Free vibration of an isotropic elastic skewed parallelepiped – A closed form study

Abstract

Abstract The free-vibration of three-dimensional non-rectangular parallelepiped is studied, aiming at providing closed form expressions for the natural frequencies by means of a systematic approximation. To this end, a kinematic approximation is formulated based on Taylor's multivariable expansion, and constitutive relations for internal forces follow from analytical integration of the weak form. Based on this approach we investigate three general classes of parallelepipeds, namely a cube, a rectangular brick, and a skewed rhombohedron. Our second order closed form solution for the cube significantly improves currently available analytical approximations derived by Cosserat point theory. The improvement is not only in accuracy, but also in the ability to provide a richer response spectrum and to capture the lowest frequencies. In addition, based on a fourth order approximation, we derive a simple explicit expression for the fundamental (lowest) frequency covering the entire range of Poisson's ratios with high accuracy – less than 1.6% error compared to FE results. In addition, we obtain, for the first time, closed form expressions, based on a second order approximation, for the fundamental frequencies of rectangular bricks and of skewed rhombohedra. These solutions cover the entire range of aspect ratios, from thin plates through a cube to slender beams, and the entire range of skew angles.