Transient Growth and Stick-Slip in Sliding Friction

[+] Author and Article Information
Norbert Hoffmann

Institute of Mechanics and Ocean Engineering,  Hamburg University of Technology, Eissendorfer Strasse 42, D-21073 Hamburg, Germanynorbert.hoffmann@tuhh.de

J. Appl. Mech 73(4), 642-647 (Nov 18, 2005) (6 pages) doi:10.1115/1.2165233 History: Received December 11, 2004; Revised November 18, 2005

In linearly stable dynamical systems the non-normality of the corresponding linear operator is known to lead to transient growth of characteristic quantities, like, e.g., energy. For sliding friction systems it is shown in the present paper that this transient growth also applies to the sliding velocity and that under certain conditions this mechanism can lead to stick-slip limit-cycles in linearly stable system configurations.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Two-degree-of-freedom model

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Figure 2

Results of eigenvalue analysis of the model system. Oscillation frequencies ω (left), growth rates σ (middle), and beating frequency (ω1−ω2)∕2 (right) as functions of the friction coefficient μk.

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Figure 6

Maximum amplification of x′, max(xamp′) versus kinetic friction coefficient μk

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Figure 5

Maximum amplification xamp′ versus time t for different values of μk (left column). Time integration results for optimal initial conditions, i.e., initial conditions leading to extremal amplification of x′, scaled such that there is a time for which x′ slightly exceeds 1. Middle column: results for linearized equations, right column: results for fully nonlinear equations.

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Figure 4

Average vibrational energy E of stick-slip limit-cycles in the subcritical regime for different ratios of μs∕μk. All solutions below μk=0.4 are subcritical.

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Figure 3

Results of direct time-integration of the nonlinear equations in the unstable regime at μk=0.42 with (x,z,x′,z′)=(0.1,0.1,0,0) as initial conditions. Both x(t), z(t) and a phase space plot of x′ versus x are shown.



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