## 3 Identification of Time-Varying Parameters
### Correction of Frequency Response Function  
The main concept of the SDMM is to take the material removal process as a structural modification operation for machining the workpiece. Discrete material is removed into many (infinite in limit case) infinitesimal mass elements along the cutting path. When an infinitesimal mass element is removed, the workpiece equivalently performs the one-time structural modification. The dynamic characteristics of the initial workpiece un-machined can be identified utilizing experimental or numerical methods. Figure 4: Material removal process for the multi-axis milling of thin-walled parts with a complex curved surface and variable thickness. The process is shown by the sequence (**a–d**). The dynamic characteristics of the workpiece during machining can then be obtained using the SDMM. It is noted that the SDMM-VM is required because the material removal amount generally changes with time for the multi-axis milling of the workpiece with a complex curved surface. Therefore, the problem is transformed into one that uses the frequency response functions (FRFs) of the initial un-machined workpiece and final machined workpiece to estimate the FRFs of the workpiece during the machining process (named as corrected FRFs). The multi-axis milling process is divided into n-cutting steps along a cutting path. Each cutting step corresponding to a mass element removed is regarded as a one-time modification to the workpiece shown in Figure 5. Thereby, the results of the dynamic characteristics identified are more accurate as the number of discrete cutting steps increases. During the multi-axis milling process of the thin-walled parts with a complex curved surface and variable thickness, the shape of material removal is irregular and the amount of material removal is nonlinear, so it is difficult to obtain the modal mass, modal damping, and modal stiffness of the structure. Here, it is assumed that the variations of the modal mass, modal damping, and modal stiffness are proportional to the amount of material removal. That is, the greater amount of material removal there is, the more the variations of modal mass, modal damping, and modal stiffness there are. Therefore, the variations of modal mass, modal damping, and modal stiffness of the structure can be expressed as  
\[\Delta m_{i}=\frac{\delta m_{i}(m_{i,n}-m_{i,0})}{m_{RM}},\ \Delta c_{i}=\frac{ \delta m_{i}(c_{i,n}-c_{i,0})}{m_{RM}},\ \Delta k_{i}=\frac{\delta m_{i}(k_{i,n}-k_{i,0})}{m_{RM}} \tag{11}\]  
where \(m_{RM}\) is the total amount of material removal during the cutting process. The subscripts '\(n\)' and '\(0\)' indicate the final machined workpiece and initial un-machined workpiece, respectively. \(m_{i,n}\), \(m_{i,0}\), \(c_{i,n}\), \(c_{i,0}\), \(k_{i,n}\), \(k_{i,0}\) are the mass, stiffness, and damping of the final machined workpiece and initial un-machined workpiece, respectively. \(\delta m_{i}\) is a mass that is removed corresponding to the \(i\)th structure modification, which can be obtained through the following procedure. The material removed during the cutting process is acquired through Figure 4, and the relationship between the material removed and the n cutting steps is presented in Figure 6. It can be known from Figure 6 that the mass of material removed in \(i\)th cutting step is expressed as  
\[\delta m_{i}=m(\mu)-m(\
u) \tag{12}\]  
where \(\mu\) and \(\
u\) indicate the starting and endpoints of the \(i\)th tool path segment, respectively. It is emphasized that \(\delta m_{i}\) is dependent on the cutting path and is generally a function of the tool position (or time), which is different from the case presented in Ref. [33] (where \(\delta m_{i}\) is constant). Here, \(\delta m_{i}\) is extracted through the CL files generated using the CAD program package. Figure 5: Material removal in multi-axis milling leads to the change of system dynamic characteristics.