Periodic Oscillations of a Truly Nonlinear Non-Natural Oscillator

Abstract

In this paper, a Lienard-type second order ordinary differential equation representing a class of non-natural oscillators with truly nonlinear periodic response was proposed and investigated. These nonlinear oscillators are characterized by position-dependent mass and velocity-dependent elastic force, and they exhibit a strong nonlinear response for all amplitudes and values of the system parameters. It was shown that the well-known Mathews-Lakshmanan oscillator, which has application in relativistic mechanics, is a special case of the present model. The exact frequency-amplitude response and periodic solution for the non-natural oscillator model were derived in closed form in terms of the Euler-Gamma and incomplete Euler-Beta functions respectively. The phase response, frequency-amplitude response and displacement response were simulated for various ranges of the index of the conservative restoring force and it was confirmed that the oscillator exhibited strong nonlinear response for all values positive real values of the index except when the index is equal to 1 or 2.