## 3 Surface Roughness Modeling Based on Residual Height
### Surface Roughness under Flank Wear Condition  
When ball end milling is used to process a titanium alloy, flank wear is the main form of failure [33]. Both the tool lifetime and flank wear are dependent on time. As time increases, the wear can indicate the decline of the tool processing performance. Moreover, the contour of the cutting edge shrinks inward as the wear progresses further. The contour contraction, caused by radial wear, can be equivalent to the increase of the offset, or row spacing, of a milling tool. This can be shown from \(O_{1}\) to \(O_{2}\), or from \(P_{initial}\) to \(p\), as in Figure 5. Meanwhile, the variation in the line spacing and corner can be used to replace  
Figure 4: Sketch map of contour height. the shrinkage of the tool wear profile to achieve an equal effect, which is the increase in the surface residual height. In this paper, the method of equalizing the shrinkage of the tool profile caused by the wear to the increase of the line spacing or the corner is proposed as the equivalent line spacing or the equivalent corner method. The equivalent angle and row spacing method can be used to establish the relationship between \(R_{a}\) and time variation. Here, the equivalent rotation method is used, and the contraction of the contour can be equivalent to the variation of the rotation angle \(\Delta\varphi\). \[\Delta\varphi=\arctan(\frac{R_{M}}{R_{o}-I_{M}})-\arctan(\frac{R_{M}-\Delta R }{R_{o}-I_{M}}) \tag{7}\]  
As shown in Figure 6, based on previous studies [11], \(\Delta R\), \(I_{M}\), and \(R_{M}\) are the radius variation, depth, and truncation radius at the maximum wear point of the ball end milling cutter, respectively. In particular, the maximum wear point was determined by parsing the traditional tool life model built based on tool wear as a position function. The study conclusion shows that the point of minimum tool life is the point of maximum wear of the cutting edge, then, the maximum wear point coordinate \(x_{2}\) and the cross-sectional circle radius \(R(x_{2})\) were derived, the correctness of which has been confirmed. Their expressions are given as follows:  
\[\left\{ \begin{array}{l}\Delta R=\frac{VB(t)\cdot\sin\alpha\cdot\cos\varphi}{ \cos(\alpha+\gamma)}\\ I_{M}=\frac{(R-\varphi_{p})\cdot[2+\rho(R-\varphi_{p})]+\rho R^{2}}{2\cdot[1+ \rho(R-\varphi_{p})]}\\ R_{M}=\sqrt{R^{2}-\{\frac{(\varphi_{p}-R)\cdot[2-\varphi_{p}(\varphi_{q}-R)]- \rho R^{2}}{2\cdot[1+\rho(R-\varphi_{q})]}\}^{2}}\end{array} \right. \tag{8}\]  
The surface roughness model can be represented by Equation (9), and all \(\varphi\) values contained in the substitutions should be replaced by \(\omega\) to calculate the surface roughness of the worn ball end milling tool. The angle variable \(\Delta\varphi\) is obtained and the angle \(\varphi\) can be expressed as (10). The \(VB(t)\)  
Figure 5: Diagram of the relationship between wear and residual height. Figure 6: Variable schematic. in \(\Delta R\) comes from wear modeling studies [11], as shown in (11), it contains tool design and cutting parameters, as well as machining time variables \(t\). In particular, as the inverse hyperbolic function is included in (11), the inverse function of (11) does not exist, but this does not affect the analysis and calculation aided by a computer program. In Equation (11), the independent variable is \(VB\), which represents the tool wear value and as a basic substitute for \(R_{a}\) study and \(t\) represents machining time. Equation (11) was established based on the principle of linear superposition, which reflects the relationship between \(VB\) and \(t\) under the influence of three kinds of wear. \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the comprehensive coefficients of abrasive, adhesive, and diffusion wear, respectively, which are affected by the processing method, environment and material properties of tool and workpiece. Research (11) obtained their values by correcting experimental data.