Free vibration analysis of three-dimensional solids with arbitrary geometries using discrete Ritz method

Abstract

A numerical method, discrete Ritz method (DRM), is proposed for the free vibration analysis of three-dimensional (3D) solids with arbitrary geometries. The problem is formulated within a cuboidal domain, and arbitrarily shaped 3D solid can be simulated by assigning cutouts within the cuboidal domain. Geometries of 3D solids are characterized by using level set functions. DRM transforms the problem into a variable stiffness system in which Chebyshev polynomials are used to approximate vibrational behavior and Gauss points are used to discretize the cuboidal domain. The vibration behaviors of arbitrarily shaped solids can be numerically simulated by setting the stiffness and thickness of Gauss points within the cutouts to zero. New formulations in DRM have resulted in completely standard energy functionals and computation procedures for arbitrarily shaped 3D solids. Free vibration behaviors of 3D solids, including polygonal cross-section cylinders, circular cylinders, rectangular plates with cutouts, perforated super-elliptical plates, ellipsoids, and pyramids, are investigated, and results are compared with those reported in the literature. DRM is shown to be capable of solving free vibration problems of complex-shaped 3D solids with high precision and stability.