As seen in the Section 3.3.5, it is possible to apply iTPA for airborne source characterisation by employing discretisation at the source receiver interface. Now we will outline the practical methodology of this application. Consider a multi-layered partition with ‘𝑙’ internal layers/elements installed between a source and receiver room as shown in Figure 3.8. The partition is treated as a ‘black box’ so that the internal details are arbitrary provided the interface to the source can be suitably discretised. Thus, the method is suitable for dealing with multi-layered or ribbed partitions.
Figure 3.8: Left graphic –A multi-layered partition installed between a source room and a receiving room with the interface in dashed red. Right graphic – Isolated view of the partition
with the interface discretised in ‘𝑗’ patches. Blue arrows denote the radiation
The interface can then be discretised into ‘𝑗’ number of patches. Once we have a discretised interface, the iTPA methodology can be applied. Accordingly, the first phase of measurement is the FRF measurement which is the passive phase. It is assumed here that the structural FRF between different patches (averaged response on patch/ point force on patch) can be represented by the measurement at the centre points of those patches. Such FRF’s have also been referred to as the Patch Transfer Function (PTF) before and their validity has been demonstrated below a certain frequency limit determined by a sampling criterion [128, 133]. The FRF measurement can be performed by impacting a patch with a
force hammer and capturing the responses at all patches by use of accelerometers. These measured accelerance functions form the accelerance matrix [𝐀].
[𝐀]𝑖𝑥𝑗 = [ 𝑎1/𝑓1 ⋯ 𝑎1/𝑓𝑗 ⋮ ⋱ ⋮ 𝑎𝑖/𝑓1 ⋯ 𝑎𝑖/𝑓𝑗 ] (3.5)
To improve the quality of blocked force measurement, the system can be overdetermined where number of response positions (i) is greater than number of force locations (𝑗). Simultaneously, the vibroacoustic FRF’s can also be measured which relate the pressure at receiver points inside the receiving room due to an impact force applied on all patches individually (𝑃𝑎/𝑁). The resulting vibroacoustic FRF matrix can be formed as,
[𝐇]𝑘𝑥𝑗= [ 𝑝1/𝑓1 ⋯ 𝑝1/𝑓𝑗 ⋮ ⋱ ⋮ 𝑝𝑘/𝑓1 ⋯ 𝑝𝑘/𝑓𝑗 ] (3.6)
[𝐇] represents the vibroacoustic FRF matrix measured for ‘𝑘’ points in the receiving room. The second measurement phase is the ‘Active’/operational test, because the active source excites the system and operational acceleration responses are measured on the patches and pressures inside the receiving cavity for validation.
{𝐚′} = {𝑎1 ′ ⋮ 𝑎𝑖′} , {𝐩 ′} = {𝑝1 ′ ⋮ 𝑝𝑘′ } (3.7)
The apostrophe (′) indicates the measurements were performed for operational conditions. With the operational accelerations over the paths measured, the blocked forces can then be calculated as
{𝐟𝐛𝐥} = [𝐀]−𝟏{𝐚′} (3.8)
‘𝐟𝐛𝐥’ represent the blocked forces on the patches which characterises the discretised airborne excitation. This is an interesting case where an acoustic sound field is mapped on the partition by measuring its vibrational characteristics. Again, the blocked forces will only be valid in a frequency range 𝐟: [0 𝑓𝑚𝑎𝑥] Hz, where ‘𝑓𝑚𝑎𝑥’ is the maximum frequency of
prediction that can be determined by a sampling criterion (derived in later sections). The contribution of a blocked force ‘𝑓𝑏𝑙,𝑛’ at a point ‘𝑘’ in receiving volume (i.e. the source contribution) can be measured as,
𝑝𝑘,𝑛𝑠 (𝐟) = 𝐻
𝑘,𝑛. 𝑓𝑏𝑙,𝑛 (3.9)
where, the superscript ‘s’ denotes that the pressure is a source contribution and not the total pressure. For brevity, the bracketed term 𝐟 will be excluded in further equations and the contributions implicitly represent the prediction in this frequency range.
Similarly the total pressure at the receiver point can be represented as a sum of 𝟏 − 𝒋 source contributions. The individual or total source contributions will represent the sound transmission through all paths which is depicted in Figure 3.9 below.
Figure 3.9: Left – Under the action of ′𝑗′ blocked forces, the receiver side pressure predicted as a sum of all source contributions in frequency range f determined by sampling. Right – Under the action of a single blocked force at patch ‘n’, the pressure measured is a source contribution (with
superscript ‘s’)
As the source characteristics and source contributions are measured by an inverse process, the methodology will be denoted by the abbreviation I-ASCA (Inverse Airborne Source Contribution Analysis).
3.4.1
Validation of I-ASCA methodology
Before we apply the I-ASCA methodology for measuring source contributions, it is necessary to validate the methodology with a verified technique. iTPA is usually validated by predicting the remote vibrational response on the receiver using the respective FRF’s and blocked forces[134, 135] (see Eq. (3.4)). Likewise, for I-ASCA where the receiver response is acoustic, a pressure response can be predicted by combining the vibroacoustic FRF’s and the blocked forces over each patch. This can be written as,
𝑝𝑝,𝑘 = {𝐇𝑘}{𝐟𝐛𝐥} (3.10)
In Eq. (3.10), 𝑝𝑝,𝑘 is the predicted pressure at the validation point ‘𝑘’, and {𝐇𝑘} represents the vibroacoustic FRF vector measured at the validation point. If it is assumed that the sound transmission takes place solely through the partition and any flanking transmission is minimal, then this predicted pressure can be compared with the measured pressure. If the prediction is equal to the measured pressure then the methodology can be considered to be validated. This is called as the ‘Pressure Validation’ test.
In practice, the accuracy of a pressure validation will depend on the quality of the measurement data such as blocked forces, FRF’s, etc. Additionally, such prediction will not be valid outside the frequency range determined by the sampling criterion which will be discussed now.
3.4.2
Sampling considerations
I-ASCA methodology employs sampling of the structure (into patches) as well as the source excitation (as equivalent point forces). As such, the discretisation imposes a frequency limit under which the vibroacoustic response of the coupled system can be predicted. In other words, the grid size will be determined by maximum frequency of the prediction. Discretisation is commonly employed by Finite Element Analysis (FEA) methods, where a sampling criterion of grid size 𝑥 ≤ 𝜆/6 is used, where 𝜆 represents the structural
wavelength in vibration. The grid size ‘𝑥’ represents the smallest dimension in the grid. As we are discretising the excitation as well as the structure, the longitudinal wavelength in air (𝜆𝑎) or bending wavelength in the structure (𝜆𝑏) will determine the grid size. For building
partitions, below the critical frequency (𝜆𝑏 < 𝜆𝑎) the sampling criterion will be based on 𝜆𝑏 whereas for supercritical frequencies (𝜆𝑏 > 𝜆𝑎), the sampling criterion will be based on 𝜆𝑎. However, there are some basic differences between FEA based discretisation and a patch based discretisation. In FEA methods, the nodes are coupled between domains while in current case, patches (much larger than nodes) are directly coupled. Also the discretisation is only limited to the source receiver interface as opposed to FEA where complete fluid and structural domains are discretised. Therefore it is possible that a strict FEA based criterion (𝑥 ≤ 𝜆/6) would not be necessary.
A quick look at the literature shows that although a FEA based criterion can be sufficient, it is not always necessary in the case of coupling patches for predicting vibroacoustic response of structures coupled to air domains. The theoretical studies in [128, 133] conclusively show that for such cases, a patch size criteria of 𝑥 ≤ 𝜆𝑏/2 is sufficient to predict the radiation till
𝑓(𝜆𝑏) Hz, where 𝜆𝑏 is the bending wavelength of the structure at frequency 𝑓. In fact, the use
of patch based coupling is touted as an advantage over FEA methods in that the coarser discretisation criteria can be used without sacrificing the accuracy and also provides computational time benefits.
For a sampling criterion of 𝑥 ≤ 𝜆𝑏/2, the maximum wavelength that can be accurately represented is 𝜆𝑏,𝑚𝑎𝑥= 2𝑥 m and accordingly the maximum frequency of prediction
is 𝑓𝑚𝑎𝑥= 𝑓(𝜆𝑏 = 2𝑥). In wavenumber terms, the criterion can also be determined as 𝑘𝑏𝑥 ≤ 𝜋, where 𝑘𝑏 = 2𝜋/𝜆𝑏 is the bending wavenumber. For a structure coupled to a
heavier fluid (e.g. water), the patch size criteria changes [136]. To verify the validity of this sampling criterion, we will now compare the prediction made by I-ASCA for different grid size with grid size/bending wavelength for a test structure.