Aalborg Universitet. Load Estimation from Natural input Modal Analysis Aenlle, Manuel López; Brincker, Rune; Canteli, Alfonso Fernández
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1 Aalbog Univesitet Load Estimation fom atual input Modal Analysis Aenlle, Manuel López; Bincke, Rune; Canteli, Alfonso Fenández Published in: Confeence Poceedings Publication date: 005 Document Vesion Publishe's PDF, also known as Vesion of ecod Link to publication fom Aalbog Univesity Citation fo published vesion (APA): Aenlle, M. L., Bincke, R., & Canteli, A. F. (005). Load Estimation fom atual input Modal Analysis. In Confeence Poceedings: IMAC-XXIII : A Confeence & Exposition on Stuctual Dynamics Society fo Expeimental Mechanics. Geneal ights Copyight and moal ights fo the publications made accessible in the public potal ae etained by the authos and/o othe copyight ownes and it is a condition of accessing publications that uses ecognise and abide by the legal equiements associated with these ights.? Uses may download and pint one copy of any publication fom the public potal fo the pupose of pivate study o eseach.? You may not futhe distibute the mateial o use it fo any pofit-making activity o commecial gain? You may feely distibute the URL identifying the publication in the public potal? Take down policy If you believe that this document beaches copyight please contact us at vbn@aub.aau.dk poviding details, and we will emove access to the wok immediately and investigate you claim. Downloaded fom vbn.aau.dk on: decembe 10, 017
2 Load Estimation fom atual input Modal Analysis OMECLATURE Manuel López Aenlle Depatment of Constuction and Manufactuing Engineeing Univesity of Oviedo, Campus de Gijón, 3303, Gijón, Spain. Rune Bincke Depatment of Building Technology and Stuctual Engineeing Aalbog Univesity, Sohngaadsholmsvej, 57, 9000, Denmak. Alfonso Fenández Canteli Depatment of Constuction and Manufactuing Engineeing Univesity of Oviedo, Campus de Gijón, 3303,Gijón, Spain. Stiffness matix [ k ] Mass matix [ m ] Damping matix [ c ] Foce vecto { f ( t) FRF matix [ H ( ω) ] Spectal matix [ S ( ω) ] atual fequency ω Scaled mode shape { φ Un-scaled mode shape { ψ Scaling facto α Damping facto ABSTRACT ζ One application of atual Input Modal Analysis consists in estimating the unknown load acting on stuctues such as wind loads, wave loads, taffic loads, etc. In this pape, a pocedue to detemine loading fom a tuncated modal model, as well as the esults of an expeimental testing pogamme, ae pesented. The method involves the invesion of the FRF matix patly solving the numeical poblems that appea because of the tuncation of the modal space. Howeve, the eo in the load estimation depends on the degee of tuncation of the modal space and on the accuacy of the modal paamete estimation. In the expeimental pogam a small stuctue subjected to vibation was used to estimate the loading fom the measuements and the expeimental modal space. The modal paametes wee estimated by atual Input Modal Analysis and the scaling factos of the mode shapes obtained by the mass change method [ ] [ 3] [ 4]. Finally, the calculated loading is compaed with the actual loading in ode to detemine the magnitude of the eos.
3 1. ITRODUCTIO When atual Input Modal testing is pefomed, the testing is nomally done by just measuing the esponses unde the natual conditions. This means that the foces ae nomally not ecoded o contolled. This epesents an impotant advantage compaed with taditional modal analysis, mainly fo big stuctues. Wind loads, wave loads, taffic loads, etc. can be consideed natual input. On the othe hand, enginees ae inteested in obtaining infomation about those loads which ae difficult to measue diectly (wind loads, wave loads, taffic loads, etc. ). They can also lean about the coelation length in the loading and thus lean how to pefom bette natual input modal analysis [ 1]. In this pape, a pocedue to detemine loading fom a tuncated modal model, as well as the esults of an expeimental testing pogamme, ae pesented. This pocedue involves the invesion of the FRF matix; also othe numeical poblems appea because of the tuncation of the modal space. An expeimental pogam is caied out to estimate the loading fom the measuements and the expeimental modal space. A steel cantileve beam ae used. The modal paametes ae obtained by natual input modal analysis and the scaling factos of the mode shapes by the mass change method [ ] [ 3] [ 4]. Finally, the estimated loading is compaed with the actual loading in ode to detemine the magnitude and the souces of the eo.. THE METHOD As well know, the equation of motion of a stuctue subjected to a foce { ( t) [ m ] { u + [ c] { u + [ k] { u = { f ( t) Tansfoming equation ( 1) in fequency domain by Fouie, yields: ( ω [ m] + jω [ c] + [ k] ) { U( ω) = { F( ω) f is given by: Defining the fequency esponse function matix (FRF) o tansfe function matix as: ( ) 1 [ H( ω )] = ω [ m] + jω [ c] + [ k] and substituting the equation ( 3) in equation ( ), it esults: { U ( ω) = [ H( ω) ] { F( ω) ( 1) ( ) ( 3) ( 4) the loading in fequency domain can be calculated by: 1 { F( ω) = [ H( ω) ] { U( ω) The foce in time domain can be obtained applying the invese Fouie tansfom of { F ( ω) schematically shown in Figue 1. ( 5). The pocess is To solve equation ( 5), the FRF matix and the esponses have to be known. If modal analysis is pefomed, the esponses ae measued and the modal paametes can be estimated. Subsequently, the FRF can be constucted fom the modal paametes. The spectal density function load matix can then be obtained fom the spectal density function esponse matix by means of the expession: 1 [ S ( ω )] = [ H( ω) ] [ S ( ω) ] [ H( ω) ] H FF UU ( 6)
4 whee the supescipt H denotes complex conjugate tanspose. 3. THE FRF MATRIX When the FRF matix is obtained fom the modal paametes, the following infomation is needed fo each mode: The natual fequencies ω, The damping factos ζ, The mass nomalized (scaled) mode shape { φ. FRF H ( ω) IVERSE FRF H ( ω ) 1 RESPOSES u ( t) FOURIER RESPOSES U ( ω) Howeve, when natual input modal analysis is pefomed, the foces ae unknown so that only the following infomation can be obtained fo each mode: The natual fequencies ω, The damping factos ζ, The un-scaled mode shape { ψ. The scaled and un-scale mode shapes ae elated by { φ = α { ψ ( 7) F FORCE 1 ( ω) = H( ω) U( ω) FORCE f ( t) IV. FOURIER Fig. 1. Pocess to estimate the loading whee α is the scaling facto of the -th mode. Theefoe, when natual input modal analysis is used, an exta method is needed to calculate the scaling factos. Recently, diffeent methods has been poposed to estimate the scaling factos involving epeated testing in which mass changes ae intoduced in the points whee the mode shapes ae known[ ] [ 3] [ 4]. In case of complex modes, the expession of the FRF matix when the modal space is used, is given by[ 5 ] : [ H ( ω) ] whee: T * { ψ { ψ Q { ψ { ψ * *T Q = + = 1 jω λ * jω λ { ψ is the -th un-scaled mode shape, = ζ ω + jω 1 λ ζ is the pole of the -th mode Q is a facto which takes into account the scale of the mode, and the supescipt * denotes complex conjugate ( 8) The facto Q α = jω Q can be elated to α though [ 5 ]: ( 9)
5 4. IVERSIO OF THE FRF MATRIX Equations ( 5) and ( 6) involve the invesion of the FRF matix fequency by fequency, but this invesion can only be pefomed using standad methods when the FRF matix is full ank, i.e., when the numbe of modes is equal to the numbe of obsevation points. Othewise, when a tuncated modal space is used, the FRF matix is singula and nomal invese does not apply anymoe. Howeve, the invesion of the FRF matix can still be done using singula value decomposition (SVD). The singula value decomposition of a complex matix [ H ] is: [ H] = [ U] [ ] [ V] H whee: [ U] and [ V ] ae unitay matices (othogonal in case of eal matices), and [ ] is a diagonal matix containing the singula values. The numbe of non-zeo singula values ae equal to the numbe of modes active at the consideed fequency. ( 10) The supescipt H denotes complex conjugate tanspose. Using equation ( 10) the invese of the matix [ H ] can be obtained as: [ ] 1 [ ] H [ ] 1 H = V [ U] 1 Taking in account the popeties of unitay matices, i.e.: [ ] H [ ] 1 [ ] H U = U and V = [ V] 1 the equation ( 11) becomes: [ ] 1 H = [ V] [ ] 1 [ U ] H ( 11) ( 1) ( 13) whee only the non-zeo singula values must be used in the calculation. Equation ( 13) povides the exact solution when all modes ae consideed, but due to the tuncation effect, the calculated FRF matix and its invese will only epesent an appoximation. As soon as moe modes ae consideed, bette accuacy will be achieved. 5. LEAKAGE REDUCTIO. Equations ( 5) and ( 6) involve the Fouie tansfom of the esponses so that the analysis of a finite time ecod can cause leakage. A method to minimize the leakage effect is to apply a window. The application of a window speads some of the enegy of the oiginal signals to the adjoining spectal components while it suppesses the enegy leaked to othe spectal components which ae fa fom the coect fequency. When a window is used to obtain the spectum, some infomation gets lost and the oiginal signal in time domain can not be ecoveed by invese Fouie tansfom of the coesponding spectum. Fo this eason, if the foce in time domain is the objective, windows can not be applied and leakage eos will be pesent in the estimated foce, but If the objective is the spectal density function foce matix, a window should be used when the Fouie tansfom of the esponses ae calculated. Howeve, leakage can still be educed using the method descibed in the next section fo eal time estimation.
6 6. REAL TIME ESTIMATIO. If the modal paametes of the stuctue ae known and the esponses ae measued in eal time, then the foce can also be estimated in eal time. Taking points of the ecoded esponse, the foce coesponding to this segment can be estimated using equation ( 5). It we want to calculate the foce coesponding to points, a way to educe the leakage is to calculate the foce fo a lage segment, i.e., a segment of β points, whee β > 1 and then select the cental points. Thus, the ( β 1) points of the estimated load on both the ight and left sides ae discaded. In Figue, the load calculation fo β = is shown. With this method the leakage will be educed. The only inconvenient is that moe numeical opeations have to be pefomed to obtain the load. / / Fig. Real time load estimation with β = RESPOSES CALCULATIOS LOAD 7. ERRORS. Thee ae fou main souces of eo when the load is estimated: Eos due to noise in the esponses Eos in the modal identification. Eos due to the tuncation of modes Eos due to leakage. When pefoming expeimental measuements, some noise is always pesent in the signals. This type of eo can be educed using bette sensos but can not be emoved. The noise effect can also be educed by filteing. A singula value decomposition of the esponses can help to decide which fequencies should be filteed. The eos in the modal paametes estimation depend on seveal factos such us the level of noise in the esponses, the method and the softwae used in the estimation, the type of excitation, etc. As mentioned befoe, when the modal space is tuncated only an appoximation of the FRF matix can be obtained so that these eos will be amplified when the invese opeation ae pefomed. Finally, as also pointed out peviously, the eos due to leakage can be educed if the objective is the calculation of the spectal density foce matix, but they will be pesent when the foce in time domain is the objective. 8. EXPERIMETAL TESTS A steel cantileve beam was used to pefom the tests. The beam was 1.85 m length, with a 80x50x4 tube ectangula section and the esponses wee measued in 8 degee of feedoms egulaly distibuted along the beam (Figue 3). Two types of excitation wee used: Stationay boad banded and impact.
7 8.1 Stationay boad banded tests In ode to detemine the modal paametes, a stationay boad banded excitation was applied to the stuctue. The loading was not measued so that a natual input modal analysis softwae was used to estimate the natual fequencies, the mode shapes and the damping. Only the fist 5 modes wee consideed in the analysis. 8 7 Section The scaling factos wee detemined by means of the mass change method [ ] [ 3] [ 4]. A popotional mass change was applied to the stuctue attaching masses of 180 gams in each degee of feedom (except at the fee bode of the cantileve beam at which a 90 gam mass was attached). Finally, a new stationay boad banded excitation was applied to the modified stuctue. L=1.85 m mm 80 mm 50 mm I=38.88 cm 4 m=7.66 The modal paametes ae shown in table 1. As can be seen, the damping is vey low. The scaling factos shown in Fig 3. Cantileve beam. the table coespond to mode shapes nomalized to unity. Table 1. Modal paametes of the cantileve beam. Mode atual fequencies (Hz) Damping (%) Scaling factos Impact tests In ode to check the accuacy of the poposed method, seveal impacts wee applied in each degee of feedom of the unmodified stuctue. Only one hit was applied evey time. An impact hamme with a ubbe tip was used to apply the hits. The esponses, togethe with the FRF matix, wee used to estimate the load with the method poposed in this pape. The eal foce in time domain applied on the 8 th degee of feedom is shown in Figue 4a wheeas the estimated foce is shown in Figue 4b. A zoom of the impact is shown in Figue 5. As can be obseved, the impacts can be detected and the eo is easonable low. Due to the noise pesent in the esponses and the eos in the estimated FRF matix (amplified in the invesion pocess) a low level foce is estimated in all channels (Figue 4b). Futhemoe, when the impact foce is applied on a degee of feedom, a small peak is estimated in the
8 adjacent degee of feedoms. In Figue 4b it can be seen that a small peak is estimated in the 8 th degee of feedom when the impact is applied on the 7 th degee of feedom. on a) b) on Fig 4. a) Real foce applied in the 8th degee of feedom. b) Estimated foce in the same degee of feedom. Due to the fact that the damping is vey low, a lage numbe of points (30000 points) had to be used to educe the leakage effect. Futhemoe, a low pass filte was applied to educe the noise effect at low fequencies pesent in the esponses. Fo this eason, the eo in the magnitude of the estimated foce in time domain is significant. The eo in the magnitude of the estimated foce is low at the fee bode of the beam (appoximately 10% of the exact magnitude) wheeas the eo inceases as the degee of feedom is neae the suppot whee the maximum eo is appoximately 40% of the exact magnitude. The foce autospectal density of both the estimated and the ecoded foce, coesponding to the 8 th degee of feedom is shown in Figue 6. Windows to educe the leakage effect wee not applied. As can be seen, the estimation is easonable good excepting the peaks in the esonances which appea due to the eos in the modal paametes estimation. Fig 5. Zoom of the eal and estimated peak in the 8th degee of feedom The foce estimation can be impoved emoving o educing the peaks in the spectum by means of a smoothing technique, but it was not used in this pape.
9 a) b) Fig 6. a) Real foce spectal density in the 8th degee of feedom. b) Estimated spectal density in the 8th degee of feedom. 9. COCLUSSIOS A pocedue is poposed to estimate the loading exciting a stuctue, fom the expeimental esponses and the modal paametes estimated by atual Input Modal Analysis. A steel cantileve beam is used to check the method poposed in this pape. The modal paametes ae estimated by natual input modal analysis which ae used togethe with the expeimental esponses to estimate the impact foces applied to the stuctue with an impact hamme. The estimated foce in fequency domain is quite good. Although the eos in the estimation of modal paametes, some peaks appea in the specta at the esonances. A low pass filte has to be used to emove the noise at low fequencies. Fo this eason, the foce in time domain can not be estimated accuately. 10. ACKOWLEDGEMETS The economic suppot given by the Spanish Ministy of Education is gatefully appeciated. 11. REFERECES [ 1] R. Bincke, C. Ventua and P.e Andesen: Why Output-Only Modal Testing is a Desiable Tool fo a Wide Range of Pactical Applications. In Poc. Of the Intenational Modal Analysis Confeence (IMAC) XXI, pape 65, Febuay, 003 [ ] R. Bincke and P. Andesen: A Way of Getting Scaled Mode Shapes in Output Only Modal Analyis. In Poc. Of the Intenational Modal Analysis Confeence (IMAC) XXI, pape 141, Febuay, 003. [ 3] M. López-Aenlle, Rune Bincke and A. Fenández-Canteli: Some Methods to Detemine Scaled Mode Shapes in atual Input Modal Analysis. In these Poceedings, Febuay, 004. [ 4] E. Paloo, P. Veboven, P. Guillaume, M. Van Ovemeie: Sensitivity- Based Opeational Mode Shape omalization. In Mech Systems and Signal Poc., 16(5), pp , 00. [ 5] W. Heylen, S. Lammens and P. Sas, Modal Analysis: Theoy and Testing. Katholieke Univesiteit Leuve, 1997.
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