Abstract
Abstract This paper deals with the stability analysis of a mass–spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick–slip phenomenon, the mass may then periodically stick to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.
Highlights
Friction appears in many mechanical systems such as drill-strings [1, 2], car steering [3] or machine positioning [4, 5]
This paper deals with the stability analysis of a mass-spring system subject to friction by taking into account that the described system presents a stickslip phenomenon, meaning that the mass is periodically stick to the ground
Stability of the linearized system 200 to the techniques used in presence of saturation or backlash nonlinearities, the fact that φvref is continuous with φvref (0) = 0, is not sufficient to study the regional stability of the origin and we must consider the stability of its linearization around an equilibrium point to apply the first Lyapunov principle [23]
Summary
Friction appears in many mechanical systems such as drill-strings [1, 2], car steering [3] or machine positioning [4, 5] This is a nonlinear force which is responsible of many undesirable effects such as stick-slip or hunting [4]. The first model was proposed by Guillaume Amontons and Charles Augustin de Coulomb [6] during 10 the eighteenth century It was studied more precisely by Stribeck [7] who experimentally observed a decrease of the friction force at low velocity. The second result studies the regional asymptotic stability of the equilibrium point and proposes an estimation of the basin of attraction of the equilibrium point. We define the operation col(u, v) = u v for any column vectors u and v.
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