# Dynamics of a Flexible Roof Test Model under Ambient Vibrations Measurements
## 3 Singular Value Decomposition Analyses  
This study was focused on a square plan roof with a hyperbolic, parabolic shape roof. Referring to Figure 2a, at the model scale, \(f_{1}\) was equal to 13.3 mm, \(f_{2}\) was equal to 26.6 mm, \(L_{1}\) and \(L_{2}\) were equal to 400 mm, \(H_{1}\) was equal to 66.7 mm and \(H_{2}\) was equal to 106.6 mm.  
In linear algebra, the Singular Value Decomposition (SVD) of a matrix is a factorization of that matrix into three matrices. This value has some interesting algebraic properties and conveys important geometrical and theoretical insights about linear transformations. It also has some important applications in data science [24; 25; 26; 27; 28; 29; 30; 31]. Singular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is an \(n\times p\) matrix) in which the \(n\) rows represent the genes, and the \(p\) columns represent the experimental conditions. The SVD theorem states:  
\[A_{nxp}=U_{nxn}\times S_{nxp}\times V_{pxp}^{T} \tag{3}\]  
where the \(U_{nxn}\) columns are the left singular vectors (gene coefficient vectors); \(S_{nxp}\) (the same dimensions as A) has singular values and is diagonal (mode amplitudes); and \(V_{pxp}^{T}\) has rows that are the right singular vectors (expression level vectors). The SVD represents an expansion of the original data in a coordinate system where the covariance matrix is diagonal. Calculating the SVD consists of finding the eigenvalues and eigenvectors of \(A_{nxp}\times A_{nxp}^{T}\) and \(A_{nxp}^{T}\times A_{nxp}\). The eigenvectors of \(A_{nxp}^{T}\times A_{nxp}\) make up the columns of \(V_{pxp}\), the eigenvectors of \(A_{nxp}\times A_{nxp}^{T}\) make up the columns of \(U_{nxp}\). The singular values in \(S_{nxp}\) are also square roots of eigenvalues from \(A_{nxp}\times A_{nxp}^{T}\) or \(A_{nxp}^{T}\times A_{nxp}\). The singular values are the diagonal entries of the \(S_{nxp}\) matrix and are arranged in descending order. The singular values are always real numbers. If matrix \(A_{nxp}\) is a real matrix, then \(U_{nxn}\) and \(V_{pxp}\) are also real.  
The singular value decomposition was applied to all 49 signals acquired by ambient vibration tests, and the trend of the singular values is shown in Figure 4. Singular value matrix \(S\) was calculated, varying the number of modes, and it was observed that the trend was very similar for the first five modes. It means that the dynamics of the first five modes are robustly defined by tests using only a few sensors. On the contrary, in order to capture six or more modes, several sensors are necessary, and they should be located close to the roof borders.  
Figures 5 and 6 show a comparison between six modes estimated by SVD (i.e., eigenvectors plot) and structural modes (i.e., test model scale) estimated by FFD.  
Figure 4: Singular values, \(S\), variation depending on the number of modes: overview (**a**), zoom view (**b**).  
Figure 5: Eigenvectors by SVD: #1 (a), #2 (b), #3 (c), #4 (d), #5 (e) and #6 (f).  
Figure 6: Structural modes by FFD: #1 (a), #2 (b), #3 (c), #4 (d), #5 (e) and #6 (f).  
It was observed that, even if some peaks are similar (for example, for mode #2, Figures 5b and 6b), the global representation of modes is different. Overall, the difference between plots by FFD and SVD is significant, highlighting that the two methods should be carefully integrated. However, several considerations should be taken into account. All plots show that, in general, peaks are located around the center of the roof. Modes are mostly asymmetric, and valleys and bumps are antipolar over the roof center. This trend confirms the deformed modal shapes given by [8] and is estimated by modal analyses performed through the FEM model of cable net roofs.