Airborne sound characterisation by Dummy Source approach

G. Pavić1, A. Lindberg1

1 INSA Lyon, Laboratoire Vibrations Acoustique25 bis avenue Jean Capelle, F-69621, Villeurbanne, Francee-mail: goran.pavic@insa-lyon.fr

Abstract A noise source which radiates sound by vibration can be characterised by techniques like Nearfield Acoustical Holography or Inverse Boundary Element Method which allow the reconstruction of normal vibration velocity across the source surface. While the former is applicable to sources of simple geometry, the latter requires detailed knowledge of source geometry. In order to circumvent the drawbacks of the two methods, the concept of "dummy" source is hereby introduced. Such a source is a rigid physical object of similar but much simpler geometry than that of the original source it models. It is equipped with a number of small drivers which have volume velocities deduced from the measurement of sound field around the original source. Since the dummy accounts for both the radiation and diffraction around it, the identified source strengths can be used to predict the sound radiation in an arbitrary space. The paper presents the physics of the approach and the selection of parameters of the dummy. A comparative measurement results are shown to illustrate the validity of the approach.

1 Introduction

Noise prediction of a built-up system, such as a vehicle or a building, requires some knowledge about the noise source(s) involved. At the present level of technology reliable characterisation of a complex source cannot be achieved entirely by modelling since many of model parameters are usually not sufficiently known. On the other hand a source of noise usually physically exists before the noise design of the entire system is made and can thus be characterised by measurement.

If the noise source is installed in a large space of known absorption area the prediction of noise level can be easily accomplished using simple laws based on Sabine acoustics [1]. In such a case the source should be characterised by its total sound power, supposed to be invariant with the surrounding acoustical space. However the relationship between the sound pressure and radiated power holds only statistically. It is usually enforced by frequency band averaging, and, in the case of a closed space, by additional spatial averaging. While the invariance of the averaged sound power approximately holds where large spaces are concerned, it does not hold necessarily in a smaller space, such as a compressor booth, engine compartment or transformer room.

To enable a comprehensive prediction of noise radiated by a source into a coupled receiver space an intrinsic source characterisation is required. One of techniques to characterise a noise source, the Equivalent Source Method (ESM), is to model it as a superposition of several simple sources, usually monopoles. The method, originated by Cremer [2], has been studied a lot from a computation point of view [3]. The strength of such equivalent sources are tuned to produce the sound field at the measurement points as close to the sound produced by the original source as possible. The method does not require any frequency averaging and can be formulated in time as well as in frequency domain. The main drawback of this simple method is that it is not intrinsic to the source: any change in the acoustical environment invalidates the source parameters. As a consequence the equivalent source model obtained in one acoustical environment cannot be applied to another one.

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Where sources which produce noise by vibration are concerned, two main types of source identification methods have been developed. The Nearfield Acoustical holography (NAH) uses spatial Fourier Transform to reconstruct the source vibration from the sound field measured by a suitably positioned array of sensors. In this way sources of simple geometry, e.g. plane or cylindrical, can be reconstructed in terms of the normal vibration velocity across the radiated surface [4-5]. To overcome the problem of NAH related to phase measurement, an alternative acoustical holographic method, the Broadband Acoustic Holography from Intensity Measurements (BAHIM), was developed [6].

The limitations of NAH and related techniques to simple geometry were circumvented by the development of Inverse Boundary Element Method (IBEM) which uses the Kirchhoff-Helmholtz integral formulation to reconstruct the normal vibration velocity of the radiator [7-8]. Satisfactory comparison between the direct computation and IBEM reconstruction has been reported for several cases of known analytical solution, [9]. Satisfactory results were further obtained with IBEM in the reconstruction of normal velocity distribution on irregularly shaped cavities [10]. Further improvement of the method has been achieved in [11] by a particular double-layer technique aimed at reconstructing the vibration velocity of the source in noisy environment.

The acoustical model of a vibrating source has to involve the source geometry as well as the vibration pattern across the radiating surface. The latter is virtually impossible to recover from measurements if the source geometry is a complex one. Moreover, if the source operates in a confined space the diffraction of sound by the source itself may be important. The idea of a dummy source is to construct a model of simple geometry and simple velocity distribution which can produce the sound field close to the original one. Such a source model can in turn be efficiently used for sound prediction in an arbitrary space.

It will be assumed that the sound is generated by source vibration and, furthermore, that the vibration is unaffected by the radiation loading. An extension of the dummy source principle to sources having low surface impedance is possible, but will not be addressed in this article.

2 The dummy source

2.1 The radiation model

The sound field radiated by a vibrating body into an acoustical space can be represented by, [12]:

SgP d)()(),()(S

snsVsrr ⋅= ∫ (1)

where g – Green's function of the acoustical space including the body, r – observation position vector, s – position vector at the body surface S, V – amplitude of vibration velocity vector, n – unit normal to the surface, dot –scalar product. The Green's function in Eq. (1) is the pressure amplitude at r divided by the amplitude of point excitation of unit volume velocity at s. Eq. (1) is formulated in frequency domain, the same as the rest of the paper. Only the (complex) amplitudes of time varying quantities will be concerned; the usual exp(jωt) multiplier is omitted.

The entire radiated surface can be subdivided into M smaller surfaces Sm , m = 1,… M. The contribution to sound radiations of mth elementary surface Sm can be then formally represented in the following form:

rmmm

Sr

mmmmmmmm VSAdS

Vg

g

SgAP

m

=⋅== ∫ ,)()(

)(

),(1)(,)()()(

snsVrsr

rrrr γγ (2)

where gm and rmV stand respectively for the Green's function and normal vibration velocity, both averaged

across Sm. The quantity Am thus represents the volume velocity of the surface Sm. If the surface Sm gets reduced, the first and second quotient in the surface integral of Eq. (2) will tend to unity, making thus the non-dimensional factor γm approach one. Thus by decomposing the radiating surface S into small enough sections, Eq. (1) assumes the following discrete form:

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∑∑ ≈=q

mmm

m gAPP )()()( rrr (3)

Eq. (3) stands for one of two simplifying assumptions on which a dummy source is founded. This assumption implies that the sound radiation from a continuously vibrating body can be approximated by radiation from a discrete set of volume velocity sources across its surface. In what follows the spatially averaged Green's function gm will be called the transfer impedance of a discrete source. This frequency function depends on the positions of both the source and the receiver points. It furthermore depends on the shape of the vibrating source and on the acoustical properties of the space into which it radiates sound.

2.2 Identifying the dummy source

A task is set here to find out whether a rigid box of simple shape – the dummy, equipped with a number of simple sound radiators – the drivers, can serve as a replacement to the original source, i.e. to produce the sound field close to the field of the original source. A dummy needs to have a body similar to that of the original source in order to simulate well enough both the radiation and the diffraction. A rigid box having the volume and overall dimensions close to those of the original source can be reasonably expected to produce sound similar to that of the original source. The principal task is to identify the required vibration pattern across its surface and to map it onto the drivers. Since direct measurement of source vibration is usually difficult, the vibration has to be obtained indirectly, from acoustical measurements.

A real source of steady-state vibration has often elaborate shape and produces complex vibration pattern across its radiating surface. In a close vicinity of such a source the radiated sound pressure will exhibit a complex pattern too, dominated as a rule by the sound nearfield. With increase in distance between the source and observation point the nearfield disappears and the radiated field takes over.

Contrary to transfer impedance gm the strength of the mth driver, Am, is independent of the position of observation point and of properties of the surrounding space. Therefore the set of Am values describes the source intrinsically. Since the transfer impedance can be obtained by measurement, the strengths of discrete drivers can be obtained from the measured data by inversion. The sound pressure created by the running source has to be measured in a number of control points around the source. Using the known transfer impedances the source strengths of individual drivers can then be fitted to the measured data for best matching.

Denoting by C the number of control points and by D the number of drivers, the described identification procedure can be condensed in the following mathematical form:

cPTA 1−= (4)

with A – D sized vector of driver amplitudes, T – (C,D) sized matrix of transfer impedances and Pc – C sized vector of control sound pressure amplitudes. All the quantities in Eq. (4) are complex, i.e. the sound pressure and the source strength are defined by the respective amplitudes and phases. If the source produces random but stationary sound both the pressure vector and the source strength vector have to be expressed by the corresponding spectral matrices Gp and Ga. By denoting conjugate transpose inverse with the superscript * Eq. (4) can be then put into an equivalent spectral form:

*1 TGTG pa−= (5)

It has been found that it is much better to avoid the determined mathematical case of having the same number of control points and drivers. While such a condition can produce a sound field which perfectly matches the pressure at the control points, it may fail to reproduce the field elsewhere. It was found that a solution is to apply an over-determined case, i.e. to use more control points than drivers, C>D. The C≈1.5D rule has been found to be a good compromise.

The key steps in creating a dummy model of a real source are thus:

• produce a rigid dummy of similar but simplified shape as the original source• equip the dummy with a number of drivers

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• measure the sound pressure of operating source in a number of control points• measure the transfer impedances between the control points and driver positions• compute the strengths of drivers by inversion.

The procedure described above produces a source model which consists of a rigid box fitted with a number of simple sources of known volume velocity. Such a model can be easily handled by currently available acoustical software.

3 Analysis of feasibility

3.1 The original source and its dummy

In order to assess the performance of the proposed dummy technique, a 2D case of sound radiation in a free space will be considered. The radiating body has an irregular "potato" shape defined by its closed contour C, Fig. 1. In this example the original source will be modelled by a 0.7m×.5m rectangular dummy of 5 cm rounded corners, shown on Fig. 1 by a dashed line. The original and dummy were made to match in area and in the positions of centres of gravity.

Figure 1: Original 2D source (grey area) and its dummy (black line).

In order to create a continuous vibration velocity distribution of the source and to compute its sound radiation, a large number of line sources, 83 in total, were employed in the interior of the contour C. The particle velocity normal to the contour line C generated by all the line sources simultaneously was then taken to be the normal vibration velocity of the original source. In this way the two sources, the original one and the array of line sources, produce exactly the same sound field outside the contour line of the original source. The sound field and the radiated power of such a source were discussed in [13].

The complex source strength amplitudes of elementary line sources were chosen at random. In order to cut down the contribution of in-phase component of line sources, which produces a dominantly symmetrical field, a complex constant was added to all the amplitudes with the purpose of reducing the in-phase component to 2% of the total RMS strength of line sources. The objective of this correction was to create a complex pattern of radiated sound. This has resulted in the total RMS source strength of 0.584×10-3 m2/s.

Given fixed strengths of the line sources, the field pressure and velocity will vary with frequency. This will produce a frequency-dependent normal velocity at the source boundary. Fig. 2 shows the magnitudes of normal velocity at four characteristic frequencies with the speed of sound taken at 343 m/s.

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Figure 2: Normal velocity profile of the source. Top: magnitude, bottom: phase. From left to right: 250

Hz, 500 Hz, 1 kHz and 2 kHz.

3.2 Comparison of sound fields

3.2.1 The radiated field

Fig. 3 compares the sound pressure field created by the two sources at four representative frequencies: 200 Hz, 500 Hz, 1 kHz and 2 kHz. The control points, indicated by black dots, were distributed uniformly and symmetrically along a rectangular line of the same shape as dummy but twice its size. The 5 cm long drivers, shown in white, were placed uniformly and symmetrically along the contour of dummy. Cosine velocity profile of the driver was assumed with a maximum at the centre and zero-slope at the extremities. To improve visual resolution the instantaneous rather than RMS pressure is displayed. In order to interpret black and white scale correctly the absolute instantaneous values are shown with the white colour set to zero. The scale maximum is adjusted to the field of original source, which makes some areas of the field of dummy source exceed the maximum. These areas were truncated to the scale maximum and thus appear in black. Remarkably good matching can be seen between the fields created by the original and its dummy. As expected the mismatch is noticeable only close to the radiating surfaces.

As a rule the reproduction of sound field by the dummy will be the better the larger the number of drivers. In the current examples the lowest number of drivers and control points were selected still producing the matching of sound pressures along the control line better than 1%. The number of control points roughly about 50% larger than the number of drivers has been found to be a suitable rule of the thumb to produce correct results providing the number of drivers is sufficient. It has been found out that working with determined cases, i.e. having equal number of drivers and control points, produced consistently a huge mismatch of the two fields. The reason for this apparent paradox is simple: a determined case produces a perfect reconstruction of the dummy field at the control points, which in turn results in very high dynamics of driver strengths. The latter effect creates in field points other than control points very high sound pressure levels and the reconstructed field becomes entirely corrupted.

The degree of matching can be better quantified by comparing the point pressure levels produced by the two sources, Fig. 4. The pressure levels were computed along the control line using 100 points per wavelength. The position 0 m corresponds to the lower left corner of the control line. The level difference of the two fields is small, typically around 0.1 dB. The extreme values of mismatch are at the positions where the level of the original field is small.

It can be shown that reducing the number of drivers leads to a considerable degradation of dummy's performance. The "per wavelength" type of criterion alone is insufficient to enable correct selection of number of drivers. At lower frequencies more drivers per wavelength are needed than at higher frequencies.

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Figure 3: Instantaneous absolute sound pressure by the original source (left) and by the dummy (right).

Figure 4: Sound pressure level. Left: by the original source; right: level difference dummy - original.

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3.2.2 The diffracted field

It has been demonstrated in §3.2.1 that a dummy equipped with a number of drivers can act as a realistic substitute to a complex source which produces sound by its vibration. The analysis was made for free field conditions for the sake of simple modelling. The identification of the required parameters of dummy, notably the number of drivers, should however work in any acoustical environment free from exterior sources of sound.

In a free field the emitted sound propagates away, making the body of the dummy responsible just for sound radiation. In a field which is not free reflections will occur and the role of dummy will consist of taking care of both the radiation and diffraction of sound. As a rule, the smaller the size of acoustical space surrounding the source the larger the effect of diffraction.

In this section a comparison will be made of the diffraction of sound by the original source and by its dummy. The same technique used for the modelling of sound radiation, described in Appendix, will be used to model sound diffraction. The body of the source, either of the original one or of its dummy, will be replaced by a (large) number of substitute sources. The role of the substitute sources, just like in the case of sound radiation, is to ensure the correct boundary conditions across the surface of the source. In the case of diffraction, the boundary condition is zero normal velocity at the surface. The strengths of substitute sources A can in this case be obtained by counteracting the normal particle velocity of the sound incident to the body, Vi, by the normal velocity produced by substitute sources, Vs. The latter can is obtained via the velocity transfer matrix H, Vs = HA. Thus:

iVAQ 1−−= (6)

Fig. 5 shows the sound fields diffracted by the original source and its dummy for 4 reference cases selected previously. In order to accommodate the current 2D representation, the incident field was created by a line source (white spot) which was offset by 0.7m horizontally and 0.5m vertically from the origin. In the close proximity of the external source the sound pressure is very high, rising to infinity at the position of this source. Here again the truncation is applied, by setting the pressure values within a circle of 10 cm diameter to the maximum pressure of the remaining field. The absolute values of instantaneous pressure are shown as before.

A fairly closed matching between the diffracted fields of the original source and its dummy can be seen. The matching is less good than the one of radiated fields obtained by large enough number of drivers. This was to be expected: the matching in radiation is achieved by adapting the shape of dummy to the original source and on top of this by fine tuning the drivers to the field at control points. In the case of diffraction there are no drivers to steer the field: the field diffracted by the dummy is entirely governed by the geometrical similarity of the two sources.

The matching of the diffracted fields along a rectangular line 1.5 larger than the contour of dummy is presented in Fig. 6. The starting position of the response line was put in the lower left corner. The point-by-point difference between the two fields is clearly visible, but the overall matching looks reasonable. Given for comparison is also the pressure along the same line due to the incident field only. This pressure is extremely different from the diffracted pressure which further confirms the weakness of equivalent source method which cannot account correctly for the diffraction.

Thus the dummy looks capable of capturing the basic features of the diffraction effect. A closer similitude in geometry between the original and the dummy than the current one would have certainly improved the matching of the diffracted fields. Improving the similitude would have most likely improved the radiation by the dummy while not bringing any increase in the measurement effort.

3.2.3 The error of measurement

The source characterisation procedure consists of two principal phases: a) measurement of operational sound pressure produced by the original source at the control points and b) measurement of transfer

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Figure 5: Diffracted field. Left column: original source; right column: dummy.

impedances between the drivers and control points. Both of these measurements are subjected to measurement errors. Of particular concern are the impedance measurements employed to create the transfer matrix T, Eq. (4-5). Various measurement imperfections, such as errors in positioning, amplitude and phase mismatch of measurement channels and instrumentation noise will distort the matrix T and thus create a measurement error which will add to the error intrinsic to the dummy. In a real situation the measurement error may easily exceed the intrinsic error by one or several orders of magnitude.

The simplest way to find out how does the matrix T resist to measurement imperfections is to look at its condition number. It can be shown that the increase in number of drivers and control points worsens the conditioning. It has been found that the increase in the number of control points C at a fixed number of drivers providing C > D, does not impact conditioning in any significant way. On the contrary, a simultaneous increase of number of drivers and control points worsens conditioning to an increasingly large extent. Moreover, increasing the distance of control points from the source worsens conditioning.

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Figure 6: Matching of diffracted fields. Clockwise from top left: 200 Hz, 500 Hz, 1 kHz, 2 kHz.

Thus it appears that the three parameters which enable the identification of dummy, C, D and the distance of the control points from the source produce conflicting effects on the accuracy of the dummy source method on the matrix condition number. It does therefore look appropriate to establish a common indicator of overall accuracy, by minimising the product of the error of characterisation method and the conditioning number of matrix T.

4 Validation by measurement

To validate the dummy source approach, an experimental characterisation of a physical source of sound in a semi-anechoic room has been carried out. The operational pressures radiated by the source were measured while the transfer functions of the dummy source as well as the dummy itself were modelled numerically. The modelling was done using the technique of substitute sources as described in [14]. The validation was then done by comparing the measured sound levels of the running source and the levels computed by the model of dummy.

4.1 The experimental set-up

The source, a 300 × 232 × 500 mm plexiglass box, was set to vibration using a shaker mounted inside of it. The vibration of the box was monitored using accelerometers mounted inside of the box. This particular experimental demonstrator was presented in [15], and further elaborated in [16]. Unlike in [15], 64 microphone positions are used in the present analysis.

The experimental set-up is shown in Fig. 7, left. The shaker was connected at two misaligned points to avoid symmetry and create a complex vibration pattern of the box. The shaker was driven by a band-pass limited white noise. A 5 × 5 planar array was moved around the controlled vibrating source, in eight positions, and the measurements were synchronised using a reference acceleration signal, Fig. 7 right.

The positions of microphones used to estimate the drivers' source strengths were selected by randomly taking 8 microphones from each of 8 array positions, in total 64 out of 200 available recordings. Note that,

AEROACOUSTICS AND FLOW NOISE 391

for simplicity, there are no microphones placed above or below the vibrating box. The sound radiated from the top and bottom plate of the box will nevertheless be present in the recordings due to the ground reflection and diffraction by the box. The top and the bottom of the dummy were equipped with drivers.

Figure 7: Left: experimental source. Right: geometry of control microphone array.

The numerical dummy source was given the shape of the original source. This has enabled the identification of transfer functions by numerical modelling instead of measurement. In total 22 drivers were distributed quasi-uniformly across the surface of dummy, which corresponds to ∼2 drivers per wavelength at 1000 Hz. The numerical model used to compute the transfer impedances has been experimentally validated in [14]. The analysis was done in 20 - 2000 Hz range; in this range the acoustic wavelength goes from much larger to comparable with the size of the source.

Since the measured operating pressure responses of the vibrating source were contaminated by measurement noise and imperfections in measurement microphone positions the source strengths of the drivers were found by using Tikhonov regularization in the matrix inversion. The comparison between the sound fields of the original source and the dummy has been done at independent microphone positions, not used to estimate the source strengths. The comparison at two points, one inside and one outside of the fictitious surface created by rotating the microphone array around the source, is shown in Fig. 8. Good matching between the measured sound field of the vibrating box and the predicted sound field of the dummy source is observed. It is seen that the dummy source very well reproduces the measured sound field between 200 Hz and 1000 Hz. The discrepancies at low frequencies result from the loss of radiation efficiency of the box, degrading the S/N ratio and coherence between channels. The poor measurement conditioning in turn leads to inversion errors. On the contrary, the discrepancy at high frequencies is due to insufficient density of drivers of the dummy.

A comparison with [15-16] has shown that the quality of sound reproduction by the dummy remains satisfactory between 200 Hz and 1000 Hz independent of the number of control microphones (32, 64, or 128). This agrees with the finding in §3.2 that the number of microphone points is of secondary importance, given that the system is overdetermined.

The results show that a dummy source with a limited number of drivers is capable of accurate enough sound reproduction of a physical source having continuous surface distribution of vibration. The

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demonstrator further shows the viability of the dummy source approach for experimental work in anechoic conditions: even using computed transfer impedances is good enough to allow for correct estimates.

Figure 8. Sound pressure levels predicted by the dummy source and recorded with a vibrating box at two points: left, interior to control surface; right, exterior to control surface.

5 Conclusions

The dummy source is a substitute to an original source which creates sound by its vibration. Its purpose is to enable prediction by computation of sound radiated by the source in an arbitrary environment. It consists of a box of simple shape into which a number of drivers are embedded. The strengths of individual drivers are identified using an inverse processing of sound signals from an array of control microphones surrounding the operating original source.

Based on a 2D modelling of sound fields created by the original source and the dummy, it has been found that at least 2 drivers per wavelength are needed for truthful reproduction of the radiated far field. It was also found that the number of control microphones has to be larger of the number of drivers by approximately 50% in order to ensure good conditioning of sound signals measured by the control array. An experimental validation of the approach in a 3D reflective acoustical environment has confirmed the theoretical findings.

Acknowledgements

Partial funding of this work by Volvo Construction Equipment is gratefully acknowledged. The Laboratoire Vibrations Acoustique is a member of the CeLyA Centre of Excellence in Acoustics.

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