We investigate longitudinal vibrations of a bar subjected to viscous boundary
conditions at each end, and an internal damper at an arbitrary point along the
bar's length. The system is described by four independent parameters and
exhibits a variety of behaviors including rigid motion, super
stability/instability and zero damping. The solution is obtained by applying
the Laplace transform to the equation of motion and computing the Green's
function of the transformed problem. This leads to an unconventional
eigenvalue-like problem with the spectral variable in the boundary conditions.
The eigenmodes of the problem are necessarily complex-valued and are not
orthogonal in the usual inner product. Nonetheless, in generic cases we obtain
an explicit eigenmode expansion for the response of the bar to initial
conditions and external force. For some special values of parameters the system
of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at
all. We thoroughly analyze physical and mathematical reasons for this behavior
and explicitly identify the corresponding parameter values. In particular, when
no eigenmodes exist, we obtain closed form solutions. Theoretical analysis is
complemented by numerical simulations, and analytic solutions are compared to
computations using finite elements.Comment: 29 pages, 6 figure