Mathematical Problems in Engineering

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

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

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

^{1}Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 888, Al-Taif, Saudi Arabia^{2}Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt^{3}Department 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 , , , , and 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]

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