Nonlinear damping and quasi-linear modelling

Nonlinear damping and quasi-linear modelling

The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a “quasi-linear” model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the n-th power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system’s response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated of the nonlinear damper, proving both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed; in microspeakers, vibration isolation, energy harvesting, and the mechanical response of the cochlea.

nonlinear damping, quasi-linear, statistical linearization

1-23

Elliott, S.J.

721dc55c-8c3e-4895-b9c4-82f62abd3567

Ghandchi Tehrani, Maryam

c2251e5b-a029-46e2-b585-422120a7bc44

Langley, R.

f46614cf-344e-4155-9f18-f3d83ed4fc1b

Elliott, S.J.

721dc55c-8c3e-4895-b9c4-82f62abd3567

Ghandchi Tehrani, Maryam

c2251e5b-a029-46e2-b585-422120a7bc44

Langley, R.

f46614cf-344e-4155-9f18-f3d83ed4fc1b

Elliott, S.J., Ghandchi Tehrani, Maryam and Langley, R.
(2015)
Nonlinear damping and quasi-linear modelling
*Philosophical Transactions of The Royal Society A*, .

## Abstract

The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a “quasi-linear” model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the n-th power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system’s response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated of the nonlinear damper, proving both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed; in microspeakers, vibration isolation, energy harvesting, and the mechanical response of the cochlea.

**Text __userfiles.soton.ac.uk_Users_slb1_mydesktop_Nonlinear damping 7 1 2015_Final.docx
- Accepted Manuscript**

## More information

Accepted/In Press date: June 2015

Keywords:
nonlinear damping, quasi-linear, statistical linearization

Organisations:
Signal Processing & Control Grp

## Identifiers

Local EPrints ID: 377831

URI: http://eprints.soton.ac.uk/id/eprint/377831

ISSN: 1364-503X

PURE UUID: 064fca6b-63e3-413d-8790-794ec89d66f4

## Catalogue record

Date deposited: 08 Jun 2015 12:26

Last modified: 15 Sep 2017 16:33

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## Contributors

Author:
R. Langley

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