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Application Notes
Measuring the Non-rigid Behaviour
of a Loudspeaker Diaphragm
using Modal Analysis
Brüel & Kjær
Measuring the Non-rigid Behaviour of a Loudspeaker Diaphragm
using Modal Analysis
by Christopher J. Struck, Brüel & Kjær
Introduction
When a loudspeaker vibrates at low
frequencies, it behaves as a rigid piston, with all parts of the diaphragm
moving in phase. Above a certain frequency, however, the diaphragm behaviour will become more complex. To
examine this phenomenon, it is necessary to make reliable measurements of
the diaphragm motion at a number of
different points.
Until recently, accurate vibration
measurements on light and delicate
structures have been troublesome, if
not impossible, to make. The reason
for this was that traditional methods
required the mounting of a transducer
on the test object. This problem
can now be solved by using the
Brüel & Kjær Laser Velocity-Transducer Set Type 3544, which avoids the
mechanical contact with the test object, and together with a Brüel & Kjær
Dual Channel Signal Analyzer Type
2032 the measurement data is easily
collected and analyzed. By applying
the techniques of Modal Analysis to
the set of measurements, a
modal
model can be derived, describing the
dynamic properties of the structure.
sented in a way which is easy to understand; for example, a single mode
shape can be animated on the computer screen.
The following assumptions are
made about the physical structure:
Modal Analysis
Modal Analysis is the process of characterizing the dynamic properties of
an elastic structure in terms of its
modes of vibration. A mode of vibramodal fretion is defined by its
quency, modal damping and mode
shape. In theory, any deflection of the
structure can be constructed as a linear combination of mode shapes.
Loosely stated, the mode shapes represent the fundamental vibration patterns of the structure.
Modal analysis is an experimental
method. Based on experience and expectations of the measurement results, a careful choice of measurement
points is made. From the set of Frequency Response Function (FRF)
measurements, the modal parameters
are extracted, giving a simple mathematical model of the structure. The
reduction of the measured data to
modal parameters is done using matrix algebra and curve-fitting algorithms, a task most suitable for a computer. Eventually, the information
from the measurements can be
pre-
With mechanical structures, the excitation is usually force applied with
an impact hammer or vibration exciter
(shaker). This is monitored with a
force transducer. The response is usually acceleration, which can be measured with an accelerometer. Either
excitation or response is fixed to a single location while moving the other
from point to point to obtain the
FRFs.
2
1. The structural motion can be adequately described by a set of linear
second order differential equations.
2. The single modes can be separated.
For each modal frequency there is
only one associated mode shape.
3. The modal frequency and damping
of each mode do not vary significantly across the structure.
4. Reciprocity between excitation and
measurement points ensures that
the transfer and system matrices
are symmetric.
Using these assumptions, it is possible to describe the FRF matrix exclusively in terms of the modal parameers[‘]~[*]. It is not the intention to give
a detailed explanation of the theory of
modal analysis here. Many excellent
subreferences are available on the
j&111v131
Measurement and analysis
techniques
Before using modal analysis to examine the behaviour of a loudspeaker
diaphragm, some practical problems
had to be considered. The first was
mass loading due to the transducer.
Several methods have been investigated including ultra low-mass accelerometers and a non-contacting probe
microphone. The effects of mass loading cannot be ignored unless the dynamic mass of the structure under in-
vestigation is several orders of magnitude greater than the transducer. For
most drivers of polyethylene or paper
construction, this mass is sufficiently
low that even with the lightest accelerometer available (0,65g), mass-loading
of the diaphragm still occurred. Although it is possible to correct for the
effects of mass loading at a single
point by using postprocessing, it is not
possible to make the correction for a
fixed excitation and a moving, mass
loading transducer. Mounting such a
transducer is also a problem, if the
test is to be non-destructive. In the
case of a loudspeaker, the excitation is
intrinsically fixed (at the electrical input terminals, exciting the loudspeaker at the center of the diaphragm), so
the transducer must move from point
to point in order to collect the data.
The effects of mass loading include a
downward shift in the resonant frequency and changes in the damping.
The probe microphone technique
also had limitations. Noise at the measured system output made it very difficult to obtain reliable data without
extremely long averaging times. Another difficulty was the limited useful
frequency range of the microphone itself. In performing such analyzes, it is
assumed that the transducers used do
not contribute to, or alter the response
of the system under investigation. Use
of the microphone also restricted the
analysis to only “relative” measurements, meaning that the transducer
output voltage could not be calibrated
in terms of some physically measured
parameter. The importance of this
limitation will be shown.
These problems were alleviated by
the use of a non-contacting laser velocity-transducer.
Test configuration
The test configuration is shown in
Fig. 1. The driver was mounted in
a baffle with the laser positioned approximately 50cm away. The measurements were made in the direction
parallel to the main axis of motion
(perpendicular to the baffle).
Response measurement
A Brüel & Kjær Dual Channel Signal
Analyzer Type 2032 having 801
fre-
HP 300 Series
Computer
Dual Channel
Slgnal Analyzer
2032
Loudspeaker
Mounted in
Baffle
Ch. A
Ch. B
Generator
Slgnal
I
Laser VelocityTransducer
8323
Power Supply
2815
Pulse Generator / Amplifier
WB 0875
w/O.1 ohm resistor
Software
Fig. 1. Test configuration
quency lines was used for the analysis.
The response was measured in channel B using a Brüel & Kjær Laser Velocity-Transducer Set Type 3544. The
excitation was simultaneously measured in channel A. The analyzer can
compute the frequency response from
these measurements. There are several steps involved in performing a complete modal analysis, from establishing a geometry to producing an animation. A procedural diagram for a
complete general modal test is shown
in Fig. 2.
When measuring a vibrating object
using the laser, a small piece of retroreflective tape is mounted on the test
object to return the source light from
the target. The laser beam is split inside the device, one beam acting as the
reference while the other is aimed at
the target. The reference is Doppler
shifted by a rotating disc and mixed
with light returned from the target. A
Doppler frequency shift also occurs in
the target beam due to the vibration
of the target. The two beams are sent
to a photodetector and heterodyned.
A detector converts this into a calibrated voltage proportional to the target velocity[41.
The response channel was calibrated by using a portable hand-held vibration calibrator, a Brüel & Kjær Calibration Exciter Type 4294. The output of the calibrator is a vibration
signal of lOmm/s at a frequency of
159,2Hz. The output of the laser
(nominally 1V/m/s) is fed to the ana-
lyzer, and the measured level is entered via the front panel as the calibration factor for channel B.
Excitation
The excitation used for the test was an
electrical signal applied at the driver
input terminals. Because loudspeaker
drivers can exhibit non-linear behaviour, random noise was used to obtain
the best approximation to the linear
response of the system. For Modal
Analysis as well as for FFT Analysis,
the structure is assumed to behave in
a linear manner or to be restricted to
its linear range of operation. A noise
excitation can also be band limited so
that the system is only excited in the
frequency range of interest, optimizing the dynamic range of the measurement instrumentation[5]. The driver
chosen for this experiment was a
Philips 10 inch woofer with a resonant
frequency of 34,5 Hz.
For test purposes, the driver was
mounted in a 1 m baffle. By comparing
a measurement of the driver’s near
field response with a measurement
made using an accelerometer mounted
on the diaphragm (see Fig.
3), it is
possible to find the frequency above
which the driver no longer behaves
like a rigid piston. The mass loading
caused by the accelerometer is not
critical in this case, as the measurement is for simple comparison purposes. The two measurements should
have the same high-pass characteristic
up to this frequency. Below this fre-
Select Measurement Points
on Structure
Make Measurements
1Calculate,~~;;es from
Fig. 2. Procedure for a general
1
modal test
quency, the near field response is proportional to accelerationL6]. This fre3
To Channel A
I(f)=10(Vg--V,(f))
Fig. 3. In the piston range of operation, the near field response is
proportional to acceleration. A comparison of the two clearly reveals
when the diaphragm no longer behaves as a rigid structure
quency could also be calculated by
finding:
k.a<l
(1)
where k is the Wave Number (~T/x)
and a is the radius of the driver. For
the driver under test, this yields
f < 430Hz
(2)
Several preliminary measurements
were also made to determine the highest frequency at which modal behaviour could be observed with adequate
resolution. Based on this, the frequency span for analysis was chosen to be
1,6kHz from 512Hz to 2112Hz.
In order to calibrate the excitation
channel, a measurement of the Thiele Small parameters of the driver was
performed. This measurement yields
* coil
the B l (magnetic flux density
length) product 17]. By monitoring the
current applied to the driver using a
small (0,l Q) series resistor (Fig. 4), the
excitation can be calibrated directly in
force units (N) where
F(f) = Bl . i(f)
= Bl . 0,l . v(f)
GND
Fig. 4. The 0,l Q resistor is used to measure the current through the
loudspeaker. This allows the excitation channel to be calibrated in
terms of force, since the force is proportional to the current
software (SMS MODAL 3.0 SE), a
geometric “wire” model resembling the
test object can be constructed using a
convenient coordinate system. For our
test object, the driver, a cylindrical coordinate system was chosen. The coordinates for our test were taken from a
driver blueprint in order to represent
the test object in a reasonable way.
The geometry does not necessarily
have to match the test object, but in
most cases it does, at least approximately. The center point in our model
represented the point of excitation or
“driving point”. The outermost diameter did not contain any measurement
data and is used to represent the driver basket and provide a plane of reference for the motion.
the form of an analytic function. This
means that the values for each FRF
throughout the s-plane can be found
from the known values along the frequency axis. The unknown modal parameters are identified by “curve-fitting” the measurements and synthetransfer
sizing
the
functions
throughout the s-plane. This provides
a mathematical definition of each
transfer function based on the poles
and residues determined for each of
the resonances in the measured
datal’]. From the curve-fit data, a
“modal model”, describing the response of the structure under investigation, can be developed.
By using a Hewlett Packard 300 Series computer and special application
Undeformed Structure
80
(3)
(4)
This calibration factor can be entered
directly from the front panel of the
analyzer in V/N.
Data handling
The amount of data required for an
accurate modal analysis of even a simple structure is large enough to justify
the need for a computer. A total of 73
FRFs were measured at points every
30 degrees at six different radii on the
driver (see Fig. 5). These FRFs are in
4
Fig. 5. Measurements were performed at 73 diff erent points around the diaphragm. No measurements were made at the points numbered from 74 to 85. These points represent the driver
frame and are used as a reference for the displacement
damped. In this case, location of the
modes is somewhat more difficult.
A typical FRF is shown in Fig. 6.
The real part of the frequency response and the measurement coherence are displayed. Coherence is a
measure of how well the output of a
system is linearly related to the input”]. As the measured output parameter was velocity, the real part of the
response (as opposed to the imaginary
or magnitude, which are also available)
is chosen for viewing at this point in
the analysis. The real part of the FRF
is proportional to displacement (if the
measured output parameter is velocity) providing some idea of the relative
amplitude of each mode.
The software also facilitates data
acquisition and storage. The software
can then take the information from
the modal model and use this to animate the geometry, giving a visual representation of the structure and its
modal behaviour. Afterwards, the processed data can be stored on disk and
used with other tools such as Structural Dynamic Modifications (SDM) and
Forced Dynamic Response (FDR).
The use of these tools will be described
later.
The measurements
Measurements were performed at 73
points including a driving point measurement at the center of the dust-cap.
This represented a reasonable compromise between required detail and
measurement time. The dust-cap was
oversized, extending approximately
1,5cm beyond the diameter of the
voice coil. This unfortunately made it
impossible to make any investigation
on the diaphragm near the voice coil
without destroying the unit. A small
retro-reflective
piece of
tape
(- 0,25cm2) was moved to each position. This had no mass loading effects.
Small positional adjustments of the laser were necessary at each position to
improve the signal-to-noise ratio of
each measurement.
Sharp, well-defined peaks appear in
the magnitude response of most mechanical structures due to low damping. For these structures, identification
of the modes is relatively simple. Besides the previously discussed problems of low mass (the total effective
moving mass of the test unit was
34,3g), a loudspeaker driver is typically designed to have its modes heavily
-i.oJ
Postprocessing of measurement data
To a large degree, the quality and accuracy of the animated shapes obtained is dependent upon the curvefit. Within the software, there are several possible types of curve-fitting
routines. A choice must be made between Single Degree Of Freedom
(SDOF) or Multiple Degree Of Freedom (MDOF) fitting, dependent upon
the structure under test and amount
of coupling between the modes. A
choice is then made between Polynomial or Circle (on a Nyquist plot) fitting. For more complex or heavily coupled MDOF systems, a Complex Exponential and a Least Squares fit
(operating on the Impulse Response)
are available. By viewing several typical measurements or an average of a
number of measurements, different
ways of curve-fitting the data, can be
attempted. The procedure is to position the cursors around one or several
resonances and select a curve-fit routine. Once a reasonable fit has been
obtained for all of the modes of interest in the sample data, an “Autofit”
routine can be engaged to apply this
curve-fit to the entire data set. This
can take a few minutes for the computer to process, depending on the
amount of data. An example of the
curve-fit for this driver is shown in
Fig. 7.
In the measured frequency range,
which extended well beyond the intended usable frequency range for this
unit, it was possible to identify seven
modes above the rigid piston region. A
list of the modal frequencies and
damping appears in Table 1. Still
views of the animated mode shapes,
including the undeformed structure,
are shown in Fig. 8 and Fig. 9.
The resulting mode shapes are
viewed in terms of displacement, regardless of the units of the measured
response. During the computer animation, it is possible to vary the speed
and amplitude of the motion as well as
the viewing angle. It can be seen that
Modes l-3 correspond well to regular
plate or stretched membrane modes, in
spite of the cone construction of the
driver. As a first approximation, these
are the shapes one would expect to
observe.
The modal frequencies for Modes
1,3,4, and 5 show good correlation to
the theoretical frequency ratios for the
fil, fzl, fozt and fo3 modes (see Table 2).
In this case, f,, is approximately
480 Hz, just beyond the theoretical rigid piston limit 1’1. The behaviour in
I
o&t
O.Bk
I
.m
1.2k
t .4k
I.&
1 .Bk
2.m
Fig. 6. A typical FRF measurement. For measurement point 24, the
real part of the FRF and the coherence are shown
Fig. 7. An example of a curve-fit to a particular mode in a single
FRF
5
Mode:
Freq :
1
Mode:
Freq :
3
783.21
Hz
Damp:
6.42
%
981.25
Hz
Damp:
2.24
%
r
Mode.:
Freq :
2
Mode:
Freq :
4
909.02
Hz
Damp
:
3.31
%
1111.46
Hz
Damp:
1.93
%
L
x
L
27
‘6
Fig. 8. Plots of the mode shapes 1, 2, 3, and 4. For each mode shape, the modal frequency and damping are listed above the structure
modes 4 and 6 is more irregular and is
not immediately obvious. Plate behaviour is again seen in Modes 5 and 7,
although the deformation near the
dust cap in mode 7 is somewhat more
severe than one would expect.
Implementation of SDM and FRS
During the design process, it is often
valuable to ask “What if . ..” questions.
The use of Structural Dynamic Modifications (SDM) and Forced Response
Simulation (FRS) allows the engineer
to do just that. For example, “What if
a mass, stiffener or tuned absorber
Mode No.
Freq.
1
703 Hz
909 Hz
981 Hz
1111 Hz
1744Hz
1839 Hz
1975 Hz
2
3
4
5
6
7
Damping
6,42 %
3.31 %
2,24 %
1,93 %
3,19%
4,42 %
3.23 %
used be obtained from calibrated inertial measurements of the force input
and the corresponding acceleration,
velocity, or displacement response[lO1.
This is why the calibration of the two
channels is so important.
An example of a mass modification
is shown in Fig. 10. This could be used
to simulate the effect of a progressive
or non-homogeneous cone material, for
example, one that is more massive toward the voice coil than at the surround. The applied mass may be either positive or negative, at one or several points. A negative mass might be
were added between two or more
points on the structure? What would
be the dynamic response of the modified structure?” It is then possible to
predict analytically the structural response to real life excitations. In addition, it is possible to compute the resulting deformation at any single frequency due to static or dynamic
loading. These simulations can then be
displayed, plotted or stored. The only
restrictions are that the mode shapes
must be previously specified from an
analysis at all points where modifications are to be made, and that the data
Measured
Mode
Measured
Freq.
783 Hz
909 Hz
981 Hz
1111 Hz
1744Hz
1839 Hz
1975 Hz
Theoretical
Mode
Theoretical
Freq.
Ratio
f 11
763 Hz
1,59
f 21
fez
f 03
-
1027 Hz
1104Hz
1728 Hz
-
2.14
2,30
3960
-
‘Ito
TO2C?LVGBO
Tablel. Modal frequencies and dampzngs
for the first 7 modes.
6
Table2. Comparison between the measured modal frequencies and the theoretical
quencies for a stretched membrane.
eigenfre-
Mode: 5
Freq :
1744.62 Hz
Damp
:
3.19
%
Mode: 7
Freq :
1975.89 Hz
Damp:
3.23
%
Mode: 6
Frea :
1839.94 Hz
Damp:
4.42
%
I-
Undeformed Structure
Fig. 9. Plots of the mode shapes 5, 6, 7 and the undeformed diaphragm.
used to compensate for a manufacturing defect where too much cone material has gathered at some point. Several different modifications could be
combined.
The use of a stiffness modification
could be used, for example, to simulate the effects of a different surround
material. This is shown in Fig. 11. AfMode:
Freq :
ter the application of the desired modifications, the response of the modal
model at any frequency can be simulated (with or without the modifications) using FRS.
The presence of modal behaviour
has a definite effect on the driver’s
acoustic performance. In general, the
primary concern in driver design is to
4
1066.63 Hz Damp:
2.06
%
t
Fig. 10. Example of an added mass simulation (5Og at point 38).
Note the downward shift in modal frequency
Hodmr
Freqr
Mode: 3
Frsq:
extend the region of rigid piston operation as high as possible with a minimum of compromise to other design
factors. To maximize the overall usable frequency range of the driver, it is
important that beyond the piston
mode, the next few modes do not cause
the acoustic efficiency to significantly
deteriorate. Excessive amounts of
4
1400.20 HI Damp:
1111.17 Hz
Damp
:
2.83
Y
2.56
%
Fig. 11. Example of a stiffness modification near the surround to
simulate a change in material. Note the upward shift in modal frequencies
7
system. These modifications can be
used to simulate and solve actual design problems.
H : 32nz ru 3.2H-k
3ETW 2
‘A: 15
References
PI 0. Døssing: “Structural Testing Part 2: Modal Analysis and Simulation”, Brüel & K j æ r March 1988
PI “MODAL 3.0 SE
- Operating
Manual”, Structural Measurement Systems, Inc., San Jose, CA,
1987
Fig. 12. A measurement showing the free field response of the driver. Structural modes contribute to the uariations in the response beyond the rigid piston range of operation
counterphase motion, where large portions of the diaphragm are out of
phase with the excitation, will produce
acoustic cancellations in the near field.
This in turn results in dips in the free
field frequency response of the driver
(see Fig. 12) and an overall lowering of
its efficiency.
Applications
It is possible to perform such a modal
analysis on a known, well behaved
driver and use this as a reference. By
performing the analysis on similar
units or perhaps slightly modified prototypes for new models, defects in construction or manufacture can quickly
be identified. In this way, improvements in design can also be studied in
detail.
If a detailed analysis of a particular
problem is required, an exact geometric model of the device under investigation can be entered. The analysis
can then be restricted to the specific
location and frequency range of interest.
Modal analysis can also be used to
verify the accuracy of analytical models constructed using Finite Element
Methods. An example of such a model
131
D. J. Ewins: “Modal Testing: Theory and Practice”, Research Studies Press, Ltd., Herts, England,
1986
141
M. Serridge: “The Laser Velocity
Transducer - Its Principles and
Applications”, Brüel & Kjær Application Note, September 1988
[51
H. Herlufsen: “Dual Channel FFT
&
2)“,
Analysis (Parts 1
Brüel & Kjær Technical Review,
Nos. 1 and 2, 1984
and its calculation is given by Kaizer
and Leeuwesteinl’].
Conclusions
Modal analysis is a powerful tool for
the analysis of the dynamic behaviour
of structures. Although normally applied to the study of relatively massive
and lightly damped mechanical structures, it can easily accommodate the
analysis of a highly damped, low-mass
device such as a loudspeaker driver.
This is made possible by the use of a
non-contacting laser velocity-transducer.
In this Application Note, a complete
step by step analysis for a typical
loudspeaker driver was carried out.
This unit was shown to exhibit modal
behaviour similar to the familiar plate
modes. The use of the laser transducer
allows calibrated response measurements to be performed. A systematic
method for calibrating the input excitation in terms of force has also been
shown. The calibration of the measurements is clearly important, as it
makes it possible to apply structural
dynamic modifications and observe
the forced dynamic response of the
“Low-Frequency
Assessment by
Loudspeaker
Nearfield Sound-Pressure MeaSoc.,
surements”, J. Audio Eng.
Vol. 22, No. 4, April 1974
WI D. B. Keele, Jr.:
[71
C. J. Struck: “Determination Of
The Thiele-Small Parameters Using Two-Channel FFT Analysis”,
presented at the AES 82nd Convention, London, l0-13 March,
1987
PI H. F. Olson: “Music, Physics and
Engineering”, Dover Publications,
Inc., New York, 1967
PI A. J. M. Kaizer
and A. Leeuwestein: “Calculation of the Sound
Radiation of a Nonrigid Loudspeaker Diaphragm Using the Finite-Element Method”, J. Audio
Eng. Soc., Vol. 36, No.
7/8 July/August 1988
WI “SDM / FRS 3.0 SE - Operating
Manual”, Structural Measurement Systems, Inc. San Jose, CA,
1987
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