## 2 Methodology
### Response Surface-Based FE Model Updating Methodology  
The overall procedure of response surface-based FE model updating is illustrated in Figure 3.  
For flexible PV support structures, their large spans and low stiffness make them susceptible to wind-induced vibrations [49]. To ensure that the updated model more accurately represents the actual dynamic characteristics of the structure, the first four modal frequencies are selected as the response features for model updating. Figure 3: Flowchart of response surface-based FE model updating. Figure 2: Illustration of the mode shape identification from modal responses. The selection of updating parameters for flexible PV support structure models differs significantly from that for buildings and bridges. The main reasons are as follows: the components of the support are mostly standardized, with minimal differences in size and material properties between them; as a steel structure, the mechanical properties of the support are more stable compared to concrete, but it is also more susceptible to corrosion; the flexible support is a prestressed cable-supported structure, and under wind loads, slippage at the cable anchor points can lead to a decrease in cable tension; PV modules consist of aluminum frames, laminates, and junction boxes, and previous modeling of PV modules often assumed them to be uniformly distributed panels, which overlooks the mass non-uniformity of the aluminum frames and laminate layers. Based on these considerations, the preliminary selection of updating parameters includes the mass of the metal frame M, cable initial tension T, and support column modeling L.  
When a complex or unknown relationship exists between the modal frequency y of a flexible PV support structure and its physical parameters, this relationship can be represented by an explicit hypersurface function, that is, a response surface model [39]:  
\[y=\eta(\mathbf{X})+\epsilon, \tag{6}\]  
where \(\mathbf{X}=[x_{1},x_{2},\ldots,x_{k}]\in\mathbb{C}^{k}\), \(\epsilon\) denotes the statistical error, and \(\eta\) is an unknown implicit function. Since \(\eta\) is unknown, researchers aim to estimate \(y\) using an approximate but explicit function, which can be achieved by the response surface model RSM(\(\bullet\)). The estimated response can be expressed as follows [39]:  
\[ \begin{array}{l}\hat{y}=\text{RSM}(\mathbf{X})\approx\eta(\mathbf{X})\\ \eta(\mathbf{X})=\text{RSM}(\mathbf{X})+\delta(\mathbf{X})^{\prime}\end{array}  \tag{7}\]  
where \(\delta(\mathbf{X})\) represents the error of the response surface, which should be minimized as much as possible. Subsequently, the response surface can be used to substitute for the physical model in numerical analysis. The selection of the response surface function type first requires that its mathematical expression can adequately describe the true relationship between the input parameters and the output response. Second, the expression should contain the minimum number of terms to reduce the required number of sampling points and enhance interpretability. For most engineering problems, a standard quadratic polynomial function can be used to represent the physical system [39]:  
\[y=\beta_{0}+\sum_{i=1}^{n}\beta_{i}x_{i}+\sum_{i=1}^{n}\beta_{ii}x_{i}^{2}+\sum _{i<j}\beta_{ij}x_{i}x_{j}+\epsilon, \tag{8}\]  
where \(y\) denotes the response feature, specifically the first four modal frequencies of the flexible PV support structure, with each frequency corresponding to a separate response surface model. \(x_{i}\in\left[x_{i}^{l},x_{i}^{u}\right]\) represents the range of values for the updating parameters, and \(\beta_{0}\), \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ij}\) are the regression coefficients. Equation (8) can be represented in matrix form as  
\[\mathbf{Y}=\mathbf{B}\mathbf{X}+\mathbf{e}, \tag{9}\]  
where \(\mathbf{Y}\) is the response vector, \(\mathbf{B}\) is the matrix of regression coefficients, and \(\mathbf{e}\) contains all errors.