Natural Response of Non-smooth Oscillators Using Homotopy Analysis Combined with Galerkin Projections

Abstract

Several problems from mechanical engineering, e.g., vibrations of a spring–mass system with unequal restraints, pendulum with impact, a gear-pair with backlash and friction, etc. are modeled using second-order differential equations involving discontinuous mathematical functions such as signum, Heaviside, modulus, etc. Several perturbation-like methods such as parameter expansion, homotopy perturbation, modified Lindstedt–Poincaré, and variational iteration have been applied successfully to get the periodic solution as well as the approximate analytical estimate of the natural frequency. The chief limitation of all the methods mentioned above is the poor approximation with the large value of the perturbation parameter.