In the paper, free vibration analysis of tapered Functionally Graded Material (FGM) plate with the inclusion of porosity has been performed. The tapered porous FGM plate is considered resting on a two-parameter (Winkler and Pasternak) elastic foundation. The displacement model of the kinematic equation for the plates in the present formulation is based on the First-order shear deformation theory (FSDT). The governing equation for free vibration analysis of FGM plates is obtained using Hamilton's principle. Simple power-law, Exponential Law, and Sigmoid law are used for tailored the material properties in the thickness direction of FGM plates. The solution of the resulting partial differential equation is obtained by using Galerkin-Vlasov's method with different boundary conditions. The solutions for uniform and uniform varying thick plates are investigated, and a comparative study is examined by comparing the results obtained with FSDT and Higher-order shear deformation theory (HSDT). The findings of the comparative study with the present approach provide pertinent outcomes for the vibration analysis of tapered FGM plates. The analytical solution for vibration analysis is presented to reveal the effects of porosity parameter, volume exponent, span ratio, aspect ratio, porosity distribution, and boundary conditions. Also, the elastic foundation parameter on tapered FGM plate increases the non-dimensional frequency, and the Pasternak foundation effect always dominates over the Winkler foundation.

在本文中，进行了带有孔隙度的锥形功能梯度材料（FGM）板的自由振动分析。锥形多孔FGM板被认为放在两个参数（Winkler和Pasternak）的弹性地基上。本公式中板的运动学方程的位移模型基于一阶剪切变形理论（FSDT）。利用汉密尔顿原理，得到了FGM板自由振动分析的控制方程。使用简单的幂律，指数律和Sigmoid律来定制FGM板厚度方向的材料特性。通过使用具有不同边界条件的Galerkin-Vlasov方法获得所得偏微分方程的解。研究了均匀且均匀变化的厚板的解决方案，并通过比较FSDT和高阶剪切变形理论（HSDT）的结果，进行了比较研究。使用本方法进行的比较研究的结果为锥形FGM板的振动分析提供了相关的结果。提出了振动分析的解析解决方案，以揭示孔隙度参数，体积指数，跨度比，纵横比，孔隙率分布和边界条件的影响。而且，锥形FGM板上的弹性基础参数会增加无量纲频率，并且Pasternak基础效应始终在Winkler基础上占主导地位。通过比较FSDT和高阶剪切变形理论（HSDT）获得的结果，进行了比较研究。使用本方法进行的比较研究的结果为锥形FGM板的振动分析提供了相关的结果。提出了振动分析的解析解决方案，以揭示孔隙度参数，体积指数，跨度比，纵横比，孔隙率分布和边界条件的影响。而且，锥形FGM板上的弹性基础参数会增加无量纲频率，并且Pasternak基础效应始终在Winkler基础上占主导地位。通过比较用FSDT和高阶剪切变形理论（HSDT）获得的结果，进行了比较研究。使用本方法进行的比较研究的结果为锥形FGM板的振动分析提供了相关的结果。提出了振动分析的解析解决方案，以揭示孔隙度参数，体积指数，跨度比，纵横比，孔隙率分布和边界条件的影响。而且，锥形FGM板上的弹性地基参数会增加无量纲频率，并且Pasternak基础效应始终在Winkler基础上占主导地位。提出了振动分析的解析解决方案，以揭示孔隙度参数，体积指数，跨度比，纵横比，孔隙率分布和边界条件的影响。而且，锥形FGM板上的弹性地基参数会增加无量纲频率，并且Pasternak基础效应始终在Winkler基础上占主导地位。提出了振动分析的解析解决方案，以揭示孔隙度参数，体积指数，跨度比，纵横比，孔隙率分布和边界条件的影响。而且，锥形FGM板上的弹性地基参数会增加无量纲频率，并且Pasternak基础效应始终在Winkler基础上占主导地位。