Natural Response of Non-smooth Oscillators Using Homotopy Analysis Combined with Galerkin Projections

Several problems from mechanical engineering, e.g., vibrations of a spring–mass system with unequal restraints, pendulum impact, gear-pair backlash and friction, etc. are modeled using second-order differential equations involving discontinuous mathematical functions such as signum, Heaviside, modulus, perturbation-like methods parameter expansion, homotopy perturbation, modified Lindstedt–Poincaré, variational iteration have been applied successfully to get the periodic solution well approximate analytical estimate natural frequency. The chief limitation all mentioned above is poor approximation large value perturbation parameter. analysis method overcomes this certain extent. It usually plagued slow convergence issue. task becomes impossible handle computationally if one includes too many terms series, especially dealing oscillators non-smooth nonlinearities. key issue now how accelerate series solution. To solution, we think convergence-control function embedding call it function. usual treatment provides an expression for frequency oscillator that also free parameters arising due Generating extra Galerkin projections solving same numerically gives response oscillators. proposed yields discontinuities type range over which differs obtained via numerical integration by less than 2 percent largest our approach compared other approaches like harmonic balance, Lindstedt–Poincaré method, temporal transform, conventional method. framework developed has extension no response, unilaterally constrained simple where solutions decaying time but scalable. superiority well-established much larger