## 2 Characterization of Surface Structure and Properties of Titanium Alloy Turning Operations
### Turning Tool Structure and Instantaneous Cutting Position
\tag{4}\]  
In order to reveal the relative motion relationship between the workpiece and the tool and to control the formation process of the machined surface, the tool, the workpiece, and the relative motions of the tool coordinate system, the cutting coordinate system, the workpiece coordinate system, and the reference coordinate system are characterized as shown in Figure 2. The dynamic cutting process variables of the turning tool are explained as shown in Table 1.  
In Figure 2, \(\alpha\)-\(xyz\) is the reference coordinate system, which is a fixed coordinate system, \(\alpha_{b}\)-\(x_{b}y_{b}\)-\(z_{b}\) is _the_ workpiece coordinate system, the origin \(\alpha_{b}\) of this coordinate system is the center point of the workpiece end face, \(x_{b}\)-axis is the direction of depth of cut, \(z_{b}\)-axis is the direction of feed rate, and \(y_{b}\)-axis is established according to the Cartesian coordinate system; \(\alpha_{c}\)-\(x_{c}\)-\(y_{c}\)-\(z_{c}\) is cutting coordinate system, where the coordinate origin \(\alpha_{c}\) is the intersection of the plane where the lowest point of the cutting tool tooth is located and the projection of the maximum radial extension point of the cutting tool tooth; the line \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Poisson** & **Hardness** & **Yield** & **Tensile** & **Compressive** & **Elastic** \\  & **(g/cm\({}^{3}\))** & **Ratio** & **(N/mm\({}^{2}\))** & **(MPa)** & **Strength** & **Strength** & **Modulus** \\ \hline TC6 & 4.51 & 0.34 & 340 & 1231 & 1300 & 1687 & 109.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: TC6 material properties. Figure 2: Turning instantaneous cutting position. between the coordinate origin and the lowest point of the cutting tool tooth is used as the direction of the \(z_{c}\)-axis, and the line between the coordinate origin and the maximum radial extension point of the cutting tool tooth is used as the direction of the \(x_{c}\)-axis. The \(y_{c}\)-axis was determined according to the principle of the Cartesian coordinate system; the origin \(o_{c}\) of the cutting coordinate system coincides with the origin \(o_{a}\) of the tool coordinate system, the \(x_{c}\)-axis coincides with the \(x_{a}\)-axis, the \(y_{a}\)-axis is in the opposite direction to the \(y_{c}\)-axis, and the \(z_{a}\)-axis is in the opposite direction to the \(z_{c}\)-axis; \(L\) is the length of the workpiece, \(R_{0}\) is the radius of the big end of the workpiece, \(R_{1}\) is the radius of the small end of the workpiece, \(R_{k}\) is the distance from the axis of the highest point of the machined surface, \(R_{p}\) is the distance of point \(p\) from the axis, \(R_{m}\) is the distance from the axis of the lowest point of the machined surface, \(v_{f}\) is the cylindrical turning tool feed rate, \(v_{f}\)' is the face turning tool feed rate, \(a_{p1}\) is the depth of cylindrical turning, \(a_{p2}\) is the end face turning depth, \(n\) is the Spindle speed, \(\theta(t)\) is the angle of rotation of the workpiece coordinate system relative to the initial position during cutting, \(z(t)\) is the feed distance of the turning tool in the direction of the feed rate during cutting, \(D(x\),\(y\),\(z)\) is the surface to be machined, and \(G(x\),\(y\),\(z)\) is the machining transition surfaces, \(K(x\),\(y\),\(z)\) is the machined surfaces. According to Figure 2, the trajectory equation of the cutting coordinate system origin \(o_{c}\) in the machine coordinate system can be expressed as follows:  
\[L_{c}(x_{c}(t),y_{c}(t),z_{c}(t))=\Phi_{0}\cdot \begin{bmatrix}0&0&0&1\end{bmatrix}  ^{T} \tag{5}\]  
where the transformation matrix \(\mathbf{\Phi_{0}}\) is \(\mathbf{T_{2}}\mathbf{T_{1}}\mathbf{M_{1}}\).