Dynamics of a 3D leader-follower system coupled via friction interface

D. Rachinskii; M. Ruderman; A. Zagvozdkin; D. Rachinskii; M. Ruderman; A. Zagvozdkin

doi:10.3934/mine.2026011

Mathematics in Engineering

2026, Volume 8, Issue 3: 340-367. doi: 10.3934/mine.2026011

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Research article

Dynamics of a 3D leader-follower system coupled via friction interface

1.
Department of Mathematical Sciences, The University of Texas at Dallas, USA
2.
Department of Engineering Sciences, University of Agder, Norway
3.
Department of Statistics, Virginia Tech University, USA

Received: 28 April 2025 Revised: 12 November 2025 Accepted: 26 March 2026 Published: 01 May 2026

We consider dynamics of an oscillatory mechanical system with 1.5 degrees of freedom. The system consists of two linear subsystems, a leader and a follower, which are coupled via a kinetic friction interface, and additionally driven by a proportional control term. The kinetic friction model accounts for the Coulomb friction and the Stribeck effect (as negative damping) at the passage from zero to nonzero relative velocity. The equations of motion are equivalent to a Lur'e system with a single sign nonlinearity. We show that at a critical value of the parameter the zero steady state blows up into an attracting continuum (segment) of equilibrium states. Moreover, two antipodal small-amplitude periodic orbits bifurcate from the end points of the segment of equilibrium states, creating stick-slip vibrations. We consider other possible attractors consisting of stick-slip solutions of the Lur'e system such as a $ \mathbb{Z}_2 $-symmetric stick-slip periodic orbit and an attractor composed of a continuum of heteroclinic loops representing stick-slip motions towards a steady state.
- Coulomb friction,
- Stribeck effect,
- Lur'e system,
- Filippov's system,
- Hopf bifurcation,
- stick-slip periodic motion,
- attractor
Citation: D. Rachinskii, M. Ruderman, A. Zagvozdkin. Dynamics of a 3D leader-follower system coupled via friction interface[J]. Mathematics in Engineering, 2026, 8(3): 340-367. doi: 10.3934/mine.2026011

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Abstract

We consider dynamics of an oscillatory mechanical system with 1.5 degrees of freedom. The system consists of two linear subsystems, a leader and a follower, which are coupled via a kinetic friction interface, and additionally driven by a proportional control term. The kinetic friction model accounts for the Coulomb friction and the Stribeck effect (as negative damping) at the passage from zero to nonzero relative velocity. The equations of motion are equivalent to a Lur'e system with a single sign nonlinearity. We show that at a critical value of the parameter the zero steady state blows up into an attracting continuum (segment) of equilibrium states. Moreover, two antipodal small-amplitude periodic orbits bifurcate from the end points of the segment of equilibrium states, creating stick-slip vibrations. We consider other possible attractors consisting of stick-slip solutions of the Lur'e system such as a $ \mathbb{Z}_2 $-symmetric stick-slip periodic orbit and an attractor composed of a continuum of heteroclinic loops representing stick-slip motions towards a steady state.

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