## 4 Matching Calculation Based on the Sensitivity Method  
The formula of the matching calculation is derived based on the sensitivity method in this section. Referring to Equation (1), the four parameters, \(G_{0}\), \(G_{\infty}\), \(\tau\), and \(\alpha\), to be identified in the fractional derivative model are taken as design variables and the differential method is used to obtain the sensitivity. Take \(G_{0}\) as an example, where the sensitivity \(S_{G_{0},i}\) of the frequency response to design variable \(G_{0}\) at the \(i\)-th frequency point can be described as:  
\[S_{G_{0},i}=\frac{\triangle x_{G_{0},i}}{\triangle G_{0}} \tag{20}\]  
where \(\triangle G_{0}\) refers to the variation of the design variable (also named as step), which is set to \(10^{-4}\times G_{0}\) here. Symbol \(\triangle x_{G_{0},i}\) is the variation of the vibration response when the change of the design variable is \(\triangle G_{0}\) and it is also named as residual. Similarly, the sensitivity \(S_{G_{\infty},i}\), \(S_{\tau,i}\), and \(S_{\alpha,i}\) of the vibration response to \(G_{\infty}\), \(\tau\), and \(\alpha\) can also be calculated, respectively. If the number of frequency points is \(P\) in the whole process of identification, then the calculated sensitivity of the design variables forms the sensitivity matrix \(\boldsymbol{S}\) and can be described as:  
\[\boldsymbol{S}=\left[ \begin{array}{cccc}S_{G_{0},1}&S_{G_{\infty},1}&S_{ \tau,1}&S_{\alpha,1}\\ S_{G_{0},2}&S_{G_{\infty},2}&S_{\tau,2}&S_{\alpha,2}\\ \vdots&\vdots&\vdots&\vdots\\ S_{G_{0},P}&S_{G_{\infty},P}&S_{\tau,P}&S_{\alpha,P}\end{array} \right] \tag{21}\]  
The objective function can be defined according to the principle of linear least square and is expressed as:  
\[\text{Min}\|\boldsymbol{W}_{\text{x}}(\boldsymbol{S}\triangle\boldsymbol{b}- \triangle\boldsymbol{x})\| \tag{22}\]  
where \(\|\cdot\|\) denotes the solution of 2-norm, \(\boldsymbol{W}_{\text{x}}\) is the weight matrix of the vibration response, \(\triangle\boldsymbol{b}\) is the vector consisting of the variation of design variables, and \(\triangle\boldsymbol{x}\) is the vector consisting of the residuals of the vibration response at each frequency point. The above vectors can be described in detail as:  
\[\triangle\boldsymbol{b}=\left[ \begin{array}{cccc}\triangle G_{0}&\triangle G _{\infty}&\triangle\tau&\triangle\alpha\end{array} \right]^{\text{T}} \tag{23a}\] \[\triangle\boldsymbol{x}=\left[\Delta x_{1},\Delta x_{2},\cdots,\Delta x_{\text {p}}\right]^{\text{T}} \tag{23b}\]  
It should be noted that the residuals in Equation (23b) correspond to the variation of response when all the design variables change and Equation (22) denotes that the 2-norm of the difference between the response values obtained from the experiment and theoretical calculations at the considered frequency points is minimal. Furthermore, if the design variable with a minimum change is also requested, then Equation (22) becomes:  
\[\text{Min}\bigg{(}\left\|\boldsymbol{W}_{\text{x}}(\boldsymbol{S}\triangle \boldsymbol{b}-\triangle\boldsymbol{x})\right\|^{2}+\left|\boldsymbol{W}_{ \text{b}}\triangle\boldsymbol{b}\right|^{2}\bigg{)} \tag{24}\]  
where \(\boldsymbol{W}_{\text{b}}\) is the weight matrix of the design variables. In Equation (24), if the derivative about \(\triangle\boldsymbol{b}\) is solved and the obtained derivative value is set to be zero, then the following iteration formula for the design variable can be yielded:\[\mathbf{b}_{j}=\mathbf{b}_{j-1}-\left(\mathbf{S}^{\mathrm{T}}\mathbf{W}_{\mathrm{x}}^{\mathrm{T}} \mathbf{W}_{\mathrm{x}}\mathbf{S}+\mathbf{W}_{\mathrm{b}}^{\mathrm{T}}\mathbf{W}_{\mathrm{b}} \right)^{-1}\mathbf{S}^{\mathrm{T}}\mathbf{W}_{\mathrm{x}}^{\mathrm{T}}\bigtriangleup\mathbf{x} \tag{25}\]  
After several iterative calculations, until the convergence conditions are satisfied, the four parameters to be identified in the fractional derivative model can be obtained.