[Paper Review] A new test for stick-slip limit cycles in dry-friction oscillators with small nonlinear friction characteristics

Oleg Makarenkov|arXiv (Cornell University)|May 17, 2016

Gear and Bearing Dynamics Analysis17 references3 citations

TL;DR

This paper develops a new perturbation-based sufficient condition for the existence of stick-slip limit cycles in dry-friction oscillators with small nonlinear friction, such as the Stribeck effect. The key result is a quantitative inequality involving the integral of the nonlinear friction's derivative, which determines when periodic stick-slip motion emerges, offering a more accurate test than divergence-based methods.

ABSTRACT

We consider a dry friction oscillator on a moving belt with both the Coulomb friction and a small nonlinear addition which can model e.g. the Stribeck effect. By using the perturbation theory, we establish a new sufficient condition for the nonlinearity to ensure the occurrence of a stick-slip limit cycle. The test obtained is more accurate compared to what one gets by building upon the divergence test.

Motivation & Objective

• To establish a rigorous mathematical condition for the existence of stick-slip limit cycles in dry-friction oscillators with small nonlinear friction characteristics.
• To overcome the limitations of existing divergence-based tests, which may fail to detect stick-slip cycles even when they exist.
• To apply perturbation theory to analyze grazing-sliding bifurcations in a piecewise-smooth system with a small nonlinear perturbation.
• To derive a specific, computable condition for the Stribeck friction model that guarantees the emergence of finite-time stable stick-slip limit cycles.
• To demonstrate the sharpness of the condition by showing that the reverse inequality implies no such cycles exist as ε→0+.

Proposed method

• Uses perturbation theory to analyze the behavior of solutions near the switching manifold $ x_2 = V $, particularly focusing on the transition from periodic motion to sliding.
• Applies Filippov's theory of differential inclusions to rigorously define sliding solutions along the discontinuity surface $ x_2 = V $.
• Derives a sufficient condition (Equation 3) based on the integral of the derivative of the nonlinear friction term $ F $ with respect to $ u $ at $ u = 0 $, involving $ au $-periodic integration over $ [0, 2 au] $.
• Constructs a grazing-sliding bifurcation scenario where a trajectory that touches $ x_2 = V $ at $ au = 2 au $ under $ u = 0 $ returns to the manifold under small $ u > 0 $, indicating a limit cycle.
• Applies the general condition to the Stribeck friction model $ F( u, u) = rac{1-eta}{1+ u heta| u|} + eta + ueta u^2 $, leading to an explicit algebraic criterion (Equation 18).
• Compares the new test with the divergence-based method, showing that the new condition is both more accurate and less restrictive in detecting stable stick-slip cycles.

Experimental results

Research questions

• RQ1 Under what conditions does a small nonlinear addition to Coulomb friction in a dry-friction oscillator lead to the emergence of a stable stick-slip limit cycle?
• RQ2 How does the perturbation-based approach improve upon the classical divergence test in detecting stick-slip oscillations?
• RQ3 What is the precise mathematical condition that ensures a grazing-sliding bifurcation leads to a finite-time stable limit cycle in this system?
• RQ4 Can the derived condition be explicitly applied to the Stribeck friction model to obtain a computable criterion for stick-slip behavior?
• RQ5 Is the derived sufficient condition sharp, such that the reverse inequality guarantees the absence of stick-slip cycles as $ u o 0^+ $?

Key findings

• A new sufficient condition for the existence of a stick-slip limit cycle is derived: $ cV au < igint_{0}^{2 au} F_{ u}'(V au ext{cos} au - V, 0) ext{cos} au d au $, which depends on the derivative of the nonlinear friction term.
• The condition is shown to be sharp: if the inequality is reversed, no stick-slip limit cycles exist as $ u o 0^+ $, indicating the boundary of the parameter space for such cycles.
• For the Stribeck friction model $ F( u, u) = rac{1-eta}{1+ u heta| u|} + eta + ueta u^2 $, the condition reduces to $ -c + heta(1-eta) - 2eta V > 0 $, providing a simple algebraic criterion.
• The new test is more accurate than the divergence-based method, as it detects cycles even when the divergence test fails due to insufficient positivity of the divergence.
• The method confirms the pivotal role of nonlinearity: pure Coulomb friction with small viscous damping cannot produce stick-slip limit cycles, highlighting the necessity of nonlinear friction for such dynamics.
• The existence of a finite-time stable stick-slip limit cycle is rigorously proven for all sufficiently small $ u > 0 $, provided the derived condition holds.