Keywords: damping, Duffing, non-viscous, bifurcation.

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. Viscous damping is the most common model for modelling of vibration damping. This model, first introduced by Lord Rayleigh [1], assumes that the instantaneous generalized velocities are the only relevant variables that determine damping. Viscous damping models are used widely for their simplicity and mathematical convenience even though the behaviour of real structural materials may not be adequately represented by viscous models. For this reason it is well recognized that in general a physically realistic model of damping will not be viscous. Damping models in which the dissipative forces depend on any quantity other than the instantaneous generalized velocities are non-viscous damping models. Mathematically, any causal model which makes the energy dissipation functional non-negative is a possible candidate for a non-viscous damping model. Clearly a wide range of choice is possible, either based on the physics of the problem, or by a priori selecting a model and fitting its parameters from experiments. Among various damping models, the 'exponential damping model' is particularly promising and has been used by many authors. With this model the damping force is expressed as

Here is the viscous damping constant, μ is the relaxation parameter and is the displacement as a function of time. In the context of viscoelastic materials, the physical basis for exponential models has been well established, see for example [2]. A selected literature review including the justifications for considering exponential damping model may be found in reference [3].

The Duffing oscillator with viscous damping has been widely studied, and comprehensive reviews of the associated literature can be found in [4]. For these type of systems the damping is nearly always assumed to be viscous. One exception is a study of study of a Duffing vibration isolator which has combined Coulomb and viscous damping [5]. In this study the authors demonstrate that the addition of a Coulomb damping term can lead to new dynamical behaviour in the system.

In Section 2 of the paper we present the mathematical model of the system, which is then partly nondimensionalised and reduced to an equivalent system of three first order differential equations. The reduced system has six system parameters, of which two relate to the external forcing, two capture the stiffness behaviour, one represents the viscous damping and one the non-viscous damping. In Section 3 we show numerical simulations of the system in both phase space, and parameter space. The reduced set of three ODE's can be simulated using a fixed step, 4th order Runge-Kutta algorithm. The numerical parameter variation is used to create bifurcation diagrams, in this case using the external forcing frequency as the bifurcation parameter -- this allows easy comparison with work on the viscous Duffing system. Structural changes in the bifurcation diagrams are then observed for increasing levels of non-viscous damping parameter.

The two cases considered relate to small and large cubic stiffness nonlinearity. In the case of small nonlinearity, hardening spring behaviour is found, and the result of adding non-viscous effects is to increase the amplitude of the resonance peak. In the case of strong nonlinearity, more complex dynamics are observed close to the first natural frequency. Here the effect of adding non-viscous damping is to cause subtle changes in the dynamics. These changes include subtle movement or shifts in the location of the different regions of dynamical behaviour along the frequency axis. Changes in the periodicity of solutions can also be observed, although it should be noted that our numerical approach is not guaranteed to capture all coexisting solutions. For both cases, significant changes only occur for large values of non-viscous damping. For general nonlinear system modelling of mechanical or structural applications, this type of non-viscous damping may give improved accuracy of the simulations when compared to experimental data.

1

L. Rayleigh, Theory of Sound (two volumes), Dover Publications, New York, 1945 re-issue, second edition, 1877.

2

L. Cremer and M. Heckl, Structure-Borne Sound, Springer-Verlag Berlin, Germany, second edition, 1973, translated by E. E. Ungar.

3

N. Wagner and S. Adhikari, "Symmetric state-space formulation for a class of non-viscously damped systems", AIAA Journal, 41(5), 951-956, 2003. doi:10.2514/2.2032

4

J. M. T. Thompson and H. B. Stewart, Nonlinear dynamics and chaos, John Wiley: Chichester, 2002.

5

B. Ravindra and A. K. Mallik, "Hard Duffing type vibration isolator with combined Coulomb and viscous damping", International Journal of Non-linear Mechanics, 28(4), 427-440, 1993. doi:10.1016/0020-7462(93)90017-F

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