Explicit solution for vibrating bar with viscous boundaries and internal damper

TL;DR

This paper derives explicit solutions for the longitudinal vibrations of a bar with viscous boundaries and an internal damper, revealing complex eigenmodes, stability conditions, and special cases with no eigenmodes, supported by theoretical and numerical analysis.

Contribution

It introduces an explicit eigenmode expansion for a vibrating bar with viscous boundaries and internal damper, including cases with no eigenmodes, advancing understanding of such systems.

Findings

Explicit eigenmode expansion derived for general parameters

Identification of parameter values where eigenmodes are incomplete or absent

Closed-form solutions obtained for cases with no eigenmodes

Abstract

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.