Dynamics of Oscillatory Systems Having A Fractional Power Damping Force

In standard mathematical models of dynamic systems the effects of dissipation/damping is represented by a linear term proportional to the velocity, i.e., if $x$ is a relevant coordinate, then this force is $F=-\lambda \dot{x},$ where $\lambda $ is a positive parameter, and the over-dot denotes differentiation in time. For a conservative system, the application of such a force produces motions for which $x$ goes to zero in an infinite time. We demonstrate that the use of a nonlinear dissipation/damping force proportional to $\dot{x}$ raised to a fractional power, i.e., \begin{equation} F=-\lambda \left[ sgn(\dot{x})\right] |\dot{x}|^{a}, \ \ 0 < a < 1, \end{equation} gives rise to dynamics for which the motion ceases in a finite time. Using the example of the Duffing equation and the method of first-order averaging, we illustrate this phenomenon. We also discuss the application of these results to the analysis of vibrations in nano-tubes and graphene sheets.