Research ArticleNonlinear Vibrations of Laminated Cross-Ply CompositeCantilever Plate in Subsonic Air Flow

Gen Liu Wei Zhang and An Xi

Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures College of Mechanical EngineeringBeijing University of Technology Beijing 100124 China

Correspondence should be addressed to Wei Zhang sandyzhang0yahoocom

Received 31 July 2019 Revised 27 December 2019 Accepted 28 February 2020 Published 22 April 2020

Academic Editor Fotios Georgiades

Copyright copy 2020Gen Liu et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e nonlinear vibrations and responses of a laminated composite cantilever plate under the subsonic air flow are investigated inthis paper e subsonic air flow around the three-dimensional cantilever rectangle laminated composite plate is considered to bedecreasing from the wing root to the wing tip According to the ideal incompressible fluid flow condition and the Kut-tandashJoukowski lift theorem the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using theVortex Lattice (VL) method e finite length flat wing is modeled as a laminated composite cantilever plate based on Reddyrsquosthird-order shear deformation plate theory and the von Karman geometry nonlinearity is introduced e nonlinear partialdifferential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamicforce are established via Hamiltonrsquos principle e Galerkin method is used to separate the partial differential equations into twononlinear ordinary differential equations and the four-dimensional nonlinear averaged equations are obtained by the multiplescale method rough comparing the natural frequencies of the linear system with different material and geometric parametersthe relationship of 1 2 internal resonance is considered Corresponding to several selected parameters the frequency-responsecurves are obtained e hardening-spring-type behaviors and jump phenomena are exhibited e influence of the force ex-citation on the bifurcations and chaotic behaviors of the laminated composite cantilever plate is investigated numerically It isfound that the system is sensitive to the exciting force according to the complicate nonlinear behaviors exhibited in this paper

1 Introduction

e vibration of the plate and shell structures owing to airflow is a matter of interest because of its significance indesign of launch vehicles and aircrafts [1] Laminatedcomposite structures are widely used in aerospace field dueto its high strength-to-weight ratio light weight and longfatigue life e dynamic behavior of the laminated com-posite plates in air flow has gained significant focus of at-tention for many researchers However there are fewresearch works dealing with the complex nonlinear dy-namics of the structure which is simplified as a laminatedcomposite cantilever plate subjected to subsonic air flowerefore the nonlinear dynamics of laminated compositeplates in subsonic flow will be worth analyzing in this work

In the 1990s the research on the vibration of plates hasbeen studied comprehensively Some literature reviews on

nonlinear vibrations of plates were given by Chia [2 3] andMehar and Panda [4] e nonlinear vibrations of laminatedcomposite spherical shell panels were also entirely investi-gated by Mahapatra et al [5ndash9] e vibration bending andbuckling behaviors about the functionally graded sandwichstructure have been investigated by Mehar et al [10ndash13] andKar and Panda [14ndash16] A third-order theory whichaccounted for a cubic variation of the in-plane displacementsthrough the plate thickness was derived by Librescu andReddy [17] Great progress has also been achieved in thestudy of the nonlinear thermoelastic frequency analysis[18ndash22] Zhang [23] studied the global bifurcations andchaotic dynamics of simply supported rectangular thinplates under parametrical excitation by introducing vonKarmanrsquos geometric nonlinearity plate theory

Nonlinear analysis considering the geometrical andmaterial nonlinearity by FEM was conducted by Panda and

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4601672 19 pageshttpsdoiorg10115520204601672

Singh [24ndash26] Onkar and Yadav [27] studied the nonlinearrandom vibrations of a simply supported cross-ply lami-nated composite plate using analytical methods and theaccuracy of response evaluation was improved in this workPark et al [28] investigated the nonlinear forced vibrationsof skew sandwich plates subjected to multiple of dynamicloads the influence of the skew angles boundary conditionsand loads on the nonlinear dynamic behaviors of compositeplates which were discussed Chien and Chen [29] studiedthe effects of initial stresses and different parameters on thenonlinear vibrations of laminated composite plates on elasticfoundations Singh et al [30] used the higher-order sheardeformation plate theory to study the dynamic responses ofa geometrically nonlinear laminated composite plate lyingon different elastic foundations Zhang et al [31 32] in-vestigated nonlinear vibration bifurcation and chaoticdynamics of laminated composite plates Alijani et al[33 34] studied nonlinear vibrations of FGM plates andshallow shells and the bifurcations and nonlinear dynamicbehaviors were analyzed in their papers Amabili [35] usedthe higher-order shear deformation theory to study thenonlinear vibrations of angle-ply laminated circular cylin-drical shells Most recently Zhang et al [36ndash39] studied thenonlinear vibrations of the laminated circular cylindricalshells based on the deployed ring truss for antenna struc-tures e complex nonlinear dynamics of the circular cy-lindrical shell were studied and the bifurcations and chaoticbehaviors were investigated by using the analytical methodsand numerical simulations

e research on nonlinear vibrations of plates andshells excited by the aerodynamic force were also widelyinvestigated by researchers In the early years Dowell[40 41] studied the nonlinear oscillations of the flutteringplate systematically Literature reviews on the nonlinearvibrations of circular cylindrical shells and panels with andwithout fluid-structure interaction were given by Amabiliand Paıumldoussis [42] e studies on the nonlinear vibra-tion of laminated plates in air flow were mostly about theplates in the supersoninc flow Singha and Ganapathi [43]investigated the effect of the system parameters on su-personic panel flutter behaviors of laminated compositeplates Haddadpour et al [44] investigated the nonlinearaeroelastic behaviors of FGM Singha andMandal [45] useda 16-node isoparametric degenerated shell element to studythe supersonic panel flutter behaviors of laminated com-posite plates and cylindrical panels Kuo [46] investigatedthe effect of variable fiber spacing on the supersonic flutterof rectangular composite plates Zhao and Zhang [47]presented the analysis of the nonlinear dynamics for alaminated composite cantilever rectangular plate subjectedto the supersonic gas flows and the in-plane excitations Ananalysis on the nonlinear dynamics of an FGM plate inhypersonic flow subjected to an external excitation anduniform temperature change was presented by Hao et al[48] e nonlinear dynamic behavior of an axially ex-tendable cantilever laminated composite plate using pie-zoelectric materials under the combined action ofaerodynamic load and piezoelectric excitation was studiedby Lu et al [49] Yao and Li [50] investigated the nonlinear

vibration of a two-dimensional laminated composite platein subsonic air flow with simply supported boundaryconditions A simple subsonic aerodynamics model wasintroduced in this work which was used to analyze two-dimensional infinite length plate

In this paper the nonlinear dynamics of the laminatedcomposite cantilever plate under subsonic air flow were in-vestigated According to the flow condition of ideal incom-pressible fluid and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensional finitelength flat wing was calculated using the Vortex Lattice (VL)method e nonlinear partial differential governing equationsof motion for the laminated composite cantilever plate sub-jected to the subsonic aerodynamic force were established viaHamiltonrsquos principle e Galerkin method was used to sep-arate the partial differential equation into two nonlinear or-dinary differential equations e numerical method wasutilized to investigate the bifurcations and periodic and chaoticmotions of the composite laminated rectangular plate enumerical results illustrate that there existed the periodic andchaotic motions of the composite laminated cantilever platee hardening-spring characteristics of the composite lami-nated cantilever plate were demonstrated by the frequency-response curves of the system

2 Derivation of the Subsonic AerodynamicForce on the Plate

21 Vortex Lattice (VL)Method In this section the subsonicair flow on the plate will be calculated by using the VLmethod e VL method is a numerical implementation onthe general 3D lifting surface problem is method dis-cretizes the vortex-sheet strength distributing on each liftingsurface by lumping it into a collection of horseshoe vorticesFigure 1 shows that a 3D lifting surface is discretized by avortex lattice of horseshoe vortices in which all contribute tothe velocity V at any field point r

For the finite wingspan the high-pressure air below thewing will turn over the low pressure air at the tip of the wingwhich will cause the pressure on the upper surface of thewing tip to be equal to that of the lower surface Unlike thetwo-dimensional flow around the airfoil the main charac-teristic of the three-dimensional flow around the wing is thevariation of the lift along the wingspan In order to calculatethe lift on the wing surface by using the vortex latticemethod the BiotndashSavart law is used to calculate the inducedvelocity on the control point e vortex strength of thevortex system is obtained and the pressure difference on theupper and lower surface of the wing surface is deduced at thesame time

e velocity induced by a vortex line with a strength of Γnand a length of dl is calculated by the BiotndashSavart law as

dV Γn(dl times r)

4πr3 (1)

As shown in Figure 2 the induced velocity is

dV Γn sin θdl

4πr2 (2)

2 Mathematical Problems in Engineering

Let AB represent a vortex segment in which the vortexvector points from A to B C is a space point and its normaldistance from the AB line is rp

We can make an integral from the point A to the point B

to find out the size of the induced velocity

V Γn

4πrp

1113946θ2

θ1sin θdθ

Γn4πrp

cos θ1 minus cos θ2( 1113857 (3)

As r0 r1and r2 respectively represent the AB AC andBC in Figure 2 the following relationships are established

rp r1 times r2

11138681113868111386811138681113868111386811138681113868

r0

cos θ1 r0 middot r1r0r1

cos θ2 r0 middot r2r0r2

(4)

en the formula calculating the induced velocitygenerated by the horseshoe vortex in the vortex latticemethod is derived

V Γn4π

r1 times r2r1 times r2

11138681113868111386811138681113868111386811138681113868

r0 middotr1r1

minusr2r2

1113888 11138891113890 1113891 (5)

From equation (5) the velocity induced by a vortexsegment AB at any point in the coordinate (x y z) can beobtained

VAB Γn4π

Fac1AB1113864 1113865 middot Fac2AB1113864 1113865 (6a)

VAinfin Γn4π

z minus z1n( 1113857j + y minus y1n( 1113857kz minus z1n( 1113857

2+ y minus y1n( 1113857

21113960 1113961

⎧⎨

⎩

⎫⎬

⎭

times 10 +x minus x1n

x minus x1n( 11138572

+ y minus y1n( 11138572

+ z minus z1n( 11138572

⎡⎣ ⎤⎦

(6b)

VBinfin minusΓn4π

z minus z2n( 1113857j + y1n minus y( 1113857kz minus z1n( 1113857

2+ y2n minus y( 1113857

21113960 1113961

⎧⎨

⎩

⎫⎬

⎭

times 10 +x minus x2n

x minus x2n( 11138572

+ y minus y2n( 11138572

+ z minus z2n( 11138572

⎡⎣ ⎤⎦

(6c)

where VAinfin is the velocity induced by vortex lines from thepoint A to infin along the x-axis and VBinfin is the velocityinduced by vortex lines from the point B to infin along thex-axis

e total velocity induced by a horseshoe vortex at apoint (x y z) representing a surface element (ie the nthpanel element) is the sum of the various components cal-culated by equations (6a)ndash(6c) erefore the velocity in-duced by 2N vortexes is obtained to get the total inducedvelocity on the first m control points It is expressed as

Vm 11139442N

n1CmnΓn (7)

e contribution of all vortices to the downwash of thecontrol point for the mth panel element

wm 11139442N

n1wmn (8)

en the lift on the rectangular cantilever plate with asweep rectangular wing surface is computed according to theVL method e vortex system is arranged on the rectan-gular wing surface in Figure 3

22 Application of VL Method on the Plate In Figure 3 thevariable b is twice the span of the wing c is the taper ratio ofthe wing surface and c is equal to 1 e sweep angle is 0∘Five panel elements are specified on the wing surface andthe vortex line is arranged at the leading edge 14 chord andthe control point is located at 34 chord e wing surface isdivided into 5 panel elements and each panel extends fromthe leading edge to the trailing edge

e coordinates of the 5 times 1 vortices on the wing surfaceare listed in Table 1

e downwash velocity of each surface element inducedat the control point is superimposed through the non-penetrating condition

wm minus Vinfinα (9)

Control points

Field point V(r)

Vinfin U

Ω

X

rY

Z

Figure 1 A 3D lifting surface discretized by a vortex lattice ofhorseshoe vortices

A

dl

C

θ2

θ

θ1

B

Vortex vector

r0

r2

r1

rp

Figure 2 e velocity induced by finite length vortex segments

Mathematical Problems in Engineering 3

e solution of vortex strength is obtained by using 5algebraic equations with unknown vortex strength

Γ1 010082 4πbVinfinα( 1113857 (10a)

Γ2 013180 4πbVinfinα( 1113857 (10b)

Γ3 013901 4πbVinfinα( 1113857 (10c)

Γ4 012496 4πbVinfinα( 1113857 (10d)

Γ5 009887 4πbVinfinα( 1113857 (10e)

According to the boundary conditions the air flow istangent to the object surface at each control point to de-termine the strength of each vortex e lift of the wing canbe calculated by satisfying the boundary conditions For anywing that does not have an upper counter angle the lift isproduced by the free flow that crosses the vortex line as thereis no side-washing speed or postwash speed

According to a finite number of elements the lift on theplate is expressed as

L0 ρinfinVinfin1113944

5

n1ΓnΔyn (11)

As the geometric relationship of each facet is Δyn 01bthe lift on the flat plate can be rewritten as

L0 ρinfinV2infinπb

2α(0238184) (12)

where ρinfin is the flow density Vinfin is the flow velocity b is halflength of the wingspan and α is the attack angle

e attack angle is considered to be affected by a periodicdisturbance α α0 + α1 cosΩ2t Taking the periodic per-turbation into the aerodynamic force and according toequation (12) the expression of the aerodynamic forcecontaining the perturbation term is obtained

L L0 + L1 cosΩ2t (13)

where L0 ρinfinV2infinπb2α0(0238184) and

L0 ρinfinV2infinπb2α1(0238184)

3 Formulation

In this section the dynamic equations of the laminatedcomposite cantilever plate are derived As shown in Figure 4the parameters x y and z are respectively the spanwisedirection the direction of the chord and the vertical di-rection of the plate e plate is clamped at the positionx 0 e ply stacking sequence is [0∘90∘]S and the layernumber is N e in-plane excitation is F F0 + F1 cosΩ1twhich is distributed along the chord direction of the platee vertical direction of the plate along the spanwise di-rection is subject to subsonic aerodynamic force L

e nonlinear governing equations are established in theCartesian coordinate system Reddyrsquos third-order sheardeformation plate theory is used

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

zw0

zx1113888 1113889

(14a)

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

zw0

zy1113888 1113889

(14b)

w(x y t) w0(x(t) y t) (14c)

e von Karman nonlinear strain-displacement relationis introduced e displacements and strain-displacementrelation are given as follows

εxx zu0

zx+12

zw0

zx1113888 1113889

2

εyy zv0

zy+12

zw0

zy1113888 1113889

2

(15a)

cxy 12

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy1113888 1113889

cyz 12

zv0

zz+

zw0

zy1113888 1113889

czx 12

zu0

zz+

zw0

zx1113888 1113889

(15b)

Table 1 e coordinates of the attachment vortex and the controlpoint in the wing surface

Panel elementnumber xm ym x1n y1n x2n y2n

1 025b 005b 0833b 0b 0833b 01b2 025b 015b 0833b 01b 0833b 02b3 025b 025b 0833b 02b 0833b 015b4 025b 035b 0833b 015b 0833b 04b5 025b 045b 0833b 04b 0833b 05b

Panel element

Control point

05b

c4

3c4

x

Oy

Figure 3 Layout of the vortex lattice on the flat wing surface

4 Mathematical Problems in Engineering

Substituting the expressions of the strain into the dis-placements we can obtain

εxx

εyy

cxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z

ε(1)x

ε(1)y

c(1)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z3

ε(3)x

ε(3)y

c(3)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(16a)

cyz

czx

1113896 1113897 c(0)

yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭ + z2 c(2)

yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ (16b)

where

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(1)xx

ε(1)yy

c(1)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(3)xx

ε(3)yy

c(3)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

minus c1

zϕx

zx+

z2w0

zx2

zϕy

zy+

z2w0

zy2

zϕx

zy+

zϕy

zx+ 2

z2w0

zxzy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

c(0)yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

c(2)yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ minus c2

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(18)

where

c1 43h2

c2 3c1

(19)

e symmetric cross-ply laminated composite plate isadopted and we can obtain

σxx

σyy

σyz

σzx

σxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q44 0 0

0 0 0 Q55 0

0 0 0 0 Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

εxx

εyy

cyz

czx

cxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)

whereQ11

Q12

Q22

Q16

Q26

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

C4 2C2S2 S2 4C2S2

C2S2 C4 + S4 C2S2 minus 4C2S2

S4 2C2S2 C4 4C2S2

C3S CS3 minus C3S minus CS3 minus 2CS C2 minus S2( 1113857

CS3 C3S minus CS3 minus C3S 2CS C2 minus S2( 1113857

C2S2 minus 2C2S2 C2S2 C2 minus S2( 11138572

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11

Q12

Q22

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(21a)

Q44

Q45

Q55

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

C2 S2

minus CS CS

S2 C2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Q44

Q551113896 1113897 (21b)

C cos θ

S sin θ(22)

where θ is the angle of laminated layer e stiffness of eachlayer is

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q44 Q55 Q66 G12

(23)

where Ei is the Young modulus G12 is the shear modulusand ]12 is Poissonrsquos ratio

According to Hamiltonrsquos principle

1113946T

0(δK minus δU + δW)dt 0 (24)

e nonlinear governing equations of motion are givenas follows

Vinfin

F

bax

y

z

a

L

Figure 4 Mechanical model of cantilever laminated composite plate

Mathematical Problems in Engineering 5

Nxxx + Nxyy I0 eurou0 + I1 minus c1I3( 1113857euroφx minus c1I3z eurow0

zx (25a)

Nyyy + Nxyx I0eurov0 + I1 minus c1I3( 1113857euroφy minus c1I3z eurow0

zy (25b)

Nyyy

zw0

zy+ Nyy

z2w0

zy2 + Nxyx

zw0

zy+ Nxyy

zw0

zx+ Nxxx

zw0

zx

+ Nxx

z2w0

zx2 + c1 Pxxxx + 2Pxyxy + Pyyyy1113872 1113873 + 2Nxy

z2w0

zxzy

+ Qxx minus c2Rxx1113872 1113873

+ Qyy minus c2Ryy1113872 1113873 + L minus c _w0 c1I3zeurou0

zx+

zeurov

zy1113888 1113889

+ c1 I4 minus c1I6( 1113857zeuroφx

zx+

zeuroφy

zy) + I0 eurow0 minus c

21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 11138891113888

(25c)

Mxxx + Mxyy minus c1Pxxx minus c1Pxyy minus Qx minus c2Rx( 1113857

I1 minus c1I3( 1113857eurou0 + I2 minus 2c1I4 + c21I61113872 1113873euroφx minus c1 I4 minus c1I6( 1113857

z eurow0

zx

(25d)

Myyy + Mxyx minus c1Pyyy minus c1Pxyx minus Qy minus c2Ry1113872 1113873

I1 minus c1I3( 1113857eurov0 + I2 minus 2c1I4 + c21I61113872 1113873euroφy minus c1 I4 minus c1I6( 1113857

z eurow0

zy

(25e)

where c is the damping coefficiente internal force is expressed as follows

Nαβ

Mαβ

Pαβ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

1113946(h2)

minus(h2)σαβ

1

z

z3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dz

Qα

Rα

⎧⎨

⎩

⎫⎬

⎭ 1113946(h2)

minus(h2)σαz

1

z2

⎧⎨

⎩

⎫⎬

⎭dz

(26)

Substituting the internal force of equation (26) intoequations (25a)ndash(25e) we can obtain the expression of thegoverning equation of motion by the displacements whichare given in Appendix

e boundary conditions of the cantilever plate areobtained at the same time as

x 0 w v u φy φx 0 (27)

x a Nxx Nxy Mxx Mxy minus c1Pxy Qx 0

(28a)

y 0 Nxy Nyy Myy Mxyminus c1Pxy Qy 0 (28b)

y b Nxy Nyy Myy Mxy minus c1Pxy Qy 0

(28c)

Qx Qx +zMxy

zyminus c2Rx + c1

zPxx

zx+

zPxy

zy1113888 1113889 (28d)

Qy Qy +zMxy

zxminus c2Ry + c1

zPyy

zy+

zPxy

zx1113888 1113889 (28e)

e dimensionless equations can be obtained by in-troducing the following parameters

w w0

h

t tπ2E

abρ1113888 1113889

(12)

Aij Aij(ab)(12)

Eh2

Bij (ab)(12)

Eh3 Bij

Dij (ab)(12)

Eh4 Dij

Eij (ab)(12)

Eh5 Eij

Fij (ab)(12)

Eh6 Fij

Hij (ab)(12)

Eh8 Hij

Ii 1

(ab)(i+12)ρIi

c (ab)2

π2h4(ρE)(12)c

x x

a

y b

y

Ωi 1π2

E

abρ1113888 1113889

(12)

Ωi

F b2

Eh3 F

(29)

4 Frequency Analysis

High-dimensional nonlinear dynamic systems containseveral types of the internal resonant cases which can lead todifferent forms of the nonlinear vibrations When the systemexists a special internal resonant relationship between twolinear modes the large amplitude nonlinear responses may

6 Mathematical Problems in Engineering

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Let AB represent a vortex segment in which the vortexvector points from A to B C is a space point and its normaldistance from the AB line is rp

We can make an integral from the point A to the point B

to find out the size of the induced velocity

V Γn

4πrp

1113946θ2

θ1sin θdθ

Γn4πrp

cos θ1 minus cos θ2( 1113857 (3)

As r0 r1and r2 respectively represent the AB AC andBC in Figure 2 the following relationships are established

rp r1 times r2

11138681113868111386811138681113868111386811138681113868

r0

cos θ1 r0 middot r1r0r1

cos θ2 r0 middot r2r0r2

(4)

en the formula calculating the induced velocitygenerated by the horseshoe vortex in the vortex latticemethod is derived

V Γn4π

r1 times r2r1 times r2

11138681113868111386811138681113868111386811138681113868

r0 middotr1r1

minusr2r2

1113888 11138891113890 1113891 (5)

From equation (5) the velocity induced by a vortexsegment AB at any point in the coordinate (x y z) can beobtained

VAB Γn4π

Fac1AB1113864 1113865 middot Fac2AB1113864 1113865 (6a)

VAinfin Γn4π

z minus z1n( 1113857j + y minus y1n( 1113857kz minus z1n( 1113857

2+ y minus y1n( 1113857

21113960 1113961

⎧⎨

⎩

⎫⎬

⎭

times 10 +x minus x1n

x minus x1n( 11138572

+ y minus y1n( 11138572

+ z minus z1n( 11138572

⎡⎣ ⎤⎦

(6b)

VBinfin minusΓn4π

z minus z2n( 1113857j + y1n minus y( 1113857kz minus z1n( 1113857

2+ y2n minus y( 1113857

21113960 1113961

⎧⎨

⎩

⎫⎬

⎭

times 10 +x minus x2n

x minus x2n( 11138572

+ y minus y2n( 11138572

+ z minus z2n( 11138572

⎡⎣ ⎤⎦

(6c)

where VAinfin is the velocity induced by vortex lines from thepoint A to infin along the x-axis and VBinfin is the velocityinduced by vortex lines from the point B to infin along thex-axis

e total velocity induced by a horseshoe vortex at apoint (x y z) representing a surface element (ie the nthpanel element) is the sum of the various components cal-culated by equations (6a)ndash(6c) erefore the velocity in-duced by 2N vortexes is obtained to get the total inducedvelocity on the first m control points It is expressed as

Vm 11139442N

n1CmnΓn (7)

e contribution of all vortices to the downwash of thecontrol point for the mth panel element

wm 11139442N

n1wmn (8)

en the lift on the rectangular cantilever plate with asweep rectangular wing surface is computed according to theVL method e vortex system is arranged on the rectan-gular wing surface in Figure 3

22 Application of VL Method on the Plate In Figure 3 thevariable b is twice the span of the wing c is the taper ratio ofthe wing surface and c is equal to 1 e sweep angle is 0∘Five panel elements are specified on the wing surface andthe vortex line is arranged at the leading edge 14 chord andthe control point is located at 34 chord e wing surface isdivided into 5 panel elements and each panel extends fromthe leading edge to the trailing edge

e coordinates of the 5 times 1 vortices on the wing surfaceare listed in Table 1

e downwash velocity of each surface element inducedat the control point is superimposed through the non-penetrating condition

wm minus Vinfinα (9)

Control points

Field point V(r)

Vinfin U

Ω

X

rY

Z

Figure 1 A 3D lifting surface discretized by a vortex lattice ofhorseshoe vortices

A

dl

C

θ2

θ

θ1

B

Vortex vector

r0

r2

r1

rp

Figure 2 e velocity induced by finite length vortex segments

Mathematical Problems in Engineering 3

e solution of vortex strength is obtained by using 5algebraic equations with unknown vortex strength

Γ1 010082 4πbVinfinα( 1113857 (10a)

Γ2 013180 4πbVinfinα( 1113857 (10b)

Γ3 013901 4πbVinfinα( 1113857 (10c)

Γ4 012496 4πbVinfinα( 1113857 (10d)

Γ5 009887 4πbVinfinα( 1113857 (10e)

According to the boundary conditions the air flow istangent to the object surface at each control point to de-termine the strength of each vortex e lift of the wing canbe calculated by satisfying the boundary conditions For anywing that does not have an upper counter angle the lift isproduced by the free flow that crosses the vortex line as thereis no side-washing speed or postwash speed

According to a finite number of elements the lift on theplate is expressed as

L0 ρinfinVinfin1113944

5

n1ΓnΔyn (11)

As the geometric relationship of each facet is Δyn 01bthe lift on the flat plate can be rewritten as

L0 ρinfinV2infinπb

2α(0238184) (12)

where ρinfin is the flow density Vinfin is the flow velocity b is halflength of the wingspan and α is the attack angle

e attack angle is considered to be affected by a periodicdisturbance α α0 + α1 cosΩ2t Taking the periodic per-turbation into the aerodynamic force and according toequation (12) the expression of the aerodynamic forcecontaining the perturbation term is obtained

L L0 + L1 cosΩ2t (13)

where L0 ρinfinV2infinπb2α0(0238184) and

L0 ρinfinV2infinπb2α1(0238184)

3 Formulation

In this section the dynamic equations of the laminatedcomposite cantilever plate are derived As shown in Figure 4the parameters x y and z are respectively the spanwisedirection the direction of the chord and the vertical di-rection of the plate e plate is clamped at the positionx 0 e ply stacking sequence is [0∘90∘]S and the layernumber is N e in-plane excitation is F F0 + F1 cosΩ1twhich is distributed along the chord direction of the platee vertical direction of the plate along the spanwise di-rection is subject to subsonic aerodynamic force L

e nonlinear governing equations are established in theCartesian coordinate system Reddyrsquos third-order sheardeformation plate theory is used

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

zw0

zx1113888 1113889

(14a)

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

zw0

zy1113888 1113889

(14b)

w(x y t) w0(x(t) y t) (14c)

e von Karman nonlinear strain-displacement relationis introduced e displacements and strain-displacementrelation are given as follows

εxx zu0

zx+12

zw0

zx1113888 1113889

2

εyy zv0

zy+12

zw0

zy1113888 1113889

2

(15a)

cxy 12

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy1113888 1113889

cyz 12

zv0

zz+

zw0

zy1113888 1113889

czx 12

zu0

zz+

zw0

zx1113888 1113889

(15b)

Table 1 e coordinates of the attachment vortex and the controlpoint in the wing surface

Panel elementnumber xm ym x1n y1n x2n y2n

1 025b 005b 0833b 0b 0833b 01b2 025b 015b 0833b 01b 0833b 02b3 025b 025b 0833b 02b 0833b 015b4 025b 035b 0833b 015b 0833b 04b5 025b 045b 0833b 04b 0833b 05b

Panel element

Control point

05b

c4

3c4

x

Oy

Figure 3 Layout of the vortex lattice on the flat wing surface

4 Mathematical Problems in Engineering

Substituting the expressions of the strain into the dis-placements we can obtain

εxx

εyy

cxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z

ε(1)x

ε(1)y

c(1)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z3

ε(3)x

ε(3)y

c(3)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(16a)

cyz

czx

1113896 1113897 c(0)

yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭ + z2 c(2)

yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ (16b)

where

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(1)xx

ε(1)yy

c(1)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(3)xx

ε(3)yy

c(3)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

minus c1

zϕx

zx+

z2w0

zx2

zϕy

zy+

z2w0

zy2

zϕx

zy+

zϕy

zx+ 2

z2w0

zxzy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

c(0)yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

c(2)yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ minus c2

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(18)

where

c1 43h2

c2 3c1

(19)

e symmetric cross-ply laminated composite plate isadopted and we can obtain

σxx

σyy

σyz

σzx

σxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q44 0 0

0 0 0 Q55 0

0 0 0 0 Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

εxx

εyy

cyz

czx

cxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)

whereQ11

Q12

Q22

Q16

Q26

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

C4 2C2S2 S2 4C2S2

C2S2 C4 + S4 C2S2 minus 4C2S2

S4 2C2S2 C4 4C2S2

C3S CS3 minus C3S minus CS3 minus 2CS C2 minus S2( 1113857

CS3 C3S minus CS3 minus C3S 2CS C2 minus S2( 1113857

C2S2 minus 2C2S2 C2S2 C2 minus S2( 11138572

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11

Q12

Q22

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(21a)

Q44

Q45

Q55

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

C2 S2

minus CS CS

S2 C2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Q44

Q551113896 1113897 (21b)

C cos θ

S sin θ(22)

where θ is the angle of laminated layer e stiffness of eachlayer is

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q44 Q55 Q66 G12

(23)

where Ei is the Young modulus G12 is the shear modulusand ]12 is Poissonrsquos ratio

According to Hamiltonrsquos principle

1113946T

0(δK minus δU + δW)dt 0 (24)

e nonlinear governing equations of motion are givenas follows

Vinfin

F

bax

y

z

a

L

Figure 4 Mechanical model of cantilever laminated composite plate

Mathematical Problems in Engineering 5

Nxxx + Nxyy I0 eurou0 + I1 minus c1I3( 1113857euroφx minus c1I3z eurow0

zx (25a)

Nyyy + Nxyx I0eurov0 + I1 minus c1I3( 1113857euroφy minus c1I3z eurow0

zy (25b)

Nyyy

zw0

zy+ Nyy

z2w0

zy2 + Nxyx

zw0

zy+ Nxyy

zw0

zx+ Nxxx

zw0

zx

+ Nxx

z2w0

zx2 + c1 Pxxxx + 2Pxyxy + Pyyyy1113872 1113873 + 2Nxy

z2w0

zxzy

+ Qxx minus c2Rxx1113872 1113873

+ Qyy minus c2Ryy1113872 1113873 + L minus c _w0 c1I3zeurou0

zx+

zeurov

zy1113888 1113889

+ c1 I4 minus c1I6( 1113857zeuroφx

zx+

zeuroφy

zy) + I0 eurow0 minus c

21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 11138891113888

(25c)

Mxxx + Mxyy minus c1Pxxx minus c1Pxyy minus Qx minus c2Rx( 1113857

I1 minus c1I3( 1113857eurou0 + I2 minus 2c1I4 + c21I61113872 1113873euroφx minus c1 I4 minus c1I6( 1113857

z eurow0

zx

(25d)

Myyy + Mxyx minus c1Pyyy minus c1Pxyx minus Qy minus c2Ry1113872 1113873

I1 minus c1I3( 1113857eurov0 + I2 minus 2c1I4 + c21I61113872 1113873euroφy minus c1 I4 minus c1I6( 1113857

z eurow0

zy

(25e)

where c is the damping coefficiente internal force is expressed as follows

Nαβ

Mαβ

Pαβ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

1113946(h2)

minus(h2)σαβ

1

z

z3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dz

Qα

Rα

⎧⎨

⎩

⎫⎬

⎭ 1113946(h2)

minus(h2)σαz

1

z2

⎧⎨

⎩

⎫⎬

⎭dz

(26)

Substituting the internal force of equation (26) intoequations (25a)ndash(25e) we can obtain the expression of thegoverning equation of motion by the displacements whichare given in Appendix

e boundary conditions of the cantilever plate areobtained at the same time as

x 0 w v u φy φx 0 (27)

x a Nxx Nxy Mxx Mxy minus c1Pxy Qx 0

(28a)

y 0 Nxy Nyy Myy Mxyminus c1Pxy Qy 0 (28b)

y b Nxy Nyy Myy Mxy minus c1Pxy Qy 0

(28c)

Qx Qx +zMxy

zyminus c2Rx + c1

zPxx

zx+

zPxy

zy1113888 1113889 (28d)

Qy Qy +zMxy

zxminus c2Ry + c1

zPyy

zy+

zPxy

zx1113888 1113889 (28e)

e dimensionless equations can be obtained by in-troducing the following parameters

w w0

h

t tπ2E

abρ1113888 1113889

(12)

Aij Aij(ab)(12)

Eh2

Bij (ab)(12)

Eh3 Bij

Dij (ab)(12)

Eh4 Dij

Eij (ab)(12)

Eh5 Eij

Fij (ab)(12)

Eh6 Fij

Hij (ab)(12)

Eh8 Hij

Ii 1

(ab)(i+12)ρIi

c (ab)2

π2h4(ρE)(12)c

x x

a

y b

y

Ωi 1π2

E

abρ1113888 1113889

(12)

Ωi

F b2

Eh3 F

(29)

4 Frequency Analysis

High-dimensional nonlinear dynamic systems containseveral types of the internal resonant cases which can lead todifferent forms of the nonlinear vibrations When the systemexists a special internal resonant relationship between twolinear modes the large amplitude nonlinear responses may

6 Mathematical Problems in Engineering

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

e solution of vortex strength is obtained by using 5algebraic equations with unknown vortex strength

Γ1 010082 4πbVinfinα( 1113857 (10a)

Γ2 013180 4πbVinfinα( 1113857 (10b)

Γ3 013901 4πbVinfinα( 1113857 (10c)

Γ4 012496 4πbVinfinα( 1113857 (10d)

Γ5 009887 4πbVinfinα( 1113857 (10e)

According to the boundary conditions the air flow istangent to the object surface at each control point to de-termine the strength of each vortex e lift of the wing canbe calculated by satisfying the boundary conditions For anywing that does not have an upper counter angle the lift isproduced by the free flow that crosses the vortex line as thereis no side-washing speed or postwash speed

According to a finite number of elements the lift on theplate is expressed as

L0 ρinfinVinfin1113944

5

n1ΓnΔyn (11)

As the geometric relationship of each facet is Δyn 01bthe lift on the flat plate can be rewritten as

L0 ρinfinV2infinπb

2α(0238184) (12)

where ρinfin is the flow density Vinfin is the flow velocity b is halflength of the wingspan and α is the attack angle

e attack angle is considered to be affected by a periodicdisturbance α α0 + α1 cosΩ2t Taking the periodic per-turbation into the aerodynamic force and according toequation (12) the expression of the aerodynamic forcecontaining the perturbation term is obtained

L L0 + L1 cosΩ2t (13)

where L0 ρinfinV2infinπb2α0(0238184) and

L0 ρinfinV2infinπb2α1(0238184)

3 Formulation

In this section the dynamic equations of the laminatedcomposite cantilever plate are derived As shown in Figure 4the parameters x y and z are respectively the spanwisedirection the direction of the chord and the vertical di-rection of the plate e plate is clamped at the positionx 0 e ply stacking sequence is [0∘90∘]S and the layernumber is N e in-plane excitation is F F0 + F1 cosΩ1twhich is distributed along the chord direction of the platee vertical direction of the plate along the spanwise di-rection is subject to subsonic aerodynamic force L

e nonlinear governing equations are established in theCartesian coordinate system Reddyrsquos third-order sheardeformation plate theory is used

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

zw0

zx1113888 1113889

(14a)

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

zw0

zy1113888 1113889

(14b)

w(x y t) w0(x(t) y t) (14c)

e von Karman nonlinear strain-displacement relationis introduced e displacements and strain-displacementrelation are given as follows

εxx zu0

zx+12

zw0

zx1113888 1113889

2

εyy zv0

zy+12

zw0

zy1113888 1113889

2

(15a)

cxy 12

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy1113888 1113889

cyz 12

zv0

zz+

zw0

zy1113888 1113889

czx 12

zu0

zz+

zw0

zx1113888 1113889

(15b)

Table 1 e coordinates of the attachment vortex and the controlpoint in the wing surface

Panel elementnumber xm ym x1n y1n x2n y2n

1 025b 005b 0833b 0b 0833b 01b2 025b 015b 0833b 01b 0833b 02b3 025b 025b 0833b 02b 0833b 015b4 025b 035b 0833b 015b 0833b 04b5 025b 045b 0833b 04b 0833b 05b

Panel element

Control point

05b

c4

3c4

x

Oy

Figure 3 Layout of the vortex lattice on the flat wing surface

4 Mathematical Problems in Engineering

Substituting the expressions of the strain into the dis-placements we can obtain

εxx

εyy

cxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z

ε(1)x

ε(1)y

c(1)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z3

ε(3)x

ε(3)y

c(3)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(16a)

cyz

czx

1113896 1113897 c(0)

yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭ + z2 c(2)

yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ (16b)

where

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(1)xx

ε(1)yy

c(1)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(3)xx

ε(3)yy

c(3)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

minus c1

zϕx

zx+

z2w0

zx2

zϕy

zy+

z2w0

zy2

zϕx

zy+

zϕy

zx+ 2

z2w0

zxzy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

c(0)yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

c(2)yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ minus c2

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(18)

where

c1 43h2

c2 3c1

(19)

e symmetric cross-ply laminated composite plate isadopted and we can obtain

σxx

σyy

σyz

σzx

σxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q44 0 0

0 0 0 Q55 0

0 0 0 0 Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

εxx

εyy

cyz

czx

cxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)

whereQ11

Q12

Q22

Q16

Q26

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

C4 2C2S2 S2 4C2S2

C2S2 C4 + S4 C2S2 minus 4C2S2

S4 2C2S2 C4 4C2S2

C3S CS3 minus C3S minus CS3 minus 2CS C2 minus S2( 1113857

CS3 C3S minus CS3 minus C3S 2CS C2 minus S2( 1113857

C2S2 minus 2C2S2 C2S2 C2 minus S2( 11138572

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11

Q12

Q22

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(21a)

Q44

Q45

Q55

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

C2 S2

minus CS CS

S2 C2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Q44

Q551113896 1113897 (21b)

C cos θ

S sin θ(22)

where θ is the angle of laminated layer e stiffness of eachlayer is

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q44 Q55 Q66 G12

(23)

where Ei is the Young modulus G12 is the shear modulusand ]12 is Poissonrsquos ratio

According to Hamiltonrsquos principle

1113946T

0(δK minus δU + δW)dt 0 (24)

e nonlinear governing equations of motion are givenas follows

Vinfin

F

bax

y

z

a

L

Figure 4 Mechanical model of cantilever laminated composite plate

Mathematical Problems in Engineering 5

Nxxx + Nxyy I0 eurou0 + I1 minus c1I3( 1113857euroφx minus c1I3z eurow0

zx (25a)

Nyyy + Nxyx I0eurov0 + I1 minus c1I3( 1113857euroφy minus c1I3z eurow0

zy (25b)

Nyyy

zw0

zy+ Nyy

z2w0

zy2 + Nxyx

zw0

zy+ Nxyy

zw0

zx+ Nxxx

zw0

zx

+ Nxx

z2w0

zx2 + c1 Pxxxx + 2Pxyxy + Pyyyy1113872 1113873 + 2Nxy

z2w0

zxzy

+ Qxx minus c2Rxx1113872 1113873

+ Qyy minus c2Ryy1113872 1113873 + L minus c _w0 c1I3zeurou0

zx+

zeurov

zy1113888 1113889

+ c1 I4 minus c1I6( 1113857zeuroφx

zx+

zeuroφy

zy) + I0 eurow0 minus c

21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 11138891113888

(25c)

Mxxx + Mxyy minus c1Pxxx minus c1Pxyy minus Qx minus c2Rx( 1113857

I1 minus c1I3( 1113857eurou0 + I2 minus 2c1I4 + c21I61113872 1113873euroφx minus c1 I4 minus c1I6( 1113857

z eurow0

zx

(25d)

Myyy + Mxyx minus c1Pyyy minus c1Pxyx minus Qy minus c2Ry1113872 1113873

I1 minus c1I3( 1113857eurov0 + I2 minus 2c1I4 + c21I61113872 1113873euroφy minus c1 I4 minus c1I6( 1113857

z eurow0

zy

(25e)

where c is the damping coefficiente internal force is expressed as follows

Nαβ

Mαβ

Pαβ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

1113946(h2)

minus(h2)σαβ

1

z

z3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dz

Qα

Rα

⎧⎨

⎩

⎫⎬

⎭ 1113946(h2)

minus(h2)σαz

1

z2

⎧⎨

⎩

⎫⎬

⎭dz

(26)

Substituting the internal force of equation (26) intoequations (25a)ndash(25e) we can obtain the expression of thegoverning equation of motion by the displacements whichare given in Appendix

e boundary conditions of the cantilever plate areobtained at the same time as

x 0 w v u φy φx 0 (27)

x a Nxx Nxy Mxx Mxy minus c1Pxy Qx 0

(28a)

y 0 Nxy Nyy Myy Mxyminus c1Pxy Qy 0 (28b)

y b Nxy Nyy Myy Mxy minus c1Pxy Qy 0

(28c)

Qx Qx +zMxy

zyminus c2Rx + c1

zPxx

zx+

zPxy

zy1113888 1113889 (28d)

Qy Qy +zMxy

zxminus c2Ry + c1

zPyy

zy+

zPxy

zx1113888 1113889 (28e)

e dimensionless equations can be obtained by in-troducing the following parameters

w w0

h

t tπ2E

abρ1113888 1113889

(12)

Aij Aij(ab)(12)

Eh2

Bij (ab)(12)

Eh3 Bij

Dij (ab)(12)

Eh4 Dij

Eij (ab)(12)

Eh5 Eij

Fij (ab)(12)

Eh6 Fij

Hij (ab)(12)

Eh8 Hij

Ii 1

(ab)(i+12)ρIi

c (ab)2

π2h4(ρE)(12)c

x x

a

y b

y

Ωi 1π2

E

abρ1113888 1113889

(12)

Ωi

F b2

Eh3 F

(29)

4 Frequency Analysis

High-dimensional nonlinear dynamic systems containseveral types of the internal resonant cases which can lead todifferent forms of the nonlinear vibrations When the systemexists a special internal resonant relationship between twolinear modes the large amplitude nonlinear responses may

6 Mathematical Problems in Engineering

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Substituting the expressions of the strain into the dis-placements we can obtain

εxx

εyy

cxy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z

ε(1)x

ε(1)y

c(1)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+ z3

ε(3)x

ε(3)y

c(3)xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(16a)

cyz

czx

1113896 1113897 c(0)

yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭ + z2 c(2)

yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ (16b)

where

ε(0)x

ε(0)y

c(0)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zu0

zx+12

zw0

zx1113888 1113889

2

zv0

zy+12

zw0

zy1113888 1113889

2

zu0

zy+

zv0

zx+

zw0

zx

zw0

zy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(1)xx

ε(1)yy

c(1)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

zϕx

zx

zϕy

zy

zϕx

zy+

zϕy

zx

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε(3)xx

ε(3)yy

c(3)xy

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

minus c1

zϕx

zx+

z2w0

zx2

zϕy

zy+

z2w0

zy2

zϕx

zy+

zϕy

zx+ 2

z2w0

zxzy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

c(0)yz

c(0)zx

⎧⎨

⎩

⎫⎬

⎭

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

c(2)yz

c(2)zx

⎧⎨

⎩

⎫⎬

⎭ minus c2

ϕy +zw0

zy

ϕx +zw0

zx

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(18)

where

c1 43h2

c2 3c1

(19)

e symmetric cross-ply laminated composite plate isadopted and we can obtain

σxx

σyy

σyz

σzx

σxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q44 0 0

0 0 0 Q55 0

0 0 0 0 Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

εxx

εyy

cyz

czx

cxy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)

whereQ11

Q12

Q22

Q16

Q26

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

C4 2C2S2 S2 4C2S2

C2S2 C4 + S4 C2S2 minus 4C2S2

S4 2C2S2 C4 4C2S2

C3S CS3 minus C3S minus CS3 minus 2CS C2 minus S2( 1113857

CS3 C3S minus CS3 minus C3S 2CS C2 minus S2( 1113857

C2S2 minus 2C2S2 C2S2 C2 minus S2( 11138572

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11

Q12

Q22

Q66

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(21a)

Q44

Q45

Q55

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

C2 S2

minus CS CS

S2 C2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Q44

Q551113896 1113897 (21b)

C cos θ

S sin θ(22)

where θ is the angle of laminated layer e stiffness of eachlayer is

Q11 E1

1 minus ]12]21

Q12 ]12E2

1 minus ]12]21

Q22 E2

1 minus ]12]21

Q44 Q55 Q66 G12

(23)

where Ei is the Young modulus G12 is the shear modulusand ]12 is Poissonrsquos ratio

According to Hamiltonrsquos principle

1113946T

0(δK minus δU + δW)dt 0 (24)

e nonlinear governing equations of motion are givenas follows

Vinfin

F

bax

y

z

a

L

Figure 4 Mechanical model of cantilever laminated composite plate

Mathematical Problems in Engineering 5

Nxxx + Nxyy I0 eurou0 + I1 minus c1I3( 1113857euroφx minus c1I3z eurow0

zx (25a)

Nyyy + Nxyx I0eurov0 + I1 minus c1I3( 1113857euroφy minus c1I3z eurow0

zy (25b)

Nyyy

zw0

zy+ Nyy

z2w0

zy2 + Nxyx

zw0

zy+ Nxyy

zw0

zx+ Nxxx

zw0

zx

+ Nxx

z2w0

zx2 + c1 Pxxxx + 2Pxyxy + Pyyyy1113872 1113873 + 2Nxy

z2w0

zxzy

+ Qxx minus c2Rxx1113872 1113873

+ Qyy minus c2Ryy1113872 1113873 + L minus c _w0 c1I3zeurou0

zx+

zeurov

zy1113888 1113889

+ c1 I4 minus c1I6( 1113857zeuroφx

zx+

zeuroφy

zy) + I0 eurow0 minus c

21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 11138891113888

(25c)

Mxxx + Mxyy minus c1Pxxx minus c1Pxyy minus Qx minus c2Rx( 1113857

I1 minus c1I3( 1113857eurou0 + I2 minus 2c1I4 + c21I61113872 1113873euroφx minus c1 I4 minus c1I6( 1113857

z eurow0

zx

(25d)

Myyy + Mxyx minus c1Pyyy minus c1Pxyx minus Qy minus c2Ry1113872 1113873

I1 minus c1I3( 1113857eurov0 + I2 minus 2c1I4 + c21I61113872 1113873euroφy minus c1 I4 minus c1I6( 1113857

z eurow0

zy

(25e)

where c is the damping coefficiente internal force is expressed as follows

Nαβ

Mαβ

Pαβ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

1113946(h2)

minus(h2)σαβ

1

z

z3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dz

Qα

Rα

⎧⎨

⎩

⎫⎬

⎭ 1113946(h2)

minus(h2)σαz

1

z2

⎧⎨

⎩

⎫⎬

⎭dz

(26)

Substituting the internal force of equation (26) intoequations (25a)ndash(25e) we can obtain the expression of thegoverning equation of motion by the displacements whichare given in Appendix

e boundary conditions of the cantilever plate areobtained at the same time as

x 0 w v u φy φx 0 (27)

x a Nxx Nxy Mxx Mxy minus c1Pxy Qx 0

(28a)

y 0 Nxy Nyy Myy Mxyminus c1Pxy Qy 0 (28b)

y b Nxy Nyy Myy Mxy minus c1Pxy Qy 0

(28c)

Qx Qx +zMxy

zyminus c2Rx + c1

zPxx

zx+

zPxy

zy1113888 1113889 (28d)

Qy Qy +zMxy

zxminus c2Ry + c1

zPyy

zy+

zPxy

zx1113888 1113889 (28e)

e dimensionless equations can be obtained by in-troducing the following parameters

w w0

h

t tπ2E

abρ1113888 1113889

(12)

Aij Aij(ab)(12)

Eh2

Bij (ab)(12)

Eh3 Bij

Dij (ab)(12)

Eh4 Dij

Eij (ab)(12)

Eh5 Eij

Fij (ab)(12)

Eh6 Fij

Hij (ab)(12)

Eh8 Hij

Ii 1

(ab)(i+12)ρIi

c (ab)2

π2h4(ρE)(12)c

x x

a

y b

y

Ωi 1π2

E

abρ1113888 1113889

(12)

Ωi

F b2

Eh3 F

(29)

4 Frequency Analysis

High-dimensional nonlinear dynamic systems containseveral types of the internal resonant cases which can lead todifferent forms of the nonlinear vibrations When the systemexists a special internal resonant relationship between twolinear modes the large amplitude nonlinear responses may

6 Mathematical Problems in Engineering

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Nxxx + Nxyy I0 eurou0 + I1 minus c1I3( 1113857euroφx minus c1I3z eurow0

zx (25a)

Nyyy + Nxyx I0eurov0 + I1 minus c1I3( 1113857euroφy minus c1I3z eurow0

zy (25b)

Nyyy

zw0

zy+ Nyy

z2w0

zy2 + Nxyx

zw0

zy+ Nxyy

zw0

zx+ Nxxx

zw0

zx

+ Nxx

z2w0

zx2 + c1 Pxxxx + 2Pxyxy + Pyyyy1113872 1113873 + 2Nxy

z2w0

zxzy

+ Qxx minus c2Rxx1113872 1113873

+ Qyy minus c2Ryy1113872 1113873 + L minus c _w0 c1I3zeurou0

zx+

zeurov

zy1113888 1113889

+ c1 I4 minus c1I6( 1113857zeuroφx

zx+

zeuroφy

zy) + I0 eurow0 minus c

21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 11138891113888

(25c)

Mxxx + Mxyy minus c1Pxxx minus c1Pxyy minus Qx minus c2Rx( 1113857

I1 minus c1I3( 1113857eurou0 + I2 minus 2c1I4 + c21I61113872 1113873euroφx minus c1 I4 minus c1I6( 1113857

z eurow0

zx

(25d)

Myyy + Mxyx minus c1Pyyy minus c1Pxyx minus Qy minus c2Ry1113872 1113873

I1 minus c1I3( 1113857eurov0 + I2 minus 2c1I4 + c21I61113872 1113873euroφy minus c1 I4 minus c1I6( 1113857

z eurow0

zy

(25e)

where c is the damping coefficiente internal force is expressed as follows

Nαβ

Mαβ

Pαβ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

1113946(h2)

minus(h2)σαβ

1

z

z3

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dz

Qα

Rα

⎧⎨

⎩

⎫⎬

⎭ 1113946(h2)

minus(h2)σαz

1

z2

⎧⎨

⎩

⎫⎬

⎭dz

(26)

Substituting the internal force of equation (26) intoequations (25a)ndash(25e) we can obtain the expression of thegoverning equation of motion by the displacements whichare given in Appendix

e boundary conditions of the cantilever plate areobtained at the same time as

x 0 w v u φy φx 0 (27)

x a Nxx Nxy Mxx Mxy minus c1Pxy Qx 0

(28a)

y 0 Nxy Nyy Myy Mxyminus c1Pxy Qy 0 (28b)

y b Nxy Nyy Myy Mxy minus c1Pxy Qy 0

(28c)

Qx Qx +zMxy

zyminus c2Rx + c1

zPxx

zx+

zPxy

zy1113888 1113889 (28d)

Qy Qy +zMxy

zxminus c2Ry + c1

zPyy

zy+

zPxy

zx1113888 1113889 (28e)

e dimensionless equations can be obtained by in-troducing the following parameters

w w0

h

t tπ2E

abρ1113888 1113889

(12)

Aij Aij(ab)(12)

Eh2

Bij (ab)(12)

Eh3 Bij

Dij (ab)(12)

Eh4 Dij

Eij (ab)(12)

Eh5 Eij

Fij (ab)(12)

Eh6 Fij

Hij (ab)(12)

Eh8 Hij

Ii 1

(ab)(i+12)ρIi

c (ab)2

π2h4(ρE)(12)c

x x

a

y b

y

Ωi 1π2

E

abρ1113888 1113889

(12)

Ωi

F b2

Eh3 F

(29)

4 Frequency Analysis

High-dimensional nonlinear dynamic systems containseveral types of the internal resonant cases which can lead todifferent forms of the nonlinear vibrations When the systemexists a special internal resonant relationship between twolinear modes the large amplitude nonlinear responses may

6 Mathematical Problems in Engineering

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

suddenly happen which is dangerous in engineering andshould be avoided

In order to study the internal resonance relationshipbetween two bending modes the finite element model of thecantilever laminated composite plate is established e plystacking sequence is [0∘90∘]S and the material parameters[51] are shown in Table 2

e natural frequencies of bending vibration of lami-nated composite cantilever plates with different span-chordratio and different layer thickness are calculated and theresults are shown in Figures 5ndash8 When the span-chord ratiois 2 1 and single thickness is 6mm the transverse vibrationmodes of the cantilever laminated composite plate withcorresponding frequencies are shown in Figure 9

In frequency analysis the first mode natural frequency ismost important Table 3 is to illustrate the 1st mode naturalfrequency for 6 (thickness values)times 4 (span-chord ratiovalues) from Figures 5ndash8

Based on the results of numerical simulations the firstsix orders natural frequencies of the laminated compositecantilever plate are obtained as shown in Figures 5ndash8 It isobviously observed that there is a proportional relationbetween the two bending modes such as relation 1 1 in areac of Figure 7 relation 1 2 in Figure 8 and relation 1 3 inFigures 5ndash7 We select 1 2 internal resonance relationshipbetween two bending modes and the nonlinear vibrations ofthe laminated composite cantilever plate are investigated inthe following analysis

5 Perturbation Analysis

ediscrete equation is derived by the Galerkin method andthe discrete function adopts the following expression

w0 w(t)1X1(x)Y1(y) + w(t)2X2(x)Y2(y) (30)

where

Xi(x) sinλi

ax minus sinh

λi

ax + αi cosh

λi

ax minus cos

λi

ax1113888 1113889

(31a)

Yj(y) sinβm

by + sinh

βm

by minus αm cosh

βm

by + cos

βm

by1113888 1113889

(31b)

where

cos λia coshλia minus 1 0

cos βmb coshβmb minus 1 0

(32a)

αi cosh λi minus cos λi

sinh λi + sin λi

αm minuscosh βm minus cos βm

sinh βm minus sin βm

(32b)

Similarly the aerodynamic force is discretized by usingthe modal function

L l1X1(x)Y1(y) + l2X2(x)Y2(y) (33)

where l1 and l2 represent the amplitudes of the aerodynamicforces corresponding to the two transvers vibration modesand they contain the perturbed items

Substituting equations (30)ndash(33) into equation (A3) thegoverning differential equations of transverse motion of thesystem are derived as follows

eurow1 + c11 _w1 + ω21w1 + c12f1 cosΩ1tw1 + c13w2 minus c14w

31

minus c15w21w2 minus c16w1w

22 minus c17w

32 c18l1

(34a)

eurow2 + c21 _w2 + ω22w2 + c22f1 cosΩ1tw2 + c23w1 minus c24w

32

minus c25w22w1 minus c26w2w

21 minus c27w

31 c28l2

(34b)

Table 2 Single-layer material setting

Material E11 E22 S12 υT300QY8911 135GPa 88GPa 45GPa 03e single layer thickness is hi(i 1 2 6) and they are specified ash1 1mm h2 2mm h6 6mm

h1

a1

a1 = 61154Hza2 = 19429Hzb1 = 24462Hzb2 = 77716Hz

b1b2

a2

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

0

200

400

600

800

1000

1200

1400

Freq

uenc

y

Span-chord ratio 11

Figure 5 Natural frequencies with different thicknesses under thespan-chord ratio 1 1

Mathematical Problems in Engineering 7

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Next the averaged equations of the system are obtainedby using the multiple scale method and the internal reso-nant relationship 1 2 is considered

ω21

14Ω2 + εσ1

ω22 Ω2 + εσ2

Ω1 Ω2 1

(35)

where ω1 and ω2 are two frequencies of the laminatedcomposite plate and σ1 and σ2 are the tuning parameters

e scale transformations are given as follows

c11⟶ εc11

c12⟶ εc12

c13⟶ εc13

c14⟶ εc14

c15⟶ εc15

c16⟶ εc16

c17⟶ εc17

c18⟶ εc18

c21⟶ εc21

c22⟶ εc22

c23⟶ εc23

c24⟶ εc24

c25⟶ εc25

c26⟶ εc26

c27⟶ εc27

c28⟶ εc28

(36)

e scale transformation can be obtained by introducingequation (36) into equations (34a) and (34b)

eurow1 + εc11 _w1 + ω21w1 + εc12f1 cosΩ1tw1 + εc13w2

minus εc14w31 minus εc15w

21w2 minus εc16w1w

22 minus εc17w

32 εc18l1

(37a)

eurow2 + εc21 _w2 + ω22w2+εc22f1 cosΩ1tw2 + εc23w1

minus εc24w32 minus εc25w

22w1 minus εc26w2w

21 minus εc27w

31 εc28l2

(37b)

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 21

0

100

200

300

400

500

600

Freq

uenc

y

h1

h2

h3

h4

h5

h6

a1

a1 = 11000Hza2 = 34115Hzb1 = 22000Hzb2 = 68229Hz

b1

b2 a2

Figure 6 Natural frequencies with different thicknesses under thespan-chord ratio 2 1

h1

h2

h3

h4

h5

h6

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 31

0

50

100

150

200

250

300

350

Freq

uenc

y

a1

c

a1 = 14839Hza2 = 52916Hzb1 = 89033Hzb2 = 31750Hz

b1

b2a2

Figure 7 Natural frequencies with different thicknesses under thespan-chord ratio 3 1

h1

h2

h3

h4

h5

h6

0

20

40

60

80

100

120

140

160

180

200

Freq

uenc

y

Mode 2 Mode 3 Mode 4 Mode 5 Mode 6Mode 1Modal order

Span-chord ratio 41

a1a1 = 19649Hza2 = 10004Hzb1 = 16374Hzb2 = 83364Hzc1 = 13099Hzc2 = 66691Hz

c1

c2b2a2

b1

Figure 8 Natural frequencies with different thicknesses under thespan-chord ratio 4 1

8 Mathematical Problems in Engineering

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

e method of multiple scales [52] is used to obtain theaveraged equations in following form

w(x t ε) w0 x T0 T1( 1113857 + εw1 x T0 T1( 1113857 (38)

where T0 t and T1 εte derivatives with respect to t becomed

dt

z

zT0

dT0

dt+

z

zT1

dT1

dt+ middot middot middot D0 + εD1 + middot middot middot (39a)

d2

dt2 D0 + εD1 + middot middot middot( 1113857

2 D

20 + 2εD0D1 + middot middot middot (39b)

where Dn (zzTn) n 0 1 2 Substituting equations (38)ndash(39b) into equations (37a)

and (37b) and eliminating the secular terms we have theaveraged equations as follows

D1A1 minus12c11A1 + iσ1A1+

12

ic12f1A1 minus 3iA21A1c13

minus 2ic16A1A2A2

(40a)

D1A2 minus12c21A2 +

12

iσ2A2 minus32

ic23A22A2 minus ic26A1A2A1

minus14

ic28L2

(40b)

In order to obtain the averaged equations in the polarcoordinate system we express A1 and A2 in the followingform

2296Hz 10234Hz

14293Hz 33000Hz

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

39990Hz 62422Hz

Figure 9 e transverse modes of the cantilever laminated composite plate when the span-chord ratio is 2 1 and single thickness is 6mm

Table 3 e 1st mode natural frequency for four span-chord ratiovalues corresponding to the thickness

Span-chord ratioFrequency (Hz)

h1 h2 h3 h4 h5 h61 1 1515 3031 4546 6061 7577 90922 1 383 765 1148 1531 1913 22963 1 169 339 508 678 847 10174 1 095 190 285 380 475 570

Mathematical Problems in Engineering 9

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

A1 T1( 1113857 12a1 T1( 1113857e

jϕ T1( )

A2 T2( 1113857 12a2 T2( 1113857e

jϕ T2( )

(41)

Substituting equation (41) into equations (40a) and (40b)and separating the real and imaginary parts the four-di-mensional averaged equations in the polar coordinate sys-tem are obtained

_a1 minus14

c11a1 sinϕ1 minus 2σ1a1 cos ϕ1 minus c12a1f1 cos ϕ1 +32c14a

31 cosϕ1 + c16a1a

22 sinϕ11113874 1113875 (42a)

a1_ϕ1 minus

14

c11a1 cos ϕ1 + 2σ1a1 sinϕ1 + c12a1f1 sinϕ1 minus32c14a

31 sinϕ1 + c16a1a

22 cosϕ11113874 1113875 (42b)

_a2 minus14

c21a2 sinϕ2 minus σ2a2 cos ϕ2 +34c24a

32 cos ϕ2 +

12c26a2a

21 sinϕ2 + c28l21113874 1113875 (42c)

a2_ϕ2 minus

14

c21a2 cos ϕ2 + σ2a2 sinϕ2 +34c24a

32 sinϕ2 +

12c26a2a

21 cos ϕ21113874 1113875 (42d)

In order to obtain the averaged equations in the Car-tesian coordinate system we rewrite A1 and A2 in thefollowing forms

A1 T1( 1113857 x1 T1( 1113857 + ix2 T1( 1113857

A2 T1( 1113857 x3 T1( 1113857 + ix4 T1( 1113857(43)

According to the same way as the above the averagedequations in the Cartesian coordinate system are obtained asfollows

_x1 minus12c11x1 minus σ1 minus

12c12f11113874 1113875x2 + 3c14x2 x

21 + x

221113872 1113873

+ 2c16x2 x23 + x

241113872 1113873

(44a)

_x2 minus12c11x2 + σ1 +

12c12f11113874 1113875x1 minus 3c14x1 x

21 + x

221113872 1113873

minus 2c16x1 x23 + x

241113872 1113873

(44b)

_x3 minus12c21x3 minus

12σ2x4 +

32c24x4 x

23 + x

241113872 1113873 + c26x4 x

21 + x

221113872 1113873

(44c)

_x4 minus12c21x4 +

12σ2x3 minus

32c24x3 x

23 + x

241113872 1113873 minus c26x3 x

21 + x

221113872 1113873

minus14c28l2

(44d)

6 Numerical Simulation

In order to study the nonlinear vibration characteristics ofthe laminated composite cantilever plate with differentmodal modes based on equations (44a) and (44d) thefrequency-response curves are used to reveal the

characteristics of this system Let _a1 _a2 _ϕ1 _ϕ2 0ϕ1 (π4) and ϕ2 (3π4)

e frequency-response functions of the system aregiven as follows

0 c11a1 minus 2σ1a1 minus c12a1f1 +32c14a

31 + c16a1a

22 (45a)

0 c21a2 + σ2a2 minus34c24a

32 +

12c26a2a

21 +

2

radicc28l2 (45b)

From equations (45a)ndash(45b) we can find that the am-plitude a1 and amplitude a2 are coupled e case of weakcoupled form is considered here rough introducing theproportional relation (a1a2) ε the frequency-responsesbetween the amplitudes and the tuning parameters areanalyzed

According to the geometries and the material propertiesof the nonlinear system the basic parameters are chosen asc11 02 c12 minus 6 c14 5 c16 minus 5 c21 06 c24 9c26 9 and c28 minus 2 e relationship between the am-plitude and the tuning parameter in different excitationconditions can be obtained

Figure 10 gives the relationship between the amplitudea1 and tuning parameter σ1 in different internal forceamplitudes f1 and Figure 11 gives the relationship betweenthe amplitude a2 and the tuning parameter σ2 in differentaerodynamic amplitudes l2

e stiffness hardening phenomenon of the system can beseen in the relationship between the tuning parameters andthe amplitude With the increase of external excitation am-plitude the stiffness hardening phenomenon is graduallystrengthenede typical jump phenomenon of the nonlinearoscillations also happened in the system e jump phe-nomenon appeared in the frequency-response curves at pointAlowast and point Blowast with the increase of the tuning parameters inFigure 11 e frequency-response curves have the widerresonance interval and the larger oscillation amplitudes underthe stronger external excitation amplitude l2

10 Mathematical Problems in Engineering

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Next the geometries and the material properties of thenonlinear system are fixed and the effects of differentdecoupling parameters ε are calculated and the relationshipbetween the amplitude and the tuning parameter can befound in Figures 12 and 13

In the following investigation the bifurcation andchaotic motions of the laminated composite cantilever platebased on the averaged equation in the case of one to twointernal resonances by using the fourth-order RungendashKuttaalgorithm are analyzed We choose the force excitation l2 as

the controlling parameter to study the complicated non-linear dynamics of the system

e tuning parameters and the initial conditions arechosen as σ1 01 σ2 11 c11 02 c12 52 c14 42c16 minus 25 c21 0529 c24 7 c26 5 c28 044 f1 09x10 044 x20 0798 x30 135 and x40 minus 05

Figure 14 presents the two-dimensional bifurcation di-agram to show the nonlinear oscillations of the laminatedcomposite plate by varying the force excitation l2 e pe-riodic and chaotic motions of the laminated compositecantilever plate system for x3 alternately occur with theincrease of the parameter l2 in the interval of [20 50]

1

2

3

4

5

6

a2

ndash200 ndash100 0 100 200 300ndash300σ2

l2 = 10

l2 = 1

l2 = 50

Alowast

Blowast

Figure 11 e relationship between amplitude a2 and tuningparameter σ2 in different aerodynamic amplitudes l2

1

2

3

4

5

6

7

a1

ndash100 0 100 200ndash200σ1

f1 = 1

f1 = 10 f1 = 20

Figure 10 e relationship between amplitude a1 and tuningparameter σ1 in different internal force amplitudes f1

25

5

75

10

125

15

a2

ndash200 ndash100 0 100 200 300ndash300σ2

ε = 10

ε = 11

ε = 09

Figure 13 e relationship between amplitude a2 and tuningparameter σ2 in different decoupling parameters ε

2

4

6

8

10

a1

ndash100 0 100 200ndash200σ1

ε = 10

ε = 09ε = 11

Figure 12 e relationship between amplitude a1 and tuningparameter σ1 in different decoupling parameters ε

Mathematical Problems in Engineering 11

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

ndash055b

a

c

ndash06

ndash065

ndash07

ndash075x 3

ndash08

ndash085

ndash09

ndash09520 25 30 25

l240 45 50

Figure 14 e bifurcation diagram of the laminated composite cantilever plate for x3 via the forcing excitation l2

04

045

05

055

06

065

07

x1

80 85 90 95 10075t

(a)

ndash03

ndash02

ndash01

0

01

02

03

x2

04 045 05 055 06 065 07 075035x1

(b)

ndash07

ndash065

ndash06

ndash055

ndash05

x3

80 85 90 95 10075t

(c)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

x4

ndash07 ndash065 ndash06 ndash055 ndash05 ndash045ndash075x3

(d)

Figure 15 Continued

12 Mathematical Problems in Engineering

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

0208

070 0605ndash02 04 x1

x2

ndash07

ndash065

ndash06

ndash055

ndash05

x3

(e)

Figure 15 e period motion of the laminated composite cantilever plate is obtained when l2 215

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash06

ndash04

ndash02

0

02

04

06

x2

ndash05 0 05 1ndash1x1

(b)

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash08x3

ndash05

ndash04

ndash03

ndash02

ndash01

0

01

02

03

x4

(d)

Figure 16 Continued

Mathematical Problems in Engineering 13

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

051

0500

ndash05ndash05 ndash1

x1

x2

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 16 e multiperiod motion of the laminated composite cantilever plate is obtained when l2 227

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

x3

80 85 90 95 10075t

(c)

ndash05

0

05

x4

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02ndash09x3

(d)

Figure 17 Continued

14 Mathematical Problems in Engineering

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

051

05x2 x1

0 0ndash05ndash05 ndash1

ndash08

ndash06

ndash04

ndash02

x3

(e)

Figure 17 e quasi-period motion of the laminated composite cantilever plate is obtained when l2 253

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x1

80 85 90 95 10075t

(a)

ndash08

ndash06

ndash04

ndash02

0

02

04

06

08

x2

ndash05 0 05 1ndash1x1

(b)

80 85 90 95 10075t

ndash08

ndash07

ndash06

ndash05

ndash04

ndash03

ndash02

x3

(c)

ndash08 ndash07 ndash06 ndash05 ndash04 ndash03 ndash02 ndash01ndash09x3

ndash05

0

05

x4

(d)

Figure 18 Continued

Mathematical Problems in Engineering 15

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

In Figures 15ndash18 (a) and (c) show the waveforms on theplanes (t x1) and (t x3) (b) and (d) depict the phase portraitson the planes (x1 x2) and (x3 x4) and (e) represents thethree-dimensional phase portrait in the space (x1 x2 x3)

From Figures 15ndash18 the periodic and chaotic motionsof the laminated composite cantilever plate occur with theincrease of the amplitude of the parameter l2 can be seenIt is noticed that both the power spectrum and thePoincare map can distinguish the periodic motions andthe chaotic motions When the l2 is in the range of [21 27]the periodic motions and chaotic motions alternatelyoccur the system experiences periodic motion multi-periodic motion quasi-periodic motion and chaoticmotion It is obvious that the nonlinear vibration char-acteristics of the system are complicated under the in-fluence of external excitation

7 Conclusions

e nonlinear dynamics of the laminated composite can-tilever plate under subsonic excitation force have been in-vestigated in this paper e aerodynamic force of a three-dimensional flat wing was calculated Unlike a two-di-mensional airfoil or an infinite length wing the aerodynamicforce of a three-dimensional flat wing is calculated Someresults are obtained

(1) According to the ideal incompressible fluid flowcondition and the KuttandashJoukowski lift theorem thesubsonic aerodynamic lift on the three-dimensionalfinite length flat wing is calculated by using thevortex lattice (VL) method

(2) e finite length flat wing is modeled as a laminatedcomposite cantilever plate based on Reddyrsquos third-order shear deformation plate theory and the vonKarman geometry nonlinearity is introduced enonlinear partial differential governing equations ofmotion for the laminated composite cantilever platesubjected to the subsonic aerodynamic force areestablished via Hamiltonrsquos principle

(3) rough comparing the natural frequencies of thelinear system with different materials and geometryparameters the relationship of 1 2 internal reso-nance is found and considered for the nonlinearvibration analysis

(4) Corresponding to several selected parameters thefrequency-response curves are obtained e hard-ening-spring-type behaviors and jump phenomenaare exhibited with the variation of the tuningparameters

(5) e influence of the force excitation on the bifur-cations and chaotic behaviors of the laminatedcomposite cantilever plate have been investigatednumerically e periodic motion multiperiodicmotion quasi-periodic motion and chaotic motionof the system occurred with the increasing of theexternal excitation

Appendix

A11z2u0

zx2 + A66z2u0

zy2 + A12 + A66( 1113857z2v0

zxzy+ A11

zw0

zx

z2w0

zx2 + A66zw0

zx

z2w0

zy2 + A12 + A66( 1113857zw0

zy

z2w0

zxzy I0 eurou0 + Jeuroφx minus c1I3

z eurow0

zx

(A1)

A66z2v0

zx2 + A22z2v0

zy2 + A21 + A66( 1113857z2u0

zxzy+ A66

zw0

zy

z2w0

zx2 + A22zw0

zy

z2w0

zy2 + A21 + A66( 1113857zw0

zx

z2w0

zxzy I0eurov0 + J1euroφy minus c1I3

z eurow0

zy

(A2)

1105 05

x1x2

0 0ndash05ndash05

ndash1

ndash1

ndash08

ndash06

ndash04

ndash02

0

x3

(e)

Figure 18 e chaotic motion of the laminated composite cantilever plate is obtained when l2 265

16 Mathematical Problems in Engineering

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

A11zu0

zx

z2w0

zx2 + A21zu0

zx

z2w0

zy2 + 2A66zu0

zy

z2w0

zxzy+ A11

z2u0

zx2zw0

zx+ A21 + A66( 1113857

z2u0

zxzy

zw0

zy+ A66

z2v0

zx2zw0

zy+ A12 + A66( 1113857

z2v0

zxzy

zw0

zx

+ A22z2v0

zy2zw0

zy+32A11

zw0

zx1113888 1113889

2z2w0

zx2 +12A21 + A661113874 1113875

zw0

zx1113888 1113889

2z2w0

zy2 + A12 + A21 + 4A66( 1113857zw0

zx

zw0

zy

z2w0

zxzy+32A22

zw0

zy1113888 1113889

2z2w0

zy2

+12A12 + A661113874 1113875

zw0

zy1113888 1113889

2z2w0

zx2 + F0 + F1 cosΩ1t( 1113857z2w0

zy2 + A55 minus 2c2D55 + c22F551113872 1113873

z2w0

zx2 + A22zv0

zy

z2w0

zy2 + A66zw0

zx

z2u0

zy2

+ 2A66zv0

zx

z2w0

zxzy+ A12

zv0

zy

z2w0

zx2 + A44 minus 2c2D44 + c22F441113872 1113873

z2w0

zy2 minus c21H11

z4w0

zx4 minus c21 H12 + H21 + 4H66( 1113857

z4w0

zx2zy2

+ A55 minus 2c2D55+c22F551113872 1113873

zφx

zxminus c

21H22

z4w0

zy4+ c1 F11 minus c1H11( 1113857

z3φx

zx3 + A44 minus 2c2D44 + c22F441113872 1113873

zφy

zy

+ c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3φx

zxzy2 + c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3φx

zx2zy

+ c1 F22 minus c1H22( 1113857z3φy

zy3 + L minus c _w0 I0 eurow0 minus c21I6

z2 eurow0

zx2 +z2 eurow0

zy21113888 1113889 + c1I3zeurou0

zx+

zeurov0

zy1113888 1113889 + c1J4

zeuroφx

zx+

zeuroφy

zy)1113888

(A3)

D11 minus 2c1F11 + c21H111113872 1113873

z2φx

zx2 + D66 minus 2c1F66 + c21H661113872 1113873

z2φx

zy2 minus c1 F11 minus c1H11( 1113857z3w0

zx3

minus A55 minus 2c2D55 + c22F551113872 1113873

zw0

zx+ D12 + D66 + c

21H66 minus 2c1F66 + c

21H12 minus 2c1F121113872 1113873

z2φy

zxzy

minus c1 F12 + 2F66 minus c1H12 minus 2c1H66( 1113857z3w0

zxzy2 + 2c2D55 minus A55 minus c22F551113872 1113873φx J1 eurou0 + K2euroφx minus c1J4

z eurow0

zx

(A4)

D66 minus 2c1F66 + c21H661113872 1113873

z2φy

zx2 minus c1 F21 + 2F66 minus c1H21 minus 2c1H66( 1113857z3w0

zx2zyminus c1 F22 minus c1H22( 1113857

z3w0

zy3

+ D21 + D66 + c21H21 minus 2c1F21 + c

21H66 minus 2c1F661113872 1113873

z2φx

zxzy+ c

21H22 + D22 minus 2c1F221113872 1113873

z2φy

zx2

minus F44c22 minus 2c2D44 + A441113872 1113873

zw0

zy+ 2c2D44 minus A44 minus c

22F441113872 1113873φy J1eurov0 + K2euroφy minus c1J4

z eurow0

zy

(A5)

where

Aij Bij Dij Eij Fij Hij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z z

2 z

3 z

4 z

61113872 1113873dz

(i j 1 2 6)

Aij Dij Fij1113872 1113873 1113944N

k11113946

zk+1

zk

Qkij 1 z

2 z

41113872 1113873dz

Ii 11139443

k11113946

zk+1

zk

ρk(z)idz (i 0 1 2 6)

Ji Ii minus c1Ii+2

K2 I2 minus 2c1I4 + c21I6

(A6)

Data Availability

e data used to support the findings of this study are in-cluded within the article All the pictures and quotations arebased on the theoretical principles and the conditions wegive in this paper

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11832002) and NationalMajor Scientific Research Instrument Research Project(Grant no 11427801)

Mathematical Problems in Engineering 17

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

References

[1] M Mehri H Asadi and M A Kouchakzadeh ldquoComputa-tionally efficient model for flow-induced instability of CNTreinforced functionally graded truncated conical curvedpanels subjected to axial compressionrdquo Computer Methods inApplied Mechanics and Engineering vol 318 pp 957ndash9802017

[2] C Y Chia Nonlinear Analysis of Plates McGraw-Hill In-ternational Book Company New York NY USA 1980

[3] C-Y Chia ldquoGeometrically nonlinear behavior of compositeplates a reviewrdquo Applied Mechanics Reviews vol 41 no 12pp 439ndash451 1988

[4] K Mehar and S K Panda ldquoElastic bending and stress analysisof carbon nanotube-reinforced composite plate experimen-tal numerical and simulationrdquo Advances in Polymer Tech-nology vol 37 no 6 pp 1643ndash1657 2018

[5] T R Mahapatra V R Kar and S K Panda ldquoLarge amplitudevibration analysis of laminated composite spherical panelsunder hygrothermal environmentrdquo International Journal ofStructural Stability and Dynamics vol 16 no 3 Article ID1450105 2016

[6] T R Mahapatra and S K Panda ldquoermoelastic vibrationanalysis of laminated doubly curved shallow panels usingnon-linear FEMrdquo Journal of ermal Stresses vol 38 no 1pp 39ndash68 2015

[7] T R Mahapatra and S K Panda ldquoNonlinear free vibrationanalysis of laminated composite spherical shell panel underelevated hygrothermal environment a micromechanical ap-proachrdquo Aerospace Science and Technology vol 49 pp 276ndash288 2016

[8] T R Mahapatra V R Kar and S K Panda ldquoNonlinear freevibration analysis of laminated composite doubly curved shellpanel in hygrothermal environmentrdquo Journal of SandwichStructures amp Materials vol 17 no 5 pp 511ndash545 2015

[9] T R Mahapatra K Mehar S K Panda et al ldquoFlexuralstrength of functionally graded nanotube reinforced sandwichspherical panel IOP conference series materials Science andEngineeringrdquo IOP Publishing vol 178 no 1 Article ID012031 2017

[10] K Mehar S K Panda and T R Mahapatra ldquoLarge defor-mation bending responses of nanotube-reinforced polymercomposite panel structure numerical and experimental an-alysesrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 233 no 5pp 1695ndash1704 2019

[11] K Mehar S K Panda and T R Mahapatra ldquoermoelasticnonlinear frequency analysis of CNT reinforced functionallygraded sandwich structurerdquo European Journal of Mechan-icsmdashASolids vol 65 pp 384ndash396 2017

[12] K Mehar and S K Panda ldquoNonlinear finite element solutionsof thermoelastic flexural strength and stress values of tem-perature dependent graded CNT-reinforced sandwich shal-low shell structurerdquo Structural Engineering and Mechanicsvol 67 no 6 pp 565ndash578 2018

[13] K Mehar and S K Panda ldquoermoelastic flexural analysis ofFG-CNTdoubly curved shell panelrdquo Aircraft Engineering andAerospace Technology vol 90 no 1 pp 11ndash23 2018

[14] V R Kar and S K Panda ldquoNonlinear flexural vibration ofshear deformable functionally graded spherical shell panelrdquoSteel and Composite Structures vol 18 no 3 pp 693ndash7092015

[15] V R Kar and S K Panda ldquoNonlinear free vibration offunctionally graded doubly curved shear deformable panels

using finite element methodrdquo Journal of Vibration andControl vol 22 no 7 pp 1935ndash1949 2016

[16] V R Kar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FGM spherical panel under nonlinearthermal loading with TD and TID propertiesrdquo Journal ofermal Stresses vol 39 no 8 pp 942ndash959 2016

[17] L Librescu and J N Reddy ldquoA few remarks concerningseveral refined theories of anisotropic composite laminatedplatesrdquo International Journal of Engineering Science vol 27no 5 pp 515ndash527 1989

[18] V K Singh and S K Panda ldquoNonlinear free vibration analysisof singledoubly curved composite shallow shell panelsrdquoin-Walled Structures vol 85 pp 341ndash349 2014

[19] K Mehar and S K Panda ldquoGeometrical nonlinear free vi-bration analysis of FG-CNT reinforced composite flat panelunder uniform thermal fieldrdquo Composite Structures vol 143pp 336ndash346 2016

[20] S K Panda and B N Singh ldquoNonlinear finite elementanalysis of thermal post-buckling vibration of laminatedcomposite shell panel embedded with SMA fibrerdquo AerospaceScience and Technology vol 29 no 1 pp 47ndash57 2013

[21] K Mehar S K Panda T Q Bui and T R MahapatraldquoNonlinear thermoelastic frequency analysis of functionallygraded CNT-reinforced singledoubly curved shallow shellpanels by FEMrdquo Journal of ermal Stresses vol 40 no 7pp 899ndash916 2017

[22] K Mehar S K Panda and T R Mahapatra ldquoNonlinearfrequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperaturefieldrdquo International Journal of Applied Mechanics vol 10no 3 Article ID 1850028 2018

[23] W Zhang ldquoGlobal and chaotic dynamics for a parametricallyexcited thin platerdquo Journal of Sound and Vibration vol 239no 5 pp 1013ndash1036 2001

[24] S K Panda and B N Singh ldquoNonlinear free vibration ofspherical shell panel using higher order shear deformationtheorymdasha finite element approachrdquo International Journal ofPressure Vessels and Piping vol 86 no 6 pp 373ndash383 2009

[25] S K Panda and B N Singh ldquoLarge amplitude free vibrationanalysis of thermally post-buckled composite doubly curvedpanel using nonlinear FEMrdquo Finite Elements in Analysis andDesign vol 47 no 4 pp 378ndash386 2011

[26] S K Panda and B N Singh ldquoNonlinear free vibration analysisof thermally post-buckled composite spherical shell panelrdquoInternational Journal of Mechanics and Materials in Designvol 6 no 2 pp 175ndash188 2010

[27] A K Onkar and D Yadav ldquoForced nonlinear vibration oflaminated composite plates with randommaterial propertiesrdquoComposite Structures vol 70 no 3 pp 334ndash342 2005

[28] T Park S-Y Lee J W Seo and G Z Voyiadjis ldquoStructuraldynamic behavior of skew sandwich plates with laminatedcomposite facesrdquo Composites Part B Engineering vol 39no 2 pp 316ndash326 2008

[29] R-D Chien and C-S Chen ldquoNonlinear vibration of lami-nated plates on an elastic foundationrdquoin-walled Structuresvol 44 no 8 pp 852ndash860 2006

[30] B N Singh A Lal and R Kumar ldquoNonlinear bending re-sponse of laminated composite plates on nonlinear elasticfoundation with uncertain system propertiesrdquo EngineeringStructures vol 30 no 4 pp 1101ndash1112 2008

[31] W Zhang X Y Guo and S K Lai ldquoResearch on periodic andchaotic oscillations of composite laminated plates with one-to-one internal resonancerdquo International Journal of Nonlinear

18 Mathematical Problems in Engineering

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19

Sciences and Numerical Simulation vol 10 pp 1567ndash15842009

[32] W Zhang Z Yao and M Yao ldquoPeriodic and chaotic dy-namics of composite laminated piezoelectric rectangular platewith one-to-two internal resonancerdquo Science in China SeriesE Technological Sciences vol 52 no 3 pp 731ndash742 2009

[33] F Alijani M Amabili K Karagiozis and F Bakhtiari-NejadldquoNonlinear vibrations of functionally graded doubly curvedshallow shellsrdquo Journal of Sound and Vibration vol 330 no 7pp 1432ndash1454 2011

[34] F Alijani F Bakhtiari-Nejad and M Amabili ldquoNonlinearvibrations of FGM rectangular plates in thermal environ-mentsrdquoNonlinear Dynamics vol 66 no 3 pp 251ndash270 2011

[35] M Amabili ldquoNonlinear vibrations of angle-ply laminatedcircular cylindrical shells skewed modesrdquo Composite Struc-tures vol 94 no 12 pp 3697ndash3709 2012

[36] W Zhang J Chen Y F Zhang and X D Yang ldquoContinuousmodel and nonlinear dynamic responses of circular meshantenna clamped at one siderdquo Engineering Structures vol 151pp 115ndash135 2017

[37] T Liu W Zhang and J F Wang ldquoNonlinear dynamics ofcomposite laminated circular cylindrical shell clamped along ageneratrix and with membranes at both endsrdquo NonlinearDynamics vol 90 no 2 pp 1393ndash1417 2017

[38] Y Sun W Zhang and M H Yao ldquoMulti-pulse chaoticdynamics of circular mesh antenna with 12 internal reso-nancerdquo International Journal of Applied Mechanics vol 9no 4 Article ID 1750060 2017

[39] W Zhang T Liu A Xi and Y NWang ldquoResonant responsesand chaotic dynamics of composite laminated circular cy-lindrical shell with membranesrdquo Journal of Sound and Vi-bration vol 423 pp 65ndash99 2018

[40] E H Dowell ldquoNonlinear oscillations of a fluttering platerdquoAIAA Journal vol 4 no 7 pp 1267ndash1275 1966

[41] E H Dowell ldquoNonlinear oscillations of a fluttering plate IIrdquoAIAA Journal vol 5 no 10 pp 1856ndash1862 1967

[42] M Amabili and M P Paıumldoussis ldquoReview of studies ongeometrically nonlinear vibrations and dynamics of circularcylindrical shells and panels with and without fluid-structureinteractionrdquo Applied Mechanics Reviews vol 56 no 4pp 349ndash381 2003

[43] M K Singha and M Ganapathi ldquoA parametric study onsupersonic flutter behavior of laminated composite skew flatpanelsrdquo Composite Structures vol 69 no 1 pp 55ndash63 2005

[44] H Haddadpour H M Navazi and F Shadmehri ldquoNonlinearoscillations of a fluttering functionally graded platerdquo Com-posite Structures vol 79 no 2 pp 242ndash250 2007

[45] M K Singha and M Mandal ldquoSupersonic flutter charac-teristics of composite cylindrical panelsrdquo Composite Struc-tures vol 82 no 2 pp 295ndash301 2008

[46] S-Y Kuo ldquoFlutter of rectangular composite plates withvariable fiber pacingrdquo Composite Structures vol 93 no 10pp 2533ndash2540 2011

[47] M H Zhao and W Zhang ldquoNonlinear dynamics of com-posite laminated cantilever rectangular plate subject to third-order piston aerodynamicsrdquo Acta Mechanica vol 225 no 7pp 1985ndash2004 2014

[48] Y X Hao W Zhang J Yang and S B Li ldquoNonlinear dy-namics of a functionally graded thin simply-supported plateunder a hypersonic flowrdquo Mechanics of Advanced Materialsand Structures vol 22 no 8 pp 619ndash632 2015

[49] S F Lu W Zhang and X J Song ldquoTime-varying nonlineardynamics of a deploying piezoelectric laminated composite

plate under aerodynamic forcerdquo Acta Mechanica Sinicavol 34 pp 1ndash12 2017

[50] G Yao and F-M Li ldquoChaotic motion of a composite lami-nated plate with geometric nonlinearity in subsonic flowrdquoInternational Journal of Non-Linear Mechanics vol 50pp 81ndash90 2013

[51] J Li Y Yan Z Liang and T Zhang ldquoExperimental andnumerical study of adhesively bonded CFRP scarf-Lap jointssubjected to tensile loadsrdquo e Journal of Adhesion vol 92no 1 pp 1ndash17 2016

[52] A H Nayfeh and D T Mook Nonlinear Oscillations OxfordUniversity Press Oxford UK 1981

Mathematical Problems in Engineering 19