  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.
