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The active minimization of harmonic enclosed sound fields, part II: A computer simulation
Journnl ofSound and Vibration (1987) 117(l), 15-33
THE
ACTIVE
SOUND
MINIMIZATION
FIELDS,
A. J. BULLMORE,
OF HARMONIC
PART II: A COMPUTER
P. A. NELSON,
ENCLOSED
SIMULATION
A. R. D. CURTIS AND S. J. ELLIOTT
Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England (Received 15 May 1986) This paper is Part II in a series of three papers on the active minimization of harmonic enclosed sound fields. In Part I it was shown that in order to achieve appreciable reductions in the total time averaged acoustic potential energy, EP, in an enclosed sound field of high modal density then the primary and secondary sources must be separated by less than one half wavelength, even when a relatively large number of secondary sources are used. In this report the same theoretical basis is used to investigate the application of active control to sound fields of low modal density. By the use of a computer model of a shallow rectangular enclosure it is demonstrated that whilst the reductions in EP which can be achieved are still critically dependent on the source locations, the criteria governing the levels of reduction are somewhat different. In particular it is shown that for a lightly damped sound field of low modal density substantial reductions in EP can be achieved by using a single secondary source placed greater than half a wavelength from the primary source, provided that the source is placed at a maximum of the primary sound field. The problems of applying this idealized form of active noise control are then discussed, and a more practical method is presented. This involves the sampling of the sound field at a number of discrete sensor locations, and then minimizing the sum of the squared pressures at these locations. Again by use of the computer model of a shallow rectangular enclosure, the effects of the number of sensors and of the locations of these sensors are investigated. It is demonstrated that when a single mode dominates the response near optimal reductions in EP can be achieved by minimizing the pressure at a single sensor, provided the sensor is at a maximum of the primary sound field. When two or three modes dominate the response it is found that if only a limited number of sensors are available then minimizing the sum of the squared pressures in the corners of the enclosure gives the best reductions in EP. The reasons for this behaviour are discussed.
1. INTRODUCTION The work presented here is a preliminary investigation of the possibilities for the active suppression of harmonically excited sound fields having a relatively low modal density. This paper is an extension of the work presented in Part I [l] in which a theoretical framework was developed for evaluating the effectiveness of active techniques for producing global reductions in internal sound pressure levels. The particular case of a high modal density sound field was dealt with theoretically in Part I and it was demonstrated that appreciable reductions in total time averaged acoustic potential energy can be obtained only if the primary and secondary sources are within half a wavelength of each other. This paper describes the results obtained by using a computer model developed, on the theoretical basis of Part I, to assess the effectiveness of active techniques for suppressing sound in a lightly damped, rectangular enclosure at frequencies for which the contained acoustic field is of low modal density. It is shown that, provided the system is being 15 0022-460X/87/160015+19
$03.00/O
0
1987 Academic Press Limited
16
A. J. BULLMORE
ET AL.
excited at an acoustic resonance, substantial reductions in the total time averaged acoustic potential energy can be obtained, even when the secondary sources are separated from the primary source by greater than half a wavelength. However, to obtain these large reductions the secondary sources must be placed at antinodes (i.e., pressure maxima) of the primary sound field. An evaluation is also presented of a possible practical technique for deriving an approximation to the total time averaged acoustic potential energy. This involves the sampling of the sound pressure field at a number of discrete locations and then adjusting the secondary source strengths to minimize the sum of the squared pressures. It is shown that, for a rectangular enclosure, placing a sensor in each of the comers can give near optimal reductions in the total time averaged acoustic potential energy, provided the acoustic response is dominated by only a few modes. The optimal locations for both sources and sensors are discussed, and the levels of reduction which can be achieved are found to be heavily dependent on these locations.
2. THE 2.1.
MINIMIZATION
OF TOTAL
TIME AVERAGED ENERGY
ACOUSTIC
POTENTIAL
THEORY
distribution of sound pressure in a steady harmonically excited enclosure can be represented as the sum of a series of modal contributions. For a time dependence of the form &“‘I, the complex pressure amplitude at any point defined by the position vector x can be written as The
p(x,o)=
; M4Gb)=*=~. n=O
The summation consists of N normal modes each having a characteristic function JI,(x) and complex amplitude a, (w ) which respectively comprise the elements of the Nth order vectors JI and a. The characteristic function normalization, as in Part I, is such that I, I+,,(nl* dV= V. A gain as in Part I, the vector a can be considered to consist of the linear superposition of contributions from some primary source distribution and a series of secondary sources. Thus one can write a=a,+Bq,,
(2)
where ap is the vector of complex mode amplitudes due to the primary source distribution, qs is an Mth order vector of complex secondary source strengths and B is the N x M matrix of modal excitation coefficients quantifying the excitation of the nth mode due to the mth secondary source. In Part I, the total time averaged acoustic potential energy was introduced as a useful single measure of the amplitude of the pressure fluctuations in an enclosed sound field. This is given by Ep = (1/4~0&
I V
bb,
~11’ d V = ( V/4pochH
8,
(3)
and it was shown that for any given primary and secondary source distributions, it is possible to calculate the unique set of secondary source strengths necessary to minimize Ep. This optimal vector of secondary source strengths is given by qso = -[B” B]-‘BH ap and results
in a corresponding
minimum
(4)
value of Ep given by
EpO = Ep,, - [ V/4poci]aFB[BH
B]-‘B” a,,
(5)
ENCLOSED
SOUND
FIELD
MINIMIZATION,
II
1’7
where EPP is the total time averaged acoustic potential energy due to the primary source distribution in the absence of the secondary sources. For the computer simulations described below, Morse’s [2] solution was again used to describe the form of the sound field in a lightly damped rectangular enclosure. Thus the normalized characteristic functions are given by
JI.(X)= Jwns,s
cos (4T%lL,)
cos(nzm/L*) cos (%~%/Ld,
(6)
The where n,, n, and n3 are integers and L, L2 and L, are the enclosure dimensions. normalization factors are given by E, = 1 if v = 0 and E, = 2 if v > 0. The complex amplitude of the nth mode is given by
an(w) = bdl where A,(w)
defines
the complex A,(w)
O%(o)
V
h(y)
mode resonance = w/&$w
sty,
~1dV,
(71
term given by
-j<&
- w2)),
(8)
s(y, w) represents the total distribution of source strength density within the enclosure and 5” and w, are the damping ratio and natural frequency of the nth mode..The latter is given by w, = ~co[(nllL1)2+(n2/L2)2+(~3/L3)211’2.
(9)
In the computer simulations, the primary source is modelled as a rectangular piston on one wall of the enclosure. Thus the complex amplitude of the nth mode due to the primary source only is given by
a,n(o)=(p,c~lV)A,(w)V,(w)
&((Y)
ds,
(10)
where V,(w) is the normal surface velocity of the piston which is assumed uniform over the surface area of the source, S,. Similarly, each of the secondary sources used is assumed to consist of a rectangular piston having a normal velocity V,,,(o) acting over an area S,,,. Thus the strength of the mth secondary source qs,,,(u) = V,(w) S,,, and the coefficients in the matrix B are given by
&m(~) = bodl VMn(~)(lISm)
I
S,,,
fin(~)
dS.
(111
Thus, with the elements of the vector a, and matrix B specified, both the optimal secondary source strengths and the maximum possible reductions in EP can be calculated for various arrangements of primary and secondary sources. 2.2.
THE
COMPUTER
SIMULATION
The model chosen for the computer simulation is that of a “two dimensional”, lightly damped, rectangular enclosure as shown in Figure 1. The enclosure dimensions are 2.264 m x 1.132 m x 0.186 m, chosen to coincide with the dimensions of an enclosure used in some preliminary experimental work undertaken by Lewers [3]. It should be noted that, although one of the dimensions is much less than the other two, the enclosure is not truly two dimensional and consequently the effects of modal contributions related to the shorter dimension have been accounted for in the model. However, at the low frequencies considered the sound field is dominated by axial and tangential modes of the two larger dimensions, the pressure distribution being essentially uniform across the shorter dimension of the enclosure. Thus the mode structure is relatively easy to visualize.
18
A. J. BULLMORE
ET AL.
Figure 1. Schematic diagram of the enclosure modelled in all computer simulations, where x, = 2.264 m, x2 = 1.132 m, xg = 0.186 m, and the positions of the centres of the sources are as follows: primary = (2.087, 0.993,0.186), Sl = (2.087,0+43,0.186), S2 = (1~892,0+96,0~186), S3 = (OTi96,0~566,0~186), S4 = (0.177,0.993, 0.186).
A damping ratio of 0.01 has been assumed for all natural frequencies, again to coincide with the plywood enclosure examined by Lewers [2], and each of the five sources shown has been modelled as a 0.15 m by 0.15 m square piston having uniform velocity distributions over their surfaces. The computed frequency response of the enclosure is shown on Figure 2. This has been evaluated as the ratio of the pressure at the centre of the primary source to the volume velocity of that source for frequencies from 50 Hz to 300 Hz. Also given are the natural frequencies of each resonance together with their associated modal integers. This curve has been computed by using a finite number of normal mode contributions. For equation (1) to give an exact representation of the sound field, an infinite number of modes must be included in the summation over the modal integers n,, n2, and n3. However, when working at low modal densities it is possible to achieve a reasonably accurate estimate of the sound pressure by truncating this infinite summation. Use of equation (1) in a computer simulation requires that a compromise be reached between including enough modes to give an accurate solution, whilst keeping processing time to
40 60
I 100
1
110
Frequency
I
200 (Hz)
I
260
I
100
Figure 2. Ratio of sound pressure evaluated on the centre of the surface of the primary source to the velocity of that source. The natural frequencies and associated modal integers are as follows: 1 = 75.3 Hz (1, 0, 0), 2 = 150.6 Hz (2,0,0)(0,1,0), 3 = 168.4 Hz (1, LO), 4= 213.0 Hz (2, 1, O), 5 = 225.9 Hz (3,0, O), 6 = 271.5 Hz (3,1,0).
ENCLOSED
SOUND
FIELD
MINIMIZATION,
19
II
within acceptable limits. In order to determine this compromise some initial computations were performed to establish the convergence of the modal summation. Figure 3 shows the convergence of equation (1) when the SPL is calculated on the surface of a source, placed at the primary source location (see Figure 1) and operating at 200 Hz. Results for both a point source and a 0.15 m square piston source are shown. Even after the inclusion of 55 000 modes the point source model does not converge, as one would anticipate in the near field of a point source [4], whereas the distributed source model requires only about 1700 modes to converge to within 1 dB of its value given by including over 400 000 modes. It should be noted that these graphs have been plotted as functions of the natural frequency of the highest order mode included in the summation, and not as a function of the number of modes. The relationship between these latter two quantities is shown in Figure 4. Figure 5 shows the dependence of the convergence of equation (1) on frequency when using a distributed source as described above. The summations at the three frequencies lying midway between resonant frequencies, 112.5 Hz, 190.5 Hz and 439 Hz, all converge at different rates, this rate of convergence decreasing with increasing frequency. The value of the pressure on the source at the resonant frequency, 336.8 Hz, converges much more rapidly, despite the fact that this frequency is higher than two of the non-resonant frequencies.
701 10 000
1000
Maximum
frequency
(Hz)
Figure 3. Convergence of the modal summation used to calculate the pressure. -O-O-, Pressure evaluated at the position of a point source; -O-O-, pressure evaluated at the centre of the surface of a 0.15 m by 0.15 m piston source. Both sources are placed at the primary source location and operating at 200 Hz.
n 60 000
-
40
-
so
000 000
-
20 000
-
X /]
/
/” NX
IO 000
-
0x /X
o-x-7 1000
I
I
I
I
I IO 000
Maximum frrquency
(Hz)
Figure 4. The relationship between the natural frequency of the highest order mode included in the summation and the number of modes having natural frequencies below this value.
20
ET AL.
A. J. BULLMORE
0.96 t
1 I
I
0.93
I
I
I
I
I
1
IO 000
2000 Morimum
trrgucncy
(Hz)
Figure 5. Dependence of the convergence of the modal summation on frequency for a 0.15 m by 0.15 m piston source at the primary source location, -+-+-, 112.5 Hz; -A-A-, 190.5 Hz; -O-O-, 336.8 Hz; -x-x-, 439 Hz. The ratio SPL (truncated)/SPL (20 000) is the ratio of the SPL evaluated by using a modal summation including all modes having a natural frequency up to that shown on the x-axis, to the SPL evaluated by using all modes having a natural frequency up to 20 000 Hz, some 413 042 modes in all.
It was thus concluded that only distributed sources should be used, and that if 7000 modes were included in the summation then the computed pressure amplitudes would be accurate to within 1 dB at all frequencies up to 300 Hz. RESULTS OF THE COMPUTER MODEL SIMULATION OF THE MINIMIZATION OF Ep The results of this section demonstrate the effects of secondary source locations on the minimization of Ep. Four different secondary source locations have been considered, and for each of these (and certain combinations of these) Ep has been minimized at each
2.3.
1 Hz interval
between
50 Hz and 300 Hz, where the source strength necessary to minimize from the solution given in equation (4). The results are presented frequency, where EpO is the minimum possible value of Ep for the given source arrangement. Figure 6 shows the effect of introducing the secondary source Sl and minimizing Ep by adjusting the gain and phase of Sl only. It would perhaps be expected from the analysis of Part I [ 11, that over the entire frequency range from 50 Hz to 300 Hz large reductions in Ep should result, as the primary and secondary sources are never separated by a distance greater than half a wavelength of the driving frequency. However, at an operating frequency of around 300 Hz, when the centre-to-centre source separation is
Ep has been computed as plots of Epo against
90
I
1
Frequency
Figure 6. The value of E,, when minimized source. -, E,, due to the primary source;
1
I
1
(Hz)
by using the single secondary
source Sl adjacent to the primary
. . ., Epo,when E,,is minimized by using secondary source Sl.
ENCLOSED
SOUND
FIELD
MINIMIZATION,
21
II
still less than half a wavelength, the reduction in Ep is less than 1 dB. Clearly this low modal density situation differs substantially from the high modal density case discussed in Part I, and the criteria governing the levels of reduction which can be achieved are also different. If source S2 is now introduced into the enclosure in the position shown in Figure 1, and Ep minimized by adjusting the source strength of S2 only, the result is as shown on Figure 7. It is clear from this graph that the introduction of just a single secondary source results in substantial reductions in Ep at a number of the acoustic resonances despite the separation of the primary and secondary sources being greater than half a wavelength for two of the frequencies involved.
40 SO
I
I
100
150
Frequency
I
I
200
250
300
(Hz)
Figure 7. The value of EP when minimized by using the single secondary source S2. -, primary source; . ., E,,, when E, is minimized by using secondary source S2.
E, due to the
However, of the six resonances considered, only half of them are successfully attenuated, with Ep at the resonances of 1506 Hz, 225.9 Hz and 271.5 Hz being uncontrollable by the source introduced at the position S2. The reason for this becomes apparent if the source’s position relative to the nodal planes of these resonances is considered (see Figure 12 to follow). For each of these three frequencies, the sound field is such that source S2 lies on, or close to, a nodal plane. If the source is centred on a nodal plane of the primary sound field then it will be unable to excite the resonant mode, and hence will not be able to set up the necessary sound field to interfere destructively with the sound field dominated by this modal contribution. If the source lies close to a nodal plane of the primary field, then, whilst it would be possible to generate the necessary sound field to destructively interfere with the dominant mode, the large volume velocity required would lead to an increased excitation of any residual modes having pressure antinodes near the source location, and consequently Ep would not necessarily be reduced by minimizing the contribution of the resonant mode. If source S3 is now introduced, and Ep minimized by adjusting the source strength of S3 only, then only two of the resonances are attenuated (see Figure 8), as S3 lies on nodal planes of all the modes with odd numbered x,-plane modal integers. If now S2 and S3 are both introduced and Ep minimized by adjusting the source strengths the results are as shown on Figure 9 and all resonances except the 271.5 Hz resonance are attenuated. Again, the reason for this is clear if the nodal planes are considered (see again Figure 12). It is interesting to note that the degenerate resonance at 150.6 Hz, consisting of the (2,0,0) and (0, 1,O) modes, is successfully attenuated when both S2 and S3 are used although neither one of these sources acting alone could appreciably reduce Ep at that frequency. This is because source S2 is positioned such that the two modal contributions are out of phase at that location and thus generation of the necessary source strength to
22
A. J. BULLMORE
I
40
50
100
I 110 Frequency
ET AL.
I
200
I 210
Figure 8. The value of E, when minimized by using the single secondary source primary source, . . . ., E,,, when E, is minimized by using secondary source S3.
40L 50
1 100
I
IS0 Frr~urncy
300
(Hz1
I
200
S3; -,
E, due to the
I
230
300
(II21
Figure 9. The value of EP when minimized by using two secondary . ’. ., E,,, when I?, is minimized by using secondary primary source;
sources S2 and S3. -, sources S2 and S3.
Ep due to the
cancel either mode would result in an increased excitation of the other mode. Source S3 is placed at a node of the (0, 1,O) mode, yet an antinode of the (2,0,0) mode. Consequently S2 can be given the correct source strength to cancel the (2,0,0) mode whilst not affecting the (0, 1,O) mode. With the effect of this (2,0,0) mode removed, Sl can be set up in phase opposition with the (0,1,0) mode and the sound field will be reduced. The resulting increased excitation of the (2,0,0) mode will be counteracted by an increase in strength of source S3, the final result being that source S2 will be of approximately equal magnitude and phase as the primary, and source S3 will be of approximately twice the magnitude of the primary and 180” out of phase with it. This does not necessarily imply, however, that this pair of resonances requires two sources to cancel them. Figure 10 shows the effect of introducing the single secondary source S4, and its position is such that all resonances up to 300 Hz are successfully attenuated including the two 150.6 Hz resonances. If the three secondary sources S2, S3 and S4 are now introduced simultaneously, the reduction in Ep shown in Figure 11 results. For this case the sources can drive several modes in antiphase with the primary field at most frequencies and consequently not only are all the resonances successfully attenuated, but many lower non-resonant frequencies are also attenuated. Figures 7-l 1 clearly show the importance of source locations on the ability to successfully reduce Ep. Provided a secondary source is placed at a maximum of each of the major contributing mode shapes, then the effects of those modal contributions can each
ENCLOSED
SOUND
FIELD
MINIMIZATION,
23
II
so 401 60
I 100
I 130
Frrqurncy
I 200
I 230
(Hz)
Figure 10. The value of EP when minimized by using the single secondary primary source, . . . . ., E,,, when E, is minimized by using secondary source
4oL 50
I 100
I 150 Frequency
I 300
1 200
I 250
source S4.
S4. -,
E,, due to the
I 300
(Hz1
Figure 11. The value of Ep when minimized by using three secondary sources S2, S3 and S4. -, the primary source; . , E,,, when E, is minimized by using secondary sources S2, S3 and S4.
E, due to
be substantially reduced without exciting the residual modes to any significant level. Also a single secondary source can reduce the contributions of more than one mode provided the primary modal contributions at the source location have the same relative phase and roughly equal amplitudes. The general conclusion which results from these observations is that the determination of the optimal strength of a given secondary source (which minimizes EP) gives the source strengths which result in the best reduction in the response of the dominant mode at a given frequency whilst giving the least excitation of the remaining “residual” modes. This is demonstrated in Figure 13 which shows Ep due to the primary source, but with the subtraction of the contribution from the dominant mode at each frequency (in the frequency range near the degenerate 150.6 Hz resonance both the (2,0,0) and (0, 1,O) modes have been subtracted). Also shown is the minimum value of E, obtained by using source S4. The 3 to 4 dB discrepancy between the two curves demonstrates that whilst source S4 successfully attenuates the contribution of the dominant mode at each frequency it also excites the residual modes and thus lessens the overall reduction in Ep. As a final example of this observation the sound pressure field at the 150.6 Hz resonance has been calculated both before and after Ep has been minimized by using S4. The resulting distributions are shown on Figures 14(a) and (b) respectively. Comparison of Figures 14(a) and (b) reveals that minimizing Ep has removed the dominant (2,0,0)(0, 1,0) mode structure to leave the effect of the nearest mode (1, 1,O) as the new dominant feature.
24
ET AL.
A. J. BULLMORE
P
•J
54
0
cl
cl
+
+
-
s2
=
+
cl
cl
SI P
d
+ 00
Cl0 75.3 Hz (I, o,o)
150.6
168.4
213-O. Hz (2, I, 0)
Hz (I, I, 0)
Figure 12. The distribution driven by the primary source
so 40
of nodal only.
Hz
+ 00
(2,0,0)
planes
225.9
271~5 Hz (3, I, 0)
Hz [3,0,(J)
for the first six resonances
of the enclosure
of Figure
1, when
-m-H
50
I
I
100
150
I
200
Frequency
I 250
300
(HZ)
Figure 13. The effect of removing the dominant mode from the modal summation used to calculate the total time averaged acoustical potential energy, E,,. -, Ep due to the primary source alone; ----, EP with the most dominant mode removed (for the 150.6 Hz resonance two modes have been removed; see the main text); . . . , E,,,, when E,, is minimized by using secondary source S4.
2.4.
THE
The
POWER
OUTPUT
power output
OF PRIMARY
of a piston
AND
SECONDARY
source having
a normal
SOURCES
surface
velocity
V(y, co) is given
by W= (l/2)
Re
P*(Y, I S
~1
WY,
w) ds,
(12)
ENCLOSED
SOUND
FIELD
MINIMIZATION,
25
Figure 14(a). The sound pressure distribution due to the primary source operating at 150.6 Hz; (b) the sound pressure distribution when E, has been minimized by using source 54 (frequency = 150.6 Hz).
where p*(y, o) is the complex conjugate of the complex acoustic pressure. By using equation (12) the power output of the primary source has been calculated both before and after E,, has been minimized by using source S4. Also, the power output of the cancelling source has been calculated, and from these results the total reduction in source power outputs due to minimizing E,, has been evaluated. These results are shown on Figures 15 and 16. It is clear that, at resonant frequencies, the power output of the primary source is substantially reduced following the minimization of Ep. It is also apparent that the power output of the cancelling source is also very much less than the original power output of the primary source, and although at some frequencies the cancelling source does absorb energy the levels involved do not account for the overall reduction of power input to the system. The mechanism by which the reduction of power is occurring is a mutual “unloading” effect, such that each source is creating an impedance condition over the other source’s surface such that the pressure and velocity are in quadrature, and thus the source power outputs are reduced.
A. J. BULLMORE
26
ET AL.
3.0
Figure 15. The power output of the primary source operating alone.
Figure 16. The power output of the primary source and secondary source S4 when Ep has been minimized by using S4. -, Power output of the primary source; . . . ., power output of source S4.
3. THE
MINIMIZATION OF THE SUM OF THE SQUARED PRESSURES DISCRETE NUMBER OF SENSOR LOCATIONS
AT
A
3.1. THEORY The foregoing approach makes possible the determination of the secondary source strengths necessary to achieve the “best possible” overall reduction in the amplitude of the sound pressure fluctuations in the enclosure. Clearly, however, sufficient detail has to be known about both the primary source distribution and the modal structure of the sound field to enable accurate determination of the complex values of up,, (w) and B,,, (w). An alternative approach is to monitor the amplitude of the pressure fluctuations at a discrete number of “sensor” locations and to adjust the strengths of the secondary sources in order to minimize the sum of the squared pressure amplitudes at these locations. Summing the squared pressures at a discrete number of locations enables an approximation to be made to the total time averaged acoustic potential energy. This.approximation can be defined as (13) where p(x,, o) is the complex pressure amplitude at the Ith sensor location, the summation being performed over L locations in total. As the number of evenly distributed locations L tends to infinity, then the value of J, tends to the value of E,, as specified by equation
ENCLOSED
(3). Now note that in vector
SOUND
notation
FIELD
equation
MINIMIZATION,
(13) can be written
Jp = ( V/4W:L)PH where the vector p is the Lth order vector whose equation (1) enables this vector to be written as
27
II
as (14)
P, Ith component
is p(x,, w). Use of
(15)
P=Ga,
where the matrix eL is of order Lx N and is the matrix of the N characteristic functions evaluated at the L sensor locations such that the n, Ith element is &(x,). Substitution of the expression for a given by equation (2) then shows that P = GT[ap +
(16)
WI.
Now let pP = JIz a, be the vector of pressures at the L sensor primary source distribution only. Similarly let Z = $1 B define ance matrix relating the complex pressure amplitudes at the complex strengths of the M secondary sources. Accordingly,
locations produced by the the Lx M transfer impedL sensor locations to the one can simply write (17)
p=pp+Zqs.
This shows that if the number of sensor locations L is equal to the number of secondary sources M then the pressure amplitude can be made equal to zero at all L sensors (i.e., p is made equal to zero) by choosing qs to be the solution of the equation pP+ Zq, = 0 such that qs = -z-‘p
P’
(18)
This of course is on the assumption that the matrix Z is non-singular. Although this result is useful in demonstrating that the pressure can be constrained to zero at a number of points in the sound field, this does not necessarily imply that the pressure at other locations will also be reduced. In fact, as will be demonstrated, the reverse may well be true. Unde,r certain conditions, driving the pressure to zero at a number of discrete locations may produce substantial increases in the pressure amplitude at other locations in the enclosure. It is thus more likely that a large number of sensors is required to produce a good approximation to Ep. Thus the situation that is more likely to be useful in practice in producing “overall” reductions in sound level is when the number of sensors is larger than the number of secondary sources (i.e., L> M). Thus one again seeks a “least squares” solution and can choose qs to minimize J, which, on combination of equations (14) and (17), is given by
Jp = ( Vhc:L)h,HZH
Zqs+d’ZH pp + P,” Zq, + p,“p,l.
(19)
This is also a quadratic function of the secondary source strengths which is of the same form as that given by equation (10) of Part I [ 11. The existence of a unique minimum value of the function can again be argued on physical grounds. Note that (V/4p,&L) qs”ZHZ qs is equal to the sum of the squared pressures at the L sensor locations produced by the secondary sources only (in the absence of any primary sound field). This quantity must be greater than zero for all non-zero values of the vector qs. This then ensures that the matrix ( V/4p&L)ZHZ (which is equivalent to the matrix A in equation (10) of Part I) is positive definite and that Jp is minimized by a unique value of the vector qE. By analogy with the solution given in Part I, the optimal vector of secondary source strengths which minimizes Jp can be written as qr, = -[Z” z]-’ ZH pp,
(20:)
ET AL.
A. J. BULLMORE
28 which results
in the fractional
reduction
in Jp being given by
Jp/ Jpp = 1- PP”Zl-Z”
Zl-‘Z” pPlp;pp,
(21)
where Jpo is the optimal value of Jp and the quantity Jpp = ( V/4p0ci) pp” pP is the value of Jp produced by the primary source only. This then specifies a technique which can be used to determine the source strength vector in practice, which, if the number of evenly distributed sensors is large enough, should give a value of q.$, which is a good approximation to the optimal value qso which minimizes Ep. Note that the evaluation of equation (20) involves the determination of the complex primary pressure amplitude at the sensor locations and the matrix of complex transfer impedances between the secondary sources and sensor locations. Both of these quantities could in principle be determined experimentally in any given enclosure without any knowledge of the nature of the primary source distribution or the modal structure of the sound field. A further detailed discussion of the use of this technique in practice will be given in Part III. 3.2.
OPTIMAL
SENSOR
LOCATIONS
IN
SOUND
FIELDS
OF
LOW
MODAL
DENSITY
The question can now be addressed of the optimal locations of sensors in order that qs, gives the best possible approximation to qso when using a limited number of sensors. Although a complete and rigorous solution this problem has as yet not been attempted, some extremely useful initial guidelines have been deduced for the case considered here of a relatively lightly damped sound field of low modal density. Firstly note that use of the expression p=@ a in equation (14) gives
Jp = ( V/%dL)a” and use of the expressions
Z = $zB
and
pP = $:a,
q,, = -LB” JIL $1
(22)
JIL + z a in equation
W’B” $4:
(20) gives
(23)
a,.
Thus if
the matrix (JIL JIl)/ L is equal to the identity matrix, then Jp will be equal to Ep (equation (3)) and qsl will equal qso (equation (4)). Confirmation that (JIL JIz)/L tends to the identity matrix as the number of evenly spaced sensors tends to infinity is given by consideration of the terms in the matrix. Expansion of the matrix (tl.r~ +:1/L shows that each term of the N x N matrix is given by the product of a pair of characteristic functions summed over the L sensor locations. Thus
i 1
[JIL JIZl/L= (l/L)
tax,)
I=1
X
,$&b,MI(xd
The orthogonality
property
i !h(x,) ILz(xr) * . . ,p, hb,) +/Nb,)
/=I
i Icl:cx,,
of the characteristic
I
. . . ,i, ~2(x,hh(x,) .
I=1
functions
(24)
shows that (25)
which demonstrates that the matrix tends to the identity matrix as L tends to infinity. The form of the matrix also suggests that a “reasonable approximation” to the identity matrix may be obtained by using a limited number of sensors placed so that the diagonal terms approach unity and the off-diagonal terms approach zero. The number of sensors
ENCLOSED
SOUND
FIELD
MINIMIZATION.
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29
has yet to be determined for enclosures required to ensure a “reasonable approximation” of an arbitrary modal density. However, attention will be initially restricted to the case dealt with here of a sound field where the response is dominated by only a few modes. Thus at frequencies close to one of the natural frequencies of the enclosure, one modal response will be dominant, except in the case of the degenerate resonance, where two modes dominate. At “off resonance” frequencies, since the enclosure is relatively lightly damped, the response is generally dominated by only a few modes whose natural frequencies lie closest to the frequency considered. Firstly consider the case where the sound field is dominated by a single mode and a single appropriately placed secondary source is used to suppress the field. Under these circumstances, equation (23) for the secondary source strength necessary to minimize Jp can be written as (26) where the nth mode has been assumed to be dominant and the terms 6, and a2 are small quantities arising from the additive contributions from all the residual modes. Note that under these circumstances, the optimal secondary source strength which minimizes Ep is given by %0(W) = -[E%
+
wlI~“12+ &I
(2711
where 6, and 6, are again small quantities arising from residual mode contributions. It to that for qs,( w) is clear that the expression for qsl(o) will give a close approximation provided that the term Cf=, tj’,(x,) is sufficiently large to ensure that the residual terms 8, and S2 remain small compared to the dominant terms in the numerator and denominator respectively of equation (26). This can easily be achieved using a single sensor located at a maximum of the dominant mode shape IL”(x). However, the location of a sensor, or number of sensors, at positions where &(x) is zero will clearly give a value of qsl(w) which bears no relationship to that of qso(w). Also note that the value of $, $i(x,) is of little relevance provided it is sufficiently large. Thus, if for example a single sensor is used at a maximum of &(x), then the value of @t(x) will be given by the square of the normalization constant J( &,r ~,~a,,~) and will therefore be equal to 2, 4 and 8 for “axial”, “tangential” and “oblique” modes respectively. Now consider the case where two modes dominate the response of the sound field. Under these circumstances, equation (23) can be written as
where it has been assumed that the nth and kth modes dominate the response and & and S6 are small residual contributions. Now note that the equivalent expression for q_(w) for the case of two dominant modes can be written as %O(~) = -IBE
a,,+B~a,k+6,1/[(B”12+IBk(*+S81,
(29)
where S, and & are again assumed small. It is clear that the major discrepancy between equations (28) and (29) arises from the presence of the term I:=, &(x1) &(x1) in equation
30
A. J. BULLMORE
ET AL.
(28). This is one of the off-diagonal terms in the matrix (JIL JIZ). If this term can be made equal to zero by the appropriate placement of sensors, and if C,“=, +,‘.(x,) and Cf=, I&X,) can be made large and equal, then qs,(w) will give a good approximation to qso(w). An arrangement of sensors which very often but not always fulfills these requirements for a given pair of modes is to place a sensor in each corner of the enclosure. For the two dimensional case considered here, examination of the mode shapes illustrated in Figure 12 shows that Et=, &(x,) &(xk) is equal to zero for these sensor placements for each successive pair of modes up to a frequency of 300 Hz. This also includes the degenerate resonance at 150.6 Hz. Corner sensor locations also ensures that a maximum of any mode shape is always detected such that Cf=, &(x1) is always large for any mode. These observations will now be illustrated by using a further computer simulation. 3.3.
RESULTS
OF THE
COMPUTER
SIMULATION
OF THE
MINIMIZATION
OF
Jp
In order to keep the visualization of the modal structure simple, the results of this section have all been obtained by using the single secondary source S4. It has already been demonstrated (Figure 10) that when this source is controlled to minimize E,, all resonances up to 300 Hz can be attenuated. Thus any inability to reduce Ep will be due to sensor locations and not source location. The foregoing analysis demonstrates that the worst possible place to locate sensors is on nodal planes of the primary sound field. Figure 17 shows a situation where three sensors have been equispaced along the x1 = 1.132 m plane of the enclosure. Even though the secondary source strength has been adjusted to minimize the sum of the squared pressures at the three sensor locations, for many frequencies an increase in Ep has resulted. Note that although the pressure has been constrained only at three discrete points in the enclosure, the plot shows the resulting Ep evaluated over the whole of the enclosed space. Reference to Figure 12 reveals that the frequencies where the greatest increase in Ep is produced are those where the sensors are placed at nodal planes of the primary sound field, and thus the contribution of the dominant mode is undetected. The analysis of section 3.2 suggests that placing a sensor in each corner of the enclosure will ensure the detection of single dominant modes, and is also the best arrangement of a limited number of sensors when two modes dominate the response. Figure 18 shows the result of minimizing Jp with four sensors positioned as described above, and close by minimization of Jp) and EPO is agreement between EPJ (th e value of Ep produced evident, particularly at the resonant frequencies, where a single mode dominates the response. Even at the 150.6 Hz degenerate resonance EPJ and E,, show good agreement, implying that the off diagonal terms of the matrix (Jlt JIT)/L (24) are small. Reference to Figure 12 confirms that the sum of the products of the two modes over the four comers of the enclosure does equal zero, and hence the matrix is diagonal. Furthermore, the two modes are both axial and thus have the same normalization constants. Therefore, apart from the residual S terms, equations (28) and (29) are equal, and qsl(o) closely resembles are at qs,(w). The frequency regions where the Epo and EP, curves differ significantly some non-resonant frequencies, where generally two modes dominate the sound field. Whilst for all of these cases the two dominant modes are such that the important off diagonal terms of the matrix (Jr= $1)/L. are all zero, the classes of the two modes are different and the diagonal terms therefore differ by the square of the normalization constants. Hence the modes are weighted and, for instance, twice as much importance is placed on the contribution of a tangential wave as on that of an axial wave, even though both modes are detected at their maxima. Consequently qs,(w) and qso(w), from equations (28) and (29) respectively, will not be equal and minimizing Jp will not result in Ep also being minimized. Where the dominant modes are of the same class, for example
ENCLOSED
501
50
FIELD
SOUND
I
I
100
IS0 Frequency
MINIMIZATION,
I
200
31
11
I
210
I
200
(Hz)
Figure 17. The value of Ep when Jp is minimized by using source 54 and three sensors equispaced along the x = 1.132 plane of the enclosure. -, E, due to the primary source; . . . ., I$, when JP is minimized by using source S4 and the three sensors.
the three axial modes between the 75 Hz and 150 Hz resonances, then equation (28) and (29) differ only by the residual terms, S, and minimizing JP does result in virtually minimizing E,.
4.
CONCLUSIONS
It has been shown that, in a low frequency, enclosed, harmonic sound field, appreciable reductions in the overall acoustic potential energy, Ep, can be achieved by the introduction of a small number of secondary sources spaced greater than half a wavelength from the primary source provided the system is being excited at, or close to, a lightly damped acoustic resonance. The locations of the secondary sources to give optimal reductions have been demonstrated to be at maxima of the primary sound field, where the major contributing modes all share the same relative phase. It has also been shown that minimizing the sum of the squared pressures at a number of discrete sensor locations can provide a good approximation to minimizing the total time averaged acoustic potential energy Ep, provided the sensors are placed at maxima
32
A. J. BULLMORE SO1
1
I
ET AL. I
I
Figure 18. The value of EP when JP is minimized by using source S4 and one sensor each in the four comers of the enclosure. -, E,, due to the primary source; . . . . ., E,,, when JP is minimized by using source S4 and the four sensors; ----, I?,,, when E,, is minimized by using source S4.
of the primary sound field. If the sensors are placed at such maxima then, for the lightly damped low modal density sound field considered, only a relatively small number of sensors are required to give a good approximation to minimizing I&. For the case of a rectangular enclosure in which the enclosed sound field is of low modal density it has been shown that positioning a sensor in each comer will ensure the detection of single dominant modes, and will thus result in near optimal reductions in Ep. It has also been shown that this is the arrangement of such a limited number of sensors most likely to result in near optimal reductions in Ep when two or three modes dominate the response.
ACKNOWLEDGMENT S. J. Elliott and P. A. Nelson are supported by the Science and Engineering Research Council under the Special Replacement Scheme; the Department of Trade and Indrlstry research grant which supports A. R. D. Curtis and A. J. Bullmore also is gratefully acknowledged.
ENCLOSED
SOUND
FIELD MINIMIZATION, II
331
REFERENCES 1. P. A. NELSON, A. R. D. CURTIS, S. J. ELLIOTT and A.J. BULLMORE 1987 JournalofSound and Vibration 107, The active minimization of harmonic enclosed sound fields, Part I: Theory. second edition. 2. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill, The active control of steady 3. T. H. LEWERS 1983 M.Sc. Dissertation, University of Souhzmpton. single frequency sound in enclosures. 4. P. M. MORSE and K. U. INGARD 1968 Methods of Theoretical Acoustics. New York: McGraw.. Hill.
THE
ACTIVE
SOUND
MINIMIZATION
FIELDS,
A. J. BULLMORE,
OF HARMONIC
PART II: A COMPUTER
P. A. NELSON,
ENCLOSED
SIMULATION
A. R. D. CURTIS AND S. J. ELLIOTT
Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England (Received 15 May 1986) This paper is Part II in a series of three papers on the active minimization of harmonic enclosed sound fields. In Part I it was shown that in order to achieve appreciable reductions in the total time averaged acoustic potential energy, EP, in an enclosed sound field of high modal density then the primary and secondary sources must be separated by less than one half wavelength, even when a relatively large number of secondary sources are used. In this report the same theoretical basis is used to investigate the application of active control to sound fields of low modal density. By the use of a computer model of a shallow rectangular enclosure it is demonstrated that whilst the reductions in EP which can be achieved are still critically dependent on the source locations, the criteria governing the levels of reduction are somewhat different. In particular it is shown that for a lightly damped sound field of low modal density substantial reductions in EP can be achieved by using a single secondary source placed greater than half a wavelength from the primary source, provided that the source is placed at a maximum of the primary sound field. The problems of applying this idealized form of active noise control are then discussed, and a more practical method is presented. This involves the sampling of the sound field at a number of discrete sensor locations, and then minimizing the sum of the squared pressures at these locations. Again by use of the computer model of a shallow rectangular enclosure, the effects of the number of sensors and of the locations of these sensors are investigated. It is demonstrated that when a single mode dominates the response near optimal reductions in EP can be achieved by minimizing the pressure at a single sensor, provided the sensor is at a maximum of the primary sound field. When two or three modes dominate the response it is found that if only a limited number of sensors are available then minimizing the sum of the squared pressures in the corners of the enclosure gives the best reductions in EP. The reasons for this behaviour are discussed.
1. INTRODUCTION The work presented here is a preliminary investigation of the possibilities for the active suppression of harmonically excited sound fields having a relatively low modal density. This paper is an extension of the work presented in Part I [l] in which a theoretical framework was developed for evaluating the effectiveness of active techniques for producing global reductions in internal sound pressure levels. The particular case of a high modal density sound field was dealt with theoretically in Part I and it was demonstrated that appreciable reductions in total time averaged acoustic potential energy can be obtained only if the primary and secondary sources are within half a wavelength of each other. This paper describes the results obtained by using a computer model developed, on the theoretical basis of Part I, to assess the effectiveness of active techniques for suppressing sound in a lightly damped, rectangular enclosure at frequencies for which the contained acoustic field is of low modal density. It is shown that, provided the system is being 15 0022-460X/87/160015+19
$03.00/O
0
1987 Academic Press Limited
16
A. J. BULLMORE
ET AL.
excited at an acoustic resonance, substantial reductions in the total time averaged acoustic potential energy can be obtained, even when the secondary sources are separated from the primary source by greater than half a wavelength. However, to obtain these large reductions the secondary sources must be placed at antinodes (i.e., pressure maxima) of the primary sound field. An evaluation is also presented of a possible practical technique for deriving an approximation to the total time averaged acoustic potential energy. This involves the sampling of the sound pressure field at a number of discrete locations and then adjusting the secondary source strengths to minimize the sum of the squared pressures. It is shown that, for a rectangular enclosure, placing a sensor in each of the comers can give near optimal reductions in the total time averaged acoustic potential energy, provided the acoustic response is dominated by only a few modes. The optimal locations for both sources and sensors are discussed, and the levels of reduction which can be achieved are found to be heavily dependent on these locations.
2. THE 2.1.
MINIMIZATION
OF TOTAL
TIME AVERAGED ENERGY
ACOUSTIC
POTENTIAL
THEORY
distribution of sound pressure in a steady harmonically excited enclosure can be represented as the sum of a series of modal contributions. For a time dependence of the form &“‘I, the complex pressure amplitude at any point defined by the position vector x can be written as The
p(x,o)=
; M4Gb)=*=~. n=O
The summation consists of N normal modes each having a characteristic function JI,(x) and complex amplitude a, (w ) which respectively comprise the elements of the Nth order vectors JI and a. The characteristic function normalization, as in Part I, is such that I, I+,,(nl* dV= V. A gain as in Part I, the vector a can be considered to consist of the linear superposition of contributions from some primary source distribution and a series of secondary sources. Thus one can write a=a,+Bq,,
(2)
where ap is the vector of complex mode amplitudes due to the primary source distribution, qs is an Mth order vector of complex secondary source strengths and B is the N x M matrix of modal excitation coefficients quantifying the excitation of the nth mode due to the mth secondary source. In Part I, the total time averaged acoustic potential energy was introduced as a useful single measure of the amplitude of the pressure fluctuations in an enclosed sound field. This is given by Ep = (1/4~0&
I V
bb,
~11’ d V = ( V/4pochH
8,
(3)
and it was shown that for any given primary and secondary source distributions, it is possible to calculate the unique set of secondary source strengths necessary to minimize Ep. This optimal vector of secondary source strengths is given by qso = -[B” B]-‘BH ap and results
in a corresponding
minimum
(4)
value of Ep given by
EpO = Ep,, - [ V/4poci]aFB[BH
B]-‘B” a,,
(5)
ENCLOSED
SOUND
FIELD
MINIMIZATION,
II
1’7
where EPP is the total time averaged acoustic potential energy due to the primary source distribution in the absence of the secondary sources. For the computer simulations described below, Morse’s [2] solution was again used to describe the form of the sound field in a lightly damped rectangular enclosure. Thus the normalized characteristic functions are given by
JI.(X)= Jwns,s
cos (4T%lL,)
cos(nzm/L*) cos (%~%/Ld,
(6)
The where n,, n, and n3 are integers and L, L2 and L, are the enclosure dimensions. normalization factors are given by E, = 1 if v = 0 and E, = 2 if v > 0. The complex amplitude of the nth mode is given by
an(w) = bdl where A,(w)
defines
the complex A,(w)
O%(o)
V
h(y)
mode resonance = w/&$w
sty,
~1dV,
(71
term given by
-j<&
- w2)),
(8)
s(y, w) represents the total distribution of source strength density within the enclosure and 5” and w, are the damping ratio and natural frequency of the nth mode..The latter is given by w, = ~co[(nllL1)2+(n2/L2)2+(~3/L3)211’2.
(9)
In the computer simulations, the primary source is modelled as a rectangular piston on one wall of the enclosure. Thus the complex amplitude of the nth mode due to the primary source only is given by
a,n(o)=(p,c~lV)A,(w)V,(w)
&((Y)
ds,
(10)
where V,(w) is the normal surface velocity of the piston which is assumed uniform over the surface area of the source, S,. Similarly, each of the secondary sources used is assumed to consist of a rectangular piston having a normal velocity V,,,(o) acting over an area S,,,. Thus the strength of the mth secondary source qs,,,(u) = V,(w) S,,, and the coefficients in the matrix B are given by
&m(~) = bodl VMn(~)(lISm)
I
S,,,
fin(~)
dS.
(111
Thus, with the elements of the vector a, and matrix B specified, both the optimal secondary source strengths and the maximum possible reductions in EP can be calculated for various arrangements of primary and secondary sources. 2.2.
THE
COMPUTER
SIMULATION
The model chosen for the computer simulation is that of a “two dimensional”, lightly damped, rectangular enclosure as shown in Figure 1. The enclosure dimensions are 2.264 m x 1.132 m x 0.186 m, chosen to coincide with the dimensions of an enclosure used in some preliminary experimental work undertaken by Lewers [3]. It should be noted that, although one of the dimensions is much less than the other two, the enclosure is not truly two dimensional and consequently the effects of modal contributions related to the shorter dimension have been accounted for in the model. However, at the low frequencies considered the sound field is dominated by axial and tangential modes of the two larger dimensions, the pressure distribution being essentially uniform across the shorter dimension of the enclosure. Thus the mode structure is relatively easy to visualize.
18
A. J. BULLMORE
ET AL.
Figure 1. Schematic diagram of the enclosure modelled in all computer simulations, where x, = 2.264 m, x2 = 1.132 m, xg = 0.186 m, and the positions of the centres of the sources are as follows: primary = (2.087, 0.993,0.186), Sl = (2.087,0+43,0.186), S2 = (1~892,0+96,0~186), S3 = (OTi96,0~566,0~186), S4 = (0.177,0.993, 0.186).
A damping ratio of 0.01 has been assumed for all natural frequencies, again to coincide with the plywood enclosure examined by Lewers [2], and each of the five sources shown has been modelled as a 0.15 m by 0.15 m square piston having uniform velocity distributions over their surfaces. The computed frequency response of the enclosure is shown on Figure 2. This has been evaluated as the ratio of the pressure at the centre of the primary source to the volume velocity of that source for frequencies from 50 Hz to 300 Hz. Also given are the natural frequencies of each resonance together with their associated modal integers. This curve has been computed by using a finite number of normal mode contributions. For equation (1) to give an exact representation of the sound field, an infinite number of modes must be included in the summation over the modal integers n,, n2, and n3. However, when working at low modal densities it is possible to achieve a reasonably accurate estimate of the sound pressure by truncating this infinite summation. Use of equation (1) in a computer simulation requires that a compromise be reached between including enough modes to give an accurate solution, whilst keeping processing time to
40 60
I 100
1
110
Frequency
I
200 (Hz)
I
260
I
100
Figure 2. Ratio of sound pressure evaluated on the centre of the surface of the primary source to the velocity of that source. The natural frequencies and associated modal integers are as follows: 1 = 75.3 Hz (1, 0, 0), 2 = 150.6 Hz (2,0,0)(0,1,0), 3 = 168.4 Hz (1, LO), 4= 213.0 Hz (2, 1, O), 5 = 225.9 Hz (3,0, O), 6 = 271.5 Hz (3,1,0).
ENCLOSED
SOUND
FIELD
MINIMIZATION,
19
II
within acceptable limits. In order to determine this compromise some initial computations were performed to establish the convergence of the modal summation. Figure 3 shows the convergence of equation (1) when the SPL is calculated on the surface of a source, placed at the primary source location (see Figure 1) and operating at 200 Hz. Results for both a point source and a 0.15 m square piston source are shown. Even after the inclusion of 55 000 modes the point source model does not converge, as one would anticipate in the near field of a point source [4], whereas the distributed source model requires only about 1700 modes to converge to within 1 dB of its value given by including over 400 000 modes. It should be noted that these graphs have been plotted as functions of the natural frequency of the highest order mode included in the summation, and not as a function of the number of modes. The relationship between these latter two quantities is shown in Figure 4. Figure 5 shows the dependence of the convergence of equation (1) on frequency when using a distributed source as described above. The summations at the three frequencies lying midway between resonant frequencies, 112.5 Hz, 190.5 Hz and 439 Hz, all converge at different rates, this rate of convergence decreasing with increasing frequency. The value of the pressure on the source at the resonant frequency, 336.8 Hz, converges much more rapidly, despite the fact that this frequency is higher than two of the non-resonant frequencies.
701 10 000
1000
Maximum
frequency
(Hz)
Figure 3. Convergence of the modal summation used to calculate the pressure. -O-O-, Pressure evaluated at the position of a point source; -O-O-, pressure evaluated at the centre of the surface of a 0.15 m by 0.15 m piston source. Both sources are placed at the primary source location and operating at 200 Hz.
n 60 000
-
40
-
so
000 000
-
20 000
-
X /]
/
/” NX
IO 000
-
0x /X
o-x-7 1000
I
I
I
I
I IO 000
Maximum frrquency
(Hz)
Figure 4. The relationship between the natural frequency of the highest order mode included in the summation and the number of modes having natural frequencies below this value.
20
ET AL.
A. J. BULLMORE
0.96 t
1 I
I
0.93
I
I
I
I
I
1
IO 000
2000 Morimum
trrgucncy
(Hz)
Figure 5. Dependence of the convergence of the modal summation on frequency for a 0.15 m by 0.15 m piston source at the primary source location, -+-+-, 112.5 Hz; -A-A-, 190.5 Hz; -O-O-, 336.8 Hz; -x-x-, 439 Hz. The ratio SPL (truncated)/SPL (20 000) is the ratio of the SPL evaluated by using a modal summation including all modes having a natural frequency up to that shown on the x-axis, to the SPL evaluated by using all modes having a natural frequency up to 20 000 Hz, some 413 042 modes in all.
It was thus concluded that only distributed sources should be used, and that if 7000 modes were included in the summation then the computed pressure amplitudes would be accurate to within 1 dB at all frequencies up to 300 Hz. RESULTS OF THE COMPUTER MODEL SIMULATION OF THE MINIMIZATION OF Ep The results of this section demonstrate the effects of secondary source locations on the minimization of Ep. Four different secondary source locations have been considered, and for each of these (and certain combinations of these) Ep has been minimized at each
2.3.
1 Hz interval
between
50 Hz and 300 Hz, where the source strength necessary to minimize from the solution given in equation (4). The results are presented frequency, where EpO is the minimum possible value of Ep for the given source arrangement. Figure 6 shows the effect of introducing the secondary source Sl and minimizing Ep by adjusting the gain and phase of Sl only. It would perhaps be expected from the analysis of Part I [ 11, that over the entire frequency range from 50 Hz to 300 Hz large reductions in Ep should result, as the primary and secondary sources are never separated by a distance greater than half a wavelength of the driving frequency. However, at an operating frequency of around 300 Hz, when the centre-to-centre source separation is
Ep has been computed as plots of Epo against
90
I
1
Frequency
Figure 6. The value of E,, when minimized source. -, E,, due to the primary source;
1
I
1
(Hz)
by using the single secondary
source Sl adjacent to the primary
. . ., Epo,when E,,is minimized by using secondary source Sl.
ENCLOSED
SOUND
FIELD
MINIMIZATION,
21
II
still less than half a wavelength, the reduction in Ep is less than 1 dB. Clearly this low modal density situation differs substantially from the high modal density case discussed in Part I, and the criteria governing the levels of reduction which can be achieved are also different. If source S2 is now introduced into the enclosure in the position shown in Figure 1, and Ep minimized by adjusting the source strength of S2 only, the result is as shown on Figure 7. It is clear from this graph that the introduction of just a single secondary source results in substantial reductions in Ep at a number of the acoustic resonances despite the separation of the primary and secondary sources being greater than half a wavelength for two of the frequencies involved.
40 SO
I
I
100
150
Frequency
I
I
200
250
300
(Hz)
Figure 7. The value of EP when minimized by using the single secondary source S2. -, primary source; . ., E,,, when E, is minimized by using secondary source S2.
E, due to the
However, of the six resonances considered, only half of them are successfully attenuated, with Ep at the resonances of 1506 Hz, 225.9 Hz and 271.5 Hz being uncontrollable by the source introduced at the position S2. The reason for this becomes apparent if the source’s position relative to the nodal planes of these resonances is considered (see Figure 12 to follow). For each of these three frequencies, the sound field is such that source S2 lies on, or close to, a nodal plane. If the source is centred on a nodal plane of the primary sound field then it will be unable to excite the resonant mode, and hence will not be able to set up the necessary sound field to interfere destructively with the sound field dominated by this modal contribution. If the source lies close to a nodal plane of the primary field, then, whilst it would be possible to generate the necessary sound field to destructively interfere with the dominant mode, the large volume velocity required would lead to an increased excitation of any residual modes having pressure antinodes near the source location, and consequently Ep would not necessarily be reduced by minimizing the contribution of the resonant mode. If source S3 is now introduced, and Ep minimized by adjusting the source strength of S3 only, then only two of the resonances are attenuated (see Figure 8), as S3 lies on nodal planes of all the modes with odd numbered x,-plane modal integers. If now S2 and S3 are both introduced and Ep minimized by adjusting the source strengths the results are as shown on Figure 9 and all resonances except the 271.5 Hz resonance are attenuated. Again, the reason for this is clear if the nodal planes are considered (see again Figure 12). It is interesting to note that the degenerate resonance at 150.6 Hz, consisting of the (2,0,0) and (0, 1,O) modes, is successfully attenuated when both S2 and S3 are used although neither one of these sources acting alone could appreciably reduce Ep at that frequency. This is because source S2 is positioned such that the two modal contributions are out of phase at that location and thus generation of the necessary source strength to
22
A. J. BULLMORE
I
40
50
100
I 110 Frequency
ET AL.
I
200
I 210
Figure 8. The value of E, when minimized by using the single secondary source primary source, . . . ., E,,, when E, is minimized by using secondary source S3.
40L 50
1 100
I
IS0 Frr~urncy
300
(Hz1
I
200
S3; -,
E, due to the
I
230
300
(II21
Figure 9. The value of EP when minimized by using two secondary . ’. ., E,,, when I?, is minimized by using secondary primary source;
sources S2 and S3. -, sources S2 and S3.
Ep due to the
cancel either mode would result in an increased excitation of the other mode. Source S3 is placed at a node of the (0, 1,O) mode, yet an antinode of the (2,0,0) mode. Consequently S2 can be given the correct source strength to cancel the (2,0,0) mode whilst not affecting the (0, 1,O) mode. With the effect of this (2,0,0) mode removed, Sl can be set up in phase opposition with the (0,1,0) mode and the sound field will be reduced. The resulting increased excitation of the (2,0,0) mode will be counteracted by an increase in strength of source S3, the final result being that source S2 will be of approximately equal magnitude and phase as the primary, and source S3 will be of approximately twice the magnitude of the primary and 180” out of phase with it. This does not necessarily imply, however, that this pair of resonances requires two sources to cancel them. Figure 10 shows the effect of introducing the single secondary source S4, and its position is such that all resonances up to 300 Hz are successfully attenuated including the two 150.6 Hz resonances. If the three secondary sources S2, S3 and S4 are now introduced simultaneously, the reduction in Ep shown in Figure 11 results. For this case the sources can drive several modes in antiphase with the primary field at most frequencies and consequently not only are all the resonances successfully attenuated, but many lower non-resonant frequencies are also attenuated. Figures 7-l 1 clearly show the importance of source locations on the ability to successfully reduce Ep. Provided a secondary source is placed at a maximum of each of the major contributing mode shapes, then the effects of those modal contributions can each
ENCLOSED
SOUND
FIELD
MINIMIZATION,
23
II
so 401 60
I 100
I 130
Frrqurncy
I 200
I 230
(Hz)
Figure 10. The value of EP when minimized by using the single secondary primary source, . . . . ., E,,, when E, is minimized by using secondary source
4oL 50
I 100
I 150 Frequency
I 300
1 200
I 250
source S4.
S4. -,
E,, due to the
I 300
(Hz1
Figure 11. The value of Ep when minimized by using three secondary sources S2, S3 and S4. -, the primary source; . , E,,, when E, is minimized by using secondary sources S2, S3 and S4.
E, due to
be substantially reduced without exciting the residual modes to any significant level. Also a single secondary source can reduce the contributions of more than one mode provided the primary modal contributions at the source location have the same relative phase and roughly equal amplitudes. The general conclusion which results from these observations is that the determination of the optimal strength of a given secondary source (which minimizes EP) gives the source strengths which result in the best reduction in the response of the dominant mode at a given frequency whilst giving the least excitation of the remaining “residual” modes. This is demonstrated in Figure 13 which shows Ep due to the primary source, but with the subtraction of the contribution from the dominant mode at each frequency (in the frequency range near the degenerate 150.6 Hz resonance both the (2,0,0) and (0, 1,O) modes have been subtracted). Also shown is the minimum value of E, obtained by using source S4. The 3 to 4 dB discrepancy between the two curves demonstrates that whilst source S4 successfully attenuates the contribution of the dominant mode at each frequency it also excites the residual modes and thus lessens the overall reduction in Ep. As a final example of this observation the sound pressure field at the 150.6 Hz resonance has been calculated both before and after Ep has been minimized by using S4. The resulting distributions are shown on Figures 14(a) and (b) respectively. Comparison of Figures 14(a) and (b) reveals that minimizing Ep has removed the dominant (2,0,0)(0, 1,0) mode structure to leave the effect of the nearest mode (1, 1,O) as the new dominant feature.
24
ET AL.
A. J. BULLMORE
P
•J
54
0
cl
cl
+
+
-
s2
=
+
cl
cl
SI P
d
+ 00
Cl0 75.3 Hz (I, o,o)
150.6
168.4
213-O. Hz (2, I, 0)
Hz (I, I, 0)
Figure 12. The distribution driven by the primary source
so 40
of nodal only.
Hz
+ 00
(2,0,0)
planes
225.9
271~5 Hz (3, I, 0)
Hz [3,0,(J)
for the first six resonances
of the enclosure
of Figure
1, when
-m-H
50
I
I
100
150
I
200
Frequency
I 250
300
(HZ)
Figure 13. The effect of removing the dominant mode from the modal summation used to calculate the total time averaged acoustical potential energy, E,,. -, Ep due to the primary source alone; ----, EP with the most dominant mode removed (for the 150.6 Hz resonance two modes have been removed; see the main text); . . . , E,,,, when E,, is minimized by using secondary source S4.
2.4.
THE
The
POWER
OUTPUT
power output
OF PRIMARY
of a piston
AND
SECONDARY
source having
a normal
SOURCES
surface
velocity
V(y, co) is given
by W= (l/2)
Re
P*(Y, I S
~1
WY,
w) ds,
(12)
ENCLOSED
SOUND
FIELD
MINIMIZATION,
25
Figure 14(a). The sound pressure distribution due to the primary source operating at 150.6 Hz; (b) the sound pressure distribution when E, has been minimized by using source 54 (frequency = 150.6 Hz).
where p*(y, o) is the complex conjugate of the complex acoustic pressure. By using equation (12) the power output of the primary source has been calculated both before and after E,, has been minimized by using source S4. Also, the power output of the cancelling source has been calculated, and from these results the total reduction in source power outputs due to minimizing E,, has been evaluated. These results are shown on Figures 15 and 16. It is clear that, at resonant frequencies, the power output of the primary source is substantially reduced following the minimization of Ep. It is also apparent that the power output of the cancelling source is also very much less than the original power output of the primary source, and although at some frequencies the cancelling source does absorb energy the levels involved do not account for the overall reduction of power input to the system. The mechanism by which the reduction of power is occurring is a mutual “unloading” effect, such that each source is creating an impedance condition over the other source’s surface such that the pressure and velocity are in quadrature, and thus the source power outputs are reduced.
A. J. BULLMORE
26
ET AL.
3.0
Figure 15. The power output of the primary source operating alone.
Figure 16. The power output of the primary source and secondary source S4 when Ep has been minimized by using S4. -, Power output of the primary source; . . . ., power output of source S4.
3. THE
MINIMIZATION OF THE SUM OF THE SQUARED PRESSURES DISCRETE NUMBER OF SENSOR LOCATIONS
AT
A
3.1. THEORY The foregoing approach makes possible the determination of the secondary source strengths necessary to achieve the “best possible” overall reduction in the amplitude of the sound pressure fluctuations in the enclosure. Clearly, however, sufficient detail has to be known about both the primary source distribution and the modal structure of the sound field to enable accurate determination of the complex values of up,, (w) and B,,, (w). An alternative approach is to monitor the amplitude of the pressure fluctuations at a discrete number of “sensor” locations and to adjust the strengths of the secondary sources in order to minimize the sum of the squared pressure amplitudes at these locations. Summing the squared pressures at a discrete number of locations enables an approximation to be made to the total time averaged acoustic potential energy. This.approximation can be defined as (13) where p(x,, o) is the complex pressure amplitude at the Ith sensor location, the summation being performed over L locations in total. As the number of evenly distributed locations L tends to infinity, then the value of J, tends to the value of E,, as specified by equation
ENCLOSED
(3). Now note that in vector
SOUND
notation
FIELD
equation
MINIMIZATION,
(13) can be written
Jp = ( V/4W:L)PH where the vector p is the Lth order vector whose equation (1) enables this vector to be written as
27
II
as (14)
P, Ith component
is p(x,, w). Use of
(15)
P=Ga,
where the matrix eL is of order Lx N and is the matrix of the N characteristic functions evaluated at the L sensor locations such that the n, Ith element is &(x,). Substitution of the expression for a given by equation (2) then shows that P = GT[ap +
(16)
WI.
Now let pP = JIz a, be the vector of pressures at the L sensor primary source distribution only. Similarly let Z = $1 B define ance matrix relating the complex pressure amplitudes at the complex strengths of the M secondary sources. Accordingly,
locations produced by the the Lx M transfer impedL sensor locations to the one can simply write (17)
p=pp+Zqs.
This shows that if the number of sensor locations L is equal to the number of secondary sources M then the pressure amplitude can be made equal to zero at all L sensors (i.e., p is made equal to zero) by choosing qs to be the solution of the equation pP+ Zq, = 0 such that qs = -z-‘p
P’
(18)
This of course is on the assumption that the matrix Z is non-singular. Although this result is useful in demonstrating that the pressure can be constrained to zero at a number of points in the sound field, this does not necessarily imply that the pressure at other locations will also be reduced. In fact, as will be demonstrated, the reverse may well be true. Unde,r certain conditions, driving the pressure to zero at a number of discrete locations may produce substantial increases in the pressure amplitude at other locations in the enclosure. It is thus more likely that a large number of sensors is required to produce a good approximation to Ep. Thus the situation that is more likely to be useful in practice in producing “overall” reductions in sound level is when the number of sensors is larger than the number of secondary sources (i.e., L> M). Thus one again seeks a “least squares” solution and can choose qs to minimize J, which, on combination of equations (14) and (17), is given by
Jp = ( Vhc:L)h,HZH
Zqs+d’ZH pp + P,” Zq, + p,“p,l.
(19)
This is also a quadratic function of the secondary source strengths which is of the same form as that given by equation (10) of Part I [ 11. The existence of a unique minimum value of the function can again be argued on physical grounds. Note that (V/4p,&L) qs”ZHZ qs is equal to the sum of the squared pressures at the L sensor locations produced by the secondary sources only (in the absence of any primary sound field). This quantity must be greater than zero for all non-zero values of the vector qs. This then ensures that the matrix ( V/4p&L)ZHZ (which is equivalent to the matrix A in equation (10) of Part I) is positive definite and that Jp is minimized by a unique value of the vector qE. By analogy with the solution given in Part I, the optimal vector of secondary source strengths which minimizes Jp can be written as qr, = -[Z” z]-’ ZH pp,
(20:)
ET AL.
A. J. BULLMORE
28 which results
in the fractional
reduction
in Jp being given by
Jp/ Jpp = 1- PP”Zl-Z”
Zl-‘Z” pPlp;pp,
(21)
where Jpo is the optimal value of Jp and the quantity Jpp = ( V/4p0ci) pp” pP is the value of Jp produced by the primary source only. This then specifies a technique which can be used to determine the source strength vector in practice, which, if the number of evenly distributed sensors is large enough, should give a value of q.$, which is a good approximation to the optimal value qso which minimizes Ep. Note that the evaluation of equation (20) involves the determination of the complex primary pressure amplitude at the sensor locations and the matrix of complex transfer impedances between the secondary sources and sensor locations. Both of these quantities could in principle be determined experimentally in any given enclosure without any knowledge of the nature of the primary source distribution or the modal structure of the sound field. A further detailed discussion of the use of this technique in practice will be given in Part III. 3.2.
OPTIMAL
SENSOR
LOCATIONS
IN
SOUND
FIELDS
OF
LOW
MODAL
DENSITY
The question can now be addressed of the optimal locations of sensors in order that qs, gives the best possible approximation to qso when using a limited number of sensors. Although a complete and rigorous solution this problem has as yet not been attempted, some extremely useful initial guidelines have been deduced for the case considered here of a relatively lightly damped sound field of low modal density. Firstly note that use of the expression p=@ a in equation (14) gives
Jp = ( V/%dL)a” and use of the expressions
Z = $zB
and
pP = $:a,
q,, = -LB” JIL $1
(22)
JIL + z a in equation
W’B” $4:
(20) gives
(23)
a,.
Thus if
the matrix (JIL JIl)/ L is equal to the identity matrix, then Jp will be equal to Ep (equation (3)) and qsl will equal qso (equation (4)). Confirmation that (JIL JIz)/L tends to the identity matrix as the number of evenly spaced sensors tends to infinity is given by consideration of the terms in the matrix. Expansion of the matrix (tl.r~ +:1/L shows that each term of the N x N matrix is given by the product of a pair of characteristic functions summed over the L sensor locations. Thus
i 1
[JIL JIZl/L= (l/L)
tax,)
I=1
X
,$&b,MI(xd
The orthogonality
property
i !h(x,) ILz(xr) * . . ,p, hb,) +/Nb,)
/=I
i Icl:cx,,
of the characteristic
I
. . . ,i, ~2(x,hh(x,) .
I=1
functions
(24)
shows that (25)
which demonstrates that the matrix tends to the identity matrix as L tends to infinity. The form of the matrix also suggests that a “reasonable approximation” to the identity matrix may be obtained by using a limited number of sensors placed so that the diagonal terms approach unity and the off-diagonal terms approach zero. The number of sensors
ENCLOSED
SOUND
FIELD
MINIMIZATION.
II
29
has yet to be determined for enclosures required to ensure a “reasonable approximation” of an arbitrary modal density. However, attention will be initially restricted to the case dealt with here of a sound field where the response is dominated by only a few modes. Thus at frequencies close to one of the natural frequencies of the enclosure, one modal response will be dominant, except in the case of the degenerate resonance, where two modes dominate. At “off resonance” frequencies, since the enclosure is relatively lightly damped, the response is generally dominated by only a few modes whose natural frequencies lie closest to the frequency considered. Firstly consider the case where the sound field is dominated by a single mode and a single appropriately placed secondary source is used to suppress the field. Under these circumstances, equation (23) for the secondary source strength necessary to minimize Jp can be written as (26) where the nth mode has been assumed to be dominant and the terms 6, and a2 are small quantities arising from the additive contributions from all the residual modes. Note that under these circumstances, the optimal secondary source strength which minimizes Ep is given by %0(W) = -[E%
+
wlI~“12+ &I
(2711
where 6, and 6, are again small quantities arising from residual mode contributions. It to that for qs,( w) is clear that the expression for qsl(o) will give a close approximation provided that the term Cf=, tj’,(x,) is sufficiently large to ensure that the residual terms 8, and S2 remain small compared to the dominant terms in the numerator and denominator respectively of equation (26). This can easily be achieved using a single sensor located at a maximum of the dominant mode shape IL”(x). However, the location of a sensor, or number of sensors, at positions where &(x) is zero will clearly give a value of qsl(w) which bears no relationship to that of qso(w). Also note that the value of $, $i(x,) is of little relevance provided it is sufficiently large. Thus, if for example a single sensor is used at a maximum of &(x), then the value of @t(x) will be given by the square of the normalization constant J( &,r ~,~a,,~) and will therefore be equal to 2, 4 and 8 for “axial”, “tangential” and “oblique” modes respectively. Now consider the case where two modes dominate the response of the sound field. Under these circumstances, equation (23) can be written as
where it has been assumed that the nth and kth modes dominate the response and & and S6 are small residual contributions. Now note that the equivalent expression for q_(w) for the case of two dominant modes can be written as %O(~) = -IBE
a,,+B~a,k+6,1/[(B”12+IBk(*+S81,
(29)
where S, and & are again assumed small. It is clear that the major discrepancy between equations (28) and (29) arises from the presence of the term I:=, &(x1) &(x1) in equation
30
A. J. BULLMORE
ET AL.
(28). This is one of the off-diagonal terms in the matrix (JIL JIZ). If this term can be made equal to zero by the appropriate placement of sensors, and if C,“=, +,‘.(x,) and Cf=, I&X,) can be made large and equal, then qs,(w) will give a good approximation to qso(w). An arrangement of sensors which very often but not always fulfills these requirements for a given pair of modes is to place a sensor in each corner of the enclosure. For the two dimensional case considered here, examination of the mode shapes illustrated in Figure 12 shows that Et=, &(x,) &(xk) is equal to zero for these sensor placements for each successive pair of modes up to a frequency of 300 Hz. This also includes the degenerate resonance at 150.6 Hz. Corner sensor locations also ensures that a maximum of any mode shape is always detected such that Cf=, &(x1) is always large for any mode. These observations will now be illustrated by using a further computer simulation. 3.3.
RESULTS
OF THE
COMPUTER
SIMULATION
OF THE
MINIMIZATION
OF
Jp
In order to keep the visualization of the modal structure simple, the results of this section have all been obtained by using the single secondary source S4. It has already been demonstrated (Figure 10) that when this source is controlled to minimize E,, all resonances up to 300 Hz can be attenuated. Thus any inability to reduce Ep will be due to sensor locations and not source location. The foregoing analysis demonstrates that the worst possible place to locate sensors is on nodal planes of the primary sound field. Figure 17 shows a situation where three sensors have been equispaced along the x1 = 1.132 m plane of the enclosure. Even though the secondary source strength has been adjusted to minimize the sum of the squared pressures at the three sensor locations, for many frequencies an increase in Ep has resulted. Note that although the pressure has been constrained only at three discrete points in the enclosure, the plot shows the resulting Ep evaluated over the whole of the enclosed space. Reference to Figure 12 reveals that the frequencies where the greatest increase in Ep is produced are those where the sensors are placed at nodal planes of the primary sound field, and thus the contribution of the dominant mode is undetected. The analysis of section 3.2 suggests that placing a sensor in each corner of the enclosure will ensure the detection of single dominant modes, and is also the best arrangement of a limited number of sensors when two modes dominate the response. Figure 18 shows the result of minimizing Jp with four sensors positioned as described above, and close by minimization of Jp) and EPO is agreement between EPJ (th e value of Ep produced evident, particularly at the resonant frequencies, where a single mode dominates the response. Even at the 150.6 Hz degenerate resonance EPJ and E,, show good agreement, implying that the off diagonal terms of the matrix (Jlt JIT)/L (24) are small. Reference to Figure 12 confirms that the sum of the products of the two modes over the four comers of the enclosure does equal zero, and hence the matrix is diagonal. Furthermore, the two modes are both axial and thus have the same normalization constants. Therefore, apart from the residual S terms, equations (28) and (29) are equal, and qsl(o) closely resembles are at qs,(w). The frequency regions where the Epo and EP, curves differ significantly some non-resonant frequencies, where generally two modes dominate the sound field. Whilst for all of these cases the two dominant modes are such that the important off diagonal terms of the matrix (Jr= $1)/L. are all zero, the classes of the two modes are different and the diagonal terms therefore differ by the square of the normalization constants. Hence the modes are weighted and, for instance, twice as much importance is placed on the contribution of a tangential wave as on that of an axial wave, even though both modes are detected at their maxima. Consequently qs,(w) and qso(w), from equations (28) and (29) respectively, will not be equal and minimizing Jp will not result in Ep also being minimized. Where the dominant modes are of the same class, for example
ENCLOSED
501
50
FIELD
SOUND
I
I
100
IS0 Frequency
MINIMIZATION,
I
200
31
11
I
210
I
200
(Hz)
Figure 17. The value of Ep when Jp is minimized by using source 54 and three sensors equispaced along the x = 1.132 plane of the enclosure. -, E, due to the primary source; . . . ., I$, when JP is minimized by using source S4 and the three sensors.
the three axial modes between the 75 Hz and 150 Hz resonances, then equation (28) and (29) differ only by the residual terms, S, and minimizing JP does result in virtually minimizing E,.
4.
CONCLUSIONS
It has been shown that, in a low frequency, enclosed, harmonic sound field, appreciable reductions in the overall acoustic potential energy, Ep, can be achieved by the introduction of a small number of secondary sources spaced greater than half a wavelength from the primary source provided the system is being excited at, or close to, a lightly damped acoustic resonance. The locations of the secondary sources to give optimal reductions have been demonstrated to be at maxima of the primary sound field, where the major contributing modes all share the same relative phase. It has also been shown that minimizing the sum of the squared pressures at a number of discrete sensor locations can provide a good approximation to minimizing the total time averaged acoustic potential energy Ep, provided the sensors are placed at maxima
32
A. J. BULLMORE SO1
1
I
ET AL. I
I
Figure 18. The value of EP when JP is minimized by using source S4 and one sensor each in the four comers of the enclosure. -, E,, due to the primary source; . . . . ., E,,, when JP is minimized by using source S4 and the four sensors; ----, I?,,, when E,, is minimized by using source S4.
of the primary sound field. If the sensors are placed at such maxima then, for the lightly damped low modal density sound field considered, only a relatively small number of sensors are required to give a good approximation to minimizing I&. For the case of a rectangular enclosure in which the enclosed sound field is of low modal density it has been shown that positioning a sensor in each comer will ensure the detection of single dominant modes, and will thus result in near optimal reductions in Ep. It has also been shown that this is the arrangement of such a limited number of sensors most likely to result in near optimal reductions in Ep when two or three modes dominate the response.
ACKNOWLEDGMENT S. J. Elliott and P. A. Nelson are supported by the Science and Engineering Research Council under the Special Replacement Scheme; the Department of Trade and Indrlstry research grant which supports A. R. D. Curtis and A. J. Bullmore also is gratefully acknowledged.
ENCLOSED
SOUND
FIELD MINIMIZATION, II
331
REFERENCES 1. P. A. NELSON, A. R. D. CURTIS, S. J. ELLIOTT and A.J. BULLMORE 1987 JournalofSound and Vibration 107, The active minimization of harmonic enclosed sound fields, Part I: Theory. second edition. 2. P. M. MORSE 1948 Vibration and Sound. New York: McGraw-Hill, The active control of steady 3. T. H. LEWERS 1983 M.Sc. Dissertation, University of Souhzmpton. single frequency sound in enclosures. 4. P. M. MORSE and K. U. INGARD 1968 Methods of Theoretical Acoustics. New York: McGraw.. Hill.