Motions of Infinite Mass-Spring Systems

In the study of physical, mechanical, and electrical systems one often encounters differentialdifference equations and recurrence relations. The sources from which these equations arise may be quite different but their mathematical forms are very similar. For example, there is an analogy between mass-spring systems and electrical systems whereby point masses correspond to inductances and springs correspond to capacitances [1]. Another area where differential-difference equations occur is in the numerical solution of the wave equations if the spatial variable is discretized [2]. It’s important to understand how these systems behave as time evolves and how changes in the parameters of the model influence this behavior. In this paper, we study systems consisting of point masses joined together by springs. In particular, we present some of the mathematical methods involved and how they are used to solve these practical problems. By obtaining the solution for these simple mass-spring systems, we indirectly obtain solutions for many similar applied problems in mechanics, physics, and engineering. Key-Words: Mass-spring system, vibration, wave, separation of variables, Laplace transform, Bessel function