Background 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-Poincare, 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. Purpose The homotopy analysis method overcomes this limitation to a certain extent. It is usually plagued with a slow convergence issue. The task becomes impossible to handle computationally if one includes too many terms of the approximate series, especially dealing with oscillators involving non-smooth nonlinearities. The key issue now is how to accelerate the convergence of the series solution. Methods To accelerate the convergence of the series solution, we think of the convergence-control parameter as a function of the embedding parameter and call it a convergence-control function. The usual treatment of the homotopy analysis provides an expression for the natural frequency of the oscillator that also includes free parameters arising due to 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. Results The proposed method yields an approximate natural response of non-smooth oscillators involving discontinuities of type Heaviside, signum, modulus, etc. The perturbation parameter range over which the approximate solution differs from the one obtained via numerical integration by less than 2 percent is the largest with our approach compared to other approaches like the method of harmonic balance, the Lindstedt-Poincare method, non-smooth temporal transform, and the conventional homotopy analysis method. The framework developed has a natural extension to oscillators with no periodic response, e.g., the unilaterally constrained simple pendulum where the solutions are decaying with time but are scalable. Conclusion The superiority of our approach is well-established over a much larger range of the perturbation parameter compared to the usual homotopy analysis method.
