In this research, the nonlinear free vibrations analysis of functionally graded (FG) rectangular plate which simply supported all edges are investigated analytically using modified Lindstedt–Poincare (MLP) method for the first time. For this purpose, with the aid of von Karman nonlinearity strain-displacement relations, the partial differential equations of motion are developed based on first-order shear deformation theory (FSDT). Afterward, by applying Galerkin method, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations. The nonlinear equation of motion is then solved analytically by MLP method to determine the nonlinear frequencies of the FG rectangular plate. The material properties are assumed to be graded through the direction of plate thickness according to power law distribution. The effects of some system parameters such as vibration amplitude, volume fraction index and aspect ratio on the nonlinear to linear frequency ratio are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

中文翻译：

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使用改进的 Lindstedt-Poincare 方法对功能梯度板进行非线性振动分析的解析解

本研究首次采用改进的 Lindstedt-Poincare (MLP) 方法对简单支撑所有边缘的功能梯度 (FG) 矩形板进行非线性自由振动分析。为此，借助von Karman非线性应变-位移关系，建立了基于一阶剪切变形理论（FSDT）的运动偏微分方程。然后，应用伽辽金方法，将非线性偏微分方程转化为时变非线性常微分方程。然后通过MLP方法解析求解非线性运动方程以确定FG矩形板的非线性频率。假设材料特性按照幂律分布沿板厚方向分级。详细讨论了振动幅度、体积分数指数和纵横比等系统参数对非线性与线性频率比的影响。为了验证分析，本文的结果与已发表的数据和数值方法进行了比较，发现了良好的一致性。