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 thelongitudinal vibrations of a bar subjected to viscous boundary conditions ateach end. The model of the system produces a non-selfadjoint eigenvalue problemwhich does not yield a self-orthogonal set of eigenfunctions with respect tothe usual inner product. Therefore, these functions cannot be used to calculatethe coefficients of expansion in the Fourier series. Furthermore, theeigenfunctions and eigenvalues are complex-valued. The eigenfunctions can beutilized if the space of the wave operator is extended and a suitable innerproduct is defined. It is further demonstrated that the series solutioncontains the solutions for free-free, fixed-damper, fixed-fixed and fixed-freebar cases. The presented procedure is applicable in general to other problemsof this type. As an illustration of the theoretical discussion, the resultsfrom numerical simulations are presented.