Lyapunov Stability of Non-smooth Non-autonomous Mechanical Systems
Heimsch, Remco Leine
The stability of non-smooth dynamical systems is a novel research field which is
receiving much attention in the mathematical as well as engineering community.
Mechanical systems with impact phenomena and unilateral constraints form an
important class of non-smooth systems as they arise in many engineering
applications. Despite the huge amount of textbooks and papers on (Lyapunov)
stability in various fields of engineering science, Lyapunov stability
properties of non-autonomous non-smooth systems described by measure
differential inclusions has not been considered so far. In order to study
Lyapunov stability criteria of equilibria in non-smooth, explicitly
time-dependent (i.e. non-autonomous) mechanical systems with unilateral
constraints, we attempt to investigate the stability of the equilibrium of an
apparently simple mechanical system meeting the requirements of non-smoothness
and explicit time-dependence. A standard problem of chaotic dynamics, which has
been extensively studied in the literature, is a ball in a constant
gravitational field which bounces inelastically on a flat vibrating table. The
governing equations of motion are highly nonlinear due to the unilateral contact
and generally do not allow for any closed form solution.
The bouncing ball system.
Global attractive stability conditions for the equilibrium of the bouncing ball system are proven
in this research project using an extension of Lyapunov’s direct method to
non-autonomous systems.Furthermore, it is proven that the attractivity of the
equilibrium is symptotic, i.e. there exists a finite time for which the solution
has converged exactly to the equilibrium. For this attraction time, an upper
bound is determined.
Trajectory of the ball bouncing on the table (left) and a Lyapunov candidate function (right).
Heimsch, T. and Leine, R.I.,
"Lyapunov stability theory for nonsmooth non-autonomous mechanical systems
applied to the bouncing ball problem", Proceedings of the ASME 2009
International Design Engineering Technical Conferences & Computers and
Information in Engineering Conference IDETC/CIE 2009, DETC/MSNDC-87185, San
Diego, USA, 2009.
Heimsch, T. "Lyapunov stability
theory for nonsmooth non-autonomous mechanical systems applied to the bouncing
ball problem", Master thesis at the Center of Mechanics, ETH Zurich, 2008.