## 3 Identification of Time-Varying Parameters
### Correction of Frequency Response Function
To identify the corrected FRFs of the workpiece during the cutting process, SDMM-EM proposed that the FRF of the workpiece is presented in excitation coordinate, response coordinate, and the modified structural positional co-ordinates, in addition to the side effects caused by the removed material at active coordinates are removed [33]. For the \(i\)th structural modifications along the tool path at the position coordinates \(\gamma\), the frequency response function \(h_{\alpha\beta,i}^{\gamma}\) at response point coordinates of \(\alpha\), excitation point coordinates of \(\beta\) can be express as  
\[h_{\alpha\beta,i}^{\gamma}=\frac{h_{\alpha\beta,i-1}^{\gamma}+\left(-\omega^{2 }\Delta m_{i}+j\omega\Delta c_{i}+\Delta k_{i}\right)\left(h_{\gamma\gamma,i-1}^ {\gamma}h_{\alpha\beta,i-1}^{\gamma}-h_{\alpha\gamma,i-1}^{\gamma}h_{\gamma \beta,i-1}^{\gamma}\right)}{1+\left(-\omega^{2}\Delta m_{i}+j\omega\Delta c_{i} +\Delta k_{i}\right)h_{\gamma\gamma,i-1}^{\gamma}} \tag{13}\]  
where \(h_{\alpha\beta,i-1}^{\gamma}\) is the FRF before the \(i\)th structure modification (or after the \((i-1)\)th modification), and \(h_{\alpha\beta,i}^{\gamma}\) is the FRF after the \(i\)th structure modification (or before the \((i+1)\)th modification). Equation (13) is the general form to calculate the frequency response after the structure modification. Since the cutting process is divided into \(n\) cutting steps along the cutting tool path, \(\gamma\) is the center position coordinates of the cutting steps before and after structure modification. Therefore, the relationship between the position coordinate \(\gamma\) and the cutting step \(i\) can be expressed as  
\[\gamma=f(i) \tag{14}\]  
The specific function expression in Equation (14) is related to the number of cutting steps. Therefore, for each modification, \(n\) FRFs corresponding to different positional coordinates \(\gamma\) can be obtained through Equation (13) with fixed the excitation coordinate and response coordinate. According to the assumption presented in SDMM, the direct FRF (\(\alpha=\beta\)) corresponding to the \(i\)th position before the \(i\)th modification is used in Equation (1) to analyze the cutting stability. Hence, Equation (13) can be expressed as  
\[h_{rr,i}=\frac{h_{rr,i-1}+\left(-\omega^{2}\Delta m_{i}+j\omega\Delta c_{i}+ \Delta k_{i}\right)\left(h_{\gamma\gamma,i-1}h_{rr,i-1}-h_{\gamma\gamma,i-1}h _{\gamma\gamma,i-1}\right)}{1+\left(-\omega^{2}\Delta m_{i}+j\omega\Delta c_{i }+\Delta k_{i}\right)h_{\gamma\gamma,i-1}}. \tag{15}\]  
In Equation (15), \(h_{rr,i}\) (\(r=1\), \(2\), \(\ldots\), \(n\); \(t=1\), \(2\), \(\ldots\), \(n\)) is instead of \(h_{\alpha\beta,i}^{f(i)}\). The structural modifications positional coordinates after the \(i\)th structure modification is \(f(i)\), thus the cutting step corresponding to the excitation point is \(t\) (i.e., \(\beta=f(t)\)), the cutting step corresponding to the response point is \(r\) (i.e., \(\alpha=f(r)\)). Equation (15) indicates how to determine the direct FRF at point \(\alpha\) (or \(r\), \(\alpha=\beta\), and \(r=t\)) using the known FRFs obtained for \((i-1)\)th modification, when the \(i\)th modification (\(\gamma=f(i)\)) is performed. When the excitation point, the response point, and the structural modification point are in  
Figure 6: The change curve of material removes with the cutting path in multi-axis milling. the same positions, that is, \(\alpha=\beta=\gamma\) (or \(r=t=i\)), as shown in Figure 5b,c, then Equation (15) can be expressed as  
Figure 5.0  
By Equation (16), the frequency response function at the structure modification position \(\gamma\) (the \(i\)th cutting step) can be obtained. Correspondingly, the original frequency response function after structure modification can be obtained easily, provided that we know the original frequency response function before structure modification and the change of modal mass, modal damping, and modal stiffness of this structure modification.