Nonlinear oscillators with a power-form restoring force: Non-isochronous and isochronous case

Abstract

This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force that comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) nonisochronous, when the system has an amplitude-dependent frequency and ii) isochronous, when the frequency of oscillations is constant (amplitude-independent). The first case is associated with the constant coefficient of the kinetic energy, while the frequency-amplitude relationship and the solution for motion need to be found. To that end, the equation of motion is solved by introducing a new small expansion parameter and by adjusting the Lindstedt-Poincare method. In the second case, the condition for the frequency of oscillations to be constant is derived in terms of the expression for the position-dependent coefficient of the kinetic energy. The corresponding solution for isochronous oscillations is obtained. Numerical verifications of the analytical results are also presented. © (2013) Trans Tech Publications, Switzerland.