## 2 Decomposition Methodology
### Ivmd-Svd  
The proposed IVMD-SVD method uses an algorithm to automatically select the number of modes (K-values) to be used in the decomposition of the impact response signal. The result, decomposed by the selected modes, is processed to obtain the Shannon entropy, which is then used in further analysis.  
The appropriate K-value is selected through consideration of the relationship between IMFs (sub-signals) and the original signal. First, SVD is used to obtain eigenvalues from both the IMFs and the original signal. Then, the ratio between eigenvalues of each IMF and the original signal is calculated. This ratio is then used as an index in the evaluation of the relationship between sub-signals and the original signal. This evaluation process selects the optimal K-value, and, hence, the number of modes to be used in the decomposition of the response signal. The process for selecting the optimal K-value is described in the following section.  
The mechanism by which the optimal K-value is selected is summarised in Figure 2.  
The steps indicated in Figure 2 are as follows:  
Step 1: Sub-signals, \(IMF_{i}(t)\), are obtained by performing VMD using a range of K-values from 2 to \(N\), i.e., VMD (\(i=1,2,\dots,N\)).  
Step 2: The IMFs calculated from each VMD are combined to form Hankel matrix representations [25] of each sub-signal using a phase-space reconstruction process, as shown in Equation (7).  
\[X=\begin{pmatrix}IMF_{i1}&&IMF_{i2}&&IMF_{in}\\ IMF_{i2}&&IMF_{i3}&\dots&IMF_{i(n-1)}\\ &\vdots&&\ddots&\vdots\\ IMF_{i(m-n+1)}&&IMF_{i(m-n+2)}&\dots&IMF_{in}\end{pmatrix} \tag{7}\]  
where \(m\) is the sample number of the signal, and \(n\) is the dimension of the matrix.  
Step 3: Eigenvalues are calculated for each Hankel matrix IMF representation using SVD, as shown in Equation (8).  
\[SVD(IMF_{i}(t))\Rightarrow\begin{pmatrix}\lambda_{i1}&0&0\\ 0&\ddots&0\\ 0&0&\lambda_{iR}\end{pmatrix} \tag{8}\]  
Figure 2: Process of selecting optimal K-value.  
At this stage, the original signal is also converted to a Hankel matrix representation and SVD is applied, as shown in Equation (9):  
\[SVD((F(t))\Rightarrow\begin{pmatrix}\lambda_{1}&0&0\\ 0&\ddots&0\\ 0&0&\lambda_{k}\end{pmatrix} \tag{9}\]  
The \(\left\{Ratio\right\}_{i}\) defined in Equation (10), represents the relationship between each sub-signal and the original signal.  
\[\left\{Ratio\right\}_{i}=\left(\frac{\sum_{j=1}^{k}\lambda_{ij}}{\sum_{j=1}^{ k}\lambda_{j}}\right)=\begin{pmatrix}Ratio_{12}&Ratio_{13}&\ldots&Ratio_{1N}\\ Ratio_{22}&Ratio_{23}&\ddots&Ratio_{2N}\\ Ratio_{33}&&\vdots\\ &&&Ratio_{NN}\end{pmatrix} \tag{10}\]  
As the number of IMFs and, thus, _ratios_ vary with K-value, the mean _ratio_ (_Relevance_) is calculated to allow direct comparison. _Relevance_ is a single measure of the relationship between the original signal and all of its sub-signals. It is generated for each K-value. This is shown in Equation (11):  
\[Relevance=\sum_{i=2}^{N}\left\{ratio\right\}_{i}/i \tag{11}\]  
Step 4: The _Relevance_ obtained from different K-values can be plotted to form a curve. The second derivative is applied to this curve in the evaluation of the optimal K-value. The optimal value is selected at the point of inflection above a specified threshold. An example is presented in Section 3.