Forced Response of Uniform n-Mass Oscillators, and an Interesting Series

The vibrations of a linear discrete mechanical system of n degrees of freedom are governed in physical space by a matrix differential equation of nth order. This means, in general, the solution of an eigenvalue problem of the dimension n. for n ≥3, the eigenvalue problems can generally be solved only numerically, by means of a computer. Only in special cases it is possible to determine the eigencharacteristics of an eigenvalue problem analytically. a uniform oscillator with n equal masses m and n equal linear springs k is an example. Such a system can be thought of, for example, as a simplified discretized model for the longitudinal vibrations of a clamped-free rod. The aim of the present study is to obtain the displacements of the n masses analytically, when the free end is subjected to a force f(t). Then the responses for special forms of f(t) such as the unit-impulse function, the unit-step function and harmonic excitation, are obtained.