Experimental Investigation of the Effect of Speckle Noise on Continuous Scan Laser Doppler Vibrometer Measurements

Transcription

1 Expeimental Investigation of the Effect of Speckle Noise on Continuous Scan Lase Dopple Vibomete Measuements Michael W. Sacic & Matthew S. Allen Univesity of Wisconsin-Madison 535 Engineeing Reseach Building 1500 Engineeing Dive Madison, WI Abstact Continuous Scan Lase Dopple Vibomety (CSLDV) sweeps a lase spot continuously ove a stuctue to measue its mode shapes at hundeds of points simultaneously. It can be used to measue accuate, detailed mode shapes using a hand-held impact hamme, while conventional point-by-point LDV can be impactical and inaccuate in that application because of stike-to-stike vaiation in the impact location and oientation. Recently pesented techniques can be used to tansfom the CSLDV measuements into a set of esponses that can be pocessed using standad system identification techniques to extact the modes. The esampling method woks best when the scan fequency is high elative to the highest natual fequency of inteest, and state of the at scanning vibometes ae capable of elatively high scan fequencies, but thee is a tadeoff between measuement quality and scan fequency due to lase-speckle noise. This wok exploes the effect of LDV measuement noise, both the peiodic and non-peiodic components, on CSLDV measuements. Of paticula inteests ae: the dependence of the signal-to-noise atio on the scan ate and geomety, the taget-to-detecto distance of the expeimental setup, the Dopple signal quality, and the sampling ate used to acquie the measuement. The esults pesented sometimes seem contay to one s intuition and to the conclusions pesented in othe woks, but they do povide a faily thoough guide fo the expeimentalist, enabling the design of CSLDV expeiments with the minimum noise level possible. Nomenclatue t : Tempoal Vaiable N o : # of Samples pe Peiod x : Spatial Vaiable N ape : # of Non-Peiodic Signal Values x m : Maximum Spatial Value N ape,lifted : # of lifted Non-Peiodic Values λ : Wavelength Vaiable T A : Signal Peiod φ i : Phase Vaiable v : Voltage Signal f samp : Sampling Fequency y : Response Signal f scan : Scanning Fequency y n : n th Pseudo Response Signal h : Suface Impefection Height y pe : Peiodic Data of Response Signal d : Taget-to-Detecto Distance Y ape (ω) : Non-Peiodic Fequency Signal θ : Lase Sweep Angle Y ape,lifted (ω) : Lifted Non-Peiodic Feq. Signal η ave : Non-Peiodic Noise Aveage η ave,lifted : Lifted Non-Peiodic Noise Aveage ρ ave : Peiodic Noise Aveage 1. Intoduction Lase Dopple Vibomety (LDV) has become inceasingly common fo vibation measuements because it povides a non-contact measuement with excellent bandwidth and allows one to measue the esponse of a suface at thousands of points using automated softwae. The conventional appoach involves illuminating a

2 single point on an object, acquiing a measuement, and then using automated scanning mios to eposition the lase spot and epeat the pocess at numeous othe points on the stuctue. This is clealy moe convenient than the time consuming setup involved with othe measuement devices such as foce tansduces and acceleometes, and is geneally moe accuate than oving hamme tests. Howeve, if each measuement ecod is long, as is the case fo a stuctue with low natual fequencies, the LDV still may equie excessive time to collect a set of measuements. Futhemoe, the measuements acquied using taditional LDV may be inconsistent if the stuctue has the potential to change with time, tempeatue o othe factos. Stanbidge, Ewins and Mataelli [1] advanced a technique, which was oiginally pesented by Siam et al. [2-4], that ovecomes some of these limitations by scanning the lase spot continuously ove the suface of a test device while acquiing a measuement, allowing one to simultaneously acquie tempoal infomation as well as spatial infomation at hundeds o thousands of points simultaneously. The shape of the scan patten is dictated by computecontolled mios, so vitually any shape can be used. A few diffeent techniques have been pesented by which one can detemine the mode shapes of a linea system fom CSLDV measuements. Oiginal effots in continuous scan lase Dopple vibomety (CSLDV) by Siam et al. used boadband excitation and sampled synchonous with the lase scan speed to geneate taditional Fequency Response Functions (FRFs) [2], although they seem to have abandoned that appoach due to high noise levels. Stanbidge et al. pesented a method wheeby the stuctue is excited nea esonance fo a cetain mode, and a Fouie analysis leads to a polynomial seies desciption of the mode shape. They have also successfully identified multiple modes of a stuctue using impact excitation and CSLDV [5, 6], using an appoach that is usually successful so long as the stuctue has lightly damped modes with elatively high natual fequencies. The authos ecently pesented a technique [7, 8] that allows one to extact the modes of a stuctue using conventional modal analysis outines, by decomposing the esponse into a set of esponses that would have been measued at each point along the scan path if the lase spot wee held stationay. The decomposition has been efeed to as a lifting technique in the context of geneal time-peiodic systems [9-12]. This technique also allows one to apply standad tools such as Mode Indicato Functions, to identify modes with close natual fequencies. Although outstanding coelation has been obseved between the identified and analytical fequencies and mode shapes in a few peliminay applications, elatively little is known egading the influence of noise and the potential thesholds fo scanning ates and othe expeimental test paametes. The occuence of noise affects all applications of lase-type tansduces. When coheent (lase) light is eflected off of an optically ough suface, ganule-like impefections on the device s suface with size on the ode of hundeds of nanometes act to dephase the light etuning fom the suface, causing bight and dak egions whee the light constuctively and destuctively intefees. Relative motion between the lase spot and the suface causes the speckle patten to evolve, poducing a spuious vibation-like signal that cannot be easily distinguished fom the tue vibation. Lase speckle has been investigated fo decades in vaious contexts, although its influence on lase vibomety has only been appeciated moe ecently. Pevious woks have noted the dependence of the speckle patten on seveal key factos. The size of the incident light wave font geatly influences the appeaance of a speckle population. Smalle diamete lase spots eflect moe developed speckle pattens; individual speckles eflected fom a smalle sized lase spots ae lage in appeaance and thee ae moe egulaized light and dak intensity pattens [13, 14]. The taget suface and any motion it undegoes also affects lase speckle. A polished metallic suface poduces a speckle patten with an intensity envelope due to the polishing diection but the fine inconsistencies of the polished suface pevents stuctued speckle pattens within that envelope. Reto-eflective tape, which is a suface teatment that consists of mico-scale glass beads, causes eflected light to scatte in a concentated cone suounding the incident light diection, and poduces well developed Aiy-function type intensity pattens [13, 14]. In-plane tanslation and out-of-plane otation of the test device have been found to significantly affect lase speckle pattens and the way they change. Rothbeg and Halkon [15] have shown that pue otation of the taget suface out of the taget plane pependicula to the optical axis of a lase setup will tend to cause a tanslation of the lase speckle patten with little o no speckle evolution. Altenatively, a pue tanslation of the taget suface within the taget plane will cause the eflected speckle patten to evolve o boil as it has been temed in the past [16]. Rothbeg [17] used Fouie optics to investigate the effects of a typical lase vibomete setup on the speckle pattens poduced in the scatteed light. He theoized that the detected speckle signals would contain time dependent intensity and phase infomation fo non-nomal suface vibations, and then obseved effects of cetain

3 taget motions on the back scatteed speckle pattens. Fo a lase vibomete incident on a otating shaft, Rothbeg noted that fequency domain amplitude peaks occued at the shaft s otation fequency and hamonics and suggested that the speckle noise is appoximately peiodic. This has been attibuted to the fact that the lase spot taces the same path on the suface, esulting in a peiodically-vaying speckle patten. Howeve, a pefectly peiodic speckle patten would equie that the lase tace the exact same path ove the ough suface, with consistency on the ode of the light wavelength (100 s o nanometes), duing each peiod. Clealy this level of consistency cannot be expected due to vibation o motion of the test device, but those factos notwithstanding, speckle noise is obseved to be pedominantly peiodic in most applications. Mataelli and Ewins [18] noted that speckle noise is also nealy peiodic in CSLDV measuements, and compaed the hamonic and non-hamonic components (sidebands) of the speckle noise between 0.2 and 20 Hz scanning fequencies. Thei esults show that noise level inceases with inceasing scan fequency, although the authos have successfully utilized scanning fequencies as high as 100 Hz, so thee is a desie to extend that wok to highe scan fequencies. Futhemoe, the analysis method used by the authos is somewhat immune to the peiodic component of the speckle noise, wheeas that was the pimay focus in [18]. Fo example, Figue 1 displays the composite FRFs, the data-fits achieved fom using the AMI algoithm [10] and the diffeence between data and the fits of the lifted o time-invaiant signals fo a CSLDV measuement of the fee esponse of an aluminum beam excited with an impact hamme. The measuement was acquied by scanning an LDV with a sinusoidal diving function at 100 Hz and the sampling fequency was 81.9 khz, poducing 820 pseudo-points pe 100 Hz cycle. The bandwidth of the lifted signal f max, is equal to half of the scanning fequency, so highe fequencies ae effectively aliased into this bandwidth, as illustated in [8]. Afte identifying the modes of the system, a noise floo emains (ed cuve) whose amplitude is compaable to that of some of the weakly excited modes. This wok seeks to discove ways of designing the CSLDV expeiment to minimize this noise level. Composite of Residual and Oiginal Data 100 Hz CSLDV, 81.9 khz Sampling Response Amplitude Data Fit Data-Fit Fequency (Hz) Figue 1: Composite FRFs of lifted CSLDV esponse with 100 Hz scanning fequency and cuve fit obtained by modal paamete identification algoithm. The ed cuve shows a composite of the diffeence between the two, which is an indication of the noise level. This pape pesents the esults of a numbe of expeiments aimed at chaacteizing both the peiodic and nonpeiodic lase speckle noise in CSLDV applications, and to detemine how it is affected by the geomety of the test setup, the sampling ate, and othe factos so that optimal paametes can be identified. The following section eviews some elevant theoy egading CSLDV and lase speckle. Section 3 pesents the esults of a seies of expeiments in which a numbe of paametes wee vaied to chaacteize how each affects the speckle noise. Conclusions ae pesented in Section 4.

4 2. Effects of Speckle Noise on CSLDV 2.1. CSLDV Theoy The signal measued fom a LDV incident on a feely vibating, linea, time-invaiant stuctue can be epesented by the summation of decaying exponentials made up of modal components as: 2N () = φ ( ) y t x C e = 1 λ = ζ ω + iω 1 ζ, λ = ζ ω iω 1 ζ N t λ (1) whee λ is the th eigenvalue, made up of the th natual fequency ω, and damping atio ζ. The coesponding mode vecto φ(x) depends on the position x of the lase beam along its path. This path could involve motion in thee dimensions in a geneal case. C is the complex amplitude of mode, which depends upon the stuctue s initial conditions, o on the impulse used to excite the stuctue if such an excitation is employed. When using CSLDV, the lase is assumed to tace out a known peiodic path of peiod T A, such that x(t) = x(t+t A ), o with the scan fequency ω A = 2π/T A, esulting in a change fom spatial to tempoal dependence of the mode vecto. 2N () = φ () y t t C e = 1 t ( x() t ) = () t = ( t+ TA ) φ φ φ This is identical to the expession fo the fee esponse of a Linea Time Peiodic (LTP) system [19]. Two methods have been poposed to detemine φ(t) and λ in eq. (2), the Fouie Seies expansion method and the lifting method Fouie Seies Expansion Method If the CSLDV scan patten is peiodic, then mode shapes φ(t) can be expanded into a Fouie Seies as follows, λ (2) φ N B () t = B exp( imω t) m= N B, m A (3) whee it is assumed that only the coefficients unning fom -N B to +N B ae significant. Substituting into eq. (2) and moving the summations to the outside esults in the following. y 2N N B () t = B (( + im )( t t ) = 1 m= N B λ ), m exp ω A 0 (4) This is mathematically equivalent to the impulse esponse of an LTI system with 2N(2N B + 1) eigenvalues λ + imω A. (5) Note that the amplitude of each hamonic B,m in eq. (4) ae the Fouie coefficients fo the th mode and the m th Fouie Tem. It is impotant to chaacteize the fequency content of the lase speckle noise so that it can be discened fom the meaningful hamonics in the esponse, because both can appea as naow-band spikes in the spectum. Stanbidge, Mataelli and Ewins use a vey simila appoach, although they allow non-peiodic scan pattens and identify a powe seies model fo the opeating shape.

5 Lifting Method The lifting method eliminates the time dependence of φ(t) in eq. (2) by sampling the signal y(t) fom the LDV discetely such that the lase s position is x(t i ) = x i (coesponding to a specific point along the lase s path each peiod). A linea time-peiodic signal, sampled as such, can be eoganized accoding to: ( ) ( A) ( A) L ( ) ( A) ( A) L ( ) ( ) ( ) y = [ y 0, y T, y 2 T, ] y = [ y Δt, y Δ+ t T, y Δ+ t 2 T, ] y = [ y 2 Δt, y 2 Δ+ t T, y2 Δ+ t 2 T, L] L (( ) ) ( ) A ( ) (( ) ) y = [ y N 1 Δt, y N 1 Δ t+ T, y N 1 Δ t+ 2 T, L] No 1 o o A o A A (6) whee N o is the atio of the sampling fequency to the peiodic signal fequency, N o =f samp /f scan. Each y i in eq. (6) (called a pseudo-esponse point because it was extacted fom the measued signal y(t)) now can be epesented by a time-invaiant system, and can be tansfomed into the fequency domain using the Discete Fouie Tansfom (DFT) and pocessed with any standad identification outine. The esults of utilizing the lifting technique to pocess fee esponse CSLDV data is thooughly pesented in [7] and [8]. Because the lifting method eoganizes CSLDV data accoding to the peiod of the scanning fequency, the peiodic component of the speckle noise is deposited on the zeo fequency line in the lifted spectum. Any sidebands to the peiodic component ae also deposited aound this zeo line as well Sampling Fequency When testing with CSLDV, sampling faste povides moe spatially dense infomation, which is beneficial fo applications such as damage detection o poviding detailed mode shapes. In tems of lase speckle, inceased sampling has the effect of allowing the detecto to see moe speckle pattens pe peiod each with a diffeent and nominally andom phase distibution. Howeve, one also obtains a lage numbe of lifted esponses, o pseudoesponse points x i as the sample ate inceases, as is eadily appaent fom eq. (6). Hence, thee is a tadeoff between identifying a lage numbe of points, which may seve to aveage out eos, and collecting moe speckle noise as the sampling ate inceases. The scenaio is somewhat diffeent fo the Fouie Seies expansion method, because the bandwidth of the CSLDV signal is detemined by both the natual fequencies of the system and by the total numbe of Fouie tems equied to epesent each mode shape. The bandwidth needs to be lage enough such that all of the Fouie tems that stand out above the noise floo ae captued, and captuing additional bandwidth adds no significant infomation. Of couse, meaningful mode shapes ae not obtained unless a lage enough numbe of Fouie tems stand out above the noise, so it impotant to take steps to minimize the nosie. One focus of this wok will be to chaacteize the effect of the sample bandwidth on speckle noise Lase Speckle Theoy Nealy all engineeing sufaces can be descibed as optically ough, which means that the impefections in the suface ae on the ode of the wavelength of light. When coheent light is incident on such a suface, the vaiation of suface featues dispeses the incident ays causing them to eflect in nealy all diections. Neighboing bumps will de-phase adjacent ays, and complex intefeence pattens then aise giving lase speckle its name because of the appeaance of speckling of bight and dak featues within such a patten. Figue 2 shows a epesentation of an incident wave font being de-phased upon eflection by an optically ough suface.

6 Dispesed Rays I = f(x,t,λ,φ 1 ) I = f(x,t,λ,φ 2 ) I = f(x,t,λ,φ 3 ) Incoming Wave Font Taget Suface Figue 2: Wave fonts incident on an optically ough suface will be eflected and dispesed due to suface featues. Adjacent eflected ays will contain diffeent phase infomation and the intensity epesented by the summation of ays on a detecto will combine adjacent ays and phase infomation will cause light and dak intefeence pattens, o speckle Lase Spot Size and Shape Matin and Rothbeg [14] have shown the stong dependence of speckle size on lase spot diamete. In fact, they obseved that fo a vaiety of sufaces, a 600 μm lase spot diamete poduces smalle sized speckles than a 100 μm lase spot diamete because the lage illuminated aea eflects geate numbes of andom phase distibuted intensity egions leading to moe destuctive intefeence pattens Most commecial scanning vibometes have focusing featues that in the standad opeation mode efocus the lase spot each time it changes position. Refocusing is not pactical when opeating in continuous scanning mode, so the focus vaies depending on the position of the spot within the taget plane. The lase spot size is at its minimum diamete in the focal plane, becoming lage as the spot stays fom this plane (See Figue 3). Note also fom Figue 3 that the diffeent spatial vaiables of the scan can be elated by the following equation: x m tan θ = (7) d Taget x m Spot Size/Shape θ d Lase Souce Focusing Lens/Detecto Position Figue 3: The size and shape of the lase spot change as it is scanned in one dimension. One school of thought suggests that speckle noise should be at a minimum if the lase souce is focused to its apetue limited focal size on the taget because this poduces only one (o a vey small numbe) of lage speckles that ae about the same size as the detecto being used. (Theoetical focal points have zeo dimensions, but eal optical systems ae pactically limited by one o moe apetue sizes [20].) As motion is

7 intoduced, this single speckle will tansition out of the detecto, but one would hope that the tansition is mild because the speckle is lage. Accoding to this theoy, as the lase spot inceases in size one obtains a lage numbe of smalle speckles on the detecto. In any instant many of these may tansition off of the detecto causing speckle noise. Figues 4 and 5 illustate this effect fo an aluminum beam. The Polytec vibomete descibed subsequently was focused on the beam s suface, and the field of lase light eflected fom the suface to a taget nea the font of the LDV was ecoded with a digital camea. When the lase beam is focused caefully on the beam s suface, a patten with elatively lage speckles is obtained at the detecto, as shown in Figue 4. This pesumably places about one speckle on the detecto of the LDV. On the othe hand, the spot size inceases as lase beam is swept away fom the best focus point, poducing smalle sized speckles, such as those shown in Figue 5. The bight vetical band in the speckle patten is pobably due to extusion lines on the suface of the beam. It has been suggested that one may educe speckle noise fo some kinds of suface motions by defocusing the lase beam [15], which would imply that one might obseve eithe an incease o decease in noise as the lase spot moves along the suface of a stuctue. This will be evaluated in Section cm 20cm 20cm Figue 4: Best focus speckle patten with smallest lase spot size. 20cm Figue 5: Speckle patten with lage lase spot size due to being slightly out of focus. 3. Expeimental Investigation of CSLDV Speckle Noise In ode to quantify the speckle noise pesent in CSLDV signals, measuements wee acquied unde nealy zeo vibation conditions, but while scanning the lase spot sinusoidally at vaious fequencies. A Polytec PSV-400 scanning lase vibomete was employed, and the same aluminum beam used in [8] and [7] was used as the taget. A numbe of paametes of the expeimental setup wee vaied to assess the effect of each on the speckle noise, including: scan length, taget-to-detecto distance and sampling ate. The qualitative esults of the speckle noise on the CSLDV pocess, in paticula the effects of suface velocity, wee noted. Quantitative measues of the aveage levels of both the peiodic and non-peiodic components of speckle noise wee found fo vaious combinations of the expeimental paametes. The following tables povide the values of the expeimental paametes used in the study with the nomenclatue fom Figue 3. Scan Length Taget-to-Detecto Sepaation Sampling Fequency 2*x m, [m] d, [m] f samp, [khz] Table 1: Expeimental paametes used in noise study.

8 Scan Fequency, f scan, [Hz] Table 2: Expeimental scanning fequencies used in noise study. The scan fequency was vaied by 10 Hz fom Hz fo scan lengths 2*x m = 0.3, 0.6, and 0.98 metes at each of taget-to-detecto sepaations d=2, 3.4 and 5.5 metes. At the closest taget-to-detecto sepaation tested, d=2 metes, scan fequencies of 5 Hz, 10 Hz, and 15 Hz wee also investigated to ovelap with othe published studies. Since suface velocity has been shown to be influential to speckle noise, it was chosen to be the independent vaiable in most of the following plots Expeimental Suface Velocity Noise Results Figue 6 shows the fequency domain vibomete signal fom a sinusoidal continuous scan at a fequency of 10 Hz on a slende aluminum beam. Shap peaks can be seen at intege multiples of 10 Hz fo the entie bandwidth of the sampling space, the majoity of which is not plotted. Based on pevious wok, one would attibute these peaks to peiodic speckle noise. The noise away fom these fequency lines is about one ode of magnitude lowe. Also, the magnitude of the fist fundamental fequency and fist few hamonics is nealy twice that of the est of the hamonics CSLDV Fequency Domain Noise Data fo 10 Hz 10-3 Noise Floo Amplitude fequency, hz Figue 6: Vibomete esponse fo a 10 Hz sinusoidal CSLDV scan on a non-vibating Aluminum beam. Peaks occu at the scan fequency and its hamonics and fill the entie bandwidth of the measuement. Figue 7 shows the same esponse in the time domain, although with each peiod of the esponse plotted with a diffeent colo make. Blue makes coespond to ealy times within the ecod, ed ones with late times. A 10 Hz scan fequency was used, and the measuements wee sampled at 12.8 khz fo a total sample time of 5.12 seconds to poduce the plots in Figue 6 and 7, so 50 full peiods of the scan wee acquied, each peiod containing the fowad and backwad path with 1280 points pe cycle. The lowe-left potion of Figue 7 contains the coesponding mio dive signal fo the peiod of the scan, which is maximum when the lase spot is nea the edges of the beam. The density plot eveals a noise signal that appeas andom besides a peiodically vaying envelope function. The envelope is maximum when the dive signal is zeo, o when the mio velocity is lagest, and minimum when the mio velocity is smallest. Howeve, a close inspection of the density plot (ight pane, showing a small peiod of time nea whee the noise was maximum) eveals an undelying stuctue in the noise; thee is significant scatte at each point, but the plot shows that the noise signal epeats itself to a significant extent each peiod. This stuctue explains the appaent peiodicity of the noise, although the vaiance at each point along the beam is cetainly significant. One should also note that the aveage spatial density in this esampled esponse is 200 mico-metes, so it is still thee odes of magnitude lage than the lase wavelength.

9 Figue 7: (left-top) Density plot of 101 peiods of a 10 Hz sinusoidal noise scan ovelaid. (left-bottom) Voltage input to scanning mios pe peiod. (ight) Zoomed egion of the density plot Discussion It appeas that the amplitude envelope seen in Figue 7 is caused by vaiation in the suface velocity of the lase spot. To confim this, a simila test was pefomed with a 10 Hz symmetic tiangle function, which has a constant deivative (except fo the discontinuity at the tiangle peak). The fequency spectum of this signal is povided in Figue CSLDV Fequency Domain Noise Data fo 10 Hz Noise Floo Amplitude fequency, hz Figue 8: Fequency domain noise signal fo a 10 Hz tiangula input signal. The fequency spectum fo the tiangula scan also has the quasi-peiodic fom that is chaacteistic of lase speckle noise, containing notable peaks at the scan fequency and its hamonics, but the fundamental fequency and fist few hamonics ae not notably lage than the emaining hamonic peaks as they wee fo the sinusoidal scan in Figue 6. Also, the hamonic peaks do not potude fom the noise floo quite as much as those fo the sinusoidal scan. Figue 9 shows a time density plot of the noise signal obtained when the tiangula mio signal was used, the deivative is equal to a negative constant a fo the fist half of the peiod and a positive constant +a fo the second half of the signal. The noise signal is theefoe modulated by a constant amplitude function, and does not show evidence of maximum o minimum levels. Howeve, the detailed view shows that thee is still consideable small-scale stuctue in the speckle noise, ageeing with othe wok [17] that also obseved that the speckle noise signal was appoximately peiodic fo a constant suface velocity of the lase spot.

10 Figue 9: (left-top) Density plot fo 50 peiods of a 10 Hz tiangula noise scan. (left-bottom) Voltage input to scanning mios pe peiod. (ight) Zoom on a egion of the density plot. As sensitive as speckle noise can be to small deviations fom an exact peiodic scan patten, the fequency spectums and time-domain esponses shown hee all contain a significant amount of peiodicity. The noise amplitude, especially components nea the fundamental scan fequency, is sensitive to the ate at which speckles tansition though the detecto and scan types should be chosen accodingly. Diving functions with time vaying velocities seem to enhance the peiodic component of noise, paticulaly at the fundamental scan fequency Expeimental Setup/Geomety Noise Results This section seeks to eveal the tends that occu when one of the test paametes effecting noise is vaied while the othes ae held constant Howeve, the vaiables that speckle noise depends on ae not easily decoupled to illustate individual influences, so it is difficult to quantify the effects of changing any individual paamete. Quantitative measues of noise wee calculated fo both the peiodic and non-peiodic components. The nonpeiodic speckle noise component was calculated in the fequency domain fo both the oiginal and the lifted signals. The aveage amplitude of the non-peiodic noise, ηave, is calculated by emoving both the hamonic peaks and thei side bands fom the fequency domain data seies, leaving the non-peiodic signal vecto Yape(ω) at Nape data points, and finding the aithmetic mean of the magnitude of the signal. N ape η ave = Y k =1 ape (ω k ) N ape (8) The lifting pocess poduces many pseudo-esponses fom a single noise signal, each of which aliases the noise into limited fequency band. Figue 1 showed the aveage magnitude, o composite of many of these pseudofrfs fo a vibation signal. The peiodic noise is aliased to the zeo fequency line and falls off to the noise floo as in the ed cuve in Figue 1. Eq. (8) can be applied to the noise floo signal, Yape,lifted(ω) compising Nape,lifted data points to give the lifted non-peiodic noise, ηave,lifted. The peiodic component of noise was quantified in the time domain by etaining those hamonic peaks that wee emoved fom the signals to calculate the non-peiodic noise and setting the emaining fequency lines to zeo. This peiodic signal was then tansfomed back into the time domain with the invese Fast Fouie tansfom (IFFT). The standad deviation of the esulting signal was then found poviding a metic of the peiodic noise level, ρave.

11 ρ ave N 1 = N 1 k = 1 y pe ( y ( t ) y ) ( ) ( t ) = IFT Y ( ω ) k pe k pe pe k (9) Taget-to-Detecto Distance and Scan Length Peiodic Speckle Noise Figue 10 povides the amplitude of the peiodic noise using eq. (9) fo the thee scan lengths on the left and fo only the scan length of 2*x m =0.6 metes on the ight, at each of the thee taget-to-detecto distances. Mataelli and Ewins povided a simila data seies in [18] fo scanning fequencies of fscan=0.2, 0.5, 1, 5, 10, and 20 Hz. Note that 5, 10, 15, and 20 Hz coesponds to the fist 4 points in the left plot of the blue cuves. Note that the suface velocity depends on both the scan fequency and scan length, so each case gives a diffeent maximum suface velocity fo a given scan fequency. Also, the blue seies fo d=2 metes and 2*x m =0.98 metes is tuncated because the mios wee not capable of scanning faste than 60 Hz at that wide of an angle. Ave Noise, mm/s 16 x 10-3 Peiodic Noise Component θ=4.4, d=2m, x =0.15m m θ=8.5, d=2m, x =0.3m m θ=13.8, d=2m, x =0.49m m θ=2.6, d=3.4m, x =0.15m m θ=5.1, d=3.4m, x =0.3m m θ=8.3, d=3.4m, x =0.49m m θ=1.6, d=5.5m, x =0.15m m θ=3.1, d=5.5m, x =0.3m m θ=5.1, d=5.5m, x =0.49m m Ave Noise, mm/s 14 x 10-3 Peiodic Noise Component, xm = 0.3 m θ=3.1, d=5.5m 12 θ=5.1, d=3.4m θ=8.5, d=2m Max Suface Velocity, m/s Max Suface Velocity, m/s Figue 10: (left) Peiodic time domain noise components fo thee taget-to-detecto sepaations and thee scan lengths. (ight) Data e-plotted fo only one scan length. The left plot in Figue 10 shows that the cuves fo a given taget-to-detecto distance cluste togethe, independent of the scan length, so the noise seems to be only a function of the suface velocity and the taget-todetecto sepaation. Fo any given data seies the peiodic component of noise inceases athe shaply with inceasing suface velocity, simila to what was obseved in [18]. The ight potion of the figue illustates thee scans in which only the taget-to-detecto distance was vaied; the scan length was held constant at 2*x m =0.6 metes. Clealy, the noise tend deceases when the taget-to-detecto distance is inceased fo a given suface velocity. This eveals that one can obtain lowe noise levels at a given maximum suface velocity simply by inceasing the sepaation between the lase and the test aticle Non-Peiodic Speckle Noise Fequency Domain The non-peiodic speckle noise as calculated by eq. (8) is plotted in Figue 11. The plot contains data seies fo the thee scan lengths at the taget-to-detecto distance d=2 metes and the shot scan length 2*x m = 0.3 metes fo the emaining two taget-to-detecto distances. Again, the noise level geneally inceases with inceasing suface velocity. Fo a fixed scan length (2*x m = 0.3 metes in the plot), the noise level is lage fo smalle tagetto-detecto distances, as was also the case fo the peiodic component of the noise. Fo a fixed taget-to-detecto distance (data in blue),longe scan lengths coesponded to inceased noise, and this tend seems even moe significant than the dependence on taget-to-detecto sepaation. This behavio is qualitatively vey diffeent than that which was obseved fo the peiodic component of the speckle noise. The lowest velocity point in the cuve fo 2*x m = 0.3 metes and d=2 metes appeas to be an outline, the cause fo which is not known.

12 4 x 10-5 Non-Peiodic Fequency Domain Noise 3.5 Ave Noise, mm/s θ=1.6, d=5.5m, x m =0.15m θ=2.6, d=3.4m, x m =0.15m θ=4.4, d=2m, x m =0.15m θ=8.5, d=2m, x m =0.3m θ=13.8, d=2m, x m =0.49m Max Suface Velocity, m/s Figue 11: Non-Peiodic fequency domain noise components Non-Peiodic Noise Lifted Fequency Domain The non-peiodic lifted noise was calculated by applying eq. (8) to the composite of the pseudo-frfs, and is shown in Figue 12. These ae the same signals shown in Figue 11, but this shows the noise level in the lifted signals wheeas Figue 11 showed the noise level in the oiginal signal. The effect of noise on the lifted signal is moe impotant when using the method in [7, 8] to extact the modes, because any modes that fall below the noise floo ae not obseved. Again, fo a fixed scan length of 2*x m = 0.3 metes, the noise level inceases fo smalle taget to detecto distances. Fo a fixed taget-to-detecto distance, the noise amplitude inceases fo inceasing scan length, and this effect is once again moe significant than the dependence on taget-to-detecto sepaation. Futhemoe, besides the outlie at 2*x m = 0.98 metes and d=2 metes, the noise level does not change appeciably with suface velocity; quite a diffeent behavio than that which was noted above fo both the peiodic and non-peiodic noise. Ave Noise, mm/s 10 x 10-4 Lifted Non-Peiodic Noise θ=1.6, d=5.5m, x m =0.15m θ=2.6, d=3.4m, x m =0.15m θ=4.4, d=2m, x m =0.15m θ=8.5, d=2m, x m =0.3m θ=13.8, d=2m, x m =0.49m Max Suface Velocity, m/s Figue 12: Non-Peiodic lifted fequency domain noise component fo thee scan lengths at each of thee taget-to-detecto sepaations Discussion Fo a given set of expeimental paametes, that is fo any taget-to-detecto sepaation d, scan length 2*x m, and fo a single suface, both the peiodic and non-peiodic noise tend to incease with maximum suface velocity in the fequency domain as seen in Figues 10 and 11. This agees with pio eseach by Mataelli and Ewins [18].

13 Howeve, Figue 12 suggests that the non-peiodic noise in the lifted signals does not vay significantly as the maximum suface velocity is inceased, suggesting that it mattes little what suface velocity (scan fequency) one uses when the lifting method is used to extact the modes. The eason fo this is not yet undestood, but one can gain some insight by obseving the noise in the lifted signals in the time domain in Figues 7 and 9. The spead in each column of points is an indicato of the noise level in one lifted esponse. The esults in Figue 12 suggest that the vaiance evealed by the height of each column of points should not change with scan fequency, even though the oveall patten of the noise signal may change. This should be exploed futhe in subsequent woks. It is impotant to note that one must emove sidebands fom the scan fequency hamonics to popely compute the non-peiodic potion of the noise. The non-peiodic noise was found to incease less damatically with the maximum suface velocity than the peiodic noise and seems to depend moe heavily on the scan length than it does on taget-to-detecto sepaation. This may be explained, in pat, by the fact that inceased scan lengths equie inceased sweep angles if the taget-to-detecto sepaation is held constant, causing the lase spot to come out of focus at the edges of the beam esulting in a speckle patten such as that shown in Figue 5. When seeking to maximize the scan fequency, as is desiable fo the lifting method, these esults show that it is advantageous to keep line scans as shot as possible and to position the LDV as fa as possible fom the test device. Moe impotantly, these esults illustate how much of an incease in noise to expect as these paametes ae vaied Expeimental Sampling Fequency Noise Results As discussed peviously, the spatial esolution of a CSLDV measuement is limited only by the bandwidth of the LDV and the scan fequency, suggesting that one could obtain any desied level of esolution by inceasing the sampling ate of the LDV. This was investigated by obseving the noise level at sampling fequencies of 12.8, 25.6, and 51.2 khz fo a fixed taget-to-detecto distance of d=2 metes and scan length of 2*x m = 0.3 metes. The noise was ecoded fo all of the scanning fequencies in Table 2 fo each sampling fequency. The top pane in Figue 13 shows the FFT of the noise signals fo a 100 Hz scan fequency at the diffeent sample ates. Both the pseudo-vibation peaks occuing at the hamonics of the 100 Hz scanning fequency, and the noise at in between these peaks extend fo the full sampling bandwidth without deceasing in amplitude. The lowe cluste of plots display the noise signal in the time domain, simila to that of Figues 7 and 9. The mio angle achieves a maximum value at about 1.2 milliseconds, whee the fist minimum of the noise envelope esides. The envelope on the noise amplitude is pesent fo all thee sampling ates but becomes moe pominent as the sampling fequency is inceased. Thee ae also a few peiodic featues in the noise pofile that ae visible at all thee sample ates, such as the peak at 8.6 ms.

14 100 Hz CSLDV Fequecy Domain Noise Signal, 12.8 khz Sampling Amplitude feq, Hz x Hz CSLDV Fequecy Domain Noise Signal, 25.6 khz Sampling Amplitude feq, Hz x Hz CSLDV Fequecy Domain Noise Signal, 51.2 khz Sampling Amplitude feq, Hz x 10 4 Figue 13: Sampling fequency effects on 100 Hz CSLDV noise signals. (top) Fequency domain spectums. (bottom) Peiodic density plots with 511 peiods supeposed. Using the same pocedues as in Section 3.2, the peiodic and lifted non-peiodic noise components wee calculated fo each test. Figue 14 povides the esults of the noise calculations fo the thee sampling fequencies. The left plot contains the peiodic noise component calculated with eq. (9), evealing that the peiodic noise inceases with the maximum suface velocity. The noise level is always highe when sampled at highe fequencies. The ight plot, which povides the lifted non-peiodic noise, also shows the noise inceasing with sampling fequency, but the lifted noise does not incease with suface velocity. The noise data sampled at 51.2 khz seems to decease in magnitude with inceasing suface velocity.

15 Ave Noise, mm/s Peiodic Noise Component f samp =12.8kHz f samp =25.6kHz f samp =51.2kHz Max Suface Velocity, m/s Ave Noise, mm/s 9 x 10-4 Lifted Non-Peiodic Noise f samp =12.8kHz f samp =25.6kHz f samp =51.2kHz Max Suface Velocity, m/s Figue 14: Noise esults fo 12.8, 25.6, and 51.2 khz sampling ates at scan length 2*x m =0.3 metes and taget to-detecto-distance d=2 metes. (left) Peiodic time domain noise component. (ight) Lifted nonpeiodic fequency domain noise Discussion These esults eveal that the lase speckle noise has seemingly infinite extent in the fequency domain. As the sample ate inceases, one seems to captue the speckle noise moe fully, so the noise level seems to incease. Howeve, this noise affects the expeiment diffeently depending on how the mode shapes ae extacted. Figue 13 shows that the amplitude of the hamonic and off-hamonic noise is basically constant with fequency, so if the method of Stanbidge et al. [5, 6] is employed, then it would seem that inceasing the bandwidth beyond some minimum value would simply add fequency lines with no meaningful infomation, because any vibation infomation caied at the added fequencies is masked by the noise. When the lifting appoach is used, one might expect to be able to incease the spatial esolution by inceasing the sampling ate, Fo example, a 100 Hz continuous scanning signal sampled at 25.6 khz will esult in N o = 256 pseudo-esponse points pe scan cycle, while the same signal sampled at 51.2 khz poduces N o = 512 pseudo-esponse points. Howeve, Figue 14 show that the noise level inceases substantially with inceasing sampling ate. To detemine the optimum sample ate, one must conside that inceasing the bandwidth doubles the point esolution, so two points ae now available to estimate the mode shape at each point in a measuement at a lowe sample ate, and the edundant points could be aveaged deceasing the vaiance by a facto of 1/2 1/2 = o about 30%. Howeve, Figue 13 eveals that doubling the sample ate inceases the non-peiodic noise by about 75%, so the best esults ae pobably obtained if the sampling ate is the minimum that captues the impotant Fouie tems descibing each mode (see eq. (4)). The non-peiodic lifted noise component inceases with the sampling fequency. The same data seies ae plotted in the left and ight potions of Figue 14, but the fist thee data points ae omitted in the lifted fequency domain because the side bands of the scan fequency hamonics filled the entie (aliased) spectum of the lifted signal. Fo example, a test with a scan fequency of 5 Hz has a lifted bandwidth of 2.5 Hz and side bands of pseudo-vibations have been obseved to suound a given hamonic by (+/-) 2 Hz o moe.) When this is the case, the noise floo cannot accuately be detemined and so it was not shown. 4. Conclusion Speckle noise is a vey complex phenomenon that affects CSLDV measuements. This wok has evealed that speckle noise in CSLDV applications is dependent on the lase spot suface velocity (o ate of speckle tansition though the detecto), the distance fom the taget device to the detecto, the length of the scan on the taget device, and the sampling ate of the data acquisition system. Pevious woks by Rothbeg and his associates suggested that speckle noise may be educed by changing the lase spot size fo cetain types of speckle evolution, but no significant dependence on the lase spot size was obseved hee.

16 The speckle noise in a CSLDV measuement has both peiodic and non-peiodic components. The peiodic component of the noise epesents the speckle pattens seen by the lase at it pesumably scans the exact same path each cycle. The magnitude of the peiodic noise inceases seveely with suface velocity, in ageement with what was obseved by Ewins and Mataelli, but can be deceased by inceasing taget-to-detecto distance. The non-peiodic noise seems to be caused by othe factos, pehaps by stay fom a tuly peiodic speckle patten due to small inconsistencies in the lase s scan path. This noise was found to incease only modestly with suface velocity and moe significantly with inceased scan lengths, and deceased as the taget-to-detecto sepaation was inceased. Howeve, when the CSLDV measuements ae decomposed using the lifting method, the noise appeas to be constant with scan fequency, advocating the use of the highest scan fequency possible. Howeve, inceasing the sampling ate inceased the noise level in the lifted signals, so one should use cae to not sample faste than necessay to captue the meaningful vibation signal. The easons fo many of the tends obseved hee ae not at all clea, and sometimes even seem counteintuitive. The decease in noise with inceased sepaation of the test device and the LDV may be due to educed Dopple signal amplitude o possibly to a change in the chaacte of the speckle motions (e.g. fom boiling motions to tanslating speckles). Inceasing the scan length could seve to incease the numbe of speckle pattens obseved in a given time inteval, so it does seem easonable that this would incease the speckle noise. Howeve, the peiodic component of the speckle noise was found to be lagely insensitive to the scan length. The lase spot size and focal point was confimed to have a significant effect on the size and natue of the speckle patten obseved at the detecto, yet this did not affect the speckle noise noticeably in any of the tests pefomed hee. It is also quite supising that the demodulato employed in the LDV seems to keep the speckle noise contained to the scan hamonics, so that the noise at othe fequencies emains essentially constant with inceasing suface velocity even though the hamonic noise is inceasing shaply. Fotunately, even though all of these factos ae not undestood completely, one can make use of the tends illustated in this wok to design successful CSLDV tests in many cicumstances. Refeences [1] A. B. Stanbidge, M. Mataelli, and D. J. Ewins, "Measuing aea vibation mode shapes with a continuous-scan LDV," Measuement, vol. 35, pp , [2] 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 , [3] P. Siam, S. Hanagud, and J. I. Caig, "Mode shape measuement using a scanning lase Dopple vibomete," Poceedings of the 9th Intenational Modal Analysis Confeence, Floence, Italy, 1991, pp [4] P. Siam, S. Hanagud, and J. I. Caig, "Mode shape measuement using a scanning lase Dopple vibomete," Intenational Jounal of Analytical and Expeimental Modal Analysis, vol. 7, pp , [5] 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, [6] A. B. Stanbidge, M. Mataelli, and D. J. Ewins, "Scanning lase Dopple vibomete applied to impact modal testing," in 17th Intenational Modal Analysis Confeence - IMAC XVII. vol. 1 Kissimmee, FL, USA: SEM, Bethel, CT, USA, 1999, pp [7] M. a. S. Allen, Michael, "Mass nomalized mode shapes using impact excitation and continuous-scan lase Dopple vibomety," in Italian Association of Lase Velocimety and non-invasive Diagnostics (AIVELA) Ancona, Italy, [8] M. S. Allen and M. W. Sacic, "A Method fo Geneating Pseudo Single-Point FRFs fom Continuous Scan Lase Vibomete Measuements," in 26th Intenational Modal Analysis Confeence (IMAC XXVI), Olando, Floida, [9] M. S. Allen, "Floquet Expeimental Modal Analysis fo System Identification of Linea Time-Peiodic Systems," in ASME 2007 Intenational Design Engineeing Technical Confeence, Las Vegas, NV, [10] 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. 20, pp , [11] L. A. Luxembug, "Fequency Analysis of Time-Vaying Peiodic Linea Systems by Using Modulo p Tansfoms and Its Applications to the Compute-Aided Analysis of Switched Netwoks," Cicuits, Systems, and Signal Pocessesing, vol. 9, pp. 3-29, 1990.

17 [12] P. a. T. Aambel, G., "Robust H oo Identification of Linea Peiodic Discete-Time Systems," Intenational Jounal of Robust and Nonlinea Contool, vol. 4, pp , [13] P. a. R. Matin, S., "Lase Vibomety: speckle noise maps fo in-plane and tilt taget motions," in OPTIMESS, Leuven, Belgium, [14] P. a. R. Matin, S., "Lase Vibomety and the secet life of speckle pattens," in Eigth Intenational Confeence on Vibation Measuements by Lase Techniques: Advances and Applications Ancona, Italy, [15] S. J. Rothbeg and B. J. Halkon, "Lase vibomety meets lase speckle," Sixth Intenational Confeence on Vibation Measuements by Lase Techniques: Advances and Applications, Ancona, Italy, 2004, pp [16] N. Takai, Iwai, T., and Asakua, T., "Coelation distance of dynamic speckles," Applied Optics, vol. 22, pp , [17] S. J. Rothbeg, "Lase vibomety. Pseudo-vibations," Jounal of Sound and Vibation, vol. 135, pp , [18] M. Mataelli and D. J. Ewins, "Continuous scanning lase Dopple vibomety and speckle noise occuence," Mechanical Systems and Signal Pocessing, vol. 20, pp , [19] M. Allen and J. H. Ginsbeg, "Floquet Modal Analysis to Detect Cacks in a Rotating Shaft on Anisotopic Suppots," in 24th Intenational Modal Analysis Confeence (IMAC XXIV), St. Louis, MO, [20] R. Guenthe, Moden Optics vol. 1. New Yok: John Wiley & Sons, Inc., 1990.