Closed form Solution to Nonlinear Equilibria and Oscillations of a Damped Eccentric Rotor

Abstract

We study an eccentric and elastically damped rotor from both a statical and dynamical point of view. The system, whose genesis is in the re-loading mechanism of an automatic watch, behaves like a generalized physical pendulum with the addition of eccentricity and damping. The static analysis is performed by settling the statical equilibria and defining their nature, whose effective computation can be done numerically. The dynamical analysis leads to a nonlinear differential initial-value problem whose integration is carried out by means of Jacobi elliptic functions. It reveals that, starting from both positional and kinetic zero initial conditions, only periodical motions, see formulae (4.8) or (4.13), are allowed and all confined inside a potential well. Closed form expressions of the oscillation period have been obtained through complete elliptic integrals of the first kind. In such a way a further treatment is added to the non-rich collection of 1-D nonlinear oscillators suitable of closed form integration.