ON THE CHANGES CAUSED BY AN UNDERLYING COUPLED ROTATIONAL MOTION ON THE NATURAL FREQUENCIES OF A MASS–SPRING SYSTEM

Abstract When a system of N masses, linked together by springs, is disturbed from its static equilibrium position, then it will vibrate in a manner characterized by the N natural frequencies of the system. Should the whole system be in rotation with constant rotation speed then these natural frequencies are all decreased by an amount depending upon the rotation rate. However, if the rotation speed is increased beyond a certain level then the motion will become “unstable”, i.e., no longer vibrational. Only rotational speeds below this level are considered here. In this work, the system is mounted upon a turntable in such a manner that the masses may move only radially and the turntable is set rotating and the masses released. As the total angular momentum is conserved then the motions of the masses are coupled with the rotation of the turntable; that is, the rotation speed is no longer constant but is intimately linked to the motions taking place upon it. The effect on the natural frequencies of this coupling, and also of the initial positions and velocities when coupling is present are investigated. Two cases are pursued, one in which the displacements from the equilibrium position are “small” and the other where the coupling is “weak”. In both cases, all the natural frequencies increase from their values at constant rotation; that is, the coupling is a stabilizing influence. Initially, a single mass system is considered in order to gain insight before the more general N -mass system is tackled. Damping is ignored throughout.