Andersen, P.

Brincker, Rune

In this paper some further work is done following the idea introduced by Parloo et all where they proposed that the scaling factor should be estimated by repeated testing introducing mass changes in different points on the structure. In this paper the approximate formula for determination of the scaling factor based on the frequency shift when introducing mass changes on the structure is derived directly from the governing equation of motion. Further the error sources are studied and a new formula introducing less approximation errors on the scaling factor is proposed. Using this new formula scaling factors can be estimated from relative large frequency shifts without introducing any approximation errors. Further it is explained how testing should be performed in order to significantly reduce approximation errors due to mode shape changes and random errors due to uncertainty on the mode shape values. It turns out that if the mass changes are well distributed over the structure, then both random errors and the approximation errors will be minimized

    Aalborg UniversitetA Way of Getting Scaled Mode Shapes in Output Only Modal TestingBrincker, Rune; Andersen, P.Published in:Proceedings of IMAC-21Publication date:2003Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Brincker, R., & Andersen, P. (2003). A Way of Getting Scaled Mode Shapes in Output Only Modal Testing. InProceedings of IMAC-21: A Conference on Structural Dynamics, February 3-6, 2003, The Hyatt Orlando,Kissimmee, Florida (pp. 141-145). Society for Experimental Mechanics.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: April 24, 2017A Way of Getting Scaled Mode Shapesin Output Only Modal TestingRune BrinckerDepartment of Building Technology and Structural EngineeringAalborg University , Sohngaardsholmsvej 57, 9000, Denmark,E-mail: i6rb@civil.auc.dkPalle AndersenStructural Vibration Solutions ApSNiels Jernes Vej 10, 9220 Aalborg East, DenmarkE-mail: pa@svibs.comAbstractIn this paper some further work is done following the ideaintroduced by Parloo et all [2] where they proposed that thescaling factor should be estimated by repeated testingintroducing mass changes in different points on thestructure. In this paper the approximate formula fordetermination of the scaling factor based on the frequencyshift when introducing mass changes on the structure isderived directly from the governing equation of motion.Further the error sources are studied and a new formulaintroducing less approximation errors on the scaling factor isproposed. Using this new formula scaling factors can beestimated from relative large frequency shifts withoutintroducing any approximation errors. Further it is explainedhow testing should be performed in order to significantlyreduce approximation errors due to mode shape changesand random errors due to uncertainty on the mode shapevalues. It turns out that if the mass changes are welldistributed over the structure, then both random errors andthe approximation errors will be minimized.NotationMass matrix MStiffness matrix KUn-scaled mode shape φScaled mode shape ψNatural frequency ωStandard deviation σNumber of DOF’s N1. IntroductionOutput-only modal testing and analysis is becoming moreand more popular due to the clear advantages of thetechnology: The testing is easier, the technology isapplicable to a wider a range of the structures, it has a widerrange of potential applications since the actual responsesare stored and can be used for instance fatigue analysis andvibration level estimation, and also some times thetechnology gives better and more reliable results in caseswhere the actual loading conditions and operating conditionsare important for the structural response.However the technology is still in the early stages ofdevelopment, and many problems still remain unsolved orpartly unsolved. One the important remaining problems isthe problem of mode shape scaling. If the identified modalmodel is going to be used for structural response simulationor for structural modification, then the scaling factors of themode shapes must be known. Also in health monitoringapplications and in cases where damage is to be identified,the scaling factors might be useful.Recently some suggestion has been given in the literaturefor solving this problem. One solution has been suggestedby Bernal and Gunes [1] based on the assumption thatpartition of the inverse of the mass matrix associated withthe measured coordinates is diagonal. However, theapproach gives exact answers only when there is a full set ofmodes, and robustness for a truncated modal space has notbeen demonstrated.  Recently Parloo et al [2] havepublished a new approach based on a more extensivetesting procedure that involves repeated testing where masschanges are introduced in the points of the structure wherethe mode shape is known. This approach seems moreappealing, since to scale a certain mode, only that particularmode has to be known.Parloo et al derived an approximate formula for thedetermination of the scaling factor from some basicsensitivity relations in linear dynamics. In this paper howeverwe will derive the same approximate formula directly fromthe equations of motion, an analysis of the random errorsand of the errors introduced by the approximations ispresented, and finally a proposal for a better formulaintroducing significantly less approximations errors will begiven.2. Approximate scaling from mass change effectsThe approximate equation that relates mass change,frequency change and the unknown scaling factor can beeasily derived from the basic equations of motion. Theclassical eigenvalue equation in case of no damping is11 KφMφ =21)1( ωwhere 1φ  is the mode shape and 1ω  is the naturalfrequency of any of the modes of the problem related to themass matrix M . Now, if we make a mass change so thatthe mass matrix becomes ∆MM + , then the eigenvalueequation becomes22 Kφ∆M)φ(M =+ 22)2( ωand 2φ , 2ω  is now the modal parameters of the modifiedproblem.Subtracting equation (1) and (2) then gives us the equationto approximate to obtain an equation for the unknownscaling factor)()()3( 222221 12221 φφKφφφM −+∆=− ωMωωIf we now assume that the mass change is so small that themode shapes does not change significantly, i.e.φφφ 21 =≅)4(and that the change of natural frequency is so small that withgood approximationωωωω ∆≅− 2)5( 2221where 21 ωωω −=∆ and 2/)( 21 ωωω += , then we obtainthe equationMφMφ ∆=∆ωω2)6(This equation holds for any mode shape scaling factor.Usually in output-only modal analysis the mode shapes arejust scaled to unity, i.e. 1=φφT , and usually in traditionalmodal analysis is preferred a scaling so that1)7( =MψψTThe desired scaling factor is the factor the relates the scaledand the un-scaled mode shapesφψ α=)8(Now using the scaled mode shapes in Eq. (4) and combiningwith Eq. (6), we directly obtain by multiplying with the samemode shape from the left∆MφφT22)9( αω=∆ωand the unknown scaling factor is obtained as∆MφφTωα∆ω2)10( =Which is the result obtained by in Parloo et all (equation (10)in [2]). The matrix ∆M  would normally be a diagonal matrixcorresponding to the case where the mass changes areplaced only at the points where the mode shapes are known.If this is case, and if all mass changes m∆  at the individualdegrees of freedom are the same, then the mass changematrix can be written D∆M m∆= , where the matrix D  is adiagonal matrix containing only ones in the diagonal. Thenequation (8) becomesDφφT=∆=ZmZ∆ω)12(2)11(ωαIn the matrix D  there is a one in the diagonal every time anadditional mass is placed in that DOF, and if a mass isremoved, there is a minus one. Thus, if a set of masses areplaced at the structure, the number Z  is a positive, and ifthe mass is removed, then the number Z  is negative. If theun-scaled mode shapes are scaled to unity, then theabsolute value of Z  is between zero and one, and in thecase the same mass is placed in all DOF’s at the structure,then Z  becomes equal to one.3. Uncertainty on the estimated scaling factorsAssuming that the uncertainty ασ on α  is controlled by theuncertainty ωσ ∆ on the frequency shift ω∆  and by theuncertainty Zσ  on the inner product Z , then linearization ofequation (9) and assuming independent stochastic variablesleads to the following approximate equation for the relativevariance on the scaling factor2222224141)13(ZZσωσασ ωα +∆=∆The uncertainty on the inner product Z  can be obtained in asimilar way toDφφT22 4)14( ϕσσ =Zwhere the uncertainty on the mode shape values areassumed all to be equal to ϕσ . Combining equation (13)and (14) then yields the final resultDφφT2222241)15( ϕωασωσασ+∆=∆From this formula it is clearly observed, that it is important tokeep the relative uncertainty of the frequency shift down to areasonable value, the same can be said about the modeshape values, however, for the mode shapes it is alsoimportant to use as many points as possible for the masschange. Thus, it can be concluded, that in order to minimizethe uncertainty on the scaling factor, the mass changesshould be distributed over as many degrees of freedom aspossible, and the mass change should be large enough toallow for a reasonably low uncertainty of the naturalfrequency shift. However, the mass changes should be smallenough for the approximations given by equation (5) and (6)to be valid. A further analysis of the errors will be given in thenext sections.4. Illustration of approximation errorsThe errors introduced by the approximate nature ofequations (4) and (5) has been studied by performingsimulation on a simple dynamic system−−−−==211...21121..10.01)16( km KMThe system had 20 degrees of freedom and only the first 5modes were used in the investigation. The exact scalingfactor α  were calculated using equation (7) and (8). Then amass change was introduced, and equation (10) was thenused to obtain an approximate value 'α  for the scalingfactor. The error on scaling factor was then calculated as1'/ −αα . Figure 1 shows the error on the scaling factor forthe first mode of the system when performing a masschange in one DOF (top plot) and five DOF’s (bottom plot).As it is observed from Figure 1, when shifting the masschanges over the structure, for some cases the error isnegative, and for some cases it is positive. However, it turnsout that the largest error is positive and corresponds to theupper envelope of the curves (the approximate linear upperbound).Taking the envelopes for the first five modes gives theresults shown in Figure 2. As it is seen from these results,when the mass change is oddly distributed (mass change in5 DOF’s), the error is significantly dependent on the mode,and when the mass changes becomes more and moreevenly distributed, the difference in error between theindividual modes tends to vanish.This is to be expected from theoretical considerations. Whenthe mass change is evenly distributed so thatM∆M β=)17(then mode shapes does not change by introducing theadditional masses, and thus, the assumption giving byequation (4) is satisfied and does not contribute to theapproximation error.Figure 1.Top: Error on scaling factor when using equation(10) for a mass change in one DOF. Each curve shows theinfluence of the mass change moving over the structure, thedifferent curves shows the influence of a mass change from1-20 % of the mass of the individual DOF’s. Bottom: Sameas the top, but now with mass change in 5 DOF’s, however,since the total  mass change is five times larger, thefrequency shift is larger.Figure 2. Maximum errors when using equation (10) on thefirst five modes of the investigated system when changingthe mass at the individual DOF’s from 1-20 %. Top: masschanges at 5 DOF’s, middle: mass changes at 10 DOF’s,bottom: mass changes at 15 DOF’s.5. A better formula for the scaling factor estimationAs it appears from the previous analysis, the errors on thescaling factor when using the approach as proposed byParloo et al have two sources of error both contributingsignificantly; a) one error source due the mode shapechanges when masses are added to the structure (equation(4)), and b) one error source due to the approximation givenby equation (5). The first error source can be minimized bydistributing the mass changes according to equation (17),however, the latter error source is inherent to application ofequation (10) and cannot be altered as long as this approachis used.However, there is no reason to perform the approximationgiven by equation (5). The resulting equation when onlyusing the approximation (4) and not approximation (5) is∆MφφT222221)18(ωωωα−=When using this formula, no approximation is introduced dueto final values of the frequency shift, and thus, largefrequency shifts can be applied without introducing any erroron the scaling factor. The error reduction when usingequation (18) in stead of equation (10) is illustrated in Figure3.Figure 3. Maximum errors when using equation (18) on thefirst five modes of the investigated system when changingthe mass at the individual DOF’s from 1-20 %. Top: masschanges at 5 DOF’s, middle: mass changes at 10 DOF’s,bottom: mass changes at 15 DOF’s.6. Error reduction strategiesIt can be concluded that the approximate equation (10)should not be used for mode shape scaling since itintroduces significant and unnecessary errors on the scalingfactor. In stead equation (18) should be used. When usingequation (18) three sources of errors are still presenta) Systematic errors due to mode shape changesb) Random errors on the frequency shiftc) Random errors on the mode shapeAs it has been shown above, the best way to reduce theerrors introduced by changes of the mode shape is to try toperform the mass changes in such a way that all DOF’s havethe same relative mass change. This approach assumes thatbefore performing the test a reasonable estimate of themass distribution can be obtained.The error on the frequency shift can be dealt with byperforming good tests and good analysis in order to keep theuncertainty on the natural frequency identification down andmaking sure that the frequency shift is significantly largerthan the uncertainty. A good natural frequency identificationcan usually be performed with a relative accuracy of theorder of 310− . Thus if we make sure that the frequency shiftis at least 1 % of the natural frequency, then the contributionto the relative error on the scaling will be within 10 %. Thisdoes not seem satisfactory, and thus higher identificationaccuracy has to be achieved and/or larger mass changedmust be applied. However, using equation (18) in stead ofequation (10) will allow for the application of large frequencyshifts without any error introduction.The influence of the random errors on the mode shape canbe reduced by introducing the mass changes in as manyDOF’s as possible. If we only perform mass change is oneDOF, then the error contribution iskϕσασ ϕα=)19(but if we introduce mass changes in all the DOF’s, then – ifwe assume that the un-scaled mode shapes are scaled tounity - the contribution becomesϕα σασ=)20(If we know the mode shape in N  DOF’s, the mode shapevalues will be of the order N/1=γ  and thus, if we haveestimated mode shapes within a relative accuracy of ε , thenthe uncertainty on the individual mode shape values will beof the order N/2ε  and thenN/)21( εασα ≅Thus, if a mode shape is known in say 100 points, and if theuncertainty on the mode shape is smaller then 10 %, thenintroducing mass changes in all 100 points, the uncertaintyon the mode shape will only contribute to the random erroron the scaling factor with less than 1 %.The main conclusion is that mode shape scaling using thestrategy proposed in this paper should be performed byusing mass changes that are as well distributed as possible,and in such a way that the resulting changes of the naturalfrequencies are large compared to the uncertainty on thenatural frequencies.AcknowledgementsWe appreciate comments and inspiration from severalcollegues. Special thanks to Aldo Sestieri and GuilianoCoppotelli at La Sapienza University, Rome, to WalterD’Ambrogio at L’Aquila University and to Dionisio Bernal atNortheastern University. Also thanks to Margaret for somenice days at the Haernaes school where the basic work wasdone.References[1] Dionisio Bernal and Burcu Gunes: DamageLocalization in Output-Only systems: A FlexibilityBased Approach. In Proc. of the International ModalAnalysis Conference (IMAC) XX, Los Angeles,California, February, 2002, p. 1185-1191.[2] E. Parloo, P. Verboven, P. Guillaume, M. VanOvermeire: Sensitivity-Based Operational ModeShape Normalization. In Mech Systems and SignalProc., 2002, 16(5), p. 757-767

A Way of Getting Scaled Mode Shapes in Output Only Modal Testing

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A Way of Getting Scaled Mode Shapes in Output Only Modal Testing

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