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Design charts for low frequency acoustic transmission through the walls of rectangular ducts
Journal of Sound and Vibration (1981) 78(2), 269-289
DESIGN CHARTS FOR LOW FREQUENCY ACOUSTIC TRANSMISSION T H R O U G H T H E WALLS OF R E C T A N G U L A R DUCTS A. CUMMINGS
Department of i~fechanical and Aerospace Engineering, University of Missouri-Rolla, Rolla Missouri 65401, U.S.A. (Received 26 November 1980, and h2 revised [onn 24 Afarch 1981) "Design charts" are presented here, from which the low frequency acoustic transmission loss of the walls of rectangular ducts may be found. The upper frequency limit on the applicability of this method is at cut-on for the first cross mode in the equivalent rigid walled duct, and it is shown how a quasi-static theory can easily be used to extend the results, as obtained from the charts, to lower frequencies. The data are given in terms of nondimensional parameters, in order to keep the number of charts to reasonable proportions. Two "design examples" are described here, and it is felt that the accuracy in the use of the charts ought to be sufficient for most practical purposes.
1. INTRODUCTION Acoustic noise transmission, at low frequencies, through the walls of rectangular ducts is of considerable importance to the engineer concerned with the acoustic design of air conditioning ductwork, since it represents a possible path whereby system noise may be radiated into interior building spaces. The author has previously carried out theoretical and experimental investigations into several features of duct wall transmission, and these are reported in references [1--6]. Guthrie [7] has also carried out an experimental study of this phenomenon, and has presented comparisons between his measurements of sound transmission through the walls of rectangular ducts, and theoretical results obtained on the basis of work described in references [1] and [2]. Generally good agreement was obtained between predictions and measurements of the duct wall transmission loss (TL,= 10 log [internal sound power/radiated sound power per unit length of duct]). In references [1, 2] and [4-7] a number of comparisons between the predicted and measured duct wall TL for both square and rectangular section ducts are given. Agreement between theory and measurement is, in the main, within about 3 dB, and some of the discrepancy, at least, can be attributed to experimental error. It is felt that the theoretical approach to calculating the wall TL of ducts, though fairly simple, is likely to give reliable results in most applications. In order to apply the theory, however, one must have recourse to a digital computer. Eight linear equations need to be solved simultaneously for each type of duct and at each frequency, and the matrix elements involve trigonometric and hyperbolic functions; additionally, a N e w t o n - R a p h s o n iteration scheme is part of the solution process. Clearly, these calculations are beyond the scope of even a programmable pocket calculator (since double precision, complex arithmetic is also required). It is doubful whether the majority of engineers involved in design work would have the time available to devote to developing a suitable computer program to carry out the calculations, and so there would appear to be 269
270
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CUMMINGS
a need for some kind of "design charts" for estimating the wall TL of rectangular ducts, whereby one could estimate the TL without having to carry out lengthy calculations. In reference [1], the author stated " A t present, there would seem to be some difficulty associated with producing design charts for duct wall transmission l o s s . . , h o w e v e r . . , this problem would merit further p u r s u i t . . . " . The author has considered this question further, and it appears that one can generate a comprehensive but moderately compact set of design curves by introducing non-dimensional quantities, as was done in reference [6]. The purpose of the present article is to give a set of design charts for the wall TL of rectangular ducts, intended to cover the range of sheet metal ducts encountered in practice, and to explain the use of the charts. 2. THEORY The theory used to calculate the TL of the duct walls is identical to that described in references [1] and [2]; it is based on the idea of a coupled wave system embodying acoustic waves travelling in the air contained within the duct, and structural waves travelling in the duct's walls; the acoustic pressure within the duct is taken to be uniform on a duct cross section. The expression for the TL given in reference [1] is (for similar fluids inside and outside the duct):
TL = 10 log (abc~/C,~ocAafiy + bfiz 12)
(1)
(see Figure 1 and the list of symbols in the Appendix; see also reference [4] for a fuller discussion of Cr). The equations which are solved to find the wall admittances contain terms involving a, b, 1", E, r/, tr, h and m (again, see the list of symbols in the Appendix).
Figure 1. Dimensions of duct.
Additionally, the length, L, of the duct is involved in calculating 6",. Clearly, this number of variables is not easily accommodated in a design procedure. One may, however, reduce the number of parameters by introducing non-dimensional quantities similar to those used in reference [6]. These are as follows: fundamental resonance frequency parameter, frequency parameter,
F = (g/m)l/2/coa,
C=fa2(m/g) 1/~,
mass per unit area parameter,
(2a) (2b)
M = m/poa,
(2c)
R = b/a.
(2d)
aspect ratio of duct cross section,
CHARTS FOR DUCT WALL TRANSMISSION
271
Here g is the fiexural rigidity of the duct wall material_~e ual to Eh3/12(1-o'2). If o- = 0 . 3 - - a typical value for metals--and one defines c = ~/E/pp, where pp is the density of the wall material, then equations (2a--c) may be rewritten as F = 0.3026 (C/co)h/a,
C = 3.305 (fa/c)a/h,
M = (pp/po)h/a.
(2e-g)
If pp, po, c and Co are constants, then equations (2e-g) would be more convenient to use than equations (2a-c), since then the wall admittances would be functions only of h/a, fa/c and b/a. The structural loss factor, r/, is incorporated by allowing the Young's modulus to be complex, and equal to E(1 + it/), in the calculation of the TL. The parameters F and C, however, are defined on the basis of the static value of E (equal to the real part of the dynamic modulus), and so these must be multiplied by ( l + i r / ) 1/2 and ( l + i r / ) -t/2, respectively, before insertion into the expression for TL in terms of non-dimensional parameters. In references [1-6], some discussion took place concerning r/. Since the TL theory (necessarily) incorporates a zero radiation impedance in estimating the duct wall response to the internal sound pressure field, ~7 must be given an appropriately large value in order to limit the wall response to realistic proportions, at and near wall resonance frequencies. A figure of 0.2 has given, overall, about the best results in cases where comparison between experiment and theory was possible, and accordingly 7/ will be assumed here to have the (constant) value of 0.2, throughout, pending information on its frequency dependence. The wall admittance may now be expressed in terms of F, C, M and R, and additionally cx/co. The dispersion equation for the coupled wave system may be written in terms of admittances and F, C and R, as well as c~/co, and hence may be solved for c~/co if F, C, M and R are specified. Because dissipation has been accounted for, this leads to a complex value of cx/co--and the imaginary part of Co/Cx represents the axial attenuation of the wave system--and so the real part of co/c~ is taken here to represent the inverse of the non-dimensional phase speed. A useful version of equation (1) may be written as
TL = T L ' - 10 l o g ] ' - 10 log C,;
(3)
here
TL'= 10 log (c2oR/2zrc~[ff~ + R/g+.]2),
(4) m
and may be expressed in terms o f F , C, M and R (since cx/co and fl~, flz are also dependent only upon F, C, M and R ) if a fixed value of Co is taken (in the present case, Co will be taken to be 343.6 m/s). If a series of curves of TL' and c,/co can be drawn for a range of values of F, C, M and R, one may find C, from cx/co and L at any given frequency, and so TL may be found at any frequency, from equation (3). In reference [6], the various ways in which the fluid-borne acoustic wave in the duct can combine with the structure-borne wave were described. It was shown that in certain types of these wave combinations, most of the energy flow was carried by the acoustic wave in the fluid, and these " m o d e s " were called A modes. It was also shown that modes existed wherein most of the power was transmitted by the structural wave, and these were termed S modes. Now the latter type of wave combination is acoustically "leaky": that is, it readily transmits sound from inside the duct to the exterior, whereas A modes transmit much less sound out through the duct walls. It was argued, however, in reference [6], that in the case of ordinary sheet metal ducts, A mode transmission should dominate over S mode transmission if the initial excitation of the duct walls occurred by internal acoustic waves. This seems to be borne out by measurements, which are generally in good agreement with
272
A. CUMMINGS
predictions made on thc basis of A mode transmission. Accordingly, all of the T L data presented in this article will bc bascd on A mode transmission only. One may sclcct A modes by simplyusing 1.0 as an initial gucss in a Newton-Raphson scheme for finding roots CJCo of the dispersion cquation. The root found by this method always corresponds to an A mode. Curves of TL', as a function of C, were produced as described, for frequencies up to the cut-on frequency of thc first acoustic higher order mode in the rigid walled duct with the same transverse dimensions as the duct in question; this limit was imposed because, above the cut-on frequency, highcr order modes may propagate, thereby violating the uniform pressure assumption on which the present theory is based. Since the dimension a of the duct is taken here as being the larger of the duct dimensions (unless the duct is square), this cut-on frequency corresponds to toa/co = ~r, o r - - a s one may s h o w - - t o C = 1/2F. The lower limit on C is 1.0, and so the frequency range is reprcsentcd by 1 < C < 1/2F. This region includes the fundamcntal transverse duct wall resonance frequency, and extends to frequencies somcwhat below this. At frcquencies whcrc C < 1-0, the simplc quasi-static formula (see later) dcscribcd in reference [6] may be used to predict TL', since the T L hcre is "stittncss controlled", and the tcmporal and axial spatial dependence, of the wave motion in the duct walls, may be ignorcd in favor of the transverse spatial dependcncc. Curves of cx--derivcd from c~/co, with Co = 343.6 m / s - - w e r e plotted on the same charts as thc TL' curves, for the same values of F, M and R, and for the same ranges of values of C. 3. USE OF THE DESIGN CHARTS The charts of TL' and cx appear in Figures 2-6. Each chart is drawn for fixed values of F and R, and embodies five curves of T L ' - - e a c h corresponding to a fixed value of M ' - - a n d five curves of cx, each of which, again, corresponds to a particular value of M. The values of M chosen are 5, 10, 20, 40 and 80, and this should cover the range of sheet metal ducts normally encountered in practice. The upper limit of the C scale corresponds in all cases to the cut-on frequency for the first higher order acoustic mode in the duct. The values of F chosen were 0.0025, 0.005, 0.01, 0.02 and 0.04, and again, this should encompass the range of values of F encountered in practice. The values of R selected were 0.2, 0.4, 0.6, 0.8, 0.9 and 1.0, and again, the aspect ratio of most practical duct systems should fall within this range. With six values of R and five values of F, one has 30 possible combinations, and it is not felt that more charts could reasonably be included in a design method. For a duct of given wall material and transverse dimensions, one could determine curves of TL' and c~ versusf (by appropriate transitions between dimensional and non-dimensional quantities). It remains still to find 6",, however, and to do this, one first calculates a quantity X, defined as Z = 2rrfL(1/cx - 1/Co),
(5)
at each frequency. Figurc 7 gives a curve of - 1 0 log C, versusx (equivalently versusf), so one may detcrminc the radiation efficiency term in equation (3) at each frequency. Thc range of values of X in Figure 7 should encompass virtually all possible situations. If X < - 6 , one may put - 1 0 log Cr = 0; physically, this means that thc duct radiates likc a line source of infinite length, carrying "peristaltic" waves of supersonic phase speed. It is unlikely that one would encountcr the situation where X > 100, but in that event, one could use an asymptotic formula, 6",
,1/rrX,
X~OO
(6a)
273
CIIARTS FOR DUCT WALL TRANSMISSION
that is - 1 0 1 o g C,
, 1 0 1 o g x + 5 dB.
(6b)
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R(MF~2c~ --~3/1-72-2-2-~R-') F Coc [ (,l + R a ) Z
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(8)
TL' and cx are found, for C = 0.5, from equations (7) and (8) and - 1 0 log C, is found from Figure 7. If this last quantity is close to 3 dB, the implication is that for C < 0.5, it will also be about 3 dB, so the correction to TL', for C,, will be effectively constant for C < 0.5. The TL is now calculated (from equation (3)) for C = 0.5, and a straight line drawn with a slope of - 9 dB/octave from the corresponding point on a TL/f plot, in the direction of decreasing frequency. This line may be connected, with a smooth curve, to the TL curve---derived from the design charts--terminating at the frequency corresponding to C=1. If - 1 0 log Cr is not close to 3 dB at C = 0.5 (and it will be greater than 3 dB), one must calculate TL (as above) at a number of values of C, less than 0.5, as desired, and complete the curve of TL versus f up to C = 0.5. Again, this curve is connected to the TL curve derived from the design charts. 3.2. INTERPOLATION AND TL CURVE PLOTrING In general, some sort of interpolation will be required, between the curves of TL' shown in Figures 2--6, since it is unlikely that the values of M, F and R for any given duct will correspond precisely to the values used in drawing the charts. Interpolation between different values of M is simple, since the adjacent curves are almost parallel to one another in most cases. Sketching in an intermediate curve corresponding to, say, M = 27, between an M = 20 curve and an M = 40 curve, is straightforward, and may be done sufficiently accurately by hand. Interpolation between fixed values of R and F is less easy.
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~.'ersus
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It is recommended here that the following procedure be adopted. First calculate the values of M, R and F for the duct in question. Secondly, find the two values of R, on the charts, between which the actual value of R lies (one may call these R1 and R2). Next, find Ft and F2, the values of F (from the charts) between which the actual value of F lies. Now there are four combinations of R and F: RI, F1; R1, F 2 ; R 2 , F I ; and R2, F 2 . Draw the TL' curves--interpolated between the two values of M--versus C, for these four combinations. Now take the curves for RI, FI and R2, FI, and overlay them on the same axes. Interpolate between the curves for R~ and R E , using the appropriate actual value of R for the duct (again, hand sketching should be adequate). The interpolation here will not be quite so straightforward as it is in the case of M, and one must take care to try to locate the maxima and minima in the interpolated curve in approximately the correct positions; this is necessary, since, for different values of R, the minima in TL' corresponding to particular duct wall resonances will be located at different values of C. Next, take the curves for R1, F2 and R2, F2 and perform exactly the same interpolation process as that described above. Finally, interpolation between FI and F2 will be necessary. Since the range of values of C will be different between the two cases, it will be necessary to take the 77_.'curve (from the above two interpolated curves) corresponding to the larger value of F, and replot it against the other curve. Then one again sketches an interpolated curve, corresponding to the actual value of F, between the curves for Ft and F2. The upper limit on C may be calculated from the actual value of F, though extrapolation, rather than interpolation, of the TL' curve to this value will be necessary for a narrow region close to the upper limit of C. Similar interpolation of the c~ curves will be required, although, perhaps, less care need be taken concerning accuracy, since 10 log C, is not extremely sensitive to changes in cx. Photocopying the charts in the present article will assist in the above interpolation processes, and so will the use of pencil and tracing paper. Step by step, the design procedure is as follows. (i) Calculate R, F, M for the duct in question. (ii) Choose four of the design charts, such that the actual values of R and F for the duct fall between the fixed values corresponding to the design charts. (iii) For the actual value of M, draw interpolated curves of both TL' and cx, on each of the four design charts. (iv) For each of the two fixed values of F, draw curves of both TL' and cx, interpolated between the two interpolated curves obtained from (iii), for the actual value of R.
CHARTS FOR DUCT WALL TRANSMISSION
285
(v) Draw the two TL' curves obtained from (iv) on the same C scale, and do likewise with the cx curves. Then draw interpolated TL' and cx curves, between the curves for the two fixed values of F, for the actual value of F. One TL' curve and one cx curve result from this. Truncate the upper ends of both curves at C = 1/2F. (vi) Draw up a table, with headings (from left to right): C, [, TL', cx, X, - 1 0 log C,, TL. (vii) From examination of the TL' curve obtained from (v), choose a series of values of C which will enable the eventual TL curve to be plotted accurately, considering the maxima and minima in particular; enter these values in the C column in the table. (viii) Calculate the value of ]', corresponding to each value of C above, as/" = CFco/a and enter the f values in the table. (ix) L o o k up the TL' and c~ values from the curves obtained in (v), for each value of C, and also enter these in the appropriate columns of the table. (x) From c~, I" and the length of the duct, calculate X at each frequency, and enter the g values in the table. (xi) From Figure 7, look up - 1 0 log C, for each value of X, and enter these values in the table. (xii) Finally, for each frequency, calculate TL as TL = T L ' - 10 log [ - 10 log C,, and enter these values in the table. (xiii) Plot a curve of TL against frequency. (xiv) If necessary, extend this curve to lower frequencies, using the quasi-static approximation previously described in this section. From start to finish, it should take in the order of two hours to obtain a TL curve, though this time may be reduced with practice. For occasional T L estimates this time is not inordinate, though if the process has to be carried out frequently, it is clearly advantageous for one to write a computer program to perform the calculations, based on the theory described in references I1] and [2].
4. EXAMPLES OF DESIGN APPLICATION In this section, two ducts are chosen as "design examples". The object here is to indicate the sort of accuracy to be expected from the design charts, by comparison with calculations made by using the coupled wave theory. 4.1. C A S E 1: A D U C T O F 582 m m x 2 5 8 mm C R O S S S E C T I O N This cross sectional size was chosen because Guthrie [7] presents measurements on an "18 gauge" galvanized steel duct of this size. The data on this duct are as follows: a = 582 mm; b = 258 mm; L = 1.2 m; h = 1-37 mm; m = 10.72 kg/m2; E = 1.82 x 1011 Pa; tr = 0.29; p0 = 1.2 kg/m3; Co = 343.6 m/s. For this duct, one finds that R = 0.443, F = 0.01 and M = 15.35. The value of F is, by coincidence, nearly enough 0.01, which is the same as one of the fixed values on the design charts. This, naturally, simplifies matters somewhat. Only two design charts are required: those for R = 0.4, F = 0.01 and R = 0.6, F = 0.01. Figure 8 shows an intermediate stage in the process of finding the TL curve, namely that of interpolating between the TL' curves corresponding to the two fixed values of R. T h e curves for the two extreme values of R are shown, and also the interpolated curve. It is important that correspondhzg maxima and minima in the two extreme curves be correctly identified (such as: 1st, 2nd, 3rd, etc., in order of increasing frequency), otherwise inaccuracies will result. The first maxima on the curves in Figure 8 are indicated, for
286
A. CUMMINGS
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F i g u r e 8. I n t e r p o l a t i o n in R of T L ' f r o m d e s i g n e x a m p l e , case 1. - - - - , R = 0.6, F = 0.01, M = 1 5 . 3 5 ; - - , i n t e r p o l a t e d c u r v e w i t h R = 0 . 4 4 3 , F = 0 . 0 1 , M = 15.35. R = 0.4, F = 0.01, M = 1 5 - 3 5 ;
example. One sees that there is some scope for error in the interpolation in R, but if care is taken, this need not be too great. Figure 9 shows the TL curve obtained f r o m the design charts, and also that obtained directly from the TL theory, together with Guthrie's measurements.
60
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Frequency(Hz) F i g u r e 9. TL of d u c t f r o m d e s i g n e x a m p l e , case 1. , TL from design charts; ----, @, G u t h r i e ' s m e a s u r e m e n t s [7]; R = 0 . 4 4 3 , F = 0 . 0 1 , M = 1 5 . 3 5 .
d i r e c t TL c a l c u l a t i o n ;
CHARTS FOR
DUCTWALL TRANSMISSION
287
O n e sees that the curve from the design charts follows the directly calculated curve closely over most of the frequency range, the m a x i m u m error being about 3 dB at 70 Hz. Elsewhere, the curves are generally much closer together. A g r e e m e n t with m e a s u r e m e n t s appears to be good except at 100 Hz. At least part of this discrepancy might be caused by m e a s u r e m e n t error, for Guthrie's m e a s u r e m e n t s were carried out in an anechoic c h a m b e r and the low frequency accuracy Was poor. It must be pointed out that the curve from the design charts was deliberately drawn before the direct calculation was carried out, s o that there was no "bias", during the interpolation process, toward making the design chart estimate agree with the direct calculation. Accordingly, the comparison here should be representative of what one might expect m o r e generally. This applies also to the curves described in the following section. O n e feels that, in this case at least, the "accuracy '' obtained from the design charts would be sufficient for most engineering purposes. 4.2. CASE 2: A DUCT OF 330 m m X 2 5 4 m m CROSS SECTION This duct size was chosen as being, perhaps, of fairly typical cross sectional size (it is actually 10 in x 13 in) and length (which is 10 ft, or 3.05 m), but otherwise arbitrarily. T h e wall material was taken, as before, to be 18 gauge galvanized steel. The appropriate data are as for the duct in section 4.1, except that a = 330 m m , b = 254 m m and L = 3.05 m. For this duct, one has R = 0.769, F = 0.0176 and M = 27.05. In this case, the full interpolation process must be gone through. (No m e a s u r e m e n t s are available on this duct.) Figure 10 shows the results. A g r e e m e n t between the T L curve found from the design chart and that obtained by direct calculation is very good in this case, and the m a x i m u m discrepancy between the curves is 1.5 dB. Again, it can be seen that the accuracy of the design charts ought to be adequate for engineering purposes.
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, TL from design charts; - - - - , direct TL calculation;
288
A. C U M M I N G S
5. DISCUSSION One feels that the design method described in this article is, although--inevitably--a little tedious, sufficiently straightforward that it can be used reasonably easily by the majority of engineers. T h e accuracy of the charts would seem to be adequate for most practical purposes, provided care is exercised in the interpolation processes (particular care would be necessary for ducts with small values of R). It is difficult to be more specific about the expected accuracy, since this would be determined partly by the individual using the charts. It was not, incidentally, considered worthwhile to encumber this article ftirther with tables of data on common duct wall materials, since these are generally easily available. In the case of inhomogeneous material such as galvanized steel, it is a simple matter to find the effective Young's modulus for the "equivalent" homogeneous material of the same thickness, by calculating the combined stiffness of the metal sheet plus coating; similarly, the mass per unit area is easily found. A n y further "collapse" of the TL' and cx curves given here would seem rather unlikely at present. It might, be tempting, for example, to collapse the TL' curves on each chart, for the five values of M, on to a common curve, but the shapes are sufficiently dissimilar, overall, that excessive inaccuracy would be incurred.
ACKNOWLEDGMENT T h e author is indebted to Noel Walkington, for his invaluable help and advice on computer graph plotting routines. REFERENCES 1. A. CUMMINGS 1978 Journal o[ Sound and Vibration 61, 327-345. Low frequency acoustic transmission through the walls of rectangular ducts. 2. A. CUMMINGS 1979 Journal of Sound and Vibration 63, 463--465. Low frequency sound transmission through the walls of rectangular ducts: further comments. 3. A. CUMMINGS 1979 Journalo[Sound and Vibration 67, 187-201. The effects of external lagging on low frequency sound transmission through the walls of rectangular ducts. 4. A. CUMMINGS 1980 Journal of Sound and Vibration 71, 201-226. Low frequency acoustic radiation from duct walls. 5. A. CUMMINGS 1980 (April) Building Services andEnvironmentalEngineer 6-7. Noise breakout from air conditioning ducts. 6. A. CUMMINGS 1981 Journal o[Sound and Vibration 74, 351-380. Stiffness control of low frequency acoustic transmission through the walls of rectangular ducts. 7. A. GWrHRIE 1979 MSc. Dissertation, Polytechnic of the South Bank. Low frequency acoustic transmission through the walls of various types of ducts.
APPENDIX: LIST OF SYMBOLS a,b C c, C Co Cx
E F
transverse duct dimensions frequency parameter radiation efficiency of duct wall adiabatic speed of sound axial phase speed of coupled wave system static Young's modulus of duct wall material fundamental resonance frequency parameter (equation (2a))
CttARTS FOR DUCT WALL TRANSMISSION
F~,F~ / g,h L M DI
R
R1, R2 TL TL'
L,L rl 17"
Po
Op o"
x
specific values of F frequency flexural rigidity, thickness of duct wall material radiating length of duct mass per unit area parameter (equation (2c)) mass per unit area of duct wall material aspect ratio of duct cross section (equation (2d)) specific values of R duct wall transmission loss (equation (1)) quantity related to TL (defined in equation (3)) duct wall admittancesaveraged across opposite pairs of walls structural loss factor of duct wall material 3.14159... density of air density of wall material Poisson's ratio of duct wall material non-dimensional parameter defined in equation (5) radian frequency
289
DESIGN CHARTS FOR LOW FREQUENCY ACOUSTIC TRANSMISSION T H R O U G H T H E WALLS OF R E C T A N G U L A R DUCTS A. CUMMINGS
Department of i~fechanical and Aerospace Engineering, University of Missouri-Rolla, Rolla Missouri 65401, U.S.A. (Received 26 November 1980, and h2 revised [onn 24 Afarch 1981) "Design charts" are presented here, from which the low frequency acoustic transmission loss of the walls of rectangular ducts may be found. The upper frequency limit on the applicability of this method is at cut-on for the first cross mode in the equivalent rigid walled duct, and it is shown how a quasi-static theory can easily be used to extend the results, as obtained from the charts, to lower frequencies. The data are given in terms of nondimensional parameters, in order to keep the number of charts to reasonable proportions. Two "design examples" are described here, and it is felt that the accuracy in the use of the charts ought to be sufficient for most practical purposes.
1. INTRODUCTION Acoustic noise transmission, at low frequencies, through the walls of rectangular ducts is of considerable importance to the engineer concerned with the acoustic design of air conditioning ductwork, since it represents a possible path whereby system noise may be radiated into interior building spaces. The author has previously carried out theoretical and experimental investigations into several features of duct wall transmission, and these are reported in references [1--6]. Guthrie [7] has also carried out an experimental study of this phenomenon, and has presented comparisons between his measurements of sound transmission through the walls of rectangular ducts, and theoretical results obtained on the basis of work described in references [1] and [2]. Generally good agreement was obtained between predictions and measurements of the duct wall transmission loss (TL,= 10 log [internal sound power/radiated sound power per unit length of duct]). In references [1, 2] and [4-7] a number of comparisons between the predicted and measured duct wall TL for both square and rectangular section ducts are given. Agreement between theory and measurement is, in the main, within about 3 dB, and some of the discrepancy, at least, can be attributed to experimental error. It is felt that the theoretical approach to calculating the wall TL of ducts, though fairly simple, is likely to give reliable results in most applications. In order to apply the theory, however, one must have recourse to a digital computer. Eight linear equations need to be solved simultaneously for each type of duct and at each frequency, and the matrix elements involve trigonometric and hyperbolic functions; additionally, a N e w t o n - R a p h s o n iteration scheme is part of the solution process. Clearly, these calculations are beyond the scope of even a programmable pocket calculator (since double precision, complex arithmetic is also required). It is doubful whether the majority of engineers involved in design work would have the time available to devote to developing a suitable computer program to carry out the calculations, and so there would appear to be 269
270
^.
CUMMINGS
a need for some kind of "design charts" for estimating the wall TL of rectangular ducts, whereby one could estimate the TL without having to carry out lengthy calculations. In reference [1], the author stated " A t present, there would seem to be some difficulty associated with producing design charts for duct wall transmission l o s s . . , h o w e v e r . . , this problem would merit further p u r s u i t . . . " . The author has considered this question further, and it appears that one can generate a comprehensive but moderately compact set of design curves by introducing non-dimensional quantities, as was done in reference [6]. The purpose of the present article is to give a set of design charts for the wall TL of rectangular ducts, intended to cover the range of sheet metal ducts encountered in practice, and to explain the use of the charts. 2. THEORY The theory used to calculate the TL of the duct walls is identical to that described in references [1] and [2]; it is based on the idea of a coupled wave system embodying acoustic waves travelling in the air contained within the duct, and structural waves travelling in the duct's walls; the acoustic pressure within the duct is taken to be uniform on a duct cross section. The expression for the TL given in reference [1] is (for similar fluids inside and outside the duct):
TL = 10 log (abc~/C,~ocAafiy + bfiz 12)
(1)
(see Figure 1 and the list of symbols in the Appendix; see also reference [4] for a fuller discussion of Cr). The equations which are solved to find the wall admittances contain terms involving a, b, 1", E, r/, tr, h and m (again, see the list of symbols in the Appendix).
Figure 1. Dimensions of duct.
Additionally, the length, L, of the duct is involved in calculating 6",. Clearly, this number of variables is not easily accommodated in a design procedure. One may, however, reduce the number of parameters by introducing non-dimensional quantities similar to those used in reference [6]. These are as follows: fundamental resonance frequency parameter, frequency parameter,
F = (g/m)l/2/coa,
C=fa2(m/g) 1/~,
mass per unit area parameter,
(2a) (2b)
M = m/poa,
(2c)
R = b/a.
(2d)
aspect ratio of duct cross section,
CHARTS FOR DUCT WALL TRANSMISSION
271
Here g is the fiexural rigidity of the duct wall material_~e ual to Eh3/12(1-o'2). If o- = 0 . 3 - - a typical value for metals--and one defines c = ~/E/pp, where pp is the density of the wall material, then equations (2a--c) may be rewritten as F = 0.3026 (C/co)h/a,
C = 3.305 (fa/c)a/h,
M = (pp/po)h/a.
(2e-g)
If pp, po, c and Co are constants, then equations (2e-g) would be more convenient to use than equations (2a-c), since then the wall admittances would be functions only of h/a, fa/c and b/a. The structural loss factor, r/, is incorporated by allowing the Young's modulus to be complex, and equal to E(1 + it/), in the calculation of the TL. The parameters F and C, however, are defined on the basis of the static value of E (equal to the real part of the dynamic modulus), and so these must be multiplied by ( l + i r / ) 1/2 and ( l + i r / ) -t/2, respectively, before insertion into the expression for TL in terms of non-dimensional parameters. In references [1-6], some discussion took place concerning r/. Since the TL theory (necessarily) incorporates a zero radiation impedance in estimating the duct wall response to the internal sound pressure field, ~7 must be given an appropriately large value in order to limit the wall response to realistic proportions, at and near wall resonance frequencies. A figure of 0.2 has given, overall, about the best results in cases where comparison between experiment and theory was possible, and accordingly 7/ will be assumed here to have the (constant) value of 0.2, throughout, pending information on its frequency dependence. The wall admittance may now be expressed in terms of F, C, M and R, and additionally cx/co. The dispersion equation for the coupled wave system may be written in terms of admittances and F, C and R, as well as c~/co, and hence may be solved for c~/co if F, C, M and R are specified. Because dissipation has been accounted for, this leads to a complex value of cx/co--and the imaginary part of Co/Cx represents the axial attenuation of the wave system--and so the real part of co/c~ is taken here to represent the inverse of the non-dimensional phase speed. A useful version of equation (1) may be written as
TL = T L ' - 10 l o g ] ' - 10 log C,;
(3)
here
TL'= 10 log (c2oR/2zrc~[ff~ + R/g+.]2),
(4) m
and may be expressed in terms o f F , C, M and R (since cx/co and fl~, flz are also dependent only upon F, C, M and R ) if a fixed value of Co is taken (in the present case, Co will be taken to be 343.6 m/s). If a series of curves of TL' and c,/co can be drawn for a range of values of F, C, M and R, one may find C, from cx/co and L at any given frequency, and so TL may be found at any frequency, from equation (3). In reference [6], the various ways in which the fluid-borne acoustic wave in the duct can combine with the structure-borne wave were described. It was shown that in certain types of these wave combinations, most of the energy flow was carried by the acoustic wave in the fluid, and these " m o d e s " were called A modes. It was also shown that modes existed wherein most of the power was transmitted by the structural wave, and these were termed S modes. Now the latter type of wave combination is acoustically "leaky": that is, it readily transmits sound from inside the duct to the exterior, whereas A modes transmit much less sound out through the duct walls. It was argued, however, in reference [6], that in the case of ordinary sheet metal ducts, A mode transmission should dominate over S mode transmission if the initial excitation of the duct walls occurred by internal acoustic waves. This seems to be borne out by measurements, which are generally in good agreement with
272
A. CUMMINGS
predictions made on thc basis of A mode transmission. Accordingly, all of the T L data presented in this article will bc bascd on A mode transmission only. One may sclcct A modes by simplyusing 1.0 as an initial gucss in a Newton-Raphson scheme for finding roots CJCo of the dispersion cquation. The root found by this method always corresponds to an A mode. Curves of TL', as a function of C, were produced as described, for frequencies up to the cut-on frequency of thc first acoustic higher order mode in the rigid walled duct with the same transverse dimensions as the duct in question; this limit was imposed because, above the cut-on frequency, highcr order modes may propagate, thereby violating the uniform pressure assumption on which the present theory is based. Since the dimension a of the duct is taken here as being the larger of the duct dimensions (unless the duct is square), this cut-on frequency corresponds to toa/co = ~r, o r - - a s one may s h o w - - t o C = 1/2F. The lower limit on C is 1.0, and so the frequency range is reprcsentcd by 1 < C < 1/2F. This region includes the fundamcntal transverse duct wall resonance frequency, and extends to frequencies somcwhat below this. At frcquencies whcrc C < 1-0, the simplc quasi-static formula (see later) dcscribcd in reference [6] may be used to predict TL', since the T L hcre is "stittncss controlled", and the tcmporal and axial spatial dependence, of the wave motion in the duct walls, may be ignorcd in favor of the transverse spatial dependcncc. Curves of cx--derivcd from c~/co, with Co = 343.6 m / s - - w e r e plotted on the same charts as thc TL' curves, for the same values of F, M and R, and for the same ranges of values of C. 3. USE OF THE DESIGN CHARTS The charts of TL' and cx appear in Figures 2-6. Each chart is drawn for fixed values of F and R, and embodies five curves of T L ' - - e a c h corresponding to a fixed value of M ' - - a n d five curves of cx, each of which, again, corresponds to a particular value of M. The values of M chosen are 5, 10, 20, 40 and 80, and this should cover the range of sheet metal ducts normally encountered in practice. The upper limit of the C scale corresponds in all cases to the cut-on frequency for the first higher order acoustic mode in the duct. The values of F chosen were 0.0025, 0.005, 0.01, 0.02 and 0.04, and again, this should encompass the range of values of F encountered in practice. The values of R selected were 0.2, 0.4, 0.6, 0.8, 0.9 and 1.0, and again, the aspect ratio of most practical duct systems should fall within this range. With six values of R and five values of F, one has 30 possible combinations, and it is not felt that more charts could reasonably be included in a design method. For a duct of given wall material and transverse dimensions, one could determine curves of TL' and c~ versusf (by appropriate transitions between dimensional and non-dimensional quantities). It remains still to find 6",, however, and to do this, one first calculates a quantity X, defined as Z = 2rrfL(1/cx - 1/Co),
(5)
at each frequency. Figurc 7 gives a curve of - 1 0 log C, versusx (equivalently versusf), so one may detcrminc the radiation efficiency term in equation (3) at each frequency. Thc range of values of X in Figure 7 should encompass virtually all possible situations. If X < - 6 , one may put - 1 0 log Cr = 0; physically, this means that thc duct radiates likc a line source of infinite length, carrying "peristaltic" waves of supersonic phase speed. It is unlikely that one would encountcr the situation where X > 100, but in that event, one could use an asymptotic formula, 6",
,1/rrX,
X~OO
(6a)
273
CIIARTS FOR DUCT WALL TRANSMISSION
that is - 1 0 1 o g C,
, 1 0 1 o g x + 5 dB.
(6b)
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R(MF~2c~ --~3/1-72-2-2-~R-') F Coc [ (,l + R a ) Z
R(1-R)(1-R2)48
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TL' and cx are found, for C = 0.5, from equations (7) and (8) and - 1 0 log C, is found from Figure 7. If this last quantity is close to 3 dB, the implication is that for C < 0.5, it will also be about 3 dB, so the correction to TL', for C,, will be effectively constant for C < 0.5. The TL is now calculated (from equation (3)) for C = 0.5, and a straight line drawn with a slope of - 9 dB/octave from the corresponding point on a TL/f plot, in the direction of decreasing frequency. This line may be connected, with a smooth curve, to the TL curve---derived from the design charts--terminating at the frequency corresponding to C=1. If - 1 0 log Cr is not close to 3 dB at C = 0.5 (and it will be greater than 3 dB), one must calculate TL (as above) at a number of values of C, less than 0.5, as desired, and complete the curve of TL versus f up to C = 0.5. Again, this curve is connected to the TL curve derived from the design charts. 3.2. INTERPOLATION AND TL CURVE PLOTrING In general, some sort of interpolation will be required, between the curves of TL' shown in Figures 2--6, since it is unlikely that the values of M, F and R for any given duct will correspond precisely to the values used in drawing the charts. Interpolation between different values of M is simple, since the adjacent curves are almost parallel to one another in most cases. Sketching in an intermediate curve corresponding to, say, M = 27, between an M = 20 curve and an M = 40 curve, is straightforward, and may be done sufficiently accurately by hand. Interpolation between fixed values of R and F is less easy.
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It is recommended here that the following procedure be adopted. First calculate the values of M, R and F for the duct in question. Secondly, find the two values of R, on the charts, between which the actual value of R lies (one may call these R1 and R2). Next, find Ft and F2, the values of F (from the charts) between which the actual value of F lies. Now there are four combinations of R and F: RI, F1; R1, F 2 ; R 2 , F I ; and R2, F 2 . Draw the TL' curves--interpolated between the two values of M--versus C, for these four combinations. Now take the curves for RI, FI and R2, FI, and overlay them on the same axes. Interpolate between the curves for R~ and R E , using the appropriate actual value of R for the duct (again, hand sketching should be adequate). The interpolation here will not be quite so straightforward as it is in the case of M, and one must take care to try to locate the maxima and minima in the interpolated curve in approximately the correct positions; this is necessary, since, for different values of R, the minima in TL' corresponding to particular duct wall resonances will be located at different values of C. Next, take the curves for R1, F2 and R2, F2 and perform exactly the same interpolation process as that described above. Finally, interpolation between FI and F2 will be necessary. Since the range of values of C will be different between the two cases, it will be necessary to take the 77_.'curve (from the above two interpolated curves) corresponding to the larger value of F, and replot it against the other curve. Then one again sketches an interpolated curve, corresponding to the actual value of F, between the curves for Ft and F2. The upper limit on C may be calculated from the actual value of F, though extrapolation, rather than interpolation, of the TL' curve to this value will be necessary for a narrow region close to the upper limit of C. Similar interpolation of the c~ curves will be required, although, perhaps, less care need be taken concerning accuracy, since 10 log C, is not extremely sensitive to changes in cx. Photocopying the charts in the present article will assist in the above interpolation processes, and so will the use of pencil and tracing paper. Step by step, the design procedure is as follows. (i) Calculate R, F, M for the duct in question. (ii) Choose four of the design charts, such that the actual values of R and F for the duct fall between the fixed values corresponding to the design charts. (iii) For the actual value of M, draw interpolated curves of both TL' and cx, on each of the four design charts. (iv) For each of the two fixed values of F, draw curves of both TL' and cx, interpolated between the two interpolated curves obtained from (iii), for the actual value of R.
CHARTS FOR DUCT WALL TRANSMISSION
285
(v) Draw the two TL' curves obtained from (iv) on the same C scale, and do likewise with the cx curves. Then draw interpolated TL' and cx curves, between the curves for the two fixed values of F, for the actual value of F. One TL' curve and one cx curve result from this. Truncate the upper ends of both curves at C = 1/2F. (vi) Draw up a table, with headings (from left to right): C, [, TL', cx, X, - 1 0 log C,, TL. (vii) From examination of the TL' curve obtained from (v), choose a series of values of C which will enable the eventual TL curve to be plotted accurately, considering the maxima and minima in particular; enter these values in the C column in the table. (viii) Calculate the value of ]', corresponding to each value of C above, as/" = CFco/a and enter the f values in the table. (ix) L o o k up the TL' and c~ values from the curves obtained in (v), for each value of C, and also enter these in the appropriate columns of the table. (x) From c~, I" and the length of the duct, calculate X at each frequency, and enter the g values in the table. (xi) From Figure 7, look up - 1 0 log C, for each value of X, and enter these values in the table. (xii) Finally, for each frequency, calculate TL as TL = T L ' - 10 log [ - 10 log C,, and enter these values in the table. (xiii) Plot a curve of TL against frequency. (xiv) If necessary, extend this curve to lower frequencies, using the quasi-static approximation previously described in this section. From start to finish, it should take in the order of two hours to obtain a TL curve, though this time may be reduced with practice. For occasional T L estimates this time is not inordinate, though if the process has to be carried out frequently, it is clearly advantageous for one to write a computer program to perform the calculations, based on the theory described in references I1] and [2].
4. EXAMPLES OF DESIGN APPLICATION In this section, two ducts are chosen as "design examples". The object here is to indicate the sort of accuracy to be expected from the design charts, by comparison with calculations made by using the coupled wave theory. 4.1. C A S E 1: A D U C T O F 582 m m x 2 5 8 mm C R O S S S E C T I O N This cross sectional size was chosen because Guthrie [7] presents measurements on an "18 gauge" galvanized steel duct of this size. The data on this duct are as follows: a = 582 mm; b = 258 mm; L = 1.2 m; h = 1-37 mm; m = 10.72 kg/m2; E = 1.82 x 1011 Pa; tr = 0.29; p0 = 1.2 kg/m3; Co = 343.6 m/s. For this duct, one finds that R = 0.443, F = 0.01 and M = 15.35. The value of F is, by coincidence, nearly enough 0.01, which is the same as one of the fixed values on the design charts. This, naturally, simplifies matters somewhat. Only two design charts are required: those for R = 0.4, F = 0.01 and R = 0.6, F = 0.01. Figure 8 shows an intermediate stage in the process of finding the TL curve, namely that of interpolating between the TL' curves corresponding to the two fixed values of R. T h e curves for the two extreme values of R are shown, and also the interpolated curve. It is important that correspondhzg maxima and minima in the two extreme curves be correctly identified (such as: 1st, 2nd, 3rd, etc., in order of increasing frequency), otherwise inaccuracies will result. The first maxima on the curves in Figure 8 are indicated, for
286
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example. One sees that there is some scope for error in the interpolation in R, but if care is taken, this need not be too great. Figure 9 shows the TL curve obtained f r o m the design charts, and also that obtained directly from the TL theory, together with Guthrie's measurements.
60
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d i r e c t TL c a l c u l a t i o n ;
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O n e sees that the curve from the design charts follows the directly calculated curve closely over most of the frequency range, the m a x i m u m error being about 3 dB at 70 Hz. Elsewhere, the curves are generally much closer together. A g r e e m e n t with m e a s u r e m e n t s appears to be good except at 100 Hz. At least part of this discrepancy might be caused by m e a s u r e m e n t error, for Guthrie's m e a s u r e m e n t s were carried out in an anechoic c h a m b e r and the low frequency accuracy Was poor. It must be pointed out that the curve from the design charts was deliberately drawn before the direct calculation was carried out, s o that there was no "bias", during the interpolation process, toward making the design chart estimate agree with the direct calculation. Accordingly, the comparison here should be representative of what one might expect m o r e generally. This applies also to the curves described in the following section. O n e feels that, in this case at least, the "accuracy '' obtained from the design charts would be sufficient for most engineering purposes. 4.2. CASE 2: A DUCT OF 330 m m X 2 5 4 m m CROSS SECTION This duct size was chosen as being, perhaps, of fairly typical cross sectional size (it is actually 10 in x 13 in) and length (which is 10 ft, or 3.05 m), but otherwise arbitrarily. T h e wall material was taken, as before, to be 18 gauge galvanized steel. The appropriate data are as for the duct in section 4.1, except that a = 330 m m , b = 254 m m and L = 3.05 m. For this duct, one has R = 0.769, F = 0.0176 and M = 27.05. In this case, the full interpolation process must be gone through. (No m e a s u r e m e n t s are available on this duct.) Figure 10 shows the results. A g r e e m e n t between the T L curve found from the design chart and that obtained by direct calculation is very good in this case, and the m a x i m u m discrepancy between the curves is 1.5 dB. Again, it can be seen that the accuracy of the design charts ought to be adequate for engineering purposes.
60
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, TL from design charts; - - - - , direct TL calculation;
288
A. C U M M I N G S
5. DISCUSSION One feels that the design method described in this article is, although--inevitably--a little tedious, sufficiently straightforward that it can be used reasonably easily by the majority of engineers. T h e accuracy of the charts would seem to be adequate for most practical purposes, provided care is exercised in the interpolation processes (particular care would be necessary for ducts with small values of R). It is difficult to be more specific about the expected accuracy, since this would be determined partly by the individual using the charts. It was not, incidentally, considered worthwhile to encumber this article ftirther with tables of data on common duct wall materials, since these are generally easily available. In the case of inhomogeneous material such as galvanized steel, it is a simple matter to find the effective Young's modulus for the "equivalent" homogeneous material of the same thickness, by calculating the combined stiffness of the metal sheet plus coating; similarly, the mass per unit area is easily found. A n y further "collapse" of the TL' and cx curves given here would seem rather unlikely at present. It might, be tempting, for example, to collapse the TL' curves on each chart, for the five values of M, on to a common curve, but the shapes are sufficiently dissimilar, overall, that excessive inaccuracy would be incurred.
ACKNOWLEDGMENT T h e author is indebted to Noel Walkington, for his invaluable help and advice on computer graph plotting routines. REFERENCES 1. A. CUMMINGS 1978 Journal o[ Sound and Vibration 61, 327-345. Low frequency acoustic transmission through the walls of rectangular ducts. 2. A. CUMMINGS 1979 Journal of Sound and Vibration 63, 463--465. Low frequency sound transmission through the walls of rectangular ducts: further comments. 3. A. CUMMINGS 1979 Journalo[Sound and Vibration 67, 187-201. The effects of external lagging on low frequency sound transmission through the walls of rectangular ducts. 4. A. CUMMINGS 1980 Journal of Sound and Vibration 71, 201-226. Low frequency acoustic radiation from duct walls. 5. A. CUMMINGS 1980 (April) Building Services andEnvironmentalEngineer 6-7. Noise breakout from air conditioning ducts. 6. A. CUMMINGS 1981 Journal o[Sound and Vibration 74, 351-380. Stiffness control of low frequency acoustic transmission through the walls of rectangular ducts. 7. A. GWrHRIE 1979 MSc. Dissertation, Polytechnic of the South Bank. Low frequency acoustic transmission through the walls of various types of ducts.
APPENDIX: LIST OF SYMBOLS a,b C c, C Co Cx
E F
transverse duct dimensions frequency parameter radiation efficiency of duct wall adiabatic speed of sound axial phase speed of coupled wave system static Young's modulus of duct wall material fundamental resonance frequency parameter (equation (2a))
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F~,F~ / g,h L M DI
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specific values of F frequency flexural rigidity, thickness of duct wall material radiating length of duct mass per unit area parameter (equation (2c)) mass per unit area of duct wall material aspect ratio of duct cross section (equation (2d)) specific values of R duct wall transmission loss (equation (1)) quantity related to TL (defined in equation (3)) duct wall admittancesaveraged across opposite pairs of walls structural loss factor of duct wall material 3.14159... density of air density of wall material Poisson's ratio of duct wall material non-dimensional parameter defined in equation (5) radian frequency
289