Frequency analysis of a linear elastic structure carrying a chain of oscillators

In this technical note we analyze the free vibration of M undamped oscillat ors attached to an arbitrarily supported, linear elastic structure. Using t he assumed-modes method with N component modes, the frequency equation gove rning the free vibration for this combined system is typically obtained as the characteristic determinant of a generalized eigenvalue problem of size (N + M) x (N + M). In this note we will show that by algebraically manipula ting the generalized eigenvalue problem associated with free vibration, we can reduce it to a simple secular equation consisting of the sum of N terms , the roots or natural frequencies of which can be obtained either numerica lly or graphically. In addition, the resultant secular equation lends itsel f to the solution of an inverse problem that cannot be easily solved by ana lyzing the original generalized eigenvalue problem.