Stick‐slip and convergence of feedback‐controlled systems with Coulomb friction

Abstract

An analysis of stick-slip behavior and convergence of trajectories in the feedback-controlled motion systems with discontinuous Coulomb friction is provided. A closed-form parameter-dependent stiction region, around an invariant equilibrium set, is proved to be always reachable and globally attractive. It is shown that only asymptotic convergence can be achieved, with at least one but mostly an infinite number of consecutive stick-slip cycles, independent of the initial conditions. Theoretical developments are supported by a number of numerical results with dedicated convergence examples.

Motion dynamics of inertial pair coupled via frictional interface

“…The following numerical results were obtained for system (11), (12). The first-order forward Euler solver was employed for numerical simulations.…”

“…The motion trajectories with an initial value x 1 (t 0 ) = 0.006 are depicted in the plot (a); the other initial values are zero. One can see that system (11), (12) starts in the stiction mode (13) and remains in this mode at all times, featuring undamped harmonic oscillations with synchronized orbits (x 1 , ẋ1 ) and (x 2 , ẋ2 ). On the other hand, when the initial value ẋ1 (t 0 ) = 0.15 is used, as shown in the plot (b), system (11), (12) starts in the mode (14), i.e.…”

“…Note that (12) does not include the acceleration-dependent switching, since the assigned example system ensures |ẍ 1 | < bm −1 2 , cf. with ( 4).…”

“…An initial condition x 1 ∨ ẋ1 = 0 ensures a finite energy storage at t = t 0 = 0 and, therefore, the onset of a relative motion of the free oscillator. The switched coupling through the frictional interface becomes particularly obvious for the second inertial mass in (12). The second mass is either synchronized with the first inertial mass and has then the same acceleration when sticking, or the second mass is sliding with respect to the first mass and dissipating energy when sgn( ż) = 0.…”

“…[4], [11]) remain also outside of our focus. Also we should stress that the Coulomb dry friction, if it only acts as a non-linear damping and adhesive element in the active (in our context bounded motion) systems with feedback control (see the recent developments in [12]), is secondary for the present analysis unless it appears as a dynamic interface between two moving bodies. Finally, we shall focus on the Coulomb type frictional interactions in their simplest form, i.e.…”