  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.
