Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid

Abstract

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths, by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamic energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramid-shaped solid bodies in which the base is clamped and the other sides free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically up to higher modes, from which the characteristics of various types of vibration arising in the bodies are studied. The results are presented in some figures and compared with those obtained by the beam theories.