The present study is dealt with the applicability of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique on free vibration of functionally graded (FG) beam with uniformly distributed porosity along the thickness of the beam. The material properties such as Young’s modulus, mass density, and Poisson’s ratio are also considered to vary along the thickness of the FG beam as per the power-law exponent model. The porous FG beam is embedded in an elastic substrate; namely, the Kerr elastic foundation and the displacement field of the beam are governed by a refined higher-order shear deformation theory. The effectiveness of the Rayleigh–Ritz method is due to the use of the shifted Chebyshev polynomials as a shape function. The orthogonality of shifted Chebyshev polynomial makes the technique more computationally efficient and avoid ill-conditioning for the higher number of terms of the polynomial. Hinged–hinged, clamped–hinged, clamped–clamped, and clamped-free boundary conditions have been taken into account for the parametric study. Validation of the present model is examined by comparing it with the existing literature in special cases showing remarkable agreement. A pointwise convergence study is also carried out for shifted Chebyshev polynomial-based Rayleigh–Ritz method, and the effect of power-law exponent, porosity volume fraction index, and elastic foundation on natural frequencies is studied comprehensively.

中文翻译：

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基于移切比雪夫多项式的瑞利-里兹方法和纳维尔技术在 Kerr 地基中功能梯度多孔梁振动分析中的应用

本研究涉及基于移动切比雪夫多项式的 Rayleigh-Ritz 方法和 Navier 技术对孔隙率沿梁厚度均匀分布的功能梯度 (FG) 梁自由振动的适用性。根据幂律指数模型，材料特性（例如杨氏模量、质量密度和泊松比）也被认为随 FG 梁的厚度而变化。多孔 FG 梁嵌入弹性基材中；即，克尔弹性地基和梁的位移场由精细的高阶剪切变形理论控制。Rayleigh-Ritz 方法的有效性是由于使用了移位的 Chebyshev 多项式作为形状函数。移位切比雪夫多项式的正交性使该技术在计算上更有效，并避免了多项式项数较多的病态。铰链铰链、夹紧铰链、夹紧夹紧和无夹紧边界条件已被考虑用于参数研究。通过在特殊情况下与现有文献进行比较来检查当前模型的有效性，显示出显着的一致性。对基于移动切比雪夫多项式的Rayleigh-Ritz方法进行了逐点收敛研究，综合研究了幂律指数、孔隙体积分数指数和弹性地基对固有频率的影响。参数化研究考虑了无钳位边界条件。通过在特殊情况下与现有文献进行比较来检查当前模型的有效性，显示出显着的一致性。对基于移动切比雪夫多项式的Rayleigh-Ritz方法进行了逐点收敛研究，综合研究了幂律指数、孔隙体积分数指数和弹性地基对固有频率的影响。参数化研究考虑了无钳位边界条件。通过在特殊情况下与现有文献进行比较来检查当前模型的有效性，显示出显着的一致性。对基于移动切比雪夫多项式的Rayleigh-Ritz方法进行了逐点收敛研究，综合研究了幂律指数、孔隙体积分数指数和弹性地基对固有频率的影响。