Free Vibration Analysis of Variable Thickness Nanoplates Using Boundary Characteristic Orthogonal Polynomials in the Rayleigh-Ritz Method

This paper presents an investigation on the vibration of nanoscale rectangular orthotropic plates considering non-uniformity in the thickness. A method based on the variational principles in conjunction with the orthogonal polynomials in the Rayleigh-Ritz method technique is applied to examine the free vibration characteristics of nanoplate. The present method can easily handle the specified boundary conditions at the edges. Based on the nonlocal continuum mechanics, equations are derived and Gram-Schmidt process is used for generating the orthogonal polynomials in two variables. Numerical solutions for the natural frequencies are obtained for different combinations of clamped and simply supported boundary conditions. Effects of thickness variation together with varying values of aspect ratio on the natural frequencies are illustrated on the first three vibration modes. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform thickness. The present study shows that the vibrating behaviors of single-layered rectangular nanoplates are sensitive to the nonlocal and non-uniform parameters.