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We propose homotopy analysis method in combination with Galerkin projectionsto approximate the natural response of non-smooth oscillators withdiscontinuities of type Heaviside, signum, modulus etc. While constructing thehomotopy, we think of convergence-control parameter as a function of theembedding parameter and call it a convergence-control function. Homotopyanalysis provides an expression for the natural frequency of the oscillatorthat also includes free parameters arising from the convergence-controlfunction. Generating extra equations using Galerkin projections and solving thesame numerically gives the approximate natural response of the non-smoothoscillators. We also seek aperiodic natural response of a unilaterallyconstrained simple pendulum. The superiority of the approach iswell-established over the method of harmonic balance, Lindstedt-Poincar\'emethod and conventional homotopy analysis method.

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