## 2 Dynamic Model of Complex Curved Surface Thin-Walled Parts with Variable Thickness  
### Time-Varying Parameters Model of Multi-Axis Milling  
The machining process of thin-walled parts with a complex curved surface and variable thickness during multi-axis milling is complicated in that the parameters and the tool movement are subject to change. Therefore, it tends to be difficult to directly apply the traditional three-axis milling stability prediction theory to the multi-axis milling process. This section establishes a dynamic model with time-varying parameters of thin-walled parts with a complex curved surface and variable thickness during multi-axis milling. In Figure 1, we consider to have a degree of freedom assuming a rigid spindle toolset because of the greater flexibility of the structure. To accurately describe the relative motion of the tool and the workpiece, three coordinate systems, Global Coordinate System (GCS), Feed Cross-feed Normal System (FCN), and Tool Coordinate System (TCS) are employed here as shown in Figure 1.  
In the milling process, the time-varying cutting force \(f_{N}(t)\) excites the workpiece, so that the workpiece has a dynamic displacement \(N(t)\) along the surface normal. Material removal affects the system dynamics, and the workpiece dynamic characteristic changes along the cutting path. Hence, the corresponding control equation of time-varying parameters system is [40]  
\[\ddot{N}(t)+2\zeta(t)\omega_{n}(t)\dot{N}(t)+\omega_{n}^{2}(t)N(t)=f_{N}(t)/m( t). \tag{1}\]  
Figure 1: Dynamic model of multi-axis milling of thin-walled parts with complex curved surface and variable thickness. where  
\(N(t)\)--the displacement of the point along the surface normal on the surface of the workpiece,  
\(\zeta(t)\)--the modal damping ratio of the workpiece and \(\zeta(t)=c/[2(mk)^{1/2}]\),  
\(\omega_{\rm n}(t)\)--the natural frequency of the workpiece and \(\omega_{\rm n}(t)=[k/m]^{1/2}\),  
\(m\), \(c\), and \(k\)- are the position-dependent parameters of the modal mass, the modal damping, and the modal stiffness of the workpiece, respectively. Due to the feed motion of the tool, the parameters \(m\), \(c\), and \(k\) are the functions of time, that is \(m(t)\), \(c(t)\), and \(k(t)\). \(f_{N}(t)\) is the cutting force exerted on the workpiece along the workpiece surface normal. Note that the numerator \(f_{N}(t)\) on the right side of Equation (1) consists of two parts, \(f_{N}(t)=p(t)+f(t)\), where \(f(t)\) is the static cutting force acting on the workpiece. This mainly causes the forced vibration of the workpiece, and \(p(t)\) is the dynamic cutting force acting on the workpiece, which is produced by the regenerative effect in the cutting process. For the multi-axis milling process, modeling the cutting model is the essential focus for further analysis, due to the complex material removal process. This problem will be discussed in the following section. During the semi-finishing and finishing of the blade, the material removal has a great influence on the blade dynamics. For a specified cutting process, the variation of the tool rake angle and roll angle is shown in Figure 2, according to [PERSON] et al. [41], by simulation method. As can be seen, the rake angle and roll angle of the tool change continuously with the cutting position in the cutting process. The roll angle stays positive while gradually decreasing from \(70^{\circ}\) to \(50^{\circ}\). Initially, the rake angle is negative but gradually increases from \(-40^{\circ}\) to \(20^{\circ}\) as the milling operation proceeds. This represents the typical characteristics of the multi-axis milling operation. In the multi-axis milling process, the dynamic cutting thickness of the \(j\)th cutter teeth is shown in Figure 3, where  
\(W\)--the static cutting thickness,  
\(n_{\rm TCS}(t)\)--the vibration displacement generated by the current cutter tooth cutting,  
\(\tau\)--the tooth passing period and \(\tau=2\pi/(N_{t}\Omega)\), and  
\(N_{t}\)--the number of cutter teeth,  
\(\Omega\)--the angular velocity of the cutting tool.