research article dynamic characteristics of

Research ArticleDynamic Characteristics of Electrostatically ActuatedMicrobeams with Slant Crack

Han Zhou Wen-Ming Zhang Zhi-Ke Peng and Guang Meng

State Key Laboratory of Mechanical System and Vibration School of Mechanical Engineering Shanghai Jiao Tong University800 Dongchuan Road Shanghai 200240 China

Correspondence should be addressed to Wen-Ming Zhang wenmingzsjtueducn

Received 14 September 2014 Accepted 18 December 2014

Academic Editor Anaxagoras Elenas

Copyright copy 2015 Han Zhou 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

An improved model of the slant crack on a microbeam is presented Based on fracture mechanics the rotation coefficient forthe slant crack is derived as a massless rotational spring accounting for the additional stress intensity factors generated by theorientation of the crack compared to the transverse crack Comparisons between microbeams with a slant crack of differentgeometry parameters (slant angle depth ratio and crack position) are investigatedwith regard to the dynamicmechanical behaviorsand nonlinear response By presenting a mathematical modeling the effects of the slant crack and the electric actuation of anelectrostatically actuated fixed-fixed microbeam on the dynamic characteristics are examined in detail It is shown that the crackposition has more significant influence on the pull-in voltage value than the slant angle or the depth ratio Approaching the slantcrack to the fixed end or enlarging the external incentives amplifies the nonlinearity of the microbeam system while the effects ofdepth ratio and slant angle are dependent on the crack positionThe resonance frequency and the resonance amplitude are affectedas well

1 Introduction

With the advantages in miniaturizing reducing cost andenergy consumption the microelectromechanical systems(MEMS) are a growing industry finding their applicationin different fields such as resonators for sensing [1 2] andelectric filtering applications [3 4] Microbeams such as can-tilever and doubly clamped beam are the major componentsof MEMS devices and the preferred actuation method isalways the electric actuation In an electrostatically actuatedmicrobeam an air gap capacitor composed of a movablemicrobeam (upper electrode) and a fixed (lower) electrodeis connected to a voltage source This generates electric fieldand electrostatic force The elastic force grows linearly withdisplacement whereas the electrostatic force grows inverselyproportional to the square of the distance when a potentialdifference is applied between the two electrodes Conse-quently the moveable beam deforms and the displacementgrows when the voltage is increased until at one point thegrowth rate of the electrostatic force exceeds than the elasticforce The system cannot reach a force balance without

a physical contact and the pull-in instability occurs Thisphenomenon is known as ldquopull-inrdquo and the critical voltageis the pull-in voltage The reason for this phenomenon is theelectrostatic force nonlinearity [5 6]

In order to avoid or use the characteristic of the phe-nomenon it is essential to study the pull-in The pull-ininstability is firstly found out in 1967 [7] and later Bernsteinet al [8] proved the existence of a bifurcation point at pull-in where the microbeam deflection becomes unstable andthen Pelesko [9] proved the uniqueness property of the pointA review including the overview of the pull-in phenomenonin electrostatically actuated MEMS and NEMS devices andthe physical principals of the pull-in instability provide acomprehensive understanding of the phenomenon [10] Inrecent decades differentmodels have been developed to solvethe pull-in problems such as the nonlinear model of a beamincluding the electrostatic force midplane stretching andapplied axial load [11 12] In the same time different analyt-ical numerical methods are improved such as the shootingmethod [11] the differential quadrature method (DQM) [13]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 208065 13 pageshttpdxdoiorg1011552015208065

2 Mathematical Problems in Engineering

and the combination of step-by-step linearization method(SSLM) and Galerkin-based reduced order model [6]

Another important problem generated during themachining process is the fatigue crack [14] The existence ofthe crack was found to induce considerable local flexibilitydue to the strain energy concentration in the vicinity of thecrack tip under load Moreover the crack will open or closein time depending on the loading conditions and vibrationamplitude which will change the dynamic characteristicsEventually the detection of the structural flaw can beavailable through measuring such changes Moreover a lotof analytical numerical and experimental investigationsare reported now Whether the crack is an open or a closedone depends on the combination of the static deflectionof the cracked beam caused by some loading componentssuch as the residual loads and the structure weights withthe vibration effect In detail the crack remains open allthe time or it opens and closes regularly when the staticdeflection is larger than the vibration amplitudes But if thestatic deflection is small then the crack will open and closein time depending on the vibration amplitude In these twocases the former system is linear while the latter one isnonlinear So most of the researchers made the assumptionthat the crack in a structural element is open and remainsopen during vibration in their work in order to avoid thecomplexities that resulted from the nonlinear characteristicspresented by introducing a breathing crack (a crack whichopens and closes during vibration) [15ndash17]

Many researchers are focused on developing suitablemodels to describe the effect of damage on the beam-likestructures and different approaches for crack modeling havebeen reported The approaches can be generally divided intothree categories [18] spring models or elastic hinges [19]local stiffness reduction [20] and finite element models [21]According to Friswell and Penny in [22] simple modelsof crack flexibility based on beam elements are adequatecomparedwith other approaches and thosemodels belong tothe category relying on the spring models In particular themodel of crack as an internal hinge endowedwith a rotationalspring connecting the two adjacent beam segments is of highaccuracy

Based on that the crack is modeled as an equivalentmassless rotational spring with a local compliance and thismethod was often used to quantify the relation between theapplied load and the strain concentration around the tip ofthe crack in a macroscopic way [23 24] With improvementof the method in recent years the crack is substituted bythe additional local compliance of a cracked beam withrelation to the strain energy concentration as well as to thestress intensity factor and a lot of results both analyticaland experimental are gained under different loading andgeometry for a number of cases Most researchers assumedthe crack to be open but not close which guaranteed aconstant stiffness and frequency shift during vibration Linet al [25] investigated the cracked beamsmodel as a functionof the crack depth as well as the consequently dynamicbehaviors and the stability characteristics FurthermoreRubio and Fernandez-Saez studied the applicability ofdifferent approximate closed-form solutions to evaluate

the natural frequencies for bending vibrations of simplysupported EulerndashBernoulli cracked beams [26] Motallebi etal [6] presented investigation of the effects of the geometryparameters such as the crack depth crack position andthe crack number of open crack on the static and dynamicpull-in voltages of the microbeams under different supportmethods Afshari and Inman presented results that theproposed crack modeling approach as a massless rotationalspring was beneficial as it provided twice-differentiablemode shapes for the cracked beam and can be used as trialfunctions in the assumed mode approximations [27]

Since resonant sensors are important components inmicromechanical devices the investigations into the fre-quency response of a resonant microbeam to an electricactuation are of big significance [28] As reported frequencyof a resonant microbeam is very sensitive to the axial straininduced by external loads such as pressure temperatureforce and acceleration and consequently causes a shift inits natural frequencies Such shift is generally converted to adigital signal related to the physical quantity being measured[29] Additionally some reports show that some other factorssuch as squeeze-film damping [30] and the elasticity of themicrobeam supports can also originate the frequency shift[31] Alsaleem et al [32] presented results of modelinganalysis and experimental investigation for nonlinear reso-nances and the dynamic pull-in instability in electrostaticallyactuated resonators Caddemi et al [18] studied the nonlineardynamic response of the EulerndashBernoulli beam in presence ofmultiple concentrated switching cracks (ie cracks that areeither fully open or fully closed) Besides the transverse crackanalyses the dynamic characteristics of slant crack attractattention of researchers But most such study is based on thevibrations analysis of rotors with slant crack for example[33 34] Little investigation of effect of slant crack on thedynamic response of microbeams is carried out

In this paper an improved model of beam with slantopen crack is presented as a massless rotational springIt is based on a continuous beam model and the stressand strain energy analysis of the vicinity of the crack Thecombinatorial method of SSLM and Galerkin-based reducedorder model is used to investigate the static response ofMEMS devices and the fixed-fixed beam model is appliedThe natural frequencies and corresponding cracked modeshapes of the microbeam are calculated with the usage of thetransfer matrix method under the boundary and patchingconditions And then the comparisons of the dynamicvibration behaviors and the pull-in voltages of beams withdifferent geometry parameters of slant crack are shown Theeffects of the crack depth crack slant angle and crack positionon the dynamic vibration behaviors and the pull-in voltagesare discussed Finally an investigation into the dynamicresponse of the microbeam is presented by using the multiplescalesmethod and the effects of the crack parameters and theelectric actuation are discussed on the vibration nonlinearity

2 Mathematical Modeling

Figure 1 shows the schematic of an electrostatically actuatedfixed-fixed microbeam with a slant crack on the surface

Mathematical Problems in Engineering 3

V

Stationary electrode

h

yx Crack bMicrobeam

d

L

a

Xc

120579

(a)

Beam

Crack

M

y

x

120579

120573

120591

120590

120573 sin120579

120573 cos120579

(b)

(c)

Figure 1 Schematic of a microbeam with a slant crack on the surface (a) The mathematical modeling of an electrostatically actuated fixed-fixed microbeam with a slant crack (b) Top view of stress components analysis of the microbeam loaded with a bending moment 119872 (c)Optical image of an actual beam specimen with a slant crack located at the center of aMEMS fatigue device and anchored to two lateral platesacting as electrostatic actuators [35]

The specimen is the small suspended fixed-fixed microbeamwith a rectangular cross section located at the centerof a MEMS device which is fabricated using the goldelectrodeposition technique [35] The beam specimen isanchored to two lateral plates acting as electrostatic actuators(Figure 1(c)) When the plates are actuated the beam struc-ture deforms and undergoes a tensile load The magnifiedsection reveals the formation of intruded and extrudedsections and the propagated crack Figure 1(a) depicts thesimplified mathematical modeling of the electrostaticallyactuated fixed-fixed microbeam including a suspended elas-tic beam with an applied electrostatic force and anotherbeam fixed at both ends to the ground conductor The upperbeam is suspended over a dielectric film deposited on topof the center conductor which will be pulled down for theelectrostatic force when a voltage is applied between theupper and lower electrodes The microbeams are consideredas Euler-Bernoulli beams of length 119871 width 119887 thickness ℎdensity 120588 Youngrsquos modulus 119864 Poissonrsquos ratio ] and 119864 =

119864(1 minus 1205922) On the surface of the beam an slant crack was

located at point119883119888 having a depth 119886 oriented at an angle of 120579relative to the normal direction of the beam the depth ratioof 120574 = 119886ℎ 119889 and 120576 are initial gap and dielectric constantrespectively 119909 and 119908(119909 ) denote the coordinate along thelength of the microbeam with its origin at the left end andthe transverse displacement of the beam respectively Nowthe entire beam is divided into two parts with length of119883119888 and 119871 minus 119883119888 respectively [6] and the global nonlinearsystem can be separated into two linear subsystems joinedby an additional local stiffness discontinuity According to

the theory of fracture mechanics the additional rotationcoefficient can be derived from the strain energy release rate119869 and the strain energy 119880 For plane strain the expression of119869 119880 and the relation of them are [36 37]

119869 =1

119864

(1198702

119868+ 1198702

119868119868+ (1 + 120592)119870

2

119868119868119868)

119880 = ∬119860

119869 119889119860

Θ =1205971198802

1205971198752

(1)

where 119860 is the area of the crack 119870119868 119870119868119868 119870119868119868119868

representthe stress intensity factors (SIFs) corresponding to openingsliding and tearingmode of crack displacement respectivelyΘ is the additional rotation in the slope due to the existenceof the crack and 119875 is the force loaded on the beam

The SIFs are derived as follows To simplify the problemthe slant-cracked beam is loaded with pure bending moment119872 As shown in Figure 1(b) compared to a transverse crackthe bending119872 leads to more numbers of stress componentsresponsible for the opening and tearing mode of crackdisplacementThe stress components include the shear stress120591 and the normal stress 120590 which cause tearing mode andopening mode of the crack respectively According to stressanalysis [38] it can be gotten that

1205900=119872

119868120573 120591

0= 0 120590

119888= 1205900cos2120579

120591119888= 1205900cos 120579 sin 120579

(2)


where 120573 is the distance from the centre point of the crackalong the crack edge 120590

0 1205910and 120590

119888 120591119888are the stress com-

ponents in the transverse crack surface and the slant cracksurface respectively And 119868 = 112119887ℎ

3 is the inertia momentof microbeam cross section To simplify the calculation wetake the largest stress which means 120590

0= 6119872119887ℎ

2Then SIFs for opening sliding and tearing mode can be

expressed as

1198700

119868= 1205900radic120587120572119865

2 119870

0

119868119868= 1198700

119868119868119868= 0

119870119888

119868= 120590119888radic120587120572119865

2 119870

119888

119868119868= 0 119870

119888

119868119868119868= 120591119888radic120587120572119865

119868119868119868

(3)

where 1198700119868 1198700119868119868 and 119870

0

119868119868119868represent SIFs related to the trans-

verse crack while 119870119888119868 119870119888119868119868 and 119870

119888

119868119868119868are related to the slant

crack and one has

1198652(119886

ℎ)

= radic2ℎ

120587119886tan(120587119886

2ℎ)(

0923 + 0199 [1 minus sin (1205871198862ℎ)]4

cos (1205871198862ℎ))

119865119868119868119868(119886

ℎ) = radic

2ℎ

120587119886tan(120587119886

2ℎ)

(4)

With the expressions of SIFs derived above synthesizing (1)to (3) adopting the correction function given by Anifantisand Dimarogonas and Taylorrsquos series expansion [23] thenondimensional rotation expression is obtained as

Θ = 6120587 (1 minus 1205922) (

ℎ

119871) [cos3 (120579) 119891

1(120574)

+ (1 + 120592) cos (120579) sin2 (120579) 1198912(120574)]

1198911(120574) = 06348120574

2minus 1035120574

3+ 37201120574

4minus 51773120574

5

+ 75531205746minus 7332120574

7+ 24909120574

8

1198912(120574) = 02026120574

2+ 003378120574

4+ 0009006120574

6+ 0002734120574

8

(5)

The microbeam is subject to a viscous damping dueto squeeze-film damping And this effect is approximatedby an equivalent damping coefficient 119888 per unit length[39 40]

119886and

119903are the stretching and residual forces

respectivelyThus the nondimensional governing equation ofthe transverse vibration of the beam can be given by [6]

1205974119908

1205971199094+1205972119908

1205971199052+ 119888

120597119908

120597119905minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881 (119905)

(1 minus 119908)]

2

(6)

The boundary conditions of the fixed-fixed microbeam are asfollows

119908 (0 119905) = 119908 (1 119905) =120597119908

120597119909(0 119905) =

120597119908

120597119909(1 119905) = 0 (7)

where the nondimensional variables and parameters are

119908 =119908

119889 119909 =

119909

119871 119905 =

119905lowast

120572 =61205761198714

119864ℎ31198893 119905

lowast= radic

120588119887ℎ1198714

119864119868

119873119903=121199031198712

119864ℎ3119887

119888 =121198881198714

119864ℎ3119887119905lowast

119873119886= 6(

119889

ℎ)

2

int

1

0

(120597119908

120597119909)

2

119889119909

(8)

By dropping the time dependence of (6) the governingequation can be rewritten as

1205974119908

1205971199094minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881

(1 minus 119908)]

2

(9)

3 Numerical Analysis

In order to study the effect of the slant crack on the localstiffness discontinuity each segment divided by the crackcan be considered as a separate beam under undamped freevibration with nondimensional equation form as follows

1205974119908119894

1205971199094+1205972119908119894

1205971199052= 0 119894 = 1 2 (10)

Using the separable solutions 119908119894(119909 119905) = 119884

119894(119909)119890119895120596119905 in (10)

leads to the associated eigenvalue problem

1205974119884119894

1205971199094minus Λ4119884119894= 0 119894 = 1 2 (11)

with patching conditions as [25]

1198841(119897119888minus) = 1198842(119897119888+) 119884

10158401015840

1(119897119888minus) = 11988410158401015840

2(119897119888+)

119884101584010158401015840

1(119897119888minus) = 119884101584010158401015840

2(119897119888+) 119884

1015840

2(119897119888+) minus 1198841015840

1(119897119888minus) = Θ119884

10158401015840

2(119897119888+)

(12)

where Λ4 = 1205962 and 119897119888 = 119883119888119871 with 120596 being the nondimen-

sional nature frequency andΛ being the frequency parameterThe general solution of the eigenvalue problem in (11)

with boundary conditions in (7) and patching conditions in(12) is written as

1198841(119909) = 119860

1sin [Λ119909] + 1198611 cos [Λ119909] + 1198621 sinh [Λ119909]

+ 1198631cosh [Λ119909] (0 le 119909 le 119897119888)

1198842(119909) = 119860

2sin [Λ (119909 minus 119897119888)] + 1198612 cos [Λ (119909 minus 119897119888)]

+ 1198622sinh [Λ (119909 minus 119897119888)] + 1198632 cosh [Λ (119909 minus 119897119888)]

(119897119888 le 119909 le 1)

(13)

where 119860119894 119861119894 119862119894 119863119894(119894 = 1 2) are constants associated with

the 119894th segment And along with the process the square of


the series value of Λ obtained corresponds to the nondimen-sional natural frequencies of associated modes

Because of the nonlinearity of the static equation thenumerical solution is complicated and time consuming andthe direct application of the Galerkin method or finite differ-ence method creates a set of nonlinear algebraic equationsIn this paper a method of two steps is used In the first stepthe SSLM is used [8 41] and in the second step the Galerkinmethod for solving the linear equation obtained is appliedUsing the SSLM it is assumed that 119908119896 is the displacement ofbeam due to the applied voltage 119881119896 Therefore by increasingthe applied voltage to a new value the displacement can bewritten as [6]

119908119896+1

119904= 119908119896

119904+ 120575119908 = 119908

119896

119904+ 120595 (119909) (14)

when

119881119896+1

= 119881119896+ 120575119881 (15)

So the equation of the static deflection of the fixed-fixedmicrobeam (equation (9)) can be rewritten in the step of 119896+1as follows

1205974119908119896+1

119904

1205971199094minus (119873119896+1

119886+ 119873119903)1205972119908119896+1

119904

1205971199092= 120572[

119881

(1 minus 119908119896+1119904

)]

2

(16)

By considering the small value of 120575119881 it is expected that120595 would be small enough Thus by using the calculus ofvariation theory and Taylorrsquos series expansion about 119908119896 in(16) and applying the truncation to its first order for suitablevalue of 120575119881 it is possible to obtain the desired accuracy Thelinearized equation for calculating 120595 can be expressed as

1198894120595

1198891199094minus (119873119896

119886+ 120575119873119886+ 119873119903)1198892120595

1198891199092minus 120575119873119886

1198892119908

1198892119909

100381610038161003816100381610038161003816100381610038161003816(119908119896 119881119896)

minus 2

120572 (119881119896)2

(1 minus 119908119896119904)3120595 minus 2

120572119881119896120575119881

(1 minus 119908119896119904)2= 0

(17)

where the variation of the hardening term based on thecalculus of variation theory can be expressed as 120575119873

119886=

int1

0120595(119909)119889

21199081198892119909|(119908119896119881119896)119889119909 Similarly because of the small

value of 120575119881 and 120595 the value of multiplying 120575119873119886by 11988921205951198891199092

would be small enough that can be neglectedThe linear differential equation obtained is solved by the

Galerkin method and 120595(119909) based on function spaces can beexpressed as

120595 (119909) =

119899

sum

119895=1

119886119895119884119895(119909) (18)

where 119884119895(119909) is selected as the 119895th undamped linear mode

shape of the cracked microbeam 120595(119909) is approximated bytruncating the summation series to a finite number 119899 in thiswork

Substituting (18) into (17) and multiplying by 119884119894(119909) as a

weight function in the Galerkin method and then integrating

the outcome from 119909 = 0 to 1 a set of linear algebraic equa-tions can be obtained as follows

119899

sum

119895=1

119870119894119895119886119895= 119865119894

(119894 = 1 2 119899) (19)

where

119870119894119895= 119870119898

119894119895+ 119870119886

119894119895minus 119870119890

119894119895 119870

119898

119894119895= int

1

0

119884119894119884(4)

119895119889119909

119870119886

119894119895= int

1

0

119884119894[ (119873119896

119886+ 119873119903) 11988410158401015840

119895

+ (int

1

0

119884119894

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896119889119909)

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896

]119889119909

119870119890

119894119895= 2120572 (119881

119896)2

int

1

0

119884119894119884119895

(1 minus 120596119896)3119889119909

119865119894= 2120572 (119881

119896+1minus 119881119896) int

1

0

119884119894(119909)

(1 minus 120596119896)2119889119909

(20)

Considering the resonant microbeam (see Figure 1) actu-ated by an electric load that is composed of a DC component(polarization voltage) 119881

119901and an AC component Vac the

voltage signal can be given by

119881 (119905) = 119881119901+ Vac cos (120596119890119905) (21)

where 120596119890is the excitation frequency and 119881

119901≫ Vac

In order to investigate the dynamic behavior of themicrobeams more simply the linearization of the right partof the governing equation (6) can be written as

[119881(119905)

(1 minus 119908)]

2

= [1198812

119901+ 2119881119901Vac cos (120596119890119905) + V2accos

2(120596119890119905)]

sdot (1 + 2119908 + 31199082sdot sdot sdot )

= 1198812

119901(1 + 2119908 + 3119908

2) + 2119881

119901Vac cos (120596119890119905)

(22)

where the term involving V2ac is dropped because typicallyVac ≪ 119881

119901in resonant sensors Using the Rayleigh-Ritz

method assuming the bending deflection of the beam to beof the form 119908(119909 119905) = 119884(119909) lowast 119906(119905) as analyzed previously andsubstituting (22) into (6) we can get the governing equationas a single degree-of-freedom Duffing equation as follows

+ 1205962

0119906 minus ℏ119906

3= 120582 + ℓ119906

2+ 120594 cos (120596

119890119905) + 120581 (23)

where 119906(119905) (119905) (119905) are the displacement speed andacceleration of the center layer of the beam And one has

1205962

0= 1198861minus 1198731199031198862minus 2120572119881

2

119901 ℏ =

611988921198863

ℎ2

120582 = minus119888 ℓ = 312057211988731198812

119901 120594 = 2120572119887

1Vac1198812

119901

120581 = 12057211988711198812

119901

(24)


where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)

To calculate the nonlinear response of the electrostaticallyactuated microbeam given by (23) the multiple scale per-turbation theory is employed The response to the primaryresonance excitation of its first mode is analyzed becauseit is the case that is used in resonator applications Stillthe analysis is general and can be used to study nonlinearresponses of the beam to a primary resonance of any of itsmodes The excitation frequency 120596

119890is usually tuned close to

the fundamental nature frequency of mechanical vibrationsnamely 120596

119890= 1205960+ 120578120590 where 120578 is a dimensionless small

parameter and 120590 is a small detuning parameter By redefiningthe parameters as ℏ = 120578ℏ 120582 = 120578120582 ℓ = 120578ℓ 120594 = 120578120594 and 120581 = 120578120581(23) can be written as

+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)

To solve (26) two time scales 1198790= 119905 and 119879

1= 120578119905 are

introduced and the first-order uniform solution is given inthe form

119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)

Substituting (27) into (26) and equating coefficients oflike powers of 120576 the linear partial differential equations oforder 1205780 and order 1205781 are obtained as

1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879

1 The general solution of

the first equation of (28) can be written as

1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879

0)119890119895120575(1198791)2 and cc denotes a com-

plex conjugate Equation (29) is then substituted into thesecond equation of (28) and the trigonometric functionsare expanded The elimination of the secular terms yieldstwo first-order nonlinear ordinary differential equations thatallow for a stability analysis Furthermore the modulation ofthe response with the amplitude 119886 and phase 120575 is given by

1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)

where 120593 = 1205901198791minus 120575 The steady-state motion occurs when

1198631119886 = 119863

1120593 = 0 which corresponds to singular points of

(30) Then the steady-state frequency response is obtained as

[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)

where 1198600is the resonance amplitude when it is steady-state

While one has 1198600le 120594(120582120596

0) a function of the independent

incentive frequency is found to be

120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)

Defining nondimensional resonance frequency Ω = 1205961198901205960

(32) can be expressed as

Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)

As indicated in (33) the nondimensional design param-eters affect the resonance frequency through changing theeffective nonlinearity of the undamped linear mode shape

The maximum resonance amplitude is reached when themagnitude under the square root is zero [42] Hence

119860max =120594

1205821205960

(34)

and the corresponding frequency is

Ω = 1 minus3ℏ

81205962

0

1198602

max (35)

The curve determined by (35) is defined as the skeletonline which manifests the relation between the peak ampli-tude and the resonance frequency dominating the shape ofthe amplitude-frequency response (AFR) The AFR curvesindicate the nonlinear characteristics of the microbeamAccording to (33) and (35) the AFR curves and the skeletonlines not only are affected by parameters of the beam structureand the slant crack but also are related to the externalincentives namely the DC and AC voltages

4 Results and Discussion

41 Frequency andMode Shapes The geometric andmaterialproperties of the microbeam are as follows 119871 = 250 120583m119887 = 50 120583m 120576 = 885PFm ℎ = 3 120583m 119889 = 1 120583m 119864 =

169GPa 120588 = 2331 kgm3 and ] = 006 [6] Figure 2 showsthe effects of the crack depth ratio for selected crack positions(119897119888 = 005 025 05) and slant angles (120579 = 0

∘ 20∘) on the

value of the first four (119899 = 1ndash4) nature frequencies along thedepth ratio It can be found that increasing the depth ratiowillgradually decrease the natural frequencies with the tendencyserving as the confirmation of the EBB theory according toHasheminejad et al [43] Some important observations arefound Increasing the crack depth ratio does not have anyimpact on the cases of 119899 = 2 (119897119888 = 05) and 119899 = 4 (119897119888 = 05)


0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)

Figure 2The first four nondimensional natural frequencies of themicrobeam along with the depth ratio of the crack for three crack positionsand two slant angles

Table 1 Values of the first four frequency parameters for a fixed-fixed beamwith different crack positions and crack slant angles Crack depthratio 120574 = 05

Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137

which exhibits the same trend with results presented byHasheminejad et al [43] This can be explained by the factthat none of the crack positions (119897119888 = 05) lie on the vibrationnodes for the beam vibrating in first and third modes (119899 = 13) while in the case of the second and fourth modes (119899 = 24) the crack position (119897119888 = 05) is exactly set on the vibrationnodes

Take the slant angle of the crack into account theincrement in angle leads to a gradual increase of the four

nature frequencies except the special cases mentioned aboveBesides the effect of the crack position on frequency differsin different modes

More detailed changes of the first four frequency param-eters for the fixed-fixed microbeam with different crack slantangles 120579 for a crack located at positions 119897119888 = 005 and 05 aregiven in Table 1 and the crack depth ratio 120574 = 05

The first-mode shapes of the cracked microbeam ofdifferent geometry characteristics are displayed in Figure 3


0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)

Figure 3 The effects of the depth ratio (a b) and the slant angle (c d) of the slant crack on the first-mode shapes of the beam at differentpositions (a) and (c) 119897119888 = 05 (b) and (d) 119897119888 = 005

The abscissa is the nondimensional distance of the crackfrom the left end of the microbeam and the ordinate is thevibration amplitude of first-mode shape It manifests thatthe effects of the slant angle and depth ratio on the first-mode shapes depend on the crack position For examplethe growth in the depth ratio will increase the maximumvibration amplitude along the beam length when the crack isnear the center point of the beam (119897119888 = 05 see Figure 3(a))but it will lead the inverse trend when the crack is locatedat the end part (119897119888 = 005 see Figure 3(b)) Similarly whenit comes to the slant angle (see Figures 3(c) and 3(d)) theangle increment causes decrease in the maximum vibrationamplitude when the crack is located near the center part ofthe beam but it increases when it is near the end The effectdifferences in vibration amplitude are in good agreementwiththe ones in the nature frequency Additionally the maximumvibration amplitude is larger when the crack position getsnear the center part

42 Pull-In Voltage Analysis In the pull-in voltage analy-sis voltages will differ when different voltage increment isapplied but the difference is small enough to be neglected

0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388

Figure 4The displacement of center point of the beamwith the DCvoltage for case of position 05 slant angle 45∘ and a static voltageincrement of 002V for three depth ratios (01 05 and 06)

Just as seen in Figure 4 the abscissa is the value of the appliedDC voltage and the ordinate is the transverse deflection ofthe middle point of the beams Figure 4 illustrates how themicrobeam resonator loses stability Before the DC voltage


35

36

37

38

39

40

120574

Vpu

ll-in lc = 025 and 075

lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘

Figure 5 Variation of the pull-in voltage of the fixed-fixedmicrobeamwith the crack depth ratio (for three crack positions andthree slant angles)

increases to a certain value it exhibits a steady increase trendin the deflection of center point of the beam Once the DCvoltage reaches a critical point it enters the pull-in instabilityzone where the deflection begins oscillating violently withthe increasing of the DC voltage This critical point indicatesthe pull-in voltage and the corresponding displacement is thepull-in displacement

By using the modified model proposed the effects of theslant crack parameters on static pull-in voltage of the fixed-fixedmicrobeam are investigatedThe variation of the pull-involtages of the cracked beam of different crack depth ratio(for three crack positions and three slant angles) is shown inFigure 5 It is shown that the crack position has a strong effecton the pull-in voltageWhen the crack is near to themiddle ofthe microbeam especially for the higher crack depth ratiosthe pull-in voltage is greatly reduced And the position of119897119888 = 025 and 075 is more ineffective on the pull-in voltagethan other positions In addition the effect of the slant angleon the pull-in voltage is appreciable The pull-in voltage isincreased slightly along with the increase of the angle Inaddition the detailed effect of the crack position on pull-involtage of the beam is investigated and shown in Figure 6 Asshown interestingly there are several extreme points locatedin 119897119888 = 025 05 075 and two endpoints As shown in thefigure when the crack is located at 119897119888 = 025 and 075 it hasthe lowest effect on the pull-in voltage and when 119897119888 = 05the effect is greater but endpoints are still lower As shownthe rate of the variation is smaller in larger crack slant angleThe results gained above show that the crack position hasmore significant effect on the pull-in voltage of the beam thanthe crack depth ratio or the slant angel And the slant angelexhibits a contrary effect on the pull-in voltage comparedwith the crack depth ratio

43 Dynamic Response Analysis The dynamic response ofthemicrobeamdescribed by (31) has been taken into accountThe set of parameters applied in all the numerical simulations

0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08

Figure 6 Comparison of the pull-in voltage of the slant-crackedmicrobeam with the crack position (for four slant angles)

Table 2 Parameters values for the solution and results values

Figure Varying parameters Resonantfrequency Max-amplitude

Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317

Figure 10(a)Vac = 001V 2204 004188Vac = 05V 2204 02094Vac = 01V 2204 04188

are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The

resonance frequency and resonance amplitude are acquiredby varying one of the parameters as listed in Table 2

As plotted in Figures 7ndash10 the dynamic response curvesof the first mode are acquired The abscissa is the nondimen-sional resonance frequency and the ordinate is the resonanceamplitude But the abscissa of the subgraphs in the graphsis the excitation frequency As plotted when the resonance


098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e

Figure 7 The effects of the crack position on the dynamic response of microbeams of different positions 119897119888

099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)

Figure 8 The effects of the slant angle of the crack on the dynamic response of microbeams (a) 119897119888 = 05 (b) 119897119888 = 005

0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)

Figure 9 The effects of the crack depth ratio of the crack on the dynamic response of microbeams (a) 119897119888 = 05 (b) 119897119888 = 005


0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)

Figure 10 The effects of the external incentives on the dynamic response of microbeams (a) DC voltage (b) AC voltage

frequency grows sufficiently large the two branches of all thecurves merge and produce a standard resonance peak with ahardening characteristic A larger degree of curvature meansthe stronger hardening behavior which points out that thenonlinearity strengthens

Figure 7 illustrates the influence of the crack position onthe frequency response of the microbeam When the crackis located at the middle point of the beam the nonlinearbehavior of the system is not evident As the crack approachesthe fixed end the influence of the nonlinearities becomesmore obvious showing a typical hardening characteristic Atmost frequencies it is noted that the fact that the crackapproaches the fixed end results in a decrease in the steady-state amplitude For case of 119897119888 = 001 the value of thenondimensional resonance frequency where the maximumamplitude occurs is bigger than the other two cases

The influence of the slant angles of crack 120579 on thedynamic response differs at different crack locations as shownin Figure 8 When the crack is located at the center partof the microbeam (119897119888 = 05 Figure 8(a)) the microbeamwith a slant crack of a bigger angle displays a strongernonlinearity but a less resonance amplitude at most fre-quencies Increasing the slant angle decreases the maximumresonance amplitude and slightly increases the correspondingfrequency But if the crack is located at the fixed end of themicrobeam (119897119888 = 005 Figure 8(b)) the result becomes thatthe microbeam with a slant crack of a smaller angle depictsthe stronger nonlinearity and the less resonance amplitudeMeanwhile increasing the slant angle of the crack has nosignificant influence on the maximum resonance amplitudebut decreases the corresponding frequency

When it comes to the influence of the crack depth ratio120574 on the dynamic response the characteristics displayedin Figure 9 are similar to what is seen in Figure 8 It isnoted that a deeper crack affects the dynamic response ofthe microbeam in the same way with how a slant crackwith a smaller slant angle does Increasing the depth ratioof the crack located at the center part of the microbeam(119897119888 = 05 Figure 9(a)) weakens the nonlinear behavior of

themicrobeam and enlarges the resonance amplitude at mostfrequencies as well as the maximum resonance amplitudebut it has no significant influence on the correspondingresonance frequency contrary to the effects of the crack depthratio on the response characteristics located at the fixed end(119897119888 = 005 Figure 9(b))

Figure 10 illustrates the influence of the external incen-tives on the dynamic response of microbeams including theDC voltage (Figure 10(a)) and the AC voltage (Figure 10(b))As plotted in the two figures it can be directly seen thatincreased external incentives result in an obvious increase inthe steady-state amplitude at most frequencies as expectedAs the value of the external incentives is decreased thehardening behavior weakens A resonance response will notbe seen if the incentive values are significantly reduced Thisresult is in agreement with that found by Younis and Nayfeh[44]

5 Conclusion

In this work the model of slant open crack is establishedand it is applied into the continuous vibration equation ofelectrostatically actuated fixed-fixed microbeams Analyticresults of the natural frequency the corresponding modeshapes and the numerical results of the pull-in voltages arepresented By comparing the calculated results in differentcases the effects of the crack depth ratio crack positionand the slant angle on the dynamic behaviors and the pull-in voltages are investigated It is shown that except for somespecial cases the first four nature frequencies decrease withthe increase of the crack depth ratio but the opposite changeis displayed when the slant angle of the crack increasesIn addition the effect of the crack position on frequencydiffers in different modes Furthermore the influences of thecrack position slant angle and the depth ratio on the pull-in voltages are similar to those on frequencies respectivelyThe pull-in voltage decreases the most when the crack islocated at the middle point of the beams and the leastwhen the crack is located at the quarter point Finally


the dynamic response of the beam is investigated by employ-ing the multiple scale perturbation theory It is found thatthe response demonstrates a hardening characteristic whichis affected by the geometry parameters of the slant crackand the electric incentives The nonlinearity of the resonanceresponse gets enlarged when the crack position approachesthe fixed end of the microbeam or the DC voltage and ACvoltage value amplifies The effects of the crack depth ratioand the slant angle on the dynamic response vary in differentpositions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge projects supported bythe National Natural Science Foundation of China (no11322215) National Program for Support of Top-notch YoungProfessionals the Fok Ying Tung Education Foundation (no141050) and the Innovation Program of Shanghai MunicipalEducation Commission (no 15ZZ010)

References

[1] J Tamayo P M Kosaka J J Ruz A San Paulo and M CallejaldquoBiosensors based on nanomechanical systemsrdquo Chemical Soci-ety Reviews vol 42 no 3 pp 1287ndash1311 2013

[2] L M Bellan D Wu and R S Langer ldquoCurrent trends innanobiosensor technologyrdquo Wiley Interdisciplinary ReviewsNanomedicine andNanobiotechnology vol 3 no 3 pp 229ndash2462011

[3] S Pourkamali and F Ayazi ldquoElectrically coupled MEMS band-pass filters part I with coupling elementrdquo Sensors andActuatorsA Physical vol 122 no 2 pp 307ndash316 2005

[4] S Pourkamali and F Ayazi ldquoElectrically coupled MEMS band-pass filters part II Without coupling elementrdquo Sensors andActuators A Physical vol 122 no 2 pp 317ndash325 2005

[5] M Fathalilou A Motallebi H Yagubizade G RezazadehK Shirazi and Y Alizadeh ldquoMechanical behavior of anelectrostatically-actuatedmicrobeam undermechanical shockrdquoJournal of Solid Mechanics vol 1 no 1 pp 45ndash57 2009

[6] A Motallebi M Fathalilou and G Rezazadeh ldquoEffect of theopen crack on the pull-in instability of an electrostaticallyactuatedmicro-beamrdquoActaMechanica Solida Sinica vol 25 no6 pp 627ndash637 2012

[7] HCNathansonW ENewell RAWickstrom and J RDavisldquoThe resonant gate transistorrdquo IEEE Transactions on ElectronDevices vol 14 no 3 pp 117ndash133 1967

[8] D Bernstein P Guidotti and J Pelesko ldquoMathematical analysisof an electrostatically actuated MEMS devicerdquo in Proceedings ofthe Modelling amp Simulation of Microsystems pp 489ndash492 2000

[9] J A Pelesko ldquoMultiple solutions in electrostatic MEMSrdquo inProceedings of the International Conference on Modeling andSimulation of Microsystems (MSM rsquo01) pp 290ndash293 March2001

[10] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014

[11] E M Abdel-Rahman M I Younis and A H Nayfeh ldquoCharac-terization of the mechanical behavior of an electrically actuatedmicrobeamrdquo Journal of Micromechanics and Microengineeringvol 12 no 6 pp 759ndash766 2002

[12] V Y Taffoti Yolong and PWoafo ldquoDynamics of electrostaticallyactuated micro-electro-mechanical systems single device andarrays of devicesrdquo International Journal of Bifurcation andChaos vol 19 no 3 pp 1007ndash1022 2009

[13] F Najar S Choura S El-Borgi E M Abdel-Rahman andA H Nayfeh ldquoModeling and design of variable-geometryelectrostatic microactuatorsrdquo Journal of Micromechanics andMicroengineering vol 15 no 3 pp 419ndash429 2005

[14] B-G Nam S Tsuchida and K Watanabe ldquoFatigue crackgrowth driven by electric fields in piezoelectric ceramics andits governing fracture parametersrdquo International Journal ofEngineering Science vol 46 no 5 pp 397ndash410 2008

[15] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[16] A D Dimarogonas ldquoVibration of cracked structures a state ofthe art reviewrdquo Engineering Fracture Mechanics vol 55 no 5pp 831ndash857 1996

[17] M Rezaee and R Hassannejad ldquoFree vibration analysis ofsimply supported beamwith breathing crack using perturbationmethodrdquo Acta Mechanica Solida Sinica vol 23 no 5 pp 459ndash470 2010

[18] S Caddemi I Calio andMMarletta ldquoThe non-linear dynamicresponse of the Euler-Bernoulli beam with an arbitrary num-ber of switching cracksrdquo International Journal of Non-LinearMechanics vol 45 no 7 pp 714ndash726 2010

[19] A Khorram F Bakhtiari-Nejad and M Rezaeian ldquoCompari-son studies between twowavelet based crack detectionmethodsof a beam subjected to a moving loadrdquo International Journal ofEngineering Science vol 51 pp 204ndash215 2012

[20] M Dona A Palmeri and M Lombardo ldquoExact closed-formsolutions for the static analysis of multi-cracked gradient-elastic beams in bendingrdquo International Journal of Solids andStructures vol 51 no 15-16 pp 2744ndash2753 2014

[21] M Kisa and M Arif Gurel ldquoFree vibration analysis of uniformand stepped cracked beams with circular cross sectionsrdquo Inter-national Journal of Engineering Science vol 45 no 2ndash8 pp 364ndash380 2007

[22] M I Friswell and J E T Penny ldquoCrack modeling for structuralhealth monitoringrdquo Structural Health Monitoring vol 1 no 2pp 139ndash148 2002

[23] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[24] S Caddemi I Calio and F Cannizzaro ldquoThe influence ofmultiple cracks on tensile and compressive buckling of sheardeformable beamsrdquo International Journal of Solids and Struc-tures vol 50 no 20-21 pp 3166ndash3183 2013

[25] H P Lin S C Chang and J D Wu ldquoBeam vibrations with anarbitrary number of cracksrdquo Journal of Sound andVibration vol258 no 5 pp 987ndash999 2002

[26] L Rubio and J Fernandez-Saez ldquoA note on the use of approx-imate solutions for the bending vibrations of simply supportedcracked beamsrdquo Journal of Vibration andAcoustics Transactionsof the ASME vol 132 no 2 2010

[27] M Afshari and D J Inman ldquoContinuous crack modeling inpiezoelectrically driven vibrations of an Euler-Bernoulli beamrdquoJournal of Vibration and Control vol 19 no 3 pp 341ndash355 2013


[28] M H Ghayesh M Amabili and H Farokhi ldquoNonlinear forcedvibrations of amicrobeambased on the strain gradient elasticitytheoryrdquo International Journal of Engineering Science vol 63 pp52ndash60 2013

[29] M H Ghayesh H Farokhi andM Amabili ldquoNonlinear behav-iour of electrically actuated MEMS resonatorsrdquo InternationalJournal of Engineering Science vol 71 pp 137ndash155 2013

[30] S-C Sun C-W Chung C-M Hsu and J-H Kuang ldquoThesqueeze film damping effect on dynamic responses of micro-electromechanical resonatorsrdquo Applied Mechanics and Materi-als vol 284-287 pp 1961ndash1965 2013

[31] H O Ekici and H Boyaci ldquoEffects of non-ideal boundaryconditions on vibrations of microbeamsrdquo Journal of Vibrationand Control vol 13 no 9ndash10 pp 1369ndash1378 2007

[32] F M Alsaleem M I Younis and H M Ouakad ldquoOn thenonlinear resonances and dynamic pull-in of electrostaticallyactuated resonatorsrdquo Journal of Micromechanics and Microengi-neering vol 19 no 4 Article ID 045013 2009

[33] Y Lin and F Chu ldquoThe dynamic behavior of a rotor systemwith a slant crack on the shaftrdquo Mechanical Systems and SignalProcessing vol 24 no 2 pp 522ndash545 2010

[34] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[35] G De Pasquale and A Soma ldquoMEMSmechanical fatigue effectof mean stress on gold microbeamsrdquo Journal of Microelectrome-chanical Systems vol 20 no 4 pp 1054ndash1063 2011

[36] A K Darpe ldquoCoupled vibrations of a rotor with slant crackrdquoJournal of Sound and Vibration vol 305 no 1-2 pp 172ndash1932007

[37] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[38] A K Darpe ldquoDynamics of a Jeffcott rotor with slant crackrdquoJournal of Sound and Vibration vol 303 no 1-2 pp 1ndash28 2007

[39] J P Lynch A Partridge K H Law T W Kenny A S Kiremid-jian and E Carryer ldquoDesign of piezoresistive MEMS-basedaccelerometer for integration with wireless sensing unit forstructuralmonitoringrdquo Journal of Aerospace Engineering vol 16no 3 pp 108ndash114 2003

[40] M I Younis E M Abdel-Rahman and A Nayfeh ldquoA reduced-ordermodel for electrically actuatedmicrobeam-basedMEMSrdquoJournal of Microelectromechanical Systems vol 12 no 5 pp672ndash680 2003

[41] G Rezazadeh M Fathalilou R Shabani S Tarverdilou and STalebian ldquoDynamic characteristics and forced response of anelectrostatically-actuated microbeam subjected to fluid load-ingrdquo Microsystem Technologies vol 15 no 9 pp 1355ndash13632009

[42] Z-Y Zhong W-M Zhang G Meng and J Wu ldquoInclinationeffects on the frequency tuning of comb-driven resonatorsrdquoJournal of Microelectromechanical Systems vol 22 no 4 pp865ndash875 2013

[43] B S M Hasheminejad B Gheshlaghi Y Mirzaei and SAbbasion ldquoFree transverse vibrations of cracked nanobeamswith surface effectsrdquo Thin Solid Films vol 519 no 8 pp 2477ndash2482 2011

[44] M I Younis and A H Nayfeh ldquoA study of the nonlinearresponse of a resonant microbeam to an electric actuationrdquoNonlinear Dynamics vol 31 no 1 pp 91ndash117 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and the combination of step-by-step linearization method(SSLM) and Galerkin-based reduced order model [6]

Another important problem generated during themachining process is the fatigue crack [14] The existence ofthe crack was found to induce considerable local flexibilitydue to the strain energy concentration in the vicinity of thecrack tip under load Moreover the crack will open or closein time depending on the loading conditions and vibrationamplitude which will change the dynamic characteristicsEventually the detection of the structural flaw can beavailable through measuring such changes Moreover a lotof analytical numerical and experimental investigationsare reported now Whether the crack is an open or a closedone depends on the combination of the static deflectionof the cracked beam caused by some loading componentssuch as the residual loads and the structure weights withthe vibration effect In detail the crack remains open allthe time or it opens and closes regularly when the staticdeflection is larger than the vibration amplitudes But if thestatic deflection is small then the crack will open and closein time depending on the vibration amplitude In these twocases the former system is linear while the latter one isnonlinear So most of the researchers made the assumptionthat the crack in a structural element is open and remainsopen during vibration in their work in order to avoid thecomplexities that resulted from the nonlinear characteristicspresented by introducing a breathing crack (a crack whichopens and closes during vibration) [15ndash17]

Many researchers are focused on developing suitablemodels to describe the effect of damage on the beam-likestructures and different approaches for crack modeling havebeen reported The approaches can be generally divided intothree categories [18] spring models or elastic hinges [19]local stiffness reduction [20] and finite element models [21]According to Friswell and Penny in [22] simple modelsof crack flexibility based on beam elements are adequatecomparedwith other approaches and thosemodels belong tothe category relying on the spring models In particular themodel of crack as an internal hinge endowedwith a rotationalspring connecting the two adjacent beam segments is of highaccuracy

Based on that the crack is modeled as an equivalentmassless rotational spring with a local compliance and thismethod was often used to quantify the relation between theapplied load and the strain concentration around the tip ofthe crack in a macroscopic way [23 24] With improvementof the method in recent years the crack is substituted bythe additional local compliance of a cracked beam withrelation to the strain energy concentration as well as to thestress intensity factor and a lot of results both analyticaland experimental are gained under different loading andgeometry for a number of cases Most researchers assumedthe crack to be open but not close which guaranteed aconstant stiffness and frequency shift during vibration Linet al [25] investigated the cracked beamsmodel as a functionof the crack depth as well as the consequently dynamicbehaviors and the stability characteristics FurthermoreRubio and Fernandez-Saez studied the applicability ofdifferent approximate closed-form solutions to evaluate

the natural frequencies for bending vibrations of simplysupported EulerndashBernoulli cracked beams [26] Motallebi etal [6] presented investigation of the effects of the geometryparameters such as the crack depth crack position andthe crack number of open crack on the static and dynamicpull-in voltages of the microbeams under different supportmethods Afshari and Inman presented results that theproposed crack modeling approach as a massless rotationalspring was beneficial as it provided twice-differentiablemode shapes for the cracked beam and can be used as trialfunctions in the assumed mode approximations [27]

Since resonant sensors are important components inmicromechanical devices the investigations into the fre-quency response of a resonant microbeam to an electricactuation are of big significance [28] As reported frequencyof a resonant microbeam is very sensitive to the axial straininduced by external loads such as pressure temperatureforce and acceleration and consequently causes a shift inits natural frequencies Such shift is generally converted to adigital signal related to the physical quantity being measured[29] Additionally some reports show that some other factorssuch as squeeze-film damping [30] and the elasticity of themicrobeam supports can also originate the frequency shift[31] Alsaleem et al [32] presented results of modelinganalysis and experimental investigation for nonlinear reso-nances and the dynamic pull-in instability in electrostaticallyactuated resonators Caddemi et al [18] studied the nonlineardynamic response of the EulerndashBernoulli beam in presence ofmultiple concentrated switching cracks (ie cracks that areeither fully open or fully closed) Besides the transverse crackanalyses the dynamic characteristics of slant crack attractattention of researchers But most such study is based on thevibrations analysis of rotors with slant crack for example[33 34] Little investigation of effect of slant crack on thedynamic response of microbeams is carried out

In this paper an improved model of beam with slantopen crack is presented as a massless rotational springIt is based on a continuous beam model and the stressand strain energy analysis of the vicinity of the crack Thecombinatorial method of SSLM and Galerkin-based reducedorder model is used to investigate the static response ofMEMS devices and the fixed-fixed beam model is appliedThe natural frequencies and corresponding cracked modeshapes of the microbeam are calculated with the usage of thetransfer matrix method under the boundary and patchingconditions And then the comparisons of the dynamicvibration behaviors and the pull-in voltages of beams withdifferent geometry parameters of slant crack are shown Theeffects of the crack depth crack slant angle and crack positionon the dynamic vibration behaviors and the pull-in voltagesare discussed Finally an investigation into the dynamicresponse of the microbeam is presented by using the multiplescalesmethod and the effects of the crack parameters and theelectric actuation are discussed on the vibration nonlinearity

2 Mathematical Modeling

Figure 1 shows the schematic of an electrostatically actuatedfixed-fixed microbeam with a slant crack on the surface


V


h

yx Crack bMicrobeam

d

L

a

Xc

120579

(a)

Beam

Crack

M

y

x

120579

120573

120591

120590

120573 sin120579

120573 cos120579

(b)

(c)






119869 =1

119864

(1198702

119868+ 1198702

119868119868+ (1 + 120592)119870

2

119868119868119868)

119880 = ∬119860

119869 119889119860

Θ =1205971198802

1205971198752

(1)




1205900=119872

119868120573 120591

0= 0 120590

119888= 1205900cos2120579

120591119888= 1205900cos 120579 sin 120579

(2)



0 1205910and 120590




0= 6119872119887ℎ


expressed as

1198700

119868= 1205900radic120587120572119865

2 119870

0

119868119868= 1198700

119868119868119868= 0

119870119888

119868= 120590119888radic120587120572119865

2 119870

119888

119868119868= 0 119870

119888

119868119868119868= 120591119888radic120587120572119865

119868119868119868

(3)

where 1198700119868 1198700119868119868 and 119870

0



119888


crack and one has

1198652(119886

ℎ)

= radic2ℎ

120587119886tan(120587119886

2ℎ)(

0923 + 0199 [1 minus sin (1205871198862ℎ)]4

cos (1205871198862ℎ))

119865119868119868119868(119886

ℎ) = radic

2ℎ

120587119886tan(120587119886

2ℎ)

(4)


Θ = 6120587 (1 minus 1205922) (

ℎ

119871) [cos3 (120579) 119891

1(120574)

+ (1 + 120592) cos (120579) sin2 (120579) 1198912(120574)]

1198911(120574) = 06348120574

2minus 1035120574

3+ 37201120574

4minus 51773120574

5

+ 75531205746minus 7332120574

7+ 24909120574

8

1198912(120574) = 02026120574

2+ 003378120574

4+ 0009006120574

6+ 0002734120574

8

(5)


119886and



1205974119908

1205971199094+1205972119908

1205971199052+ 119888

120597119908

120597119905minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881 (119905)

(1 minus 119908)]

2

(6)


119908 (0 119905) = 119908 (1 119905) =120597119908

120597119909(0 119905) =

120597119908

120597119909(1 119905) = 0 (7)


119908 =119908

119889 119909 =

119909

119871 119905 =

119905lowast

120572 =61205761198714

119864ℎ31198893 119905

lowast= radic

120588119887ℎ1198714

119864119868

119873119903=121199031198712

119864ℎ3119887

119888 =121198881198714

119864ℎ3119887119905lowast

119873119886= 6(

119889

ℎ)

2

int

1

0

(120597119908

120597119909)

2

119889119909

(8)


1205974119908

1205971199094minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881

(1 minus 119908)]

2

(9)



1205974119908119894

1205971199094+1205972119908119894

1205971199052= 0 119894 = 1 2 (10)


119894(119909)119890119895120596119905 in (10)


1205974119884119894

1205971199094minus Λ4119884119894= 0 119894 = 1 2 (11)


1198841(119897119888minus) = 1198842(119897119888+) 119884

10158401015840

1(119897119888minus) = 11988410158401015840

2(119897119888+)

119884101584010158401015840

1(119897119888minus) = 119884101584010158401015840

2(119897119888+) 119884

1015840

2(119897119888+) minus 1198841015840

1(119897119888minus) = Θ119884

10158401015840

2(119897119888+)

(12)




1198841(119909) = 119860

1sin [Λ119909] + 1198611 cos [Λ119909] + 1198621 sinh [Λ119909]

+ 1198631cosh [Λ119909] (0 le 119909 le 119897119888)

1198842(119909) = 119860



(119897119888 le 119909 le 1)

(13)






119908119896+1

119904= 119908119896

119904+ 120575119908 = 119908

119896

119904+ 120595 (119909) (14)

when

119881119896+1

= 119881119896+ 120575119881 (15)


1205974119908119896+1

119904

1205971199094minus (119873119896+1

119886+ 119873119903)1205972119908119896+1

119904

1205971199092= 120572[

119881

(1 minus 119908119896+1119904

)]

2

(16)


1198894120595

1198891199094minus (119873119896

119886+ 120575119873119886+ 119873119903)1198892120595

1198891199092minus 120575119873119886

1198892119908

1198892119909

100381610038161003816100381610038161003816100381610038161003816(119908119896 119881119896)

minus 2

120572 (119881119896)2

(1 minus 119908119896119904)3120595 minus 2

120572119881119896120575119881

(1 minus 119908119896119904)2= 0

(17)


119886=

int1

0120595(119909)119889





120595 (119909) =

119899

sum

119895=1

119886119895119884119895(119909) (18)






119899

sum

119895=1

119870119894119895119886119895= 119865119894

(119894 = 1 2 119899) (19)

where

119870119894119895= 119870119898

119894119895+ 119870119886

119894119895minus 119870119890

119894119895 119870

119898

119894119895= int

1

0

119884119894119884(4)

119895119889119909

119870119886

119894119895= int

1

0

119884119894[ (119873119896

119886+ 119873119903) 11988410158401015840

119895

+ (int

1

0

119884119894

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896119889119909)

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896

]119889119909

119870119890

119894119895= 2120572 (119881

119896)2

int

1

0

119884119894119884119895

(1 minus 120596119896)3119889119909

119865119894= 2120572 (119881

119896+1minus 119881119896) int

1

0

119884119894(119909)

(1 minus 120596119896)2119889119909

(20)




119881 (119905) = 119881119901+ Vac cos (120596119890119905) (21)


119901≫ Vac


[119881(119905)

(1 minus 119908)]

2

= [1198812

119901+ 2119881119901Vac cos (120596119890119905) + V2accos

2(120596119890119905)]


= 1198812

119901(1 + 2119908 + 3119908

2) + 2119881

119901Vac cos (120596119890119905)

(22)




+ 1205962

0119906 minus ℏ119906

3= 120582 + ℓ119906

2+ 120594 cos (120596

119890119905) + 120581 (23)


1205962

0= 1198861minus 1198731199031198862minus 2120572119881

2

119901 ℏ =

611988921198863

ℎ2

120582 = minus119888 ℓ = 312057211988731198812

119901 120594 = 2120572119887

1Vac1198812

119901

120581 = 12057211988711198812

119901

(24)


where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)






+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)


1= 120578119905 are


119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)


1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879



1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879



1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)


1198631119886 = 119863



[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)


While one has 1198600le 120594(120582120596



120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)



Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)



119860max =120594

1205821205960

(34)


Ω = 1 minus3ℏ

81205962

0

1198602

max (35)





∘ 20∘) on the



0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










V


h

yx Crack bMicrobeam

d

L

a

Xc

120579

(a)

Beam

Crack

M

y

x

120579

120573

120591

120590

120573 sin120579

120573 cos120579

(b)

(c)






119869 =1

119864

(1198702

119868+ 1198702

119868119868+ (1 + 120592)119870

2

119868119868119868)

119880 = ∬119860

119869 119889119860

Θ =1205971198802

1205971198752

(1)




1205900=119872

119868120573 120591

0= 0 120590

119888= 1205900cos2120579

120591119888= 1205900cos 120579 sin 120579

(2)



0 1205910and 120590




0= 6119872119887ℎ


expressed as

1198700

119868= 1205900radic120587120572119865

2 119870

0

119868119868= 1198700

119868119868119868= 0

119870119888

119868= 120590119888radic120587120572119865

2 119870

119888

119868119868= 0 119870

119888

119868119868119868= 120591119888radic120587120572119865

119868119868119868

(3)

where 1198700119868 1198700119868119868 and 119870

0



119888


crack and one has

1198652(119886

ℎ)

= radic2ℎ

120587119886tan(120587119886

2ℎ)(

0923 + 0199 [1 minus sin (1205871198862ℎ)]4

cos (1205871198862ℎ))

119865119868119868119868(119886

ℎ) = radic

2ℎ

120587119886tan(120587119886

2ℎ)

(4)


Θ = 6120587 (1 minus 1205922) (

ℎ

119871) [cos3 (120579) 119891

1(120574)

+ (1 + 120592) cos (120579) sin2 (120579) 1198912(120574)]

1198911(120574) = 06348120574

2minus 1035120574

3+ 37201120574

4minus 51773120574

5

+ 75531205746minus 7332120574

7+ 24909120574

8

1198912(120574) = 02026120574

2+ 003378120574

4+ 0009006120574

6+ 0002734120574

8

(5)


119886and



1205974119908

1205971199094+1205972119908

1205971199052+ 119888

120597119908

120597119905minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881 (119905)

(1 minus 119908)]

2

(6)


119908 (0 119905) = 119908 (1 119905) =120597119908

120597119909(0 119905) =

120597119908

120597119909(1 119905) = 0 (7)


119908 =119908

119889 119909 =

119909

119871 119905 =

119905lowast

120572 =61205761198714

119864ℎ31198893 119905

lowast= radic

120588119887ℎ1198714

119864119868

119873119903=121199031198712

119864ℎ3119887

119888 =121198881198714

119864ℎ3119887119905lowast

119873119886= 6(

119889

ℎ)

2

int

1

0

(120597119908

120597119909)

2

119889119909

(8)


1205974119908

1205971199094minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881

(1 minus 119908)]

2

(9)



1205974119908119894

1205971199094+1205972119908119894

1205971199052= 0 119894 = 1 2 (10)


119894(119909)119890119895120596119905 in (10)


1205974119884119894

1205971199094minus Λ4119884119894= 0 119894 = 1 2 (11)


1198841(119897119888minus) = 1198842(119897119888+) 119884

10158401015840

1(119897119888minus) = 11988410158401015840

2(119897119888+)

119884101584010158401015840

1(119897119888minus) = 119884101584010158401015840

2(119897119888+) 119884

1015840

2(119897119888+) minus 1198841015840

1(119897119888minus) = Θ119884

10158401015840

2(119897119888+)

(12)




1198841(119909) = 119860

1sin [Λ119909] + 1198611 cos [Λ119909] + 1198621 sinh [Λ119909]

+ 1198631cosh [Λ119909] (0 le 119909 le 119897119888)

1198842(119909) = 119860



(119897119888 le 119909 le 1)

(13)






119908119896+1

119904= 119908119896

119904+ 120575119908 = 119908

119896

119904+ 120595 (119909) (14)

when

119881119896+1

= 119881119896+ 120575119881 (15)


1205974119908119896+1

119904

1205971199094minus (119873119896+1

119886+ 119873119903)1205972119908119896+1

119904

1205971199092= 120572[

119881

(1 minus 119908119896+1119904

)]

2

(16)


1198894120595

1198891199094minus (119873119896

119886+ 120575119873119886+ 119873119903)1198892120595

1198891199092minus 120575119873119886

1198892119908

1198892119909

100381610038161003816100381610038161003816100381610038161003816(119908119896 119881119896)

minus 2

120572 (119881119896)2

(1 minus 119908119896119904)3120595 minus 2

120572119881119896120575119881

(1 minus 119908119896119904)2= 0

(17)


119886=

int1

0120595(119909)119889





120595 (119909) =

119899

sum

119895=1

119886119895119884119895(119909) (18)






119899

sum

119895=1

119870119894119895119886119895= 119865119894

(119894 = 1 2 119899) (19)

where

119870119894119895= 119870119898

119894119895+ 119870119886

119894119895minus 119870119890

119894119895 119870

119898

119894119895= int

1

0

119884119894119884(4)

119895119889119909

119870119886

119894119895= int

1

0

119884119894[ (119873119896

119886+ 119873119903) 11988410158401015840

119895

+ (int

1

0

119884119894

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896119889119909)

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896

]119889119909

119870119890

119894119895= 2120572 (119881

119896)2

int

1

0

119884119894119884119895

(1 minus 120596119896)3119889119909

119865119894= 2120572 (119881

119896+1minus 119881119896) int

1

0

119884119894(119909)

(1 minus 120596119896)2119889119909

(20)




119881 (119905) = 119881119901+ Vac cos (120596119890119905) (21)


119901≫ Vac


[119881(119905)

(1 minus 119908)]

2

= [1198812

119901+ 2119881119901Vac cos (120596119890119905) + V2accos

2(120596119890119905)]


= 1198812

119901(1 + 2119908 + 3119908

2) + 2119881

119901Vac cos (120596119890119905)

(22)




+ 1205962

0119906 minus ℏ119906

3= 120582 + ℓ119906

2+ 120594 cos (120596

119890119905) + 120581 (23)


1205962

0= 1198861minus 1198731199031198862minus 2120572119881

2

119901 ℏ =

611988921198863

ℎ2

120582 = minus119888 ℓ = 312057211988731198812

119901 120594 = 2120572119887

1Vac1198812

119901

120581 = 12057211988711198812

119901

(24)


where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)






+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)


1= 120578119905 are


119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)


1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879



1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879



1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)


1198631119886 = 119863



[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)


While one has 1198600le 120594(120582120596



120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)



Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)



119860max =120594

1205821205960

(34)


Ω = 1 minus3ℏ

81205962

0

1198602

max (35)





∘ 20∘) on the



0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1205910and 120590




0= 6119872119887ℎ


expressed as

1198700

119868= 1205900radic120587120572119865

2 119870

0

119868119868= 1198700

119868119868119868= 0

119870119888

119868= 120590119888radic120587120572119865

2 119870

119888

119868119868= 0 119870

119888

119868119868119868= 120591119888radic120587120572119865

119868119868119868

(3)

where 1198700119868 1198700119868119868 and 119870

0



119888


crack and one has

1198652(119886

ℎ)

= radic2ℎ

120587119886tan(120587119886

2ℎ)(

0923 + 0199 [1 minus sin (1205871198862ℎ)]4

cos (1205871198862ℎ))

119865119868119868119868(119886

ℎ) = radic

2ℎ

120587119886tan(120587119886

2ℎ)

(4)


Θ = 6120587 (1 minus 1205922) (

ℎ

119871) [cos3 (120579) 119891

1(120574)

+ (1 + 120592) cos (120579) sin2 (120579) 1198912(120574)]

1198911(120574) = 06348120574

2minus 1035120574

3+ 37201120574

4minus 51773120574

5

+ 75531205746minus 7332120574

7+ 24909120574

8

1198912(120574) = 02026120574

2+ 003378120574

4+ 0009006120574

6+ 0002734120574

8

(5)


119886and



1205974119908

1205971199094+1205972119908

1205971199052+ 119888

120597119908

120597119905minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881 (119905)

(1 minus 119908)]

2

(6)


119908 (0 119905) = 119908 (1 119905) =120597119908

120597119909(0 119905) =

120597119908

120597119909(1 119905) = 0 (7)


119908 =119908

119889 119909 =

119909

119871 119905 =

119905lowast

120572 =61205761198714

119864ℎ31198893 119905

lowast= radic

120588119887ℎ1198714

119864119868

119873119903=121199031198712

119864ℎ3119887

119888 =121198881198714

119864ℎ3119887119905lowast

119873119886= 6(

119889

ℎ)

2

int

1

0

(120597119908

120597119909)

2

119889119909

(8)


1205974119908

1205971199094minus (119873119886+ 119873119903)1205972119908

1205971199092= 120572 [

119881

(1 minus 119908)]

2

(9)



1205974119908119894

1205971199094+1205972119908119894

1205971199052= 0 119894 = 1 2 (10)


119894(119909)119890119895120596119905 in (10)


1205974119884119894

1205971199094minus Λ4119884119894= 0 119894 = 1 2 (11)


1198841(119897119888minus) = 1198842(119897119888+) 119884

10158401015840

1(119897119888minus) = 11988410158401015840

2(119897119888+)

119884101584010158401015840

1(119897119888minus) = 119884101584010158401015840

2(119897119888+) 119884

1015840

2(119897119888+) minus 1198841015840

1(119897119888minus) = Θ119884

10158401015840

2(119897119888+)

(12)




1198841(119909) = 119860

1sin [Λ119909] + 1198611 cos [Λ119909] + 1198621 sinh [Λ119909]

+ 1198631cosh [Λ119909] (0 le 119909 le 119897119888)

1198842(119909) = 119860



(119897119888 le 119909 le 1)

(13)






119908119896+1

119904= 119908119896

119904+ 120575119908 = 119908

119896

119904+ 120595 (119909) (14)

when

119881119896+1

= 119881119896+ 120575119881 (15)


1205974119908119896+1

119904

1205971199094minus (119873119896+1

119886+ 119873119903)1205972119908119896+1

119904

1205971199092= 120572[

119881

(1 minus 119908119896+1119904

)]

2

(16)


1198894120595

1198891199094minus (119873119896

119886+ 120575119873119886+ 119873119903)1198892120595

1198891199092minus 120575119873119886

1198892119908

1198892119909

100381610038161003816100381610038161003816100381610038161003816(119908119896 119881119896)

minus 2

120572 (119881119896)2

(1 minus 119908119896119904)3120595 minus 2

120572119881119896120575119881

(1 minus 119908119896119904)2= 0

(17)


119886=

int1

0120595(119909)119889





120595 (119909) =

119899

sum

119895=1

119886119895119884119895(119909) (18)






119899

sum

119895=1

119870119894119895119886119895= 119865119894

(119894 = 1 2 119899) (19)

where

119870119894119895= 119870119898

119894119895+ 119870119886

119894119895minus 119870119890

119894119895 119870

119898

119894119895= int

1

0

119884119894119884(4)

119895119889119909

119870119886

119894119895= int

1

0

119884119894[ (119873119896

119886+ 119873119903) 11988410158401015840

119895

+ (int

1

0

119884119894

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896119889119909)

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896

]119889119909

119870119890

119894119895= 2120572 (119881

119896)2

int

1

0

119884119894119884119895

(1 minus 120596119896)3119889119909

119865119894= 2120572 (119881

119896+1minus 119881119896) int

1

0

119884119894(119909)

(1 minus 120596119896)2119889119909

(20)




119881 (119905) = 119881119901+ Vac cos (120596119890119905) (21)


119901≫ Vac


[119881(119905)

(1 minus 119908)]

2

= [1198812

119901+ 2119881119901Vac cos (120596119890119905) + V2accos

2(120596119890119905)]


= 1198812

119901(1 + 2119908 + 3119908

2) + 2119881

119901Vac cos (120596119890119905)

(22)




+ 1205962

0119906 minus ℏ119906

3= 120582 + ℓ119906

2+ 120594 cos (120596

119890119905) + 120581 (23)


1205962

0= 1198861minus 1198731199031198862minus 2120572119881

2

119901 ℏ =

611988921198863

ℎ2

120582 = minus119888 ℓ = 312057211988731198812

119901 120594 = 2120572119887

1Vac1198812

119901

120581 = 12057211988711198812

119901

(24)


where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)






+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)


1= 120578119905 are


119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)


1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879



1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879



1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)


1198631119886 = 119863



[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)


While one has 1198600le 120594(120582120596



120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)



Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)



119860max =120594

1205821205960

(34)


Ω = 1 minus3ℏ

81205962

0

1198602

max (35)





∘ 20∘) on the



0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119908119896+1

119904= 119908119896

119904+ 120575119908 = 119908

119896

119904+ 120595 (119909) (14)

when

119881119896+1

= 119881119896+ 120575119881 (15)


1205974119908119896+1

119904

1205971199094minus (119873119896+1

119886+ 119873119903)1205972119908119896+1

119904

1205971199092= 120572[

119881

(1 minus 119908119896+1119904

)]

2

(16)


1198894120595

1198891199094minus (119873119896

119886+ 120575119873119886+ 119873119903)1198892120595

1198891199092minus 120575119873119886

1198892119908

1198892119909

100381610038161003816100381610038161003816100381610038161003816(119908119896 119881119896)

minus 2

120572 (119881119896)2

(1 minus 119908119896119904)3120595 minus 2

120572119881119896120575119881

(1 minus 119908119896119904)2= 0

(17)


119886=

int1

0120595(119909)119889





120595 (119909) =

119899

sum

119895=1

119886119895119884119895(119909) (18)






119899

sum

119895=1

119870119894119895119886119895= 119865119894

(119894 = 1 2 119899) (19)

where

119870119894119895= 119870119898

119894119895+ 119870119886

119894119895minus 119870119890

119894119895 119870

119898

119894119895= int

1

0

119884119894119884(4)

119895119889119909

119870119886

119894119895= int

1

0

119884119894[ (119873119896

119886+ 119873119903) 11988410158401015840

119895

+ (int

1

0

119884119894

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896119889119909)

1198892120596

1198891199092

100381610038161003816100381610038161003816100381610038161003816120596119896119881119896

]119889119909

119870119890

119894119895= 2120572 (119881

119896)2

int

1

0

119884119894119884119895

(1 minus 120596119896)3119889119909

119865119894= 2120572 (119881

119896+1minus 119881119896) int

1

0

119884119894(119909)

(1 minus 120596119896)2119889119909

(20)




119881 (119905) = 119881119901+ Vac cos (120596119890119905) (21)


119901≫ Vac


[119881(119905)

(1 minus 119908)]

2

= [1198812

119901+ 2119881119901Vac cos (120596119890119905) + V2accos

2(120596119890119905)]


= 1198812

119901(1 + 2119908 + 3119908

2) + 2119881

119901Vac cos (120596119890119905)

(22)




+ 1205962

0119906 minus ℏ119906

3= 120582 + ℓ119906

2+ 120594 cos (120596

119890119905) + 120581 (23)


1205962

0= 1198861minus 1198731199031198862minus 2120572119881

2

119901 ℏ =

611988921198863

ℎ2

120582 = minus119888 ℓ = 312057211988731198812

119901 120594 = 2120572119887

1Vac1198812

119901

120581 = 12057211988711198812

119901

(24)


where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)






+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)


1= 120578119905 are


119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)


1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879



1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879



1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)


1198631119886 = 119863



[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)


While one has 1198600le 120594(120582120596



120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)



Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)



119860max =120594

1205821205960

(34)


Ω = 1 minus3ℏ

81205962

0

1198602

max (35)





∘ 20∘) on the



0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1198861= int

1

0

119884(4)(119909) 119884 (119909) 119889119909 119886

2= int

1

0

119884(2)(119909) 119884 (119909) 119889119909

1198863= 1198862int

1

0

[1198841015840(119909)]2

119889119909 119887119894= int

1

0

119884119894(119909) 119889119909

(119894 = 1 2 3) 1198872= 1

(25)






+ 1205962

0119906 = 120578 [ℏ119906

3+ 120582 + ℓ119906

2+ 120594 cos (120596

0+ 120576120590) 119905 + 120581] (26)


1= 120578119905 are


119906 (119905 120578) = 1199060(1198790 1198791) + 120578119906

1(1198790 1198791) (27)


1198632

01199060+ 1205962

01199060= 0

1198632

01199061+ 1205962

01199061= minus2119863

011986311199060+ ℏ1199063

0+ 12058211986301199060+ ℓ1199062

0

+ 120594 cos (1205960+ 120578120590) 119879

0+ 120581

(28)

where 1198630= 120597120597119879

0and 119863

1= 120597120597119879



1199060(1198790 1198791) = 119886 (119879

0) cos [120596

01198790+ 120575 (119879

1)] = 119860 (119879

1) 11989011989512059601198790 + cc

(29)

where 119860(1198791) = 119886(119879



1198631119886 =

120582119886

2+

120594

21205960

sin120593

1198861198631120575 = minus(

3ℏ1198863

81205960

+120594

21205960

cos120593) (30)


1198631119886 = 119863



[(120596119890minus 1205960+3ℏ1198602

0

81205960

)

2

+ (120582

2)

2

]1198602

0= (

120594

21205960

)

2

(31)


While one has 1198600le 120594(120582120596



120596119890= 1205960minus

3ℏ

81205960

1198602

0plusmn radic

1205942

41205962

01198602

0

minus1

41205822 (32)



Ω = 1 minus3ℏ

81205962

0

1198602

0plusmn radic

1205942

41205964

01198602

0

minus1205822

41205962

0

(33)



119860max =120594

1205821205960

(34)


Ω = 1 minus3ℏ

81205962

0

1198602

max (35)





∘ 20∘) on the



0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1445

455

465

475

Λ

120574

n = 1

02 04 06 08

47

46

45

lc = 025

lc = 05

lc = 005

(a)

0 1

755

765

775

785

Λ

120574

n = 2

02 04 06 08

79

78

77

76

75

lc = 025

lc = 05 lc = 005

(b)

0 1

104

106

108

11

Λ

120574

n = 3

02 04 06 08

120579 = 0∘

120579 = 20∘

lc = 025

lc = 05

lc = 005

(c)

1404

1406

1408

141

1412

1414

0 1

Λ

120574

lc = 025

lc = 05

lc = 005

n = 4

02 04 06 08

120579 = 0∘

120579 = 20∘

(d)



Λ119897119888 = 005 119897119888 = 05

120579 = 0∘ 120579 = 15∘ 120579 = 30∘ 120579 = 45∘ 120579 = 0∘ 120579 = 45∘ 120579 = 30∘ 120579 = 45∘

119899 = 1 46392 46452 46616 46836 46673 46719 46840 46995119899 = 2 77647 77702 77855 78065 78532 78532 78532 78532119899 = 3 10929 10933 10944 10960 10806 10819 10854 10901119899 = 4 14100 14102 14108 14117 14137 14137 14137 14137







0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1minus05

0

05

1

15

x

Y

04 05 06

16165

17

120574 = 04

120574 = 02 120574 = 06

02 04 06 08

(a)

15

16

0 1minus05

0

05

1

15

x

Y 120574 = 04

120574 = 02

120574 = 06

02 04

04 05

06 08

(b)

16

18

0 1minus05

0

05

1

15

x

Y

120579 = 0∘

120579 = 45∘ 120579 = 40∘

02 04

04 05

06 08

(c)

045 055

14

15

16

0 1minus05

0

05

1

15

x

Y 120579 = 0∘

120579 = 45∘

120579 = 40∘

02 04

04 05

06 08

(d)




0 10 20 30 40

w 3885

041

38 39 40

035

045

39

038039

120574 = 06120574 = 01 120574 = 05

Vp

05

04

03

02

01

0

04

395

0404

388




35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










35

36

37

38

39

40

120574

Vpu


lc = 01 and 09

01 02 03 04 05 06 07 08 09

lc = 05

120579 = 0∘

120579 = 15∘

120579 = 30∘





0 136

37

38

39

Vpu

ll-in

120579 = 0∘120579 = 15∘120579 = 30∘120579 = 45∘

lc

395

385

375

365

02 04 06 08




Figure 7119897119888 = 005 2163 04317119897119888 = 025 2234 04150119897119888 = 05 2206 04193

Figure 8(a)120574 = 01 119897119888 = 05 2234 04141120574 = 05 119897119888 = 05 2204 04188120574 = 07 119897119888 = 05 2143 04289

Figure 8(b)120574 = 01 119897119888 = 005 2233 04143120574 = 05 119897119888 = 005 2188 04235120574 = 07 119897119888 = 005 2109 04407

Figure 9(a) 120579 = 0∘ 119897119888 = 05 1829 04399120579 = 23∘ 119897119888 = 05 1853 04377

Figure 9(a) 120579 = 45∘ 119897119888 = 05 2088 04453120579 = 70∘ 119897119888 = 05 2156 04305

Figure 9(b)

120579 = 0∘ 119897119888 = 005 2041 04561120579 = 23∘ 119897119888 = 005 2053 04532120579 = 45∘ 119897119888 = 005 2088 04453120579 = 70∘ 119897119888 = 005 2156 04305

Figure 10(a)119881119901= 10V 2206 007172

119881119901= 60V 2204 02513

119881119901= 60V 2199 04317


are 120579 = 45∘ 119897119888 = 05 120574 = 05 119881

119901= 35V and Vac = 01V The




098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










098 1 102 104

18 20 220

05

lc = 001

lc = 05

lc = 025

05

04

03

02

01

0

A0

Ω

120596e


099 1 101 102 103

19 20 210

05

04

03

02

04

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(a)

099 1 101 102 103 104

185 195 2050

0505

04

03

02

01

0

A0

Ω

120596e

120579 = 0∘

120579 = 23∘

120579 = 45∘

120579 = 70∘

(b)


0995 1 1005 101

20 21 230

0504

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(a)

099 1 101 102

20 21 22 230

0505

04

03

02

01

0

A0

Ω

120596e

120574 = 05

120574 = 01

120574 = 07

(b)



0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0995 1 1005 101 1015 102

219 22 2210

04

03

02

01

02

01

0

A0

Ω

120596e

Vp = 60V

Vp = 35VVp = 10V

(a)

1 1005 1010

01

02

03

04

22 2220

02

04

A0

Ω

120596e

ac = 01V

ac = 005Vac = 001V

(b)








5 Conclusion






Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article dynamic characteristics of

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