Proceedings of the Institute of Acoustics

 

 

Towards a low cost method for measuring the airborne sound insulation of partitions

 

Gabriel Whittle¹, iKoustic Soundproofing and the Acoustics Research Centre, University of Salford, Wetherby, UK

Andrew Elliott, Acoustics Research Centre, University of Salford, Salford, UK

Joshua Meggitt, Acoustics Research Centre, University of Salford, Salford, UK

Daniel Wong-McSweeney, Acoustics Research Centre, University of Salford, Salford, UK

Rick Parsons, iKoustic Soundproofing, Wetherby, UK

 

ABSTRACT

 

BS EN ISO 10140-2 outlines a method for estimating the airborne sound insulation of a partition using adjacent rooms. In this paper, a new method is presented by which the airborne sound insulation is obtained without the need for a transmission suite. Using a stand-alone partition, a representative blocked pressure field is applied numerically to a measured frequency response function matrix characterising the partition. The incident and transmitted powers are determined from the blocked pressure and velocity field on the source and receiver sides, respectively. Results of several partitions measured using the proposed methodology will be compared against results obtained according to BS EN ISO 10140-2.

 

1. INTRODUCTION

 

The design and optimisation of new soundproofing products requires the ability to extensively test the partition or system the product is designed for. A method for measuring the sound reduction index R of a partition is outlined in the standard BS EN ISO 10140-2 [1], and has the advantage that the method has been validated and used for many years, giving repeatable results to a given level of repeatability, and a standardised, universal method of gaining R. The method requires two adjacent reverberant rooms, one a source room with more than one omnidirectional or moving sound source and one a receiver room with several microphones, whereby the difference in sound pressure level between the rooms is compared. The disadvantage of this method is that for companies looking to optimise their products and get a range of performance data, these acoustic laboratories are few and far between, and getting testing completed regularly is often not a feasible option. This paper is concerned with creating a novel testing methodology which will substantially reduce the size and cost of testing while getting comparable results to a full size acoustics laboratory. The approach involves utilising frequency response function (FRF) measurements to estimate the radiated sound power of a partition. This means that with the correct testing protocol, the sound reduction index R of complex partitions can be tested without the need to utilise facilities elsewhere. To facilitate the collection of comparable data, a test rig is designed to replicate the aperture within standard acoustic laboratories. The incident blocked pressure field in a laboratory is also measured with a bespoke microphone rig, meaning that a comparison can be drawn between a synthesised diffuse acoustic field and a measured acoustic field.

 

Section 2 introduces the theory behind the research conducted in this paper, section 3 explains the methodology for designing the test rig and the bespoke microphone rig, and section 4 offers results, comparing laboratory measurements with results from the novel methodology.

 

2. THEORY

 

The standardised method to discover the airborne sound reduction index R of a partition within an acoustic laboratory is defined in BS ISO 10140-2, with the equations:

 

 

Where L_I is the sound pressure level in the source room, L_R is the sound pressure level in the receiver room, S is the area of the specimen under test and A is the equivalent sound absorption area in the receiving room. Throughout this paper the subscript I will signify incident or source room side, and the subscript R will signify radiating or receiver room side of the partition. The aim of this research is to find an alternative method to the rating of airborne sound insulation partitions without the use of reverberant test rooms. This leads to the equation utilising sound power:

 

 

Where Π_I is the incident sound power and Π_R is the radiated sound power. The following sections outline the methods used for gaining the incident and radiated sound power.

 

2.1. Radiated power using FRFs

 

The calculation of radiated sound power requires velocity data gathered at discrete points across the surface of a radiating partition [2].

 

 

where v_R is the radiating velocity on the receiver side of the partition, f_R is a force vector on the radiating/receiver side of the partition, H indicates Hermitian transpose, and Tr() indicates that the answer is the trace of the matrix within the brackets. ω shows the frequency dependency of each term, which will be assumed in later equations. Radiating velocity is obtained by the equation:

 

 

where H_RI  is an FRF matrix of size N_R × N_I  and f_I is a force vector of size N_I × 1 which can be related to pressure with the following equation:

 

 

where S_I is a diagonal matrix containing the area allocated to the input force on the incident/source side of the partition and p_I is the sound pressure incident on the source side surface. The radiating force can similarly be related to:

 

 

The radiated pressure can be obtained from radiating velocity using a radiation matrix, as defined in equation 9:

 

 

Combining all of the above equations gives:

 

 

where G_pp = is the acoustic field (either synthesised or measured) to be applied to the calculation, as outlined in section 2.1.1 and 2.1.2.

 

To evaluate the radiated power the radiation impedance matrix Z_RR must be computed:

 

 

where ρ₀ is the pressure of air, c is the speed of sound in air, and S_Rii  is the vibrating area of each element on the radiating side of the partition.

 

The results section will compare the influence of both a synthesised diffuse acoustic field (DAF) and an acoustic field (AF) that is measured within a UKAS accredited laboratory specifically designed to be used with BS ISO 10140.

 

2.1.1 Synthesised diffuse acoustic field

 

A diffuse acoustic field can be synthesised with the equations [3]:

 

 

where k₀ =s the acoustic wavenumber and R_ij  is the distance between the points x_i and x_j. In lieu of pressure measurements,  is assumed to be 0.5 for all frequencies.

 

2.1.2 Measured acoustic field

 

If sound pressure measurements are available (the methodology for measuring pressures within a reverberant acoustic chamber utilising a bespoke microphone rig is described in section 3.2), the cross spectral density can be computed with the equation:

 

 

where p_I  is a vector containing the pressure at discrete points across the incident/source room side of a partition installed within a UKAS accredited acoustic laboratory.

 

2.2. Radiated sound power using radiation efficiency

 

An alternative method for calculating radiated sound power is found in ISO 7849, using average velocity measurements and a radiation efficiency term [4]:

 

 

where ϵ_R is radiation efficiency. The standard specifies that either a radiation efficiency dependent on frequency can be used, or the assumption of a radiation efficiency of 1 can be used. A method for calculating radiation efficiency is available in Cremer [5]:

 

 

where f_c is the coincidence frequency of the radiating plate and η is the loss factor of the plate. This equation is making an infinite plate assumption, where the area S is the power radiated from a partial area of an infinitely large plate. The advantage of using this equation is that it requires fewer response points, as it is a more general derivation using an average of the vibration velocity data. The use of equations 14 and 8 will be compared in section 3 to evaluate the best method for calculating R.

 

2.3. Incident power

 

The incident power on the source side can be expressed as [6]:

 

 

To put this theory into practice, a test rig with the specific purpose of measuring airborne sound insulation through partitions was designed and constructed. As per the construction measurements used in the acoustic laboratory, the aperture where the sound insulating partition is to be installed must be 3.6m × 2.4m. The structure surrounding the aperture is two brick columns with a concrete lintel over the top, as shown in figures 1 and 2. This construction gives a representative aperture to install partitions without the need for the surrounding reverberant rooms.

 

3.2. Measurement of the blocked pressures across partition within an acoustic laboratory

 

A microphone rig was designed to measure a representative blocked pressure field across a lightweight partition installed within a UKAS accredited acoustic laboratory. The Schroeder frequency is the limit between modes dominating in the low frequencies and a diffuse field in the high frequencies, and is calculated:

 

 

In the source room of the laboratory, RT60 = 5s is the reverberation time of the room in seconds and V=136m³ is the volume. f_s = 383Hz, so to allow for error, this is made 600Hz and the wavelength can be calculated:

 

 

To have a microphone placed at every quarter wavelength, the spacing is 0.57/4 = 0.143m, meaning that the microphone rig must have a spacing of 0.14m to capture 612Hz as the highest non-diffuse frequency.

 

 

Figure 1: Test rig construction.

 

 

Figure 2: Test rig with simple plasterboard partition installed.

 

3.3. Test rig measurement protocol

 

The FRF matrix to be used in equation 8 is mobility measured across the entirety of the source room side of the test rig. This data will be gathered with an impact hammer and accelerometers. There are 442 blocked pressure measurements across the surface and 15 accelerometers available, which means that the H matrix will be N₁₅ × N₄₄₂. In papers such as [3], a laser vibrometer was utilised to gather the response points. The advantage of using a laser vibrometer is that you can have many more response points, meaning that the spatial sampling will be improved and there will be more accuracy in high frequencies. However, one of the purposes of this testing methodology is to create a simple test with a reduced cost. This will come at the cost of fidelity in high frequencies.

 

4. RESULTS

 

4.1. Validation of the methodology using a finite element steel plate

 

To validate the approach numerically, mobility data for a steel plate is gathered through the use of a finite element model. The plate is a 0.63mm thick sheet of galvanised steel, with a total of 442 points of interest specified in the model. The sound reduction index using both a synthesised DAF and a measured AF is compared with measurements obtained within the acoustic laboratory. The results in 1/3rd octave bands demonstrated in figure 3 show that from 100-300Hz both a synthesised and measured field give comparable values to that of the one measured under laboratory conditions. The results drop off at 320Hz which is where the spatial sampling is too sparse to capture meaningful data. Figure 4 shows that if the spatial sampling is reduced to 111 elements on the radiating side, the results drop off at 200Hz, showing that the number of response points is a contributing factor to the functioning of this method.

 

4.2. Double-leaf and air cavity partition measurements

 

As per the measurement methodology outlined in section 3.3, a double leaf partition with two single layers of plasterboard installed on timber stud with an air-cavity is tested. Figure 5 shows that, similarly to figures 3 and 4, the measured DAF gives a higher R value, by an average of 4dB. Utilising measured pressures also gives a more representative fit to the BS EN ISO 10140 measurements within a lab. The resonance exhibited by the air cavity between 50-100Hz is captured, but at 125Hz the lack of spatial sampling makes the measurements incomparable to the measurements completed within an acoustic laboratory. It is also worth noting that the results using the measured AF is more accurate than the synthesised field, where up to 125Hz the results are within 2dB of the ISO measured results and follow the trend of the curve, as opposed to the synthesised field which is between 2-7dB out from the ISO measured results.

 

 

Figure 3: Finite element steel plate using 442 elements and equation 8.

 

 

Figure 4: Finite element steel plate using 111 elements and equation 8.

 

 

Figure 5: Transmission loss of a 3.6m x 2.4m double-leaf plasterboard partition, using the hammer and accelerometer testing methodology and equation 8 to calculate radiated power.

 

The coupling of the vibration velocity field in the plate to the acoustic field is achieved by the use of a Rayleigh integral and radiation efficiency of the plate. This method requires a large amount of response points to capture the detail in the higher frequencies, so it can be assumed that given enough sensors, a full frequency range for transmission loss can be estimated using the mobility measurement method. However, this methodology is designed to be cost sensitive and allow for the methodology to be carried out without a large cost. The alternative method in equation 14 allows for a less accurate result but gives a view into the the R values in the higher frequencies with the use of fewer sensors. Using just the measured AF the result can be seen in figure 6.

 

Figure 6 shows that using the simplified equation 14, the transmission loss of the partition in a broadband frequency range can be computed to a reasonable degree, although there are obvious issues with the accuracy. The detail in the low frequencies is captured, but as the frequency increases there

 

Figure 6: Transmission loss of a 3.6m x 2.4m double-leaf plasterboard partition using the hammer and accelerometer testing methodology and equation 14 to calculate radiated power.

 

is a gradual under-estimation of the transmission loss. It can also be observed that a coincidence dip of sorts is captured around 1600Hz, which is captured in the laboratory at 2500Hz. Given the accuracy of the equation 8 results in the low frequencies and the ball-park result given in high frequencies for equation 14, the solution for a result encompassing a broadband frequency range will be a hybrid method joining the two at a select crossover frequency. Further work will be carried out by testing a greater range of airborne sound insulating partitions to determine the optimum crossover frequency point.

 

5. CONCLUSIONS

 

The comparison of the synthesised diffuse acoustic field and measured acoustic field shows the synthesised field leads to a slight underestimation and the measured field leads to a slight overestimation. When utilising the new testing methodology, the measured acoustic field gave better results than the synthesised field, validating the need for the measurement.

 

The method outlined in equation 8 couples the vibrational velocity matrix of the plate to the acoustic field using the Rayleigh integral - the research conducted on this method shows that it is more accurate than equation 14. This method requires a significant amount of response points to capture detail in high frequencies, meaning it will be limiting if a large frequency range is required.

 

Utilising ISO 7849 and equation 14 gives a less accurate but more complete view of the sound reduction over a broader frequency band. In light of the research conducted, it can be assumed that a hybrid method combining the two will allow for the detail captured in the low frequencies, with an estimation of the result in higher frequencies.

 

The work that has gone into designing and building the test rig and measuring the acoustic field offers a new and exciting look into what the future of soundproofing testing outside of UKAS accredited laboratories may be like.

 

ACKNOWLEDGEMENTS

 

The work was carried out at part of the KTP project between iKoustic Ltd and the University of Salford, with funding from iKoustic and Innovate UK.

 

REFERENCES

 

1. BS EN ISO 10140-2:2021 Acoustics - Laboratory measurement of sound insulation of building elements - Part 2: Measurement of airborne sound insulation.

2. B. Mingsian and T. Mingchun. Estimation of sound power of baffled planar sources using radiation matrices. The Journal of the Acoustical Society of America, 112:876–83, 2002.

3. N. B. Roozen, Q. Leclère, D. Urbán, T. Méndez Echenagucia, P. Block, M. Rychtáriková, and C. Glorieux. Assessment of the airborne sound insulation from mobility vibration measurements; a hybrid experimental numerical approach. Journal of Sound and Vibration, 432:680–698, 2018.

4. DD ISO/TS 7849 - 1:2009 Acoustics. Determination of airborne sound power levels emitted by machinery using vibration measurement. Survey method using a fixed radiation factor, 2009.

5. L Cremer, M Heckl, and B Petersson. Structure-Borne Sound. Springer, 3rd edition, 2010.

6. O. Robin A. Berry. Estimating the sound transmission loss of a single partition using vibration measurements. Applied Acoustics, 141:301–306, 2018.

 

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¹ gabriel@ikoustic.com