The dynamics of a Duffing oscillator with non-viscous damping

Wagg D. J., Adhikari, S.,
The Eighth International Conference on Computational Structures Technology, Las Palmas de Gran Canaria, Spain, 2006.

In this paper a Duffing oscillator with non-viscous damping function is considered. The non-viscous damping function is an exponential damping model as described which has been studied by a range of authors - see for example [] and references therein. This type of damping model adds a decaying memory property to the damping term of the oscillator. The governing equation for the system can be expressed as
m
××
x
 
 +c ó
õ 		t

0 
 me-m(t-t)
×
x
 
 (t)dt-k1x1+k2 x13=Acos(Wt),
(1)
where x represents the displacement of mass m and the stiffnesses coefficients are given by k1, k2. The damping coefficients are c and m. The harmonic forcing functions is Acos(wt).
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. Methods for solving such systems have been studied by a wide range of authors - see, for example, [1] and references therein. 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 [1]. As a result the method chosen to compute solutions for this work is a second order approximation to the exact solution based on trapiziodal approximation.
The Duffing oscillator with viscous damping has been widely studied, and a comprehensive review of the associated literature can be found in [2]. For these type of systems the viscous damping coefficient can be used as a bifurcation parameter. For example a supercritical Hopf bifurcation can be observed as the damping coefficient passes through zero from positive to negative [2]. The effect of the non-viscous damping model on this transition will be studied in this paper.

References

[1]
P. Linz. Analytical and numerical methods for Volterra equations. SIAM Studies in Applied Mathematics, 1985.
[2]
J. M. T. Thompson and H. B. Stewart. Nonlinear dynamics and chaos. John Wiley: Chichester, 2002.
BiBTeX Entry
@INPROCEEDINGS{cp20,
    AUTHOR={D. J. Wagg and S. Adhikari},
    TITLE = {On the Dynamics of a Duffing Oscillator with an Exponential Non-Viscous Damping Model},
    BOOKTITLE = {Proceedings of the Eighth International Conference on Computational Structures Technology},
    EDITOR = {Topping, B. H. V. and Montero, G. and Montenegro, R.},
    PUBLISHER = {Civil-Comp Press},
    ADDRESS = {Stirlingshire, United Kingdom},
    NOTE = {paper 75},
    YEAR = 2006
}
by Sondipon Adhikari