Research Papers

On the Robustness of Instabilities in Friction–Induced Vibration

[+] Author and Article Information
Sebastian Kruse

Audi AG,
Development Foundation Brake,
Ingolstadt 85045, Germany;
Structural Dynamics Group,
Mechanical Engineering,
Hamburg University of Technology,
Hamburg 21073, Germany
e-mail: sebastian.kruse@audi.de

Norbert P. Hoffmann

Structural Dynamics Group,
Mechanical Engineering,
Hamburg University of Technology,
Hamburg 21073, Germany;
Dynamics of Machines and Structures Group,
Imperial College London,
London SW7 2AZ, UK
e-mail: norbert.hoffmann@tuhh.de

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received December 8, 2012; final manuscript received June 30, 2013; published online August 6, 2013. Assoc. Editor: Steven W Shaw.

J. Vib. Acoust 135(6), 061013 (Aug 06, 2013) (8 pages) Paper No: VIB-12-1342; doi: 10.1115/1.4024939 History: Received December 08, 2012; Revised June 30, 2013

Friction-induced vibration and the noise or wear it causes are everlasting problems in the design of dynamical mechanical systems. The most common way to analyze friction-induced vibration is to determine the borders of linear stability. In that framework, the present study focuses on robustness concepts of systems prone to friction-induced vibration. Here, robustness is defined on two different levels. First, robustness will be considered in a global design perspective, giving an answer to the question of how many realizations within an overall ensemble of possible designs will show instability and if a given stability characteristic remains robust under parameter variations. Second, robustness will be understood with respect to the sensitivity of the system’s eigenvalues against parameter variations in general, focusing on the questions of how single eigenvalues react to parameter variation and if the real parts of the system’s eigenvalues give a measure for changes of stability characteristics under parameter variation. To answer the posed questions, dynamical model systems subject to friction-induced vibration are generated on the basis of specified random processes and evaluated in statistical terms. It shows that the size of the real parts of the eigenvalues, i.e., the growth or decay rates of the linear modes, which are in practice often used as decisive values in the interpretation of stability calculations, cannot be used as a well defined indicator for any kind of the considered robustness concepts. We, thus, suggest a novel measure taking into account variance properties to rate the robustness of systems subject to friction-induced vibration.

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Akay, A., 2002, “Acoustics of Friction,” J. Acoustic Society of America, Vol. J. Acoust. Soc. Am., 111(4), pp. 1525–1548. [CrossRef]
Turrin, S., Hanss, M., and Gaul, L., 2006, “Fuzzy Arithmetical Vibration Analysis of a Windshield With Uncertain Parameters,” Proc. of the IX International Conference on Recent Advances in Structural Dynamics—RASD 2006, Southampton, UK, July 17–19.
Gauger, U., Hanss, M., and Gaul, L., 2006, “On the Inclusion of Uncertain Parameters in Brake Squeal Analysis,” IMAC-XXIV: Conference & Exposition on Structural Dynamics, St. Louis, MO, January 30–February 2.
Butlin, T., and Woodhouse, J., 2009, “Sensitivity Studies of Friction-Induced Vibration,” Int. J. Vehicle Design, 51(1/2), pp. 238–257. [CrossRef]
Butlin, T., and Woodhouse, J., 2010, “Friction-Induced Vibration: Quantifying Sensitivity and Uncertainty,” J. Sound Vib., 329, pp. 509–526. [CrossRef]
Bathe, K.-J., 2002, Finite Elemente Methoden, Springer-Verlag, Berlin.
Müller, P. C., 1977, Stabilität und Matrizen, Springer-Verlag, Berlin.
Voss, H., 2010, Iterative Projection Methods for Large Scale Nonlinear Eigenvalue Problems, Computational Technology Review, Vol. 1, Saxe-Coburg Publications, Stirlingshire, UK.
Kinkaid, N. M., O’Reilly, O. M., and Papadopoulos, P., 2003, “Automotive Disc Brake Squeal,” J. Sound Vib., 267, pp. 105–166. [CrossRef]
AbuBakar, A. R., and Ouyang, H., 2006, “Complex Eigenvalue Analysis and Dynamic Transient Analysis in Predicting Disc Brake Squeal,” Int. J. Vehicle Noise Vib., 2(2), pp. 143–155. [CrossRef]
Ghazaly, N. M., Mohammed, S., and Abd-El-Tawwab, A. M., 2012, “Understanding Mode-Coupling Mechanism of Brake Squeal Using Finite Element Analysis,” Int. J. Eng. Res. Appl., 2(1), pp. 241–250.
Kung, S W., Dunlap, K. B., and Ballinger, R. S., 2000, “Complex Eigenvalue Analysis for Reducing Low Frequency Brake Squeal,” SAE, Warrendale, PA, Technical Report 2000-01-0444.
Popp, K., and Rudolph, M., 2004, “Vibration Control to Avoid Stick-Slip Motion,” J. Vib. Contr., 10, pp. 1585–1600. [CrossRef]
Hoffmann, N., Fischer, M., Allgaier, R., and Gaul, L., 2002, “A Minimal Model for Studying Properties of the Mode-Coupling Instability in Friction Induced Oscillations,” Mech. Res. Commun., 29(4), pp. 197–205. [CrossRef]
Hoffmann, N., and Gaul, L., 2003, “Effects of Damping on Mode-Coupling Instability in Friction Induced Oscillations,” ZAMM, 83(8), pp. 524–534. [CrossRef]
Hetzler, H., 2008, Zur Stabilität von Systemen Bewegter Kontinua mit Reibkontakten am Beispiel des Bremsenquietschens, Schriftenreihe des Instituts für Technische Mechanik, Band 8, Universitätsverlag Karlsruhe, Karlsruhe, Germany.
Madras, N., 2002, “Lectures on Monte Carlo Methods,” Fields Institute Monographs 16, American Mathematical Society, Providence, RI.
Huang, J. C., Krousgrill, C. M., and Bajaj, A. K., 2009, “An Efficient Approach to Estimate Critical Value of Friction Coefficient in Brake Squeal Analysis,” ASME J. Appl. Mech., 74, pp. 534–541. [CrossRef]


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Fig. 1

System design for statistical experimentation

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Fig. 2

System destabilization due to rising friction coefficient μ

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Fig. 3

Fraction of unstable systems with varying cd

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Fig. 4

Maximum Re(λi) of 1000 randomly generated systems

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Fig. 5

Sorted maximum Re(λi) of all unstable systems

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Fig. 6

Probability distribution of size of Re(λi)

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Fig. 7

Sorted maximum Re(λi) of all originally stable systems after 100 perturbations

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Fig. 8

Fraction of unstable systems after 1000 perturbations among all originally stable systems

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Fig. 9

Fraction of nonrobust systems overdamping factor

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Fig. 10

Distribution of maximum Re(λi) for systems that stay unstable

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Fig. 11

Distribution of maximum Re(λi) for systems that become stable

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Fig. 12

Probability distribution of a change of stability for size of eigenvalues’ real part

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Fig. 13

Example for a robust instability

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Fig. 14

Example for a sensitive instability

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Fig. 15

Real part’s standard deviation of first modes of 100 systems subjected to perturbations

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Fig. 16

Robustness rev of the first modes of 100 systems subjected to perturbations

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Fig. 17

Probability distribution of eigenvalue robustness rλi

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Fig. 18

Example for a negative noncritical eigenvalue

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Fig. 19

Example for a negative but critical eigenvalue



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