3-Dimensional Laser Doppler Vibration Analysis of Stradivarius Violins

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1 3-Dimensiona Laser Dopper Vibration Anaysis of Stradivarius Vioins D.E. Oiver and V. Paan Poytec, Inc., Hopkinton, MA G. Bissinger and D. Rowe East Caroina University Greenvie, NC NOMENCLATURE ( 1x, 1y, 1z ) unit vector in the direction of aser i (i =1, 2, 3) in the object coordinate system (V x, V y, V z ) 3D veocity component at a particuar point in the object coordinate system V i (i=1, 2, 3) instantaneous veocity component in the direction of aser beam (or sensor head) i R(ω) radiativity = pressure/force Y(ω) mobiity ABSTRACT A Poytec PSV-400-3D scanning Dopper vibrometer system was used to scan three famous od Itaian vioins to measure 3-dimensiona mobiity profies as we as simutaneousy extracting accurate geometries for each vioin substructure scanned. Speciaized software "stitching" capabiities aow scans of various portions of the vioin to be combined into a true 3-dimensiona object viewabe from a directions. These scans showed significant in-pane motions in the top and back pates that shoud not be capabe of significant radiation and thus not show up in accompanying acoustic scans. Such a 3-dimensiona scan over the object aso permits computation of voume changes over a cyce, which is reated at ow frequencies to the air forced out of the dua ports (f-hoes) of the vioin. Such body-motion-forced air motion through the f-hoes had previousy been shown to be a new radiation mechanism for the vioin that contributes a major fraction of the overa radiation from the whoe vioin for the 1st corpus bending modes near 500 Hz. Another indirect radiation mechanism A1 cavitymode-forced body motion was seen quite strongy in one Stradivarius and one Guarnerius de Gesu. INTRODUCTION Moda anaysis has greaty expanded our understanding of vioin dynamics over the ast 2 decades. For exampe: 1) the vioin itsef is a inear device so norma mode representations are warranted [1], 2) in any reasonaby constructed and setup vioin a of the norma modes faing in the open string pitch region are observed in vioins good or bad [2], surface norma aser scans combined with acoustic scans have ed to radiation efficiencies and effective critica frequencies [3], radiation damping for good and bad vioins is different but their interna damping is the same [4]. Yet there are sti important experimenta matters not propery addressed, such as in-pane vibrations of the vioin top and back pates (impied from previous scans of the rib substructures). Whie these are not expected to radiate much or show up in acoustics scans they might be quite informative in showing how the vioin fexes in-pane, and thus possiby of some vaue in understanding the (mechanica) payabiity issue, as we as being quite usefu in vaidating soid modes. Of course utimatey a soid mode with reiabe density and eastic modui particuar to a payabe vioin is what woud be most vauabe for what if experiments, yet certain intractabe probems seem to keep us from reay creating this soid mode. For exampe, the empirica, ergonomic deveopment of the vioin eading to its iconic, but non-anaytic, shape, a structure where the was encose a cavity whose modes are wa-driven but quite substantiay back-coupe to the body of the vioin, the very materia itsef - anisotropic, inhomogeneous, orthotropic wood, the substantia mode couping (even when not hed), four important direct and indirect radiation mechanisms, and then add to this the paying/aging effects on the structure and its materias a conspire to make it a compact horror to simuate. Into this mix we introduce a new experimenta technoogy, tri-aser scans capabe of creating 3-dimensiona mobiity FRFs over a surface, combined with software to stitch these into other surface measurements thereby creating into a true 3-D scan of an object. Here we present some preiminary resuts from appying this advanced technoogy for the very first time to moda anaysis of three fine od Itaian vioins: two Stradivarius vioins, the 1715 Titian (from the so-caed goden period) and the 1734 Wiemotte (made just a few years

2 before his death in 1737 at age of 93) by the most famous of a vioin makers, and an equay fine 1735 Powden Guarnerius de Gesu. 3-D Laser Dopper Vibrometry Laser Dopper Vibrometry takes advantage of the optica Dopper-effect in which ight backscattered from a moving surface is shifted in frequency. The frequency shift is proportiona to the reative surface veocity. This effect is anaogous to the acoustic Dopper-effect that is experienced in everyday ife when the tone of sound from a vehice (such as a poice siren) changes as it passes the observer. An interferometer as shown schematicay in Figure 1 beow, together with frequency demoduation eectronics, can be used to provide an eectrica signa proportiona to instantaneous veocity of the measurement object in the direction of the aser. A Bragg ce introduces a frequency shift in order to distinguish the direction that the surface is moving with respect to the aser. A signa proportiona to instantaneous dispacement can aso be obtained by anayzing the reative phase moduation with respect to the carrier signa driving the Bragg ce. Figure 1. Mach-Zehnder interferometer commony used in aser vibrometers. Athough the groundwork for Laser Dopper Vibrometry was aid in the eary 1960s, [5-8] scanning aser Dopper vibrometers (SLDVs) were not deveoped unti the eary-1980s. A homodyne SLDV, incorporating a gimbamounted gavanometer-driven mirror, deveoped by the Atomic Energy Research Estabishment in the UK was used from about 1983 for the deveopment of oudspeakers. This design, the basis of which can be found in L. E. Drain s book The Laser Dopper Technique [9] was icensed to Ometron Ltd by AERE in 1985 and was eventuay reeased as the first commerciay avaiabe SLDV. In parae, the German car manufacturer Vokswagen AG deveoped an SLDV, which was reported in 1985 [10]. Poytec further refined the SLDV by adding a video camera for heping define scan points, FFT anaysis, aser autofocus and incorporating many sophisticated software features for data acquisition, dispay, post-processing and export. One imitation of these SLDVs was that they coud ony measure vibration in one direction at a time, the aser direction. Scanning the aser beam over a compex shaped surface offered a particuary demanding scenario in which the direction of the measured vibration changes with the scan ange reative to the SLDV and reative to the measurement surface. Traditiona moda anaysis requires measurements in mutipe degrees of freedom, in a known coordinate system reative to the geometry of the structure being tested. These imitations can be overcome by measuring a vibration components simutaneousy and knowing or measuring the 3-D geometry. A prototype 3-D SLDV was therefore deveoped by Poytec in 2002 in cose cooperation with Bosch, Germany, in an effort to bring a the advantages of aser vibrometry to moda anaysis. The 3-D SLDV used for this study is shown in Figure 2. It incudes three independent scan heads, each controed by a centra data management system. One of the scan heads has a aser distance sensor buit-in that uses the time-of-fight principe to map the geometry of the test part. Aternativey, a geometry fie can be imported into the system.

3 In order to scan each seected point on the structure simutaneousy from three aser directions and convert the measurements into the orthogona coordinate system of the measured object, the system requires the foowing information: The position and orientation of the scan heads reative to one another The coordinates of the scan points Figure 2. 3-D Scanning Laser Vibrometer. The foowing procedure is carried out in order to define the goba coordinate system and determine the reative position and orientation of the scan heads: The aser distance sensor is directed to at east three points with known coordinates and the distance between the scan-head and the object is measured accuratey. These points coud for exampe be at the origin of our coordinate system, somewhere on the x-axis and somewhere on the x/y pane. Lasers from the three scan heads are merged at a minimum of four arbitrariy chosen points in order to determine the reative positions of the three scan heads to each other. From the measured coordinates, distances and scanner anges, a non-inear optimization routine cacuates the position and orientation of the scan heads. The procedure is aso possibe if no coordinates of the object are known, in this case the coordinate system can be set deiberatey. With an accurate caibration, the beams can be positioned to independent test points within an accuracy of better than mm, depending on the object size. This accuracy has proven to be more than adequate for a the appications investigated by the authors to date. Once the positions and orientations of the scan heads are known, the scan points can be defined as in the 1D-SLDV using specia software toos together with the ive video. The aser distance sensor then scans the seected points to determine the coordinates for a scan points. From the positions and orientations of the scan heads and the coordinates of the scan points, the vectors of the aser-beams can be cacuated (eq. 1). Vector of aser beam 1: ( ) L 1x 1y 1z 1 = (1) Using the vectors of the aser-beams, the measured vibration data are transferred into the goba coordinate system (eq. 2). v v v x y z = 1x 2x 3x 1y 2 y 3 y 1z 2z 3z 1 v v v (2)

4 Experimenta Setup The experimenta setup consists of a vioin mounted on a specia fixture. The fixture was designed such that the mounted vioin can be rotated 360 degrees aong the vertica axis. This feature is usefu since it gives the abiity to scan both the top-pate and the back-pate without moving the sensor heads. Figure 3 shows the positions of the three sensor heads with respect to the vioin surface being measured. It shoud be noted that the distance between the vioin surface and the sensor heads, known as stand-off distance, was chosen such that it corresponds to one of the visibiity maxima of the aser vibrometer. The ight source of the vibrometers is a Heium-Neon aser. This muti-mode aser is designed such that a maximum of two modes can exist. The interference of these two modes makes the intensity of the resuting optica signa vary periodicay with the stand-off distance. The stand-off distances at which the intensity increases to a maximum are caed visibiity maxima. These visibiity maxima occur at mutipes of the aser cavity ength, which in this case is 204 mm. Figure 3 - Tri-aser system setup for vioin mounted free-free in rotating, transating support fixture with sound absorbent foam behind the vioin. Three different surfaces were scanned on each vioin under consideration top-pate, back-pate and the ribs on one side. In addition to these measurements, a singe point measurement was conducted at drive point. The top-pate and the back-pate were scanned such that these two data sets can be stitched together ater on. This is done with the hep of the stitching feature in the Poytec PSV software. The vioin is excited with an automated force hammer mounted on the test fixture. This arrangement ensured that the excitation was provided in the same direction, at the same point and of the same magnitude for a the measurement ocations. The abiity to strike at the same ocation becomes important in identifying the driving point. The excitation point chosen to best represent actua energy fow through the vioin was the G-string bridge corner bridge based on past studies [11]. Common settings for a measurements were: a) frequency bandwidth - 5 khz, b) number of averages -3, c) type of averaging - compex, d) number of frequency ines , e) frequency resoution: Hz, f) reference: force sensor associate with PCB 086C80 force hammer, g) ow pass fiter - 20 khz. Compex averaging was performed in the software using the reation: S N N = = + N 1 1 S n Re( S n ) i Im( S N n= 1 N n= 1 n= 1 n ) (3)

5 where S n is the measured ampitude and N is the number of averages seected in the software. The parameter that varied sighty from one measurement to other was the number of scan points. The number of scan points, which corresponds to the number of measurement ocations, was chosen such that it provides enough spatia resoution in the frequency range of interest whie keeping the measurement time to a minimum. In addition to the above mentioned parameters, a specia hardware option caed tracking fiter was used. This hardware option, which can be activated via PSV software is usefu in conditions where a considerabe in-pane motion is observed. The impact direction in the experiments under consideration is such that it causes unavoidabe significant in-pane motion. This in-pane motion occasionay causes oss of signa at the detector in the interferometer, causing sudden spikes, known as signa dropouts, in the time domain signa. The tracking fiter option reduces the occurrence of these dropouts, and hence improves the signa-to-noise ratio of the input signa of the sensor head. Simpy put, the tracking fiter can be considered an eectronic fywhee, in that it bridges brief dropouts which occur due to the specke nature of the ight scattered back from the measurement surface. top pate back pate stitched X Y Y Z X Z YZ Figure 4 - Peak magnitudes for Titian Stradivarius at 1 st corpus bending mode B1+ mode at 545 Hz. Noda ine shapes (green) for out-of-pane Y motions for this mode are consistent with those measured for a other vioins.

6 Resuts Using the experimenta setup and the acquisition settings described above, a series of measurements were conducted. Figure 4 shows the peak magnitudes for a representative measurement. In the figure, the vibration patterns shown are for one of the dominant frequencies for the Titian Stradivarius. A few observations that one woud make from the figure are given beow: 1. Note that the Poytec software provides the feature to resove the in-pane and out-of-pane motion. The in-pane is in X and Z, whereas Y constitutes the out-of-pane motion for the top and back pates. 2. As is cear from the figure, the out-of-pane motion dominates the response. 3. Aso, note that the top and the back pate measurement coud be stitched together to get a better picture of the overa response. A pre-requisite for stitching is that the scan points on the surfaces to be stitched shoud be defined in the same coordinate system. 4. Specia care was taken for measurements on the top pate to ensure that the three aser beams woud coincide on the vioin surface. Obvious obstructions on the top-pate are the strings, finger-board, F- hoes and the bridge. Another important obstruction was the moda hammer, which was carefuy avoided to acquire cean and meaningfu data. In case of a 1-D system, one is deaing with just one aser beam and one sensor head. This makes it easier to pace the sensor head norma to the measurement surface. In case of a 3-D system, one has to be carefu and check the aser beams from a the three sensor heads and make sure that a these heads can see the surface under consideration. SOME INITIAL RESULTS A compete scan of the top pate of a modern vioin (2006) by the renowned vioin maker Joseph Curtin was made in addition to the od Itaian vioin measurements. Such modern vs. od Itaian comparisons are usefu in terms of understanding mode frequency pacement, tota damping and reative mode ampitudes, pus any reative trends. Such a comparison is presented in Figure 5 for the Curtin vioin and the Powden Guarnerius de Gesu where the rms mobiity over the top pates is overaid. Ceary there are different detais here but this woud be true in comparison of any two vioins. Generay the od and new vioin have overa simiar enveopes with high mobiities at the owest frequencies, then a brief drop off foowed by a rise to a nomina maximum at 1-2 khz and then a sow taiing off. Figure 5 Comparison of od Itaian vs. modern vioin rms mobiities (top pate ony).

7 The moda anaysis fitting resuts ceary showed the presence of the 1 st ongitudina cavity air mode A1 through its effect on the motiona profie of the top and back pates. A1 cannot radiate through the f-hoes because these create a node and the resutant voume fows constitute an acoustic short in the f-hoe region. Thus the observed radiation must be from cavity-mode-induced surface motion radiation. Both the A1 vibration and radiation eves were quite high for the Titian and Powden. More interesting initia resuts are those that expoit the 3-dimensionaity of the measurements. For exampe, the ratio of the averaged out-of-pane surface norma mobiity Y to the in-pane Y mobiity ratio. An exampe of this (for the top pate ony) is presented in Figure 6 for three vioins the Titian and Powden as we as a modern vioin (assembed in 2006) by Joseph Curtin. As expected this ratio is greater than 1 but surprisingy not much greater than 1 maximum ratio is about and the Curtin vioin is actuay trending beow 1 near 5 khz. The Y /Y ratio curves for the two superative vioins the Titian and Powden are quite different with the Powden amost a factor of 2 ower than the Titian above 3 khz, whie the Curtin curve is quite simiar to the Powden over most of the range. Yet the Powden averaged-over-sphere, far-fied rms radiativity curve is quite simiar to the Titian in magnitude. Such variations are one of the perpexing aspects of anayzing vioins, since the curves show a the detais whie the ear treats the sound produced by these vibrations in a hoistic way. Figure 6 Surface norma rms mobiity Y divided by in-pane rms mobiity Y for the Titian Stradivarius, Powden Guarnerius de Gesu, and Curtin modern vioin. CONCLUSIONS The Poytec 3-D aser scans offer an entirey new way to examine vioin vibrations. Yet there were some significant imitations on what coud be measured. The requirement that a three asers be abe to see every point on the structure meant that certain portions of the top pate - considered the singe most important substructure for ensuring vioin quaity - were shieded from measurement by the raised fingerboard and taipiece substructures, more so even than in a 1-D scan. Such imitations must be baanced against the abiity

8 to measure in-pane motion. Whie previous 1-D scans on the vioin ribs certainy created expectations for significant in-pane vibrations in the top and back pates for some ow-ying modes, these 3-D measurements exposed a whoe new eve of compexity in the vibrationa behavior at higher frequencies. The 3-D resuts showed that the in-pane vibrations coud be surprisingy arge compared to surface norma vibrations, even though the former are not expected to produce significant acoustic radiation. Yet accompanying averaged acoustic radiativity measurements showed itte difference between the vioins. What is not so surprising is that two od Itaian vioins, considered among the finest of their kind, measure quite differenty. At this stage it is not cear how significant a this new in-pane motion information is. As in any technoogica advance the advantages of a new way of ooking at things might not be immediatey apparent. Since the vibrationa data wi be used at some future time to update a soid mode with shape and density information from 600+ sice CT scans (1 mm spacing), the unique in-pane data can be used to hep back out the important eastic modui for various substructures in these od Itaian vioins that cannot be obtained in any other way. We woud ike to express our gratitude to the owners of these exceptiona od Itaian vioins for their oan, and to acknowedge the aid of Joseph Curtin, Sam Zygmuntowicz, Fan Tao, Joseph Regh and Christina Fan in various stages of this experiment. REFERENCES 1. K.D. Marsha, "Moda anaysis of a vioin, J. Acoust. Soc. Am.77, pp (1985). 2. G. Bissinger and A. Gregorian, "Reating norma mode properties of vioins to overa quaity: signature modes," Catgut Acoust. Soc. J., 4, No. 8 (Series II), pp (2003). 3. G. Bissinger and J. C. Keiffer, "Radiation damping, efficiency, and directivity for vioin norma modes beow 4 khz," Acoust. Res. Lett. Onine, 4, 7-12 (2003) onine at 4. G. Bissinger, G., Extracting interna damping from tota damping and radiation efficiency measurements, Proc. 21 st Intern. Moda Anaysis Conf.- Soc. Exp. Mechanics (CD ony), Bethe, CT, 2003, paper # Y. Yeh and H.Z. Cummins, Locaized Fuid Fow Measurements with an He-Ne Laser Spectrometer, Appied Physics Letters, 4, No.10, 1964, pp R. D. Kroeger, Motion Sensing by Optica Heterodyne Dopper Detection form Diffuse Surfaces, Proc. IEEE, Feb. 1965, pp H. A. Deferrari, R. A. Darby and F. A. Andrews, Vibrationa Dispacement and Mode-Shape Measurement by a Laser Interferometer, J. Acoust. Soc. Am., 42, No. 5, 1967 pp F. J. Eberhardt and F. A. Andrews, Laser Heterodyne System for Measurement and Anaysis of Vibration, J. Acoust. Soc. Am., 48, No. 3, pt. 1, 1970, pp L. E. Drain, The Laser Dopper Technique, Wiey, A.C. Lewin, Non-contact surface vibration anaysis using a monomode fiber optic interferometer, J. Physics: Scientific Instruments, 18, No. 7, 1985, pp G. Bissinger and K. Ye, "Automated hammer-impact moda anaysis with a scanning aser vibrometer: working exampe - a vioin", Proc. 18 th Intern. Moda Anaysis Conf.- Soc. Exp. Mechanics, Bethe, CT, 2000, pp