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 [4]. 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 [5], 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 [2].
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.