Journal of Sound and Vibration "0887# 104"2#\ 436458Article No[ sv870698

LINEAR DAMPING MODELS FOR STRUCTURALVIBRATION

J[ WOODHOUSEDepartment of Engineering\ University of Cambridge\ Trumpington Street\

Cambridge CB1 0PZ\ England

"Received 01 January 0887\ and in _nal form 13 April 0887#

Linear damping models for structural vibration are examined] _rst the familiardissipation!matrix model\ then the general linear model[ In both cases\ an approximationof small damping is used to obtain simple expressions for damped natural frequencies\complex mode shapes\ and transfer functions[ Results for transfer functions can beexpressed in the form of very direct extensions of the familiar expression for the undampedcase[ This allows a detailed discussion of the implications of the various damping modelsfor the interpretation of measured transfer functions\ especially in the context ofexperimental modal analysis[ In the case of a dissipation!matrix model\ it would be possiblein principle to determine all the model parameters from measurements[ In the case of thegeneral model\ however\ there is a fundamental ambiguity which prevents fulldetermination of the model from measurements on a single structure[

7 0887 Academic Press

0[ INTRODUCTION

All structures exhibit vibration damping\ but despite a large literature on the subjectdamping remains one of the least well!understood aspects of general vibration analysis[The major reason for this is the absence of a universal mathematical model to representdamping forces[ There are excellent reasons why the sti}ness and inertia properties of ageneral discrete system\ executing small vibrations around a position of stable equilibrium\may be approximated via the familiar sti}ness and mass matrices[ These simply representthe _rst non!trivial terms which do not vanish when the potential and kinetic energyfunctions are Taylor!expanded for small amplitudes of motion 0[ Nothing so simple canbe done to represent damping\ because it is not in general clear which state variables thedamping forces will depend on[ A commonly!used model\ originated by Rayleigh 0\supposes that instantaneous generalized velocities are the only relevant state variables[Taylor expansion then leads to a model which encapsulates damping behaviour in adissipation matrix\ directly analogous to the mass and sti}ness matrices[ This model willbe examined in some detail in this paper[

However\ it is important to avoid the widespread misconception that this is the onlylinear model of vibration damping[ It is perfectly possible for damping forces to dependon values of other quantities\ or equivalently to depend\ via convolution integrals\ on thepast history of the motion[ Any such model which guarantees that the energy dissipationrate is non!negative can be a potential candidate to represent the damping of a givenstructure[ The dissipation!matrix model is just one among many such models[

The appropriate choice of model depends\ of course\ on the detailed mechanism"s# ofdamping[ Unfortunately\ these mechanisms are more varied and less well!understood thanthe physical mechanisms governing sti}ness and inertia[ A very brief review is useful to

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set the scene[ In broad terms\ structural damping mechanisms can be divided into threeclasses] "0# energy dissipation distributed throughout the bulk material making up thestructure\ which can generically be called {{material damping||^ "1# dissipation associatedwith junctions or interfaces between parts of the structure\ generically {{boundarydamping||^ and "2# dissipation associated with a ~uid in contact with the structure\involving either local viscous e}ects or radiation away into the ~uid[

Material damping can arise from a variety of microstructural mechanisms "see e[g[\reference 1# but for small strains it is often adequate to represent it through an equivalentlinear\ viscoelastic continuum model of the material[ Damping can then be taken intoaccount via the {{viscoelastic correspondence principle||\ which leads to the conceptof complex moduli 2[ For sinusoidal motion at a given frequency v\ the e}ects ofviscoelasticity can be exactly represented by replacing each of the real elastic constantsof the material by a suitable complex value[ The number of independent elastic constantsfor a given material is governed by its microscopic symmetries as usual*an isotropicmaterial requires two\ a general orthotropic material requires nine and so on[

It is important to keep in mind that complex moduli must be de_ned in the frequencydomain[ Although it is often found empirically that complex moduli are almostindependent of frequency in the low audio range\ considerations of causality 3 show thatit is not possible for the complex moduli to be entirely independent of frequency[ If inverseFourier transformation is used to obtain results in the time domain\ this inevitablefrequency dependence of the complex moduli must be taken into account[ Failure to dothis properly leads to the notion of {{a di}erential equation with frequency!dependentcoef_cients||\ which is all too often encountered in the literature[ As has been pointed outforcibly by Crandall 3\ this is a mathematical nonsense because it mixes time!domain andfrequency!domain concepts\ and is likely to lead to fallacious results[ In particular\ itdisguises the distinction between linear and non!linear models] the cases with constantcoef_cients and frequency!dependent coef_cients are often described as {{linear|| and{{non!linear||\ respectively\ when in reality both are linear models[

{{Boundary damping|| is less easy to model than material damping\ but it is of crucialimportance in most engineering structures[ When damping is measured on a built!upstructure like a ship or a building\ it is commonly found to be at least an order ofmagnitude higher than the intrinsic material damping of the main components of thestructure[ This di}erence is attributed to e}ects such as frictional micro!slipping at jointsand air!pumping in riveted seams\ but this attribution is usually based on negative evidence"{{what else could it be

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mechanisms of material or boundary damping it is far from obvious that instantaneousgeneralized velocities will be the only state variables determining the rate of energydissipation\ even to a _rst approximation[ In this paper the dissipation!matrix model isconsidered\ and then the results are compared with those from the most general linearmodel of damping[ In both cases\ a small!damping approximation is used to obtain simpleexpressions for complex frequencies and mode shapes\ and for transfer functions[

Within the scope of this approximation\ it is possible to address some general questions[Given a particular physical system\ how could one determine experimentally whether aparticular damping model "such as a dissipation!matrix model# is appropriate< If aparticular model can be _tted to a set of measurements to satisfactory accuracy\ does thatmean that the underlying physical mechanisms have been well represented< Will the e}ectof structural modi_cations be well predicted< Could an entirely di}erent damping modelbe _tted equally well to the same set of measurements< Would that matter for the accuracyof prediction of system behaviour\ or of the ef_cacy of vibration!control measures