D'Alembert sums for vibrating bar with viscous ends

TL;DR

This paper introduces d'Alembert sums, a new analytic method for solving certain vibrating bar problems with viscous ends, providing efficient solutions especially at small times and when eigenmodes are incomplete.

Contribution

The paper presents a novel d'Alembert sum method for solving initial-boundary PDE problems, extending classical wave solutions to systems with viscous damping and incomplete eigenmodes.

Findings

D'Alembert sums effectively solve vibrating bar problems with viscous ends.

The method simplifies and generalizes classical wave formulas.

Computer simulations confirm accuracy and efficiency.

Abstract

We describe a new method for finding analytic solutions to some initial-boundary problems for partial differential equations with constant coefficients. The method is based on expanding the denominator of the Laplace transformed Green's function of the problem into a convergent geometric series. If the denominator is a linear combination of exponents with real powers one obtains a closed form solution as a sum with finite but time dependent number of terms. We call it a d'Alembert sum. This representation is computationally most effective for small evolution times, but it remains valid even when the system of eigenmodes is incomplete and the eigenmode expansion is unavailable. Moreover, it simplifies in such cases. In vibratory problems d'Alembert sums represent superpositions of original and partially reflected traveling waves. They generalize the d'Alembert type formulas for the wave equation, and reduce to them when original waves can undergo only finitely many reflections in the entire course of evolution. The method is applied to vibrations of a bar with dampers at each end and at some internal point. The results are illustrated by computer simulations and comparisons to modal and FEM solutions.