Periodic solutions of non-smooth friction oscillators

Abstract

We study the differential equation $x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t)$ with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation $x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t)$ and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on μ we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the “linearized” differential equation is an impulsive Hill equation.