A Fourier series solution for the longitudinal vibrations of a bar with viscous boundary conditions at each end

摘要

This paper presents the generalized Fourier series solution for the longitudinal vibrations of a bar subjected to viscous boundary conditions at each end. The model of the system produces a non-self-adjoint eigenvalue problem which does not yield a self-orthogonal set of eigenfunctions with respect to the usual inner product. Therefore, these functions cannot be used to calculate the coefficients of expansion in the Fourier series. Furthermore, the eigenfunctions and eigenvalues are complex-valued. The eigenfunctions can be utilized if the space of the wave operator is extended and a suitable inner product is defined. It is further demonstrated that the series solution contains the solutions for free-free, fixed-damper, fixed-fixed, and fixed-free bar cases. The presented procedure is applicable in general to other problems of this type. As an illustration of the theoretical discussion, the results from numerical simulations are presented.