Optimizing Shape Functions in the Rayleigh-Ritz Method for Efficient Free Vibration Analysis of Functionally Graded Nanobeams

Abstract

This study explores the effectiveness of different shape functions in solving the free vibration problem of functionally graded (FG) nanobeams using the Rayleigh-Ritz method. The structural properties of the nanobeam vary continuously through its thickness, following a power law distribution of material volume fractions. The FG nanobeam is modeled using the Euler-Bernoulli beam theory, while small-scale effects are incorporated through Eringen’s nonlocal elasticity theory. Various shape functions are examined within the Rayleigh-Ritz framework to assess their computational efficiency. The primary objective is to identify the optimal shape function for this method. To achieve this, the mass and stiffness matrices are computed, and a generalized eigenvalue problem is formulated to determine the non-dimensional frequency parameter. The study evaluates the convergence behavior and computational time of each shape function to identify the most effective option. The results are validated against existing literature for specific cases, demonstrating the performance of the optimal shape function under different boundary conditions, small-scale parameters, and power law exponents. Additionally, new insights are provided into the vibrational behavior of FG nanobeams across various boundary conditions. This focus on shape function optimization enhances computational methodologies in FG nanobeam vibration analysis.