I-PCA – methodology

Consider a multi-layered partition discretised into ‘𝑗’ paths (patches) on the source side panel of Figure 3.8. In operational conditions, the velocities on each path can be easily

measured and the volume velocity can be found as per Eq. (3.11). Under PCA methodology, the total pressure at point ‘k’ in the receiver volume for this source ‘1’ can be written as,

𝑝_𝑘 = {𝑈11 … 𝑈1𝑛} {

𝑣₁′

⋮ 𝑣𝑛′

} 𝑑𝑆 (3.14)

where, 𝑑𝑆 is the patch area. To maintain phase between all the operational quantities, the operational responses 𝑣 and 𝑝 can be referenced to the source. The source can be airborne (loudspeaker) or structure borne (shaker or impact hammer). In the case of an impact hammer hit at patch ‘1’, the operational quantities can be referenced to the force 𝑓₁ and Eq. (3.14) can be written as,

𝑝_𝑘 𝑓1 = {𝑈𝑘1 … 𝑈_𝑘𝑛} { 𝑣1′ 𝑓₁ ⋮ 𝑣𝑛′ 𝑓1} 𝑑𝑆 (3.15)

The ‘𝑣₁/𝑓₁’ quantity is nothing but the mobility of the patch ‘1’ for a source acting on patch ‘1’. Similarly the quantity ‘𝑝_𝑘/𝑓₁’ is the vibroacoustic FRF ‘𝐻_𝑘1’. The above Eq. (3.15) can now be concisely written as,

𝐻_𝑘1 = {𝑈𝑘1 … 𝑈𝑘𝑛} {

𝑌11

⋮

𝑌_𝑛1} 𝑑𝑆 (3.16)

In the same way, the hammer can now be hit on every patch centre position from 1 to n, and the mobilities and vibroacoustic FRF’s that are measured can be written in a matrix formulation as below, {𝐻𝑘,1 … 𝐻𝑘,𝑛} = {𝑈𝑘,1 … 𝑈𝑘,𝑛} [ 𝑌₁₁ … 𝑌_1𝑛 ⋮ ⋱ ⋮ 𝑌𝑛1 … 𝑌𝑛𝑛 ] 𝑑𝑆 ∴ {𝐇} = {𝐔}[𝐘]𝑑𝑆 = 1 𝑗𝜔{𝐔}[𝐀]𝑑𝑆 (3.17)

Note that for each excitation, only the velocity and pressure responses change while the acoustic FRF’s remains the same. This is because the acoustic FRF is an invariant property of vibroacoustic system. Post-multiplying by the inverse of the accelerance matrix and dividing by 𝑑𝑆 on both sides of Eq. (3.17), we get,

𝑗𝜔

𝑑𝑆{𝐇}[𝐀]−𝟏= {𝐔} (3.18)

Thus the acoustic FRF vector can be measured inversely from accelerance and vibroacoustic FRF’s. Therefore the first step of I-PCA is measuring the accelerances and vibroacoustic FRF’s similar to I-ASCA methodology outlined in Section 3.4 and utilising Eq. (3.18) to get the acoustic FRF’s.

Next, for the operational phase, the airborne source can be activated and the operational accelerations and pressures can be measured as per Eq. (3.7). The FRF and the operational measurements together constitute the I-PCA measurement phase. Then the path contribution of interest can then be measured as per Eq. (3.13) which represents the radiation from a single patch under the given airborne excitation. Likewise, the total pressure at a receiver point can be represented as a sum of all such path contributions under the airborne excitation as shown below.

Figure 3.23: Left – Under the action of source excitation, the receiver side pressure at ‘𝑘’ predicted as a sum of all path contributions from the vibrating partition. Right – Under the action of source excitation, the radiation from a single patch ‘n’ is the path contribution (with

3.6.1.1

I-PCA validation

To validate the I-PCA methodology outlined in previous section, a pressure validation test can be conducted where the total pressure at a validation point ‘𝑘’ can be written as a sum of contributions from all the patches characterised as equivalent volume velocity sources. The pressure predicted in this case would be,

𝑝_𝑝,𝑘 = {𝐔_𝑘}{𝐐′_{} = {𝐔}

𝑘}{𝐯′}𝑑𝑆 =

1

𝑗𝜔{𝐔𝑘}{𝐚′}𝑑𝑆 (3.19)

If the predicted pressure is equal to the measured pressure then the methodology can be said to be valid in the frequency range 𝐟. This would also confirm the validity of the inversely measured acoustic FRF’s. We can also see that by substituting Eq. (3.18) into Eq. (3.19), we get, 𝑝𝑝,𝑘 = 1 𝑗𝜔 𝑗𝜔 𝑑𝑆{𝐇𝑘}[𝐀]−𝟏{𝐚′}𝑑𝑆 = {𝐇𝑘}[𝐀]−𝟏{𝐚′} = {𝐇𝑘}{𝐟𝐛𝐥} (3.20) Thus using the I-PCA methodology, we have reached a result (Eq. 3.20) which has been proven and validated in Section 3.5. This result is also not surprising as the total receiver pressure can be conceptualised to be either a sum of source contributions or sum of path contributions. The result also provides confidence in the I-PCA methodology to predict the airborne sound transfer through the partition and diagnose the path contributions. These path contributions would allow us to diagnose the sound transfer locally/spatially through the partition. Using these contributions, the weak regions of sound insulation can be diagnosed as regions of high path contribution.

We can also see that, using the same measurements from I-ASCA, we can measure the acoustic FRF’s (Eq. 3.18) and path contributions (Eq. 3.13), without performing a single extra measurement. This proves the versatility of the analysis in that both source and path contributions can be obtained from a single set of data with the same accuracy. Additionally, this analysis presents a novel measurement approach for acoustic FRF’s where a direct

measurement cannot be performed due to space restrictions so the path contributions can be measured for source side panels as well.