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

Huang¹presented 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 closed-form 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 Huang¹, we can obtain only frequencies for the flexural 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.