On Some Property of the Natural Frequencies of Elastic Chains

In this paper we consider plane oscillations of a chain consisting of an elastic spring with concentrated mass points attached to it. It is shown that the sum of squared reciprocal natural frequencies does not depend on the number and distribution of the attached masses. For an arbitrary chain, an equivalent simple oscillator with one or two degrees of freedom can be constructed having the same sum of squared reciprocal natural frequencies as the original chain. Results are also specialized to regular chains with equal masses at equidistant spacing.