A three dimensional elasticity based solution for free vibrations of simultaneously skewed and twisted cantilever parallelepipeds

Abstract

This work is the first 3-D continuum study of the free vibration of skewed and simultaneously skewed and twisted, cantilever parallelepipeds. The purpose of this study is to make available in the literature an enlarged data base of natural frequencies of these practical problems for researchers and design engineers to draw upon. The Ritz method is used to determine approximate natural frequency data. The total potential energy of the parallelepipeds is formulated using the three-dimensional theory of elasticity. The three orthogonal displacement components (u,v,w) are each approximated by finite triple series of simple algebraic polynomials with arbitrary coefficients (which are determined by applying the Ritz method). All terms of the series are constructed to satisfy the geometric boundary conditions at the fixed end of the parallelepiped. No other kinematic constraints are imposed in this analysis. Hence, the finite series of algebraic polynomials are both admissible and "mathematically complete" (75). Several convergence studies of natural frequencies are conducted on cantilever parallelepipeds. Effects of geometrical parameters such as side ratio, thickness ratio, skew angle, and twisted angle are presented in the form of nondimensional tables and graphs. Accuracy of solution method is substantiated through comparison with existing rectangular, skewed, and twisted plate results. The central focus of these comparisons are to verify the correctness and accuracy of free vibration data obtained by investigators using classical plate theories and two- and three-dimensional finite element methods.