2+ ๐2
1+๐2
1๐1
4๐1
โฮ1๐1
๐1
] ๐2
1
โ๐2
1๐2
1
64๐4
1
โ ๐ 2
1+๐ 1๐ผ2
4๐1
๐2
1๐3+๐ 1๐ผ6
2๐1
๐2๐2
3+๐ 1๐ผ4
4๐1
๐2
2๐3
+3๐ 1๐ผ8
4๐1
๐3
2+3๐ 1๐ผ1
4๐1
๐2
1๐2+๐ 1๐ผ10
4๐1
๐1๐2๐3โ๐ผ2
6
16๐2
1
๐2
2๐4
3
-
Mathematical Problems in Engineering 11A
mpl
itude
u3
0 50 100 150 200โ2
โ1
0
1
2
Timeโ2 โ1 0 1 2
โ10
0
10
Velo
city
0 50 100 150 200โ2
โ1
0
1
2
Time
โ2 โ1 0 1 2โ10
โ5
0
5
10
Velo
city
0 50 100 150 200โ0.5
0
0.5
Timeโ0.4 โ0.2 0 0.2 0.4
โ10
โ5
0
5
10Ve
loci
ty
Am
plitu
deu1
Am
plitu
deu2
u1
u2
u3
Figure 3: Time response and phase-plane diagrams of the system at simultaneous resonance case, ฮฉ3โ ๐1, ๐2โ ๐1, and ๐
3โ 3๐1.
โ๐ผ4๐ผ6
16๐2
1
๐3
2๐3
3โ๐ผ2
2
64๐2
1
๐4
1๐2
3โ9๐ผ2
1
64๐2
1
๐4
1๐2
2โ9๐ผ1๐ผ8
32๐2
1
๐2
1๐4
2
โ9๐ผ2
8
64๐2
1
๐6
2โ3๐ผ4๐ผ8
32๐2
1
๐5
2๐3โ3๐ผ8๐ผ10
32๐2
1
๐1๐4
2๐3โ๐ผ2๐ผ10
32๐2
1
๐3
1๐2๐2
3
โ๐ผ4๐ผ10
32๐2
1
๐1๐3
2๐2
3โ๐ผ6๐ผ10
16๐2
1
๐1๐2
2๐3
3โ๐ผ2๐ผ6
16๐2
1
๐2
1๐2๐3
3
โ3๐ผ1๐ผ10
32๐2
1
๐3
1๐2
2๐3โ3๐ผ1๐ผ2
32๐2
1
๐4
1๐2๐3โ ๐ 4๐2
1๐2
2๐2
3
โ ๐ 5๐4
2๐2
3โ ๐ 6๐3
2๐2
1๐3= 0,
9๐ฝ2
8
64๐2
2
๐6
2+ [๐3+3๐ฝ8(๐2โ ๐1)
4๐2
] ๐4
2
+ [๐2+ (๐2โ ๐1)2โ๐2
2(๐2โ ๐1)
4๐2
+ฮ2(๐2โ ๐1)
๐2
] ๐2
2
โ๐2
2๐2
2
64๐4
2
โ ๐2
1+3๐1๐ฝ3
4๐2
๐2
2๐1+3๐1๐ฝ7
4๐2
๐3
1+๐1๐ฝ1
4๐2
๐2
1๐2
+๐1๐ฝ4
4๐2
๐2
2๐3+๐1๐ฝ5
2๐2
๐1๐2
3+๐1๐ฝ2
4๐2
๐2
1๐3+๐1๐ฝ10
4๐2
๐1๐2๐3
โ3๐ฝ1๐ฝ7
32๐2
2
๐5
1๐2โ9๐ฝ2
3
64๐2
2
๐4
2๐2
1โ9๐ฝ2
7
64๐2
2
๐6
1โ๐ฝ2
4
64๐2
2
๐4
2๐2
3
โ3๐ฝ1๐ฝ3
32๐2
2
๐3
1๐3
2โ๐ฝ2
5
16๐2
2
๐2
1๐4
3โ๐ฝ2๐ฝ5
16๐2
2
๐3
1๐3
3โ3๐ฝ3๐ฝ4
32๐2
2
๐1๐4
2๐3
โ3๐ฝ2๐ฝ7
32๐2
2
๐5
1๐3โ๐ฝ4๐ฝ5
16๐2
2
๐1๐2
2๐3
3โ๐ฝ5๐ฝ10
16๐2
2
๐2
1๐2๐3
2โ๐ฝ4๐ฝ10
32๐2
2
๐1๐3
2๐2
3
โ ๐4๐2
1๐2
2๐2
3โ ๐5๐4
1๐2
2โ ๐6๐4
1๐2
3โ ๐7๐3
1๐2
2๐3โ ๐8๐2
1๐3
2๐3
โ ๐9๐3
1๐2๐2
3โ ๐10๐4
1๐2๐3= 0,
9๐พ2
9
64๐2
3
๐6
3+ [๐พ2+3๐พ9(๐3โ 3๐1)
4๐3
] ๐4
3
+[๐พ1+ (๐3โ3๐1)2โ๐2
3(๐3โ 3๐1)
4๐3
+ฮ3(๐3โ 3๐1)
๐3
] ๐2
3
-
12 Mathematical Problems in Engineering
0 50 100 150 200โ2
โ1
0
1
2
3
Time0 50 100 150 200
โ3
โ2
โ1
0
1
2
3
Time
0 50 100 150 200
โ0.4
โ0.2
0
0.2
0.4
0.6
0.8
Time
Numerical solution๐1(๐ก) perturbation solution
Numerical solution๐2(๐ก) perturbation solution
Numerical solution๐3(๐ก) perturbation solution
Am
plitu
deu1
Am
plitu
deu2
Am
plitu
deu3
Figure 4: Comparison between numerical solution (using RKM) and analytical solution (using perturbation method) of the system atresonance case, ฮฉ
3โ ๐1, ๐2โ ๐1, and ๐
3โ 3๐1.
โ๐พ2
8
64๐2
3
๐6
2โ
๐พ2
7
64๐2
3
๐6
1โ๐พ1๐พ7
32๐2
3
๐5
1๐2โ๐พ3๐พ8
32๐2
3
๐1๐5
2
โ ๐พ3๐4
1๐2
2โ ๐พ4๐2
1๐4
2โ ๐พ5๐3
1๐2
2= 0,
(29)
where
๐ 1= [
๐1
2๐1
โ๐1๐1
4๐2
1
] ,
๐ 2= [
๐2
1
4+๐4
1
64๐2
1
+ฮ2
1
4๐2
1
โฮ1๐2
1
8๐2
1
] ,
๐ 3= [
3๐ผ7๐2
1
32๐2
1
โ3๐ผ7ฮ1
8๐2
1
] ,
๐ 4= [
๐ผ2
10
64๐2
1
+3๐ผ1๐ผ6
16๐2
1
+๐ผ2๐ผ4
32๐2
1
] ,
๐ 5= [
3๐ผ6๐ผ8
16๐2
1
+๐ผ2
4
64๐2
1
] ,
๐ 6= [
3๐ผ1๐ผ4
32๐2
1
+3๐ผ2๐ผ8
32๐2
1
] ,
๐1= [
๐2
2๐2
โ(๐1โ ๐2) ๐2
4๐2
2
] ,
-
Mathematical Problems in Engineering 13
200 200.5 201 201.5 202 202.5 203โ0.5
0
0.5
Time200 200.5 201 201.5 202 202.5 203
โ0.2
โ0.1
0
0.1
0.2
Time
200 200.5 201 201.5 202 202.5 203โ0.2
โ0.1
0
0.1
0.2
Time
RKMFDM
RKMFDM
RKMFDM
Am
plitu
deu1
Am
plitu
deu2
Am
plitu
deu3
Figure 5: Time response of composite laminated rectangular plate using RKM and FDM.
๐2= [
๐2
2
4+๐4
2
64๐2
2
+ฮ2
2
4๐2
2
โฮ2๐2
2
8๐2
2
] ,
๐3= [
3๐ฝ8ฮ2
8๐2
2
โ3๐ฝ8๐2
2
32๐2
2
] ,
๐4= [
๐ฝ2
10
64๐2
2
+3๐ฝ3๐ฝ5
16๐2
2
+๐ฝ2๐ฝ4
32๐2
2
] ,
๐5= [
๐ฝ2
1
64๐2
2
+9๐ฝ3๐ฝ7
32๐2
2
] ,
๐6= [
๐ฝ2
2
64๐2
2
+3๐ฝ5๐ฝ7
16๐2
2
] ,
๐7= [
3๐ฝ2๐ฝ3
32๐2
2
+3๐ฝ4๐ฝ7
32๐2
2
+๐ฝ1๐ฝ10
32๐2
2
] ,
๐8= [
3๐ฝ3๐ฝ10
32๐2
2
+๐ฝ1๐ฝ4
32๐2
2
] ,
๐9= [
๐ฝ2๐ฝ10
32๐2
2
+๐ฝ1๐ฝ5
16๐2
2
] ,
๐10= [
3๐ฝ7๐ฝ10
32๐2
2
+๐ฝ1๐ฝ2
32๐2
2
] ,
๐พ1= [
๐2
3
4+๐4
3
64๐2
3
+ฮ2
3
4๐2
3
โฮ3๐2
3
8๐2
3
] ,
๐พ2= [
3๐พ9ฮ3
8๐2
3
โ3๐พ9๐2
3
32๐2
3
] ,
๐พ3= [
๐พ2
1
64๐2
3
+๐พ3๐พ7
32๐2
3
] ,
๐พ4= [
๐พ2
3
64๐2
3
+๐พ1๐พ8
32๐2
3
] ,
๐พ5= [
๐พ1๐พ3
32๐2
3
+๐พ7๐พ8
32๐2
3
] .
(30)
-
14 Mathematical Problems in Engineering
โ1.5 โ1 โ0.5 0 0.5 1 1.5
2
4
6
8
10
Analytical solutionNumerical solution
๐1a 1
Figure 6: Comparison between analytical prediction using multiple time scale and numerical integration of the first mode.
โ1 โ0.5 0 0.5 1
5
10
15
20
0.2
0.1
0.3
a 1
๐1
Figure 7: Effects of the linear damping ๐1.
โ1.5 โ1 โ0.5 0 0.5 1 1.5
2
4
6
8
10
12
141.7
2.3
4.3
a 1
๐1
Figure 8: Effects of the natural frequency ๐1.
โ1.5 โ1 โ0.5 0 0.5 1 1.5
5
10
15
20
4
8
3
a 1
๐1
Figure 9: Effects of the external excitation ๐1.
-
Mathematical Problems in Engineering 15
โ1.5 โ1 โ0.5 0 0.5 1 1.5
2
4
6
8
10
12
0.06
โ0.06
0.01
a 1๐1
Figure 10: Effects of the nonlinear parameter ๐ผ7.
โ1.5 โ1 โ0.5 0 0.5 1 1.5
2
4
6
8
10
0.5510
a 1
๐1
Figure 11: Effects of the parametric excitation ๐14.
โ1.5 โ1 โ0.5 0 0.5 1 1.5
123456
Analytical solutionNumerical solution
a 2
๐2
Figure 12: Comparison between analytical prediction using multiple time scale and numerical integration of the second mode.
โ1.5 โ1 โ0.5
โ0.08
0
0.2
0.5 1 1.50
2
0.01
4
6
a 2
๐2
Figure 13: Effects of the nonlinear parameter ๐ฝ8.
-
16 Mathematical Problems in Engineering
โ1 โ0.5 0 0.5
0.1
0.2
0.3
10
4
8
a 2
๐2
Figure 14: Effects of the linear damping ๐2.
โ1.5 โ1 โ0.5 0
4
2.2
0.5 1 1.50
2
4
6
8
a 2
๐2
Figure 15: Effects of the natural frequency ๐2.
โ1.5 โ1 โ0.5 0 0.5 1 1.50
2
4
6
8
2
4
6
a 2
๐2
Figure 16: Effects of the external excitation ๐2.
In the frequency response curves, the stable (unstable)steady-state solutions have been represented by solid (dotted)lines.
4. Numerical Results and Discussion
To study the behavior of the system, the Runge-Kutta fourth-order method (RKM) was applied to (4a), (4b), and (4c)
governing the oscillating system. A good criterion of bothstability and dynamic chaos is the phase plane trajectories.Figure 2 illustrates the response and the phase-plane for thenonresonant system at some practical values of the equationparameters: ๐
1= 0.05, ๐
2= 0.05, ๐
3= 0.05, ๐ผ
1= ๐ฝ1= ๐พ1=
1.5, ๐ผ2= ๐ฝ2= ๐พ2= 1.9, ๐ผ
3= ๐ฝ3= ๐พ3= 0.4, ๐ผ
4= ๐ฝ4= ๐พ4=
0.6, ๐ผ5= ๐ฝ5= ๐พ5= 0.6, ๐ผ
6= ๐ฝ6= ๐พ6= 0.2, ๐ผ
7= ๐ฝ7= ๐พ7=
0.01, ๐ผ8= ๐ฝ8= ๐พ8= 0.01, ๐ผ
9= ๐ฝ9= ๐พ9= 0.4, ๐ผ
10= ๐ฝ10=
๐พ10= 1.8, ฮฉ
1= 6.4, ฮฉ
2= 6.1, ฮฉ
3= 5.7, ฮฉ
4= 2, ๐
1= 4.7,
-
Mathematical Problems in Engineering 17
โ2 โ1 0 1 20
2
4
6
Am
plitu
des
๐1
a1
a1
a2a2
a3a3
Figure 17: Effects of the detuning parameter ๐1.
โ1 0 1
4
6
1
โ1 0 10.5
1.5
2.5
1
โ1 0 1
0.2
0.6
1
1.4
1
f1 = 10
f1 = 10
f1 = 10
a 1
a 2
a 3
๐1
๐1๐1
Figure 18: Effects of the external excitation force ๐1.
๐2= 2.7, ๐
3= 13.1, ๐
1= ๐2= 4, ๐
3= 20.5, ๐
11= ๐21= ๐31=
0.1,๐12= ๐22= ๐32= 0.2, and๐
14= ๐24= ๐34= 0.5. Figure 3
shows the steady-state amplitudes and phase plane of thesystem at simultaneous resonance caseฮฉ
3โ ๐1,๐2โ ๐1, and
๐3โ 3๐1. It is clear from Figure 3 that the steady-state ampli-
tude of the first, second, and thirdmodes is increased to about
370%, 890%, and 260%, respectively, of its value shown inFigure 2. Also, it can be seen that the time response of thesystem is tuned with multilimit cycle.
It is quite clear that such case is undesirable in thedesign of such system because it represents one of the worstbehaviors of the system. Such case should be avoided as
-
18 Mathematical Problems in Engineering
โ2 โ1 0 1 22
3
4
5
6
7
2.5
1.5
โ2 0 10.5
1.5
2.5
3.5
4.5
1.5
2.5
0 1
0.2
0.6
1.2
2.5
1.5
โ1
โ1
a 2a 1
a 3
๐1
๐1๐1
๐ผ1 = 0.5
๐ฝ1 = 0.5
๐พ1 = 0.5
Figure 19: Effects of the nonlinear parameters (๐ผ1, ๐ฝ1, and ๐พ
1coupling terms).
working condition for the system. It is advised for such systemnot to have ๐
2= ๐1or ๐3= 3๐1. Figure 4 shows the com-
parison between numerical integration for the system equa-tion (4a), (4b), and (4c) solid lines and the amplitude-phase modulating equation (20) dashed lines. We foundthat all predictions from analytical solutions dashed linesare in good agreement with the numerical simulation solidlines.
4.1. FDM with Approximation O(๐2). The infinite equations(2a)โ(2e) will be solved via a finite differencemethod (FDM).We briefly describe the procedure here. More details areavailable in [23, 26].The infinite dimensional equations (2a)โ(2e) can be reduced to the finite dimensional one via the finitedifference method with second-order approximation ๐(๐2).Namely, at each mesh node the following system of ordinary
differential equations is obtained:
๐ฟ1,๐(๐๐ฅ๐ฅ, ๐๐ฅ๐ฆ) = ๐ผ0(๏ฟฝฬ๏ฟฝ0)๐,๐+ ๐ฝ1( ฬ๐๐ฅ)๐,๐
โ2
3โ2๐๐ผ3((๏ฟฝฬ๏ฟฝ0)๐+1,๐
โ (๏ฟฝฬ๏ฟฝ0)๐โ1,๐
) ,
๐ฟ2,๐(๐๐ฅ๐ฆ, ๐๐ฆ๐ฆ) = ๐ผ0(Vฬ0)๐,๐+ ๐ฝ1( ฬ๐๐ฆ)๐,๐
โ2
3โ2๐๐ผ3((๏ฟฝฬ๏ฟฝ0)๐,๐+1
โ (๏ฟฝฬ๏ฟฝ0)๐,๐โ1
) ,
๐ฟ3,๐(๐๐ฅ, ๐๐ฆ, ๐๐ฅ๐ฅ, ๐ค0, ๐๐ฅ๐ฆ)
= ๐ผ0(๏ฟฝฬ๏ฟฝ0)๐,๐โ
16
9โ4๐2๐ผ6((๏ฟฝฬ๏ฟฝ0)๐+1,๐
โ 2(๏ฟฝฬ๏ฟฝ0)๐,๐+ (๏ฟฝฬ๏ฟฝ0)๐+1,๐
+(๏ฟฝฬ๏ฟฝ0)๐,๐+1
โ2(๏ฟฝฬ๏ฟฝ0)๐,๐+(๏ฟฝฬ๏ฟฝ0)๐,๐+1
)
-
Mathematical Problems in Engineering 19
โ2 โ1 0 1
3
4
5
6
10
5
โ1 0 10.5
1
1.5
2
2.5
5
10
โ2 โ1 0 1
0.2
0.6
1
5
10
๐1
๐1๐1
a 2a1
a 3
๐ผ2 = 0.1๐ฝ2 = 0.1
๐พ2 = 0.1
Figure 20: Effects of the nonlinear parameters (๐ผ2, ๐ฝ2, and ๐พ
2coupling terms).
+4
3โ2๐2๐ผ3((๏ฟฝฬ๏ฟฝ0)๐+1,๐
โ (๏ฟฝฬ๏ฟฝ0)๐โ1,๐
+ (Vฬ0)๐,๐+1
โ (Vฬ0)๐,๐โ1
)
+๐ฝ4
2๐(( ฬ๐๐ฅ)๐+1,๐
โ ( ฬ๐๐ฅ)๐โ1,๐
+ ( ฬ๐๐ฆ)๐,๐+1
โ ( ฬ๐๐ฆ)๐,๐โ1
) ,
๐ฟ4,๐(๐๐ฅ๐ฅ,๐๐ฅ๐ฆ) = ๐ฝ1(๏ฟฝฬ๏ฟฝ0)๐,๐+ ๐2( ฬ๐๐ฅ)๐,๐
โ2๐ฝ4
3โ2๐((๏ฟฝฬ๏ฟฝ0)๐+1,๐
โ (๏ฟฝฬ๏ฟฝ0)๐โ1,๐
) ,
๐ฟ5,๐(๐๐ฅ๐ฆ,๐๐ฆ๐ฆ) = ๐ฝ1(Vฬ0)๐,๐+ ๐2( ฬ๐๐ฆ)๐,๐
โ2๐ฝ4
3โ2๐((๏ฟฝฬ๏ฟฝ0)๐,๐+1
โ (๏ฟฝฬ๏ฟฝ0)๐,๐โ1
) ,
(๐ = 1, 2, . . . , ๐) , (๐ = 1, 2, . . . , ๐) ,
(31)
where ๐ denotes the partition number regarding a spatialcoordinate, ๐ is the computational step regarding spatialcoordinate, and ๐ฟ
1,๐(โ ), ๐ฟ2,๐(โ ), ๐ฟ3,๐(โ ), ๐ฟ4,๐(โ ), and ๐ฟ
5,๐(โ ) are
the difference operators as follows:
๐ฟ1,๐(๐๐ฅ๐ฅ, ๐๐ฅ๐ฆ) =
1
2๐((๐๐ฅ๐ฅ)๐+1,๐
โ (๐๐ฅ๐ฅ)๐โ1,๐
+ (๐๐ฅ๐ฆ)๐,๐+1
โ (๐๐ฅ๐ฆ)๐,๐โ1
) ,
๐ฟ2,๐(๐๐ฅ๐ฆ, ๐๐ฆ๐ฆ) =
1
2๐((๐๐ฅ๐ฆ)๐+1,๐
โ (๐๐ฅ๐ฆ)๐โ1,๐
+ (๐๐ฆ๐ฆ)๐,๐+1
โ (๐๐ฆ๐ฆ)๐,๐โ1
) ,
-
20 Mathematical Problems in Engineering
๐ฟ3,๐(๐๐ฅ, ๐๐ฆ, ๐๐ฅ๐ฅ, ๐ค0, ๐๐ฅ๐ฆ) =
1
2๐((๐๐ฅ)๐+1,๐
โ (๐๐ฅ)๐โ1,๐
+ (๐๐ฆ)๐,๐+1
โ (๐๐ฆ)๐,๐โ1
)
+ ๐๐ฅ๐ฅ(
(๐ค0)๐โ1,๐
โ 2(๐ค0)๐,๐+ (๐ค0)๐+1,๐
๐2) + (
(๐ค0)๐+1,๐
โ (๐ค0)๐โ1,๐
2๐)(
(๐๐ฅ๐ฅ)๐+1,๐
โ (๐๐ฅ๐ฅ)๐โ1,๐
2๐)
+ ๐๐ฅ๐ฆ(
((๐ค0)๐+1,๐+1
โ 2(๐ค0)๐+1,๐โ1
) ((๐ค0)๐โ1,๐+1
โ 2(๐ค0)๐โ1,๐โ1
)
4๐2)
+ (
(๐ค0)๐,๐+1
โ (๐ค0)๐,๐โ1
2๐)(
(๐๐ฅ๐ฆ)๐+1,๐
โ (๐๐ฅ๐ฆ)๐โ1,๐
2๐)
+ ๐๐ฅ๐ฆ(
((๐ค0)๐+1,๐+1
โ (๐ค0)๐+1,๐โ1
) โ ((๐ค0)๐โ1,๐+1
โ (๐ค0)๐โ1,๐โ1
)
8๐3)
+ (
(๐ค0)๐+1,๐
โ (๐ค0)๐โ1,๐
2๐)(
(๐๐ฆ๐ฆ)๐,๐+1
โ (๐๐ฆ๐ฆ)๐,๐โ1
2๐) + ๐
๐ฆ๐ฆ(
(๐ค0)๐,๐โ1
โ 2(๐ค0)๐,๐+ (๐ค0)๐,๐+1
๐2)
+ (
(๐ค0)๐,๐+1
โ (๐ค0)๐,๐โ1
2๐)(
(๐๐ฆ๐ฆ)๐,๐+1
โ (๐๐ฆ๐ฆ)๐,๐โ1
2๐) +
4
3โ2(
(๐๐ฅ๐ฅ)๐โ1,๐
โ 2(๐๐ฅ๐ฅ)๐,๐+ (๐๐ฅ๐ฅ)๐+1,๐
๐2)
+4
3โ2(2(
((๐๐ฅ๐ฆ)๐+1,๐+1
โ (๐๐ฅ๐ฆ)๐โ1,๐+1
) โ ((๐๐ฅ๐ฆ)๐+1,๐โ1
โ (๐๐ฅ๐ฆ)๐โ1,๐โ1
)
8๐3)+
(๐๐ฆ๐ฆ)๐,๐โ1
โ 2(๐๐ฆ๐ฆ)๐,๐+ (๐๐ฆ๐ฆ)๐,๐+1
๐2)+ ๐,
๐ฟ4,๐(๐๐ฅ๐ฅ,๐๐ฅ๐ฆ) =
1
2๐((๐๐ฅ๐ฅ)๐+1,๐
โ (๐๐ฅ๐ฅ)๐โ1,๐
+ (๐๐ฅ๐ฆ)๐,๐+1
โ (๐๐ฅ๐ฆ)๐,๐โ1
) โ ๐๐ฅ,
๐ฟ5,๐(๐๐ฅ๐ฆ,๐๐ฆ๐ฆ) =
1
2๐((๐๐ฅ๐ฆ)๐+1,๐
โ (๐๐ฅ๐ฆ)๐โ1,๐
+ (๐๐ฆ๐ฆ)๐,๐+1
โ (๐๐ฆ๐ฆ)๐,๐โ1
) โ ๐๐ฆ.
(32)
The obtained system of (31) with the supplemented boundaryconditions equation and the initial conditions equation issolved by the fourth-order Runge-Kutta method. Figure 5shows a comparison between the time responses of the systemequations (4a), (4b), and (4c) using the Runge-Kutta offourth-order method and the time response of the problems(31), using the finite difference method at the same values ofthe parameters shown in Figure 2.
4.2. Frequency Response Curves. When the amplitudeachieves a constant nontrivial value, a steady-state vibrationexists. Using the frequency response equations we can assessthe influence of the damping coefficients, the nonlinear para-meters, and the excitation amplitude on the steady-stateamplitudes. The frequency response equations (24)โ(29) arenonlinear equations in ๐
1, ๐2, and ๐
3which are solved
numerically.Thenumerical results are shown in Figures 6โ25,and in all figures the region of stability of the nonlinearsolutions is determined by applying the Routh-Hurwitz
criterion.The solid lines stand for the stable solution, and thedotted lines stand for the unstable solution. From the geo-metry of the figures, we observe that each curve is continuousand has stable and unstable solutions.
4.2.1. Response Curve of Case 1. To check the accuracy ofthe analytical solution derived by the multiple time scalein predicting the amplitude of the first mode, we comparethe amplitude of the first mode obtained from frequencyresponse equation of Case 1 with values obtained fromnumerical integration of (4a). Figure 6 shows a comparisonof these outputs for the first mode.The effects of the detuningparameter ๐
1on the steady-state amplitude of the first mode
for the stability first case, where ๐1ฬธ= 0, ๐2= 0, and ๐
3= 0, for
the parameters ๐1= 0.2, ๐ผ
7= 0.01, ๐
1= 2.3, ๐
1= 4, ๐
11=
0.1, ๐12= 0.2, ๐
14= 0.5, ฮฉ
1= 1, ฮฉ
2= 1.2, and ฮฉ
4= 1.4, as
shown in Figure 6.Figures 7โ11, show the effects of the damping coeffi-
cient ๐1, the first mode natural frequency ๐
1, the external
-
Mathematical Problems in Engineering 21
โ1 0 12
4
63
8
โ2 0 2
1.5
2.5
3.5
8
3
โ2 0 2
0.2
0.6
1
8
3
๐1
๐1๐1
a 2
a 1
a 3
๐ผ8 = 1.5
๐ฝ8 = 1.5
๐พ8 = 1.5
Figure 21: Effects of the nonlinear parameters (๐ผ8, ๐ฝ8, and ๐พ
8coupling terms).
excitation amplitude๐1, the nonlinear spring stiffness ๐ผ
7, and
the parametric excitation ๐14. Figures 7 and 8 show that the
steady-state amplitude ๐1is inversely proportional to ๐
1and
๐1, and also for decreasing๐
1or๐1the curve is bending to the
left. It is clear from Figure 9 that the steady-state amplitude๐1is increasing for increasing value of external excitation
force ๐1, and the zone of instability is increased. Figure 10
shows that as the nonlinear spring stiffness ๐ผ7is increased;
the continuous curve ismoved downwards. Also, the negativeand positive values of ๐ผ
7produce either hard or soft spring,
respectively, as the curve is either bent to the right or to theleft, leading to the appearance of the jump phenomenon.Theregion of stability is increased for increasing value of๐ผ
7. From
Figure 11, we observe that for increasing value of parametricexcitation amplitude ๐
14, the steady-state amplitude of the
first mode is increased, and the curve is shifted to the left.
4.2.2. Response Curve of Case 2. Figures 12โ16, show thefrequency response curves for the stability of the second case,where ๐
2ฬธ= 0, ๐
1= 0, and ๐
3= 0. Figure 12 shows a com-
parison of these outputs for the second mode. The effects ofthe detuning parameter ๐
2on the steady-state amplitude of
the second mode for the stability second case, where ๐2ฬธ= 0,
๐1= 0, ๐
3= 0, for the parameters: ๐
2= 0.2, ๐ฝ
8= 0.01,
๐2= 4, ๐
2= 4, ๐
21= 0.1, ๐
22= 0.2, ๐
24= 0.5, ฮฉ
1= 1, ฮฉ
2=
1.2, and ฮฉ4= 1.4, as shown in Figure 12. It can be seen from
the figure thatmaximum steady-state amplitude occurs when๐2โ ๐1. Figure 13 shows that as the nonlinear spring stiffness
๐ฝ8is increased, the continuous curve is moved downwards.
Also, the positive and negative values of ๐ฝ8produce either
soft or hard spring, respectively, as the curve is either bent tothe left or to the right, leading to the appearance of the jumpphenomenon. Figures 14, 15, and 16 show that the steady-state amplitude ๐
2is inversely proportional to ๐
2, ๐2and
directly proportional to the external excitation ๐2. Also, for
decreasing ๐2, ๐2the curve is bending to the left.
4.2.3. Response Curve of Case 6. Figures 17, 18, 19, 20, 21,22, 23, 24, and 25 show that the frequency response curvesfor practical case stability, where ๐
1ฬธ= 0, ๐2ฬธ= 0, and ๐
3ฬธ= 0.
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22 Mathematical Problems in Engineering
โ2 0 22
4
6
8
10
1
โ2 0 2
0.5
1.5
2.5
3.5
4.5
0.01
โ2 0 2
0.2
0.6
1
1.4
0.01
a 3
a 2
a 1
๐1๐1
๐1
๐ผ7 = 0.01๐ฝ7 = 1
๐พ7 = 1
Figure 22: Effects of the nonlinear parameters (๐ผ7, ๐ฝ7, and ๐พ
7coupling terms).
Figure 17 shows that the effects of the detuning parameter๐1on the amplitudes of the three modes. From this figure,
we observe that these modes intersect, and for positive valueof the detuning parameter ๐
1the amplitudes are stable. For
negative value of the detuning parameter ๐1down to and
including โ0.5 the system becomes unstable.Figure 18 shows that the steady-state amplitudes of the
three modes ๐1, ๐2, and ๐
3are directly proportional to
the external excitation force ๐1. Also, form this figure we
show that the stability region is decreased for increasing ๐1.
Figures 19โ21 show that the steady-state amplitudes andstability of the threemodes ๐
1, ๐2, and ๐
3are inversely propor-
tional to the nonlinear parameters (๐ผ1, ๐ผ2, ๐ผ8), (๐ฝ1, ๐ฝ2, ๐ฝ8),
and (๐พ1, ๐พ2, ๐พ8), respectively.
Figures 22โ24, show the effects of the nonlinear param-eters (๐ผ
7, ๐ฝ7, ๐พ7), (๐ผ3, ๐ฝ3, ๐พ3), and (๐ผ
5,๐ฝ5, ๐พ5) on the steady-
state amplitudes of the three modes. It is clear that thestability regions are increased for increasing these nonlinear
parameters. For increasing value of linear viscous dampingcoefficients, we note that the steady-state amplitudes areincreasing or decreasing for the first, second, and thirdmodes, respectively, as shown in Figure 25. The region ofstability system is increased for decreasing value of dampingcoefficients.
4.3. Comparison Study. In the previous work [11], the chaoticdynamics of a six-dimensional nonlinear system whichrepresents the averaged equation of a composite laminatedpiezoelectric rectangular plate subjected to the transverse, in-plane excitations and the excitation loaded by piezoelectriclayers are analyzed. The case of 1 : 2 : 4 internal resonances isconsidered.
In our study, the nonlinear analysis and stability ofa composite laminated piezoelectric rectangular thin plateunder simultaneous external and parametric excitation forces
-
Mathematical Problems in Engineering 23
โ2 โ1 0 1 20
2
4
6
8
5
10
1.5
โ2 โ1 0 1 20.5
1.5
2.5
3.5
5
โ1
โ1 โ0.5 0 0.50
0.5
1
1.5
2
5
1.5-1
๐1
๐1๐1
a 3
a 2
a 1
๐ผ3 = โ1
๐ฝ3 = 10
๐พ3 = 10
Figure 23: Effects of the nonlinear parameters (๐ผ3, ๐ฝ3, and ๐พ
3coupling terms).
are investigated.The second-order approximation is obtainedto consider the influence of the cubic terms on nonlin-ear dynamic characteristics of the composite laminatedpiezoelectric rectangular plate using the multiple scalemethod. All possible resonance cases are extracted at thisapproximation order. The case of 1 : 1 : 3 internal resonanceand primary resonance is considered. The stability of thesystem and the effects of different parameters on systembehavior have been studied using phase plane and frequencyresponse curves. The analytical results given by the methodof multiple time scale are verified by comparison with resultsfrom numerical integration of the modal equations. Reliabil-ity of the obtained results is verified by comparison betweenthe finite difference method (FDM) and Runge-Kuttamethod (RKM). Variation of the some parameters leads tomultivalued amplitudes and hence to jump phenomena. It isquite clear that some of the simultaneous resonance cases areundesirable in the design of such system. Such cases shouldbe avoided as working conditions for the system.
5. Conclusions
Multiple time scale perturbation method is applied to deter-mine second-order approximate solutions for rectangularsymmetric cross-ply laminated composite thin plate sub-jected to external and parametric excitations. Second-orderapproximate solutions are obtained to study the influenceof the cubic terms on nonlinear dynamic characteristics ofthe composite laminated piezoelectric rectangular plate. Allpossible resonance cases are extracted at this approximationorder. The study is focused on the case of 1 : 1 : 3 primary res-onance and internal resonance, whereฮฉ
3โ ๐1, ๐2โ ๐1, and
๐3โ 3๐1.The analytical results given by themethod ofmulti-
ple time scale are verified by comparison with results fromnumerical integration of the modal equations. Reliability ofthe obtained results is verified by comparison between thefinite difference method (FDM) and Runge-Kutta method(RKM). The stability of a composite laminated piezoelectricrectangular thin plate is investigated. The phase-plane
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24 Mathematical Problems in Engineering
โ2 โ1 0 12
3
4
5
6
40
โ2 โ1 0 10.5
1.5
2.5
3.5
20
โ1 0 10.2
0.6
1
40
๐1
๐1๐1
a 3
a 2a 1
๐ผ5 = 20
๐ฝ5 = 40
๐พ5 = 20
Figure 24: Effects of the nonlinear parameters (๐ผ5, ๐ฝ5, and ๐พ
5coupling terms).
method and frequency response curves are applied to studythe stability of the system. From the previous study thefollowing may be concluded.
(1) The simultaneous resonance case ฮฉ3โ ๐1, ๐2โ ๐1,
and ๐3โ 3๐
1is one of the worst case, and they
should be avoided in design of such system.Of course,the excitation frequencyฮฉ
3is out of control. But this
case should be avoided through having ๐2ฬธ= ๐1or
๐3ฬธ= 3๐1.
(2) A comparison between the solutions obtained numer-ically with that prediction from the multiple scalesshows an excellent agreement.
(3) Reliability of the obtained results is verified by com-parison between the finite difference method (FDM)and Runge-Kutta method (RKM).
(4) Variation of the parameters ๐1, ๐2, ๐ผ7, ๐ฝ8, ๐1, ๐2, ๐1,
๐2leads tomultivalued amplitudes and hence to jump
phenomena.
(5) For the first and second modes, the steady-stateamplitudes ๐
1and ๐2are directly proportional to the
excitation amplitude ๐1and ๐
2and inversely propor-
tional to the linear damping ๐1and ๐
2, respectively.
(6) The negative and positive values of nonlinear stiffness๐ผ7, ๐ฝ8produce either hard or soft spring, respectively,
as the curve is either bent to the right or to the left.(7) The region of instability increase, which is undesir-
able, for increasing excitation amplitudes ๐1, ๐2and
for negative values of nonlinear stiffness ๐ผ7, ๐ฝ8.
(8) The multivalued solutions are disappeared forincreasing linear damping coefficients ๐
1, ๐2.
(9) The region of stability increases, which is desirablefor decreasing excitation amplitude ๐
1, nonlinear
parameters (๐ผ1, ๐ฝ1, ๐พ1), (๐ผ2, ๐ฝ2, ๐พ2), and for increasing
nonlinear stiffness (๐ผ7, ๐ฝ7, ๐พ7).
(10) The steady-state amplitudes of the three modes ๐1,
๐2, and ๐
3are directly proportional to the exci-
tation amplitude ๐1and inversely proportional to
-
Mathematical Problems in Engineering 25
โ1 0 13
4
5
6
0.5
0.2
โ1 0 1
0.5
1.5
2.5
0.5
0.2
โ2 โ1 0 1
0.4
0.8
1.2
1.6
0.5
0.2
๐1
๐1๐1
a 3
a 2
a 1
๐1 = 1.2
๐2 = 1.2
๐3 = 1.2
Figure 25: Effects of the linear damping coefficients.
the nonlinear parameters (๐ผ1, ๐ผ2, ๐ผ8), (๐ฝ1, ๐ฝ2, ๐ฝ8), and
(๐พ1, ๐พ2, ๐พ8), respectively.
For further work, we intend to extend our work to explorethe modeling formulation by investigating the role staticloading (in addition to the dynamic component).
Acknowledgment
The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions for improving thequality of this paper.
References
[1] M. Ye, J. Lu,W. Zhang, andQ.Ding, โLocal and global nonlineardynamics of a parametrically excited rectangular symmetriccross-ply laminated composite plate,โChaos, Solitons&Fractals,vol. 26, no. 1, pp. 195โ213, 2005.
[2] W. Zhang, โGlobal and chaotic dynamics for a parametricallyexcited thin plate,โ Journal of Sound and Vibration, vol. 239, no.5, pp. 1013โ1036, 2001.
[3] X. Y. Guo, W. Zhang, and M. Yao, โNonlinear dynamics ofangle-ply composite laminated thin plate with third-order sheardeformation,โ Science China Technological Sciences, vol. 53, no.3, pp. 612โ622, 2010.
[4] W. Zhang, Z. Liu, and P. Yu, โGlobal dynamics of a parametri-cally and externally excited thin plate,โNonlinear Dynamics, vol.24, no. 3, pp. 245โ268, 2001.
[5] W. M. Tien, N. S. Namachchivaya, and A. K. Bajaj, โNon-linear dynamics of a shallow arch under periodic excitation-I. 1:2 internal resonance,โ International Journal of Non-LinearMechanics, vol. 29, no. 3, pp. 349โ366, 1994.
[6] M. Sayed and A. A. Mousa, โSecond-order approximation ofangle-ply composite laminated thin plate under combined exci-tations,โ Communications in Nonlinear Science and NumericalSimulation, vol. 17, no. 12, pp. 5201โ5216, 2012.
[7] W. Zhang, C. Song, and M. Ye, โFurther studies on nonlinearoscillations and chaos of a symmetric cross-ply laminated thin
-
26 Mathematical Problems in Engineering
plate under parametric excitation,โ International Journal ofBifurcation and Chaos in Applied Sciences and Engineering, vol.16, no. 2, pp. 325โ347, 2006.
[8] W. Zhang, J. Yang, and Y. Hao, โChaotic vibrations of anorthotropic FGM rectangular plate based on third-order sheardeformation theory,โ Nonlinear Dynamics, vol. 59, no. 4, pp.619โ660, 2010.
[9] W. Zhang, Z. Yao, and M. Yao, โPeriodic and chaotic dynamicsof composite laminated piezoelectric rectangular plate withone-to-two internal resonance,โ Science in China, Series E, vol.52, no. 3, pp. 731โ742, 2009.
[10] W. Zhang and S. B. Li, โResonant chaotic motions of a buckledrectangular thin plate with parametrically and externally exci-tations,โ Nonlinear Dynamics, vol. 62, no. 3, pp. 673โ686, 2010.
[11] W. Zhang, M. Gao, M. Yao, and Z. Yao, โHigher-dimensionalchaotic dynamics of a composite laminated piezoelectric rect-angular plate,โ Science in China, Series G, vol. 52, no. 12, pp.1989โ2000, 2009.
[12] W. Zhang and W. L. Hao, โMulti-pulse chaotic dynamicsof six-dimensional non-autonomous nonlinear system for acomposite laminated piezoelectric rectangular plate,โNonlinearDynamics, 2013.
[13] X. Y. Guo and W. Zhang, โNonlinear dynamics of compositelaminated thin plate with 1:2:3 inner resonance,โ in Proceedingsof the 2nd International Conference on Mechanic Automationand Control Engineering (MACE โ11), pp. 7479โ7482, July 2011.
[14] M. Eissa and M. Sayed, โA comparison between active andpassive vibration control of non-linear simple pendulum. I.,โMathematical & Computational Applications, vol. 11, no. 2, pp.137โ149, 2006.
[15] M. Eissa and M. Sayed, โA comparison between active andpassive vibration control of non-linear simple pendulum. II.Longitudinal tuned absorber and negative ๐บ๏ฟฝฬ๏ฟฝ and ๐บ๐๐ feed-back,โ Mathematical & Computational Applications, vol. 11, no.2, pp. 151โ162, 2006.
[16] M. Eissa and M. Sayed, โVibration reduction of a three DOFnon-linear spring pendulum,โ Communications in NonlinearScience and Numerical Simulation, vol. 13, no. 2, pp. 465โ488,2008.
[17] M. Sayed, Improving the mathematical solutions of nonlineardifferential equations using different control methods [Ph.D.thesis], Menoufia University, Menoufia, Egypt, 2006.
[18] M. Sayed and M. Kamel, โStability study and control ofhelicopter blade flapping vibrations,โ Applied MathematicalModelling, vol. 35, no. 6, pp. 2820โ2837, 2011.
[19] M. Sayed and M. Kamel, โ1:2 and 1:3 internal resonance activeabsorber for non-linear vibrating system,โ Applied Mathemati-cal Modelling, vol. 36, no. 1, pp. 310โ332, 2012.
[20] M. Sayed andY. S.Hamed, โStability and response of a nonlinearcoupled pitch-roll ship model under parametric and harmonicexcitations,โ Nonlinear Dynamics, vol. 64, no. 3, pp. 207โ220,2011.
[21] Y. A. Amer,H. S. Bauomy, andM. Sayed, โVibration suppressionin a twin-tail system to parametric and external excitations,โComputers and Mathematics with Applications, vol. 58, no. 10,pp. 1947โ1964, 2009.
[22] Y. S. Hamed, M. Sayed, D.-X. Cao, and W. Zhang, โNonlinearstudy of the dynamic behavior of a string-beam coupled systemunder combined excitations,โActaMechanica Sinica, vol. 27, no.6, pp. 1034โ1051, 2011.
[23] J. Awrejcewicz, V. A. Krysko, I. V. Papkova, and A. V. Krysko,โRoutes to chaos in continuous mechanical systems. Part 1:mathematical models and solution methods,โ Chaos, Solitons &Fractals, vol. 45, pp. 687โ708, 2012.
[24] A. V. Krysko, J. Awrejcewicz, I. V. Papkova, and V. A. Krysko,โRoutes to chaos in continuous mechanical systems. Part 2:modelling transitions from regular to chaotic dynamics,โChaos,Solitons & Fractals, vol. 45, pp. 709โ720, 2012.
[25] J. Awrejcewicz, A. V. Krysko, I. V. Papkova, and V. A. Krysko,โRoutes to chaos in continuous mechanical systems. Part 3: theLyapunov exponents, hyper, hyper-hyper and spatial-temporalchaos,โ Chaos, Solitons & Fractals, vol. 45, pp. 721โ736, 2012.
[26] J. Awrejcewicz, A. V. Krysko, J. Mrozowski, O. A. Saltykova, andM. V. Zhigalov, โAnalysis of regular and chaotic dynamics of theEuler-Bernoulli beams using finite difference and finite elementmethods,โ Acta Mechanica Sinica, vol. 27, no. 1, pp. 36โ43, 2011.
[27] J. N. Reddy, โA refined nonlinear theory of plates with transverseshear deformation,โ International Journal of Solids and Struc-tures, vol. 20, no. 9-10, pp. 881โ896, 1984.
[28] Z. G. Yao, W. Zhang, and L. H. Chen, โPeriodic and chaoticoscillations of laminated composite piezoelectric rectangularplate with 1:2:3 internal resonances,โ in Proceedings of the 5thInternational Conference on Nonlinear Mechanics, pp. 720โ725,Shanghai, China, 2007.
[29] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience, New York, NY, USA, 1981.
[30] A. H. Nayfeh, Nonlinear Interactions: Analytical, Computa-tional, and Experimental Methods, Wiley Series in NonlinearScience, Wiley-Interscience, New York, NY, USA, 2000.
[31] A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, NewYork, NY, USA, 1973.
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