Modal Analysis Comparison of New Violin Before and After -250 Hours of Playing George Bissinger East Carolina University Greenvtlie, NC 27858 ABSTRACT Violin lore has it that the playing quality of a recently assembled violin changes for the better after it is played vigorously by the musician for a substantial period of time, a circumstance that implies changes in modal parameters. To investigate this, hammer-impact modal analysis was performed over a 0 - -1 kHz range on a recently completed violin before any significant playing had occurred. After a professional violinist had played the instrument for -250 hours the measurements were repeated. The effects on the violin’s mechanical modes of vibration were intercompared using multiple criteria frequency, damping, mode shape (via MAC values), violin part with maximum vibration and relative strength in the Modal Peaks spectrum - to match the modes before and after playing. In general after playing the modes showed: i) a average 2.1 % drop in frequency cxccpt for the lowest tailpiece modes and the Helmholtz cavity resonance A0 at 275 Hz, ii) similar overall %critical dampings, although with substantial fluctuations, iii) increased amplitude after the violin had been played as noted both in Modal Peaks spectra and in maximum cigcnvector values for each mode, and iv) substantial chaages in some mode shapes. NOMENCLATUR8/ABBREVIATIONS < = % critical damping TP = violin top plate BP = viotio hack plate TPC = violill tailpiece NKFB = violin neck-fingerboard /Ihotc = s-shaped slots in violin top plate ;d:tIF = acceleration frequency response function STDEV = standard deviation of mean INTRODUCTION Musicians have felt for a very long period that the violin’s sound - and feel generally benefits from vigorous playing, without any real quantitative evidence to back up this claim. Recently however there have been experiments where the violin was subjected to acoustically induced vibrations for extended periods and measurable differences in vibration frequency and response for modes between 400 - 600 Hz were seen - changes partially reversible if the instrument was not played subsequently [l]. This interesting and important empirical effect appears to offer an excellent application for modal analysis techniques to determine the frequency, damping, and mode shape and amplitude changes associated with vigorous vibrations of a violin. In this work a newly assembled violin has been subjected to tbe vigorous vibrations produced during the actual playing by a professional violinist for -250 hrs, and modal analysis performed before and after this period. EXPERIMENT I. Aooaratus The modal analysis measurements were made on a varnished Vuillamme pattern violin (saris chinrest) with a typical Stradivarius-style arching built by Deena Spear. The wood had been air dried for over 25 years. The geometry at 270 points including the ribs was put into a coordinate table in the SMS STAR modal analysis program. After tuning all the strings (A = 440 Hz) the violin was set up on foam cutout pads very similar to those used in previous violin studies [21. A very thin foam strip was then threaded through the strings to damp their vibrations. This damping was considered very important here because the strings are essentially decoupled from the violin corpus and, with much lower damping, will vibrate long after any typical corpus mode has died out. The strings parcel energy to the corpus and thus extend the decay time of the modes. This quite possibly could affect the very shape of the decay curve and hence the cxtractcd damping values. 822 11. Data Acauisition A PCB model 086C80 minihammer with nylon tip and no mass extender, mounted on a handheld electromechanical striker 131, was used for excitation of the violin. A 0.0006 kg PCB 309x accelerometer was placed immediately in front of-but not touching - the bridge G-suing foot. The calibrated (via pendulum mass system) hammer and accelerometer voltage outputs were routed to inputs of a Zonics WCA dual FFT signal analysis system. The initial aFRF measurements were 800 line, 0 - 1 kHz range, average-of-5.impact measurements, with rectangular window on the excitation and response inputs. Remeasurement at -250 hrs was similar except it covered the 0 2 kHz range with 1600 lines. (Note that the acquisition system aFRF had to be transferred in ASCI format for analysis, resulting in 1024 and 2048 lines, respectively, even though analysis of the 0 hrs data set was limited to the lower 80% of the range.) Acquisition was triggered at -0.1% of maximum hammer signal. For the initial tests aFRFs were taken at all 270 points; however, some points had to be dropped in the remeasurement at -250 hrs because of severe time constraints on the period that the violin was available between performances. III. Cavitv Mode Freauencies Some of the cavity modes of the violin below 1 kHz are known to contribute either directly or indirectly to the overall acoustic output of the violin. To determine the frequencies of these modes, labeled A0 (the Helmholtz mode) and Al (the first longitudinal cavity mode), a small acoustic driver. with a ramped sine wave input, was slipped in through one f-hole of the violin into the upper bout of the violin and a small microphone was slipped in through the otherf-hole to measure acoustic response inside the violin. The A0 mode frequency was measured at 276 Hz while the Al mode fell at 477 Hz. IV. Data Analvsis The FRF Xllmagl (sum of absolute values of aFRF) Modal Peaks capability of the SMS STAR modal analysis program was used to aid in identifvine and correlatine imoortant peaks in the response spectra. -These spectra w&e valuable also in evaluating the average strength of modes. The Modal Peaks spectra summed over the entire violin before and after playing are shown for reference in Figure 1. The cavity air modes are labeled as well as regions where the 1st corpus bending modes are located . While there are clear similarities between the spectra, a noticeable droop in peak frequencies across the entire frequency range was obvious. along with substantial differences in relative peak strength in some regions. v. EBEEits The data were tit two ways: 1) Preliminary the Advanced Curve Fit (ACF) capability of the SMS STAR program was used to find stable mode frequency and damping values from analysis of a large fraction of the violin’s aFRFs in a semiautomatic way. TIE modes with stable frequency and damping values were then chosen for the global fit to determine the eigenvectors for each mode at each response point. / / t 0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0 f(W Figure 1 Sum of llmagl of (entire violin) FRFs Modal Peaks spectra before (thick solid line) and after (thin solid line) -250 brs of playing. (Labels are shown for A0 and Al cavity modes and acoustically important 1st corpus bending modes Bl- and Bl+.) 823 2) the mode frequency and damping values were then extracted by a Polynomial fit over one band over all the FRFs to ascertain the minimum number of peaks that would provide a good tit all of the peaks inside the band. Then a Global Frequency and Damping (GFD) tit over all the spectra followed by a Global Residues (GRBS) tit was examined to determine the quality of fit across all FRFs. If the tit was not of good quality the ACF mode shapes were used to locate the parts of the violin with the largest response for each mode and a GFD fit over just these parts was followed by a GRES fit over all the FRFs. This procedure was followed in an interactive way until acceptableexcellent tits were achieved for all aFRFs. The heavy modal coupling for a substantial number of the modes, or the weakness of the mode, made the ACF results less reliable than the GFD/GRES results (based on onscreen judgments of quality of all fits during the computer analysis). RESULTS I. Mode Freouencv Change Figure 1 shows the general decrease in mode frequencies for the violin after -250 hrs of vigorous playing in a graphic way. By comparing frequencies, shapes and cross-MAC values (hereafter labeled X-MAC) between the mode shapes before and after playing, it was possible to correlate the great majority of the modes in the violin at -250 hrs with those at 0 hrs. These correlated mode parameters (frequency, damping, X-MAC values) are accompanied by a listing of the violin part with maximum motion (taken from the pertinent mode shape table in STAR) in Table 1. A ratio of correlated mode frequencies ( -250 hr/O hr) listed in Table 1 was calculated to quantify the overall trend observed in Figure 1. This ratio, presented in Figure 2, shows an almost uniform drop in frequency with no trends as mode frequency increases. The overall averaged f-ratio is 0.979*0.0046 (STDEV), a statistically signiticsnt drop. Table 1 - Frequencies, dampings, and X-MAC values for correlated modes of violin UDI at 0 hrs and after -250 hrs playing. The maximum motion of various parts of the violin are noted with abbreviations as follows: TPC = tailpiece, NKFB = neck-fingerboard, TP = top plate, BP = back plate. If the maximum is similar for multiple parts, they are presented in order of decreasing amplitude. f(W 121.4 135.5 158.8 189.4 201.7 246.4 255.4 266.0 269.4 274.0 373.1 415.8 450.2 460.8 542.0 581.6 665.0 680.1 702.0 730.9 765.0 799.1 848.2 897.8 927.7 1049.0 1082.0 %critical Damping 0.09 0.38 0.71 0.65 0.24 0.08 0.87 0.69 0.88 2.01 0.79 1.50 1.45 0.81 1.00 0.31 0.30 1.17 1.12 0.83 0.74 1.31 0.84 1.14 1.10 0.85 0.93 Maximum Motion Part TPC TPC NKFB TPC lF TPC TPC NKFB TPC TPcn-P lP l? BPm IP lT l7 IF BP TP/BP TP NKFB lF BP lP TPCYBP/TP TP/BP TP KW 122.3 133.6 156.7 186.2 201.3 229.4 229.4 261.7 266.9 273.5 365.3 383.7 442.8 450.7 533.1 560.8 655.5 668.1 691.9 720.9 754.5 793.9 836.0 878.0 941.6 1024.0 1056.0 824 %cfitical Damping 0.11 0.50 0.41 0.33 0.25 0.71 0.71 0.74 1.30 1.33 0.84 0.22 2.33 1.34 0.94 0.81 2.02 0.97 0.70 0.94 0.81 1.58 0.88 1.07 1.15 1.10 1.08 MXh”Ut” Motion Part TFC m NKFB TPC TPOTP TK TPC NKFB NKFB l? TP Tp TP IF TP IF lF BP/IF’ NKFB/TFJ TP NKFB TP BP TPcrrP l? BP IP X-MAC Value 0.98 0.00 0.16 0.08 0.33 0.91 0.21 0.90 0.23 0.53 0.76 0.62 0.92 0.55 0.97 0.58 0.30 0.98 0.62 0.38 0.92 0.90 0.89 0.46 0.51 0.76 0.87 II. Damninr Chanu Another imporlant property of all musical inslrumcnts is lhe damping of modes of vibration. The Critical analysis rcsults from Tahlc 1 wcrc used to calculate a ratio of damping values (-250 hr/O hr) as shown in Figure 3. No obvious ucnd can he seen in the figure. The overall average %critical damping ratio of 1.582 indicated an increase after playing hut the standard deviation of 51.884 was so high that it could not hc considcrcd significant. After using standard statistical techniques to evaluate the possibility of lit error , which led to rejection of the two highest points, the mean dropped to 1.08+0.09, still indicating no significant change in overall damping after -250 hrs of playing. and -250 hr modes. These X-MAC graphs arc shown in Figure 4a and 4b. Note that the modes covcred in each ligure were overlapped so as not to miss a corrclatiotl arising from a possible ma.jor mode frequency change although no such change was observed. A 1.100 1.050 0 1.000 z 2 0 10 20 30 Mode number 0.950 Figure 3 - %critical-ratio of correlated mode frequencies (-250/O) showing the effect of playing on the normal mode dampings. 0.900 0.850 0 10 20 30 Mode number Figure 2 f-ratio of correlated mode frequencies (-250/O) showing the effect of playing on the normal mode frequencies. Ill. Mode Shaoe Comoarison The mode shapes resulting from the analysis were all viewed in animation. Unfortunately the large number of points, the complex shape of the violin and the asymmetric mode shapes made it very difficult to intercompare mode shapes. It was possible to note however if the modes showed any complex behavior. Only in the case of modes where the motion was dominated by a relatively isolated small part such as the tailpiecc (small triangular part behind the bridge) was out-of-phase motion (relative to the corpus) ohscrvcd. Since the tailpiece is entirely supported by strings, essentially decoupling it from the corpus, this was intcrprctcd as a tailpiece local mode. llx hcst quantitative indicator of mode shape similarity is prohahly the MAC values (X-MAC) calculated between 0 hr 825 Finally, the effects of the A0 and Al cavity modes on the corpus mode motion were significant for this violin. The mode noted in Table 1 at 274 Hz is associated with the A0 cavity mode interacting with the compliant corpus walls and directly exciting corpus motion; this mode had an X-MAC value of 0.53 indicating tbe shapes were similar. After -250 hrs of playing the amplitude of the AO-induced corpus motion had grown by an factor of -2. Note that the violin top plate was clearly the most active part of the violin after playing whereas the tailpiece was the most active before Al excited corpus motion, although weak in the 0 hrs data set, clearly showed up in the mode at 477 Hz (shown in Figure 5). The violin top and back plate motion is quite small along the edges of the plates at the position where the ribs we glued to the plates. The violin top and back plate both showed motions consistent with the Al mode’s 180” phase difference between lower bout and upper bout regions of the cavity as well as a nodal line running across the bridge region of the violin. Moreover the top and back plate motion in the upper or lower bouts are also 1800 out 01 phase. There was no clear evidence for such an Al-related mode for the violin after playing even though a special effort was made to extract such, hence it is missing from Table 1. Modet 3 (0 W (a) 2 ‘ry L” hr) (b) Figure 4 - a) X-MAC graph for lowest 25 modes of violin UDI before and after -250 hrs of playing b) as above but for approximately 10 additional higher modes (note some overlap with part a) IV. Mode The Modal Peaks spectra (ZIImagl over the entire violin) in Figure 1 were based on calibrated measurements, hence their relative behavior can be compared reliably. The ratio of Modal Peak amplitudes (-250/O) calculated for each correlated mode is shown in Figure 6; the average ratio is 1.92 ti.27, clearly a significant increase. There are only two low-f modes where the ratio is ~1, both below the range of the violin (frequency of lowest string is typically 196 Hz). The same increase was observed in ratios of the maximum eigenvector in the shape table for each mode although individual mode maxima increases were more or less that the averaged Modal Peaks ratio presented in Figure 6. because the instrument’s radiation efficiency and directionality must be taken into account also . DISCUSSION The overall analysis results presented here indicate that playing a violin does indeed affect the modal response of the violin substantiallv. The aeneral results associated with playing a violin that can-be extracted from our modal analysis were that: i) the mode frequencies dropped with playing in our case by an average of 2.1%. ii) little average effect on damping values was observed, iii) correlated modes can change shape - in our case sometimes so much so that the X-MAC value - 0, and iv) mode amplitudes increased - in OUT case by an average factor of 1.92. The purely mechanical aspects of the effect of playing on the violins response are now starting to become clearer. UnfOrtUnatdy these measurements cannot yet be related to what the musician hears - only how the instrument feels, Figure 5 0 hrs corpus mode at 478 Hz excited by cavity mode Al (at 477 Hz). No equivalent mode found at -250 hrs. Relative phases marked by +/-. 826 My thanks to Deena Spear for allowing me to mark and beat her violin, to Robert Spear for aid in cavity resonance measurements, and to Sylvia Ouelette for playing the violin into “shape”. 10.0 0 = c Y 8 P REFERENCES 1. C.M. Hutchins and 0. Rodgers, “Methods of changing the frequency spacing (delta) between the Al and Bl modes of the violin”, Catgut Acoust. Sot. J. Vol. 2, #l(Sefies II), 13-19 (1992). 1.0 z 2. G. Bissinger, “The influence of the soundpost on the mechanical motions of the violin”, Proc. 12th Intern. Modal Analysis Conf., Sot. Exp. Mechanics, Bethel, CT, 1994, pp. 294-300. si 0.1 0 10 20 30 Mode number Figure 6 Modal peaks-ratio of correlated mode amplitudes in Modal Peaks spectrum (-250/O) showing the effect of playing on the normal mode average aFRF response. 827 3 . K. Jacobs, “Comprehensive investigation of the violin’s acoustically important corpus, cavity, and cotpus-cavity modes of vibration below 2 kHz”, MS Thesis, East Carolina University, Greenville, NC, 1993. 4. K. D. Marshall, “Modal analysis of a violin”, J. Acoust. Sot. Am. 77,695709 (1985). 5. E.M. Pugh and G.H. Winslow, The Analysis ofPhysical Measurements, Addison-Wesley (Reading, MA 1966). p. 109.
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