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

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 𝑃, such as Young’s modulus 𝐸, the coefficient of thermal expansion 𝛼, thermal conductivity 𝜅, and mass density 𝜌, can be expressed as a function of temperature as [11]𝑃_𝑖=𝑃₀(𝑃₋₁𝑇⁻¹+1+𝑃₁𝑇+𝑃₂𝑇²+𝑃₃𝑇³), (2.1) where 𝑃₀, 𝑃₋₁, 𝑃₁, 𝑃₂, and 𝑃₃ are temperature coefficients.

The effective material properties 𝑃 of the FGM plate can be expressed as𝑃=𝑃_𝑡𝑉_𝑐+𝑃_𝑏𝑉_𝑚, (2.2) where subscripts “𝑡” and “𝑏” represent the top and bottom surfaces of the FGMs plate, respectively, and 𝑉_𝑐 and 𝑉_𝑚 are the volume fraction of ceramic and metal which add to unity𝑉_𝑐+𝑉_𝑚=1. (2.3)

The metal volume fraction 𝑉_𝑚 is defined as𝑉_𝑚(𝑧)=(2𝑧+ℎ)2ℎ^𝑁, (2.4) where exponent 𝑁 is a real number that characterizes the material profile along plate thickness.

From (2.2)–(2.4), the effective values of 𝐸, 𝛼, 𝜌, and 𝜅 at an arbitrary point of the plate can be expressed as𝐸=(𝐸_𝑏−𝐸_𝑡)𝑉_𝑚+𝐸_𝑡,𝛼=(𝛼_𝑏−𝛼_𝑡)𝑉_𝑚+𝛼_𝑡,𝜌=(𝜌_𝑏−𝜌_𝑡)𝑉_𝑚+𝜌_𝑡,𝜅=(𝜅_𝑏−𝜅_𝑡)𝑉_𝑚+𝜅_𝑡. (2.5)

It is also assumed that the plate is initially stress free at 𝑇₀ and is subjected to a uniform temperature variation Δ𝑇=𝑇−𝑇₀ that is constant in the 𝑥𝑦 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: −𝑑𝑑𝑧[𝜅(𝑧)𝑑𝑇𝑑𝑧]=0. (2.6)

This equation is solved by imposing boundary condition of 𝑇=𝑇_𝑏 at 𝑧=ℎ/2 and 𝑇=𝑇_𝑡 at 𝑧=−ℎ/2. As a special case, the solution of (2.6) for isotropic homogeneous material, may be expressed as𝑇𝑇(𝑧)=^𝑡+𝑇_𝑏2+𝑇_𝑏−𝑇_𝑡ℎ𝑧. (2.7)

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 𝑦 direction at 𝑥=0 and 𝑥=𝑎 and is of the form 𝑝₀−𝑝₁cosΩ₂𝑡. The transversal excitation subject to the FGMs plate is represented by 𝐹(𝑥,𝑦)cosΩ₁𝑡. Here the Ω₁ and Ω₂ are the frequencies of the transversal excitation and the in-plane excitation, respectively.

Figure 1 

The model of a FGMs rectangular plate and the coordinate system.

As usual, the coordinate 𝑂𝑥𝑦𝑧 has its origin at the corner of the plate on the middle plane. Assume that (𝑢,𝑣,𝑤) and (𝑢₀,𝑣₀,𝑤₀) represent the displacements of an arbitrary point and a point in the middle surface of the FGMs rectangular plate in the 𝑥, 𝑦 and 𝑧 directions, respectively. It is also assumed that 𝜙_𝑥 and 𝜙_𝑦, respectively, represent the mid-plane rotations of two transverse normals about the 𝑥 and 𝑦 axes. With Reddy’s third-order shear deformation plate theory [12], the displacements of the FGM plate can be expressed as follows:𝑢(𝑥,𝑦,𝑡)=𝑢₀(𝑥,𝑦,𝑡)+𝑧𝜙_𝑥(𝑥,𝑦,𝑡)−𝑧³43ℎ²(𝜙_𝑥+𝜕𝑤₀𝜕𝑥),𝑣(𝑥,𝑦,𝑡)=𝑣₀(𝑥,𝑦,𝑡)+𝑧𝜙_𝑦(𝑥,𝑦,𝑡)−𝑧³43ℎ²(𝜙_𝑦+𝜕𝑤₀𝜕𝑦),𝑤(𝑥,𝑦,𝑡)=𝑤₀(𝑥,𝑦,𝑡) (2.8)

Based on the nonlinear strains-displacement relation and the above displacement field, we obtain𝜀_𝑥𝑥=𝜕𝑢+1𝜕𝑥2(𝜕𝑤)𝜕𝑥²,𝜀_𝑦𝑦=𝜕𝑣+1𝜕𝑦2(𝜕𝑤)𝜕𝑦²,𝛾_𝑥𝑦=12(𝜕𝑢+𝜕𝑥𝜕𝑣+𝜕𝑦𝜕𝑤𝜕𝑥𝜕𝑤𝛾𝜕𝑦),_𝑦𝑧=12(𝜕𝑣+𝜕𝑧𝜕𝑤𝜕𝑦),𝛾_𝑧𝑥=12(𝜕𝑢+𝜕𝑧𝜕𝑤𝜀𝜕𝑥),_𝑥𝑥𝜀_𝑦𝑦𝛾_𝑥𝑦=𝜀⁽⁰⁾_𝑥𝑥𝜀⁽⁰⁾_𝑦𝑦𝛾⁽⁰⁾_𝑥𝑦𝜀+𝑧𝜀^{(1)𝑥𝑥(1)}_𝑦𝑦𝛾⁽¹⁾_𝑥𝑦+𝑧³𝜀⁽³⁾_𝑥𝑥𝜀⁽³⁾_𝑦𝑦𝛾⁽³⁾_𝑥𝑦,𝛾_𝑦𝑧𝛾_𝑧𝑥=𝛾⁽⁰⁾_𝑦𝑧𝛾⁽⁰⁾_𝑧𝑥+𝑧²𝛾⁽²⁾_𝑦𝑧𝛾⁽²⁾_𝑧𝑥, (2.9)(2.10) where𝛾⁽⁰⁾_𝑦𝑧𝛾⁽⁰⁾_𝑧𝑥=𝜙_𝑦+𝜕𝑤₀𝜙𝜕𝑦_𝑥+𝜕𝑤₀,𝛾𝜕𝑥^(2)𝑦𝑧𝛾⁽²⁾_𝑧𝑥=−𝑐₂𝜙_𝑦+𝜕𝑤₀𝜙𝜕𝑦_𝑥+𝜕𝑤₀,𝜀𝜕𝑥_(0)𝑥𝑥𝜀⁽⁰⁾_𝑦𝑦𝛾⁽⁰⁾_𝑥𝑦=𝜕𝑢₀+1𝜕𝑥2(𝜕𝑤₀)𝜕𝑥²𝜕𝑣₀+1𝜕𝑦2(𝜕𝑤₀)𝜕𝑦²𝜕𝑢₀+𝜕𝑦𝜕𝑣₀+𝜕𝑥𝜕𝑤₀𝜕𝑥𝜕𝑤₀,𝜀𝜕𝑦^(1)𝑥𝑥𝜀⁽¹⁾_𝑦𝑦𝛾⁽¹⁾_𝑥𝑦=−𝑐₁𝜕𝜙_𝑥𝜕𝑥𝜕𝜙_𝑦𝜕𝑦𝜕𝜙_𝑥+𝜕𝑦𝜕𝜙_𝑦,𝜀𝜕𝑥_(3)𝑥𝑥𝜀⁽³⁾_𝑦𝑦𝛾⁽³⁾_𝑥𝑦=−𝑐₁𝜕𝜙_𝑥+𝜕𝜕𝑥²𝑤₀𝜕𝑥²𝜕𝜙_𝑦+𝜕𝜕𝑦²𝑤₀𝜕𝑦²𝜕𝜙_𝑥+𝜕𝑦𝜕𝜙_𝑦𝜕𝜕𝑥+2²𝑤₀,𝑐𝜕𝑥𝜕𝑦₂=3𝑐₁,𝑐₁=43ℎ². (2.11)

Taking into account the thermal effects, the linear stress-strain constitutive relationship is 𝜎_𝑥𝑥𝜎_𝑦𝑦𝜎_𝑦𝑧𝜎_𝑧𝑥𝜎_𝑥𝑦=𝑄₁₁𝑄₁₂𝑄000₂₁𝑄₂₂00000𝑄₄₄00000𝑄₅₅00000𝑄₆₆{𝜀_𝑥𝑥𝜀_𝑦𝑦𝛾_𝑦𝑧𝛾_𝑧𝑥𝛾_𝑥𝑦−𝛼_𝑥𝑥𝛼_𝑦𝑦002𝛼_𝑥𝑦Δ𝑇}, (2.12) where 𝑄 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]𝑁_{𝑥𝑥,𝑥}+𝑁_{𝑥𝑦,𝑦}=𝐼₀𝑢̈₀+(𝐼₁−𝑐₁𝐼₃)𝜙̈_𝑥−𝑐₁𝐼₃𝜕𝑤̈₀,𝑁𝜕𝑥_{𝑦𝑦,𝑦}+𝑁_{𝑥𝑦,𝑥}=𝐼₀𝑣̈₀+(𝐼₁−𝑐₁𝐼₃)𝜙̈_𝑦−𝑐₁𝐼₃𝜕𝑤̈₀,𝑁𝜕𝑦_{𝑦𝑦,𝑦}𝜕𝑤₀𝜕𝑦+𝑁_𝑦𝑦𝜕²𝑤₀𝜕𝑦²+𝑁_{𝑥𝑦,𝑥}𝜕𝑤₀𝜕𝑦+𝑁_{𝑥𝑦,𝑦}𝜕𝑤₀𝜕𝑥+2𝑁_𝑥𝑦𝜕²𝑤₀𝜕𝑦𝜕𝑥+𝑁_{𝑥𝑥,𝑥}𝜕𝑤₀𝜕𝑥+𝑁_𝑥𝑥𝜕²𝑤₀𝜕𝑥²+𝑐₁(𝑃_{𝑥𝑥,𝑥𝑥}+2𝑃_{𝑥𝑦,𝑥𝑦}+𝑃_{𝑦𝑦,𝑦𝑦})+(𝑄_𝑥,𝑥−𝑐₂𝑅_𝑥,𝑥)+(𝑄_𝑦,𝑦−𝑐₂𝑅_𝑦,𝑦)+𝐹−𝛾𝑤̇₀=𝐼₀𝑤̈₀+𝑐₁𝐼₃(𝜕𝑢̈₀+𝜕𝑣̈𝜕𝑥𝜕𝑥⁰)+𝑐₁(𝐼₄−𝑐₁𝐼₆𝜕𝜙̈)(^𝑥+𝜕𝜙̈𝜕𝑥^𝑦𝑀𝜕𝑦),_{𝑥𝑥,𝑥}+𝑀_{𝑥𝑦,𝑦}−𝑐₁𝑃_{𝑥𝑥,𝑥}−𝑐₁𝑃_{𝑥𝑦,𝑦}−(𝑄_𝑥−𝑐₂𝑅_𝑥)=(𝐼₁−𝑐₁𝐼₃)𝑢̈₀+(𝐼₂−2𝑐₁𝐼₄+𝑐²₁𝐼₆)𝜙̈_𝑥−𝑐₁(𝐼₄−𝑐₁𝐼₆)𝜕𝑤̈₀,𝑀𝜕𝑥_{𝑦𝑦,𝑦}+𝑀_{𝑥𝑦,𝑥}−𝑐₁𝑃_{𝑦𝑦,𝑦}−𝑐₁𝑃_{𝑥𝑦,𝑥}−(𝑄_𝑦−𝑐₂𝑅_𝑦)=(𝐼₁−𝑐₁𝐼₃)𝑣̈₀+(𝐼₂−2𝑐₁𝐼₄+𝑐²₁𝐼₆)𝜙̈_𝑦−𝑐₁(𝐼₄−𝑐₁𝐼₆)𝜕𝑤̈₀,𝜕𝑦 (2.13) where 𝛾 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 by𝐼_𝑖=^ℎ/2_−ℎ/2𝑧^𝑖𝑝(𝑧)𝑑𝑧,(𝑖=0,1,2,3,4,6). (2.14) the stress resultants are represented as follows𝐍𝐌𝐏=𝐀𝐁𝐄𝜺𝐁𝐃𝐅𝐄𝐅𝐇^(𝟎)𝜺^(𝟏)𝜺^(𝟑)+𝐍^𝑇𝐌^𝑇𝐏^𝑇,𝐐𝐑=𝐀𝐃𝛾𝐃𝐅𝛾^(𝟎)(𝟐), (2.15) where the membrane stress resultants, moments, higher-order moments, transverse shear stress resultants, and their higher-order counterparts are represented as follows:𝐍=[𝑁_𝑥𝑥,𝑁_𝑦𝑦,𝑁_𝑥𝑦]^𝐓,𝐌=[𝑀_𝑥𝑥,𝑀_𝑦𝑦,𝑀_𝑥𝑦]^𝐓,𝐏=[𝑃_𝑥𝑥,𝑃_𝑦𝑦,𝑃_𝑥𝑦]^𝐓,𝐐=[𝑄_𝑦𝑦,𝑄_𝑥𝑥]^𝐓,𝐑=[𝑅_𝑦𝑦,𝑅_𝑥𝑥]^𝐓. (2.16) The stiffness elements of the FGMs plate are denoted by (𝐴_𝑖𝑗,𝐵_𝑖𝑗,𝐷_𝑖𝑗,𝐸_𝑖𝑗,𝐹_𝑖𝑗,𝐻_𝑖𝑗)=^ℎ/2_−ℎ/2𝑄_𝑖𝑗(1,𝑧,𝑧²,𝑧³,𝑧⁴,𝑧⁶)𝑑𝑧,(𝑖,𝑗=1,2,6),(𝐴_𝑖𝑗,𝐷_𝑖𝑗,𝐹_𝑖𝑗)=^ℎ/2_−ℎ/2𝑄_𝑖𝑗(1,𝑧²,𝑧⁴)𝑑𝑧,(𝑖,𝑗=4,5). (2.17)

And the thermal stress resultants in (2.16) can be represented as{𝐍^𝑇,𝐌^𝑇,𝐏^𝑇𝑁}=^𝑇𝑥𝑥𝑀^𝑇_𝑥𝑥𝑃^𝑇_𝑥𝑥𝑁^𝑇_𝑦𝑦𝑀^𝑇_𝑦𝑦𝑃^𝑇_𝑦𝑦𝑁^𝑇_𝑥𝑦𝑀^𝑇_𝑥𝑦𝑃^𝑇_𝑥𝑦=^ℎ/2_−ℎ/2[𝐴_𝑥𝑥,𝐴_𝑦𝑦,𝐴_𝑥𝑦]^𝐓(1,𝑧²,𝑧³)Δ𝑇𝑑𝑧, (2.18) where𝐴_𝑥𝑥𝐴_𝑦𝑦𝐴_𝑥𝑦𝑄=−¹¹𝑄₁₂0𝑄₂₁𝑄₂₂000𝑄₆₆𝛼𝛼100100. (2.19) The nonlinear governing equations of motion for the FGM rectangular plate can be expressed in ters of displacements (𝑢₀,𝑣₀,𝑤₀,𝜙_𝑥,𝜙_𝑦) 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 𝑥=0 and 𝑥=𝑎,𝑤=𝜙_𝑦=𝑀_𝑥𝑥=𝑃_𝑥𝑥=𝑁_𝑥𝑦=0, (2.20)

at 𝑦=0 and 𝑦=𝑏, 𝑤=𝜙_𝑥=𝑀_𝑦𝑦=𝑃_𝑦𝑦=𝑁_𝑥𝑦=0,𝑁_𝑦𝑦|_𝑦=0,𝑏=0,_𝑏0𝑁_𝑥𝑥|_𝑥=0,𝑎𝑑𝑦=−^𝑏₀(𝑝₀−𝑝₁cosΩ₂𝑡)𝑑𝑦. (2.21) 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:𝑤(𝑥,𝑦,𝑡)=𝑤₁(𝑡)sin𝜋𝑥𝑎sin3𝜋𝑦𝑏+𝑤₂(𝑡)sin3𝜋𝑥𝑎sin𝜋𝑦𝑏, (2.22) where 𝑤₁ and 𝑤₂ are the amplitudes of two modes, respectively.

The transverse excitation can be represented as𝐹(𝑥,𝑦,𝑡)=𝐹₁(𝑡)sin𝜋𝑥𝑎sin3𝜋𝑦𝑏+𝐹₂(𝑡)sin3𝜋𝑥𝑎sin𝜋𝑦𝑏, (2.23) where 𝐹₁ and 𝐹₂ 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 𝑢, 𝑣, 𝜙_𝑥, and 𝜙_𝑦 in (2.13), we can obtain the displacements 𝑢, 𝑣, 𝜙_𝑥, and 𝜙_𝑦 with respect to w. Then by the Galerkin procedure, the governing differential equations of transverse motion of the FGMs rectangular plate are obtained𝑤̈₁+𝜔²₁𝑤₁+𝑎₁𝑤̇₁+𝑎₂𝑤₁cosΩ₂𝑡+𝑎₃𝑤²₁+𝑎₄𝑤²₂+𝑎₅𝑤₁𝑤²₂+𝑎₆𝑤³₁+𝑎₇𝑤₁𝑤₂=𝑓₁cosΩ₁𝑤̈𝑡,₂+𝜔²₂𝑤₂+𝑏₁𝑤̇₂+𝑏₂𝑤₂cosΩ₂𝑡+𝑏₃𝑤₁𝑤₂+𝑏₄𝑤²₁+𝑏₅𝑤²₂+𝑏₆𝑤₂𝑤²₁+𝑏₇𝑤³₂=𝑓₂cosΩ₁𝑡, (2.24) where 𝑤₁ and 𝑤₂ are the vibration amplitudes of the first two modes, respectively. 𝑓₁ and 𝑓₂ are the amplitudes of the transverse excitation force corresponding to the two nonlinear modes. The lengthy expressions of constants 𝑎₁−𝑎₇, 𝑏₁−𝑏₇ and the transverse excitation force 𝑓₁ and 𝑓₂ 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 𝜔²₁𝑚=−⁰⁰⁷+𝑝₀𝑚₀₀₈𝑚₀₀₁,𝜔²₂𝑛=−⁰⁰⁷+𝑃₀𝑛₀₀₈𝑛₀₀₂, (2.25) where 𝑝₀ 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 𝑝₀ can change the type of internal resonance.

When 𝜔₁ is close to 𝜔₂, the one-to-one internal resonance occurs and 𝑝₀ is as follows:𝑝₀₁=𝑚₀₀₇𝑛₀₀₂−𝑚₀₀₁𝑛₀₀₇𝑚₀₀₁𝑛₀₀₈−𝑚₀₀₈𝑛₀₀₂. (2.26)

When 𝜔₂≈2𝜔₁ or 𝜔₂≈3𝜔₁, the one-to-two or one-to-three internal resonance occurs. The in-plane forces in these cases are given by (2.27) 𝑝₀₂=𝑚₀₀₇𝑛₀₀₂−4𝑚₀₀₁𝑛₀₀₇4𝑚₀₀₁𝑛₀₀₈−𝑚₀₀₈𝑛₀₀₂,𝑝₀₃=𝑚₀₀₇𝑛₀₀₂−9𝑚₀₀₁𝑛₀₀₇9𝑚₀₀₁𝑛₀₀₈−𝑚₀₀₈𝑛₀₀₂. (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 (𝑎=𝑏=1 m, ℎ=𝑎/20). The volume fraction exponent is 𝑛=0.2. The transverse load amplitude is −10⁶N/m². 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 𝑇₀=300K are as follows.

Ti-6Al-4V:
𝐸=105.7GPa,𝜈=0.2981,𝜌=4429kgm³. (3.1)

Aluminum oxide:
𝐸=320.24GPa,𝜈=0.2600,𝜌=3750kgm³. (3.2)

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 is𝜔₁=Ω₁2,Ω₂=Ω₁. (3.3)

(a)

(b)

Figure 2 

Effect of in-plane excitation on the dynamic response of the FGM plate with 1  :  1 internal resonance.

(a)

(b)

Figure 3 

Effect of in-plane excitation on the dynamic response of the FGM plate with 1  :  2 internal resonance.

(a)

(b)

Figure 4 

Effect of in-plane excitation on the dynamic response of the FGM plate with 1  :  3 internal resonance.

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 𝑝₀ is as 7.33×10⁹N/m. Figures 3 and 4 show that the beat vibration and quasiperiod dynamic response take place at 1  :  2 internal resonance when 𝑝₀ is as 6.24×10¹⁰N/m and 1  :  3 internal resonance when 𝑝₀ is as 1.11×10¹¹N/m, 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.