A Fourier series solution for the transverse vibration of a clamped beam with a torsional damper at the boundary

Abstract

In this paper we present the generalized Fourier series solution for the transverse vibration of a beam subjected to a viscous end condition in the form of a torsional damper. The model of the system produces a nonself-adjoint eigenvalue-like problem that does not yield orthogonal eigenfunctions; therefore, such functions cannot be used to calculate the coefficients of expansion in the Fourier series. Furthermore, the eigenfunctions and eigenvalues are complex valued. Nevertheless, the eigenfunctions can be utilized if the space of the operator is extended and a suitable inner product is defined. The methodology presented in the paper utilizes Hilbert space methods and is applicable in general to other problems of this type. As an adjunct to the theoretical discussion, the results from numerical simulations are presented.