A Lifting Algorithm for Output-only Continuous Scan Laser Doppler Vibrometry

Transcription

1 53d AIAA/ASME/ASCE/AHS/ASC Stuctues Stuctual Dynamics and Mateials Confeence<BR>2th AI Apil 22 Honolulu Hawaii AIAA 22- A Lifting Algoithm fo Output-only Continuous Scan Lase Dopple Vibomety Shifei Yang Matthew S. Allen 2 Univesity of Wisconsin-Madison Madison Wisconsin 5376 USA Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- Continuous Scan Lase Dopple Vibomety (CSLDV) can geatly acceleate modal testing by continuously sweeping the measuing lase ove the stuctue effectively captuing the esponse of the stuctue at tens o even hundeds of points simultaneously. The authos ecently extended this technique to the case whee the input foces ae unmeasued and andom using hamonic powe spectum. This pape pesents a vaiant on the poposed method that combines lifting a esampling appoach with the output only algoithm. Lifting causes all of the peaks in the hamonic powe spectum to collapse onto a single peak fo each mode geatly simplifying modal paamete estimation. The poposed appoach woks by estimating and then lifting the hamonic coelation function which is analogous to the impulse esponse of the system. The poposed algoithm is evaluated on a simulated beam and compaed with the pevious output only methods indicating that the new appoach gives compaable esults to those of the pevious methods but the data eduction is fa simple. The algoithm is then used to identify the fist seveal modes of a paked wind tubine unde wind excitation captuing the defomation shape along one blade in detail. A new long ange Remote Sensing Vibomete (RSV) fom PolyTec was employed fo these measuements. This new vibomete allows the fist seveal modes of the tubine to be captued fom a standoff distance of 77 metes without the eto-eflective tape applied to the tubine. The speckle noise in the measuements is found to be emakably small allowing a 36 Hz scan fequency to be employed which coesponds to a suface velocity of the lase spot of moe than 5 m/s. Nomenclatue CSLDV = Continuous-Scan Lase Dopple Vibomety EMP = Exponentially Modulated Peiodic HTF = Hamonic Tansfe Function HCF = linea Hamonic Coelation Function phcf = positive linea Hamonic Coelation Function HPSD = Hamonic Powe Spectal Density phpsd = positive Hamonic Powe Spectal Density LTI = Linea Time Invaiant LTP = Linea Time Peiodic State Space LTI System Λ = diagonal matix of system poles B = contol o input matix C = output matix P = matix of state space eigenvectos q = state of the uncoupled system o modal paticipation facto Gaduate Assistant Engineeing Physics Depatment 53 Engineeing Reseach Building 5 Engineeing Dive Madison WI AIAA Membe syang66@wisc.edu. 2 Assistant Pofesso Engineeing Physics Depatment 535 Engineeing Reseach Building 5 Engineeing Dive Madison WI AIAA Membe msallen@eng.wisc.edu. Copyight 22 by the Inc. All ights eseved.

2 u = state space input y = state space output x = position of lase spot = intege efeing to a paticula mode ψ = th mode shape of the undelying LTI system ω = th natual fequency of the undelying LTI system ζ = th damping atio of the undelying LTI system λ = th eigenvalue of the undelying LTI system Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- LTP system l = intege descibing the offset of a hamonic peak fo a paticula mode n = intege giving the ode of a hamonic in a Fouie seies expansion o an EMP signal T A = fundamental peiod of the LTP system ω A = fundamental fequency of the LTP system o scan fequency fo CSLDV ω A = T A /2π U n (ω) = nth hamonic of the EMP input in the fequency domain Y n (ω) = nth hamonic of the EMP output in the fequency domain Y ( ) = collection of EMP output signals U ( ) = collection of EMP input signals G ( ) = hamonic tansfe function matix C = EMP mode vecto at the lth hamonic of the th mode C n = nth Fouie coefficient fo the th time vaing mode shape A = Residue of the lth hamonic of the th mode in HPSD S YY (ω) = autospectum of EMP output o HPSD H YY (ω) = phpsd R[n] = linea hamonic coelation function R m [k] = lifted linea hamonic coelation function at the mth point ( ) = FFT of phcf fo the mth point on the lase path m Res m = esidue matix of the th mode identified fom the lifted phcf at mth point. = modal contibution constant of the lth hamonic fo th mode I. Intoduction n continuous-scan lase Dopple vibomety (CSLDV) the lase spot continuously sweeps ove a Istuctue while ecoding the esponse along the scan path educing the time equied to measue the stuctue s mode shapes. Many eseaches have investigated CSLDV since it was fist intoduced in 99s []. Among them Ewin Stanbidge et al. have modeled the opeating deflection shape as a continuous polynomial function of the lase spot position. The polynomial coefficients ae obtained fom the sideband peaks in the spectum of measued esponse and the opeating shape can then be econstucted with these coefficients. This method has been successfully applied with sinusoidal [2] impact [3] and pseudo-andom excitation []. On the othe hand Allen et al. poposed a lifting appoach whee the esponses at the same location ae gouped togethe to fom a set of pseudo tansduces along the lase path [5]. The measued specta fom the pseudo tansduces ae the same as would be obtained with an aay of conventional sensos except that thee is a constant time delay between these pseudo sensos and the sampling ate of each senso becomes the lase scan fequency. So the natual fequencies highe than the half of the scan fequency will be aliased accoding the Nyquist-Shannon sampling theoem. The authos applied the lifting method to a fee-fee beam unde impact excitation and the unaliased natual fequencies and mass nomalized mode shapes wee identified [6]. The advantage of the lifting appoach is that it poduces a set of specta that ae mathematically equivalent to a collection of fequency esponse functions at a set of points. Hence the stuctue s modes ae eadily extacted fom the measuements using standad softwae. Howeve this method is moe suitable fo stuctues with low natual fequencies because speckle noise and the mio inetia limit the maximum pactical scan fequency. 2

3 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- All of these methods equie that the foce exciting the stuctue be eithe impulsive o some known caefully contolled function (e.g. sinusoidal). Howeve sometimes it is difficult o even impossible to diectly measue the dynamic load on a stuctue fo example wind tubines o aicaft wings excited by fluctuations in the flow field. CSLDV is especially attactive fo these applications because it can povide spatially detailed mode shapes with as few as one single measuement befoe the excitation conditions o the stuctue changing appeciably. In the authos pevious wok [7] the measued esponse using CSLDV was teated as a the output of a linea time peiodic (LTP) system and an output-only methodology was poposed based Weeley s hamonic tansfe function (HTF) concept [8]. The HTF fo an LTP system is equivalent to the tansfe function of linea time invaiant (LTI) system and was used to define a new type of spectum dubbed the hamonic powe spectum (HPSD) which is fomed fom the CSLDV measuement. The stuctue s mode shapes natual fequencies and damping atios can then be obtained fom the HPSD using peak-picking o conventional opeational modal analysis cuve fitting outines. The method was used to identify seveal modes of a paked wind tubine blade unde wind excitation [7]. While the method pesented in [7] has poved effective the identification pocedue is somewhat labo intensive since a multitude of peaks ae pesent in the HPSD fo each mode of the system. The esulting mode shapes can also vay depending on which peaks in the HPSD ae used to estimate them. On the othe hand the authos lifting method [5 6] allowed one to extact a set of mode shapes fom CSLDV measuements almost automatically. This wok seeks to extend the lifting method to output only measuements. This is accomplished by using the positive hamonic coelation function (phcf) which is analogous to an impulse esponse function and can be estimated fom the HPSD. The appoach used is basically an extension of the positive powe spectum concept developed by Caubeghe fo LTI systems [9] to linea time peiodic systems. In Caubeghe s woks the positive coelation function was tansfomed into a positive powe spectum which is a FRF-like function that can be teated with conventional cuve-fitting outines. That method was extended to LTP systems in [] evealing that one could obtain simila esults with the HPSD o phpsd although the latte ae moe convenient to cuve fit. Howeve both of those specta contain seveal peaks fo each mode so quite a bit of effot is equied to pefom system identification. In this wok the phcf is lifted using the appoach in [5 6] to compute specta that ae analogous to a set of Single-Input Multi-Output (SIMO) fequency esponse functions simila to what would be obtained fom an aay of stationay sensos. The esulting spectum is much simple to intepet than the HPSD and can be cuve-fit with vitually any modal paamete identification outine to identify the natual fequencies damping atios and mode shapes of the stuctue. The est of this pape is oganized as follows. Section II biefly intoduces the hamonic powe spectum concept the hamonic coelation function and the poposed lifting appoach. In Section III the poposed algoithm is demonstated on a simulated beam and compaed with the HPSD and phpsd methods. In Section IV the algoithm is tested on a eal wind tubine unde ambient excitation using measuements fom a new long ange lase vibomete the Remote Sensing Vibomete (RSV) fom PolyTec with a customized mio system. Section V pesents the conclusions. II. Theoy A. Hamonic Tansfe Function and Hamonic Powe Spectum When applying CSLDV with a closed peiodic scan patten to a LTI stuctue the output appeas to be fom a LTP system. Following the deivation in [7] the equation of motion fo the system can be witten in uncoupled state space fom as q qp Bu () y CP() t q 2 whee is a diagonal matix containing the eigenvalues j of the system with the modal damping atio and the natual fequency and P is a matix of state space eigenvectos. The only peiodic tem in the LTP system is the output matix CP(t) which is a ow vecto containing the shape of each mode of the system at the cuent time instant. Following the deivation in [7] 3

4 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- this is denoted * * CP( t) ( x( t)) N( x( t)) ( x( t)) N( x( t)) (2) whee x(t)= x(t+t A ) denotes the position of the LDV measuement point at time t and x denotes the state space mode shape at location x and in the diection sensed by the lase. The fundamental peiod of the scan patten has been denoted T A and it is undestood that the lase path x(t) could involve motion in thee dimensions. It is well known that a single fequency input to a LTI system leads to an output at the same fequency. In contast the esponse of a LTP system will be at the input fequency and also at an infinite numbe of hamonics each sepaated by the fundamental fequency ω A of the LTP system A TA /2. The LTP identification stategy makes use of the hamonic tansfe function concept [] which elates the input and output of LTP system by intoducing the exponentially modulated peiodic (EMP) signal. The EMP signal is composed of the input o output at a collection of fequencies sepaated by ω A. Fo example if the esponse measued with continuously scanning lase vibomete is denoted y(t) and the scan fequency (fundamental fequency) is ω A then one could compute the EMP signal Y ( ) by taking the Fouie tansfom of y(t) and then shifting the spectum by n A. These steps can actually be combined as follows. ( jj A ) ( ) ( ) n t Yn y t e dt The EMP signal is the collection of the fequency shifted copies of Y (). (3) Y( ) Y ( ) Y ( ) Y( ) () Weeley pefomed a simila opeation on the input foces u(t) and used the geneal solution of the state space equation and a hamonic balance appoach to elate the input and output with a hamonic tansfe function (HTF) G ( ) Details of this deivation can be found in [8] Chapte 3. Y G U (5) ( ) ( ) Notice that the HTF is a matix even in the case whee we only have a single input and a single output because it elates the esponse of LTP system at and its hamonics + A A etc to the input at the same fequencies. In this wok we ae concened with the case whee the output is measued using a single beam CSLDV and theefoe it is a scala and the input is unknown white noise and potentially applied to many points along the stuctue. In [7] the authos showed that the hamonic output autospectum (HPSD) of the LTP system can be witten as follows. N H H C W( ) Cl SYY ( ) EY( ) Y ( ) (6) H l j ( j l ) j ( j l ) T A A The equation on the ight is an appoximation because it neglects coss tems between pais of modes whee one mode is esonant and the othe is not. Fo a stuctue unde uncoelated andom white noise input W ( ) is a scala elated to the net excitation of the th mode and does not depend on l so it is witten as W ( ) fom this point fowad [7]. The elements in the vectos C ae not the usual mode shapes (i.e. the amplitudes of motion at vaious points on the stuctue) but ae Fouie coefficients that descibe the th time vaying mode shape and ae defined below. jn At () ( ()) n n Ct xt C e C C l C l C l T (7)

5 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- Of couse the stuctue s th mode shape is time invaiant but it appeas to be time vaying in the CSLDV measuement because the lase spot is continuously moving. The HPSD has the same fom as the output autospectum of an LTI system; it is a sum of modal contibutions. Howeve the time vaying mode shapes of the system give ise to peaks in the CSLDV esponse nea each natual fequency and also at the fequencies l A fo any intege l. Hence one can obtain an estimate of each mode vecto C fom a numbe of diffeent peaks although the tems in each vecto C contain the Fouie coefficients but shifted at diffeent locations as explained in [7]. Fo example the fundamental tem in the Fouie seies C is found in the cente of C but that tem appeas l tems below the cente in C. Eq. (6) can be decomposed into a convenient fom by patial faction expansion. This esults in * tems that ae esonant at both the stable poles ( j la) ( j la) which have negative eal * pats and unstable poles ( j l ) ( j l ). The HPSD then can be witten as follows S YY A A N /2 * * A A A Al ( ) (8) * * lj ( j la) j ( j ) j ( j l A) l A j ( j la) whee the esidue matix at the lth hamonic fo the th mode is defined as follows. H C W( ) Cl A (9) Hence one can extact the Fouie coefficients of the time vaying mode shapes by cuve fitting the measued HPSD to a standad modal model with both stable and unstable poles at each peak. The time vaying mode shapes can then be econstucted with the identified Fouie coefficient vectos using Eq. (7) and since the lase scan path is known this can be used to detemine the stuctue s mode shapes along the scan path. This appoach was used in [7] and found to wok quite well although thee ae potentially many peaks to be fit in the powe spectum even if the stuctue has only a few modes. B. Positive Hamonic Linea Coelation and Positive Hamonic Powe Spectum The HPSD in Eq. (8) includes each of the system s poles twice one set having stable poles and the othe having unstable poles so one must use a cuve fitting outine that is specialized to OMA measuements (most common cuve fitting outines ae deived fo fequency esponse function measuements whee the unstable poles ae not needed). The equivalent issue fo the case of LTI measuements has been addessed in conventional opeational modal analysis using the positive powe spectum [9]. Allen et al. extended this concept to LTP systems defining the positive Hamonic Powe Spectum []. Fist each block of the exponentially modulated time signal is zeo-padded to twice its length and the HPSD is computed. The linea hamonic coelation function is the one side invese DFT of the HPSD in Eq (8) as shown below N /2 l * ( j la) nts * ( j la) nts Al Rn [ ] e e A * * ( j la)( nns) Ts ( j la) ( nns) Ts e e n Ns A A whee T s is the lase sampling peiod and N s is numbe of samples in the HPSD. When n is small the fist two tems in Eq. () dominate the esponse so the hamonic coelation function takes on the fom of an impulse esponse function that has only stable poles. Howeve as n inceases the fist two tems damp out and the last two tems with positive eal pats become dominant. Often the tems fo the last N/2 samples ae shifted to the left and used as an estimate of the impulse esponse function fo negative time lags. Howeve Caubeghe suggested instead that a ectangula window be used to delete the pat of the HCF that coesponds to negative time lags. This leaves only the phcf which is analogous to a () 5

6 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- one-sided impulse esponse as might be found in a hamme modal test. Then the phpsd H YY ( ) is found by computing the FFT of the phcf and the esult is a spectum that contains only the positively damped poles. H YY ( ) A N /2 * * lj ( j la) j ( j la) A The phpsd has the same mathematical fom as a fequency esponse function and can be cuve fit using standad methods. This phpsd appoach was applied to data fom an LTP system in []. C. Lifting the Positive Hamonic Coelation This wok exploes a diffeent analysis pocedue which damatically simplifies the post pocessing of the CSLDV measuements. The phcf is simila to the impulse esponse of the system. If thee ae exactly N A samples pe scan peiod then one can define N A points on the stuctue at mth point samples have been acquied at time instants mt s +kt A. The esponse of the mth point at its kth time instant can then be obtained fom the fist two tems in Eq. () as N /2 * * ( jla) mts kta * ( jl A) mts kta m[ ] l l l R k e e e e () A A (2) Lifted esponses exist fo m...( N A ) and each lifted esponse contains one sample at each instant kt A ove the span of the measuement. This esampling causes the eigenvalues jla to collapse to a single fequency and hence the lifted esponse contains only one exponential tem pe mode of the undelying LTI system. The numbe of points N A obtained pe scan cycle (and hence the numbe of lifted esponses) is elated to the sample incement of the lase T s and the scan peiod T A by T A =N A T s. The FFT of the lifted linea coelation function at the mth point then becomes N * Resm Resm m( ) * i i (3) ( jl A) mts e Res A m l This expession has the same mathematical fom as a fequency esponse function with a single esonance at Imag( ) fom each mode. Note that some modes may be aliased so they would appea to occu at <<ω A /2. When this occus the tue (unaliased) can be found by adding the coect intege multiple of ω A as discussed in [5 6]. Hence the multitude of peaks fo each intege l in eq. () have collapsed onto a single fequency ω between [ A /2]. We can then use a least squae appoach to ecove the mode vecto fom the identified esidue at each esonance. Suppose n = -p p is used to modulate the EMP signal and that significant hamonics ae pesent fo the th mode fo l = -q q in HPSD (one must select p>q to obtain meaningful esults). Then the th identified esidue at the mth measuement point Res m is a column vecto of (2p+) elements that sums the contibutions of each sideband q ( jl A) mts Al e lq path theefoe the esidue fo the th mode has dimensions (2p+)N A as follows.. Thee ae N A measuement point along the scan 6

7 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- q q q ( jla) Ts ( jla)( N A) Ts Res A l A le A le (2p) NA lq lq lq (2p) NA ( jqa) Ts ( jqa) N ATs e e C C q q q q (2p) (2q) ( jqa) Ts ( jqa) NATs e e (2q) N A Χ E H Whee q is a constant scala that epesents W ( ) C q /( ). A least squaes poblem can then be fomed to obtain the Χ C q q C q q matix as H H Χ =ResE (EE ) and then singula value decomposition can be used to extact C afte shifting each column in Χ accoding to the position of each sideband with espect to the unaliased natual fequency as was elaboated in [7]. Then the mode shapes can be econstucted fom the Fouie coefficients in C using eq. (7). D. Signal Pocessing Pocedue Resampling Filteing C y(t) Least Squae Poposed Lifting Appoach Exponential modulating Res AMI m ( ) Windowing Y ( ) S ( YY ) n Aveaging Figue outlines the signal pocessing pocedue fo the poposed CSLDV method with lifting and shows how the poposed appoach is elated to the authos pevious output-only methods fo CSLDV and linea time peiodic systems. The steps involved in the poposed algoithm ae explained in moe detail below. Recod the esponse y(t) of an LTI system to a white noise andom input using CSLDV with a peiodic scan path. 2. Resample the esponse y(t) accoding to the scan fequency ω A such that thee ae pecisely N A samples pe scanning peiod A method fo doing this is discussed in [5]. 3. Build the EMP signals with n=-p p in time domain by defining the nth modulated time signal as -jn At yn() t y() t e. Beak these EMP signals into sub blocks apply a Hanning window with ovelap if desied zeo-pad each block to twice its length and compute the discete Fouie tansfom. Compute the pimay column of the HPSD matix using the usual technique (e.g. H S YY ( ) n E Y( ) Y ( ) whee the expectation opeato denotes the aveage ove all available estimates of the Y ( ). Windowing Lifting IDFT HCF R[k] phcf Figue. Output only algoithms fo CSLDV FFT HPSD S. Yang et al. MSSP 2 phpsd M. S. Allen et al. IMAC XXIX 2 () 7

8 Downloaded by Matthew Allen on Decembe 22 DOI:.25/ Take invese FFT of the HPSD to obtain the linea HCF. Use a ectangula window to delete the negative pat of the HCF. 6. Lift the phcf by gouping the esponses at the same location along the lase path accoding to Eq. (2) and then take the FFT to obtain a spectum that is descibed by the modal model in Eq. (3). 7. Identify the eigenvalues of the stuctue and the coesponding mode shapes using a modal paamete identification outine such as AMI [2] o a simple appoach such as peak picking could be used if the system is vey lightly damped. 8. The natual fequencies of the stuctue can be obtained fom the identified eigenvalues using a vaiant on the unaliasing algoithm descibed in [5]. 9. The identified esidue is used to fom the least squae poblem in Eq. (). Afte shifting to align the Fouie coefficients the singula value decomposition method descibed in [7] is used to find the best estimate of the Fouie coefficient vecto C.. The th mode shape can be econstucted fom the Fouie coefficient vecto using Eq. (7). The time vaying shape Ct () is plotted vesus the lase scan path to obtain the mode shapes of the undelying LTI system. Notice that all of these steps except fo the identification in step 7 ae eadily automated so the use only need be concened with intepeting the lifted spectum and extacting any modes that ae pesent. These steps ae fa simple using the lifted spectum than they ae using the full HPSD as will be shown in the examples that follow. III. Simulation Results The poposed algoithm was fist evaluated using simulated measuements fom a fee-fee beam. This povided flexibility in vaying the paametes used to test the beam and the accuacy of the method could be assessed since the exact solution is available. Table lists the physical paametes of the beam that simulates a eal beam tested in [6]. The simulated esponse is also pocessed with the HPSD and phpsd algoithm and the esults ae evaluated and the advantages and disadvantages of each method ae discussed. Table : Paametes fo the simulated CSLDV test Beam geomety L 97.6 mm H 25. mm W 3.2 mm Density 27 kg/m 3 Elastic modulus 66 GPa Lase scan fequency 28 Hz Lase sampling fequency 2 Hz Simulated duation 88 s The fist 5 bending modes with.5% modal damping ae used to constuct the mass damping and stiffness matices by means of the Ritz method [3]. The esponse of the beam unde andom excitation is obtained using the lsim function in Matlab with the simulated model. The mode shape is vaied peiodically to simulate a case whee a lase scans a line patten fo 88s at the fequency of 28Hz with the sampling fequency of 2Hz; these paametes ae moe than feasible with the lase vibomete used in [5]. These acquisition settings esult in 8 pseudo measuement points along the scan path. The simulated time histoy is then used to built the EMP output signal accoding to step 3 with n=-. Each of the 2 exponentially modulated time histoies ae then divided into 6.s sub-blocks with 5% ovelap and a Hanning window is applied. Each block is then zeo-padded to twice its oiginal length and the auto and coss specta between the modulated signals and the oiginal time histoy (unmodulated histoy) ae computed and aveaged ove 5 blocks to obtain the pimay column of the HPSD SYY ( ) n. The esulting HPSD is shown in Figue 2. 8

9 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- Mag PSD Output Spectum Syy fo LTP System S k Fequency (Hz) Figue 2. Pimay column of the HPSD SYY ( ) n fo the simulated fee-fee beam. The index n anges fom - to as shown in the legend. Thee ae seveal hamonics in the specta up to 3Hz due to each of the beam s 5 modes and all of thei sidebands accoding Eq. (6). The sidebands of each mode also spead ove a wide ange of fequency since the sidebands ae sepaated by the 28Hz scan fequency. In addition folding happens when any of the sidebands eaches the negative plane futhe complicating the modal identification. In [7] the authos descibed a semi-automatic pocedue that can be used to locate the sidebands of each mode. An estimate of the mode vecto can then be obtained fom each peak using peak-picking This pocedue was used and the physical mode shape was econstucted although the pocedue did equie quite a bit of use input to discad peaks whee the esponse was noisy and the mode shapes wee found to vay depending on which peaks wee included in the peak picking. The identified natual fequencies and damping atios ae listed in Table 2 along with the modal assuance citeion (MAC) between the identified shapes and the tue mode shapes. The damping atios wee obtained by cuve fitting a few of the dominant peaks in the HPSD. Next the phpsd and new lifting method wee used. This was accomplished by applying a two sided invese DFT to the spectum SYY ( ) n to obtain the coesponding linea hamonic coelation function. A ectangula window was then used to zeo out the negative HCF. The phcf has the same length as the oiginal 6.s measuement blocks since the hamonic powe spectum was computed with linea coelation. Hence the phcf has 8 pseudo measuement points in each scan cycle as well. The esponses at the same measuement point ae then gouped to fom a single-input multi-output system that has 8 pseudo sensos fo each of the 2 phcf fo a total of 68 outputs. Figue 3 shows the composite esponse of the lifted phcf which is the aveage ove all 68 measuements. As mentioned peviously the lifting method aliases all of the sidebands of each mode to fequencies between and ω A /2. The lifted esponse contains only one peak fo each mode and when the measuements fom the vaious pseudo-points ae aveaged the esulting spectum is vey clean. The AMI modal identification outine [2] was able to pocess this set of measuements semi-automatically to identify the natual fequencies and damping atios. The cuve fit to the measuements is shown in Figue 3 with a dotted line. The ed line shows the eo between the fitted esponse and oiginal data whee we can obseve that the fitting is actually vey accuate. It is impotant to note that AMI teats the entie 68-output set of measuements; plots of the aveage specta such as that shown hee ae used only fo visualization puposes. Hence the esidue vecto etuned by AMI had 68 elements. These wee pocessed accoding to step 8 and an optimal estimate of the Fouie coefficient vecto was then extacted and used to econstuct the physical mode shape by plotting the time vaying shapes vesus the lase scan path. Figue shows the mode shape extacted fo the fist mode at the 8 pseudo measuement points (ed cicles) and compaes it with the analytical fist bending mode. The MAC values between all of the identified mode vectos and the analytical shapes ae given in Table 2. 9

10 Composite of Residual Afte Mode Isolation & Refinement Data Fit Eo 3 2 Downloaded by Matthew Allen on Decembe 22 DOI:.25/ Fequency (Hz) Figue 3: Lifted phcf and AMI cuve fitting Elastic Mode Shapes at 7. Hz Lift-pHCF Analytical Position (m) Figue : Fist bending mode obtained via lift phcf In ode to make the compaison complete the phpsd method descibed in [] which uses the phpsd in place of the HPSD was also employed. Recall that the phpsd is the FFT of the un-lifted phcf []. The phpsd ae simila in appeaance to the HPSD in Figue 2 and the identification pocedue used hee was identical to that descibed ealie fo the HPSD. The pimay diffeence between the phpsd and HPSD is that the latte is a squaed spectum and hence does not captue the evolution of each mode s phase nea esonance. Hence the HPSD equies a specialized modal paamete modal identification outine while vitually any method can be applied to the phpsd. The modal paametes extacted fom phpsd ae listed in Table 2. Table 2 compaes the identified natual fequencies and damping atios fom the thee algoithms with the exact solution. The mode shapes obtained with each method ae compaed using the MAC between the identified shapes and the tue analytical shapes. We can see that the natual fequencies fom the thee methods ae almost identical to the exact values and the MAC values ae geate than.99 fo all of the modes. The identified damping fo the fist mode shows significant eo most likely due to distotion caused by the Hanning window []. The HPSD phpsd and the lifting appoach all have vey simila accuacy fo this example. Howeve the lifting appoach povides a much simple use inteface that geatly educes the effot equied in modal identification. Moeove the lifting algoithm might also be advantageous if the stuctue of inteest contained modes with close natual fequencies whee MIMO measuements ae needed to detemine the numbe of modes pesent and to identify thei paametes.

11 Mode Fequency (Hz) Table 2. Compaison of identified mode fom diffeent method Analytical HPSD phcf-lifitng phpsd Damping (%) Fequency Damping (Hz) (%) MAC Fequency Damping (Hz) (%) MAC Fequency Damping MAC (Hz) (%) Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- IV. Application to Wind Tubine Blade using Remote Sensing Vibomete The poposed method was then used to identify the modes of a wind tubine blade mounted on the towe as depicted in Figue 5. The wind tubine is the same as in [7] except with a diffeent set of blades installed. Duing the tests the tubine oto was locked to pevent otation and the blade of inteest was pitched so that the lase was nominally pependicula to the chod of the blade (i.e. measuing in the flapwise diection). A pototype of Polytec s new Remote Sensing Vibomete was used in this wok which incopoates a lage wavelength lase (55 nm) and highe lase powe ( mw) than pevious LDVs. This lase is designed fo long standoff distances and hence was able to acquie easonable measuements without any suface teatment. The standoff distance fom the vibomete to the tubine blade was 77m. (In the authos pio wok [7] a Polytec PSV- (633nm lase) vibomete was used and it was noted that easonable measuements could not be obtained at that distance unless eto-eflective tape was applied along the length of the scan aea.) A customized x-y mio system was used to ediect the lase of the RSV to scan ove as much of the.5m long blade as was possible. The blade was excited puely by the wind whose maximum speed was about 9m/s duing the tests. Figue 5. Schematic of expeimental setup. The photogaph on the left shows a geneic photo of an RSV vibomete by Polytec. The vibomete used was a pototype with nominally identical specifications.

12 Figue 6. Photogaphs showing the position of the CSLDV measuement point at the extemes of its tavel. The CSLDV scan path was a line connecting these two points. Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- The lase was scanned a line ove the blade fo seconds with a scan fequency of 36Hz and a sampling fequency of 256Hz. The lase was not visible and thee was no guide lase in the pototype so the scan path was defined by detemining what voltages to apply to the mio system to position the lase at the tip and oot of the beam as shown in Figue 6. These voltages wee then used to define a scan path that coesponded to a line between these two points. This pocedue esults in fa less uncetainty in the position of the measuement points than that used in the authos pevious wok [7]. Figue 7 shows the time signal ove a few scan cycles (the whole time signal would have the appeaance of andom noise due to the andom natue of the input). The signal is dominated by a 36Hz fequency; howeve thee ae clealy seveal fequencies pesent in the esponse pesumably due to the vibation modes of the tubine. Velocity (mm/s) RSV-CSLDV output signal Time (s) Figue 7: RSV-CSLDV output signal unde andom excitation (36Hz scan fequency) The whole measued time histoy was esampled at 2592Hz to geneate 72 samples pe scan cycle. The esampled signal was low pass filteed and the fequency component coesponding to the 36Hz scan fequency was deleted since it is dominated by speckle noise. The esampled signal was then exponentially modulated with hamonics n= -3 3 esulting in 7 modulated time histoies. Each of these modulated signals was decomposed into 25.6s sub-blocks with 5% ovelap and a Hanning window was applied to each of the 3 blocks. A HPSD of dimensions [ ] whee is the numbe of fequency lines was then obtained as descibed peviously. The phcf was then obtained using the invese FFT and ectangula window. Figue 8 pesents positive HCF geneated fom S ( ) YY. The phcf has the appeaance of a standad impulse esponse function with a dominant low fequency mode which pesists fo moe than 2s and seveal highe fequency components that disappea afte about s. 2

13 3 x -5 positive Hamonic Coelation Function Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- The phcf was then lifted and a SIMO system was fomed. Figue 9 shows the aveage of the lifted esponses (solid gey line) and the aveage of the AMI cuve fit (dots). Only two peaks ae pominent in the plot but a close inspection eveals a few othe peaks. Specifically the esidual (ed line) shows two peaks at 2.3 and 3.3 Hz. These peaks which ae only baely visible in the aveage spectum ae actually faily pominent in the subtaction esidual and a mode was identified nea each of these peaks. Similaly thee appea to be othe close modes nea 3.3 Hz and this is to be expected as thee fist blade bending modes typically appea at thee close fequencies fo tubines such as this. A mode was fit to the stongest peak at 3.33 Hz and then natual fequencies and damping atios of each identified mode ae epoted in Table 3. Othe modes could be fit nea 3.3 Hz as well but it was difficult to be sue that they wee meaningful so they ae not epoted. In any event measuements would be needed on each blade to obtain meaningful estimates of the vaious fist blade bending modes as they tend to diffe pimaily in the elative amplitudes of the thee blades Time (s) Figue 8: phcf R[ n ] geneated fom S ( ) YY Composite of Residual Afte Mode Isolation & Refinement Data Fit Eo Fequency (Hz) Figue 9: Aveage of the spectum ( ) of the lifted esponse AMI fit and the diffeence between the two. The mode shapes wee econstucted fom the Fouie coefficients that wee identified using eq. () and they ae shown in Figue. The eal and imaginay pats of the mode shapes ae shown as AMI fits m 3

14 a complex mode model to the measuements. Howeve a lightly damped stuctue such as this is expected to have eal modes so the imaginay pats most likely aise due to inaccuacy in the measuements. They ae vey small fo the fist two modes and easonably small fo modes 3 and consideing that the esponse of those modes was two odes of magnitude lowe than the dominant mode. The blade appeas to move as a igid body in the fist mode at.8hz suggesting that this mode pimaily involves bending of the towe. The mode at 3.33Hz appeas to be the fist flap-wise bending mode of the blade. The.8 and 3.33 Hz modes both have vey small imaginay pats and the shape estimated as the lase spot moved fom oot to tip agees vey well with the shape estimated as lase spot etuned fom tip to oot suggesting that the shape is quite accuate. The thid and fouth identified fequencies ae second bending modes of the blade. These modes have elatively lage imaginay pats but the oot-tip and tip-oot shapes ae quite consistent again suggesting that they have been accuately identified. Table 3: Modes identified fom the lifted esponse Downloaded by Matthew Allen on Decembe 22 DOI:.25/ Mode Natual fequency Damping Towe Bending.8Hz.6% Flap Wise Bending 3.33Hz.52% Flap Wise Bending 2.8 Hz Hz 3 2 Position (m) 2.2Hz 3.Hz %.7% 3.33 Hz Hz 3 2 Position (m) Figue : Real (dots) and imaginay pats (dashed-line) of the mode shapes identified fom the CSLDV measuements..5 Fo compaison puposes a standad OMA test was also pefomed and used to estimate the blade s mode shapes. Fo this test a patch of eto-eflective tape was applied to the tip of the blade and a PSV- LDV was diected towads this point and used as a efeence. The RSV lase was then positioned sequentially at five diffeent points along the length of the blade. The positions of the points wee detemined by measuing the angle of the RSV lase head and using the known length of the blade. The auto and coss specta between the two lases was then used to detemine the natual fequencies and modes shapes of the tubine fom these ten specta (five autospecta fo the efeence and five coss specta between the RSV and PSV-). The esulting mode shapes ae shown in Figue. Thee diffeent mode shapes wee extacted nea 3.3 Hz but as was discussed peviously thee is little evidence that the additional modes ae meaningful. Similaly a thid second blade bending mode was identified at 3. Hz. It is infomative to compae these esults with the CSLDV esults. Fist one should note that the standad OMA appoach equied a second lase adding tens of thousands of dollas to the cost of the equipment needed. Second the standad OMA test equied acquisition of five time histoies which

15 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- would nominally incease the measuement time by a facto of five. Howeve since time was limited a smalle numbe of aveages wee used fo the standad OMA test and each one of these ecods was acquied in 5.3 minutes esulting in a total test time of 26.5min fo standad OMA and 6.7 min fo CSLDV. Thid the measuements obtained by standad OMA agee faily well qualitatively with those obtained by CSLDV but thee ae seveal points which appea to be questionable. This could possibly be explained by the fact that the wind conditions may have changed fom one point to the next o the eflectivity of the blade suface may have been infeio at some points leading to inceased noise. In any event thee is little that can be done to assess the eliability of each measuement point without epeating the test. Fouth the mode shapes obtained by classical OMA seem to be fa less detailed than those obtained by CSLDV On the othe hand because speckle noise was educed in the standad OMA test two additional peaks wee visible in the spectum at.6 and 5.3 Hz. The second towe bending mode is thought to eside nea these fequencies so these shapes ae thought to eflect the motion of the blades in the second towe bending mode(s). Since these fequencies ae close to the fist blade bending mode it is not supising that the blades have essentially the same defomation shape as they do in the fist bending modes. Although this towe mode was weakly excited its pesence could be detected in the standad OMA measuements while it was buied by speckle noise and the aliased contibutions of the dominant modes in the CSLDV measuement Elastic Mode Shapes Position (m) Figue : Mode shapes at five points obtained using a standad OMA technique with a second lase seving as a efeence Hz 3.9 Hz 3.3 Hz 3.69 Hz.63 Hz 5.3 Hz 2.38 Hz 3.3 Hz 3.38 Hz V. Conclusion This pape combined output only continuous-scan lase Dopple vibomety with a lifting appoach simplifying the post pocessing equied to extact the mode shapes fom CSLDV measuements. As with conventional OMA the method assumes that the foces exciting the system ae andom white and that they sufficiently excite all of the modes of inteest. The measued CSLDV signal is exponentially modulated to estimate the hamonic powe spectum and the theoetical development eveals that each mode then appeas at seveal peaks in the HPSD. An invese FFT is then pefomed on the HPSD to obtain the hamonic coelation function. The positive HCFs ae then lifted and aanged to fom a single input multiple output system. A conventional model identification outine such as AMI can then be used to extact modal paametes fom the lifted esponses. The identified esidues have a moe complicated definition than they did in [5] but a least squaes poblem is eadily fomulated to extact the Fouie coefficients of the mode shapes fom the identified esidue vectos. The mode shapes can then be 5

16 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- econstucted by plotting the time vaying shapes vesus the lase path. The poposed method was fist applied to simulated measuements fom a fee-fee beam. The modes extacted fom the measuements using the HPSD (method in [7]) phpsd (method in []) and the new lifting appoach showed that simila esults could be obtained with any of the appoaches. Howeve the lifting appoach povides a much simple use inteface that geatly educed the effot equied to extact the modal paametes. The methodology pesented hee was futhe exploed by applying it to measue the modes of a paked wind tubine with a new long ange vibomete called the emote sensing vibomete (RSV). In this application the RSV lase was capable of extacting accuate measuements at a lage standoff distance (77 metes) without any suface teatment. The authos pevious wok used a standad vibomete and easonable measuements wee not possible unless eto-eflective tape was fist applied to the suface of the blade. Futhemoe the speckle noise in the measuements with the RSV was elatively small so high scan fequencies wee possible inceasing the attactiveness of the lifting appoach. This is quite emakable since at a 36 Hz scan fequency the peak suface velocity of the lase spot was about 5 m/s. Hence it appeas that this methodology would be feasible fo lage wind tubines whee the natual fequencies ae lowe and hence the time equied to obtain opeational modal analysis data can often be excessive. Fo the esults pesented in this wok a single 6.7-min. CSLDV time histoy was used to extact seveal modes of vibation including mode shapes with good epeatability. Duing the post-pocessing it was noted that it would have been pefeable to have a longe time histoy since the numbe of aveages (3) was somewhat maginal. Howeve the measuements wee still adequate to obtain qualitatively easonable esults fo the fist seveal modes of the tubine. The CSLDV esults wee compaed with those fom a standad OMA test using a second lase as a efeence evealing the elative meits of the two appoaches. When two lases ae available the authos ecommend a blended appoach whee CSLDV is used to captue spatially detailed shapes ove citical sufaces and point measuements ae used to veify the CSLDV esults captue additional discete points of inteest and to identify weakly excited modes. Acknowledgements This mateial is based on wok suppoted by the National Science Foundation unde Gant No. CMMI The authos wish to thank Renewegy LLC fo making thei facilities available fo the testing descibed hee. The authos also wish to thank Vikant Palan and Aend von de Lieth fom PolyTec fo thei assistance with the Remote Sensing Vibomete. Refeences [] P. Siam J. I. Caig and S. Hanagud "Scanning lase Dopple vibomete fo modal testing" Intenational Jounal of Analytical and Expeimental Modal Analysis vol. 5 pp [2] A. B. Stanbidge and D. J. Ewins "Modal testing using a scanning lase Dopple vibomete" Mechanical Systems and Signal Pocessing vol. 3 pp [3] R. Ribichini D. Di Maio A. B. Stanbidge and D. J. Ewins "Impact Testing With a Continuously-Scanning LDV" in 26th Intenational Modal Analysis Confeence (IMAC XXVI) Olando Floida 28. [] D. D. Maio G. Caloni and D. J. Ewins "Simulation and validation of ODS measuements made using a Continuous SLDV method on a beam excited by a pseudo andom signal" in 28th Intenational Modal Analysis Confeence (IMAC XXVIII) Jacksonville Floida 2. [5] M. S. Allen and M. W. Sacic "A New Method fo Pocessing Impact Excited Continuous-Scan Lase Dopple Vibomete Measuements" Mechanical Systems and Signal Pocessing vol. 2 pp [6] S. Yang M. W. Sacic and M. S. Allen "Two algoithms fo mass nomalizing mode shapes fom impact excited continuous-scan lase Dopple vibomety" Jounal of Vibation and Acoustics vol. 3 Apil 22. [7] S. Yang and M. S. Allen "Output-Only Modal Analysis Using Continuous-Scan Lase Dopple Vibomety and Application to a 2kW Wind Tubine" Mechanical Systems and Signal Pocessing vol. Submitted Apil

17 Downloaded by Matthew Allen on Decembe 22 DOI:.25/6.22- [8] N. M. Weeley "Analysis and Contol of Linea Peiodically Time Vaying Systems" PhD Thesis Depatment of Aeonautics and Astonautics Cambidge: Massachusetts Institute of Technology 99. [9] B. Caubeghe "Applied Fequency-Domain System Identification in the Field of Expeimental and Opeational Modal Analysis" PhD Thesis Faculteit Tegepaste Wetenschappen Vakgoep Wektuigkunde Bussels Belgium: Vije Univesiteit Bussel 2. [] M. S. Allen S. Chauhan and M. H. Hansen "Advanced Opeational Modal Analysis Methods fo Linea Time Peiodic System Identification" in 29th Intenational Modal Analysis Confeence (IMAC XXIX) Jacksonville Floida 2. [] N. M. Weeley and S. R. Hall "Fequency esponse of linea time peiodic systems" Honolulu HI USA 99 pp [2] M. S. Allen and J. H. Ginsbeg "A Global Single-Input-Multi-Output (SIMO) Implementation of The Algoithm of Mode Isolation and Applications to Analytical and Expeimental Data" Mechanical Systems and Signal Pocessing vol. 2 pp [3] J. H. Ginsbeg Mechanical and Stuctual Vibations Fist ed. New Yok: John Wiley and Sons 2. [] Wang Jintian Qu Shuying Hou Xingmin and Z. Jia "Influences of Window Function on the Subsoil Damping Ratio" in 2 Intenational Confeence on ICEEE pp. - 7