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

TL;DR

This paper develops a Fourier series method for solving longitudinal vibrations in a bar with viscous boundary conditions, addressing complex eigenvalues and non-orthogonal eigenfunctions, and demonstrating its applicability through numerical examples.

Contribution

It introduces a generalized Fourier series approach for non-selfadjoint eigenvalue problems with viscous boundary conditions, extending the applicability of Fourier methods to complex eigenfunctions.

Findings

The series solution encompasses various boundary conditions such as free-free and fixed-fixed.

Eigenfunctions are complex-valued and require an extended inner product.

Numerical simulations validate the theoretical approach.

Abstract

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-selfadjoint 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.