Gibbs-like phenomenon inherent in a lumped element model of a rod

Abstract

The underlying assumption of a lumped element model is that a spatially distributed physical system can be approximated by a topology of discrete entities. The impact of this assumption is illustrated by a model of a finitely long elastic rod with uniform cross section. The model involves a cascade of masses and springs, where the boundary itself is driven by a step function. Previous authors have found closed-form solutions to related problems using the Laplace transform, while in this article we obtain closed-form solutions by eigenvalue decomposition. This means that the extension to a rod with non-uniform cross section is further illuminated. The closed-form solution is compared to a closed-form solution of a distributed parameter model. Both solutions involve a sum of a forward and a backward moving wave that travels with the speed of sound. In the case of the distributed parameter model, these waves are perfect square waves, while in the case of the lumped element model, these waves are imperfect square waves that are subjected to ‘‘Gibbs-like’’ ringing. Properties of this phenomenon are described. It is also shown that this phenomenon disappears, when using a continuous step function and a model with sufficiently many elements.