estimation of modal dampings for unmeasured modes

F. PÁPAI, S. ADHIKARI, B. WANG

ESTIMATION OF MODAL DAMPINGS FOR UNMEASURED MODES

KEY WORDS

• Modal damping ratio,• Modal decay rate,• Regression,• Damping matrix.

ABSTRACT

Damping effects are of great interest for structural analysis and evaluations. Structural modal damping characteristics can be obtained from experiments. This paper introduces new possibilities for the modelling of the damping of a dynamic system with classical normal modes and provides an overview of the known methods for formulating a damping matrix base with experimental modal damping values. The proposed method offers an opportunity to extrapolate modal damping values for unmeasured modes by a regression method based on the measured modal properties. The points of view on the choice of an analytical form for damping regression functions are examined. An analytical form of regression functions can be chosen as the modal decay rate versus the square of the frequency or the modal damping ratio versus the frequency. Damping regressions can be performed based on a group of typical vibration modes, such as bending, torsion and lateral, symmetrical or anti-symmetrical modes. The regression data obtained for the damping constants can then be applied in a finite element model for further structural analysis.

Ferenc PÁPAIemail:[email protected]

Research field: experimental modal analysis, identi-fication of damping mechanism, characterization ofproportionaldamping,structuredynamicmodification,structural diagnostic, damage detection, localisation,parameterization,monitoring

Address:Department of BuildingMachines,Materials HandlingMachines and Manufacturing Logistic, Faculty ofTransportation Engineering, Budapest University ofTechnologyandEconomics,Budapest,Hungary.

Sondipon ADHIKARIemail:[email protected]

Research field:uncertaintyquantification in computa-tionalmechanics, bio and nanomechanics (nanotubes,graphene,cellmechanics,nano-biosensors),dynamicsof complex systems, inverse problems for linear andnon-linear dynamics, renewable energy (wind energy,vibrationenergyharvesting)

Address:AerospaceEngineering,CollegeofEngineering,SwanseaUniversity,SingletonPark,SwanseaSA28PP,UnitedKingdom.

Bor-Tsuen WANGemail:[email protected]

Researchfield:vibrationnoisecontrol,computeraidedengineering

Address:DepartmentofMechanicalEngineering,NationalPingtungUniversityofScienceandTechnology,Pingtung,912,Taiwan.

Vol. XX, 2012, No. 4, 17 – 27

1. INTRODUCTION

Theequationofmotionofaviscouslydampedsystemisexpressedby

, (1)

where M, C, Kare the mass, damping and stiffness matricesrespectively; is the dynamic displacement response.Thesystemmatrixof(1)is

. (2)

2012 SLOVAK UNIVERSITY OF TECHNOLOGY 17

18 ESTIMATION OF MODAL DAMPINGS FOR UNMEASURED MODES

2012/4 PAGES 17 — 27

The eigenvalues are complexquantities. are damping, and is the damped naturalfrequency. It is known that for symmetrical undamped systems,the eigenvalues of the system matrix are pure imaginary quantities, and due to the

⇒ , thereareneigenvectors ,whicharelinearlyindependent.Theycanbemade forvectorswith realelementsbyappropriatenormalisation.The equivalent transformation, which is performed by amodalmatrix constructed from eigenvectors , simultaneouslydiagonalizesM,K. Suchmodal shapes of undamped systems arecallednormal modes orclassical normal modesintheliterature.Inthe case of normalmodes all the displacement coordinates of thestructurereachtheextremepositionatthesametime(purephase/outphase;inotherwords,astandingwaveinamodalshape).The concept of decaying systems of classical normal modes wasintroduced by Lord Rayleigh (1877), who proved that classicalmodescanexistindampedsystemstoo.Suchacaseis,forexample,theso-calledproportionaldamping,wherethedampingmatrixCisalinearcombinationofMandK,thatis:

(Rayleighdamping) (3)

wherethea,bmaterialpendants(usuallyreal)arescalarconstants.

Caughey and O’Kelly (1965)[1]provedthatanecessary and sufficient condition for the existence ofclassicalnormalmodesistosatisfy

, (4)

theso-called commutative relationship .Ifthe commutative relationship shown in Eq.(4) is satisfied for thecoefficient matrices of system Eq.(1), the real modal coordinatetransformation

(5)canbeperformed. Inthismodalspacetheequationofthemotionofaviscouslydampedsystemisexpressedby

(6a)

where, , ,

(6b)(6c)(6d)(6e)notation meansadiagonalmatrix.

Caughey (1960)[2]verifiedthatthesenormalmodesofadamped

system are equivalent to the modes of an undamped system. “Inthe case of adamped linear system, classical normalmodes existif and only if the damping matrix is diagonalized by the sametransformation, which makes the undamped system diagonal.”He formulated asufficient condition for coefficient matrices: “Indamped systems, the condition for classical normalmodes is thattheproductM–1CcanbeexpressedasthepowerofproductM–1K,intheformofthefollowingseries:”

(7)

Heproposedthefollowingexpressionforthepossibleanalyticalformofadampingmatrixthatsatisfiestheconditionmentionedabove:

. (8)

ThisformiscalledtheCaugheyseriesintheliterature.ThedampingmatrixformedbyEq.(8)isalsoknownastheproportionaldampingmatrixafter Rayleigh.ThetraditionalRayleighdampinginEq.(3)isaspecialcaseoftheexpressioninEq.(8),becausewiththesubstitutiona0=a,a1=b,a2=a3=...=an-1=0,wehave:

. (9)

Fawzy(1977)[3]gavethefollowinganalyticalformforadampingmatrix: ,whichsatisfiedthecondition

in Eq.(4). Sas et al. (1998) [4] composed the commutativerelationshipintheformof .

Adhikari (2000) [5] suggested afurther generalization foraproportional damping matrix. He proved that if the dampingmatrixcanbeexpressedas , (10) , (11) , (12) , (13)

where are analytical functions, the systempossessesclassicalnormalmodes.

1.1 The relationship between modal characteristics and the damping matrix

LetusstudyAdhikari’s relationshipasgivenbyEq.(10)

. (14)

2012/4 PAGES 17 — 27

19ESTIMATION OF MODAL DAMPINGS FOR UNMEASURED MODES

In this relationship replace with their spectraldecompo-sitiontoexpressthemodalmatricesfromEqs.(6b)and(6c)

, (15)

andthenthespecifiedCdampingmatrixtransformsintothemodalspace

. (16)

Bring forwardmatrices from this spectral decomposition[6,p.249],

. (17)

SubstitutingthemodaldampingEq.(6c) relation-shipintoEq.(17),wefinallyget

, (18)

asasimpleequation.Thisexpressionmeansthefollowing:„Modal dampings of symmetrical, single eigenvalued systems with classical normal modes are equal to the return values of a generalized Rayleigh damping matrix generator function at undamped eigenfrequencysquare locations.”

InthecaseofclassicRayleighdampingthefunctioninEq.(18)hasbeen known for along time: ,which is seeninFig. 1.

ThegeneratorfunctionoftheCaugheyseriesdefinedinEq.(8)is

or asascalarfunction.

Akey objective of this paper is to exploit the generic functionalrelationship in Eq.(18) and investigate whether this can be usedeffectivelytomodeladampingmechanisminpracticalengineeringproblems.Anovelideabasedonthegroupingofsimilarmodeshasbeenproposedinthiscontext.

2. IDENTIFICATION OF THE PARAMETERS OF A DAMPING MECHANISM

Therelationship inEq.(18)canbeappropriate for identifying theparametersof adampingmechanism. If the regression function isknown, the following procedure and possible application can beused(Fig. 2):

1.Anexperimentalmodalanalysis iscarriedout todetermine theeigenfrequencies and damping values in the frequency rangeinvestigated.

2.These value pairs are drawn according to achart (Fig. 2), andthentheregressionfunction isdetermined.

3.Extrapolatetheregressionfunction totheoutbandmodes(tothehigherfrequencyrange).

4.The results are available for FEM programs as damping datainput.

5.The damping matrix can be written in the explicit form.

Thefollowingquestionsorproblemsoccurrelatingtoregression:a)Undamped eigenfrequencies cannot be established by experi-mentalmeasurements.

b)W h a t kind of quantities should be represented foraregression?

c)Is the regression function considered global in thesensethatitisvalidforallthemodaltypes?

d)Whatshouldtheanalyticalformoftheregressionfunctionbe?

Asiswellknown,undampedeigenfrequenciescannotbedeterminedby experimental measurements. The undamped system eigen-

Fig. 1 Generator function of the proportional (Rayleigh) damping matrix.

Fig. 2 Regression of generalised Rayleigh damping.


2012/4 PAGES 17 — 27

frequencies areincludedinEq.(18);however,theexperimentalmeasurements define the damped eigenfrequencies . Theapproximation is widely accepted in the case of smalldampedsystems.Forplottingtherelationshipbetweendampingvs.aneigenfrequency,the (undampedeigenfrequencyvs.Lehrdamping) diagram is used in the literature, e.g., [10]-[14]. In thecaseofthemeasurementdata pointsareplotted,wherethedamped eigenfrequencies are . If the undampedeigenfrequency squares , as determined by FEM or anyanalyticalmethods,areavailable,itisadvisabletousethem.As to what kind of quantities should be represented for theregression, we propose the selection of the regression functionfor plotting instead of , because the former isappropriateforthefunctiondefinedinEq.(18).

3. CASE STUDIES

Inthefollowingsectionsweintroduceaselectionofgraphrepresentationssuitableforevaluatingtheexperimentalmeasurementresults.

3.1. Experimental investigations: I. Cantilever beam

MeasurementsoftheEMA(ExperimentalModalAnalysis)cantileverbeam,SISO (Single Input –SingleOutput), andFRF (FrequencyResponseFunction)wereperformed(Fig. 3).Aschematicdiagramoftheexperimentalsetupisshownintheillustration.The excitation and response measurements were performedhorizontally (xy). The total number of the measured locationswas153.N=5modesweredetectedintheinvestigatedfrequencyrange[0–200Hz].TheLSFD (Least Squares Frequency Domain)methodwasapplied to identify theeigenfrequenciesanddampingparameters[8](Fig. 4).TheresultsareshowninTable 1.Thenwecomposedamodalmodelof thestructureandvisualizeditsmodalshapes.Thefirstfourbendingmodesandthefirsttorsionmode appeared in the investigated frequency band. The thirdbendingandtheshapeofthefirsttorsionmodesareshowninFig. 5.Theeigenvalueparameters seen inTable 1 areplotted inFig. 6aby atraditional representation. The features of the curve can beidentifiedwithdifficultyonthischart.In the proposed representation shown inFig. 6b, the relationshipofthevaluespresentsasquarerootfunction.Aregressionwiththeanalytical form was carried out, whichcorresponds toalinearregressionon the values(Fig. 6c).The resultsof the regression for function canbe seeninFig. 7,which is proper for the extrapolationof damping in anoutbandfrequencyrange.

Accordingtotheexperimentalresults,inthecaseoftheinvestigatedprismaticbeam,theregressionfunction isglobalforboththebending and torsionmodes.Only one torsionmodewas detectedinthefrequencyrangeinvestigated.Wethinkmoretestsshouldbecarriedoutinahigherfrequencybandforhighertorsionmodes.The problem of determining the shape of acurve in regressionanalysis isrelatedto thequestionofwhether thespecificfunction

isglobal.Theglobal means the function isvalidforallthemodes.Weconductedsomeexperimentalmeasurements,wherewefoundthat different types of modes could have different regressionfunctions.Someofthemareshownbelow.

Fig. 3. Schematic representation of the experimental setup of the cantilever beam [8] (EMA, SISO, FRF measurements).

Fig. 4 LSFD regression of the measured FRF function [8].

Table 1 Eigenfrequency and damping values of the cantilever beam [8].Modei 1 2 3 4 5fi[Hz] 4.671 30.93 87.36 125.5 171.5ξi[%] 5.49 0.918 0.373 0.282 0.224Type BXZ1 BXZ2 BXZ3 TZ1 BXZ4

BXZbendingTtorsion

2012/4 PAGES 17 — 27


Fig. 5 Modal shapes of the cantilever beam [8].

Fig. 6 Methods representating of the cantilever beam’s eigenvalue parameters.

Fig. 7 Regression of the eigenvalue parameters.


2012/4 PAGES 17 — 27

3.2. Experimental investigation II: Damping Identification in a Clamped Plate With Slots

In this section we consider atwo-dimensional structure [7].Aschematic model of the test structure is shown inFig. 8. Thisis fabricated by making slots in amild steel rectangular plate ofa2mmthickness,resultinginthree-cantileverbeamsjoinedattheirbasebyarectangularplate.Thesourceofthedampinginthisteststructureisthewedgedfoambetweentheforkbenches1and2.

AschematicdiagramofthetestrigisshowninFig. 9.AclassicEMA SISO FRFmeasurementwasperformed.Themodalshapesfromthe4thtothe9thareshowninFig. 10.Ontheleftsidemodes4th,5thand6thare thesecondbending, torsionandmixedmodes.Ontherightmodes7th,8thand9tharethethirdbending,torsionandmixedmodesrespectively.

Themeasurednaturalfrequenciesanddampingfactorsforthefirstnine modes are shown inTable 2. The three mode groups wereseparatedaccordingtothetypeofmodeshape:abending,atorsionandamixedone.Eachmodegroupconsistsofthreemodes.

Table 2 Measured natural frequencies, damping factors and modal types of the clamped plate with slots [7].Mode: 1 2 3 4 5 6 7 8 9f[Hz] 12.46 14.36 15.01 75.60 88.94 93.97 232.74 243.37 261.93ξ[%] 0.1032 0.0969 0.1159 0.1404 0.1389 0.1254 0.1494 0.0953 0.1260Type B T M B T M B T M

ModetypesB:Bending,T:Torsion,M:Mixed

Fig. 8 Geometric shape of the plate with slots [7].

Fig. 10 The mode shapes from the 4th to the 9th [7].

Fig. 9 Schematic representation of the experimental setup of the clamped plate with slots [7].

2012/4 PAGES 17 — 27


ThemeasuredvaluesonFig. 11aareseeninthreedifferentviewingplots.TheresultsoftheregressionbymodegroupsonFig. 11b areshown.Theanalyticalformoftheregressionfunctionineachgroupis withdifferentcoefficientsof .

3.3. Experimental investigation III: Ambient Response Analysis of the Heritage Court Tower Building

LetusseeasanexampletheregressionofthemodaldampingtakenfromtheliteraturebasedonthemeasurementsofBrincker et al. [9].

ThemodalshapesoftheTowerBuildingwereexamined.Thephotoof this building is seen inFig.12. Its size is 20mx25mx50m.Thebuildingstructurehasclassicalnormalmodes.TheeigenvalueparametersandtypesofmodalshapesaresummarizedinTable 3.Inprocessingthepublisheddata,wegotthefollowingresults.ThedampingandeigenfrequencyvalueswereplottedontraditionalandproposeddiagramsasshowninFig. 13.Forthisdataseriestheregressionwasperformedbymodegroupsasinthepreviouschapter.Onemodegroupisofthebendingmodes;theotherisofthetorsionones.TheresultsareshowninFig. 14.

Fig 11. Regression of the eigenvalue parameters of the plate.


2012/4 PAGES 17 — 27

Theseresultsraisethefollowingquestions:• Could classical normalmodes exist in the case of anot globalgeneratorfunction?

• Canregressionbedoneper-mode-groupwithout thelossof theclassicalnormalmodes?

Thefollowingsectiondescribesthemethodofgroupingthemodes.

4. GROUPING OF THE MODES

Thespectraldecompositionofmatrix ,usingtheexpressionsinEqs.(6b)and(6d),is:

, (19)

whichisthesumofthennumberofdyads(columnsof ,androwsof ,andthematrix isadiagonalmatrix).Let usmakem number of mode groups.According to themodegroups,mperformsapartitionof thecolumnsof themodalmatrix

, the elements of the spectral matrix , and the rows of theinverseofmodalmatrix ,

Fig. 12 Picture of the investigated building.

Fig. 13 Eigenvalue parameters of the investigated Tower Building [9].

Table 3 Building’s EMA measurement results [9]. Modei 1 2 3 4 5 6 7 8 9 10 11fi[Hz] 1.23 1.28 1.45 3.85 4.25 5.34 6.39 7.47 7.58 8.22 9.26ξi[%] 2.12 1.77 1.2 1.16 1.52 1.69 1.54 2.2 2.31 2.66 1.95Type BXZ1 T1 BYZ1 T2 BXZ2 BYZ2 T3 BXZ BXZ BXZ T4

BXZ:BendinginXZplane BYZ:BendinginYZplane T:TorsionaroundZaxis

2012/4 PAGES 17 — 27


, ,

(20)

where aretheeigenvectorsofmodegroupp; aretherowvectorsof ,whichbelongtomodegroupp,andfinally aretheelementsofthespectralmatrixbelongingtomodegroupp.Thespectraldecompositionof intermsofmodegroupsis

. (21)

Letfunctions betheregressionfunctionsofthemodegroups.Itisverifiablethatthedampingmatrix

(22)

generatedbyfunctions satisfiesthecommutative relationshipinEq.(4), so thedampedsystemhasclassicalnormalmodes, the

proof is seen in the Appendix. Using based onEq.(6b)inEq.(22),wecanobtainanexplicitformoftheclassicaldampingmatrix

(23)

(seeLancasterdampingmatrix).Iftheregressionfunctionisnotglobal,butthegroupingofthemodesissecure,youcanusethefollowingprocedureandapplicationfield:1.Theexperimentalmodalanalysisiscarriedoutintheinvestigatedfrequencyrange,andthedampingandeigenfrequencyvaluesareidentified.

2.The eigenfrequency-damping values are plotted.The damping-eigenfrequency points are grouped according to the types ofmodalshapes.

3.The regression functions are fitted on points by the group ofmodes.

4.The fitted smooth continuous functions used as extrapolationfunctions can give the value of the damping for the outbandmodes, which can be used with FEM programs to input thedampingvaluesfortheunmeasuredmodes.

5.ThegeneralizedproportionaldampingmatrixcanbemadeinanexplicitformbyapplyingEq.(23).

5. SUMMARY, ADDITIONAL TASKS

In thispaper thepossibilitiesof identifyingdampingmechanismshave been reviewed. After our performed measurements and ananalysisoftheliterature,themainconclusionsare:

Fig. 14 Eigenvalue parameters of the Tower Building investigated [9] Regression of damping by mode group.


2012/4 PAGES 17 — 27

Constructionofadampingmatrixbyaglobalgeneratorfunctionisnotalwayspossible.However, it may be constructed by functions defined in modegroups,withoutthelossofclassicalnormalmodes.

Additional tasks

Thegroupingofmodesisshouldbesolved.Suchamethodshould

bedevelopedbywhich theclassificationofamodalshapecanbeperformed automatically. The effects that cause variations in thedampingvalueswithanEMAmethodshouldbeanalysed.

AcknowledgmentF.P. acknowledges the support of the New Széchenyi Plan Új Széchenyi TervTÁMOP-4.2.1/B-09/1/KMR-2010-0002.

REFERENCES

[1]Caughey,T.K.,O’Kelly,M.E.J.: ClassicalNormalModesinDampedLinearDynamicSystem.Transactionof theASME.JournalofAppliedMechanics32,1965.pp:583-588.

[2] Caughey, T. K.: Classical Normal Modes in Damped LinearDynamic System. Transaction of the ASME. Journal ofAppliedMechanicsJune.1960.pp:269-271.

[3] Fawzy, T. K.: ATheorem on the Free Vibration of DampedSystem. Transaction of the ASME. Journal of AppliedMechanicsMarc.1977.pp:132-134.

[4] Sas, P.,Ward, H., Lammens, S.:ModalAnalysis Theory andTesting.KatholikeUniversiteitLeuven.ISBN90-73802-61-X.Heverlee(Belgium)1998.pp:1-300.

[5]Adhikari,S.: Dampingmodellingusinggeneralizedproportionaldamping.JournalofSoundandVibration.2006,Vol.293,Nos.1-2,pp:156-170.

[6]Rózsa,P.:LinearAlgebraandApplications.Műszakikönyvkiadó,Budapest,1976.p.685(inHungarian)

[7]Adhikari,S.,Phani,S.: ExperimentalIdentificationofGeneralizedProportional Viscous Damping Matrix. Transaction of theASME Journal of Vibration andAcoustics. February 2009,Vol.131,pp:1-12.

[8] Pápai, F.: Structural Diagnostic of Building and MaterialsHandling Machines by Application of Experimental ModalAnalysis.BudapestUniversityofTechnologyandEconomics.FacultyofTransportationEngineering.DepartmentofBuilding

Machines,MaterialsHandlingMachines andManufacturingLogistic.PhDThesis,2007.p.179,(inHungarian).

[9] Brincker, R., Andersen, P.: Ambient Response Analysis ofthe Heritage Court Tower Building Structure, IMAC XVIII: Proceedings of the International Modal Analysis Conference (IMAC), San Antonio, USA, February 7-10, 2000,pp.1081-1087.

[10] Benedettini, F., Gentile, C.: Operational Modal Testing andFE Model Tuning of Cable-stayed Bridge. EngineeeringStructures(2011)

[11]Allen,M. S.: Global andMulti-Input-Multi-Output (MIMO)Extension of theAlgorithm of Mode Isolation (AMI). PhDThesis,2005,GeorgiaInstituteofTechnology.

[12] Siringoringo, D. M., Fujino, Y.: System Identification ofSuspension Bridge from Ambient Vibration Response.EngineeringStructures30(2008)462–477

[13] Doebling, S.W., Farrar, C. R., Goodman, R. S.: Effects ofMeasurement Statistics on the Detection of Damage in theAlamosa Canyon Bridge. Proc. of the 15th InternationalModalAnalysisConference,Orlando,FL,Feb.3-6,1997,pp.919-929.

[14] Cunha, A., Caetano, E., Brincker, R. - Andersen, P.:Identification from theNaturalResponseofVascodaGamaBridge.ProceedingsofIMAC-22:AConferenceonStructuralDynamics, January26 -29,2004,HyattRegencyDearborn,Dearborn,Michigan,USA.pp.202-209.

2012/4 PAGES 17 — 27


APPENDIX

Theorem:Thedampingmatrixgeneratedby

(A-1)satisfiesthecommutativerelationship.

Proof:Considerthe commutativerelationship.Substitutethespectraldecompositionofallthematricesinit.Theleftsideofthisrelationshipis

Therightsideofthisrelationshipis:

Here

Consequently,

(A-2)

(A-3)

TherightsideofEq.(A-2)equalstherightsideofEq.(A-3),becausea and arediagonalmatrices,sotheyarecommutative.Consequently,theleftsideofEq.(A-2)andtheleftsideofEq.(A-3)areequal,sothecommutativerelationshipissatisfied.

estimation of modal dampings for unmeasured modes

Documents