Research ArticleNonlinear Dynamics Modeling and Subharmonic ResonancesAnalysis of a Laminated Composite Plate

Ting Ma1 Xiao Juan Song2 and Shu Feng Lu 1

1Department of Mechanics Inner Mongolia University of Technology Hohhot 010051 China2College of Mechanical Engineering Inner Mongolia University of Technology Hohhot 010051 China

Correspondence should be addressed to Shu Feng Lu shufenglu163com

Received 28 November 2019 Revised 28 January 2020 Accepted 18 May 2020 Published 13 August 2020

Academic Editor Mohammad Rafiee

Copyright copy 2020 TingMa et al-is is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e nonlinear subharmonic resonance of an orthotropic rectangular laminated composite plate is studied Based on the theory ofhigh-order shear laminates von Karmanrsquos geometric relation for the large deformation of plates and Hamiltonrsquos principle thenonlinear dynamic equations of a rectangular orthotropic composite laminated plate subjected to the transverse harmonicexcitation are established According to the displacement boundary conditions the modal functions that satisfy the boundaryconditions of the rectangular plate are selected -e two-degree-of-freedom ordinary differential equations that describe thevibration of the rectangular plate are obtained by the Galerkin method -e multiscale method is used to obtain an approximatesolution to the resonance problem Both the amplitude-frequency equation and the average equations in the Cartesian coordinateform are obtained -e amplitude-frequency curves bifurcation diagrams phase diagrams and time history diagrams of therectangular plate under different parameters are obtained numerically -e influence of relevant parameters such as excitationamplitude tuning parameter and damping coefficient on the nonlinear dynamic response of the system is analyzed

1 Introduction

Composite laminates have many advantages such as highspecific strength high specific stiffness and good fatigueresistance Because laminated materials are often made intothin-walled structures they are prone to large deformationunder various external loads resulting in nonlinear dynamiccharacteristics that exhibit complex geometries Undercertain excitation conditions harmonic resonance mayoccur which also has a significant impact on the accuracy ofthe structure -erefore it is necessary to analyze thenonlinear harmonic resonance characteristics of laminatedcomposite plates

Although the nonlinear vibration characteristics ofcomposite structures have been studied for many yearsresearch on the free vibration of laminated plates and shellstructure still retains the attention of many scholars Wanget al [1] quoted a refined plate theory (RPT) with a newpolynomial shape function to establish the displacementfield of the middle core layer of sandwich laminates and the

free vibration frequency was obtained Nonlinear free vi-bration of symmetrical magneto-electro-elastic laminatedrectangular plates under the simply supported boundarycondition has been studied by Razavi and Shooshtari [2]Sadri and Younesian [3] analyzed the nonlinear free vi-bration of a plate-cavity system using the harmonic balancemethod Aranda-Iiglesias et al [4] studied the large am-plitude axisymmetric free vibration of incompressible elasticcylinder structures Based on the asymmetric mode shapeformula of annular plates Torabi and Ansari [5] used apseudo-arclength continuation method to study the freevibration of carbon nanotube (CNT) reinforced compositeannular plates under thermal loads

Analytical and numerical methods have been widelyused in the study of the dynamic characteristics of compositestructures [6ndash11] However the solution of nonlinear forcedvibration of laminated plates remains a difficult problemFor example the reliability and convergence of the resultsfor solving complex boundary conditions are unsatisfactoryand it is not easy to satisfy all the displacement boundary

HindawiShock and VibrationVolume 2020 Article ID 7913565 16 pageshttpsdoiorg10115520207913565

conditions and static boundary conditions Khdeir andReddy [12] solved the vibration equation of laminated platesby using the state variable method Litewka and Lew-andowski [13] studied the nonlinear vibration of Zenerviscoelastic plates Eslami and Kandil [14] studied the forcedvibration of rectangular laminated composite plates sub-jected to harmonic excitation Amabili [15] derived thenonlinear vibration equation of a rectangular plate by usingthe Lagrange equation Delapierre et al [16] studied thetransverse nonlinear vibration of isotropic uniform annularthin films subjected to uniform transverse loads Kumar et al[17] carried out the nonlinear forced vibration analysis of anaxially functionally graded inhomogeneous plate Chen et al[18] put forth the numerical solution of the nonlinear vi-bration of an arbitrary prestressed plate-e large amplitudeforced vibration of thin rectangular plates made of differentrubber materials was studied experimentally and theoreti-cally by Balasubramanian et al [19]

Nonlinear vibration of the laminated plates exhibitsdifferent characteristics with the change in boundary con-ditions Studies on thin plates having different boundaryconditions provide the following results An analysis of thenonlinear dynamics of a clamped-clamped FGM circularcylindrical shell subjected to an external excitation anduniform temperature change was presented by Zhang et al[20] Bennett [21] studied the nonlinear vibration of anti-symmetric angle-ply laminated plates Rafiee et al [22]studied the nonlinear vibration characteristics of simplysupported functionally graded material shells under com-bined electrical thermal mechanical and aerodynamicloading Kattimani [23] studied the nonlinear vibration ofcomposite plates and hyperbolic shells with simply sup-ported or clamped boundary conditions -e nonlineardynamic response of piezoelectric functionally graded ma-terial plates with different boundary conditions resting onPasternak-type elastic foundations in the thermal environ-ment was studied by Duc et al [24] Mohamed et al [25]used a new numerical method to study the effects of axialloads imperfections and nonlinear elastic foundations onthe natural frequencies and forced vibration characteristicsof beams Cho et al [26] studied the vibration of rectangularplates with circular holes and stiffeners mounted on elasticdevices by using the energy-based assumed mode method

Internal resonance is also unique to nonlinear systemsand is different from linear systems Internal resonance willoccur when the two natural frequencies of the system satisfya certain relationship -e unique internal resonance phe-nomenon of the nonlinear system will excite the originalnonexcited modes due to the energy transfer between themodes Nayfeh and Mook [27] studied the dynamic char-acteristics of discrete and continuous systems under dif-ferent resonance conditions Nonlinear vibration of acomposite laminated cantilever rectangular plate with one-to-one internal resonance under in-plane and transverseexcitations was studied by Zhang and Zhao [28] -enonlinear vibration behavior of carbon nanotube reinforcedcomposite plates and piezoelectric rectangular composite

laminates under parametric and forced excitations wasstudied by Zhang et al [29 30] Chang et al [31] studied thesubharmonic responses of rectangular plates which areharmonically excited with one-to-one internal resonanceZhang et al [32] studied the nonlinear transverse vibrationsof in-plane accelerating viscoelastic plates in the presence ofprincipal parametric and 3 1 internal resonance

Secondary resonance is a phenomenon particular to thenonlinear system which includes superharmonic and sub-harmonic resonance Many scholars have studied the sec-ondary resonance of nonlinear systems -e subharmonicresonance of FGM truncated conical shell under aerody-namics and in-plane force is investigated by the method ofmultiple scales by Yang et al [33] Nonlinear subharmonicresonances of the current-conducting thin plate in elec-tromagnetic field are studied by Hu and Li [34] Li and Guo[35] studied the subharmonic resonance of both ends of acomposite laminated circular cylindrical shell in a subsonicair flow under radial harmonic excitation by using themethod of multiple scales Jomehzadeh et al [36] investi-gated the nonlinear subharmonic resonances of graphene-matrix composite to the harmonic -e primary sub-harmonic and superharmonic responses of a fractionalviscoelastic plate are studied by Permoon et al [37] and asimilar research has been conducted for cylindrical shells byAhmadi and Foroutan [38] Hosseini et al [39] investigatedthe nonlinear forced vibrations of a viscoelastic piezoelectriccantilever in the cases of primary resonance and non-resonance hard excitation including subharmonic andsuperharmonic Naprstek and Fischer [40] studied the superand subharmonic synchronization effects of the van der Polequation on harmonic excitation It is found in the litera-tures that most of the researches on subharmonic resonancesare focused on the conservative systems having a single-degree-of-freedom system

In this study the subharmonic resonance characteristicsof a two-degree-of-freedom laminated composite platesubjected to transverse harmonic excitations are investi-gated -e innovation of this paper lies in that the nonlinearvibration modeling of the thin plates with arbitraryboundary shapes and boundary conditions and the non-linear vibration of the plates under different boundaryconditions can be studied by assuming the correspondingmode function -e rectangular plate with simply supportedboundary condition studied in this study is only a specificcase when the boundary shape is determined and theboundary is acted on by no external force In the absence ofinternal resonance the low-order and high-order modes areuncoupled and so they are studied separately Based on thetheory of higher-order shear deformation plate and vonKarmanrsquos geometric relationship the nonlinear dynamicequations are established by using Hamiltonrsquos principle -eordinary differential equations for the vibration of therectangular plate were derived by two-order discretizationusing the Galerkin method-emultiscale method is appliedto obtain an approximate solution to the resonance problemBoth the amplitude-frequency response equation and the

2 Shock and Vibration

average equations in rectangular coordinates are obtainedIn addition the nonlinear dynamic responses of the two-order modes with system parameters are comparedconcretely

2 Governing Equations of Motion

-e mechanical model of the special orthotropic symmetricrectangular laminated plate that is simply supported on foursides is shown in Figure 1 Assume the length width andthickness of the rectangular laminated plate to be a b and hrespectively and a uniformly distributed harmonic excita-tion force q q0 cosΩt is applied in the transverse planewhere q0 is the amplitude of excitation

-e linear constitutive relation of each laminate is asfollows

σxx

σyy

σxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k

Q11 Q12 0

Q12 Q22 0

0 0 Q66

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k εxx

εyy

cxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k

σyz

σxz

1113888 1113889 Q44 0

0 Q551113888 1113889

kcyz

cxz

1113888 1113889

k

(1)

where k represents the number of layers of the laminatedplate and

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q66 G12

Q44 G23

Q55 G13

(2)

Based on the higher-order shear deformation platetheory the displacement fields are

u(x y z t) u0(x y t) + zϕx(x y t) minus z3 43h2 ϕx +

zw0

zx1113888 1113889

v(x y z t) v0(x y t) + zϕy(x y t) minus z3 43h2 ϕy +

zw0

zy1113888 1113889

w(x y z t) w0(x y t)

(3)

where (u0 v0 w0) represent the displacement of a point onthe midplane and (ϕx ϕy) are the rotations of a transversenormal about the y and x axes respectively

According to the von Karman nonlinear geometricrelation

εxx zu

zx+12

zw

zx1113888 1113889

2

εxz 12

zu

zz+

zw

zx1113888 1113889

εxy 12

zu

zy+

zv

zx+

zw

zx

zw

zy1113888 1113889

εyy zv

zy+12

zw

zy1113888 1113889

2

εyz 12

zv

zz+

zw

zy1113888 1113889

(4)

For the assumed displacement field in (3) the strains in(4) can be expressed as

εxx

εyy

cxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ z

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ z3

czϕx

zx+ c

z2w0

zx2

czϕy

zy+ c

z2w0

zy2

czϕx

zy+

z2w0

zxzy1113888 1113889 + c

zϕy

zx+

z2w0

zxzy1113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(5a)

cyz

cxz

1113888 1113889

ϕy +zw0

zy

ϕx +zw0

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+ z

2

3c ϕy +zw0

zy1113888 1113889

3c ϕx +zw0

zx1113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (5b)

where c minus (43h2)-e governing equations describing the vibration of

rectangular plate are obtained by the Hamilton principle

1113946t2

t1

(δT minus δU + δW)dt 0 (6)

where the variation in potential energy δU the variation inkinetic energy δT and the virtual work done by the externalforce δW are given by

Shock and Vibration 3

δU C σxxδεxx + σyyδεyy + σxyδcxy + σxzδcxz + σyzδcyz1113872 1113873dV

(7a)

δT Cρ _w0δ _w0dV + Cρ _u0 + z _ϕx + cz3 _ϕx +

z _w0

zx1113888 11138891113890 1113891

middot δ _u0 + zδ _ϕx + cz3 δ _ϕx +

zδ _w0

zx1113888 11138891113890 1113891dV

+ Cρ _v0 + z _ϕy + cz3 _ϕy +

z _w0

zy1113888 11138891113890 1113891

middot δ _v0 + zδ _ϕy + cz3 δ _ϕy +

zδ _w0

zy1113888 11138891113890 1113891dV

(7b)

δW minus Bμ _w0δ _w0dx dy + Bq0 cosΩtδw0dx dy

+ 1113929 1113946 1113954σnn δu0n + zδϕn + cz3 δϕn +

zδw0

zn1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σns δu0s + zδϕs + cz3 δϕs +

zδw0

zs1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σnzδw0dz ds

1113929 1113954Nnnδu0n + 1113954Mnnδϕn + c1113954Pnnδϕn + c1113954Pnn

zδw0

zn1113888

+ 1113954Nnsδu0s + 1113954Mnsδϕs + c1113954Pnsδϕs

+c1113954Pns

zδw0

zs+ 1113954Qnδw0ds1113889

+ Bqδw0dx dy minus Bc _w0δw0dx dy

(7c)

where μ is the damping coefficient a superposed dot on avariable indicates its time derivative ρ is the density of theplate (1113954σnn 1113954σns 1113954σnz) are the stress components on theboundary (δu0n δu0s) are the virtual displacements alongthe normal and tangential direction respectively on theboundary (nx ny) are the direction cosines of the outwardnormal with respect to the x- and y-axis at a point on theplate boundary and

Ii 1113946h2

minus h2ρz

idz

Nxx

Nyy

Nxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dz

Mxx

Myy

Mxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

zdz

Pxx

Pyy

Pxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

z3dz

(8a)

Qy

Qx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭dz

Ry

Rx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭z2dz

1113954Nnn

1113954Nns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭dz

1113954Mnn

1113954Mns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭zdz

(8b)

1113954Qn 1113946h2

minus h21113954σnzdz

1113954Pnn

1113954Pns

1113896 1113897 1113946h2

minus h2

1113954σnn

1113954σns

1113896 1113897z3dz

ϕx

ϕy

1113890 1113891 nx minus ny

ny nx

1113890 1113891ϕn

ϕs

1113890 1113891

(8c)

In this study all the applied forces on the boundary arezero-at is to say 1113954Nnn 1113954Nns 1113954Mnn 1113954Mns 1113954Pnn 1113954Pnn are all zerosSubstituting (7a) (7b) (7c) and (8a) (8b) (8c) into (6) thevibration equations are obtained as follows

q = q0cosΩty

z

x

(a)

y

Z

h

(b)

Figure 1 Mechanical model of a composite rectangular laminated plate

4 Shock and Vibration

zNxx

zx+

zNxy

zy I0eurou0 + I1 + cI3( 1113857euroϕx + cI3

zeurow0

zx (9a)

zNyy

zy+

zNxy

zx I0eurov0 + I1 + cI3( 1113857euroϕy + cI3

zeurow0

zy (9b)

z Nxx zw0zx( 1113857( 1113857

zxminus c

z2Pxx

zx2 +z Nyy zw0zy( 11138571113872 1113873

zyminus c

z2Pyy

zy2 +z Nxy zw0zy( 11138571113872 1113873

zx+

z Nxy zw0zx( 11138571113872 1113873

zy

minus 2cz2Pxy

zxzy+

zQx

zx+ 3c

zRx

zx+

zQy

zy+ 3c

zRy

zy

I0 eurow0 minus cI3zeurou0

zxminus cI3

zeurov0zy

minus cI4 + c2I61113872 1113873

zeuroϕx

zxminus cI4 + c

2I61113872 1113873

zeuroϕy

zyminus c

2I6

z2 eurow0

zx2 minus c2I6

z2 eurow0

zy2 minus q0 cosΩt + μ _w0

(9c)

zMxx

zx+ c

zPxx

zx+

zMxy

zy+ c

zPxy

zyminus Qx minus 3cRx

I1 + cI3( 1113857eurou0 + I2 + 2cI4 + c2I61113872 1113873euroϕx + cI4 + c

2I61113872 1113873

zeurow0

zx

(9d)

zMyy

zy+ c

zPyy

zy+

zMxy

zx+ c

zPxy

zxminus Qy minus 3cRy

I1 + cI3( 1113857eurov0 + I2 + 2cI4 + c2I61113872 1113873euroϕy + cI4 + c

2I61113872 1113873

zeurow0

zy

(9e)

Equations (9a)ndash(9e) can be written in form of general-ized displacements (u0 v0 w0 ϕx ϕy) and dimensionlessparameters are introduced as u0 (u0a) v0 (v0b)ϕx ϕx ϕy ϕy with the dimensionless parameter forms of

other physical quantities being the same as those of [24]-en the dimensionless partial differential equations ofvibration of the rectangular plates are obtained as

a10z2u0

zx2 + a11zw0

zx

z2w0

zx2 + a12z2v0zxzy

+ a13z2w0

zxzy

zw0

zy+ a14

z2u0

zy2 + a15z2w0

zy2zw0

zx

a16eurou0 + a17euroϕx + a18

zeurow0

zx

(10a)

b10z2u0

zxzy+ b11

z2w0

zxzy

zw0

zx+ b12

z2v0

zy2 + b13z2w0

zy2zw0

zy+ b14

z2v0

zx2 + b15z2w0

zx2zw0

zy

b16eurov0 + b17euroϕy + b18

zeurow0

zy

(10b)

Shock and Vibration 5

c10z2u0

zx2zw0

zx+ c11

zw0

zx1113888 1113889

2z2w0

zx2 + c12z2v0zxzy

zw0

zx+ c13

z2w0

zxzy

zw0

zx

zw0

zy+ c14

z2w0

zx2zu0

zx

+ c15z2w0

zx2zv0

zy+ c16

z2w0

zx2zw0

zy1113888 1113889

2

+ c17z3ϕx

zx3 + c18z3ϕx

zx2zy+ c19

z4w0

zx4 + c20z4w0

zx2zy2

+ c21z2u0

zxzy

zw0

zy+ c22

z2v0

zy2zw0

zy+ c23

zw0

zy1113888 1113889

2z2w0

zy2 + c24z2w0

zy2zu0

zx+ c25

z2w0

zy2zv0

zy

+ c26z3ϕx

zxzy2 + c27z3ϕy

zy3 + c28z4w0

zy4 + c29z2v0zx2

zw0

zy+ c30

z2w0

zxzy

zu0

zy+ c31

z2w0

zxzy

zv0zx

+ c32z2u0

zy2zw0

zx+ c33

z2w0

zy2zw0

zx1113888 1113889

2

+ c34zϕx

zx+ c35

z2w0

zx2 + c36zϕy

zy+ c37

z2w0

zy2

+ c38q0 cosΩt + c39 _w0

c40 eurow0 + c41zeurou0

zx+ c42

zeurov0zy

+ c43zeuroϕx

zx+ c44

zeuroϕy

zy+ c45

z2 eurow0

zx2 + c46z2 eurow0

zy2

(10c)

d10z2ϕx

zx2 + d11z2ϕy

zxzy+ d12

z3w0

zx3 + d13z2ϕx

zy2 + d14ϕx + d15zw0

zx+ d16

z3w0

zxzy2

d17eurou0 + d18euroϕx + d19

zeurow0

zx

(10d)

e10z2ϕx

zxzy+ e11

z2ϕy

zy2 + e12z3w0

zx2zy+ e13

z3w0

zy3 + e14z2ϕy

zx2 + e15ϕy + e16zw0

zy

e17eurov0 + e18euroϕy + e19

zeurow0

zy

(10e)

For the sake of convenience in writing the transverselines above the physical quantities are omitted -eboundary conditions of the simply supported plate can beexpressed as

x 0 andx a v w Nxx Mxx ϕy Pxx 0

(11a)

y 0 andy b u w Nyy Myy ϕx Pyy 0

(11b)

Due to the fact that the higher-order modes are not easilyexcited in structural vibration the first two modes are takenfor truncation analysis Based on the displacement boundaryconditions the first two-order modal functions are selectedas follows

u0(x y t) u1(t)cosπx

asin

πy

b+ u2(t)cos

3πx

asin

πy

b

(12a)

v0(x y t) v1(t)sinπx

acos

πy

b+ v2(t)sin

3πx

acos

πy

b

(12b)

w0(x y t) w1(t)sinπx

asin

πy

b+ w2(t)sin

3πx

asin

πy

b

(12c)

ϕx(x y t) ϕ1(t)cosπx

asin

πy

b+ ϕ2(t)cos

3πx

asin

πy

b

(12d)

ϕy(x y t) ϕ3(t)sinπx

acos

πy

b+ ϕ4(t)sin

3πx

acos

πy

b

(12e)

Since the out-of-plane vibration is dominant in thevibration system the in-plane vibrations are ignored inthis study -e inertia term is ignored and the modalfunctions (12a)ndash(12e) are substituted into the vibrationequations (10a)ndash(10e) -e Galerkin method is used toseparate the space-time variables and the two-degree-of-freedom ordinary differential dynamic equations areobtained as

eurow1 + ω102

w1 μ _w1 + β11w23

+ β22w1w22

+ β33w12w2

+ β44w13

+ P1 cos(Ωt)(13a)

eurow2 + ω202

w2 μ _w2 + β66w13

+ β77w12w2 + β88w1w2

2

+ β99w23

+ P2 cos(Ωt)(13b)

where the coefficients ω102 β44 β11 β22 β33 P1 ω20

2 β99β66 β77 β88 and P2 are constants related to the system

6 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

average equations in rectangular coordinates are obtainedIn addition the nonlinear dynamic responses of the two-order modes with system parameters are comparedconcretely

2 Governing Equations of Motion

-e mechanical model of the special orthotropic symmetricrectangular laminated plate that is simply supported on foursides is shown in Figure 1 Assume the length width andthickness of the rectangular laminated plate to be a b and hrespectively and a uniformly distributed harmonic excita-tion force q q0 cosΩt is applied in the transverse planewhere q0 is the amplitude of excitation

-e linear constitutive relation of each laminate is asfollows

σxx

σyy

σxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k

Q11 Q12 0

Q12 Q22 0

0 0 Q66

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k εxx

εyy

cxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

k

σyz

σxz

1113888 1113889 Q44 0

0 Q551113888 1113889

kcyz

cxz

1113888 1113889

k

(1)

where k represents the number of layers of the laminatedplate and

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q66 G12

Q44 G23

Q55 G13

(2)

Based on the higher-order shear deformation platetheory the displacement fields are

u(x y z t) u0(x y t) + zϕx(x y t) minus z3 43h2 ϕx +

zw0

zx1113888 1113889

v(x y z t) v0(x y t) + zϕy(x y t) minus z3 43h2 ϕy +

zw0

zy1113888 1113889

w(x y z t) w0(x y t)

(3)

where (u0 v0 w0) represent the displacement of a point onthe midplane and (ϕx ϕy) are the rotations of a transversenormal about the y and x axes respectively

According to the von Karman nonlinear geometricrelation

εxx zu

zx+12

zw

zx1113888 1113889

2

εxz 12

zu

zz+

zw

zx1113888 1113889

εxy 12

zu

zy+

zv

zx+

zw

zx

zw

zy1113888 1113889

εyy zv

zy+12

zw

zy1113888 1113889

2

εyz 12

zv

zz+

zw

zy1113888 1113889

(4)

For the assumed displacement field in (3) the strains in(4) can be expressed as

εxx

εyy

cxy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ z

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ z3

czϕx

zx+ c

z2w0

zx2

czϕy

zy+ c

z2w0

zy2

czϕx

zy+

z2w0

zxzy1113888 1113889 + c

zϕy

zx+

z2w0

zxzy1113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(5a)

cyz

cxz

1113888 1113889

ϕy +zw0

zy

ϕx +zw0

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+ z

2

3c ϕy +zw0

zy1113888 1113889

3c ϕx +zw0

zx1113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (5b)

where c minus (43h2)-e governing equations describing the vibration of

rectangular plate are obtained by the Hamilton principle

1113946t2

t1

(δT minus δU + δW)dt 0 (6)

where the variation in potential energy δU the variation inkinetic energy δT and the virtual work done by the externalforce δW are given by

Shock and Vibration 3

δU C σxxδεxx + σyyδεyy + σxyδcxy + σxzδcxz + σyzδcyz1113872 1113873dV

(7a)

δT Cρ _w0δ _w0dV + Cρ _u0 + z _ϕx + cz3 _ϕx +

z _w0

zx1113888 11138891113890 1113891

middot δ _u0 + zδ _ϕx + cz3 δ _ϕx +

zδ _w0

zx1113888 11138891113890 1113891dV

+ Cρ _v0 + z _ϕy + cz3 _ϕy +

z _w0

zy1113888 11138891113890 1113891

middot δ _v0 + zδ _ϕy + cz3 δ _ϕy +

zδ _w0

zy1113888 11138891113890 1113891dV

(7b)

δW minus Bμ _w0δ _w0dx dy + Bq0 cosΩtδw0dx dy

+ 1113929 1113946 1113954σnn δu0n + zδϕn + cz3 δϕn +

zδw0

zn1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σns δu0s + zδϕs + cz3 δϕs +

zδw0

zs1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σnzδw0dz ds

1113929 1113954Nnnδu0n + 1113954Mnnδϕn + c1113954Pnnδϕn + c1113954Pnn

zδw0

zn1113888

+ 1113954Nnsδu0s + 1113954Mnsδϕs + c1113954Pnsδϕs

+c1113954Pns

zδw0

zs+ 1113954Qnδw0ds1113889

+ Bqδw0dx dy minus Bc _w0δw0dx dy

(7c)

where μ is the damping coefficient a superposed dot on avariable indicates its time derivative ρ is the density of theplate (1113954σnn 1113954σns 1113954σnz) are the stress components on theboundary (δu0n δu0s) are the virtual displacements alongthe normal and tangential direction respectively on theboundary (nx ny) are the direction cosines of the outwardnormal with respect to the x- and y-axis at a point on theplate boundary and

Ii 1113946h2

minus h2ρz

idz

Nxx

Nyy

Nxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dz

Mxx

Myy

Mxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

zdz

Pxx

Pyy

Pxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

z3dz

(8a)

Qy

Qx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭dz

Ry

Rx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭z2dz

1113954Nnn

1113954Nns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭dz

1113954Mnn

1113954Mns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭zdz

(8b)

1113954Qn 1113946h2

minus h21113954σnzdz

1113954Pnn

1113954Pns

1113896 1113897 1113946h2

minus h2

1113954σnn

1113954σns

1113896 1113897z3dz

ϕx

ϕy

1113890 1113891 nx minus ny

ny nx

1113890 1113891ϕn

ϕs

1113890 1113891

(8c)

In this study all the applied forces on the boundary arezero-at is to say 1113954Nnn 1113954Nns 1113954Mnn 1113954Mns 1113954Pnn 1113954Pnn are all zerosSubstituting (7a) (7b) (7c) and (8a) (8b) (8c) into (6) thevibration equations are obtained as follows

q = q0cosΩty

z

x

(a)

y

Z

h

(b)

Figure 1 Mechanical model of a composite rectangular laminated plate

4 Shock and Vibration

zNxx

zx+

zNxy

zy I0eurou0 + I1 + cI3( 1113857euroϕx + cI3

zeurow0

zx (9a)

zNyy

zy+

zNxy

zx I0eurov0 + I1 + cI3( 1113857euroϕy + cI3

zeurow0

zy (9b)

z Nxx zw0zx( 1113857( 1113857

zxminus c

z2Pxx

zx2 +z Nyy zw0zy( 11138571113872 1113873

zyminus c

z2Pyy

zy2 +z Nxy zw0zy( 11138571113872 1113873

zx+

z Nxy zw0zx( 11138571113872 1113873

zy

minus 2cz2Pxy

zxzy+

zQx

zx+ 3c

zRx

zx+

zQy

zy+ 3c

zRy

zy

I0 eurow0 minus cI3zeurou0

zxminus cI3

zeurov0zy

minus cI4 + c2I61113872 1113873

zeuroϕx

zxminus cI4 + c

2I61113872 1113873

zeuroϕy

zyminus c

2I6

z2 eurow0

zx2 minus c2I6

z2 eurow0

zy2 minus q0 cosΩt + μ _w0

(9c)

zMxx

zx+ c

zPxx

zx+

zMxy

zy+ c

zPxy

zyminus Qx minus 3cRx

I1 + cI3( 1113857eurou0 + I2 + 2cI4 + c2I61113872 1113873euroϕx + cI4 + c

2I61113872 1113873

zeurow0

zx

(9d)

zMyy

zy+ c

zPyy

zy+

zMxy

zx+ c

zPxy

zxminus Qy minus 3cRy

I1 + cI3( 1113857eurov0 + I2 + 2cI4 + c2I61113872 1113873euroϕy + cI4 + c

2I61113872 1113873

zeurow0

zy

(9e)

Equations (9a)ndash(9e) can be written in form of general-ized displacements (u0 v0 w0 ϕx ϕy) and dimensionlessparameters are introduced as u0 (u0a) v0 (v0b)ϕx ϕx ϕy ϕy with the dimensionless parameter forms of

other physical quantities being the same as those of [24]-en the dimensionless partial differential equations ofvibration of the rectangular plates are obtained as

a10z2u0

zx2 + a11zw0

zx

z2w0

zx2 + a12z2v0zxzy

+ a13z2w0

zxzy

zw0

zy+ a14

z2u0

zy2 + a15z2w0

zy2zw0

zx

a16eurou0 + a17euroϕx + a18

zeurow0

zx

(10a)

b10z2u0

zxzy+ b11

z2w0

zxzy

zw0

zx+ b12

z2v0

zy2 + b13z2w0

zy2zw0

zy+ b14

z2v0

zx2 + b15z2w0

zx2zw0

zy

b16eurov0 + b17euroϕy + b18

zeurow0

zy

(10b)

Shock and Vibration 5

c10z2u0

zx2zw0

zx+ c11

zw0

zx1113888 1113889

2z2w0

zx2 + c12z2v0zxzy

zw0

zx+ c13

z2w0

zxzy

zw0

zx

zw0

zy+ c14

z2w0

zx2zu0

zx

+ c15z2w0

zx2zv0

zy+ c16

z2w0

zx2zw0

zy1113888 1113889

2

+ c17z3ϕx

zx3 + c18z3ϕx

zx2zy+ c19

z4w0

zx4 + c20z4w0

zx2zy2

+ c21z2u0

zxzy

zw0

zy+ c22

z2v0

zy2zw0

zy+ c23

zw0

zy1113888 1113889

2z2w0

zy2 + c24z2w0

zy2zu0

zx+ c25

z2w0

zy2zv0

zy

+ c26z3ϕx

zxzy2 + c27z3ϕy

zy3 + c28z4w0

zy4 + c29z2v0zx2

zw0

zy+ c30

z2w0

zxzy

zu0

zy+ c31

z2w0

zxzy

zv0zx

+ c32z2u0

zy2zw0

zx+ c33

z2w0

zy2zw0

zx1113888 1113889

2

+ c34zϕx

zx+ c35

z2w0

zx2 + c36zϕy

zy+ c37

z2w0

zy2

+ c38q0 cosΩt + c39 _w0

c40 eurow0 + c41zeurou0

zx+ c42

zeurov0zy

+ c43zeuroϕx

zx+ c44

zeuroϕy

zy+ c45

z2 eurow0

zx2 + c46z2 eurow0

zy2

(10c)

d10z2ϕx

zx2 + d11z2ϕy

zxzy+ d12

z3w0

zx3 + d13z2ϕx

zy2 + d14ϕx + d15zw0

zx+ d16

z3w0

zxzy2

d17eurou0 + d18euroϕx + d19

zeurow0

zx

(10d)

e10z2ϕx

zxzy+ e11

z2ϕy

zy2 + e12z3w0

zx2zy+ e13

z3w0

zy3 + e14z2ϕy

zx2 + e15ϕy + e16zw0

zy

e17eurov0 + e18euroϕy + e19

zeurow0

zy

(10e)

For the sake of convenience in writing the transverselines above the physical quantities are omitted -eboundary conditions of the simply supported plate can beexpressed as

x 0 andx a v w Nxx Mxx ϕy Pxx 0

(11a)

y 0 andy b u w Nyy Myy ϕx Pyy 0

(11b)

Due to the fact that the higher-order modes are not easilyexcited in structural vibration the first two modes are takenfor truncation analysis Based on the displacement boundaryconditions the first two-order modal functions are selectedas follows

u0(x y t) u1(t)cosπx

asin

πy

b+ u2(t)cos

3πx

asin

πy

b

(12a)

v0(x y t) v1(t)sinπx

acos

πy

b+ v2(t)sin

3πx

acos

πy

b

(12b)

w0(x y t) w1(t)sinπx

asin

πy

b+ w2(t)sin

3πx

asin

πy

b

(12c)

ϕx(x y t) ϕ1(t)cosπx

asin

πy

b+ ϕ2(t)cos

3πx

asin

πy

b

(12d)

ϕy(x y t) ϕ3(t)sinπx

acos

πy

b+ ϕ4(t)sin

3πx

acos

πy

b

(12e)

Since the out-of-plane vibration is dominant in thevibration system the in-plane vibrations are ignored inthis study -e inertia term is ignored and the modalfunctions (12a)ndash(12e) are substituted into the vibrationequations (10a)ndash(10e) -e Galerkin method is used toseparate the space-time variables and the two-degree-of-freedom ordinary differential dynamic equations areobtained as

eurow1 + ω102

w1 μ _w1 + β11w23

+ β22w1w22

+ β33w12w2

+ β44w13

+ P1 cos(Ωt)(13a)

eurow2 + ω202

w2 μ _w2 + β66w13

+ β77w12w2 + β88w1w2

2

+ β99w23

+ P2 cos(Ωt)(13b)

where the coefficients ω102 β44 β11 β22 β33 P1 ω20

2 β99β66 β77 β88 and P2 are constants related to the system

6 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

δU C σxxδεxx + σyyδεyy + σxyδcxy + σxzδcxz + σyzδcyz1113872 1113873dV

(7a)

δT Cρ _w0δ _w0dV + Cρ _u0 + z _ϕx + cz3 _ϕx +

z _w0

zx1113888 11138891113890 1113891

middot δ _u0 + zδ _ϕx + cz3 δ _ϕx +

zδ _w0

zx1113888 11138891113890 1113891dV

+ Cρ _v0 + z _ϕy + cz3 _ϕy +

z _w0

zy1113888 11138891113890 1113891

middot δ _v0 + zδ _ϕy + cz3 δ _ϕy +

zδ _w0

zy1113888 11138891113890 1113891dV

(7b)

δW minus Bμ _w0δ _w0dx dy + Bq0 cosΩtδw0dx dy

+ 1113929 1113946 1113954σnn δu0n + zδϕn + cz3 δϕn +

zδw0

zn1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σns δu0s + zδϕs + cz3 δϕs +

zδw0

zs1113888 11138891113890 1113891dz ds

+ 1113929 1113946 1113954σnzδw0dz ds

1113929 1113954Nnnδu0n + 1113954Mnnδϕn + c1113954Pnnδϕn + c1113954Pnn

zδw0

zn1113888

+ 1113954Nnsδu0s + 1113954Mnsδϕs + c1113954Pnsδϕs

+c1113954Pns

zδw0

zs+ 1113954Qnδw0ds1113889

+ Bqδw0dx dy minus Bc _w0δw0dx dy

(7c)

where μ is the damping coefficient a superposed dot on avariable indicates its time derivative ρ is the density of theplate (1113954σnn 1113954σns 1113954σnz) are the stress components on theboundary (δu0n δu0s) are the virtual displacements alongthe normal and tangential direction respectively on theboundary (nx ny) are the direction cosines of the outwardnormal with respect to the x- and y-axis at a point on theplate boundary and

Ii 1113946h2

minus h2ρz

idz

Nxx

Nyy

Nxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dz

Mxx

Myy

Mxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

zdz

Pxx

Pyy

Pxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

1113946h2

minus h2

σxx

σyy

σxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

z3dz

(8a)

Qy

Qx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭dz

Ry

Rx

⎧⎨

⎩

⎫⎬

⎭ 1113946h2

minus h2

σyz

σxz

⎧⎨

⎩

⎫⎬

⎭z2dz

1113954Nnn

1113954Nns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭dz

1113954Mnn

1113954Mns

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113946

h2

minus h2

1113954σnn

1113954σns

⎧⎨

⎩

⎫⎬

⎭zdz

(8b)

1113954Qn 1113946h2

minus h21113954σnzdz

1113954Pnn

1113954Pns

1113896 1113897 1113946h2

minus h2

1113954σnn

1113954σns

1113896 1113897z3dz

ϕx

ϕy

1113890 1113891 nx minus ny

ny nx

1113890 1113891ϕn

ϕs

1113890 1113891

(8c)

In this study all the applied forces on the boundary arezero-at is to say 1113954Nnn 1113954Nns 1113954Mnn 1113954Mns 1113954Pnn 1113954Pnn are all zerosSubstituting (7a) (7b) (7c) and (8a) (8b) (8c) into (6) thevibration equations are obtained as follows

q = q0cosΩty

z

x

(a)

y

Z

h

(b)

Figure 1 Mechanical model of a composite rectangular laminated plate

4 Shock and Vibration

zNxx

zx+

zNxy

zy I0eurou0 + I1 + cI3( 1113857euroϕx + cI3

zeurow0

zx (9a)

zNyy

zy+

zNxy

zx I0eurov0 + I1 + cI3( 1113857euroϕy + cI3

zeurow0

zy (9b)

z Nxx zw0zx( 1113857( 1113857

zxminus c

z2Pxx

zx2 +z Nyy zw0zy( 11138571113872 1113873

zyminus c

z2Pyy

zy2 +z Nxy zw0zy( 11138571113872 1113873

zx+

z Nxy zw0zx( 11138571113872 1113873

zy

minus 2cz2Pxy

zxzy+

zQx

zx+ 3c

zRx

zx+

zQy

zy+ 3c

zRy

zy

I0 eurow0 minus cI3zeurou0

zxminus cI3

zeurov0zy

minus cI4 + c2I61113872 1113873

zeuroϕx

zxminus cI4 + c

2I61113872 1113873

zeuroϕy

zyminus c

2I6

z2 eurow0

zx2 minus c2I6

z2 eurow0

zy2 minus q0 cosΩt + μ _w0

(9c)

zMxx

zx+ c

zPxx

zx+

zMxy

zy+ c

zPxy

zyminus Qx minus 3cRx

I1 + cI3( 1113857eurou0 + I2 + 2cI4 + c2I61113872 1113873euroϕx + cI4 + c

2I61113872 1113873

zeurow0

zx

(9d)

zMyy

zy+ c

zPyy

zy+

zMxy

zx+ c

zPxy

zxminus Qy minus 3cRy

I1 + cI3( 1113857eurov0 + I2 + 2cI4 + c2I61113872 1113873euroϕy + cI4 + c

2I61113872 1113873

zeurow0

zy

(9e)

Equations (9a)ndash(9e) can be written in form of general-ized displacements (u0 v0 w0 ϕx ϕy) and dimensionlessparameters are introduced as u0 (u0a) v0 (v0b)ϕx ϕx ϕy ϕy with the dimensionless parameter forms of

other physical quantities being the same as those of [24]-en the dimensionless partial differential equations ofvibration of the rectangular plates are obtained as

a10z2u0

zx2 + a11zw0

zx

z2w0

zx2 + a12z2v0zxzy

+ a13z2w0

zxzy

zw0

zy+ a14

z2u0

zy2 + a15z2w0

zy2zw0

zx

a16eurou0 + a17euroϕx + a18

zeurow0

zx

(10a)

b10z2u0

zxzy+ b11

z2w0

zxzy

zw0

zx+ b12

z2v0

zy2 + b13z2w0

zy2zw0

zy+ b14

z2v0

zx2 + b15z2w0

zx2zw0

zy

b16eurov0 + b17euroϕy + b18

zeurow0

zy

(10b)

Shock and Vibration 5

c10z2u0

zx2zw0

zx+ c11

zw0

zx1113888 1113889

2z2w0

zx2 + c12z2v0zxzy

zw0

zx+ c13

z2w0

zxzy

zw0

zx

zw0

zy+ c14

z2w0

zx2zu0

zx

+ c15z2w0

zx2zv0

zy+ c16

z2w0

zx2zw0

zy1113888 1113889

2

+ c17z3ϕx

zx3 + c18z3ϕx

zx2zy+ c19

z4w0

zx4 + c20z4w0

zx2zy2

+ c21z2u0

zxzy

zw0

zy+ c22

z2v0

zy2zw0

zy+ c23

zw0

zy1113888 1113889

2z2w0

zy2 + c24z2w0

zy2zu0

zx+ c25

z2w0

zy2zv0

zy

+ c26z3ϕx

zxzy2 + c27z3ϕy

zy3 + c28z4w0

zy4 + c29z2v0zx2

zw0

zy+ c30

z2w0

zxzy

zu0

zy+ c31

z2w0

zxzy

zv0zx

+ c32z2u0

zy2zw0

zx+ c33

z2w0

zy2zw0

zx1113888 1113889

2

+ c34zϕx

zx+ c35

z2w0

zx2 + c36zϕy

zy+ c37

z2w0

zy2

+ c38q0 cosΩt + c39 _w0

c40 eurow0 + c41zeurou0

zx+ c42

zeurov0zy

+ c43zeuroϕx

zx+ c44

zeuroϕy

zy+ c45

z2 eurow0

zx2 + c46z2 eurow0

zy2

(10c)

d10z2ϕx

zx2 + d11z2ϕy

zxzy+ d12

z3w0

zx3 + d13z2ϕx

zy2 + d14ϕx + d15zw0

zx+ d16

z3w0

zxzy2

d17eurou0 + d18euroϕx + d19

zeurow0

zx

(10d)

e10z2ϕx

zxzy+ e11

z2ϕy

zy2 + e12z3w0

zx2zy+ e13

z3w0

zy3 + e14z2ϕy

zx2 + e15ϕy + e16zw0

zy

e17eurov0 + e18euroϕy + e19

zeurow0

zy

(10e)

For the sake of convenience in writing the transverselines above the physical quantities are omitted -eboundary conditions of the simply supported plate can beexpressed as

x 0 andx a v w Nxx Mxx ϕy Pxx 0

(11a)

y 0 andy b u w Nyy Myy ϕx Pyy 0

(11b)

Due to the fact that the higher-order modes are not easilyexcited in structural vibration the first two modes are takenfor truncation analysis Based on the displacement boundaryconditions the first two-order modal functions are selectedas follows

u0(x y t) u1(t)cosπx

asin

πy

b+ u2(t)cos

3πx

asin

πy

b

(12a)

v0(x y t) v1(t)sinπx

acos

πy

b+ v2(t)sin

3πx

acos

πy

b

(12b)

w0(x y t) w1(t)sinπx

asin

πy

b+ w2(t)sin

3πx

asin

πy

b

(12c)

ϕx(x y t) ϕ1(t)cosπx

asin

πy

b+ ϕ2(t)cos

3πx

asin

πy

b

(12d)

ϕy(x y t) ϕ3(t)sinπx

acos

πy

b+ ϕ4(t)sin

3πx

acos

πy

b

(12e)

Since the out-of-plane vibration is dominant in thevibration system the in-plane vibrations are ignored inthis study -e inertia term is ignored and the modalfunctions (12a)ndash(12e) are substituted into the vibrationequations (10a)ndash(10e) -e Galerkin method is used toseparate the space-time variables and the two-degree-of-freedom ordinary differential dynamic equations areobtained as

eurow1 + ω102

w1 μ _w1 + β11w23

+ β22w1w22

+ β33w12w2

+ β44w13

+ P1 cos(Ωt)(13a)

eurow2 + ω202

w2 μ _w2 + β66w13

+ β77w12w2 + β88w1w2

2

+ β99w23

+ P2 cos(Ωt)(13b)

where the coefficients ω102 β44 β11 β22 β33 P1 ω20

2 β99β66 β77 β88 and P2 are constants related to the system

6 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

zNxx

zx+

zNxy

zy I0eurou0 + I1 + cI3( 1113857euroϕx + cI3

zeurow0

zx (9a)

zNyy

zy+

zNxy

zx I0eurov0 + I1 + cI3( 1113857euroϕy + cI3

zeurow0

zy (9b)

z Nxx zw0zx( 1113857( 1113857

zxminus c

z2Pxx

zx2 +z Nyy zw0zy( 11138571113872 1113873

zyminus c

z2Pyy

zy2 +z Nxy zw0zy( 11138571113872 1113873

zx+

z Nxy zw0zx( 11138571113872 1113873

zy

minus 2cz2Pxy

zxzy+

zQx

zx+ 3c

zRx

zx+

zQy

zy+ 3c

zRy

zy

I0 eurow0 minus cI3zeurou0

zxminus cI3

zeurov0zy

minus cI4 + c2I61113872 1113873

zeuroϕx

zxminus cI4 + c

2I61113872 1113873

zeuroϕy

zyminus c

2I6

z2 eurow0

zx2 minus c2I6

z2 eurow0

zy2 minus q0 cosΩt + μ _w0

(9c)

zMxx

zx+ c

zPxx

zx+

zMxy

zy+ c

zPxy

zyminus Qx minus 3cRx

I1 + cI3( 1113857eurou0 + I2 + 2cI4 + c2I61113872 1113873euroϕx + cI4 + c

2I61113872 1113873

zeurow0

zx

(9d)

zMyy

zy+ c

zPyy

zy+

zMxy

zx+ c

zPxy

zxminus Qy minus 3cRy

I1 + cI3( 1113857eurov0 + I2 + 2cI4 + c2I61113872 1113873euroϕy + cI4 + c

2I61113872 1113873

zeurow0

zy

(9e)

Equations (9a)ndash(9e) can be written in form of general-ized displacements (u0 v0 w0 ϕx ϕy) and dimensionlessparameters are introduced as u0 (u0a) v0 (v0b)ϕx ϕx ϕy ϕy with the dimensionless parameter forms of

other physical quantities being the same as those of [24]-en the dimensionless partial differential equations ofvibration of the rectangular plates are obtained as

a10z2u0

zx2 + a11zw0

zx

z2w0

zx2 + a12z2v0zxzy

+ a13z2w0

zxzy

zw0

zy+ a14

z2u0

zy2 + a15z2w0

zy2zw0

zx

a16eurou0 + a17euroϕx + a18

zeurow0

zx

(10a)

b10z2u0

zxzy+ b11

z2w0

zxzy

zw0

zx+ b12

z2v0

zy2 + b13z2w0

zy2zw0

zy+ b14

z2v0

zx2 + b15z2w0

zx2zw0

zy

b16eurov0 + b17euroϕy + b18

zeurow0

zy

(10b)

Shock and Vibration 5

c10z2u0

zx2zw0

zx+ c11

zw0

zx1113888 1113889

2z2w0

zx2 + c12z2v0zxzy

zw0

zx+ c13

z2w0

zxzy

zw0

zx

zw0

zy+ c14

z2w0

zx2zu0

zx

+ c15z2w0

zx2zv0

zy+ c16

z2w0

zx2zw0

zy1113888 1113889

2

+ c17z3ϕx

zx3 + c18z3ϕx

zx2zy+ c19

z4w0

zx4 + c20z4w0

zx2zy2

+ c21z2u0

zxzy

zw0

zy+ c22

z2v0

zy2zw0

zy+ c23

zw0

zy1113888 1113889

2z2w0

zy2 + c24z2w0

zy2zu0

zx+ c25

z2w0

zy2zv0

zy

+ c26z3ϕx

zxzy2 + c27z3ϕy

zy3 + c28z4w0

zy4 + c29z2v0zx2

zw0

zy+ c30

z2w0

zxzy

zu0

zy+ c31

z2w0

zxzy

zv0zx

+ c32z2u0

zy2zw0

zx+ c33

z2w0

zy2zw0

zx1113888 1113889

2

+ c34zϕx

zx+ c35

z2w0

zx2 + c36zϕy

zy+ c37

z2w0

zy2

+ c38q0 cosΩt + c39 _w0

c40 eurow0 + c41zeurou0

zx+ c42

zeurov0zy

+ c43zeuroϕx

zx+ c44

zeuroϕy

zy+ c45

z2 eurow0

zx2 + c46z2 eurow0

zy2

(10c)

d10z2ϕx

zx2 + d11z2ϕy

zxzy+ d12

z3w0

zx3 + d13z2ϕx

zy2 + d14ϕx + d15zw0

zx+ d16

z3w0

zxzy2

d17eurou0 + d18euroϕx + d19

zeurow0

zx

(10d)

e10z2ϕx

zxzy+ e11

z2ϕy

zy2 + e12z3w0

zx2zy+ e13

z3w0

zy3 + e14z2ϕy

zx2 + e15ϕy + e16zw0

zy

e17eurov0 + e18euroϕy + e19

zeurow0

zy

(10e)

For the sake of convenience in writing the transverselines above the physical quantities are omitted -eboundary conditions of the simply supported plate can beexpressed as

x 0 andx a v w Nxx Mxx ϕy Pxx 0

(11a)

y 0 andy b u w Nyy Myy ϕx Pyy 0

(11b)

Due to the fact that the higher-order modes are not easilyexcited in structural vibration the first two modes are takenfor truncation analysis Based on the displacement boundaryconditions the first two-order modal functions are selectedas follows

u0(x y t) u1(t)cosπx

asin

πy

b+ u2(t)cos

3πx

asin

πy

b

(12a)

v0(x y t) v1(t)sinπx

acos

πy

b+ v2(t)sin

3πx

acos

πy

b

(12b)

w0(x y t) w1(t)sinπx

asin

πy

b+ w2(t)sin

3πx

asin

πy

b

(12c)

ϕx(x y t) ϕ1(t)cosπx

asin

πy

b+ ϕ2(t)cos

3πx

asin

πy

b

(12d)

ϕy(x y t) ϕ3(t)sinπx

acos

πy

b+ ϕ4(t)sin

3πx

acos

πy

b

(12e)

Since the out-of-plane vibration is dominant in thevibration system the in-plane vibrations are ignored inthis study -e inertia term is ignored and the modalfunctions (12a)ndash(12e) are substituted into the vibrationequations (10a)ndash(10e) -e Galerkin method is used toseparate the space-time variables and the two-degree-of-freedom ordinary differential dynamic equations areobtained as

eurow1 + ω102

w1 μ _w1 + β11w23

+ β22w1w22

+ β33w12w2

+ β44w13

+ P1 cos(Ωt)(13a)

eurow2 + ω202

w2 μ _w2 + β66w13

+ β77w12w2 + β88w1w2

2

+ β99w23

+ P2 cos(Ωt)(13b)

where the coefficients ω102 β44 β11 β22 β33 P1 ω20

2 β99β66 β77 β88 and P2 are constants related to the system

6 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

c10z2u0

zx2zw0

zx+ c11

zw0

zx1113888 1113889

2z2w0

zx2 + c12z2v0zxzy

zw0

zx+ c13

z2w0

zxzy

zw0

zx

zw0

zy+ c14

z2w0

zx2zu0

zx

+ c15z2w0

zx2zv0

zy+ c16

z2w0

zx2zw0

zy1113888 1113889

2

+ c17z3ϕx

zx3 + c18z3ϕx

zx2zy+ c19

z4w0

zx4 + c20z4w0

zx2zy2

+ c21z2u0

zxzy

zw0

zy+ c22

z2v0

zy2zw0

zy+ c23

zw0

zy1113888 1113889

2z2w0

zy2 + c24z2w0

zy2zu0

zx+ c25

z2w0

zy2zv0

zy

+ c26z3ϕx

zxzy2 + c27z3ϕy

zy3 + c28z4w0

zy4 + c29z2v0zx2

zw0

zy+ c30

z2w0

zxzy

zu0

zy+ c31

z2w0

zxzy

zv0zx

+ c32z2u0

zy2zw0

zx+ c33

z2w0

zy2zw0

zx1113888 1113889

2

+ c34zϕx

zx+ c35

z2w0

zx2 + c36zϕy

zy+ c37

z2w0

zy2

+ c38q0 cosΩt + c39 _w0

c40 eurow0 + c41zeurou0

zx+ c42

zeurov0zy

+ c43zeuroϕx

zx+ c44

zeuroϕy

zy+ c45

z2 eurow0

zx2 + c46z2 eurow0

zy2

(10c)

d10z2ϕx

zx2 + d11z2ϕy

zxzy+ d12

z3w0

zx3 + d13z2ϕx

zy2 + d14ϕx + d15zw0

zx+ d16

z3w0

zxzy2

d17eurou0 + d18euroϕx + d19

zeurow0

zx

(10d)

e10z2ϕx

zxzy+ e11

z2ϕy

zy2 + e12z3w0

zx2zy+ e13

z3w0

zy3 + e14z2ϕy

zx2 + e15ϕy + e16zw0

zy

e17eurov0 + e18euroϕy + e19

zeurow0

zy

(10e)

For the sake of convenience in writing the transverselines above the physical quantities are omitted -eboundary conditions of the simply supported plate can beexpressed as

x 0 andx a v w Nxx Mxx ϕy Pxx 0

(11a)

y 0 andy b u w Nyy Myy ϕx Pyy 0

(11b)

Due to the fact that the higher-order modes are not easilyexcited in structural vibration the first two modes are takenfor truncation analysis Based on the displacement boundaryconditions the first two-order modal functions are selectedas follows

u0(x y t) u1(t)cosπx

asin

πy

b+ u2(t)cos

3πx

asin

πy

b

(12a)

v0(x y t) v1(t)sinπx

acos

πy

b+ v2(t)sin

3πx

acos

πy

b

(12b)

w0(x y t) w1(t)sinπx

asin

πy

b+ w2(t)sin

3πx

asin

πy

b

(12c)

ϕx(x y t) ϕ1(t)cosπx

asin

πy

b+ ϕ2(t)cos

3πx

asin

πy

b

(12d)

ϕy(x y t) ϕ3(t)sinπx

acos

πy

b+ ϕ4(t)sin

3πx

acos

πy

b

(12e)

Since the out-of-plane vibration is dominant in thevibration system the in-plane vibrations are ignored inthis study -e inertia term is ignored and the modalfunctions (12a)ndash(12e) are substituted into the vibrationequations (10a)ndash(10e) -e Galerkin method is used toseparate the space-time variables and the two-degree-of-freedom ordinary differential dynamic equations areobtained as

eurow1 + ω102

w1 μ _w1 + β11w23

+ β22w1w22

+ β33w12w2

+ β44w13

+ P1 cos(Ωt)(13a)

eurow2 + ω202

w2 μ _w2 + β66w13

+ β77w12w2 + β88w1w2

2

+ β99w23

+ P2 cos(Ωt)(13b)

where the coefficients ω102 β44 β11 β22 β33 P1 ω20

2 β99β66 β77 β88 and P2 are constants related to the system

6 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

3 Perturbation Analyses

-e multiscale method is used for the approximate solutionof the vibration equations Firstly the small parameter ε isintroduced and the original equations (13a) and (13b) aretransformed into

eurow1 + ω102w1 εμ _w1 + εβ44w1

3+ εβ11w2

3+ εβ22w1w2

2

+ εβ33w12w2 + P1 cos(Ωt)

(14a)

eurow2 + ω202w2 εμ _w2 + εβ99w2

3+ εβ66w1

3+ εβ77w1

2w2

+ εβ88w1w22

+ P2 cos(Ωt)

(14b)

-e approximate solutions of (14a) and (14b) can beexpressed as follows

w1 w10 T0 T1( 1113857 + εw11 T0 T1( 1113857 (15a)

w2 w20 T0 T1( 1113857 + εw21 T0 T1( 1113857 (15b)

where T0 t T1 εt-e operators can be defined as

ddt

z

zT0

zT0

zt+

z

zT1

zT1

zt+ D0 + εD1 + (16a)

d2

dt2 D0

2+ 2εD0D1 + 1113872 1113873

(16b)

where D0 (zzT0) D1 (zzT1)By substituting the expressions (15a) (15b) and (16a)

(16b) into (14a) and (14b) and equating the coefficients of thesame power of ε on both sides of (14a) and (14b) one canobtain order ε0

D02w10 + ω10

2w10 P1 cos(Ωt) (17a)

D02w20 + ω20

2w20 P2 cos(Ωt) (17b)

Order ε1

D02w11 + ω10

2w11 minus 2D0D1w10 + μD0w10 + β44w10

3+ β11w20

3+ β22w10w20

2+ β33w10

2w20 (18a)

D02w21 + ω20

2w21 minus 2D0D1w20 + μD0w20 + β99w20

3+ β66w10

3+ β77w10

2w20 + β88w10w20

2 (18b)

-e solutions of (17a) and (17b) are written in thecomplex form

w10 A1 T1( 1113857eiω10T0 + A1 T1( 1113857e

minus iω10T0 + A0eiΩT0 + A0e

minus iΩT0

(19a)

w20 A2 T1( 1113857eiω20T0 + A2 T1( 1113857e

minus iω20T0 + A3eiΩT0 + A3e

minus iΩT0

(19b)

where A0 (P12(ω102 minus Ω2)) A3 (P22(ω20

2 minus Ω2))

Subharmonic resonance occurs when the relationshipbetween the excitation frequency and the first natural fre-quency of the system satisfies the relation

Ω 3ω10 + εσ1 (20)

where σ1 quantitatively describes the relationship betweenthe frequency of the external excitation and the naturalfrequency of the derived system

Substituting (19a) (19b) and (20) into (18a) and (18b)and eliminating the secular terms the following expressionsare obtained

dA1

dT1 minus

i

2ω10

ω10μA1i + +6A1A0A0β44 + 2A1A3A0β33 + 3β44A12A1 + 2β22A1A2A2

+2β33A1A3A0 + 2β22A1A3A3 + 3β44A12A0 + A3A1

2β331113874 1113875eiσ1T1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21a)

dA2

dT1 minus

i

2ω20

ω20μA2i + +6A2A3A3β99 + 2A2A3A0β88 + 2β88A2A3A0

+2β77A2A0A0 + 3β99A2A22 + 2β77A2A1A1

1113890 1113891 (21b)

and A1 and A2 are written in the polar form

Shock and Vibration 7

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

A1 T1( 1113857 12a1 T1( 1113857e

iθ1 T1( ) (22a)

A2 T1( 1113857 12a2 T1( 1113857e

iθ2 T1( ) (22b)

By substituting (22a) and (22b) into (21a) and (21b) weseparate the resulting equations into the real and imaginarypart thus yielding

_a1 μ2a1 +

(34)β44a12A0 +(14)A3a1

2β33( 1113857

ω10sinφ1 (23a)

a1_θ1 minus

3β44ω10

A02a1 minus

2β33ω10

A0A3a1 minus3β448ω10

a13

minusβ224ω10

a22a1 minus

β22ω10

A32a1

minus(34)β44a1

2A0 +(14)A3a12β33( 1113857

ω10cosφ1

(23b)

_a2 μ2a2 (23c)

a2_θ2 minus

β77ω20

A02a2 minus

2β88ω20

A0A3a2 minusβ774ω20

a12a2 minus

3β998ω20

a23

minus3β99ω20

A32a2 (23d)

where φ1 σ1T1 minus 3θ1 equation (23c) shows that whenT1⟶infin a2 decays to zero-at is to say the second-ordermode is not excited and so for the steady-state motion ofthe system there is _a1 _φ1 0 By substituting _a1 _φ1 0

and a2 0 into (23a) (23b) (23c) and (23d) the amplitude-frequency response equation of subharmonic resonance forthe first-order mode is concluded

μ2a11113874 1113875

2+

13a1σ1 +

3β44ω10

A02a1 +

2β33ω10

A0A3a1 +3β448ω10

a13

+β22ω10

A32a11113888 1113889

2

(34)β44a1

2A0 +(14)A3a12β33( 1113857

2

ω102

(24)

One can also write A1 in the form of rectangular co-ordinates as A1 x1 + x2i Substituting A1 x1 + x2i and

A2 0 into (21a) and (21b) the average equations in theform of rectangular coordinates are obtained as

_x1 μ2x1 +

3A02β44

ω10x2 +

2β33A0A3

ω10x2 +

3β44 x12 + x2

2( 1113857x2

2ω10+

A32β22ω10

x2

minusA3β33x1x2

ω10cos σ1T1( 1113857 minus

3A0β44x22

2ω10sin σ1T1( 1113857 +

A3β33x12

2ω10sin σ1T1( 1113857

minusA3β33x2

2

2ω10sin σ1T1( 1113857 minus

3A0β44x1x2

ω10cos σ1T1( 1113857 +

3A0β44x12

2ω10sin σ1T1( 1113857

(25a)

_x2 μ2x2 minus

3A02β44

ω10x1 minus

2β33A0A3

ω10x1 minus

3β44 x12 + x2

2( 1113857x1

2ω10minus

A32β22ω10

x1

minus3β44A0x1x2

ω10sin σ1T1( 1113857 minus

β33A3x1x2

ω10sin σ1T1( 1113857 minus

3β44A0x12

2ω10cos σ1T1( 1113857

+3β44A0x2

2

2ω10cos σ1T1( 1113857 minus

β33A3x12

2ω10cos σ1T1( 1113857 +

β33A3x22

2ω10cos σ1T1( 1113857

(25b)

8 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

-e external excitation frequency and second naturalfrequency of the system satisfy the following relationship

Ω 3ω20 + εσ1 (26)

Similar to the first order the amplitude-frequency re-sponse equation of subharmonic resonance for the second-mode mode is given by

μ2a21113874 1113875

2+

13a2σ1 +

β77ω20

A02a2 +

2β88ω20

A0A3a2 +3β998ω20

a23

+3β99ω20

A32a21113888 1113889

2

(14)β88A0a2

2 +(34)β99A3a22( 1113857

2

ω202

(27)

-e average equations in the form of rectangular co-ordinates of the subharmonic resonance for the secondmode can be obtained as follows

_x3 minusβ88A0x3x4

ω20cos σ1T1( 1113857 +

A02β77ω20

x4 +3A3

2β99ω20

x4 +2β88A0A3

ω20x4

+3β99 x3

2 + x42( 1113857x4

2ω20+μ2x3 minus

3A3β99x42

2ω20sin σ1T1( 1113857 +

3A3β99x32

2ω20sin σ1T1( 1113857

minusA0β88x4

2

2ω20sin σ1T1( 1113857 +

A0β88x32

2ω20sin σ1T1( 1113857 minus

3β99A3x3x4

ω20cos σ1T1( 1113857

(28a)

_x4 minus3A3β99x3x4

ω20sin σ1T1( 1113857 minus

3β99 x32 + x4

2( 1113857x3

2ω20+β88A0x4

2

2ω20cos σ1T1( 1113857

minusA0

2β77ω20

x3 minus2β88A0A3

ω20x3 minus

3A32β99

ω20x3 +

μ2x4 +

3A3β99x42

2ω20cos σ1T1( 1113857

minusA0β88x3x4

ω20sin σ1T1( 1113857 minus

3A3β99x32

2ω20cos σ1T1( 1113857 minus

β88A0x32

2ω20cos σ1T1( 1113857

(28b)

4 Results and Discussion

-e values of parameters related to the laminated materialsare respectively β11 5307 β22 minus 293 β33 984β44 minus 8036 β66 minus 636 β77 1207 β88 2005β99 minus 18057 ω10 512 ω20 106 Based on the ampli-tude-frequency equations (24) and (27) of the two modesthe effects of damping excitation amplitude and excitationfrequency on the 13 subharmonic resonance characteristicsare studied as shown in Figures 2ndash4

Figure 2 is the amplitude-frequency response curve ofthe two-order modes of the rectangular laminated plate fordifferent excitation amplitudes when 13 subharmonicresonance occurs -e vertical coordinate represents theamplitude of two-order modes and the abscissa is thedetuning parameter P1 and P2 are the amplitudes of ex-ternal excitation It can be seen from Figure 2 that there aretwo steady-state solutions for the two-order modes re-spectively at the same excitation frequency When the ex-citation amplitudes are the same the amplitudes of the two

steady-state solutions increase gradually with the increase inexcitation frequency When the excitation frequency is thesame the influence of the excitation amplitude on thesteady-state solutions with small vibration amplitude of thetwo-order modes is significant -e solution with smallamplitude of the first-order mode increases with the increasein the excitation amplitude Contrary to the first-ordermode the solution with small amplitude of the second-ordermode decreases with the increase in the excitation ampli-tude For the same excitation frequency the influence of theexcitation amplitude on steady-state solutions with the largeamplitude of the two-order modes is not obvious

Figure 3 shows the amplitude-frequency response curvesof the two-order modes of the rectangular laminated platefor different damping values when 13 subharmonic reso-nance occurs -e vertical coordinate is the vibration am-plitude of the two-order modes and the abscissa is thedetuning parameter μ is a physical quantity that charac-terizes the damping coefficient of laminated materialsObviously it can be seen from Figure 3 that the damping

Shock and Vibration 9

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

affects the range of the steady-state solution (ie distributionregion of the curve) As the damping value increases therange within which the solution of the subharmonic vi-bration exists becomes smaller while the two modal am-plitudes are also affected by the damping coefficient Withthe change in damping the overall change trend of theamplitude-frequency response curve of the two-ordermodesof the laminated plates is consistent

Figure 4 shows ptthe force-amplitude response curves ofthe two-order modes of a rectangular laminated plate underdifferent external excitation frequencies when 13 subharmonicresonance occurs -e vertical coordinate represents the vi-bration amplitude of two-order modes and the abscissa is theexternal excitation amplitude It can be seen from Figure 4 that

there are also two nonzero steady-state solutions for the twomodes respectively Under the same external excitation fre-quency the steady-state solutions associated with small am-plitudes first decrease and then increase with the increase in theexcitation amplitude and the solutions with large amplitudedecrease monotonously as the excitation amplitude increasesFor the same external excitation amplitude as the excitationfrequency increases the vibration amplitudes of the two modesalso increase -e trend of the force-amplitude response curvesof the two-ordermodes of the laminated plate remains identicalBesides from the analysis of the frequency-resonance curves itcan be found that the results of this paper are consistent with thequalitative results given by Hu and Li [28] and Permoon et al[37]

0 20 40ndash20σ1

0

1

2

3a 1

P1 = 10P2 = 25P3 = 40

(a)

0 50 1000

05

1

15

2

σ1

a 2

P2 = 200P2 = 300P2 = 400

(b)

Figure 2 Amplitude-frequency response curves of the first-order and second-order mode for different excitation amplitudes

0 50 1000

1

2

3

a 1

σ1

μ = 005μ = 1μ = 5

(a)

0 50 100 150 2000

1

2

3

4

a 2

σ1

μ = 05μ = 5μ = 10

(b)

Figure 3 Amplitude-frequency response curves of the first-order and the second-order mode for different damping coefficients

10 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

Based on the average (25a) and (25b) the bifurcationdiagram for the first-order vibration is simulated numeri-cally by the RungendashKutta method -e amplitude of theexternal excitation is selected as the control parameter of thesystem and the influence of the amplitude of the externalexcitation on the nonlinear vibration characteristics of thesystem is studied -e initial conditions are x10 08x20 11 and the other parameters related to the laminatematerials are the same as those in the above amplitude-frequency equation (24) -e global bifurcation diagram ofthe subharmonic vibration of the rectangular laminatedplate is obtained as shown in Figure 5

In Figure 5 the abscissa is the amplitude of the ex-ternal excitation and the vertical coordinate is the ab-stract physical quantity that represents the transversedeflection of the rectangular plate It can be seen fromFigure 5 that for a small amplitude of the external ex-citation the vibration response of the first mode of the

subharmonic resonance of the laminated plate is period-2motion As the amplitude of external excitation increasesthe vibration response of the system changes from pe-riodic motion to chaotic motion With the continualincrease in the amplitude of the external excitation theresponse of the system changes from chaotic motion toquasiperiodic motion and subsequently to period-2motion Finally the vibration response of the systemchanges to haploid periodic motion In the brief intervalwhen the amplitude of external excitation changes thefirst-order mode undergoes all the forms of vibration ofthe nonlinear response It shows a variety of dynamiccharacteristics At the same time the phase and waveformdiagrams of different vibration states are also obtainedFigures 6ndash10 show the waveforms and phase diagrams ofthe subharmonic vibration responses of rectangularlaminated plates under different vibration stages inFigure 5

0 50 100P1

0

05

1

15

2a 1

σ1 = 6σ1 = 16σ1 = 26

(a)

P2

0 200 4000

05

1

15

2

a 2

σ1 = 6σ1 = 16σ1 = 26

(b)

Figure 4 Force-amplitude response curves of the first-order and second-order mode for different excitation frequencies

x 1

45 50 60 6540 55P1

ndash4

ndash2

0

2

4

Figure 5 Bifurcation diagram for the first-order mode via external excitation

Shock and Vibration 11

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4x 2

(a)

ndash4

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 6 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-3motion when P1 42

0ndash2 2 4ndash4x1

ndash4

ndash2

0

2

4

x 2

(a)

12 13 14 1511t

ndash4

ndash2

0

2

4x 1

(b)

Figure 7 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with chaotic motionwhen P1 46

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

ndash2

0

2

4

x 1

(b)

Figure 8 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with quasiperiodicmotion when P1 52

12 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

Similarly the RungendashKutta algorithm is utilized for thenumerical simulation of the average equations (28) of thesecond-order response and the initial value is selected asx30 14 x40 175 By choosing the excitation amplitudeas the control parameter the global bifurcation diagram ofthe second mode is obtained in Figure 11

Figure 11 is the global bifurcation diagram of thesecond mode of the subharmonic vibration of rectangularlaminated plates It can be seen from the figure that as theamplitude of the external excitation increases the

vibration response of the second mode also presentsdifferent vibration forms When the amplitude of externalexcitation is small the vibration response of the system ischaotic With the increase in the amplitude of externalexcitation the system changes from chaotic motion toperiod-3 motion and then from period-3 motion tochaotic motion again Figures 12ndash14 show the waveformand phase diagrams of the second-order modes of thesubharmonic vibration of laminated plates with differentexcitation amplitudes

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

12 13 14 1511t

x 1

ndash2

0

2

4

(b)

Figure 9 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-2motion when P1 57

ndash3

ndash2

ndash1

0

1

2

0ndash2 2 4x1

x 2

(a)

ndash2

0

2

4

x 1

12 13 14 1511t

(b)

Figure 10 Phase portrait on plane (x1 x2) and the waveforms on the plane (t x1) of the first-order mode of the system with periodic-1motion when P1 63

Shock and Vibration 13

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

0ndash2 2 4ndash4x3

ndash4

ndash2

0

2

4

x 4

(a)

ndash4

ndash2

0

2

4x 3

12 13 14 1511t

(b)

Figure 12 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 32

ndash3

ndash2

ndash1

0

2

1

x 4

0ndash2 ndash1 1 2x3

(a)

ndash2

ndash1

0

2

1

x 3

12 13 14 1511t

(b)

Figure 13 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with periodic-3motion when P1 42

30 35 40 45 50 55P2

ndash3

ndash2

ndash1

0

1

2

x 3

Figure 11 Bifurcation diagram for the second-order mode via external excitation

14 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

5 Conclusions

In this work the subharmonic resonance of a rectangularlaminated plate under harmonic excitation is studied -e vi-bration equations of the system are established using vonKarmanrsquos nonlinear geometric relation and Hamiltonrsquos prin-ciple -e amplitude-frequency equations and the averageequations in rectangular coordinates are obtained by using themultiscale method -e amplitude-frequency equations andaverage equations are numerically simulated in order to obtainthe influence of system parameters on the nonlinear charac-teristics of vibration-e following conclusions can be obtained

(1) -e results show that there are many similar char-acteristics in the nonlinear response of the uncoupledtwo-order modes which are excited separately whensubharmonic resonance occurs Under the sameamplitude of external excitation the amplitudes of thetwo-order modes increase as the excitation frequencyincreases along with both the subharmonic domainsFor the same excitation frequency the steady-statesolutions corresponding to the large amplitudes arealmost independent of the external excitation -edifference is that the steady-state solution with smalleramplitude of the first-order mode increases with theincrease in the excitation amplitude while that of thesecond-order mode decreases with the increase in theexcitation amplitude

(2) From the bifurcation diagram it can be concluded thatthe systemmay experiencemany kinds of vibration stateswith the change in the amplitude of external excitationsuch as quasiperiodic periodic and chaotic motion -evibration of laminated plates can be controlled byadjusting the amplitude of the external excitation

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interest

Acknowledgments

-e authors sincerely acknowledge the financial support ofthe National Natural Science Foundation of China (Grantsnos 11862020 11962020 and 11402126) and the InnerMongolia Natural Science Foundation (Grants nos2018LH01014 and 2019MS05065)

References

[1] M Wang Z-M Li and P Qiao ldquoVibration analysis ofsandwich plates with carbon nanotube-reinforced compositeface-sheetsrdquo Composite Structures vol 200 pp 799ndash8092018

[2] S Razavi and A Shooshtari ldquoNonlinear free vibration ofmagneto-electro-elastic rectangular platesrdquo CompositeStructures vol 119 pp 377ndash384 2015

[3] M Sadri and D Younesian ldquoNonlinear harmonic vibrationanalysis of a plate-cavity systemrdquo Nonlinear Dynamicsvol 74 no 4 pp 1267ndash1279 2013

[4] D Aranda-Iiglesias J A Rodrıguez-Martınez andM B Rubin ldquoNonlinear axisymmetric vibrations of ahyperelastic orthotropic cylinderrdquo International Journal ofNon Linear Mechanics vol 99 pp 131ndash143 2018

[5] J Torabi and R Ansari ldquoNonlinear free vibration analysis ofthermally induced FG-CNTRC annular plates asymmetricversus axisymmetric studyrdquo Computer Methods in AppliedMechanics and Engineering vol 324 pp 327ndash347 2017

[6] P Ribeiro and M Petyt ldquoNon-linear free vibration of iso-tropic plates with internal resonancerdquo International Journal ofNon-linear Mechanics vol 35 no 2 pp 263ndash278 2000

[7] H Asadi M Bodaghi M Shakeri and M M AghdamldquoNonlinear dynamics of SMA-fiber-reinforced compositebeams subjected to a primarysecondary-resonance excita-tionrdquo Acta Mechanica vol 226 no 2 pp 437ndash455 2015

[8] Y Zhang and Y Li ldquoNonlinear dynamic analysis of a doublecurvature honeycomb sandwich shell with simply supported

ndash4

ndash2

0

2

4

0ndash2 2 4ndash4x3

x 4

(a)

ndash4

ndash2

0

2

4

x 3

12 13 14 1511t

(b)

Figure 14 Phase portrait on plane (x3 x4) and the waveforms on the plane (t x3) of the second-order mode of the system with chaoticmotion when P1 52

Shock and Vibration 15

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration

boundaries by the homotopy analysis methodrdquo CompositeStructures vol 221 p 110884 2019

[9] J N ReddyMechanics of Laminated Composite Plates 6eoryand Analysis CRC Press Boca Raton FL USA 2004

[10] Y-C Lin and C-C Ma ldquoResonant vibration of piezoceramicplates in fluidrdquo Interaction and Multiscale Mechanics vol 1no 2 pp 177ndash190 2008

[11] I D Breslavsky M Amabili and M Legrand ldquoNonlinearvibrations of thin hyperelastic platesrdquo Journal of Sound andVibration vol 333 no 19 pp 4668ndash4681 2014

[12] A A Khdeir and J N Reddy ldquoOn the forced motions ofantisymmetric cross-ply laminated platesrdquo InternationalJournal of Mechanical Sciences vol 31 no 7 pp 499ndash5101989

[13] P Litewka and R Lewandowski ldquoNonlinear harmonicallyexcited vibrations of plates with zener materialrdquo NonlinearDynamics vol 89 pp 1ndash22 2017

[14] H Eslami and O A Kandil ldquoTwo-mode nonlinear vibrationof orthotropic plates using method of multiple scalesrdquo AiaaJournal vol 27 pp 961ndash967 2012

[15] M Amabili ldquoNonlinear damping in nonlinear vibrations ofrectangular plates derivation from viscoelasticity and ex-perimental validationrdquo Journal of the Mechanics and Physicsof Solids vol 118 pp 275ndash292 2018

[16] M Delapierre S H Lohaus and S Pellegrino ldquoNonlinearvibration of transversely-loaded spinning membranesrdquoJournal of Sound and Vibration vol 427 pp 41ndash62 2018

[17] S Kumar A Mitra and H Roy ldquoForced vibration response ofaxially functionally graded non-uniform plates consideringgeometric nonlinearityrdquo International Journal of MechanicalSciences vol 128-129 pp 194ndash205 2017

[18] C-S Chen C-P Fung and R-D Chien ldquoA further study onnonlinear vibration of initially stressed platesrdquo AppliedMathematics and Computation vol 172 no 1 pp 349ndash3672006

[19] P Balasubramanian G Ferrari M Amabili and Z J G N delPrado ldquoExperimental and theoretical study on large ampli-tude vibrations of clamped rubber platesrdquo InternationalJournal of Non-linear Mechanics vol 94 pp 36ndash45 2017

[20] W Zhang Y X Hao and J Yang ldquoNonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edgesrdquoComposite Structures vol 94 no 3 pp 1075ndash1086 2012

[21] J A Bennett ldquoNonlinear vibration of simply supported angleply laminated platesrdquo AIAA Journal vol 9 no 10pp 1997ndash2003 1971

[22] M Rafiee M Mohammadi B Sobhani Aragh andH Yaghoobi ldquoNonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectricfunctionally graded laminated composite shellsrdquo CompositeStructures vol 103 pp 188ndash196 2013

[23] S C Kattimani ldquoGeometrically nonlinear vibration analysisof multiferroic composite plates and shellsrdquo CompositeStructures vol 163 pp 185ndash194 2017

[24] N D Duc P H Cong and V D Quang ldquoNonlinear dynamicand vibration analysis of piezoelectric eccentrically stiffenedFGM plates in thermal environmentrdquo International Journal ofMechanical Sciences vol 115-116 pp 711ndash722 2016

[25] NMohamedM A Eltaher S AMohamed and L F SeddekldquoNumerical analysis of nonlinear free and forced vibrations ofbuckled curved beams resting on nonlinear elastic founda-tionsrdquo International Journal of Non-linear Mechanicsvol 101 pp 157ndash173 2018

[26] D S Cho J-H Kim T M Choi B H Kim and N VladimirldquoFree and forced vibration analysis of arbitrarily supported

rectangular plate systems with attachments and openingsrdquoEngineering Structures vol 171 pp 1036ndash1046 2018

[27] A H Nayfeh and D T Mook Nonlinear Oscillations WileyHoboken NJ USA 1979

[28] W Zhang and M H Zhao ldquoNonlinear vibrations of acomposite laminated cantilever rectangular plate with one-to-one internal resonancerdquo Nonlinear Dynamics vol 70 no 1pp 295ndash313 2012

[29] Y F Zhang W Zhang and Z G Yao ldquoAnalysis on nonlinearvibrations near internal resonances of a composite laminatedpiezoelectric rectangular platerdquo Engineering Structuresvol 173 pp 89ndash106 2018

[30] X Y Guo and W Zhang ldquoNonlinear vibrations of a rein-forced composite plate with carbon nanotubesrdquo CompositeStructures vol 135 pp 96ndash108 2016

[31] S I Chang J M Lee A K Bajaj and C M KrousgrillldquoSubharmonic responses in harmonically excited rectangularplates with one-to-one internal resonancerdquo Chaos Solitons ampFractals vol 8 no 4 pp 479ndash498 1997

[32] D-B Zhang Y-Q Tang H Ding and L-Q Chen ldquoPara-metric and internal resonance of a transporting plate with avarying tensionrdquo Nonlinear Dynamics vol 98 no 4pp 2491ndash2508 2019

[33] S W Yang W Zhang Y X Hao and Y Niu ldquoNonlinearvibrations of FGM truncated conical shell under aerody-namics and in-plane force along meridian near internalresonancesrdquo 6in-walled Structures vol 142 pp 369ndash3912019

[34] Y D Hu and J Li ldquo-e magneto-elastic subharmonic res-onance of current-conducting thin plate in magnetic fieldrdquoJournal of Sound amp Vibration vol 319 pp 1107ndash1120 2009

[35] F-M Li and G Yao ldquo13 Subharmonic resonance of anonlinear composite laminated cylindrical shell in subsonicair flowrdquo Composite Structures vol 100 pp 249ndash256 2013

[36] E Jomehzadeh A R Saidi Z Jomehzadeh et al ldquoNonlinearsubharmonic oscillation of orthotropic graphene-matrixcompositerdquo Computational Materials Science vol 99pp 164ndash172 2015

[37] M R Permoon H Haddadpour and M Javadi ldquoNonlinearvibration of fractional viscoelastic plate primary sub-harmonic and superharmonic responserdquo InternationalJournal of Non-linear Mechanics vol 99 pp 154ndash164 2018

[38] H Ahmadi and K Foroutan ldquoNonlinear vibration of stiffenedmultilayer FG cylindrical shells with spiral stiffeners rested ondamping and elastic foundation in thermal environmentrdquo6in-Walled Structures vol 145 pp 106ndash116 2019

[39] S M Hosseini A Shooshtari H Kalhori andS N Mahmoodi ldquoNonlinear-forced vibrations of piezo-electrically actuated viscoelastic cantileversrdquo Nonlinear Dy-namics vol 78 no 1 pp 571ndash583 2014

[40] J Naprstek and C Fischer ldquoSuper and sub-harmonic syn-chronization in generalized van der Pol oscillatorrdquo Computersamp Structures vol 224 Article ID 106103 2019

16 Shock and Vibration