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Measurement of transmission loss using vibration transducers
j? Sound
vib. (1967) 6 (3), 419-423
MEASUREMENT
OF TRANSMISSION
VIBRATION
LOSS USING
TRANSDUCERS
W. A. UTLEY AND Ii. A. MULHOLLAND Department
of Building Science, The University Liverpool 3, England (Received
23 May
of Liverpool,
1967)
An alternative method of measuring sound transmission loss using accelerometers is described. The discrepancies between the new method and the standard method are discussed and it is shown that by using suitable corrections to the new method good agreement can be obtained. Finally, it is shown that the values of vibration amplitude derived from the mass law do not agree with those found experimentally.
I. INTRODUCTION
There are situations in which it is not possible to use the standard method of transmissionloss measurements as laid down in B.S. 2750: for example, when measurements of transmission loss are required in the presence of unavoidable air flanking paths. In these circumstances it would be useful to be able to measure the transmission loss by a method that does not require a quiet reverberant reception room. Such a method has been described briefly by Ward [I]. In this method the receptionroom microphone is replaced by an accelerometer which measures the actual vibration of the panel transmitting the sound energy. The transmission loss of the panel is then deduced from the sound energy level as measured by the transmission-room microphone and the panel vibration amplitude as measured by a transducer. The critical point in this method is the theory used to relate the free field sound pressure level reading of the microphone to the panel vibration amplitude as measured by the accelerometer in a way that yields directly the transmission loss. The simplest assumption is to take the sound energy falling on unit area per set as being equal to the sound energy recorded by the microphone. Sound energy falling on unit area per set is Ef=E,antilog(L,/Io), where L, is the sound level observed and E,, is the reference energy level. The amount of energy transmitted by the panel is then taken as being ET= ippcU& where UO is the peak velocity amplitude of the panel from the accelerometer reading [2]. The transmission coefficient T is found from Efr = $pc .?IJzand thus QCU; T =
E, antilog (LT)
*
The agreement found between the two methods using the above expressions is found to be quite fair. Both Ward’s experiments and the ones carried out at Liverpool (see Figure I) show moderate agreement. There are, however, two main discrepancies. At high frequencies (above IOOO Hz) the accelerometer curve lies above the B.S. 2750 curve and at low frequencies (below about 400 Hz) the opposite effect occurs. Because of this it was decided to examine the accelerometer method more closely with 419
W. A. UTLEY AND K. A. MULHOLLAND
420
a view to eliminating the discrepancies. A final section is added in which it is shown that using the mass law theory to predict the vibration amplitude of panels gives results that are not in agreement with measured values of vibration amplitude.
z. THE HIGH-FREQUENCY
DISCREPANCY
In the method used to measure the vibration amplitude of the panel a crystal accelerometer was fixed to the panel and the velocity amplitude of the panel caused a fluctuating voltage to be produced by the transducer. As the panels used were light in weight, being of the order of 0.5 g/cm2, and the mass of the transducer was 30 g, it was thought that the mass of the transducer would be loading the panel and reducing the amplitude of vibration at the point of measurement. To overcome this difficulty it was decided to use two transducers of different mass, from which it was hoped to deduce the velocity amplitude of the unloaded panel in the following way. If the surface mass of the unloaded panel is m. and if the masses of the two transducers are ml and m2 then the two measured velocities of the panels using these transducers would be k1 and k2 given by
F = jw(mo+m~)~~, F =jw(mo+m,)~2, where F is the force on the panel due to the sound field and the amplitude of the unloaded panel, 20, would be given by F=jwmo3io. The above expressions are based on the assumption of a constant pressure on the panel whether it is loaded or not, which seems reasonable in view of the large difference in impedance between the panel and the air.
01 100
400
200
000 Frequency
Figure
I.
1600
3150
(Hz)
Lead panel.
From these three equations k. may be found in terms of the measurable ml, m2, kl and c+~: ji’o
=
quantities
(m-m2)4*2 (ml 3il - m2 3i2)*
When measurements were made on the lead panel (see Figure I) it was noticed that the heavier accelerometer predicted a higher transmission loss than the lighter accelerometer at high frequencies. This is in agreement with the idea of the panel being “ loaded ”
TRANSMISSION
LOSS FROM
ACCELEROMETERS
421
by the accelerometers. When the predicted vibration amplitude of the unloaded panel, &, was used the agreement between this curve and the B.S. 2750 curve was much improved at the high frequencies. Figure z shows the transmission-loss curves for a lighter panel (aluminium). Agreement between the two methods of measurement is very good at high frequencies. 3.
LOW-FREQUENCY
DISCREPANCY
It will be seen from Figures I and 2 that the transmission-loss curves obtained using the accelerometer lie below the B.S. 2750 curve at low frequencies. In order to try to explain this discrepancy the effect of the panel’s radiation efficiency was considered.
I
I
I
200
400
800 Frequency
I
1600
I
3150
(Hz)
Figure 2. o*o35-in. aluminiumpanel.
Any practical wall has dimensions that are of the order of several meters so that, considered as a whole, the wall will have a radiation efficiency equal to unity for all frequencies greater than about 20 Hz. However, if the waves in passing through the panel have a coherent area much less than the size of the panel then the radiation efficiency involved in the receipt and re-radiation of the sound energy will be small, thus causing a higher vibration amplitude and a smaller predicted transmission loss. If it is assumed that the sound incident on the panel is only self-coherent over a radius a, then the impedance seen by such an area is not pc but pc(li +jX), where R=(r_!.?k$))
and
X=q_
Here J1 is a Bessel function and HI is a Struve function. The intensity produced by a vibration of the panel of amplitude U, will thus be &Re(pv*)
= +RpcU$
This is equal to the simple expression Qc Ui used previously, multiplied by the radiation efficiency term R. Since the term R 2 I the predicted transmission loss would be greater than that deduced from the simple expression which would give a better fit with the B.S. 2750 curve. If the radius of coherence, a, is invariant with frequency then the increase in predicted transmission loss will only occur at low frequencies since for large values of ka the value of R is unity.
W. A. UTLEY AND K. A. MULHOLLAND
422
In order to check the validity of this radiation efficiency idea it would be necessary to obtain values of correlation between different points on the panel. Unfortunately this information is not at present available.
4. MASS
LAW
AND TRANSMISSION
LOSS
It has been shown that it is possible to measure the transmission loss of a panel by using suitable transducers to determine its velocity amplitude. A simple ad hoc formula, I=&pcU& is used together with the two corrections mentioned above. The random incidence mass law may be used to predict the vibration amplitude of the panel as follows. If a wave of potential $I is incident on a panel and as a result waves of potential 4R and & are transmitted by the panel then we have from consideration of continuity and impedance +I-+R P(41f$J?-&)
= +T,
(I)
= jw?%-case.
(2)
incidence) and jwm represents the impedance velocity amplitude of the panel, Ua, is
13is the angle of incidence (0 = o for normal of the panel. We also note that the normal
Ua = jk+, cos 0
(3)
but since & = &/( I + jwm cos ejzpc)
(4)
we find U0 = jkq5, cos or, since jkq51= V, (V, being the velocity
e/(
I + jwm cos ejzpc),
amplitude
(5)
of the incident
wave),
Ua = V, cos e/( I + jwrn cos e/2Pc).
(6)
In a diffuse field a large number of waves are incident on the panel and we shall assume that N waves are incident from one steradian. To find the net velocity amplitude of the panel in a diffuse field we must sum the squares of the individual velocity amplitudes: 82
s
V$N~~~~esinedB 2?T o [r+( wm cos e/2pcy]
G=
The quantity V:N cannot be directly waves are plane waves then we find
measured
(7)
’
but if we assume
that the individual
PI = @VI* Thus the net pressure over a free field condenser the squares of these pressures : p p
is the measured
quantity
= 4rrNp:
corresponding
(8)
microphone
can be found
= ~T&c)~NV+. to an observed
by summing
(9) sound
level, L,
L = 1olog(~/p$),
(10)
= pi antilog (L/10)
(11)
= 3/47~(pc)~ = pg antilog (L/Io)/~+x)~.
(12)
whence T and so
NV;
TRANSMISSION
We ought,
therefore,
to measure
423
LOSS FROM ACCELEROMETERS
a vibration
amplitude er
u2
_
cos* %sin %d%
antilog (L/IO)
25-p;
0
oI [I+( W?lzcos %/2/x)‘]
457(f#
*
(13)
Measurements of L and lJ,2 were made for an aluminium panel of mass 8.6 kg/m2. The results are shown in Figure 3. It can be seen from the figure that measured values of panel
0 100
I
200
I
I
400
800 Frequency
I
1600 (Hz
y-Y_] 3150
I
Figure 3. i-in. aluminium panel.
vibration amplitude are very much higher than the values predicted by the mass law theory. We have at present no explanation for the unsatisfactory state of affairs. It is possible that the radiation efficiency idea discussed earlier may have a bearing on the problem. 5. CONCLUSION
,4n alternative method of transmission-loss measurement to that used in B.S. 2750 has been described. The new method has the advantage that it may be used to determine transmission loss in the presence of flanking paths. Agreement between measurements using the new method and the reverberant room method is good provided that (i) a suitable correction is used for the loading of the panel by the transducers at high frequencies, and (ii) a correction is applied at low frequencies for the finite coherence area of the waves on the panel. Finally, it was shown that the vibration amplitude predicted by the mass law is considerably lower than that actually measured. REFERENCES I. F. WARD 1963 BBC 2.
Report No. B-078. The accelerometer pick-up as a diagnostic tool in noise studies in buildings. P. M. MORSE 1936 Vibration and Sound. Second edition 1948, page 338. New York: McGrawHill Book Company.
vib. (1967) 6 (3), 419-423
MEASUREMENT
OF TRANSMISSION
VIBRATION
LOSS USING
TRANSDUCERS
W. A. UTLEY AND Ii. A. MULHOLLAND Department
of Building Science, The University Liverpool 3, England (Received
23 May
of Liverpool,
1967)
An alternative method of measuring sound transmission loss using accelerometers is described. The discrepancies between the new method and the standard method are discussed and it is shown that by using suitable corrections to the new method good agreement can be obtained. Finally, it is shown that the values of vibration amplitude derived from the mass law do not agree with those found experimentally.
I. INTRODUCTION
There are situations in which it is not possible to use the standard method of transmissionloss measurements as laid down in B.S. 2750: for example, when measurements of transmission loss are required in the presence of unavoidable air flanking paths. In these circumstances it would be useful to be able to measure the transmission loss by a method that does not require a quiet reverberant reception room. Such a method has been described briefly by Ward [I]. In this method the receptionroom microphone is replaced by an accelerometer which measures the actual vibration of the panel transmitting the sound energy. The transmission loss of the panel is then deduced from the sound energy level as measured by the transmission-room microphone and the panel vibration amplitude as measured by a transducer. The critical point in this method is the theory used to relate the free field sound pressure level reading of the microphone to the panel vibration amplitude as measured by the accelerometer in a way that yields directly the transmission loss. The simplest assumption is to take the sound energy falling on unit area per set as being equal to the sound energy recorded by the microphone. Sound energy falling on unit area per set is Ef=E,antilog(L,/Io), where L, is the sound level observed and E,, is the reference energy level. The amount of energy transmitted by the panel is then taken as being ET= ippcU& where UO is the peak velocity amplitude of the panel from the accelerometer reading [2]. The transmission coefficient T is found from Efr = $pc .?IJzand thus QCU; T =
E, antilog (LT)
*
The agreement found between the two methods using the above expressions is found to be quite fair. Both Ward’s experiments and the ones carried out at Liverpool (see Figure I) show moderate agreement. There are, however, two main discrepancies. At high frequencies (above IOOO Hz) the accelerometer curve lies above the B.S. 2750 curve and at low frequencies (below about 400 Hz) the opposite effect occurs. Because of this it was decided to examine the accelerometer method more closely with 419
W. A. UTLEY AND K. A. MULHOLLAND
420
a view to eliminating the discrepancies. A final section is added in which it is shown that using the mass law theory to predict the vibration amplitude of panels gives results that are not in agreement with measured values of vibration amplitude.
z. THE HIGH-FREQUENCY
DISCREPANCY
In the method used to measure the vibration amplitude of the panel a crystal accelerometer was fixed to the panel and the velocity amplitude of the panel caused a fluctuating voltage to be produced by the transducer. As the panels used were light in weight, being of the order of 0.5 g/cm2, and the mass of the transducer was 30 g, it was thought that the mass of the transducer would be loading the panel and reducing the amplitude of vibration at the point of measurement. To overcome this difficulty it was decided to use two transducers of different mass, from which it was hoped to deduce the velocity amplitude of the unloaded panel in the following way. If the surface mass of the unloaded panel is m. and if the masses of the two transducers are ml and m2 then the two measured velocities of the panels using these transducers would be k1 and k2 given by
F = jw(mo+m~)~~, F =jw(mo+m,)~2, where F is the force on the panel due to the sound field and the amplitude of the unloaded panel, 20, would be given by F=jwmo3io. The above expressions are based on the assumption of a constant pressure on the panel whether it is loaded or not, which seems reasonable in view of the large difference in impedance between the panel and the air.
01 100
400
200
000 Frequency
Figure
I.
1600
3150
(Hz)
Lead panel.
From these three equations k. may be found in terms of the measurable ml, m2, kl and c+~: ji’o
=
quantities
(m-m2)4*2 (ml 3il - m2 3i2)*
When measurements were made on the lead panel (see Figure I) it was noticed that the heavier accelerometer predicted a higher transmission loss than the lighter accelerometer at high frequencies. This is in agreement with the idea of the panel being “ loaded ”
TRANSMISSION
LOSS FROM
ACCELEROMETERS
421
by the accelerometers. When the predicted vibration amplitude of the unloaded panel, &, was used the agreement between this curve and the B.S. 2750 curve was much improved at the high frequencies. Figure z shows the transmission-loss curves for a lighter panel (aluminium). Agreement between the two methods of measurement is very good at high frequencies. 3.
LOW-FREQUENCY
DISCREPANCY
It will be seen from Figures I and 2 that the transmission-loss curves obtained using the accelerometer lie below the B.S. 2750 curve at low frequencies. In order to try to explain this discrepancy the effect of the panel’s radiation efficiency was considered.
I
I
I
200
400
800 Frequency
I
1600
I
3150
(Hz)
Figure 2. o*o35-in. aluminiumpanel.
Any practical wall has dimensions that are of the order of several meters so that, considered as a whole, the wall will have a radiation efficiency equal to unity for all frequencies greater than about 20 Hz. However, if the waves in passing through the panel have a coherent area much less than the size of the panel then the radiation efficiency involved in the receipt and re-radiation of the sound energy will be small, thus causing a higher vibration amplitude and a smaller predicted transmission loss. If it is assumed that the sound incident on the panel is only self-coherent over a radius a, then the impedance seen by such an area is not pc but pc(li +jX), where R=(r_!.?k$))
and
X=q_
Here J1 is a Bessel function and HI is a Struve function. The intensity produced by a vibration of the panel of amplitude U, will thus be &Re(pv*)
= +RpcU$
This is equal to the simple expression Qc Ui used previously, multiplied by the radiation efficiency term R. Since the term R 2 I the predicted transmission loss would be greater than that deduced from the simple expression which would give a better fit with the B.S. 2750 curve. If the radius of coherence, a, is invariant with frequency then the increase in predicted transmission loss will only occur at low frequencies since for large values of ka the value of R is unity.
W. A. UTLEY AND K. A. MULHOLLAND
422
In order to check the validity of this radiation efficiency idea it would be necessary to obtain values of correlation between different points on the panel. Unfortunately this information is not at present available.
4. MASS
LAW
AND TRANSMISSION
LOSS
It has been shown that it is possible to measure the transmission loss of a panel by using suitable transducers to determine its velocity amplitude. A simple ad hoc formula, I=&pcU& is used together with the two corrections mentioned above. The random incidence mass law may be used to predict the vibration amplitude of the panel as follows. If a wave of potential $I is incident on a panel and as a result waves of potential 4R and & are transmitted by the panel then we have from consideration of continuity and impedance +I-+R P(41f$J?-&)
= +T,
(I)
= jw?%-case.
(2)
incidence) and jwm represents the impedance velocity amplitude of the panel, Ua, is
13is the angle of incidence (0 = o for normal of the panel. We also note that the normal
Ua = jk+, cos 0
(3)
but since & = &/( I + jwm cos ejzpc)
(4)
we find U0 = jkq5, cos or, since jkq51= V, (V, being the velocity
e/(
I + jwm cos ejzpc),
amplitude
(5)
of the incident
wave),
Ua = V, cos e/( I + jwrn cos e/2Pc).
(6)
In a diffuse field a large number of waves are incident on the panel and we shall assume that N waves are incident from one steradian. To find the net velocity amplitude of the panel in a diffuse field we must sum the squares of the individual velocity amplitudes: 82
s
V$N~~~~esinedB 2?T o [r+( wm cos e/2pcy]
G=
The quantity V:N cannot be directly waves are plane waves then we find
measured
(7)
’
but if we assume
that the individual
PI = @VI* Thus the net pressure over a free field condenser the squares of these pressures : p p
is the measured
quantity
= 4rrNp:
corresponding
(8)
microphone
can be found
= ~T&c)~NV+. to an observed
by summing
(9) sound
level, L,
L = 1olog(~/p$),
(10)
= pi antilog (L/10)
(11)
= 3/47~(pc)~ = pg antilog (L/Io)/~+x)~.
(12)
whence T and so
NV;
TRANSMISSION
We ought,
therefore,
to measure
423
LOSS FROM ACCELEROMETERS
a vibration
amplitude er
u2
_
cos* %sin %d%
antilog (L/IO)
25-p;
0
oI [I+( W?lzcos %/2/x)‘]
457(f#
*
(13)
Measurements of L and lJ,2 were made for an aluminium panel of mass 8.6 kg/m2. The results are shown in Figure 3. It can be seen from the figure that measured values of panel
0 100
I
200
I
I
400
800 Frequency
I
1600 (Hz
y-Y_] 3150
I
Figure 3. i-in. aluminium panel.
vibration amplitude are very much higher than the values predicted by the mass law theory. We have at present no explanation for the unsatisfactory state of affairs. It is possible that the radiation efficiency idea discussed earlier may have a bearing on the problem. 5. CONCLUSION
,4n alternative method of transmission-loss measurement to that used in B.S. 2750 has been described. The new method has the advantage that it may be used to determine transmission loss in the presence of flanking paths. Agreement between measurements using the new method and the reverberant room method is good provided that (i) a suitable correction is used for the loading of the panel by the transducers at high frequencies, and (ii) a correction is applied at low frequencies for the finite coherence area of the waves on the panel. Finally, it was shown that the vibration amplitude predicted by the mass law is considerably lower than that actually measured. REFERENCES I. F. WARD 1963 BBC 2.
Report No. B-078. The accelerometer pick-up as a diagnostic tool in noise studies in buildings. P. M. MORSE 1936 Vibration and Sound. Second edition 1948, page 338. New York: McGrawHill Book Company.