Natural response of non-smooth oscillators using homotopy analysis combined with Galerkin projections

We propose homotopy analysis method in combination with Galerkin projections to approximate the natural response of non-smooth oscillators with discontinuities of type Heaviside, signum, modulus etc. While constructing the homotopy, we think of convergence-control parameter as a function of the embedding parameter and call it a convergence-control function. Homotopy analysis provides an expression for the natural frequency of the oscillator that also includes free parameters arising from the convergence-control function. Generating extra equations using Galerkin projections and solving the same numerically gives the approximate natural response of the non-smooth oscillators. We also seek aperiodic natural response of a unilaterally constrained simple pendulum. The superiority of the approach is well-established over the method of harmonic balance, Lindstedt-Poincar\'e method and conventional homotopy analysis method.