Vibration , Stability , and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plate under parametric and external excitations is presented.Themethod of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the systemup to and including the second-order approximation. All possible resonance cases are extracted at this approximation order.The case of 1 : 1 : 3 primary and internal resonance, whereΩ 3 ≅ ω 1 ,ω 2 ≅ ω 1 , andω 3 ≅ 3ω 1 , is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influences of the cubic terms onnonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied. The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integration of themodal equations. Reliability of the obtained results is verified by comparison between the finite differencemethod (FDM) and Runge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system. Such cases should be avoided as working conditions for the system. Variation of the parameters μ 1 , μ 2 , α 7 , β 8 , ω 1 , ω 2 , f 1 , f 2 leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.


Introduction
Composite laminated plates that are widely used in several engineering fields such as machinery, shipbuilding, aircraft, automobiles, robot arm, watercraft-hydropower, and wings of helicopters are made of the angle-ply composite laminated plates.Several researchers have focused their attention on studying the nonlinear dynamics, bifurcations, and chaos of the composite laminated plates.
Internal resonance has been found in many engineering problems in which the natural frequencies of the system are commensurable.Ye et al. [1] investigated the local and global nonlinear dynamics of a parametrically excited symmetric cross-ply composite laminated rectangular thin plate under parametric excitation.The study is focused on the case of 1 : 1 internal resonance and primary parametric resonance.Zhang [2] dealt with the global bifurcations and chaotic dynamics of a parametrically excited, simply supported rectangular thin plate.The method of multiple scales is used to obtain the averaged equations.The case of 1 : 1 internal resonance and primary parametric resonance is considered.Guo et al. [3] studied the nonlinear dynamics of a four-edge simply supported angle-ply composite laminated rectangular thin plate excited by both the in-plane and transverse loads.The asymptotic perturbation method is used to derive the four averaged equations under 1 : 1 internal resonance.Zhang et al. [4] investigated the local and global bifurcations of a parametrically and externally excited simply supported rectangular thin plate under simultaneous transversal and in-plane excitations.The studies are focused on the case of 1 : 1 internal resonance and primary parametric resonance.Tien et al. [5] applied the averaging method and Melnikov technique to study local, global bifurcations and chaos of a two-degreesof-freedom shallow arch subjected to simple harmonic excitation for case of 1 : 2 internal resonances.Sayed and Mousa [6] investigated the influence of the quadratic and cubic terms on nonlinear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations.Two cases of the subharmonic resonances cases in the presence of 1 : 2 internal resonances are considered.The method of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the system up to and including the secondorder approximation.Zhang et al. [7] gave further studies on the nonlinear oscillations and chaotic dynamics of a parametrically excited simply supported symmetric cross-ply laminated composite rectangular thin plate with the geometric nonlinearity and nonlinear damping.Zhang et al. [8] dealt with the nonlinear vibrations and chaotic dynamics of a simply supported orthotropic functionally graded material (FGM) rectangular plate subjected to the in-plane and transverse excitations together with thermal loading in the presence of 1 : 2 : 4 internal resonance, primary parametric resonance, and subharmonic resonance of order 1/2.Zhang et al. [9] investigated the bifurcations and chaotic dynamics of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate subject to the transverse, inplane excitations and the excitation loaded by piezoelectric layers.Zhang and Li [10] analyzed the resonant chaotic motions of a simply supported rectangular thin plate with parametrically and externally excitations using exponential dichotomies and an averaging procedure.Zhang et al. [11] analyzed the chaotic dynamics of a six-dimensional nonlinear system which represents the averaged equation of a composite laminated piezoelectric rectangular plate subjected to the transverse, in-plane excitations and the excitation loaded by piezoelectric layers.The case of 1 : 2 : 4 internal resonances is considered.Zhang and Hao [12] studied the global bifurcations and multipulse chaotic dynamics of the composite laminated piezoelectric rectangular plate by using the improved extended Melnikov method.The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.Guo and Zhang [13] studied the nonlinear oscillations and chaotic dynamics for a simply supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations.The case of 1 : 2 : 3 internal resonance is considered.The method of multiple scales is employed to obtain the six-dimensional averaged equation.The numerical method is used to investigate the periodic and chaotic motions of the composite laminated rectangular thin plate.Eissa and Sayed [14][15][16] and Sayed [17] investigated the effects of different active controllers on simple and spring pendulum at the primary resonance via negative velocity feedback or its square or cubic.Sayed and Kamel [18,19] investigated the effects of different controllers on the vibrating system and the saturation control to reduce vibrations due to rotor blade flapping motion.The stability of the system is investigated using both phase-plane method and frequency response curves.Sayed and Hamed [20] studied the response of a two-degreeof-freedom system with quadratic coupling under parametric and harmonic excitations.The method of multiple scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to and including the second-order approximations.Amer et al. [21] investigated the dynamical system of a twin-tail aircraft, which is described by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities under different controllers.Best active control of the system has been achieved via negative acceleration feedback.The stability of the system is investigated applying both frequency response equations and phaseplane method.Hamed et al. [22] studied the nonlinear dynamic behavior of a string-beam coupled system subjected to external, parametric, and tuned excitations that are presented.The case of 1 : 1 internal resonance between the modes of the beam and string, and the primary and combined resonance for the beam is considered.The method of multiple scales is applied to obtain approximate solutions up to and including the second-order approximations.All resonance cases are extracted and investigated.Stability of the system is studied using frequency response equations and the phaseplane method.Awrejcewicz et al. [23][24][25] studied the chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells.The considered problems are solved by the Bubnov-Galerkin, Ritz method with higher approximations, and finite difference method.Convergence and validation of those methods are studied.Awrejcewicz et al. [26] investigated the chaotic vibrations of flexible nonlinear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions.Reliability of the obtained results is verified by the finite difference method and finite element method with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes.
In this paper, the perturbation method and stability of the composite laminated piezoelectric rectangular plate under simultaneous transverse and in-plane excitations are investigated.The method of multiple scales are applied to obtain the second-order uniform asymptotic solutions.All possible resonance cases are extracted at this approximation order.The study is focused on the case of 1 : 1 : 3 internal resonance and primary resonance.The stability of the system and the effects of different parameters on system behavior have been studied using frequency response curves.Stability is performed of figures by solid and dotted lines.The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integration of the modal equations.It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system.Such cases should be avoided as working conditions for the system.Some recommendations regarding the different parameters of the system are reported.Comparison with the available published work is reported.

Mathematical Analysis
Consider a simply supported four edges composite laminated piezoelectric rectangular plate of lengths ,  and thickness ℎ, as shown in Figure 1.The composite laminated piezoelectric rectangular plate is considered as regular symmetric crossply laminates with  layers.A Cartesian coordinate system is y b q cos Ω 3 t q 0 + q x cos Ω 1 t q 1 + q y cos Ω 2 t located in the middle surface of the plate.Assume that , V, and  represent the displacements of an arbitrary point of the composite laminated piezoelectric rectangular plate in the , , and  directions, respectively.The in-plane excitations are loaded along the direction at  = 0 in the form  0 +   cos Ω 1 , and the excitations are loaded along the  direction at  = 0 in the form  1 +   cos Ω 2 .The transverse excitation subjected to the composite laminated piezoelectric rectangular plate is represented by  cos Ω 3 .The dynamic electrical loading is expressed as   cos Ω 4 .Based on Reddy's third-order shear deformation plate theory [27], the displacement field at an arbitrary point in the composite laminated plate is expressed as [13]  (, , ) =  0 (, , ) +   (, , ) −  3 where  0 , V 0 , and  0 are the original displacement on the midplane of the plate in the ,, and  directions, respectively.Let   and   represent the midplane rotations of transverse normal about the  and  axes, respectively.From the van Karman-type plate theory and Hamilton's principle, the nonlinear governing equations of motion of the composite laminated piezoelectric rectangular plate are given as follows [12]: The boundary conditions of the simply supported rectangular composite laminated plate are expressed as follows: at  = 0,  =  : Applying the Galerkin procedure, we obtain that the dimensionless differential equations of motion for the simply supported symmetric cross-ply rectangular thin plate are shown as follows [11,28]: where  1 ,  2 , and  3 are the vibration amplitudes of the composite laminated piezoelectric rectangular plate for the firstorder, second-order, and the third-order modes, respectively,  1 ,  2 , and  3 are the linear viscous damping coefficients,  1 ,  2 , and  3 are the natural frequencies of the rectangular plate, and Ω 1 , Ω 2 , Ω 3 , and Ω 4 are the excitations frequencies. 1 ,  2 ,  3 , and   ( = 1, 2, 3) are the amplitudes of parametric and external excitation forces corresponding to the three nonlinear modes, and   ,   , and   ( = 1, 2, . . ., 10) are the nonlinear coefficients.The linear viscous damping and exciting forces are assumed to be where  is a small perturbation parameter and 0 <  ≪ 1.
The external excitation forces f are of the order 2, and the linear viscous damping μ , parametric exciting forces f1 , f2 , and f3 are of the order 1.
To consider the influence of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate, we need to obtain the secondorder approximate solution of (4a), (4b), and (4c).Method of multiple scales [29][30][31] is applied to obtain a secondorder approximation for the system.For the second-order approximation, we introduce three time scales defined by In terms of these scales, the time derivatives become where   = /  ,  = 0, 1, 2. We seek a uniform approximation to the solution of (4a), (4b), and (4c) in the form: Terms of ( 4 ) and higher orders are neglected.Substituting ( 5) and (7a)-( 8) into (4a), (4b), and (4c) and equating the coefficients of like powers of , we obtain the following. Order Order  3 The general solutions of (9a), (9b), and (9c) can be written in the form where  1 ,  2 , and  3 are a complex function in  1 ,  2 which can be determined from eliminating the secular terms at the next approximation and  stands for the complex conjugate of the preceding terms.Substituting (12a), (12b), and (12c) into (10a), (10b), and (10c) and eliminating the secular terms, then the first-order approximations are given by where   ( = 1, 2, . . ., 22) are the complex functions in  1 and  2 .From (12a)-(13c) into (11a), (11b), and (11c) and eliminating the secular terms, the second-order approximation is given by where   ( = 1, 2, . . ., 75) are the complex functions in  1 and  2 .From the above derived solutions, the reported resonance cases are the following.
Any combination of the previous resonance cases is considered as simultaneous resonance.
Case 1.In this case, where  2 = 0,  3 = 0, the frequency response equation is given by Case 2. In this case, where  1 = 0,  3 = 0, the frequency response equation is given by Case 3. In this case, where  3 = 0, the frequency response equations are given by Case 4. In this case, where  2 = 0, the frequency response equations are given by Case 5.In this case, where  1 = 0, the frequency response equations are given by Case 6.In this case, where  1 ̸ = 0,  2 ̸ = 0, and  3 ̸ = 0, this is the practical case, the frequency response equations are given by  Comparison between numerical solution (using RKM) and analytical solution (using perturbation method) of the system at resonance case, Ω 3 ≅  1 ,  2 ≅  1 , and where ] , Mathematical Problems in Engineering ] .In the frequency response curves, the stable (unstable) steady-state solutions have been represented by solid (dotted) lines.

Numerical Results and Discussion
To study the behavior of the system, the Runge-Kutta fourthorder method (RKM) was applied to (4a), (4b), and (4c) governing the oscillating system.A good criterion of both stability and dynamic chaos is the phase plane trajectories.Figure 2 illustrates the response and the phase-plane for the nonresonant system at some practical values of the equation parameters:   3 shows the steady-state amplitudes and phase plane of the system at simultaneous resonance case Ω 3 ≅  1 ,  2 ≅  1 , and  3 ≅ 3 1 .It is clear from Figure 3 that the steady-state amplitude of the first, second, and third modes is increased to about 370%, 890%, and 260%, respectively, of its value shown in Figure 2. Also, it can be seen that the time response of the system is tuned with multilimit cycle.
It is quite clear that such case is undesirable in the design of such system because it represents one of the worst behaviors of the system.Such case should be avoided as Mathematical Problems in Engineering working condition for the system.It is advised for such system not to have  2 =  1 or  3 = 3 1 .Figure 4 shows the comparison between numerical integration for the system equation (4a), (4b), and (4c) solid lines and the amplitudephase modulating equation (20) dashed lines.We found that all predictions from analytical solutions dashed lines are in good agreement with the numerical simulation solid lines.

FDM with Approximation O(𝑐 2
).The infinite equations (2a)-(2e) will be solved via a finite difference method (FDM).We briefly describe the procedure here.More details are available in [23,26].The infinite dimensional equations (2a)-(2e) can be reduced to the finite dimensional one via the finite difference method with second-order approximation ( 2 ).Namely, at each mesh node the following system of ordinary differential equations is obtained: where  denotes the partition number regarding a spatial coordinate,  is the computational step regarding spatial coordinate, and  1, (⋅),  2, (⋅),   (32) The obtained system of (31) with the supplemented boundary conditions equation and the initial conditions equation is solved by the fourth-order Runge-Kutta method.Figure 5 shows a comparison between the time responses of the system equations (4a), (4b), and (4c) using the Runge-Kutta of fourth-order method and the time response of the problems (31), using the finite difference method at the same values of the parameters shown in Figure 2.

Frequency Response Curves.
When the amplitude achieves a constant nontrivial value, a steady-state vibration exists.Using the frequency response equations we can assess the influence of the damping coefficients, the nonlinear parameters, and the excitation amplitude on the steady-state amplitudes.The frequency response equations ( 24)-( 29) are nonlinear equations in  1 ,  2 , and  3 which are solved numerically.The numerical results are shown in Figures 6-25, and in all figures the region of stability of the nonlinear solutions is determined by applying the Routh-Hurwitz criterion.The solid lines stand for the stable solution, and the dotted lines stand for the unstable solution.From the geometry of the figures, we observe that each curve is continuous and has stable and unstable solutions.

Response Curve of Case 1.
To check the accuracy of the analytical solution derived by the multiple time scale in predicting the amplitude of the first mode, we compare the amplitude of the first mode obtained from frequency response equation of Case 1 with values obtained from numerical integration of (4a). Figure 6 shows a comparison of these outputs for the first mode.The effects of the detuning parameter  1 on the steady-state amplitude of the first mode for the stability first case, where  1 ̸ = 0,  2 = 0, and  3 = 0, for the parameters  1 = 0.2,  7 = 0.01,  1 = 2.3,  1 = 4,  11 = 0.1,  12 = 0.2,  14 = 0.5, Ω 1 = 1, Ω 2 = 1.2, and Ω 4 = 1.4,as shown in Figure 6.excitation amplitude  1 , the nonlinear spring stiffness  7 , and the parametric excitation  14 .Figures 7 and 8 show that the steady-state amplitude  1 is inversely proportional to  1 and  1 , and also for decreasing  1 or  1 the curve is bending to the left.It is clear from Figure 9 that the steady-state amplitude  1 is increasing for increasing value of external excitation force  1 , and the zone of instability is increased.Figure 10 shows that as the nonlinear spring stiffness  7 is increased; the continuous curve is moved downwards.Also, the negative and positive values of  7 produce either hard or soft spring, respectively, as the curve is either bent to the right or to the left, leading to the appearance of the jump phenomenon.The region of stability is increased for increasing value of  7 .From Figure 11, we observe that for increasing value of parametric excitation amplitude  14 , the steady-state amplitude of the first mode is increased, and the curve is shifted to the left.12-16, show the frequency response curves for the stability of the second case, where  2 ̸ = 0,  1 = 0, and  3 = 0. Figure 12 shows a com-parison of these outputs for the second mode.The effects of the detuning parameter  2 on the steady-state amplitude of the second mode for the stability second case, where  2 ̸ = 0,  1 = 0,  3 = 0, for the parameters:  2 = 0.2,  8 = 0.01,  2 = 4,  2 = 4,  21 = 0.1,  22 = 0.2,  24 = 0.5, Ω 1 = 1, Ω 2 = 1.2, and Ω 4 = 1.4,as shown in Figure 12.It can be seen from the figure that maximum steady-state amplitude occurs when  2 ≅  1 .Figure 13 shows that as the nonlinear spring stiffness  8 is increased, the continuous curve is moved downwards.Also, the positive and negative values of  8 produce either soft or hard spring, respectively, as the curve is either bent to the left or to the right, leading to the appearance of the jump phenomenon.Figures 14,15,and 16 show that the steadystate amplitude  2 is inversely proportional to  2 ,  2 and directly proportional to the external excitation  2 .Also, for decreasing  2 ,  2 the curve is bending to the left.Case 6. Figures 17,18,19,20,21,22,23,24,and 25 show that the frequency response curves for practical case stability, where  1 ̸ = 0,  2 ̸ = 0, and  3 ̸ = 0. Figure 17 shows that the effects of the detuning parameter  1 on the amplitudes of the three modes.From this figure, we observe that these modes intersect, and for positive value of the detuning parameter  1 the amplitudes are stable.For negative value of the detuning parameter  1 down to and including −0.5 the system becomes unstable.Figure 18 shows that the steady-state amplitudes of the three modes  1 ,  2 , and  3 are directly proportional to the external excitation force  1 .Also, form this figure we show that the stability region is decreased for increasing  1 .Figures 19-21 show that the steady-state amplitudes and stability of the three modes  1 ,  2 , and  3 are inversely proportional to the nonlinear parameters ( 1 ,  2 ,  8 ), ( 1 ,  2 ,  8 ), and ( 1 ,  2 ,  8 ), respectively.

Response Curve of
Figures 22-24, show the effects of the nonlinear parameters ( 7 ,  7 ,  7 ), ( 3 ,  3 ,  3 ), and ( 5 , 5 ,  5 ) on the steadystate amplitudes of the three modes.It is clear that the stability regions are increased for increasing these nonlinear parameters.For increasing value of linear viscous damping coefficients, we note that the steady-state amplitudes are increasing or decreasing for the first, second, and third modes, respectively, as shown in Figure 25.The region of stability system is increased for decreasing value of damping coefficients.

Comparison Study.
In the previous work [11], the chaotic dynamics of a six-dimensional nonlinear system which represents the averaged equation of a composite laminated piezoelectric rectangular plate subjected to the transverse, inplane excitations and the excitation loaded by piezoelectric layers are analyzed.The case of 1 : 2 : 4 internal resonances is considered.
In our study, the nonlinear analysis and stability of a composite laminated piezoelectric rectangular thin plate under simultaneous external and parametric excitation forces  are investigated.The second-order approximation is obtained to consider the influence of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate using the multiple scale method.All possible resonance cases are extracted at this approximation order.The case of 1 : 1 : 3 internal resonance and primary resonance is considered.The stability of the system and the effects of different parameters on system behavior have been studied using phase plane and frequency response curves.The analytical results given by the method of multiple time scale are verified by comparison with results from numerical integration of the modal equations.Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM).Variation of the some parameters leads to multivalued amplitudes and hence to jump phenomena.It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system.Such cases should be avoided as working conditions for the system.

Conclusions
Multiple time scale perturbation method is applied to determine second-order approximate solutions for rectangular symmetric cross-ply laminated composite thin plate subjected to external and parametric excitations.Second-order approximate solutions are obtained to study the influence of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate.All possible resonance cases are extracted at this approximation order.The study is focused on the case of 1 : 1 : 3 primary resonance and internal resonance, where Ω 3 ≅  1 ,  2 ≅  1 , and  3 ≅ 3 1 .The analytical results given by the method of multiple time scale are verified by comparison with results from numerical integration of the modal equations.Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM).The stability of a composite laminated piezoelectric rectangular thin plate is investigated.The phase-plane method and frequency response curves are applied to study the stability of the system.From the previous study the following may be concluded.
(1) The simultaneous resonance case Ω 3 ≅  1 ,  2 ≅  1 , and  3 ≅ 3 1 is one of the worst case, and they should be avoided in design of such system.Of course, the excitation frequency Ω 3 is out of control.But this case should be avoided through having  2 ̸ =  1 or  3 ̸ = 3 1 .
(2) A comparison between the solutions obtained numerically with that prediction from the multiple scales shows an excellent agreement.
(3) Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM).
(5) For the first and second modes, the steady-state amplitudes  1 and  2 are directly proportional to the excitation amplitude  1 and  2 and inversely proportional to the linear damping  1 and  2 , respectively.
(6) The negative and positive values of nonlinear stiffness  7 ,  8 produce either hard or soft spring, respectively, as the curve is either bent to the right or to the left.(7) The region of instability increase, which is undesirable, for increasing excitation amplitudes  1 ,  2 and for negative values of nonlinear stiffness  7 ,  8 .
For further work, we intend to extend our work to explore the modeling formulation by investigating the role static loading (in addition to the dynamic component).

Figure 1 :
Figure 1: The model of the composite laminated piezoelectric rectangular plate.

3 Figure 2 :
Figure 2: Time response and phase-plane diagrams of the system for nonresonance case.

3 Figure 5 :
Figure 5: Time response of composite laminated rectangular plate using RKM and FDM.

2 Figure 25 :
Figure 25: Effects of the linear damping coefficients.