Prediction of Far Field Sound Radiation Using Blocked Pressures and Reciprocally Measured Vibro-Acoustic Frequency Response Functions Lucy Barton 1 University of Salford The Crescent, Salford M5 4WT, UK Andrew Elliott 2 University of Salford The Crescent, Salford M5 4WT, UK John Smith 3 DSTL Porton Down, Salisbury, Wiltshire, SP4 0JQ, UK

ABSTRACT

In industry, the use of blocked forces for the characterisation of structure borne sound sources is

now common practice, whereas the sound power is typically used for airborne sound sources. It should also be possible to use blocked forces (or blocked pressures) to predict airborne sound radi-

ation using a similar approach and measurement methodology as the in-situ blocked force method.

This is achieved by discretising the surface of the source into small unit cells, and measuring the vibration velocity whilst operational. To convert these velocities to blocked forces, the mobility of the source for each of the positions must also be known: and to predict the radiated sound, a meas-

ured vibro-acoustic frequency response function is also required. Typically, vibro-acoustic FRFs are measured by exciting the structure with an instrumented hammer or shaker, but they may also be measured reciprocally using a volume velocity source. Described in the paper is an experiment

conducted in a fully anechoic chamber to investigate the feasibility of the above approach, i.e. us- ing blocked pressures and reciprocally measured vibro-acoustic FRFs to characterise and predict airborne sound. The method is validated by comparing the directly measured radiated sound pres-

sure to the predicted pressures, according to the above.

1 L.S.Barton@edu.salford.ac.uk

2 A.S.Elliott@salford.ac.uk

3 Not Supplied

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1. INTRODUCTION

Structure-borne radiated noise is an important factor to consider in a multitude of settings. This in- cludes domestic appliances such as a washing machines, industrial equipment in factories, and vehi- cles such as cars and trains. Noise issues can present a range of problems, from causing annoyance, to structural damage. In order to predict the radiated noise, the structure which is radiating the noise must be characterised [1]. A now common method is to use the blocked forces of the system in the In-Situ Transfer Path Analysis approach [2]. Whereas the vibroacoustic FRFs are usually measured using an impact hammer, another option could be to measure the FRFs reciprocally, using a volume velocity source. Contained in this paper is a description of the Blocked Force iTPA method, and an experiment con- ducted in an anechoic chamber for the purpose of predicting radiated sound from a simple baffled plate test rig. In addition to a prediction using an impact hammer for the measured vibro-acoustic frequency response function, a volume velocity source is also used to reciprocally measure the FRF. The two predictions are compared to the measured sound radiated by the test rig when excited by a shaker mounted to the rear of the plate, producing white noise.

2. THEORY The in-situ blocked force method allows the source to remain in position, coupled to the receiver, throughout the analysis procedure entirely. This is in contrast to the classical Transfer Path Analysis matrix inversion approach [3], which requires the source to be uncoupled from the receiver at certain measurement stages.

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Figure 1: Schematic of source receiver structure, consisting of an arbitrary vibration source ‘A’, which is coupled to a receiver structure ‘B’. When operational, the source creates multiple forces at ‘(a)’, which are then transmitted through the interface ‘(c)’ to the receiver structure ‘B’. The receiver ‘(d)’ is the point at which a combination of the radiated sound from the receiver ‘B’ and airborne

radiated sound from the source ‘A’ are measured as sound pressures. ‘C’ denotes the assembly as a coupled structure, and ‘(b)’ the vibration velocities on the plate. The blocked forces of the system can be found by:

−1 𝑣ሶ 𝐶,𝑐 (1)

𝑓 ҧሶ

𝐴,𝑐 = 𝑌 𝐶,𝑐𝑐

+ 𝑣ሶ 𝐶,𝑏 (2)

𝑓 ҧሶ

𝐴,𝑐 = 𝑌 𝐶,𝑏𝑐

+

𝑣 ሶ 𝐶,𝑐 𝑣ሶ 𝐶,𝑏 ൠ (3)

𝑌 𝐶,𝑐𝑐 𝑌 𝐶,𝑏𝑐

𝑓 ҧሶ

𝐴,𝑐 = ൤

𝑇 ൨

൜

Where:

𝑓 ҧሶ

𝐴,𝑐 = −𝑓ሶ 𝐴,𝑐 ห 𝑣ሶ 𝐴,𝑐 =0 (4)

Upper-case subscripts are the structure elements A- source, B- receiver, C – coupled elements; lower case are measurement locations (a), (b), (c), (d); dot denotes that the source is operational, and bar denotes the blocked force. 𝑣ሶ 𝐶,𝑐 and 𝑣ሶ 𝐶,𝑏 are the vectors of complex velocities at measure- ment locations (c) and (b), on assembly C, whilst the source A is operational. Subscripts are ar- ranged as row (response) then column (excitation): 𝑌 𝐶,𝑐𝑐 and 𝑌 𝐶,𝑏𝑐 are matrices of frequency re- sponse functions of velocities due to force/ mobility [4], [5] . Assuming the direct sound pressure at (d) radiated from source A during measurement is negligible, the sound pressure due to the structure-borne noise radiated by the coupled assembly is defined as:

𝑃 ሶ 𝐶,𝑑 = 𝐻 𝐶,𝑑𝑐 𝐹 തሶ 𝐴,𝑐 (5) Where 𝐻 𝐶,𝑑𝑐 is the measured transfer function for the coupled assembly C. This transfer function can be measured in multiple ways. In this paper, the transfer function for 𝐻 𝐶,𝑑𝑐 will be measured in two ways: using a force hammer, which will be referred to as 𝐻 𝑝𝑓 - sound pressure due to force; and reciprocally with a volume velocity source, which will be referred to as 𝐻 𝑣𝑞 - surface velocities due to volume velocity.

𝑃 ሶ 𝐶,𝑑 = 𝐻 𝑝𝑓 𝐹 തሶ 𝐴,𝑐 (6) The reciprocal volume velocity transfer function is obtained by utilizing a reference microphone lo- cated a distance away from the assembly, and is calculated using:

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𝑗𝜌 0 𝑒 𝑗(𝜔𝑡−𝑘𝑟)

𝑎

𝑎

2𝜆𝑟 (7)

𝐻 𝑎𝑞 =

𝑞 1 =

𝑃 2 ×

𝐻 𝑣𝑞 =

𝐻 𝑎𝑞

𝑗𝜔 (8)

𝑃 ሶ 𝐶,𝑑 = 𝐻 𝑣𝑞 𝐹 തሶ 𝐴,𝑐 (9) Where

𝑎

𝑞 1 is the transfer function between the accelerometers and the volume velocity source, 𝑃 2 is

𝑎

the sound pressure at the reference microphone,

𝑃 2 is a transfer function between the accelerometers

and the pressure at the reference microphone, 𝜌 0 = air density, 𝑒 = 2.71, 𝜔 = 2 * π * frequency, 𝑡 = time = 1, 𝑘 = wavenumber, 𝑟 is the distance between the volume velocity source and the reference microphone, and λ = wavelength [6]. The volume velocity source is activated at each of the microphone positions to gain a transfer func- tion for each: these positions are equivalent to multiple (d) as shown in figure 1. 3. METHODOLOGY

An aluminium plate of dimensions 0.7 x 0.9 x 0.025m, mounted in a lightweight frame, was installed in an anechoic chamber. The plate had 2 shakers mounted on the rear side, with the entire back side of the plate enclosed by a baffle. The plate was instrumented with a grid of 26 accelerometers, 24 distributed evenly over the rear surface, and 1 accelerometer at each of the shaker positions, mounted on the front surface.

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Figure 2: Plate and baffle instrumented for testing. The aluminium plate can be seen on the left, with the wooden baffle on the right. On the rear side of the plate, the side which will be enclosed by the baffle, 24 accelerometers are mounted, along with a small shaker. Between the shaker and the plate, a force sensor is mounted, to measure the shaker and be a reference to calculate the transfer functions. Inside the baffle, acoustic foam tiles have been added to help baffle the plate and absorb direct noise from the shaker. This is necessary in order to preserve the direct transfer function between the plate and microphones when the shakers operational.

Figure 3: schematic diagram showing a rear view of the plate. Numbers denote accelerometer posi- tions, and the shaker is shown in red and black. An array of measurement microphones were mounted on a metal arch, measuring the radiation of the plate at varying intervals between 0° and 90°. The plate was excited using the shakers for the meas- ured radiated noise, and an impact hammer/ volume velocity source for the measurement of the vibro-

acoustic frequency response functions of the plate. The impact hammer was used to make an excita- tion at each of the accelerometer positions on the plate, to collect the mobility matrix of the plate. The impacts were repeated to measure the transfer functions between each of the accelerometer po- sitions and the measurement microphones.

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Figure 4: Experiment set up inside anechoic chamber. Beginning on the left, an arch with micro- phones mounted from 0° to 90° in relation to the plate-visible lower right. In order to measure the reciprocal volume velocity transfer functions, a reference microphone was mounted 2.3m away from the test rig, in a corner of the room. The measurement microphones were removed, and the volume velocity source was activated at each of the measurement microphone po- sitions. The response was measured at the reference microphone, for the use of Equation 7, and at the accelerometer positions.

3. RESULTS

Presented in this section is a comparison between the transfer functions collected using the impact hammer, 𝐻 𝑝𝑓 , and the transfer functions collected using the volume velocity source, 𝐻 𝑣𝑞 , with re- spect to an arbitrary selection of accelerometer positions on the plate. Following this, the results of the predictions made using these transfer functions and blocked forces are shown in comparison with the directly measured radiated noise signal produced by the plate when the excitation source is made operational. The excitation source consists of a single shaker generating white noise.

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Figure 5: Comparison between 𝐻 𝑣𝑞 , shown in black; and 𝐻 𝑝𝑓 , shown in red, between a selection of accelerometer positions on the plate and the measurement microphones for a frequency range of 50Hz-5kHz. The quantities have been averaged across the 8 measurement microphones. Figure 5 shows a selection of comparisons of 𝐻 𝑝𝑓 and 𝐻 𝑣𝑞 with respect to accelerometer positions on the plate. Referring to a rear view of the plate, as in Figure 3, it can be seen that accelerometer 1 is located in the upper right corner of the plate, accelerometer 10 is located in the right of centre region, Accelerometer 14 is in the upper left area, and 22 is to the centre of the right side. The agreement between the transfer functions is relatively close.

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Figure 6: Mean average sound pressure at the measurement microphones: predicted using 𝐻 𝑣𝑞 , shown in dashed black; 𝐻 𝑝𝑓 , shown in red; and measured transfer function using the force reference, shown in solid black; over a frequency range of 200Hz to 8kHz.

Figure 6 shows the predictions for the radiated sound pressure using equations 6 and 9. In solid black, the direct measured sound pressure from the plate to the measurement microphones when the plate is excited by a shaker is shown. In red, the prediction using the hammer transfer function 𝐻 𝑝𝑓 ; and in dashed black, the prediction using the volume velocity source transfer function 𝐻 𝑣𝑞 . In general, the agreement is fairly accurate. There are some instances, such as at 768Hz, where there is a distinct anti-resonance in the prediction which is not present in the measured signal. Encouragingly, however, the large peak at around 2900Hz is correctly predicted with both methods. 5. CONCLUSIONS

In conclusion, the transfer functions 𝐻 𝑝𝑓 and 𝐻 𝑣𝑞 have shown to produce similarly accurate results for the prediction of the radiated noise from the test rig. This indicates that the volume velocity may be used reciprocally as an alternative to the impact hammer for the measurement of vibroacoustic FRFs. In practice, this could be helpful for measuring structures for which physical access is difficult. 6. REFERENCES

[1] A. T. Moorhouse, A. S. Elliott, and T. A. Evans, “In situ measurement of the blocked force of structure-borne sound sources,” J. Sound Vib. , 2009. [2] A. S. Elliott, A. T. Moorhouse, T. Huntley, and S. Tate, “In-situ source path contribution analysis of structure borne road noise,” J. Sound Vib. , 2013. [3] A. S. Elliott, J. W. R. Meggitt, and A. T. Moorhouse, “Blocked forces for the characterisation of structure borne noise,” in Conference Proceedings of Internoise and Noise-Con Congress , 2015, vol. 250, no. 1. [4] D. J. Ewins, Modal Testing: Theory, Practice, and Application . Research Studies Press Ltd, 2000. [5] L. Barton, A. Elliott, A. Moorhouse, and J. Smith, “In-situ transfer path analysis of multiple vibration sources in a complex source- receiver assembly,” Proc. Int. Congr. Acoust. , vol. 2019-Septe, no. September, pp. 435–442, 2019. [6] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, “Fundamentals of Acoustics (4th Edition),” John Wiley Sons Inc. , p. 560, 1999.

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