## 4 Procedures
### Ill-Conditioning in Coupling Methods  
Noise in measuring the FRFs of subsystems poses significant challenges to coupling methods, leading to ill-conditioned matrices that compromise the stability and accuracy of matrix inversion operations, especially in nearly singular matrices, where errors are amplified during inversion. A rank deficiency may occur due to linear dependence among rows or columns, frequently due to insufficient measurements. Disparities in the dynamic characteristics of coupled subsystems, such as differences in stiffness or damping, can cause wide-ranging eigenvalues and numerical instability. High-frequency effects further exacerbate sensitivity to minor data inaccuracies, while poorly chosen interface DoFs might fail to capture essential dynamics, resulting in ill-conditioned matrices. Additionally, closely spaced natural frequencies can render matrices nearly singular, complicating the inversion process. To address these challenges, recent studies have explored methods to enhance the accuracy and stability of matrix inversions. [PERSON] et al. [36] proposed a regularization method specifically designed for FRF-based substructuring, which complements the use of Singular Value Decomposition (SVD) in mitigating noise-related issues. Similarly, [PERSON] et al. [37] applied Tikhonov regularization alongside the pseudo-inverse method to improve inverse force identification, illustrating ongoing efforts to refine mathematical tools in the face of real-world measurement inaccuracies. In coupling methods, non-square matrices may arise, particularly when subsystems with different degrees of freedom are involved. These matrices, representing relationships between input and output variables such as \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Nodes** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Freq. (HZ) & 0 & 0 & 5.13 & 14.17 & 31.09 & 55.81 & 108.06 & 132.94 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The natural frequencies for the free-free beam system ABC. receptance matrices or transfer functions, often lack invertibility. For an ideal scenario with n\({}_{1}\) excitations and corresponding responses, the receptance matrix \(\left[\mathrm{H}(\omega)\right]\) is square and invertible. The inversion equation is expressed as follows:  
\[\left\{\mathrm{F}\right\}_{\mathrm{n}_{1\times 1}}\mathrm{e}^{\mathrm{i} \mathrm{e}\mathrm{i}\mathrm{e}\mathrm{t}}=\left[\mathrm{H}(\omega)\right]_{ \mathrm{n}_{1\times 1}}^{-1}\left\{\mathrm{X}\right\}_{\mathrm{n}_{1\times 1}} \mathrm{e}^{\mathrm{i}\mathrm{e}\mathrm{t}} \tag{41}\]  
However, when dealing with n\({}_{1}\) excitation forces and n\({}_{2}\) measurement responses, the challenge involves due to inverting a rectangular, rank-deficient receptance or FRF matrix \(\left[\mathrm{H}(\omega)\right]_{\mathrm{n}_{1\times 2}}^{-1}\). In this case, the pseudo-inverse matrix \(\left[\mathrm{H}(\omega)\right]_{\mathrm{n}_{1\times 2}}^{+}\) approximates the inverse:  
\[\left\{\mathrm{F}\right\}_{\mathrm{n}_{1\times 1}}\mathrm{e}^{\mathrm{i} \mathrm{e}\mathrm{t}\mathrm{t}}=\left[\mathrm{H}(\omega)\right]_{\mathrm{n}_{ 1\times 2}}^{+}\left\{\mathrm{X}\right\}_{\mathrm{n}_{2\times 1}} \mathrm{e}^{\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{t}} \tag{42}\]  
The pseudo-inverse matrix is computed as follows:  
\[\left[\mathrm{H}(\omega)\right]_{\mathrm{n}_{1\times 2}}^{+}=\left(\left[ \mathrm{H}(\omega)\right]^{\mathrm{T}}\left[\mathrm{H}(\omega)\right]\right)^{ -1}\left[\mathrm{H}(\omega)\right]^{\mathrm{T}} \tag{43}\]  
To address noise in measured data, techniques such as regularization or SVD are often preferred, which decompose the data matrix into singular vectors and singular values, filtering out noise by focusing on dominant modes of variation. The core principle of SVD involves expressing matrix H as \(\mathrm{H}=\mathrm{U}\mathrm{S}\mathrm{V}^{\mathrm{T}}\), where U and V are orthogonal matrices, and S is a diagonal matrix containing singular values \(\left\{s_{i}\right\}=\left\{s_{1},s_{2},\ldots\right\}\).