On the Dynamics of a Duffing Oscillator with an Exponential Non-Viscous Damping Model

Abstract

In this paper a Duffing oscillator with non-viscous damping function is considered. The non-viscous damping function is an exponential damping model with a decaying memory property to the damping term of the oscillator. Introducing a non-viscous damping term with a decaying memory kernel means that the governing equation is an integro-differential equation of the Volterra type. Many methods used for the numerical solution of ordinary differential equations (ODE’s) can be extended to this type of integro-differential equation. Naturally the solution of the integro-differential equation is more expensive in terms of computational time compared with methods for ODE’s. In this system we can exploit the form of exponential damping function to recast the system as a set of three ODE’s. Our numerical simulations show that structural changes in the bifurcation diagrams are observed for increasing levels of non-viscous damping parameter. The two cases considered relate to small and large cubic stiffness nonlinearity. The non-viscous damping parameter is found to have an effect on the bifurcation structure in both cases. However, significant changes only occur for large values of non-viscous damping.