- Email: [email protected]
BOUNDARY LAYER INDUCED NOISE IN AIRCRAFT, PART II: THE TRIMMED FLAT PLATE MODEL
Journal of Sound and Vibration (1996) 192(1), 121–138
BOUNDARY LAYER INDUCED NOISE IN AIRCRAFT, PART II: THE TRIMMED FLAT PLATE MODEL W. R. G Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England (Received 7 February 1995, and in final form 5 July 1995) The influence of the cabin interior treatment on boundary layer noise levels is known to be significant, and we thus extend the model of Part I [1] to include it. The extended model consists of the boundary layer excited flat plate, with its internal surface covered by two dissipative layers (representing insulation) and an elastic plate (the trim panel). By comparison with the bare plate, the trimmed plate has much higher modal damping, due to the ability of the insulation to absorb energy at the wavenumbers associated with resonant modes, and greatly reduced radiation efficiency, due to the insulation’s attenuating effect on supersonic wavenumber disturbances. Most of the conclusions of Part I are essentially unaffected by this behavioural modification, but structural damping treatments are now expected to have a negligible effect on the sound radiated by the plate. 7 1996 Academic Press Limited
1. INTRODUCTION
In Part I of this work [1] we introduced a simple model for boundary layer induced noise in aircraft, based on the sound radiated by a single flat plate set in an infinite rigid baffle and driven by a turbulent boundary layer. After a numerical study of this model, we concluded that increases in structural damping and decreases in skin stiffness and the number of fuselage reinforcements were likely to have a beneficial effect on cabin sound levels. However, while the model was carefully specified in the light of the physical characteristics of the problem, we noted that it neglected one important component: the cabin interior treatment. In this paper we address that deficiency. In order to reduce cabin noise and heat loss, and for decorative reasons, passenger aircraft cabins are trimmed. Insulation bags containing a glass-fibre material are placed in each frame bay and covered with plastic trim panels, as shown schematically in Figure 1. This treatment is known to have a significant effect not only on interior sound levels, but also the fundamental behaviour of the problem [2], and our model must therefore be extended to include it as an integral part, rather than a later addition. Such an extension will require a description of wave propagation in the insulating material, a complicated phenomenon which has been the subject of much research. According to Bies [3], sound propagation in porous materials may be described by using the usual (single frequency) acoustic equations with a frequency-dependent complex mean density and sound speed to account for dissipative effects such as viscosity and heat transfer. However, this approach is only strictly valid if there is one wave type in the vibration, and in fact there may be several. Beranek [4] and Kosten and Janssen [5] presented theories that allow for two (one air-borne and one structure-borne), while the more sophisticated model of Bolton and Shiau [6] exhibits three (one air-borne and two 121 0022–460X/96/160121 + 18 $18.00/0
7 1996 Academic Press Limited
122
. .
Figure 1. Aircraft cabin interior treatment.
structure-borne). The method chosen to represent the insulation will thus require consideration. In this paper, we first consider the extension of the existing model to account for insulation and trim (section 2), and then, in section 3, present the results of a parallel numerical study to that of Part I. The conclusions of the work, and the extent to which they differ from those of Part I, are discussed in section 4.
2. EXTENSION OF THE FLAT PLATE MODEL
2.1. In considering this extension, we first re-examine the validity of representing a curved panel by a flat plate. In Part I we argued that the important factor is the acoustic dissipation of the interior surface, which must be low enough that the surrounding baffle should appear rigid, but high enough to reduce returning sound waves to negligible levels. Since we shall now be modelling the surface acoustic dissipation more carefully, the former requirement is no longer relevant, but the latter is still important. However, a trimmed and furnished aircraft contains many scatterers and absorbers of sound, so the sound returning to the plate will be only a small fraction of its radiated field, which will therefore be essentially the same as if the plate were radiating to infinity. The flat plate approximation will thus be a satisfactory representation of the actual problem. The geometry of the model must now be specified. The insulation bag contains two layers of differing insulation material, and is separated from the plastic trim panel by an air gap. We neglect the effects of finiteness in the insulation layers and trim panel (essentially assuming that the system is sufficiently damped that the wave behaviour appears to be travelling rather than reverberant), and define the model configuration of Figure 2. Here the insulation is represented by two dissipative layers, covered by a limp sheet (the insulation bag facing) and separated from an elastic plate (the trim panel) by
Figure 2. The trimmed flat plate model.
, II
123
Figure 3. A generic insulation layer.
an air gap. The mass of the insulation bag facing in contact with the plate is neglected in comparison to the plate mass. In the forthcoming analysis, we shall find the complete vibro-acoustic solution to this model, rather than taking the usual approach whereby the effect of insulation and trim is added to the bare structure solution in a more or less ad hoc manner (see, e.g., references [7, 8]). It remains to choose a model for wave propagation in the dissipative layers. Here we take the approach of Bies [3], and consider one wave type only. This approximation is supported by Beranek’s work [4], which showed that the longitudinal structure-borne wave is highly attenuated in flexible materials, such as aircraft insulation, and only the air-borne wave is important. It also has the advantage of simplicity, yielding results capable of physical interpretation and containing no more empirical parameters than may be determined by a straightforward experiment. However, it is an untested assumption, and later refinements of the model may prove necessary. 2.2. The plate is driven by the turbulent boundary layer pressure, pt , the external acoustic pressure, p0 , and the first dissipative layer pressure, pi1 . The plate response equation thus becomes B94v − Nz 1 2v/1z 2 − Ny 1 2v/1y 2 − Mv 2v = iv[pt + p0 − pi1 ]
(1)
(cf., equation (1) of Part I; note that the harmonic time dependence e−ivt of all the variables has been suppressed). Once this equation is solved, the inwardly radiated power is found by integrating pressure and velocity over the trim surface, and we therefore require a description of the pressures and velocities not only on the plate, but also on the trim panel. In principle, these may be obtained as part of a complete solution for the vibration in the layered system, but it is much simpler to concentrate attention on the behaviour at layer boundaries, an approach which retains all the necessary information. It is then possible to use impedance methods (see, e.g., references [9–12]) to obtain the desired results. To demonstrate the general approach we consider the layer shown in Figure 3, with pressures and velocities pa , va and pb , vb at its boundaries. The material in the layer obeys the usual Helmholtz equation (see equation (2) in Part I), and has free wavenumber ks ( = v/cs , cs being the sound speed) and characteristic impedance Zs ( = rs cs , rs being the mean density). In our case the description applies to the air gap and to the dissipative layers (with rs , cs complex), and the subscript s will take the form ag (air gap) i1 (dissipative layer 1) or i2 (dissipative layer 2) as appropriate. As previously, we consider variables Fourier transformed with respect to the y and z
. .
124
directions, in which case the solution for the pressure in the layer takes the one-dimensional form p˜s (x, ky , kz , v) = A eiGs x + B e−iGs x,
(2)
where p˜s (x, ky , kz , v) =
g g a
a
−a
−a
ps (x, y, z, v) e−iky y e−ikz z dy dz,
Gs = (ks2 − ky2 − kz2 )1/2 (3, 4)
and Gs is taken to lie in the first quadrant for definiteness. (This requirement is consistent with our previous definitions for such x-direction wavenumbers, where ks was real rather than potentially complex.) The velocity in the layer may be calculated by application of the linearized momentum equation, and the quantities p˜a , v˜a , p˜b , v˜b thus found in terms of the constants A and B. We obtain the ratios p˜b /v˜b = Z s tanh (iGs d + Ca )
and v˜b /v˜a = cosh (iGs d + Ca )/cosh Ca ,
(5, 6)
where Z s is the effective one-dimensional impedance; Z s = Zs ks /Gs
(7)
and the ‘‘phase angle’’ Ca is defined in terms of Z s and the end impedance Z a ( = p˜a /v˜a ): tanh Ca = Z a /Z s .
(8)
Thus, if the impedance at one side of the layer, Z a , is known, the impedance at the other side and the velocity ratio across the layer may be found, and the solutions for the multi-layered structure built up by repeated application of the above formulae. Starting at the trim panel, we find p˜ag /v˜ag =x = −(d 1 + d 2 + d 3 ) = −(Z 1 + Z t ),
(9)
where Z 1 and Z t are the effective impedances of the interior air space and the trim panel respectively. Z 1 is defined by equation (7), while for a trim panel of bending stiffness Bt and surface density Mt , Z t = i[Bt (ky2 + kz2 )2 − Mt v 2 ]/v.
(10)
Application of equations (5), (6) and (8) then yields the conditions on the other side of the air gap: p˜ag v˜ag
b
= Z 1 tanh (iG1 d3 + C3 )
and v˜ag =x = −(d 1 + d 2 )
x = −(d 1 + d 2 )
=
cosh (iG1 d3 + C3 ) v˜ag =x = −(d 1 + d 2 + d 3 ) , cosh C3
(11, 12)
where tanh C3 = −(Z 1 + Z t )/Z 1 .
(13)
(The conditions in the air gap are those of the interior air space, so Z ag = Z 1 and Gag = G1 .) The limp air bag facing transmits normal velocity continuously, but has an associated
, II
125
p˜i2 /v˜i2 =x = −(d 1 + d 2 ) = p˜ag /v˜ag =x = −(d 1 + d 2 ) + ivMf ,
(14)
pressure drop, so that
where Mf is the facing surface density. Two more applications of the generic layer equations then yield p˜i1 /v˜i1 =x = 0 = Z i1 tanh (iGi1 d1 + C1 ), v˜i1 =x = 0 =
(15)
cosh (iGi1 d1 + C1 ) cosh (iGi2 d2 + C2 ) cosh (iG1 d3 + C3 ) v˜ag =x = −(d 1 + d 2 + d 3 ) , cosh C1 cosh C2 cosh C3 (16)
where tanh C1 =
Z i2 tanh (iGi2 d2 + C2 ) , Z i1
tanh C2 =
Z 1 tanh (iG1 d3 + C3 ) + ivMf . Z i2
(17,18)
Equations (15) and (16), along with the relationship between pressure and velocity on the cabin side of the trim panel, p˜1 = −Z 1 v˜1 ,
(19)
provide a succinct summary of the information we need to solve our problem. First, using the customary modal approach, the relationship between the Fourier transformed pressure, p˜i1 (0, ky , kz , v), and velocity, v˜i1 (0, ky , kz , v), at the plate surface yields if pi1mn (v) = −ri1 ci1 s Zmnpq vpq (v),
(20)
p,q
where if Zmnpq =
−1 (2p)2
g g a
−a
a
ki1 tanh (iGi1 d1 + C1 )S* mn (ky , kz )Spq (ky , kz ) dky dkz . G i1 −a
(21)
Since the densities and sound speeds in the insulation are of the same order of magnitude as those in the air space, modal coupling may again be neglected, and equation (20) then reduces to the single term form used previously. The solution for the response is thus the same as in Part I, i.e., Mvvmn = −ptmn /dmn , with the exception that the dimensionless impedance dmn is replaced by
6
d'mn = i
7
Br (km2 + kn2 )2 + Nzr km2 + Nyr kn2 0f if (1 − ios ) − 1 +of 0 Z mnmn + ofi Zmnmn , Mv 2
(22)
where ofi = ri1 ci1 /Mv.
(23)
The expressions for the dimensionless spectra of boundary layer input, outwardly radiated and structurally dissipated power (equations (19)–(21) of Part I) follow through, with d'mn exchanged for dmn , while equation (17) becomes if S i (v) = 2 s Re(ofi Zmnmn ) m,n
F mn =d'mn =2
(24)
. .
126
and now represents not the inwardly radiated power, but the power entering the insulation. The inwardly radiated power spectrum is given by the integral 2pd(v − v')S1 (v) = −2
g g a
a
−a
−a
Re [p1 (−(d1 + d2 + d3 ), y z, v)v* 1 (−(d1 + d2 + d3 ), y, z, v')] dy dz, (25)
which covers the entire trim surface. This integral may be expressed in terms of the Fourier transformed pressure and velocity, p˜1 and v˜1 , and the relationships (19) and (16) used to phrase it in terms of the plate velocity. On writing the plate velocity as the usual modal sum, employing the response equation and applying the non-dimensionalization of Part I, the spectrum of the inwardly radiated power is found to be S 1 (v) = 2of 1 s Re(Imn ) m,n
F mn , =d'mn =2
(26)
b
(27)
where Imn =
1 (2p)2
g g b a
−a
a
k1 v˜ag =x = −(d 1 + d 2 + d 3 ) =Smn (ky , kz )=2 dky dkz G v ˜ 1 i1 =x = 0 −a 2
and the velocity ratio is given by equation (16). Equation (26) is formally similar to the 1f bare plate expression (17) of Part I, with the radiation efficiency Re(Zmnmn ) replaced by the real part of the integral Imn . 2.3. I The analysis presented in the previous section yields equations similar in form to those of Part I. However, there are differences, and these have an important effect on the physical behaviour of the model. In the bare plate case, the modal damping due to the interior air and the inwardly radiated sound were essentially two aspects of the same phenomenon. Here this is not the case, and they are effectively decoupled. The complex form of the insulation densities and sound speeds means that energy may propagate in these materials for all ky and kz wavenumbers, not just those which correspond to a supersonic trace speed, if so that all wavenumbers contribute to the Re(ofi Zmnmn ) damping term in d'mn . In practice, this will mean that the term is determined by the integrand value at the (m, n) modal wavenumber, where the shape function product =Smn =2 peaks†, even if that wavenumber is subsonic (as is the case for all the resonant modes in our frequency range). The ‘‘radiation efficiency’’, Re(Imn ), on the other hand, arises only from wavenumbers with supersonic trace speed in the interior air space (G1 real), which form a relatively small part of the energy in a subsonic mode. This decoupling phenomenon changes the characteristics of the system significantly. It is now possible for a mode simultaneously to have high ‘‘radiation’’ damping, but a low radiation efficiency, the discrepancy being accounted for by dissipation in the insulation. The influence of the insulation and trim may thus be split into two essentially distinct effects—alteration of the structural response, largely by the provision of increased †To a first approximation =Smn =2 acts as a delta function centred on the modal wavenumbers—equation (28). This behaviour can clearly be seen in the expressions for the modal excitation term derived in the Appendix to Part I.
, II
127
T 1 Typical aircraft panel parameters Free-stream velocity Ua (m/s) Convection velocity Uc (m/s) Friction velocity Ut (m/s) Boundary layer thickness d (m) Plate mass/unit area M (kg/m2) Plate bending stiffness Br (N/m) Plate longitudinal tension Nzr (N/m) Plate lateral tension Nyr (N/m) Structural damping factor os Plate length a (m) Plate width b (m) External air density r0 (kg/m3) External sound speed c0 (m/s) Internal air density r1 (kg/m3) Internal sound speed c1 (m/s)
225 0·7Ua 0·03Ua 0·1 2·77 6·0 29·3 × 103 62·1 × 103 0·01 0·5 0·2 0·44 300 1·2 340
damping, and attenuation of the supersonic components of the resulting vibration as they propagate from the plate to the interior air space. This demonstrates the importance of including insulation and trim as an integral part of the model rather than taking an ad hoc approach. For example, a simple ‘‘transmission loss’’ prediction applied to the bare plate results would describe only the attenuation effect, and would completely neglect the damping effect—an important feature of the problem. Clearly this change in the characteristics of our model will affect our previous results. The extent to which this is so will become apparent in the following section, where we describe a numerical study parallelling that of Part I.
3. NUMERICAL STUDY
3.1. As in Part I, we commence our numerical study of the model with the solution for parameters appropriate to the aircraft case (see Tables 1 and 2). The characteristic impedances and free wavenumbers quoted in Table 2 were determined experimentally, from an insulation sample, and an account of this work is given in the Appendix. We shall first describe novel features of the numerical implementation and then present the reference T 2 Insulation and trim parameters Layer 1 characteristic impedance Layer 1 free wavenumber Layer 2 characteristic impedance Layer 2 free wavenumber Layer 1 thickness d1 (m) Layer 2 thickness d2 (m) Air gap thickness d3 (m) Insulation bag facing mass/unit area Mf (kg/m2) Trim panel bending stiffness Bt (N/m) Trim panel mass/unit area Mt (kg/m2)
equation (A4a) equation (A4b) equation (A5a) equation (A5b) 0·009 0·016 0·062 0·044 1·4 2·37
128
. .
Figure 4. Contribution to the (5,4) modal damping from the insulation.
case results. Subsequent sections will consider the physical characteristics of the model, and the effect of insulation and trim on our previous parameter study. In the numerical implementation of the model, features corresponding to the bare plate case were treated identically, leaving the evaluation of the insulation fluid-loading term, if ofi Zmnmn , and the radiation integral, Imn , as outstanding problems. An analytical expression for the former was found by using the delta function approximation to the shape functions [13, 14], =Smn (ky , kz )=2 1 (2p)2d(kz − km )d(ky − kn ),
(28)
which corresponds to taking only the leading order, pole contribution in the asymptotic evaluation of the modal impedances presented by Graham [15]. Here, the resulting term is complex, and thus provides leading order real and imaginary parts without the need for a more complicated analysis. The result, valid as long as the rest of the integrand in equation (21) is slowly varying in comparison to the shape functions, is if ofi Zmnmn 0 −ofi [(ki1 /Gi1 ) tanh (iGi1 d1 + C1 )]kz = km ,ky = kn .
(29)
As noted previously, the real part of this term will have a significant effect on modal damping levels, and this is demonstrated in Figure 4, which shows its contribution to the
Figure 5. Radiation integral for mode (3,2).
, II
129
Figure 6. Inwardly radiated power for the reference case: – – –, bare plate; ——, trimmed plate.
Figure 7. Boundary layer input power for the reference case: – – –, bare plate; ——, trimmed plate.
damping in mode (5, 4) for the parameter values specified above.† (Figure 4 should be compared with a typical structural damping factor of 0·01–0·02.) A similar approximation is not possible for Imn , and it was evaluated numerically, by using a Gaussian quadrature with weighting functions corresponding to the rapidly varying components of the shape function product in equation (27). The numerical integration was checked by comparison with results obtained by using NAG routines, and found to be both accurate and efficient. A typical example of the radiation integral is shown in Figure 5, where the real part of I32 is plotted against dimensionless frequency, for the reference case parameters. As the dimensionless frequency increases towards one (coincidence), Re(Imn ) first falls, then grows, reflecting the competing effects of the increasing acoustic attenuation associated with the insulation and the rising supersonic wavenumber content of the mode shape. Above coincidence, when most of the modal energy is already in supersonic wavenumbers, only the attenuation increases significantly with frequency, and Re(Imn ) decreases rapidly as a result. The solution for the reference case may now be considered, and compared with the corresponding results for the bare plate. Figures 6–9 show the spectra for inwardly radiated †The significance of the real part of the insulation impedance term also provides further justification for the neglect of modal coupling in evaluating the plate response, as coupling is only potentially important when the acoustic impedance has small real part [15].
130
. .
Figure 8. Structurally dissipated power for the reference case: – – –, bare plate; ——, trimmed plate.
Figure 9. Power dissipated in the insulation for the reference case.
power, boundary layer input power, structurally dissipated power and insulation dissipated power, S id (v) ( = S i (v) − S 1 (v)). The resonance peaks of the bare plate case are completely absent from the trimmed plate model and the results are typical of a heavily damped system, with the radiated power also showing the attenuating effect of the insulation and trim. The majority of the input power is once again dissipated, but now in the insulation rather than the plate, demonstrating the importance of insulation damping in the trimmed plate model. 3.2. The basic physical characteristics of the bare model remain. Thus modal resonance frequencies are determined by Im(d'mn ) = 0, modes are inefficient or efficient acoustic radiators depending on whether they are below or above coincidence† (with respect to the interior acoustic wavenumber) and the boundary layer excitation peaks for modes with longitudinal trace speed equal to the convection velocity. Equally, the three features identified in Part I—critical frequency (which determines whether or not resonant modes are acoustically efficient), hydrodynamic coincidence (peak excitation for resonant modes) and wavenumber scattering (energy conversion from subsonic to supersonic †At a given frequency, above coincidence modes will still be much more efficient radiators than those below coincidence, Figure 5 notwithstanding.
, II
131
Figure 10. Effect of hydrodynamic coincidence on the radiated power: – – –, reference case; ——, (a) case 1, (b) case 2.
wavenumbers)—are still present in our model. However, as the resonant modes no longer obviously dominate the power spectra, non-resonant, above coincidence modes may also contribute to the radiated sound. Since both hydrodynamic coincidence and the critical frequency are associated with modes at resonance, and wavenumber scattering is similarly most marked there, we may find some ‘‘dilution’’ of the effects seen in Part I. We thus proceed to revisit the parameter study presented there.
Figure 11. Effect on the radiated power of lowering the critical frequency: – – –, reference case; ——, case 3.
132
. .
Figure 12. Effect of structural inhomogeneity on the radiated power: – – –, reference case; ——, large plate.
3.3. The non-dimensionalizing constant dependences discussed in Part I are unchanged, so it remains to consider the effects of the four physical factors investigated there, namely structural damping, hydrodynamic coincidence, ciritical frequency and structural inhomogeneity. The cases considered for the latter three are shown in Figures 10–12, from which it is clear that there is no significant change in behaviour compared with Part I. Cases 1 and 2, chosen to exhibit more hydrodynamic coincidence than the reference case, again show little difference from it, while case 3 demonstrates that the stiffer plate still radiates more sound. Equally, the scaled output from the large plate shows that reducing the degree of structural inhomogeneity again reduces noise. As the range plotted is greater than in Part I, the differences seem less marked, but in fact they are hardly reduced, implying that the ‘‘dilution’’ effect discussed above is not great: i.e., that non-resonant, above coincidence modes do not significantly contribute to these spectra. Contrasting with these essentially unchanged results, Figure 13 shows that the effect of doubling the structural damping is almost negligible in comparison to the significant reduction it caused in the sound radiated by the bare plate. Whereas previously the modal vibration at resonance was controlled by structural damping, it is now a relatively trivial factor, with insulation damping dominating (especially at low frequencies).
Figure 13. Effect of increased damping on the radiated power: – – –, reference case; ——, increased damping.
, II
133
3.4. As predicted in section 2, the trimmed flat plate model has been found to have much higher damping levels than the bare plate model, as well as the expected attenuating effect on radiated sound. This increased damping, arising from the ability of the insulation to dissipate energy, significantly alters the behaviour of the system, and modifies one of the conclusions of the original parameter study. While the hydrodynamic coincidence, critical frequency and structural inhomogeneity effects are found to be essentially unchanged, increasing structural damping is found to have little influence on the acoustic output of the plate, particularly at low frequencies. The practical conclusions of this work remain largely unchanged from Part I—it is still likely to be advantageous to build homogeneous, floppy aircraft, although perhaps somewhat less so than suggested previously. However, the use of structural damping treatments in trimmed aircraft will probably be futile, given the results presented here. Of course, these predictions are based on an unvalidated model, and remain to be confirmed. Of the previous work described in Part I, only Cooper presented experimental comparisons between bare and trimmed fuselage structures [2], and there is limited qualitative agreement between our model predictions and his results. Cooper did find that structural damping treatments had a negligible effect on sound levels (or even raised them in some cases) when the cabin was trimmed, but his panel vibration measurements, with and without damping treatments, show little evidence of being affected by the presence of insulation and trim. As few details of the interior configuration are given in the report, it is difficult to draw concrete conclusions from this; however the ineffectiveness of the damping treatments raises the possibility of ‘‘flanking’’ transmission paths contributing significantly to the cabin noise levels in this case.
4. CONCLUSIONS
In this paper we have described the extension of the flat plate model presented in Part I to account for the influence of the cabin interior treatment—insulation and trim panels. As expected, these components significantly modify the behaviour of the system via two related, but distinct, effects: insulation damping, which arises from the ability of the dissipative insulation materials to absorb energy at all wavenumbers; and attenuation, whereby supersonic wavenumber components are greatly weakened before they can be radiated to the cabin. As far as the general implications for design discussed in Part I are concerned, conclusions (ii)–(iv) stand: i.e., it will be acoustically advantageous to build aircraft with floppy, homogeneous structures, accepting that hydrodynamic coincidence will occur. However, the efficacy of structural damping treatments in reducing trimmed cabin noise levels is likely to be minimal. Instead, it will be much more productive to concentrate on optimizing the acoustic characteristics of the interior treatment. Future work in this area should be directed at validating the predictions and implications of this model. A preliminary step in this direction has already been taken, by using a room acoustics approach to compare predicted and measured third-octave cabin noise levels in two aircraft [16]. The results showed good agreement at low frequencies, but at higher frequencies the experimental data were underpredicted. Given the number of assumptions necessary for this study and the limited experimental data available, no definite causes for the discrepancy can be identified, and further progress will be dependent on comparison with much more complete data sets. This will allow detailed testing of the model assumptions, from the boundary layer excitation through to the wave propagation in the
134
. .
insulation layers. Once these issues are resolved, the basic form of this model, and in particular the integrated approach to describing the cabin interior treatment, is expected to result in an accurate prediction and design tool.
ACKNOWLEDGMENTS
As in Part I of this work, it is a pleasure to acknowledge the financial support of British Aerospace (Regional Aircraft) Ltd, the guidance of my supervisor, Professor Ann Dowling, and the value of numerous discussions with Frank Ogilvie and Clive Florentin of British Aerospace. I would also like to thank the referees for their perceptive and helpful comments on both parts of this paper.
REFERENCES 1. W. R. G 1996 Journal of Sound and Vibration 192, 101–120. Boundary layer induced noise in aircraft, part I: the flat plate model. 2. G. J. C 1980 British Aerospace Acoustics Report 624. Airframe vibration, internal noise and the effects of damping treatment. 3. D. A. B 1971 in Noise and Vibration Control (L. L. Beranek, editor). New York: McGraw-Hill. Chapter 10: Acoustical properties of porous materials. 4. L. L. B 1947 Journal of the Acoustical Society of America 19(4), 556–568. Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets. 5. C. W. K and J. H. J 1957 Acustica 7(6), 372–378. Acoustic properties of flexible and porous materials. 6. J. S. B and N.-M. S 1987 AIAA-87-2660. Oblique incidence sound transmission through multi-panel structures lined with elastic porous materials. 7. K. H. H 1977 Ph.D. Thesis, Institute of Sound and Vibration Research, University of Southampton. Boundary layer induced cabin noise. 8. J. S. M and J. F. W 1991 in Aeroacoustics of Flight Vehicles: Theory and Practice (NASA Reference Publication 1258) (H. H. Hubbard, editor), two vol. Chapter 16: Interior noise. 9. L. L. B and G. A. W 1949 Journal of the Acoustical Society of America 21(4), 419–428. Sound transmission through multiple structures containing flexible blankets. 10. R. A. M 1963 Journal of the Acoustical Society of America 35(7), 1023–1029. Optimization of the mass distribution and the air spaces in multiple-element soundproofing structures. 11. P. H. W and A. P 1966 Journal of the Acoustical Society of America 40(4), 821–832. Transmission of random sound and vibration through a rectangular double wall. 12. K. A. M, A. J. P and H. D. P 1968 Journal of the Acoustical Society of America 43(6), 1432–1435. Transmission loss of multiple panels in a random incidence field. 13. H. G. D 1971 Journal of the Acoustical Society of America 49(3), 878–889. Sound from turbulent-boundary-layer-excited panels. 14. Y. M. C and P. L 1979 Journal of Sound and Vibration 64(2), 243–256. Acoustic impedance of rectangular panels. 15. W. R. G 1995 Philosophical Transactions of the Royal Society of London, Series A 352, 1–43. High frequency vibration and acoustic radiation of fluid-loaded plates. 16. W. R. G 1993 Proceedings of NOISE-CON 93. Evaluation of a model for boundary layer induced noise in aircraft. New York: Noise Control Foundation. 17. M. E. D and E. N. B 1969 NPL Aerodynamics Division Report Ac37 . Acoustical characteristics of fibrous materials.
, II
135
APPENDIX: EXPERIMENTAL DETERMINATION OF THE INSULATION PARAMETERS
A1. In the model used for the dissipative layers, knowledge of two parameters, the complex density and sound speed, is sufficient to determine the characteristics of all wave propagation; normally incident, obliquely incident and evanescent. It is therefore only necessary to determine these two quantities, or equivalents, for normal incidence waves, and this may be achieved by testing samples in an appropriate wave tube. For any sample whose behaviour is linear, the pressure and velocity on one side, pa and va , are related to those on the other, pb and vb by a transfer matrix,
$ % $
pa z z = 11 12 r1 c1 va z21 z22
%$ % pb r1 c1 vb
(A1)
(where r1 and c1 are the ambient density and sound speed), and knowledge of this matrix fully characterizes the plane wave behaviour of the sample. Clearly, the elements of the transfer matrix are not uniquely determined by a single set of pressure and velocity measurements, and two independent experiments are required to find them. With the two runs denoted by 1 and 2, equation (A1) yields z11 = (pa1 vb2 − pa2 vb1 )/(pb1 vb2 − pb2 vb1 ),
(A2)
with similar expressions applying for z12 , z21 and z22 . A2. The apparatus used is shown in Figure A1. The wave tube is of diameter 100 mm (which gives a maximum useful frequency of about 1·8 kHz), and consists of two sections divided by a sampler holder. Plane acoustic waves were excited by loudspeakers at one end, and pressure measurements taken using four 1/4 in (6·4 mm) microphones, two either side of the sample holder. The geometry of the wave tube was altered by inserting a cap at the open end, so that two independent sets of pressures and velocities could be found. For all the tests described below, a band-limited (100–2500 Hz) white noise excitation was input to the loudspeaker, and this signal, along with those from the microphones, was anti-alias filtered (cut-off frequency 2340 Hz) and digitized at a sampling frequency of 6 kHz. Only two microphones were available, so each test was conducted in two parts, with the unused microphone positions blocked off. Ambient pressure and temperature were
Figure A1. Experimental apparatus for the insulation tests.
. .
136
measured at regular intervals, and the microphone calibrations were checked before and during the test. The sample of insulation used for the experiments consisted of two layers of absorptive material inside an insulation bag, with transfer matrices required for each layer, and for the bag facing material. First, a piece of the facing material was inserted at the rear end of the sample holder (see Figure A1) and tested. Then the two absorptive materials were tested separately, with the bag facing still in position (to counteract a tendency for samples to move towards the rear of the wave tube). Finally, data were also taken for both possible configurations of the absorptive materials in combination, as a check on the individual results. A3. For each test, transfer and coherence functions between the input signal and the four microphone positions were calculated by averaging cross-spectra. Single frequency checks during testing had shown that the system was behaving linearly, and this was confirmed by generally high levels of coherence. Pressure and velocity on either side of the sample were then found in terms of the measured transfer functions, by decomposing them into travelling wave components, and the elements of the transfer matrix calculated. As these are expressed in terms of ratios of transfer functions (see equation (A2)), common factors such as the loudspeaker response cancel out. The measured matrix elements are thus solely functions of the pressures and velocities on either side of the sample, as they should be. For the insulation bag facing, the transfer matrix ought to be of the form
$ % 1 0
Zf 1
and this was found to be the case. Figure A2 shows the real and imaginary parts of Zf , and the least squares lines fitted to them, defined by Re(Zf ) = 0·10 + 8·8 × 10−5f,
Im(Zf ) = −6·6 × 10−4f,
(A3a,b)
where f is the frequency in Hz. These equations represent a sheet with a small amount of damping, and mass per unit area 0·044 kg/m2 (a value which is in agreement with that obtained by weighing the facing). The fitted matrix was used to obtain the transfer matrices for the insulation layers from the combined insulation and facing measurements. Figure A3 shows the resulting absolute values of the transfer matrix components for layer 2. For such a layer, the transfer matrix should be of the form
$
cos(ks d) −ir1 c1 sin (ks d)/Zs
%
−iZs sin (ks d)/r1 c1 , cos (ks d)
where, as previously, d is the thickness, ks the free wavenumber and Zs the characteristic impedance of the material in the layer. This matrix satisfies two conditions: its diagonal elements are equal and it has determinant one. The measured matrix approximately satisfies both conditions, but inaccuracies in the data are apparent. To determine the required parameters, the matrix was first corrected to the expected form by using a least squares technique†, and ks and Zs then extracted. †The corrected matrix has =z12 = and =z21 = almost unchanged, and =z11 =, =z22 = approximately equal to the average of the raw values shown in Figure A3.
, II
137
Figure A2. Insulation bag facing impedance and least squares fits.
Delany and Bazley [17] found that the free wavenumber and characteristic impedance of a fibrous material are functions of the dimensionless parameter (r1 f/s), where s is the flow resistance of the material. Assuming a dependency of this form (with s unknown) led to the following curve fits: layer 1,
layer 2,
Zi1 /r1 c1 = 1 + 110(r1 f/r1r )−0·65 + 1·6i(r1 f/r1r )−0·07,
(A4a)
ki1 /k1 = 1 + 400(r1 f/r1r )−0·77 + 5·7i(r1 f/r1r )−0·18,
(A4b)
−0·82
Zi2 /r1 c1 = 1 + 300(r1 f/r1r )
+ 8·2i(r1 f/r1r )
−0·34
ki2 /k1 = 1 + 50(r1 f/r1r )−0·50 + 3·8i(r1 f/r1r )−0·17.
,
(A5a) (A5b)
Here r1r is a reference density, equal to 1·2 kg/m . The fits for layer 2 are plotted, with the raw data, in Figure A4, and the smooth variation of free wavenumber with frequency shown here should be noted, as it lends further weight to the single wave type assumption (at least for plane wave excitation). Interestingly, both these results and those for layer 1 differ significantly in frequency dependence from the fits given by Delany and Bazley [17], so that attempting to deduce the insulation parameters from a measurement of flow resistance, s, would lead to quite erroneous predictions. 3
Figure A3. Transfer matrix components for layer 2.
138
. .
Figure A4. Dimensionless wavenumber and characteristic impedance for layer 2.
Figure A5. Transfer matrix components for layers 1 and 2 combined—measured data and predictions from individual least squares fits.
A final check on the validity of the results is a comparison between the measured and predicted values of the elements of the transfer matrix for the combined, two-layer, case. This comparison is shown in Figure A5, which exhibits reasonable, albeit not perfect, agreement between theory and experiment.
BOUNDARY LAYER INDUCED NOISE IN AIRCRAFT, PART II: THE TRIMMED FLAT PLATE MODEL W. R. G Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England (Received 7 February 1995, and in final form 5 July 1995) The influence of the cabin interior treatment on boundary layer noise levels is known to be significant, and we thus extend the model of Part I [1] to include it. The extended model consists of the boundary layer excited flat plate, with its internal surface covered by two dissipative layers (representing insulation) and an elastic plate (the trim panel). By comparison with the bare plate, the trimmed plate has much higher modal damping, due to the ability of the insulation to absorb energy at the wavenumbers associated with resonant modes, and greatly reduced radiation efficiency, due to the insulation’s attenuating effect on supersonic wavenumber disturbances. Most of the conclusions of Part I are essentially unaffected by this behavioural modification, but structural damping treatments are now expected to have a negligible effect on the sound radiated by the plate. 7 1996 Academic Press Limited
1. INTRODUCTION
In Part I of this work [1] we introduced a simple model for boundary layer induced noise in aircraft, based on the sound radiated by a single flat plate set in an infinite rigid baffle and driven by a turbulent boundary layer. After a numerical study of this model, we concluded that increases in structural damping and decreases in skin stiffness and the number of fuselage reinforcements were likely to have a beneficial effect on cabin sound levels. However, while the model was carefully specified in the light of the physical characteristics of the problem, we noted that it neglected one important component: the cabin interior treatment. In this paper we address that deficiency. In order to reduce cabin noise and heat loss, and for decorative reasons, passenger aircraft cabins are trimmed. Insulation bags containing a glass-fibre material are placed in each frame bay and covered with plastic trim panels, as shown schematically in Figure 1. This treatment is known to have a significant effect not only on interior sound levels, but also the fundamental behaviour of the problem [2], and our model must therefore be extended to include it as an integral part, rather than a later addition. Such an extension will require a description of wave propagation in the insulating material, a complicated phenomenon which has been the subject of much research. According to Bies [3], sound propagation in porous materials may be described by using the usual (single frequency) acoustic equations with a frequency-dependent complex mean density and sound speed to account for dissipative effects such as viscosity and heat transfer. However, this approach is only strictly valid if there is one wave type in the vibration, and in fact there may be several. Beranek [4] and Kosten and Janssen [5] presented theories that allow for two (one air-borne and one structure-borne), while the more sophisticated model of Bolton and Shiau [6] exhibits three (one air-borne and two 121 0022–460X/96/160121 + 18 $18.00/0
7 1996 Academic Press Limited
122
. .
Figure 1. Aircraft cabin interior treatment.
structure-borne). The method chosen to represent the insulation will thus require consideration. In this paper, we first consider the extension of the existing model to account for insulation and trim (section 2), and then, in section 3, present the results of a parallel numerical study to that of Part I. The conclusions of the work, and the extent to which they differ from those of Part I, are discussed in section 4.
2. EXTENSION OF THE FLAT PLATE MODEL
2.1. In considering this extension, we first re-examine the validity of representing a curved panel by a flat plate. In Part I we argued that the important factor is the acoustic dissipation of the interior surface, which must be low enough that the surrounding baffle should appear rigid, but high enough to reduce returning sound waves to negligible levels. Since we shall now be modelling the surface acoustic dissipation more carefully, the former requirement is no longer relevant, but the latter is still important. However, a trimmed and furnished aircraft contains many scatterers and absorbers of sound, so the sound returning to the plate will be only a small fraction of its radiated field, which will therefore be essentially the same as if the plate were radiating to infinity. The flat plate approximation will thus be a satisfactory representation of the actual problem. The geometry of the model must now be specified. The insulation bag contains two layers of differing insulation material, and is separated from the plastic trim panel by an air gap. We neglect the effects of finiteness in the insulation layers and trim panel (essentially assuming that the system is sufficiently damped that the wave behaviour appears to be travelling rather than reverberant), and define the model configuration of Figure 2. Here the insulation is represented by two dissipative layers, covered by a limp sheet (the insulation bag facing) and separated from an elastic plate (the trim panel) by
Figure 2. The trimmed flat plate model.
, II
123
Figure 3. A generic insulation layer.
an air gap. The mass of the insulation bag facing in contact with the plate is neglected in comparison to the plate mass. In the forthcoming analysis, we shall find the complete vibro-acoustic solution to this model, rather than taking the usual approach whereby the effect of insulation and trim is added to the bare structure solution in a more or less ad hoc manner (see, e.g., references [7, 8]). It remains to choose a model for wave propagation in the dissipative layers. Here we take the approach of Bies [3], and consider one wave type only. This approximation is supported by Beranek’s work [4], which showed that the longitudinal structure-borne wave is highly attenuated in flexible materials, such as aircraft insulation, and only the air-borne wave is important. It also has the advantage of simplicity, yielding results capable of physical interpretation and containing no more empirical parameters than may be determined by a straightforward experiment. However, it is an untested assumption, and later refinements of the model may prove necessary. 2.2. The plate is driven by the turbulent boundary layer pressure, pt , the external acoustic pressure, p0 , and the first dissipative layer pressure, pi1 . The plate response equation thus becomes B94v − Nz 1 2v/1z 2 − Ny 1 2v/1y 2 − Mv 2v = iv[pt + p0 − pi1 ]
(1)
(cf., equation (1) of Part I; note that the harmonic time dependence e−ivt of all the variables has been suppressed). Once this equation is solved, the inwardly radiated power is found by integrating pressure and velocity over the trim surface, and we therefore require a description of the pressures and velocities not only on the plate, but also on the trim panel. In principle, these may be obtained as part of a complete solution for the vibration in the layered system, but it is much simpler to concentrate attention on the behaviour at layer boundaries, an approach which retains all the necessary information. It is then possible to use impedance methods (see, e.g., references [9–12]) to obtain the desired results. To demonstrate the general approach we consider the layer shown in Figure 3, with pressures and velocities pa , va and pb , vb at its boundaries. The material in the layer obeys the usual Helmholtz equation (see equation (2) in Part I), and has free wavenumber ks ( = v/cs , cs being the sound speed) and characteristic impedance Zs ( = rs cs , rs being the mean density). In our case the description applies to the air gap and to the dissipative layers (with rs , cs complex), and the subscript s will take the form ag (air gap) i1 (dissipative layer 1) or i2 (dissipative layer 2) as appropriate. As previously, we consider variables Fourier transformed with respect to the y and z
. .
124
directions, in which case the solution for the pressure in the layer takes the one-dimensional form p˜s (x, ky , kz , v) = A eiGs x + B e−iGs x,
(2)
where p˜s (x, ky , kz , v) =
g g a
a
−a
−a
ps (x, y, z, v) e−iky y e−ikz z dy dz,
Gs = (ks2 − ky2 − kz2 )1/2 (3, 4)
and Gs is taken to lie in the first quadrant for definiteness. (This requirement is consistent with our previous definitions for such x-direction wavenumbers, where ks was real rather than potentially complex.) The velocity in the layer may be calculated by application of the linearized momentum equation, and the quantities p˜a , v˜a , p˜b , v˜b thus found in terms of the constants A and B. We obtain the ratios p˜b /v˜b = Z s tanh (iGs d + Ca )
and v˜b /v˜a = cosh (iGs d + Ca )/cosh Ca ,
(5, 6)
where Z s is the effective one-dimensional impedance; Z s = Zs ks /Gs
(7)
and the ‘‘phase angle’’ Ca is defined in terms of Z s and the end impedance Z a ( = p˜a /v˜a ): tanh Ca = Z a /Z s .
(8)
Thus, if the impedance at one side of the layer, Z a , is known, the impedance at the other side and the velocity ratio across the layer may be found, and the solutions for the multi-layered structure built up by repeated application of the above formulae. Starting at the trim panel, we find p˜ag /v˜ag =x = −(d 1 + d 2 + d 3 ) = −(Z 1 + Z t ),
(9)
where Z 1 and Z t are the effective impedances of the interior air space and the trim panel respectively. Z 1 is defined by equation (7), while for a trim panel of bending stiffness Bt and surface density Mt , Z t = i[Bt (ky2 + kz2 )2 − Mt v 2 ]/v.
(10)
Application of equations (5), (6) and (8) then yields the conditions on the other side of the air gap: p˜ag v˜ag
b
= Z 1 tanh (iG1 d3 + C3 )
and v˜ag =x = −(d 1 + d 2 )
x = −(d 1 + d 2 )
=
cosh (iG1 d3 + C3 ) v˜ag =x = −(d 1 + d 2 + d 3 ) , cosh C3
(11, 12)
where tanh C3 = −(Z 1 + Z t )/Z 1 .
(13)
(The conditions in the air gap are those of the interior air space, so Z ag = Z 1 and Gag = G1 .) The limp air bag facing transmits normal velocity continuously, but has an associated
, II
125
p˜i2 /v˜i2 =x = −(d 1 + d 2 ) = p˜ag /v˜ag =x = −(d 1 + d 2 ) + ivMf ,
(14)
pressure drop, so that
where Mf is the facing surface density. Two more applications of the generic layer equations then yield p˜i1 /v˜i1 =x = 0 = Z i1 tanh (iGi1 d1 + C1 ), v˜i1 =x = 0 =
(15)
cosh (iGi1 d1 + C1 ) cosh (iGi2 d2 + C2 ) cosh (iG1 d3 + C3 ) v˜ag =x = −(d 1 + d 2 + d 3 ) , cosh C1 cosh C2 cosh C3 (16)
where tanh C1 =
Z i2 tanh (iGi2 d2 + C2 ) , Z i1
tanh C2 =
Z 1 tanh (iG1 d3 + C3 ) + ivMf . Z i2
(17,18)
Equations (15) and (16), along with the relationship between pressure and velocity on the cabin side of the trim panel, p˜1 = −Z 1 v˜1 ,
(19)
provide a succinct summary of the information we need to solve our problem. First, using the customary modal approach, the relationship between the Fourier transformed pressure, p˜i1 (0, ky , kz , v), and velocity, v˜i1 (0, ky , kz , v), at the plate surface yields if pi1mn (v) = −ri1 ci1 s Zmnpq vpq (v),
(20)
p,q
where if Zmnpq =
−1 (2p)2
g g a
−a
a
ki1 tanh (iGi1 d1 + C1 )S* mn (ky , kz )Spq (ky , kz ) dky dkz . G i1 −a
(21)
Since the densities and sound speeds in the insulation are of the same order of magnitude as those in the air space, modal coupling may again be neglected, and equation (20) then reduces to the single term form used previously. The solution for the response is thus the same as in Part I, i.e., Mvvmn = −ptmn /dmn , with the exception that the dimensionless impedance dmn is replaced by
6
d'mn = i
7
Br (km2 + kn2 )2 + Nzr km2 + Nyr kn2 0f if (1 − ios ) − 1 +of 0 Z mnmn + ofi Zmnmn , Mv 2
(22)
where ofi = ri1 ci1 /Mv.
(23)
The expressions for the dimensionless spectra of boundary layer input, outwardly radiated and structurally dissipated power (equations (19)–(21) of Part I) follow through, with d'mn exchanged for dmn , while equation (17) becomes if S i (v) = 2 s Re(ofi Zmnmn ) m,n
F mn =d'mn =2
(24)
. .
126
and now represents not the inwardly radiated power, but the power entering the insulation. The inwardly radiated power spectrum is given by the integral 2pd(v − v')S1 (v) = −2
g g a
a
−a
−a
Re [p1 (−(d1 + d2 + d3 ), y z, v)v* 1 (−(d1 + d2 + d3 ), y, z, v')] dy dz, (25)
which covers the entire trim surface. This integral may be expressed in terms of the Fourier transformed pressure and velocity, p˜1 and v˜1 , and the relationships (19) and (16) used to phrase it in terms of the plate velocity. On writing the plate velocity as the usual modal sum, employing the response equation and applying the non-dimensionalization of Part I, the spectrum of the inwardly radiated power is found to be S 1 (v) = 2of 1 s Re(Imn ) m,n
F mn , =d'mn =2
(26)
b
(27)
where Imn =
1 (2p)2
g g b a
−a
a
k1 v˜ag =x = −(d 1 + d 2 + d 3 ) =Smn (ky , kz )=2 dky dkz G v ˜ 1 i1 =x = 0 −a 2
and the velocity ratio is given by equation (16). Equation (26) is formally similar to the 1f bare plate expression (17) of Part I, with the radiation efficiency Re(Zmnmn ) replaced by the real part of the integral Imn . 2.3. I The analysis presented in the previous section yields equations similar in form to those of Part I. However, there are differences, and these have an important effect on the physical behaviour of the model. In the bare plate case, the modal damping due to the interior air and the inwardly radiated sound were essentially two aspects of the same phenomenon. Here this is not the case, and they are effectively decoupled. The complex form of the insulation densities and sound speeds means that energy may propagate in these materials for all ky and kz wavenumbers, not just those which correspond to a supersonic trace speed, if so that all wavenumbers contribute to the Re(ofi Zmnmn ) damping term in d'mn . In practice, this will mean that the term is determined by the integrand value at the (m, n) modal wavenumber, where the shape function product =Smn =2 peaks†, even if that wavenumber is subsonic (as is the case for all the resonant modes in our frequency range). The ‘‘radiation efficiency’’, Re(Imn ), on the other hand, arises only from wavenumbers with supersonic trace speed in the interior air space (G1 real), which form a relatively small part of the energy in a subsonic mode. This decoupling phenomenon changes the characteristics of the system significantly. It is now possible for a mode simultaneously to have high ‘‘radiation’’ damping, but a low radiation efficiency, the discrepancy being accounted for by dissipation in the insulation. The influence of the insulation and trim may thus be split into two essentially distinct effects—alteration of the structural response, largely by the provision of increased †To a first approximation =Smn =2 acts as a delta function centred on the modal wavenumbers—equation (28). This behaviour can clearly be seen in the expressions for the modal excitation term derived in the Appendix to Part I.
, II
127
T 1 Typical aircraft panel parameters Free-stream velocity Ua (m/s) Convection velocity Uc (m/s) Friction velocity Ut (m/s) Boundary layer thickness d (m) Plate mass/unit area M (kg/m2) Plate bending stiffness Br (N/m) Plate longitudinal tension Nzr (N/m) Plate lateral tension Nyr (N/m) Structural damping factor os Plate length a (m) Plate width b (m) External air density r0 (kg/m3) External sound speed c0 (m/s) Internal air density r1 (kg/m3) Internal sound speed c1 (m/s)
225 0·7Ua 0·03Ua 0·1 2·77 6·0 29·3 × 103 62·1 × 103 0·01 0·5 0·2 0·44 300 1·2 340
damping, and attenuation of the supersonic components of the resulting vibration as they propagate from the plate to the interior air space. This demonstrates the importance of including insulation and trim as an integral part of the model rather than taking an ad hoc approach. For example, a simple ‘‘transmission loss’’ prediction applied to the bare plate results would describe only the attenuation effect, and would completely neglect the damping effect—an important feature of the problem. Clearly this change in the characteristics of our model will affect our previous results. The extent to which this is so will become apparent in the following section, where we describe a numerical study parallelling that of Part I.
3. NUMERICAL STUDY
3.1. As in Part I, we commence our numerical study of the model with the solution for parameters appropriate to the aircraft case (see Tables 1 and 2). The characteristic impedances and free wavenumbers quoted in Table 2 were determined experimentally, from an insulation sample, and an account of this work is given in the Appendix. We shall first describe novel features of the numerical implementation and then present the reference T 2 Insulation and trim parameters Layer 1 characteristic impedance Layer 1 free wavenumber Layer 2 characteristic impedance Layer 2 free wavenumber Layer 1 thickness d1 (m) Layer 2 thickness d2 (m) Air gap thickness d3 (m) Insulation bag facing mass/unit area Mf (kg/m2) Trim panel bending stiffness Bt (N/m) Trim panel mass/unit area Mt (kg/m2)
equation (A4a) equation (A4b) equation (A5a) equation (A5b) 0·009 0·016 0·062 0·044 1·4 2·37
128
. .
Figure 4. Contribution to the (5,4) modal damping from the insulation.
case results. Subsequent sections will consider the physical characteristics of the model, and the effect of insulation and trim on our previous parameter study. In the numerical implementation of the model, features corresponding to the bare plate case were treated identically, leaving the evaluation of the insulation fluid-loading term, if ofi Zmnmn , and the radiation integral, Imn , as outstanding problems. An analytical expression for the former was found by using the delta function approximation to the shape functions [13, 14], =Smn (ky , kz )=2 1 (2p)2d(kz − km )d(ky − kn ),
(28)
which corresponds to taking only the leading order, pole contribution in the asymptotic evaluation of the modal impedances presented by Graham [15]. Here, the resulting term is complex, and thus provides leading order real and imaginary parts without the need for a more complicated analysis. The result, valid as long as the rest of the integrand in equation (21) is slowly varying in comparison to the shape functions, is if ofi Zmnmn 0 −ofi [(ki1 /Gi1 ) tanh (iGi1 d1 + C1 )]kz = km ,ky = kn .
(29)
As noted previously, the real part of this term will have a significant effect on modal damping levels, and this is demonstrated in Figure 4, which shows its contribution to the
Figure 5. Radiation integral for mode (3,2).
, II
129
Figure 6. Inwardly radiated power for the reference case: – – –, bare plate; ——, trimmed plate.
Figure 7. Boundary layer input power for the reference case: – – –, bare plate; ——, trimmed plate.
damping in mode (5, 4) for the parameter values specified above.† (Figure 4 should be compared with a typical structural damping factor of 0·01–0·02.) A similar approximation is not possible for Imn , and it was evaluated numerically, by using a Gaussian quadrature with weighting functions corresponding to the rapidly varying components of the shape function product in equation (27). The numerical integration was checked by comparison with results obtained by using NAG routines, and found to be both accurate and efficient. A typical example of the radiation integral is shown in Figure 5, where the real part of I32 is plotted against dimensionless frequency, for the reference case parameters. As the dimensionless frequency increases towards one (coincidence), Re(Imn ) first falls, then grows, reflecting the competing effects of the increasing acoustic attenuation associated with the insulation and the rising supersonic wavenumber content of the mode shape. Above coincidence, when most of the modal energy is already in supersonic wavenumbers, only the attenuation increases significantly with frequency, and Re(Imn ) decreases rapidly as a result. The solution for the reference case may now be considered, and compared with the corresponding results for the bare plate. Figures 6–9 show the spectra for inwardly radiated †The significance of the real part of the insulation impedance term also provides further justification for the neglect of modal coupling in evaluating the plate response, as coupling is only potentially important when the acoustic impedance has small real part [15].
130
. .
Figure 8. Structurally dissipated power for the reference case: – – –, bare plate; ——, trimmed plate.
Figure 9. Power dissipated in the insulation for the reference case.
power, boundary layer input power, structurally dissipated power and insulation dissipated power, S id (v) ( = S i (v) − S 1 (v)). The resonance peaks of the bare plate case are completely absent from the trimmed plate model and the results are typical of a heavily damped system, with the radiated power also showing the attenuating effect of the insulation and trim. The majority of the input power is once again dissipated, but now in the insulation rather than the plate, demonstrating the importance of insulation damping in the trimmed plate model. 3.2. The basic physical characteristics of the bare model remain. Thus modal resonance frequencies are determined by Im(d'mn ) = 0, modes are inefficient or efficient acoustic radiators depending on whether they are below or above coincidence† (with respect to the interior acoustic wavenumber) and the boundary layer excitation peaks for modes with longitudinal trace speed equal to the convection velocity. Equally, the three features identified in Part I—critical frequency (which determines whether or not resonant modes are acoustically efficient), hydrodynamic coincidence (peak excitation for resonant modes) and wavenumber scattering (energy conversion from subsonic to supersonic †At a given frequency, above coincidence modes will still be much more efficient radiators than those below coincidence, Figure 5 notwithstanding.
, II
131
Figure 10. Effect of hydrodynamic coincidence on the radiated power: – – –, reference case; ——, (a) case 1, (b) case 2.
wavenumbers)—are still present in our model. However, as the resonant modes no longer obviously dominate the power spectra, non-resonant, above coincidence modes may also contribute to the radiated sound. Since both hydrodynamic coincidence and the critical frequency are associated with modes at resonance, and wavenumber scattering is similarly most marked there, we may find some ‘‘dilution’’ of the effects seen in Part I. We thus proceed to revisit the parameter study presented there.
Figure 11. Effect on the radiated power of lowering the critical frequency: – – –, reference case; ——, case 3.
132
. .
Figure 12. Effect of structural inhomogeneity on the radiated power: – – –, reference case; ——, large plate.
3.3. The non-dimensionalizing constant dependences discussed in Part I are unchanged, so it remains to consider the effects of the four physical factors investigated there, namely structural damping, hydrodynamic coincidence, ciritical frequency and structural inhomogeneity. The cases considered for the latter three are shown in Figures 10–12, from which it is clear that there is no significant change in behaviour compared with Part I. Cases 1 and 2, chosen to exhibit more hydrodynamic coincidence than the reference case, again show little difference from it, while case 3 demonstrates that the stiffer plate still radiates more sound. Equally, the scaled output from the large plate shows that reducing the degree of structural inhomogeneity again reduces noise. As the range plotted is greater than in Part I, the differences seem less marked, but in fact they are hardly reduced, implying that the ‘‘dilution’’ effect discussed above is not great: i.e., that non-resonant, above coincidence modes do not significantly contribute to these spectra. Contrasting with these essentially unchanged results, Figure 13 shows that the effect of doubling the structural damping is almost negligible in comparison to the significant reduction it caused in the sound radiated by the bare plate. Whereas previously the modal vibration at resonance was controlled by structural damping, it is now a relatively trivial factor, with insulation damping dominating (especially at low frequencies).
Figure 13. Effect of increased damping on the radiated power: – – –, reference case; ——, increased damping.
, II
133
3.4. As predicted in section 2, the trimmed flat plate model has been found to have much higher damping levels than the bare plate model, as well as the expected attenuating effect on radiated sound. This increased damping, arising from the ability of the insulation to dissipate energy, significantly alters the behaviour of the system, and modifies one of the conclusions of the original parameter study. While the hydrodynamic coincidence, critical frequency and structural inhomogeneity effects are found to be essentially unchanged, increasing structural damping is found to have little influence on the acoustic output of the plate, particularly at low frequencies. The practical conclusions of this work remain largely unchanged from Part I—it is still likely to be advantageous to build homogeneous, floppy aircraft, although perhaps somewhat less so than suggested previously. However, the use of structural damping treatments in trimmed aircraft will probably be futile, given the results presented here. Of course, these predictions are based on an unvalidated model, and remain to be confirmed. Of the previous work described in Part I, only Cooper presented experimental comparisons between bare and trimmed fuselage structures [2], and there is limited qualitative agreement between our model predictions and his results. Cooper did find that structural damping treatments had a negligible effect on sound levels (or even raised them in some cases) when the cabin was trimmed, but his panel vibration measurements, with and without damping treatments, show little evidence of being affected by the presence of insulation and trim. As few details of the interior configuration are given in the report, it is difficult to draw concrete conclusions from this; however the ineffectiveness of the damping treatments raises the possibility of ‘‘flanking’’ transmission paths contributing significantly to the cabin noise levels in this case.
4. CONCLUSIONS
In this paper we have described the extension of the flat plate model presented in Part I to account for the influence of the cabin interior treatment—insulation and trim panels. As expected, these components significantly modify the behaviour of the system via two related, but distinct, effects: insulation damping, which arises from the ability of the dissipative insulation materials to absorb energy at all wavenumbers; and attenuation, whereby supersonic wavenumber components are greatly weakened before they can be radiated to the cabin. As far as the general implications for design discussed in Part I are concerned, conclusions (ii)–(iv) stand: i.e., it will be acoustically advantageous to build aircraft with floppy, homogeneous structures, accepting that hydrodynamic coincidence will occur. However, the efficacy of structural damping treatments in reducing trimmed cabin noise levels is likely to be minimal. Instead, it will be much more productive to concentrate on optimizing the acoustic characteristics of the interior treatment. Future work in this area should be directed at validating the predictions and implications of this model. A preliminary step in this direction has already been taken, by using a room acoustics approach to compare predicted and measured third-octave cabin noise levels in two aircraft [16]. The results showed good agreement at low frequencies, but at higher frequencies the experimental data were underpredicted. Given the number of assumptions necessary for this study and the limited experimental data available, no definite causes for the discrepancy can be identified, and further progress will be dependent on comparison with much more complete data sets. This will allow detailed testing of the model assumptions, from the boundary layer excitation through to the wave propagation in the
134
. .
insulation layers. Once these issues are resolved, the basic form of this model, and in particular the integrated approach to describing the cabin interior treatment, is expected to result in an accurate prediction and design tool.
ACKNOWLEDGMENTS
As in Part I of this work, it is a pleasure to acknowledge the financial support of British Aerospace (Regional Aircraft) Ltd, the guidance of my supervisor, Professor Ann Dowling, and the value of numerous discussions with Frank Ogilvie and Clive Florentin of British Aerospace. I would also like to thank the referees for their perceptive and helpful comments on both parts of this paper.
REFERENCES 1. W. R. G 1996 Journal of Sound and Vibration 192, 101–120. Boundary layer induced noise in aircraft, part I: the flat plate model. 2. G. J. C 1980 British Aerospace Acoustics Report 624. Airframe vibration, internal noise and the effects of damping treatment. 3. D. A. B 1971 in Noise and Vibration Control (L. L. Beranek, editor). New York: McGraw-Hill. Chapter 10: Acoustical properties of porous materials. 4. L. L. B 1947 Journal of the Acoustical Society of America 19(4), 556–568. Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets. 5. C. W. K and J. H. J 1957 Acustica 7(6), 372–378. Acoustic properties of flexible and porous materials. 6. J. S. B and N.-M. S 1987 AIAA-87-2660. Oblique incidence sound transmission through multi-panel structures lined with elastic porous materials. 7. K. H. H 1977 Ph.D. Thesis, Institute of Sound and Vibration Research, University of Southampton. Boundary layer induced cabin noise. 8. J. S. M and J. F. W 1991 in Aeroacoustics of Flight Vehicles: Theory and Practice (NASA Reference Publication 1258) (H. H. Hubbard, editor), two vol. Chapter 16: Interior noise. 9. L. L. B and G. A. W 1949 Journal of the Acoustical Society of America 21(4), 419–428. Sound transmission through multiple structures containing flexible blankets. 10. R. A. M 1963 Journal of the Acoustical Society of America 35(7), 1023–1029. Optimization of the mass distribution and the air spaces in multiple-element soundproofing structures. 11. P. H. W and A. P 1966 Journal of the Acoustical Society of America 40(4), 821–832. Transmission of random sound and vibration through a rectangular double wall. 12. K. A. M, A. J. P and H. D. P 1968 Journal of the Acoustical Society of America 43(6), 1432–1435. Transmission loss of multiple panels in a random incidence field. 13. H. G. D 1971 Journal of the Acoustical Society of America 49(3), 878–889. Sound from turbulent-boundary-layer-excited panels. 14. Y. M. C and P. L 1979 Journal of Sound and Vibration 64(2), 243–256. Acoustic impedance of rectangular panels. 15. W. R. G 1995 Philosophical Transactions of the Royal Society of London, Series A 352, 1–43. High frequency vibration and acoustic radiation of fluid-loaded plates. 16. W. R. G 1993 Proceedings of NOISE-CON 93. Evaluation of a model for boundary layer induced noise in aircraft. New York: Noise Control Foundation. 17. M. E. D and E. N. B 1969 NPL Aerodynamics Division Report Ac37 . Acoustical characteristics of fibrous materials.
, II
135
APPENDIX: EXPERIMENTAL DETERMINATION OF THE INSULATION PARAMETERS
A1. In the model used for the dissipative layers, knowledge of two parameters, the complex density and sound speed, is sufficient to determine the characteristics of all wave propagation; normally incident, obliquely incident and evanescent. It is therefore only necessary to determine these two quantities, or equivalents, for normal incidence waves, and this may be achieved by testing samples in an appropriate wave tube. For any sample whose behaviour is linear, the pressure and velocity on one side, pa and va , are related to those on the other, pb and vb by a transfer matrix,
$ % $
pa z z = 11 12 r1 c1 va z21 z22
%$ % pb r1 c1 vb
(A1)
(where r1 and c1 are the ambient density and sound speed), and knowledge of this matrix fully characterizes the plane wave behaviour of the sample. Clearly, the elements of the transfer matrix are not uniquely determined by a single set of pressure and velocity measurements, and two independent experiments are required to find them. With the two runs denoted by 1 and 2, equation (A1) yields z11 = (pa1 vb2 − pa2 vb1 )/(pb1 vb2 − pb2 vb1 ),
(A2)
with similar expressions applying for z12 , z21 and z22 . A2. The apparatus used is shown in Figure A1. The wave tube is of diameter 100 mm (which gives a maximum useful frequency of about 1·8 kHz), and consists of two sections divided by a sampler holder. Plane acoustic waves were excited by loudspeakers at one end, and pressure measurements taken using four 1/4 in (6·4 mm) microphones, two either side of the sample holder. The geometry of the wave tube was altered by inserting a cap at the open end, so that two independent sets of pressures and velocities could be found. For all the tests described below, a band-limited (100–2500 Hz) white noise excitation was input to the loudspeaker, and this signal, along with those from the microphones, was anti-alias filtered (cut-off frequency 2340 Hz) and digitized at a sampling frequency of 6 kHz. Only two microphones were available, so each test was conducted in two parts, with the unused microphone positions blocked off. Ambient pressure and temperature were
Figure A1. Experimental apparatus for the insulation tests.
. .
136
measured at regular intervals, and the microphone calibrations were checked before and during the test. The sample of insulation used for the experiments consisted of two layers of absorptive material inside an insulation bag, with transfer matrices required for each layer, and for the bag facing material. First, a piece of the facing material was inserted at the rear end of the sample holder (see Figure A1) and tested. Then the two absorptive materials were tested separately, with the bag facing still in position (to counteract a tendency for samples to move towards the rear of the wave tube). Finally, data were also taken for both possible configurations of the absorptive materials in combination, as a check on the individual results. A3. For each test, transfer and coherence functions between the input signal and the four microphone positions were calculated by averaging cross-spectra. Single frequency checks during testing had shown that the system was behaving linearly, and this was confirmed by generally high levels of coherence. Pressure and velocity on either side of the sample were then found in terms of the measured transfer functions, by decomposing them into travelling wave components, and the elements of the transfer matrix calculated. As these are expressed in terms of ratios of transfer functions (see equation (A2)), common factors such as the loudspeaker response cancel out. The measured matrix elements are thus solely functions of the pressures and velocities on either side of the sample, as they should be. For the insulation bag facing, the transfer matrix ought to be of the form
$ % 1 0
Zf 1
and this was found to be the case. Figure A2 shows the real and imaginary parts of Zf , and the least squares lines fitted to them, defined by Re(Zf ) = 0·10 + 8·8 × 10−5f,
Im(Zf ) = −6·6 × 10−4f,
(A3a,b)
where f is the frequency in Hz. These equations represent a sheet with a small amount of damping, and mass per unit area 0·044 kg/m2 (a value which is in agreement with that obtained by weighing the facing). The fitted matrix was used to obtain the transfer matrices for the insulation layers from the combined insulation and facing measurements. Figure A3 shows the resulting absolute values of the transfer matrix components for layer 2. For such a layer, the transfer matrix should be of the form
$
cos(ks d) −ir1 c1 sin (ks d)/Zs
%
−iZs sin (ks d)/r1 c1 , cos (ks d)
where, as previously, d is the thickness, ks the free wavenumber and Zs the characteristic impedance of the material in the layer. This matrix satisfies two conditions: its diagonal elements are equal and it has determinant one. The measured matrix approximately satisfies both conditions, but inaccuracies in the data are apparent. To determine the required parameters, the matrix was first corrected to the expected form by using a least squares technique†, and ks and Zs then extracted. †The corrected matrix has =z12 = and =z21 = almost unchanged, and =z11 =, =z22 = approximately equal to the average of the raw values shown in Figure A3.
, II
137
Figure A2. Insulation bag facing impedance and least squares fits.
Delany and Bazley [17] found that the free wavenumber and characteristic impedance of a fibrous material are functions of the dimensionless parameter (r1 f/s), where s is the flow resistance of the material. Assuming a dependency of this form (with s unknown) led to the following curve fits: layer 1,
layer 2,
Zi1 /r1 c1 = 1 + 110(r1 f/r1r )−0·65 + 1·6i(r1 f/r1r )−0·07,
(A4a)
ki1 /k1 = 1 + 400(r1 f/r1r )−0·77 + 5·7i(r1 f/r1r )−0·18,
(A4b)
−0·82
Zi2 /r1 c1 = 1 + 300(r1 f/r1r )
+ 8·2i(r1 f/r1r )
−0·34
ki2 /k1 = 1 + 50(r1 f/r1r )−0·50 + 3·8i(r1 f/r1r )−0·17.
,
(A5a) (A5b)
Here r1r is a reference density, equal to 1·2 kg/m . The fits for layer 2 are plotted, with the raw data, in Figure A4, and the smooth variation of free wavenumber with frequency shown here should be noted, as it lends further weight to the single wave type assumption (at least for plane wave excitation). Interestingly, both these results and those for layer 1 differ significantly in frequency dependence from the fits given by Delany and Bazley [17], so that attempting to deduce the insulation parameters from a measurement of flow resistance, s, would lead to quite erroneous predictions. 3
Figure A3. Transfer matrix components for layer 2.
138
. .
Figure A4. Dimensionless wavenumber and characteristic impedance for layer 2.
Figure A5. Transfer matrix components for layers 1 and 2 combined—measured data and predictions from individual least squares fits.
A final check on the validity of the results is a comparison between the measured and predicted values of the elements of the transfer matrix for the combined, two-layer, case. This comparison is shown in Figure A5, which exhibits reasonable, albeit not perfect, agreement between theory and experiment.