Four Types of Strongly Nonlinear Oscillators: Generalization of a Perturbation Procedure

This study is concerned with a general perturbation method for the quantitative analysis of one degree-of-freedom nonlinear oscillators governed by the equations of motion that contain a coefficient of the nonlinear term which need not be small. Four classes of conservative oscillators with odd restoring forces are considered simultaneously: purely nonlinear, bistable, hardening and softening. By introducing a new small parameter as a function of the original parameter of nonlinearity, power of nonlinearity and the amplitude, the original equations of motion are transformed into the perturbed equation of motion of a linear oscillator, but its frequency depends on three parameters mentioned previously. Approximate solutions obtained for the frequency of vibration and periodic motion around the origin are compared with other available analytical and numerical results and good accuracy is demonstrated.