On the analytical approximation of damped oscillations of autonomous single degree of freedom oscillators

In this paper, we discuss the generalization of a recently developed perturbation method for conservative single degree of freedom systems to the cases of damping forces. We show that by explicitly considering the solution as a function of the amplitude parameter, frequency and phase of oscillation, one can merge the classical Krylov–Bogoliubov–Mitropolsky method and a modified Lindstedt–Poincaré method to derive better accurate slow-flow for damped oscillation of single degree of freedom systems.