Natural response of non-smooth oscillators using homotopy analysis combined with Galerkin projections
For researchers studying non-smooth oscillators, this method provides a more accurate approximation of natural responses, though it is an incremental improvement over existing techniques.
The paper combines homotopy analysis with Galerkin projections to approximate natural responses of non-smooth oscillators, achieving superior accuracy over harmonic balance, Lindstedt-Poincaré, and conventional homotopy methods.
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-Poincaré method and conventional homotopy analysis method.