Exact Characteristic Equations in Closed-Form for Vibration of Completely Free Timoshenko Beams

Abstract

Huang(1) presented first in 1961 the characteristic equations and normal mode equations for all six common types of simple, finite Timoshenko beams in closed-form. Unfortunately, there exist several errors, not typographical, in the frequency in the characteristic equations for a Timoshenko beam free at both ends. In this paper, the exact characteristic equations in closedform for completely free Timoshenko beams are derived based on Ref. 1. In order to justify the solutions obtained by amending herein Huang's equations, both the closed-form exact solution and the one obtained by the Ritz method will be adopted as the references. Using the characteristic equations derived by Huang, 1 we can obtain only frequencies for the exural modes, while using the closed-form exact method and Ritz method, we can obtain frequencies for the thickness-shear modes, as well as for the bending modes. The purpose of the present study is to identify the errors in Ref. 1, correct them, and provide some numerical results.