Nonlinear Vibrations of Laminated Cross-Ply Composite Cantilever Plate in Subsonic Air Flow

,e nonlinear vibrations and responses of a laminated composite cantilever plate under the subsonic air flow are investigated in this paper. ,e subsonic air flow around the three-dimensional cantilever rectangle laminated composite plate is considered to be decreasing from the wing root to the wing tip. According to the ideal incompressible fluid flow condition and the Kutta–Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using the Vortex Lattice (VL) method. ,e finite length flat wing is modeled as a laminated composite cantilever plate based on Reddy’s third-order shear deformation plate theory and the von Karman geometry nonlinearity is introduced. ,e nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force are established via Hamilton’s principle. ,e Galerkin method is used to separate the partial differential equations into two nonlinear ordinary differential equations, and the four-dimensional nonlinear averaged equations are obtained by the multiple scale method. ,rough comparing the natural frequencies of the linear system with different material and geometric parameters, the relationship of 1 : 2 internal resonance is considered. Corresponding to several selected parameters, the frequency-response curves are obtained. ,e hardening-spring-type behaviors and jump phenomena are exhibited. ,e influence of the force excitation on the bifurcations and chaotic behaviors of the laminated composite cantilever plate is investigated numerically. It is found that the system is sensitive to the exciting force according to the complicate nonlinear behaviors exhibited in this paper.


Introduction
e vibration of the plate and shell structures owing to air flow is a matter of interest because of its significance in design of launch vehicles and aircrafts [1]. Laminated composite structures are widely used in aerospace field due to its high strength-to-weight ratio, light weight, and long fatigue life. e dynamic behavior of the laminated composite plates in air flow has gained significant focus of attention for many researchers. However, there are few research works dealing with the complex nonlinear dynamics of the structure which is simplified as a laminated composite cantilever plate subjected to subsonic air flow. erefore, the nonlinear dynamics of laminated composite plates in subsonic flow will be worth analyzing in this work.
In the 1990s, the research on the vibration of plates has been studied comprehensively. Some literature reviews on nonlinear vibrations of plates were given by Chia [2,3] and Mehar and Panda [4]. e nonlinear vibrations of laminated composite spherical shell panels were also entirely investigated by Mahapatra et al. [5][6][7][8][9]. e vibration, bending, and buckling behaviors about the functionally graded sandwich structure have been investigated by Mehar et al. [10][11][12][13] and Kar and Panda [14][15][16]. A third-order theory which accounted for a cubic variation of the in-plane displacements through the plate thickness was derived by Librescu and Reddy [17]. Great progress has also been achieved in the study of the nonlinear thermoelastic frequency analysis [18][19][20][21][22]. Zhang [23] studied the global bifurcations and chaotic dynamics of simply supported rectangular thin plates under parametrical excitation by introducing von Karman's geometric nonlinearity plate theory.
Nonlinear analysis considering the geometrical and material nonlinearity by FEM was conducted by Panda and Singh [24][25][26]. Onkar and Yadav [27] studied the nonlinear random vibrations of a simply supported cross-ply laminated composite plate using analytical methods, and the accuracy of response evaluation was improved in this work. Park et al. [28] investigated the nonlinear forced vibrations of skew sandwich plates subjected to multiple of dynamic loads, the influence of the skew angles, boundary conditions, and loads on the nonlinear dynamic behaviors of composite plates which were discussed. Chien and Chen [29] studied the effects of initial stresses and different parameters on the nonlinear vibrations of laminated composite plates on elastic foundations. Singh et al. [30] used the higher-order shear deformation plate theory to study the dynamic responses of a geometrically nonlinear laminated composite plate lying on different elastic foundations. Zhang et al. [31,32] investigated nonlinear vibration, bifurcation, and chaotic dynamics of laminated composite plates. Alijani et al. [33,34] studied nonlinear vibrations of FGM plates and shallow shells, and the bifurcations and nonlinear dynamic behaviors were analyzed in their papers. Amabili [35] used the higher-order shear deformation theory to study the nonlinear vibrations of angle-ply laminated circular cylindrical shells. Most recently, Zhang et al. [36][37][38][39] studied the nonlinear vibrations of the laminated circular cylindrical shells based on the deployed ring truss for antenna structures. e complex nonlinear dynamics of the circular cylindrical shell were studied and the bifurcations and chaotic behaviors were investigated by using the analytical methods and numerical simulations. e research on nonlinear vibrations of plates and shells excited by the aerodynamic force were also widely investigated by researchers. In the early years, Dowell [40,41] studied the nonlinear oscillations of the fluttering plate systematically. Literature reviews on the nonlinear vibrations of circular cylindrical shells and panels with and without fluid-structure interaction were given by Amabili and Paı¨doussis [42]. e studies on the nonlinear vibration of laminated plates in air flow were mostly about the plates in the supersoninc flow. Singha and Ganapathi [43] investigated the effect of the system parameters on supersonic panel flutter behaviors of laminated composite plates. Haddadpour et al. [44] investigated the nonlinear aeroelastic behaviors of FGM. Singha and Mandal [45] used a 16-node isoparametric degenerated shell element to study the supersonic panel flutter behaviors of laminated composite plates and cylindrical panels. Kuo [46] investigated the effect of variable fiber spacing on the supersonic flutter of rectangular composite plates. Zhao and Zhang [47] presented the analysis of the nonlinear dynamics for a laminated composite cantilever rectangular plate subjected to the supersonic gas flows and the in-plane excitations. An analysis on the nonlinear dynamics of an FGM plate in hypersonic flow subjected to an external excitation and uniform temperature change was presented by Hao et al. [48]. e nonlinear dynamic behavior of an axially extendable cantilever laminated composite plate using piezoelectric materials under the combined action of aerodynamic load and piezoelectric excitation was studied by Lu et al. [49]. Yao and Li [50] investigated the nonlinear vibration of a two-dimensional laminated composite plate in subsonic air flow with simply supported boundary conditions. A simple subsonic aerodynamics model was introduced in this work, which was used to analyze twodimensional infinite length plate.
In this paper, the nonlinear dynamics of the laminated composite cantilever plate under subsonic air flow were investigated. According to the flow condition of ideal incompressible fluid and the Kutta-Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing was calculated using the Vortex Lattice (VL) method. e nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force were established via Hamilton's principle. e Galerkin method was used to separate the partial differential equation into two nonlinear ordinary differential equations. e numerical method was utilized to investigate the bifurcations and periodic and chaotic motions of the composite laminated rectangular plate. e numerical results illustrate that there existed the periodic and chaotic motions of the composite laminated cantilever plate. e hardening-spring characteristics of the composite laminated cantilever plate were demonstrated by the frequencyresponse curves of the system.

Derivation of the Subsonic Aerodynamic
Force on the Plate In this section, the subsonic air flow on the plate will be calculated by using the VL method. e VL method is a numerical implementation on the general 3D lifting surface problem. is method discretizes the vortex-sheet strength distributing on each lifting surface by lumping it into a collection of horseshoe vortices. Figure 1 shows that a 3D lifting surface is discretized by a vortex lattice of horseshoe vortices, in which all contribute to the velocity V at any field point r.
For the finite wingspan, the high-pressure air below the wing will turn over the low pressure air at the tip of the wing, which will cause the pressure on the upper surface of the wing tip to be equal to that of the lower surface. Unlike the two-dimensional flow around the airfoil, the main characteristic of the three-dimensional flow around the wing is the variation of the lift along the wingspan. In order to calculate the lift on the wing surface by using the vortex lattice method, the Biot-Savart law is used to calculate the induced velocity on the control point. e vortex strength of the vortex system is obtained, and the pressure difference on the upper and lower surface of the wing surface is deduced at the same time.
e velocity induced by a vortex line with a strength of Γ n and a length of dl is calculated by the Biot-Savart law as As shown in Figure 2, the induced velocity is Let AB represent a vortex segment in which the vortex vector points from A to B. C is a space point, and its normal distance from the AB line is r p .
We can make an integral from the point A to the point B to find out the size of the induced velocity: As r 0 , r 1 ,and r 2 , respectively, represent the AB, AC, and BC in Figure 2, the following relationships are established: en, the formula calculating the induced velocity generated by the horseshoe vortex in the vortex lattice method is derived: From equation (5), the velocity induced by a vortex segment AB at any point in the coordinate (x, y, z) can be obtained: where V A∞ is the velocity induced by vortex lines from the point A to ∞ along the x-axis and V B∞ is the velocity induced by vortex lines from the point B to ∞ along the x-axis. e total velocity induced by a horseshoe vortex at a point (x, y, z) representing a surface element (i.e., the nth panel element) is the sum of the various components calculated by equations (6a)-(6c). erefore, the velocity induced by 2N vortexes is obtained to get the total induced velocity on the first m control points. It is expressed as e contribution of all vortices to the downwash of the control point for the mth panel element: en, the lift on the rectangular cantilever plate with a sweep rectangular wing surface is computed according to the VL method. e vortex system is arranged on the rectangular wing surface in Figure 3. Figure 3, the variable b is twice the span of the wing, c is the taper ratio of the wing surface, and c is equal to 1. e sweep angle is 0 ∘ . Five panel elements are specified on the wing surface, and the vortex line is arranged at the leading edge 1/4 chord, and the control point is located at 3/4 chord. e wing surface is divided into 5 panel elements and each panel extends from the leading edge to the trailing edge. e coordinates of the 5 × 1 vortices on the wing surface are listed in Table 1.

Application of VL Method on the Plate. In
e downwash velocity of each surface element induced at the control point is superimposed through the nonpenetrating condition: Control points Figure 1: A 3D lifting surface discretized by a vortex lattice of horseshoe vortices. Mathematical Problems in Engineering e solution of vortex strength is obtained by using 5 algebraic equations with unknown vortex strength: According to the boundary conditions, the air flow is tangent to the object surface at each control point to determine the strength of each vortex. e lift of the wing can be calculated by satisfying the boundary conditions. For any wing that does not have an upper counter angle, the lift is produced by the free flow that crosses the vortex line as there is no side-washing speed or postwash speed.
According to a finite number of elements, the lift on the plate is expressed as As the geometric relationship of each facet is Δy n � 0.1b, the lift on the flat plate can be rewritten as where ρ ∞ is the flow density, V ∞ is the flow velocity, b is half length of the wingspan, and α is the attack angle. e attack angle is considered to be affected by a periodic disturbance α � α 0 + α 1 cos Ω 2 t. Taking the periodic perturbation into the aerodynamic force and according to equation (12), the expression of the aerodynamic force containing the perturbation term is obtained: where

Formulation
In this section, the dynamic equations of the laminated composite cantilever plate are derived. As shown in Figure 4, the parameters x, y, and z are, respectively, the spanwise direction, the direction of the chord, and the vertical direction of the plate. e plate is clamped at the position which is distributed along the chord direction of the plate. e vertical direction of the plate along the spanwise direction is subject to subsonic aerodynamic force L.
e nonlinear governing equations are established in the Cartesian coordinate system. Reddy's third-order shear deformation plate theory is used: e von Karman nonlinear strain-displacement relation is introduced. e displacements and strain-displacement relation are given as follows:  Substituting the expressions of the strain into the displacements, we can obtain where where e symmetric cross-ply laminated composite plate is adopted, and we can obtain where where θ is the angle of laminated layer. e stiffness of each layer is 21 , 21 , where E i is the Young modulus, G 12 is the shear modulus, and ] 12 is Poisson's ratio. According to Hamilton's principle, e nonlinear governing equations of motion are given as follows: Mathematical Problems in Engineering 5 where c is the damping coefficient. e internal force is expressed as follows: Substituting the internal force of equation (26) into equations (25a)-(25e), we can obtain the expression of the governing equation of motion by the displacements, which are given in Appendix.
e boundary conditions of the cantilever plate are obtained at the same time as x � a: e dimensionless equations can be obtained by introducing the following parameters:

Frequency Analysis
High-dimensional nonlinear dynamic systems contain several types of the internal resonant cases which can lead to different forms of the nonlinear vibrations. When the system exists a special internal resonant relationship between two linear modes, the large amplitude nonlinear responses may suddenly happen which is dangerous in engineering and should be avoided. In order to study the internal resonance relationship between two bending modes, the finite element model of the cantilever laminated composite plate is established. e ply stacking sequence is [0 ∘ /90 ∘ ] S and the material parameters [51] are shown in Table 2.
e natural frequencies of bending vibration of laminated composite cantilever plates with different span-chord ratio and different layer thickness are calculated, and the results are shown in Figures 5-8. When the span-chord ratio is 2 : 1 and single thickness is 6 mm, the transverse vibration modes of the cantilever laminated composite plate with corresponding frequencies are shown in Figure 9.
In frequency analysis, the first mode natural frequency is most important. Table 3 is to illustrate the 1st mode natural frequency for 6 (thickness values) × 4 (span-chord ratio values) from Figures 5-8.
Based on the results of numerical simulations, the first six orders natural frequencies of the laminated composite cantilever plate are obtained, as shown in Figures 5-8. It is obviously observed that there is a proportional relation between the two bending modes, such as relation 1 : 1 in area c of Figure 7, relation 1 : 2 in Figure 8, and relation 1 : 3 in Figures 5-7. We select 1 : 2 internal resonance relationship between two bending modes and the nonlinear vibrations of the laminated composite cantilever plate are investigated in the following analysis.

Perturbation Analysis
e discrete equation is derived by the Galerkin method, and the discrete function adopts the following expression: where where cos λ i a coshλ i a − 1 � 0, Similarly, the aerodynamic force is discretized by using the modal function: where l 1 and l 2 represent the amplitudes of the aerodynamic forces corresponding to the two transvers vibration modes and they contain the perturbed items. Substituting equations (30)-(33) into equation (A.3), the governing differential equations of transverse motion of the system are derived as follows:  Next, the averaged equations of the system are obtained by using the multiple scale method, and the internal resonant relationship 1 : 2 is considered: where ω 1 and ω 2 are two frequencies of the laminated composite plate and σ 1 and σ 2 are the tuning parameters. e scale transformations are given as follows: e scale transformation can be obtained by introducing equation (36) into equations (34a) and (34b): € w 2 + εc 21 _ w 2 + ω 2 2 w 2 +εc 22 f 1 cos Ω 1 tw 2 + εc 23 w 1 − εc 24     8 Mathematical Problems in Engineering e method of multiple scales [52] is used to obtain the averaged equations in following form: where T 0 � t and T 1 � εt. e derivatives with respect to t become d dt where D n � (z/zT n ), n � 0, 1, 2, . . .. Substituting equations (38)-(39b) into equations (37a) and (37b) and eliminating the secular terms, we have the averaged equations as follows: In order to obtain the averaged equations in the polar coordinate system, we express A 1 and A 2 in the following form: Substituting equation (41) into equations (40a) and (40b) and separating the real and imaginary parts, the four-dimensional averaged equations in the polar coordinate system are obtained: In order to obtain the averaged equations in the Cartesian coordinate system, we rewrite A 1 and A 2 in the following forms: According to the same way as the above, the averaged equations in the Cartesian coordinate system are obtained as follows: (44d)
e frequency-response functions of the system are given as follows: From equations (45a)-(45b), we can find that the amplitude a 1 and amplitude a 2 are coupled. e case of weak coupled form is considered here. rough introducing the proportional relation (a 1 /a 2 ) � ε, the frequency-responses between the amplitudes and the tuning parameters are analyzed.
e relationship between the amplitude and the tuning parameter in different excitation conditions can be obtained. Figure 10 gives the relationship between the amplitude a 1 and tuning parameter σ 1 in different internal force amplitudes f 1 , and Figure 11 gives the relationship between the amplitude a 2 and the tuning parameter σ 2 in different aerodynamic amplitudes l 2 .
e stiffness hardening phenomenon of the system can be seen in the relationship between the tuning parameters and the amplitude. With the increase of external excitation amplitude, the stiffness hardening phenomenon is gradually strengthened. e typical jump phenomenon of the nonlinear oscillations also happened in the system. e jump phenomenon appeared in the frequency-response curves at point A * and point B * with the increase of the tuning parameters in Figure 11. e frequency-response curves have the wider resonance interval and the larger oscillation amplitudes under the stronger external excitation amplitude l 2 .
Next, the geometries and the material properties of the nonlinear system are fixed, and the effects of different decoupling parameters ε are calculated and the relationship between the amplitude and the tuning parameter can be found in Figures 12 and 13.
In the following investigation, the bifurcation and chaotic motions of the laminated composite cantilever plate based on the averaged equation in the case of one to two internal resonances by using the fourth-order Runge-Kutta algorithm are analyzed. We choose the force excitation l 2 as the controlling parameter to study the complicated nonlinear dynamics of the system. e tuning parameters and the initial conditions are chosen as σ 1 � 0.1, σ 2 � 1.  Figure 14 presents the two-dimensional bifurcation diagram to show the nonlinear oscillations of the laminated composite plate by varying the force excitation l 2 . e periodic and chaotic motions of the laminated composite cantilever plate system for x 3 alternately occur with the increase of the parameter l 2 in the interval of [20,50].   From Figures 15-18, the periodic and chaotic motions of the laminated composite cantilever plate occur with the increase of the amplitude of the parameter l 2 can be seen. It is noticed that both the power spectrum and the Poincaré map can distinguish the periodic motions and the chaotic motions. When the l 2 is in the range of [21,27], the periodic motions and chaotic motions alternately occur, the system experiences periodic motion, multiperiodic motion, quasi-periodic motion, and chaotic motion. It is obvious that the nonlinear vibration characteristics of the system are complicated under the influence of external excitation.

Conclusions
e nonlinear dynamics of the laminated composite cantilever plate under subsonic excitation force have been investigated in this paper. e aerodynamic force of a threedimensional flat wing was calculated. Unlike a two-dimensional airfoil or an infinite length wing, the aerodynamic force of a three-dimensional flat wing is calculated. Some results are obtained: (1) According to the ideal incompressible fluid flow condition and the Kutta-Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using the vortex lattice (VL) method.
(2) e finite length flat wing is modeled as a laminated composite cantilever plate based on Reddy's thirdorder shear deformation plate theory and the von Karman geometry nonlinearity is introduced. e nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force are established via Hamilton's principle.
(3) rough comparing the natural frequencies of the linear system with different materials and geometry parameters, the relationship of 1 : 2 internal resonance is found and considered for the nonlinear vibration analysis. (4) Corresponding to several selected parameters, the frequency-response curves are obtained. e hardening-spring-type behaviors and jump phenomena are exhibited with the variation of the tuning parameters. (5) e influence of the force excitation on the bifurcations and chaotic behaviors of the laminated composite cantilever plate have been investigated numerically. e periodic motion, multiperiodic motion, quasi-periodic motion, and chaotic motion of the system occurred with the increasing of the external excitation.