Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 418374, 26 pages

http://dx.doi.org/10.1155/2013/418374

Research Article

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

M. Sayed1,2 and A. A. Mousa1,3

¹Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 888, Al-Taif, Saudi Arabia

²Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt

³Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shibin El-Kom, Egypt

Received 10 November 2012; Revised 15 April 2013; Accepted 1 May 2013

Academic Editor: Dane Quinn

Copyright © 2013 M. Sayed and A. A. Mousa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plate under parametric and external excitations is presented. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the system up 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, where , , and , is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influences of the cubic terms on nonlinear 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 the modal equations. Reliability of the obtained results is verified by comparison between the finite difference method (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 , , , , 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.

1. 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-degrees-of-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 second-order 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, in-plane 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 [14–16] 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-degree-of-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 phase-plane 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 phase-plane method. Awrejcewicz et al. [23–25] 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.

2. Mathematical Analysis

Consider a simply supported four edges composite laminated piezoelectric rectangular plate of lengths ,   and thickness , as shown in Figure 1. The composite laminated piezoelectric rectangular plate is considered as regular symmetric cross-ply laminates with layers. A Cartesian coordinate system is located in the middle surface of the plate. Assume that ,  , 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 in the form , and the excitations are loaded along the direction at in the form . The transverse excitation subjected to the composite laminated piezoelectric rectangular plate is represented by . The dynamic electrical loading is expressed as . 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]

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

where , , and 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.

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 , and are the vibration amplitudes of the composite laminated piezoelectric rectangular plate for the first-order, second-order, and the third-order modes, respectively, , , and are the linear viscous damping coefficients, , , and are the natural frequencies of the rectangular plate, and , , , and are the excitations frequencies. , , , and are the amplitudes of parametric and external excitation forces corresponding to the three nonlinear modes, and , , and are the nonlinear coefficients. The linear viscous damping and exciting forces are assumed to be

where is a small perturbation parameter and .

The external excitation forces are of the order 2, and the linear viscous damping , parametric exciting forces , , and 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 second-order approximate solution of (4a), (4b), and (4c). Method of multiple scales [29–31] is applied to obtain a second-order 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 , . We seek a uniform approximation to the solution of (4a), (4b), and (4c) in the form:

Terms of 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  

Order  

The general solutions of (9a), (9b), and (9c) can be written in the form

where , , and are a complex function in , 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 are the complex functions in and . From (12a)–(13c) into (11a), (11b), and (11c) and eliminating the secular terms, the second-order approximation is given by

where are the complex functions in and . From the above derived solutions, the reported resonance cases are the following.

(i)Primary resonance: , , , , and .(ii)Subharmonic resonance: , , , and .(iii)Internal or secondary resonance: , , , , , ,  , , and .(iv)Combined resonance: , , , , , , , , , , , , ,  , and   .(v)Simultaneous or incident resonance.

Any combination of the previous resonance cases is considered as simultaneous resonance.

3. Stability Analysis

The behavior of such a system can be very complex, especially when the natural frequencies and the forcing frequency satisfy certain internal and external resonance conditions. The study is focused on the case of 1 : 1 : 3 primary resonance and internal resonance, where , , and . To describe how close the frequencies are to the resonance conditions, we introduce detuning parameters as follows:

where and , are called the external and internal detuning parameters, respectively. Eliminating the secular terms leads to solvability conditions for the first- and second-order expansions as follows:

From (7a), multiplying both sides by , we get

To analyze the solutions of (16a)–(17c), we express in the polar form as follows:

where and are the steady-state amplitudes and phases of the motion, respectively. Substituting (16a)–(17c) and (19) into (18) and equating the real and imaginary parts, we obtain the following equations describing the modulation of the amplitudes and phases of the response:

where

Steady-state solutions of the system correspond to the fixed points of (20), which in turn correspond to

Hence, the fixed points of (20) are given by

There are six possibilities besides the trivial solution as follows:

(1), , (single mode), (2), , (single mode), (3), , (two modes), (4), , (two modes),(5), , (two modes), (6), , (three modes).

Case 1. In this case, where , , the frequency response equation is given by

Case 2. In this case, where , , the frequency response equation is given by

Case 3. In this case, where , the frequency response equations are given by

Case 4. In this case, where , the frequency response equations are given by

Case 5. In this case, where , the frequency response equations are given by

Case 6. In this case, where , , and , this is the practical case, the frequency response equations are given by

where

In the frequency response curves, the stable (unstable) steady-state solutions have been represented by solid (dotted) lines.

4. Numerical Results and Discussion

To study the behavior of the system, the Runge-Kutta fourth-order 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: , , , , , , , , , , , , , , , , , , , , , , , , and . Figure 3 shows the steady-state amplitudes and phase plane of the system at simultaneous resonance case , , and . 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.

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

Figure 3: Time response and phase-plane diagrams of the system at simultaneous resonance case, , , and .

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 working condition for the system. It is advised for such system not to have or . Figure 4 shows the comparison between numerical integration for the system equation (4a), (4b), and (4c) solid lines and the amplitude-phase 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.

Figure 4: Comparison between numerical solution (using RKM) and analytical solution (using perturbation method) of the system at resonance case, , , and .

4.1. FDM with Approximation

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 . 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 , , , , and are the difference operators as follows:

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.

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

4.2. 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 , , and 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.

Figure 6: Comparison between analytical prediction using multiple time scale and numerical integration of the first mode.

Figure 7: Effects of the linear damping .

Figure 8: Effects of the natural frequency .

Figure 9: Effects of the external excitation .

Figure 10: Effects of the nonlinear parameter .

Figure 11: Effects of the parametric excitation .

Figure 12: Comparison between analytical prediction using multiple time scale and numerical integration of the second mode.

Figure 13: Effects of the nonlinear parameter .

Figure 14: Effects of the linear damping .

Figure 15: Effects of the natural frequency .

Figure 16: Effects of the external excitation .

Figure 17: Effects of the detuning parameter .

Figure 18: Effects of the external excitation force .

Figure 19: Effects of the nonlinear parameters (, , and coupling terms).

Figure 20: Effects of the nonlinear parameters (, , and coupling terms).

Figure 21: Effects of the nonlinear parameters (, , and coupling terms).

Figure 22: Effects of the nonlinear parameters (, , and coupling terms).

Figure 23: Effects of the nonlinear parameters (, , and coupling terms).

Figure 24: Effects of the nonlinear parameters (, , and coupling terms).

Figure 25: Effects of the linear damping coefficients.

4.2.1. 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 on the steady-state amplitude of the first mode for the stability first case, where , , and , for the parameters , , , ,  , , , , , and , as shown in Figure 6.

Figures 7–11, show the effects of the damping coefficient , the first mode natural frequency , the external excitation amplitude , the nonlinear spring stiffness , and the parametric excitation . Figures 7 and 8 show that the steady-state amplitude is inversely proportional to and , and also for decreasing or the curve is bending to the left. It is clear from Figure 9 that the steady-state amplitude is increasing for increasing value of external excitation force , and the zone of instability is increased. Figure 10 shows that as the nonlinear spring stiffness is increased; the continuous curve is moved downwards. Also, the negative and positive values of 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 . From Figure 11, we observe that for increasing value of parametric excitation amplitude , the steady-state amplitude of the first mode is increased, and the curve is shifted to the left.

4.2.2. Response Curve of Case 2

Figures 12–16, show the frequency response curves for the stability of the second case, where , , and . Figure 12 shows a comparison of these outputs for the second mode. The effects of the detuning parameter on the steady-state amplitude of the second mode for the stability second case, where , , , for the parameters: , , , , , , , , , and , as shown in Figure 12. It can be seen from the figure that maximum steady-state amplitude occurs when . Figure 13 shows that as the nonlinear spring stiffness is increased, the continuous curve is moved downwards. Also, the positive and negative values of 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 steady-state amplitude is inversely proportional to , and directly proportional to the external excitation . Also, for decreasing , the curve is bending to the left.

4.2.3. Response Curve of Case 6

Figures 17, 18, 19, 20, 21, 22, 23, 24, and 25 show that the frequency response curves for practical case stability, where , , and . Figure 17 shows that the effects of the detuning parameter on the amplitudes of the three modes. From this figure, we observe that these modes intersect, and for positive value of the detuning parameter the amplitudes are stable. For negative value of the detuning parameter down to and including −0.5 the system becomes unstable.

Figure 18 shows that the steady-state amplitudes of the three modes , , and are directly proportional to the external excitation force . Also, form this figure we show that the stability region is decreased for increasing . Figures 19–21 show that the steady-state amplitudes and stability of the three modes , , and are inversely proportional to the nonlinear parameters , , and , respectively.

Figures 22–24, show the effects of the nonlinear parameters , , and on the steady-state 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.

4.3. 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, in-plane 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.

5. 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 , , and . 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 , and is one of the worst case, and they should be avoided in design of such system. Of course, the excitation frequency is out of control. But this case should be avoided through having or .(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).(4)Variation of the parameters , , , , , , , leads to multivalued amplitudes and hence to jump phenomena.(5)For the first and second modes, the steady-state amplitudes and are directly proportional to the excitation amplitude and and inversely proportional to the linear damping and , respectively.(6)The negative and positive values of nonlinear stiffness , 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 , and for negative values of nonlinear stiffness , .(8)The multivalued solutions are disappeared for increasing linear damping coefficients , .(9)The region of stability increases, which is desirable for decreasing excitation amplitude , nonlinear parameters , , and for increasing nonlinear stiffness .(10)The steady-state amplitudes of the three modes , , and are directly proportional to the excitation amplitude and inversely proportional to the nonlinear parameters , , and , respectively.

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).

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions for improving the quality of this paper.

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