EXPLORING FREQUENCY RESPONSE
IN ACOUSTIC GUITARS
Kirsten A. Bender
Gibson Guitar Corporation
2560 9th Street, Suite 212
Berkeley, CA 94710
and relative humidity are recorded and normally do
not change during the test process.
ABSTRACT
This paper summa&es a method of data acquisition
and analysis employed by Gibson Guifar Corporation in
normal mode analysis of acoustic guitars. A fved point
random excifation source and a roving accelerometer
and are used to measure frequency response functions
for the surface of a &far. The data exhibits many
lightly damped, well-coupled modes.
2.2 Data Acquisition
1.
The signal source is random noise output through a
graphic equalizer and amplifier to the excitation
device. The excitation device, a mini-speaker mounted
on a wood dowel and peg, is inserted in the third hole
of the bridge and adheres to the instrument through
friction, as shown below in figure (1).
INTRODUCTION
In the interest of linking a guitar’s performance to its
mechanical behavior, Gibson Guitar Corporation has
used modal analysis to identify natural frequencies
and normal mode shapes in a variety of guitars.
Although the company considers most of this project
confidential, it would like to share certain aspects
related to test procedure with the modal research
community. The hope is that this gesture will spark
discussion of test methods appropriate for stringed
instruments as well as possible improvements to these
methods.
2.
EXPERIMENTAL.
Data acquisition is performed in the frequency domain
using the Data Physics FFT Analyzer, an excitation
device mounted at the instrument’s bridge, and a
receiver that roves over a variety of locations on the
instrument’s surface. An equipment scheme is shown
in tigure (2).
The receiver is a mini accelerometer by PCB
Piezotronics, model 352B22.
PROCEDURE
2.1 set-up
The goal in setting up a guitar for one type of modal
test is a configuration free of variables that a player
might introduce to alter the instrument’s performance.
Strings are removed, and the guitar is suspended at
its tail pin and tuning pegs by shock cords that hang
from a relatively rigid frame. Ambient temperature
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Frequency response functions are recorded for each
receiver location using two channels of input to the
FFT Analyzer. The signal input to channel one is the
signal of approximately 0.35 Volts RMS measured
across a one ohm
resistor placed in the circuit
between the amplifier and excitation device.
The
signal input to channel two is the signal generated by
the receiver after Xl0 gain.
characterize input on two channels over a total of 8
seconds.
Input to both channels is 16 bit resolution with zero
pre-triggering and is multiplied by a Hanning window.
Frequency resolution is 984 lines over a range of 1250
Hz, or one line per 1.27 Hz. Sampling resolution is
approximately 5 kHz, or one sample every 0.2
milliseconds, over a time record length of 0.8 seconds.
For each receiver location or measurement, a
frequency response function and a coherence function
are computed from the three power spectra.
For each 0.8 second time record, the spectrum on
each channel is multiplied by its complex conjugate,
and the spectrum on one channel is multiplied by the
complex conjugate of the spectrum on the other. Ten
such products are averaged to produce auto power
spectra for each channel and a cross power spectrum
That is, each
from channel 1 to channel 2 [3].
that
meaS”reme”t consists of
three
spectra
2.3 Nodal Analysis
Modal Analysis is performed on the frequency
response functions using bands within a frequency
range of 80 to 1250 Hz.
Use of this range reflects
some practical considerations. First, our FFT Analyzer
is not equipped with zoom and provides a maximum of
1000 lines per spectrum, so an increase in frequency
range is an automatic decrease in spectral resolution.
Also, our excitation device exhibits mechanical
resonances outside this frequency range, one main
resonance at approximately 50 Hz and smaller
resonances above 2500 Hz.
A second consideration is the fact that most
fundamental frequencies introduced to a guitar as its
strings are plucked fall within the 80 to 1250 Hz
range. The lowest plucked note on an acoustic guitar
strung to standard Spanish tuning is approximately
82 Hz, the pitch of the open low E string, and the
highest plucked note is approximately 1047 Hz, the
pitch of the high E string barred at fret 20. While
excitation by a plucked string may also include
Prior to data collection, the excitation device and
receiver are checked as a unit for uneven response.
Frequency response and coherence functions are
computed from power spectra obtained while the
receiver is mounted on the back or magnet of the
excitation device.
The coherence. function always
displays a uniform value of one for frequencies above
60 Hz. The frequency response function is relatively
flat in amplitude for frequencies that range from
approximately 80 Hz through 2500 Hz.
harmonic frequencies beyond the 80 to 1250 Hz range,
those within this range would most often be largest in
amplitude, and modes within this range would be most
likely to figure in the instrument’s tone. The 80 to
1250 Hz range, while not a complete picture of an
acoustic guitar’s behavior, offers a logical starting
place for a study.
Modal analysis is implemented in several stages using
the STARModal
software package by SMS. First, a
cursory analysis is performed. Modal peaks functions
are computed for the entire set of frequency response
functions, bands are set to include frequencies that
display maximum amplitude in plots of the modal
peaks functions, and a “peak fit” operation is applied
to the data to obtain a residue value for each band for
each frequency response function. The goal of the first
stage is a quick display of modes. Damping values, tit
error, and closely spaced modes are ignored for sake of
brevity, and computation of the modal assurance
criterion (MAC) often shows relatively high values.
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During a second, more rigorous stage of analysis, the
data is divided into three groups: data obtained while
the receiver was located on the top plate of the guitar,
on the back plate of the guitar, and on the neck or
head. For each data group, cuwe fitting bands are
“set” to include frequencies that display maximum
amplitude in individual frequency response functions,
rather than maximum amplitude in a plot of a modal
The group’s frequency response
peaks function.
functions are “fit” for a frequency and damping value
using a global method that is based on a least-squares
approximation to a rational fraction polynomial [2].
Most of the bands are set to span one peak in each of
the measured frequency response functions, and the
data is approximated using a quadratic polynomial
that represents a single-degree-of-freedom system.
Four extra terms are usually included in the
numerator to compensate for modes outside the band
in question. Bands that span more than one peak in
most of the group’s frequency response functions are
tit using a polynomial for a system with multiple
degrees of freedom with four extra terms supplied in
the numerator. After frequency and damping values
have been determined, residue values for each band
are obtained from each frequency response function by
T
/
805
again applying a global method [l]. The second stage
yields more modes than the first and MAC values that
are below 0.1. Most error values are below one on a
scale from zero to one hundred.
For a third analysis stage, the bands obtained during
stage two are used to fit the entire data set, rather
than one data group. That is, global methods are
applied to all frequency response functions to obtain
first frequency and damping values, and second,
residue values. Like stage two, stage three yields MAC
values that are below 0.1. Error values for stage three
are higher than those obtained during stage hvo,
however.
3. EXAMPLE
As an example, nearly three hundred frequency
response functions have been collected for a single
guitar.
Each of the frequency response functions
plotted below in figures (31, (4), and (5) contains data
from a different group.
4. RESULTS
The plots provide an idea of the number of modes
visible within the frequency range of testing and the
degree to which the modes are coupled. The vertical
line cursors highlight a frequency band that
contains two closely spaced modes.
The procedure outlined in section 2.3 above results
in a large variety of modes and natural frequencies.
Figures (6) through (9) below depict some of the
modes obtained for the sample guitar. The modes
are not displayed in order of increasing frequency.
fqure
(6): mode a
606
figure (9,: mode d
5.
DISCUSSION
is list of bands that contain the number of modes
most appropriate to the entire body of data.
Because our frequency response functions exhibit
both local modes and modes that are heavily
coupled, the division of data into the three groups
proves essential during analysis.
Within the
frequency range of a local mode, magnitude plots of
data from two of the groups may contain no peak,
and the mode may be overlooked until the third is
analyzed.
In general, the frequency and damping values
obtained for a mode do not change as data files are
fit. When most single files are Iit for frequency and
damping values, the results match those obtained
when an entire group is fit, or those obtained when
all of the data is tit.
Occasionally, an exception
arises when a single data tile displays a “saddle” or a
rounded peak rather than a sharp peak within a
particular band. In such a case, the frequency and
damping values obtained by fitting the single data
file may differ by a few Hertz from the values
obtained by fitting the entire data set.
In the case of a frequency range that includes
several heavily coupled modes, data within one
group may present one peak between cursors,
indicating that a fit to a single mode is appropriate.
Although such a fit might register low error between
the data and its polynomial approximation, the iit
would lead to significant error in parameter
estimation. Frequency and damping values would
differ as various data files were fit. When analysis is
performed separately for each data group, the three
sets of curye fit bands, the number of modes they
contain, and the amount of error resulting between
the data and each polynomial approximation can be
compared
and
then
used to
produce a
comprehensive set of curve fitting bands. The result
Our results usually include on the order of thirty
lightly damped but well-coupled modes.
6.
CONCLUSIONS
Our test method has proven to be effective in
identifying natural frequencies and normal mode
shapes of an acoustic guitar. Some aspects could be
improved, however.
In particular, our excitation
device limits our study to the 80 to 1250 Hz range.
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8. REFERENCES
A device that has a flat response over a wider range
and that, like our current excitation device,
contributes little mas.s to a guitar’s top plate while
adhering to the bridge through friction, would be to
our advantage. Suggestions are welcome.
[l] Richardson, Mark H., Global Frequencv
and
Dampinp. Estimates from Fresuencv Res!~onse
Measurements, SMS Technical Note 85-7,
Structural
Measurement
Systems,
Incorporated, 645 River Oaks Parkway, San
Jose, California 95134, 1985, p. 3.
7. ACKNOWLEDGMENTS
The author would like to acknowledge Response
Dynamics of Oakland, California for guidance and
support in the development of the test method
Special thanks are to
described in this paper.
Reuben Hale, Bill Nelson, Bob Rost, and Randy
Allemang for answers to technical questions, and to
Carl Goy and Marie Baudot for assistance in editing
the final draft.
[Z] Richardson, Mark H. and David L. Formenti,
from
FR!qUE2lCV
Parameter
Estimation
RWIOnSe
Measurements
Using R a t i o n a l
Fraction Polvnomials,
SMS Technical Note 85.
Structural
Measurement
Systems,
3,
Incorporated, 645 River Oaks Parkway, San
Jose, California 95134, 1985, p. 4.
[31
80%
~, DP420 Multi-Channel FFT Analvzer
Operatine Manual, Data Physics Corporation,
1190 S. Bascom Ave. Suite 220, San Jose,
California 95128, 1994, p. 4.18.