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Research Papers

# Self-Excited Vibrations and Damping in Circulatory Systems

[+] Author and Article Information
Peter Hagedorn

Professor
Fellow ASME
Dynamics and Vibrations Group,
Numerical Methods in Mechanical Engineering,
Dolivostr. 15,

Manuel Eckstein

Dynamics and Vibrations Group,
Numerical Methods in Mechanical Engineering,
Dolivostr. 15,

Eduard Heffel

Dynamics and Vibrations Group,
Numerical Methods in Mechanical Engineering,
Dolivostr. 15,

Andreas Wagner

Dynamics and Vibrations Group,
Numerical Methods in Mechanical Engineering,
Dolivostr. 15,

Clearly the MKN-system with $M=I,K=[k1000k2000k3],N=[0ν0ν00000]$ and $k1≠k2=k3$ is stable, since the system uncouples into one with n = 2 and no repeated eigenvalues, and a trivial MKN-system with n = 1. Hence the term generically in the previous statement.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 15, 2013; final manuscript received August 7, 2014; accepted manuscript posted August 11, 2014; published online August 27, 2014. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(10), 101009 (Aug 27, 2014) (9 pages) Paper No: JAM-13-1469; doi: 10.1115/1.4028240 History: Received November 15, 2013; Revised August 07, 2014; Accepted August 11, 2014

## Abstract

Self-excited vibrations in mechanical engineering systems are in general unwanted and sometimes dangerous. There are many systems exhibiting self-excited vibrations which up to this day cannot be completely avoided, such as brake squeal, the galloping vibrations of overhead transmission lines, the ground resonance in helicopters and others. These systems have in common that in the linearized equations of motion the self-excitation terms are given by nonconservative, circulatory forces. It has been well known for some time, that such systems are very sensitive to damping. Recently, several new theorems concerning the effect of damping on the stability and on the self-excited vibrations were proved by some of the authors. The present paper discusses these new mathematical results for practical mechanical engineering systems. It turns out that the structure of the damping matrix is of utmost importance, and the common assumption, namely, representing the damping matrix as a linear combination of the mass and the stiffness matrices, may give completely misleading results for the problem of instability and the onset of self-excited vibrations. The authors give some indications on improving the description of the damping matrix in the linearized problems, in order to enhance the modeling of the self-excited vibrations. The improved models should lead to a better understanding of these unwanted phenomena and possibly also to designs oriented toward their avoidance.

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## Figures

Fig. 1

Stability map in the ξ–d-plane; dashed line: stability boundary for k1 = k2, ①: flutter instability, ②: asymptotic stability

Fig. 2

Stability map in the α–β-plane; ①: flutter instability, ②: asymptotic stability

Fig. 3

Disk brake model with wobbling disk [30]

Fig. 6

Rotor in contact with friction pin

Fig. 4

Stability map in the dt-Ω-plane; ①: flutter instability, ②: asymptotic stability

Fig. 5

Model of a roller pair in a paper calender

Fig. 9

FEM model of a disk brake

Fig. 7

Stability map for dp = 0, ①: flutter instability, ②: asymptotic stability, all imaginary parts are nonzero, ③: stable, some imaginary parts are zero

Fig. 8

Stability map for d2 = 4300 Ns/m, ①: flutter instability, ②: asymptotic stability, all imaginary parts are nonzero, ③: stable, some imaginary parts are zero

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