Modal Assurance Criterion

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1 vailable online at Pocedia Engineeing 48 (2012 ) MMaMS 2012 Modal ssuance Citeion Mioslav Pasto a, Michal Binda a *, omáš Ha aik a a echnical Univesity of Košice, Faculty of Mechanical Engineeing, Letná 9, Košice, Slovak Republic bstact his aticle eviews the using of the oiginal Modal ssuance Citeion (MC). he Modal ssuance Citeion is a statistical indicato that is most sensitive to lage diffeences and elatively insensitive to small diffeences in the mode shapes. his yields a good statistic indicato and a degee of consistency between mode shapes. he MC consides only modal shapes which mean that a sepaate feuency compaison must be used in conjunction with the MC values to detemine the coelated mode pais. he MC is often to used to pai modes shapes deived fom analytical models with those obtained expeimentally. It is easy to apply and does not euie an estimate of the system matices. It is bounded between 0 and 1, with 1 indicating fully consistent mode shapes. It can only indicate consistency and does not indicate validity o othogonality. value nea 0 indicates that the modes ae not consistent he uthos. Published by Elsevie Ltd Published by Elsevie Ltd.Selection and/o pee-eview unde esponsibility of the Banch Office of Slovak Metallugical Society at Faculty Selection of and/o Metallugy pee-eview and Faculty unde of Mechanical esponsibility Engineeing, of the Banch echnical Office Univesity of Slovak of Košice Metallugical Open access Society unde at CC Faculty BY-NC-ND of Metallugy license. and Faculty of Mechanical Engineeing, echnical Univesity of Košice. Keywods: Modal esting, Modal ssuance Citeion, Expeimental Modal nlaysis Nomenclatue { } { } { ϕ } tanspose of { } ϕ test modal vecto, mode ϕ compatible analytical modal vecto, mode ϕ { ϕ } tanspose of { ϕ } { } { } { ψ } * complex conjugate of { ψ } ψ eigenvecto defined the complex damped mode of vibation ψ eigenvecto defined the complex damped mode of vibation { ψ } tanspose of { ψ } { ψ } * complex conjugate of { ψ } * Coesponding autho. el.: addess: michal.binda@gmail.com Published by Elsevie Ltd.Selection and/o pee-eview unde esponsibility of the Banch Office of Slovak Metallugical Society at Faculty of Metallugy and Faculty of Mechanical Engineeing, echnical Univesity of Košice Open access unde CC BY-NC-ND license. doi: /j.poeng

2 544 Mioslav Pasto et al. / Pocedia Engineeing 48 ( 2012 ) Compaison of modal popeties Modal popeties that ae compaed usually include natual feuencies, eal mode shape vectos, modal masses, modal kinetic and stain enegies. Fo system with complex modes of vibation one can add modal damping atios and complex mode shapes. Compaison of modal vectos can be done at the educed ode o at the full ode of the FEM. Reduction of the physical mass matix o expansion of test modal vectos bing inheent appoximations in the compaison citeia. test-analysis compaison is meaningful only fo matched modes. hese ae estimates of the same physical mode shape and thei enties coespond one-fo-one with thei countepats. Mode matching is an essential step befoe any compaison can be undetaken [1], [10]. It is useful to compae: Measued mode shapes against vectos detemined by analytical model Estimates of the same test modal vecto obtained fom diffeent excitation locations Estimates of the same modal vecto obtained fom diffeent modal paamete identification pocesses using the same test data One test mode shape befoe and afte a change in the physical stuctue caused by a wanted modification [8], [10] 2. Modal assuance citeion One of the most popula tools fo the uantitative compaison of modal vectos is the Modal ssuance Citeion (MC). he development of the MC was modeled afte the development of the odinay coheence calculation associated with computation of the feuency esponse function. he MC is a statistical indicato, just like odinay coheence. his least suaes based fom of linea egession analysis yields an indicato that is most sensitive to the lagest diffeence between compaative values and esults in modal assuance citeion that is insensitive to small changes o small magnitudes. he MC was oiginally intoduced in modal testing in connection with he Modal Scale Facto, as an additional confidence facto in the evaluation of modal vecto fom diffeent excitation locations. When an FRF matix is expessed in the patial faction expansion fom, the numeato of each tem epesents the matix of esidues o modal constants [5], [6]. he MC is calculated as the nomalized scala poduct of the two sets of vectos { ϕ } and { ϕ } scalas ae aanged into the MC matix (, ) MC { ϕ} { ϕ } { } { } ( ϕ ϕ )({ ϕ} { ϕ} ) 2. he esulting =, (1) whee the fom of coheence function can be ecognized, indicating the casual elationship between { ϕ } and { } ϕ. In this case the modulus in the numeato is taken afte the vecto multiplication, so that the absolute value of the sum of poduct elements is suaed. nd euivalent fomulation is (, ) MC = n { ϕ } { } 1 ϕ j= j j n n ( { ϕ } ) { } j 1 ϕ = j j= 1 ( j) 2 2. (2) he MC has been used as a Mode Shape Coelation Constant to uantify the accuacy of identified mode shapes. Fo complex modes of vibation (, ) MC { ψ } { ψ } { } { } * 2 =. (2) * * ( ψ ψ )({ ψ } { ψ } ) and is clealy a eal uantity, even if the mode shape data ae complex.

3 Mioslav Pasto et al. / Pocedia Engineeing 48 ( 2012 ) he MC takes value between 0 (epesenting no consistent coespondence) and 1 (epesenting a consistent coespondence). Values lage than 0.9 indicate consistent coespondence wheeas small values indicate poo esemblance of the two shapes. he MC can take on value nea zeo fo the following easons: he system is non-stationay esulting fom changes in the mass, stiffness and damping popeties duing testing he system is non-linea hee is noise on the efeence mode shape he paamete extaction techniue is invalid fo the measued data set he mode shapes ae linealy independent. While the MC is not a tue othogonality check since the mass o stiffness matices have not been included in the calculation, it can be used as an appoximation of an othogonality check Obviously, if the fist fou easons can be eliminated, the MC can be intepeted in a simila way as an othogonality calculation. he MC can take on a value nea unity fo the following easons: he numbe of esponse degees of feedom is insufficient to distinguish between independent mode shapes he mode shapes ae a esult of unmeasued foces to the system he mode shapes ae pimaily coheent noise he mode shapes epesent the same motion diffeent only by a scala heefoe, if the fist thee easons can be eliminated, he MC can be used to indicate consistence shapes theeby indicating the validity of the modal scale facto. he ideal MC matix cannot be a unit matix because the modal vectos ae not diectly othogonal but mass othogonal. Howeve, the MC matix indicates which individual modes fom the two sets elate to the each othe. If two vectos ae switched in one set, then the lagest enties of the MC matix ae no moe on the leading diagonal and it esembles a pemutation matix. he MC can only indicate consistency, not validity, so it is mainly used in pe-test mode paiing. he MC is incapable of distinguishing between systematic eos and local discepancies. It cannot identify whethe the vectos ae othogonal o incomplete [5], [6] MC pesentation fomats One of the big changes in the application of the MC ove the last yeas in the way the infomation is pesented. oday most compute systems outinely utilize colo to pesent magnitude data like MC using 2D and 3D plot, see Fig. 1. It is impotant to emembe that MC is a discete calculation and what appeas as a colo contou plot eally only epesents the discete mode to mode compaison. Nevetheless, a colo plot does allow fo moe data to be pesented in an undestandable fom in minimum space [2]. Fig. 1 2D MC plot example

4 546 Mioslav Pasto et al. / Pocedia Engineeing 48 ( 2012 ) Expeimental pocedue and esults s a test case fo the poposed pocedue, the modal analysis of a steel beam with fee edges has been pefomed, see Fig. 2. he steel beam was excited with modal hamme type 8206 with plastic tip by Buel&Kjae at appopiate points. n analysis was pefomed using analyze Buel&Kjae Pulse 3560C and laptop PC unning PULSE softwae. he esponse to excitation of stuctue was measued in two steps. Fist, system esponse was measued by acceleomete Buel&Kjae 4506 B. he second step, the system esponse was detemined using by optical vibomete PDV100. Fig. 2 measuement chain Natual feuencies of the steel beam wee detemined by using PULSE Reflex softwae. In the Fig. 2 thee ae feuency esponse functions inetances. Natual feuencies wee estimated using softwae algoithms using method Polyefeence ime, see able 1. (a) (b) Fig. 3. (a) the maximum feuency esponse function - maximum inetance measued by Buel&Kjae 4506 B (b) the maximum feuency esponse function - maximum inetance measued by PDV100.

5 Mioslav Pasto et al. / Pocedia Engineeing 48 ( 2012 ) able 1. Natual feuencies Numbe of Mode Feuency - Buel&Kjae 4506 B (Hz) Feuency PDV100 (Hz) 3.1. utomc citeion utomc uantitatively compaes all the possible combinations of test and analysis mode shape pais fo only one set of mode shapes. he calculation assigns a value of 1.0 to test and analysis mode shape pais that exactly match. value of 0 is given to those pais that ae completely independent o unelated, and can indicate a poblem. hee could be spatialaliasing, which indicates that not enough esponse points wee measued to uniuely identify the individual modes. lso, one of the modes could be computational and could, theefoe, look vey much like a stuctual mode. matix pesents the esults of this MC calculation showing which test mode shapes match which analysis mode shapes and the degee of coelation. Values between 0 and 1.0 ae popotional to the degee of coelation between the mode shapes. n example of an oiginal utomc plot fo a simple steel beam is shown in able 2. he use of the diffeent colofilled to indicate the MC values ae shown to be uite useful. able 2. utomc matix CossMC citeion CossMC is simila to utomc, howeve, it is detemined by using two diffeent sets of modes fo the ows and columns of the MC matix. CossMC can be used to: Calculate the MC between two diffeent test models Calculate the MC between a test model and a Finite Element Model (FEM) Calculate the MC between two analysis models developed using two diffeent modal analysis algoithms Values close to one mean that the two mode shapes ae highly coelated and almost identical. able. 3 shows the compaison of natual feuencies obtained by using of two diffeent acceleometes (Buel&Kjae 4506 B vs. PDV100) via CossMC. able 3. CossMC matix

6 548 Mioslav Pasto et al. / Pocedia Engineeing 48 ( 2012 ) Conclusion In this pape, we have eviewed a method fo the compaison and uantitative coelation between the dynamic popeties measued in a modal test. It is possible to detemine the coelated modal pai. Duing last two decades, the Modal ssuance Citeion has been demonstated as a simple statistical concept, which can become an extemely useful tool in expeimental modal analysis. he use of the Modal ssuance Citeion and the development and use of a significant numbe of elated citeia has been emakable and is most likely due to the oveall simplicity of the concept. New uses of the MC will be developed ove the next yeas as uses moe fully undestand the limitations of the cuent citeia. In the next few yeas, the inceased uses of othe statistical methods as well as futhe development of singula value/vecto methods ae elated aeas. Since the MC gives no infomation about the feuency coespondence and can sometimes show good coelation between modes that have significant feuency sepaation, a natual feuency compaison plot is also euied. cknowledgements he pape has been ceated ealization of poject Centum výskumu iadenia technických enviomentálnych a humánnych izík pe tvalý ozvoj podukcie a výobkov v stojástve (IMS: ), based on opeating pogam suppot Reseach and Development financed fom Euopean Regional Development Fund and by Scientific Gant gency VEG MŠ SR fo the suppot of this wok unde Pojects No. 1/0289/11 and No.1/0265/10. Refeences [1] Rades, M., Mechanical Vibations II Stuctual Dynamic Modelling. Editua PRINECH. [2] Randall, J., he Modal ssuance Citeion wenty Yeas of Use and buse, Sound and Vibation, ugust 2003.,pp [3] Ewins, D., J Model Validation: Coelation fo updating. Sadhana, Vol. 25, pat 3, June 2000, pp [4] Ewins, D., J Modal esting: heoy, Pactice and pplication, Reseach Studies Pess Ltd., US. [5] Fotsch, D., Ewins, D., J. pplication of MC in the Feuency Domain, Mechanical Engineeing Depatment, Impeial College of Science,echnology and Medicine, London, UK. [6] vilés, R. 2009, he Modal ssuance Citeion (MC). Bilbao. [7] ebu a, F., Šim ák, F Píu ka expeimentálnej mechaniky. ypopess, Košice. [8] Bilošová, Expeimentální modální analýza. Spectis Paha spol. s..o., Paha. [9] Kožešník, J Kmitání mechanických soustav. cademica. [10] Milá ek, S Modální analýza mechanických kmitú. VU, Paha.