## 3 Identification of Time-Varying Parameters
### Implementation Procedure  
In the multi-axis milling process of thin-walled parts, as shown in Figure 1, the coordinates of the excitation point, the response point, and the structural change point are the same (\(\alpha=\beta=\gamma\)). The direction of excitation and response are in the normal direction along the workpiece surface, which is perpendicular to the direction of the tool path. Due to the multi-axis milling, the path is divided into n cutting steps; the coordinates and the original frequency response function of different cutting steps are different. The main work here is to study the dynamic characteristics of the original function with the change of the cutting path. The FEM or experimental modal analysis (EMA) can be used to obtain the original frequency response function and the cross-point frequency response function at the n-cutting step midpoint coordinates of the initial un-machined workpiece (\(i=0\)) and final machined workpiece (\(i=n\)). According to the obtained frequency response function, by Equation (11), the change amount of modal mass, modal damping, and modal stiffness can be obtained in each structural modification process after the \(i\)th (\(i=0,\,1,\,2,\,\ldots,\,n\)) structure modification. By Equation (16), the frequency response function between any two points can be obtained. The flowchart for the calculation of the frequency response function at the original position point of each cutting step is shown in Figure 7. In the figure, the black solid arrow represents the material removal, which is the forward method; the red solid arrow represents the addition of material, which is the reverse method. Both methods generate the same results. The specific steps and the verification process of both methods are described as follows:  
Step 1: Divide the cutting process into \(n\) cutting steps (or cutting positions) along the cutting tool path, \(\gamma=f(i)\) is expressed as the position coordinates of the \(i\)th cutting step (\(i=0,\,1,\,2,\,\ldots,\,n\)). The \(\beta=f(t)\) (\(t=1,\,2,\,\ldots,\,n\)) is defined as excitation point coordinates and the \(\alpha=\text{f}(\text{r})\) (\(r=1,\,2,\,\ldots,\,n\)) is defined as the response point coordinate along the tool path. The above formula indicates that the excitation point coordinate corresponds to the \(t\) cutting step, and the response point coordinate corresponds to the \(r\) cutting step;  
Step 2: For \(i=0\), that is the initial system with no structural modification at each cutting position, modal analyses (using EMA or FEM) are conducted to obtain the origin-point and the cross-point FRFs \(\overline{h}_{rt,0}\);  
Step 3: For \(i=n\), that is the final system after the \(i\)th structure modification at each cutting position, modal analyses (using EMA or FEM) are conducted to obtain the origin-point and the cross-point FRFs \(\overline{h}_{rt,n}\);  
Step 4: Eliminate the accelerometer mass loading effect by Equation (15), obtain the corrected FRFs of initial and final system structure \(h_{rt,0}\), \(h_{tr,n}\) if the modal parameters are obtained by the EMA method in steps 2 and 3. This step is not necessary if the modal parameters are obtained by FEM;  
Step 5: Extract the modal mass (\(m_{r,0}\), \(m_{r,n}\)), modal damping(\(c_{r,0}\), \(c_{r,n}\)), and modal stiffness (\(k_{r,0}\), \(k_{r,n}\)) parameters from the FRFs, \(h_{rt,0}\), \(h_{tr,n}\). Using Equation (11) calculate the \(\Delta m_{r}\), \(\Delta c_{r}\), and \(\Delta k_{r}\). It is worth noting that the subscript r here and the subscript i in Equation (11) have the same meaning, that is, \(\Delta m_{i}=\Delta m_{r}\), \(\Delta c_{i}=\Delta c_{r}\), \(\Delta k_{i}=\Delta k_{r}\);  
Step 6: For \(i=1,\,2,\,\ldots,\,n-1\), calculate the origin-point and the cross-point FRFs at the midpoint position of each cutting step using Equation (15);Step 7: For \(i=1,\,2,\,\ldots,\,n\), calculate the original frequency response function \(h_{ii,i\text{-}1}\) from Step 6 at the midpoint of each cutting step, and the modal parameters are obtained.