Mathematical Problems in Engineering

Volume 2010 (2010), Article ID 738648, 12 pages

http://dx.doi.org/10.1155/2010/738648

## Nonlinear Dynamic Response of Functionally Graded Rectangular Plates under Different Internal Resonances

^{1}College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing 100192, China^{2}College of Mechanical Engineering, Beijing University of Technology, Beijing 100142, China^{3}National Key Laboratory of Mechatronics Engineering and Control, Beijing Institute of Technology, Beijing 100081, China

Received 17 November 2009; Revised 29 April 2010; Accepted 1 May 2010

Academic Editor: Carlo Cattani

Copyright © 2010 Y. X. Hao et al. 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

The nonlinear dynamic response of functionally graded rectangular plates under combined transverse and in-plane excitations is investigated under the conditions of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent and vary along the thickness direction. The thermal effect due to one-dimensional temperature gradient is included in the analysis. The governing equations of motion for FGM rectangular plates are derived by using Reddy's third-order plate theory and Hamilton's principle. Galerkin's approach is utilized to reduce the governing differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms, which are then solved numerically by using 4th-order Runge-Kutta algorithm. The effects of in-plane excitations on the internal resonance relationship and nonlinear dynamic response of FGM plates are studied.

#### 1. Introduction

Functionally graded materials (FGMs) are new engineering materials. Due to their advantages of being able to withstand severe high-temperature gradient while maintaining structural integrity, FGMs are considered to be advanced composite materials in high temperature and vibration environments [1, 2].

With the increasing use of FGMs, it is important to understand the nonlinear vibration behavior of FGM structures. Quite a few studies in this area have been conducted. Praveen and Reddy [3] analyzed the nonlinear static and dynamic response of functionally graded ceramic-metal plates in a steady temperature field based on the first-order shear deformation plate theory.Sundararajanet al. [4] carriedout finite element analysis of nonlinear-free vibration of both rectangular and skew FGM plates. Yang et al. [5] investigated the large amplitude vibration of pre-stressed FGM plates composed of a functionally graded layer and two surface-mounted piezoelectric actuator layers.

A semi analytical method and Galerkin technique were employed to predict the nonlinear vibration behavior of FGM-laminated plates. The parametric resonance of functionally graded rectangular plates under harmonic in-plane loading was investigated by Ng et al. [6]. Using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless local Petrov-Galerkin (MLPG) method, Qian et al. [7] analyzed the static deformation, and free and forced vibrations of a thick rectangular functionally graded plate.Veland Batra [8] gave a three-dimensional exact solution for the linear free and forced vibration of simply supported FGM rectangular plates. Woo and Meguid [9] studied the nonlinear deflection of FGM plates and shells under transverse mechanical loads and a temperature field. Hao et al. [10] reported a nonlinear dynamic analysis of a simply supported FGM rectangular plate subjected to transversal and in-plane excitations. The resonant case considered in their work is 1 : 1 internal resonance and principal parametric resonance. The asymptotic perturbation method is used to obtain four-dimensional nonlinear averaged equation. It was found that periodic, and quasiperiodic solutions and chaotic motions occur under some conditions. It is known that for a two-degree-of-freedom nonlinear vibration system, different internal resonance between two modes, such as 1 : 1, 1 : 2, and 1 : 3 internal resonances, can exist in some cases. To the best of the authors’ knowledge, there is still no literature concerning nonlinear dynamic behavior of FGM plates with different cases of internal resonances.

The present work aims to investigate the nonlinear dynamic response of a simply supported FGM rectangular plate subjected to transversal and in-plane excitations in a thermal environment. The cases considered in this paper include 1 : 1, 1 : 2, and 1 : 3 internal resonances and principal parametric resonance-1/2 subharmonic resonance. It is assumed that the material properties of the plate are graded in the thickness direction according to a power-law distribution. The analysis is based on the nonlinear dynamic governing equations derived in our previous work [10]. The influences of the in-plane excitations on the internal resonance relationship and nonlinear dynamic response of the FGM plate are studied in numerical examples.

#### 2. Theoretical Formulation

##### 2.1. Material Properties

It is assumed that the bottom surface of the plate is metal rich, whereas the top surface is ceramic rich. The material properties , such as Young’s modulus , the coefficient of thermal expansion , thermal conductivity , and mass density , can be expressed as a function of temperature as [11] where , , , , and are temperature coefficients.

The effective material properties of the FGM plate can be expressed as where subscripts “” and “” represent the top and bottom surfaces of the FGMs plate, respectively, and and are the volume fraction of ceramic and metal which add to unity

The metal volume fraction is defined as where exponent is a real number that characterizes the material profile along plate thickness.

From (2.2)–(2.4), the effective values of , , , and at an arbitrary point of the plate can be expressed as

It is also assumed that the plate is initially stress free at and is subjected to a uniform temperature variation that is constant in the plane of the plate while varies in the thickness direction only. In this case, the temperature distribution along plate thickness can be obtained from a steady-state heat transfer equation:

This equation is solved by imposing boundary condition of at and at . As a special case, the solution of (2.6) for isotropic homogeneous material, may be expressed as

##### 2.2. Equations of Motion

A simply supported four-edges FGMs rectangular plate of length *a*, width *b* and thickness *h*, which is subjected to the in-plane and transversal excitations is considered, as shown in Figure 1. The in-plane excitation of the FGMs plate is distributed along the direction at and and is of the form . The transversal excitation subject to the FGMs plate is represented by . Here the and are the frequencies of the transversal excitation and the in-plane excitation, respectively.

As usual, the coordinate has its origin at the corner of the plate on the middle plane. Assume that and represent the displacements of an arbitrary point and a point in the middle surface of the FGMs rectangular plate in the , and directions, respectively. It is also assumed that and , respectively, represent the mid-plane rotations of two transverse normals about the and axes. With Reddy’s third-order shear deformation plate theory [12], the displacements of the FGM plate can be expressed as follows:

Based on the nonlinear strains-displacement relation and the above displacement field, we obtain where

Taking into account the thermal effects, the linear stress-strain constitutive relationship is where are elastic stiffness elements [12].

According to Reddy’s third-order shear deformation theory and Hamilton’s principle, the nonlinear governing equations of motion for the FGM rectangular plate are given as [10] where is the damping coefficient, a comma denotes the partial differentiation with respect to a specified coordinate, and a super dot implies the partial differentiation with respect to time.

All kinds of inertias in (2.13) are calculated by the stress resultants are represented as follows where the membrane stress resultants, moments, higher-order moments, transverse shear stress resultants, and their higher-order counterparts are represented as follows: The stiffness elements of the FGMs plate are denoted by

And the thermal stress resultants in (2.16) can be represented as where The nonlinear governing equations of motion for the FGM rectangular plate can be expressed in ters of displacements by substituting for the force and moments resultants. The equations of motion are very complicate nonlinear partial differential equations that can be seen in the conference [10].

The boundary conditions for the simply supported FGM plate requires that

at and ,

at and , The present study focuses on the nonlinear transverse oscillations of FGM plates in the first two modes. It is then reasonable to construct deflection functions as a combination of the first two vibration mode shapes as follows: where and are the amplitudes of two modes, respectively.

The transverse excitation can be represented as where and represent the amplitudes of the transverse forcing excitation corresponding to the two nonlinear modes.

Based on research given in [13, 14], neglecting all inertia terms on , , , and in (2.13), we can obtain the displacements , , , and with respect to *w*. Then by the Galerkin procedure, the governing differential equations of transverse motion of the FGMs rectangular plate are obtained
where and are the vibration amplitudes of the first two modes, respectively. and are the amplitudes of the transverse excitation force corresponding to the two nonlinear modes. The lengthy expressions of constants , and the transverse excitation force and are not given here for brevity.

The present study focuses on the transverse nonlinear oscillations of a simply supported FGM rectangular plate in the first two modes.

The first two linear frequencies of this nonlinear dynamic system can be rewritten as where is the static component in the in-plane excitation. The other coefficients in (2.13) are functions of geometric and physical parameters, in-plane excitations, and temperature field. That means that under different conditions, the system can have different internal resonance and exhibit different dynamic response.

It is seen that the in-plane stationary excitation can change the type of internal resonance.

When is close to , the one-to-one internal resonance occurs and is as follows:

When or , the one-to-two or one-to-three internal resonance occurs. The in-plane forces in these cases are given by (2.27)

#### 3. Numerical Results

The influence of in-plane stationary excitation on internal resonance is studied. The fourth-order Runge-Kutta algorithm is employed to numerically solve (2.11) and (2.12) to obtain the nonlinear dynamic response of the FGM rectangular plate subjected to thermal and mechanical loads with various internal resonance and primary parametric resonance.

Aluminum oxide and Ti-6Al-4V are chosen to be the constituent materials of the plate ( m, ). The volume fraction exponent is . The transverse load amplitude is . In addition, the plate is subjected to a temperature field where the aluminum oxide rich top surface is held at 900 K and the Ti-6Al-4V rich bottom surface is held at 300 K. Their temperature-dependent material properties evaluated at are as follows.

*Ti-6Al-4V:*

*Aluminum oxide:*

Figures 2–4 depict, respectively, nonlinear dynamic response of FGM plates. The plots of phase portrait for the cases of 1 : 1, 1 : 2 and 1 : 3 internal resonance with different in-plane stationary loading are shown in Figures 2(a), 3(a), and 4(a) and the central deflection versus time curve is displayed in Figures 2(b), 3(b), and 4(b). The combinational resonance of the additive type is

It is observed that the central deflections are reduced by increasing the ratio of the two frequencies. In the case of 1 : 2 internal resonance the amplitude of the central deflection is larger than the one at other two frequency ratios. The case of internal resonance can be controlled by changing the in-plane excitation force, indicating that in the different case of internal resonance there is a different fundamental frequency.

Obviously, Figure 2 illustrates that the periodic response of the FGM rectangular plate occurs at 1 : 1 internal resonance when the is as . Figures 3 and 4 show that the beat vibration and quasiperiod dynamic response take place at 1 : 2 internal resonance when is as and 1 : 3 internal resonance when is as , respectively.

#### 4. Conclusions

The nonlinear dynamics response of FGM rectangular plates under combined transverse and in-plane excitations is investigated in the cases of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent. Based on Reddy’s third-order shear deformation plate theory, the governing equations of motion for the FGMs rectangular plate are derived using Hamilton’s principle. Galerkin’s approach is used to reduce the governing equations of motion to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. 1 : 1, 1 : 2 and 1 : 3 internal resonance and principal parametric resonance-1/2 subharmonic resonance are considered and solutions are obtained by using fourth-order Runge-Kutta method.

Numerical results show that plate geometry parameter, in-plane excitation and temperature field play important role in the internal resonance relationship and the nonlinear dynamic behavior of the FGM plate. In the case of 1 : 2 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the vibration amplitude at the plate center is much greater than the one at other two cases of internal resonance. So in the actual condition, it is necessary to analyze what kinds of internal resonance may occur and how to control them.

#### Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant nos. 10732020 and 10972026 and the Science Foundation of Beijing Municipal Education Commission through Grant nos. KM200910772004 and KM201010772003.

#### References

- J. Yang and X.-L. Huang, “Nonlinear transient response of functionally graded plates with general imperfections in thermal environments,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 196, no. 25–28, pp. 2619–2630, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C.-S. Chen and A.-H. Tan, “Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates,”
*Composite Structures*, vol. 78, no. 4, pp. 529–536, 2007. View at Publisher · View at Google Scholar · View at Scopus - G. N. Praveen and J. N. Reddy, “Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates,”
*International Journal of Solids and Structures*, vol. 35, no. 33, pp. 4457–4476, 1998. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. Sundararajan, T. Prakash, and M. Ganapathi, “Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments,”
*Finite Elements in Analysis and Design*, vol. 42, no. 2, pp. 152–168, 2005. View at Publisher · View at Google Scholar · View at Scopus - J. Yang, S. Kitipornchai, and K. M. Liew, “Large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 192, no. 35-36, pp. 3861–3885, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Y. Ng, K. Y. Lam, and K. M. Liew, “Effects of FGM materials on the parametric resonance of plate structures,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 190, no. 8–10, pp. 953–962, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L. F. Qian, R. C. Batra, and L. M. Chen, “Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method,”
*Composites Part B*, vol. 35, no. 6–8, pp. 685–697, 2004. View at Google Scholar - S. S. Vel and R. C. Batra, “Three-dimensional exact solution for the vibration of functionally graded rectangular plates,”
*Journal of Sound and Vibration*, vol. 272, no. 3–5, pp. 703–730, 2004. View at Google Scholar · View at Scopus - J. Woo and S. A. Meguid, “Nonlinear analysis of functionally graded plates and shallow shells,”
*International Journal of Solids and Structures*, vol. 38, no. 42-43, pp. 7409–7421, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. X. Hao, L. H. Chen, W. Zhang, and J. G. Lei, “Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate,”
*Journal of Sound and Vibration*, vol. 312, no. 4-5, pp. 862–892, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. C. Pradhan, C. T. Loy, K. Y. Lam, and J. N. Reddy, “Vibration characteristics of functionally graded cylindrical shells under various boundary conditions,”
*Applied Acoustics*, vol. 61, no. 1, pp. 111–129, 2000. View at Publisher · View at Google Scholar · View at Scopus - J. N. Reddy, “A refined nonlinear theory of plates with transverse shear deformation,”
*International Journal of Solids and Structures*, vol. 20, no. 9-10, pp. 881–896, 1984. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Bhimaraddi, “Large amplitude vibrations of imperfect antisymmetric angle-ply laminated plates,”
*Journal of Sound and Vibration*, vol. 162, no. 3, pp. 457–470, 1993. View at Publisher · View at Google Scholar · View at Scopus - A. Nosier and J. N. Reddy, “A study of non-linear dynamic equations of higher-order shear deformation plate theories,”
*International Journal of Non-Linear Mechanics*, vol. 26, no. 2, pp. 233–249, 1991. View at Google Scholar · View at Scopus