## 3 Surface Roughness Modeling Based on Residual Height  
### Surface Roughness without Wear  
Assume that a thin wall convex surface part was machined with a ball end milling tool by one-way row cutting, as shown in Figure 3. The instantaneous curvature along the feed direction is defined as \(\rho\)' and the curvature orthogonal thereto as \(\rho\), as shown in Figure 2(a),b. As this assumption is based on machining in a five-axis CNC machining center, \(\rho\)' was not introduced in the derivation process and experimental sample design. The milling part is shown in section A-A of Figure 2(a),b, \(O_{1}\) denotes the center of the ball end milling tool in processing, and \(O_{2}\) denotes the center of the ball end milling tool on the adjacent tool path that has just been milled. The residual height is expressed as Equation (1) due to the change in curvature, and is related to the curvature \(\rho\), where \(R\) is the radius of the ball end milling and \(p\) is the row spacing. Additionally, \(h(\theta)\) is the contour height function, as shown in Figure 4. The \(h(\theta)\) can be deduced in an analytical form. \[h(\theta)=R_{\theta}\cdot[\cos(\varphi-\theta)-1]+R-\sqrt{R^{2}-\sin^{2}( \varphi-\theta)\cdot R_{\theta}}^{2}\Bigg{\{} \begin{array}{l}\varphi=arc\sin \frac{\rho\cdot P_{initial}}{2\cdot(1-\rho\cdot q_{p})}\\ R_{\theta}=\frac{1}{\rho}[1+\rho\cdot(R-q_{p})]\end{array} , \tag{1}\]  
Figure 3: Surface milling residual height diagram. Here, \(\theta\)is the center angle of the sample length for the integral mean value. To distinguish the subsequent line spacing changes, the initial value of the line spacing is expressed as \(P_{initial}\) and the surface roughness is the average arithmetical deviation of the contour [32]. This can be represented as (2) when applied to surface milling. To calculate the roughness by the average arithmetical deviation of the contour, the midline of the outline can be obtained according to \(S_{1}=S_{2}\); at this point, \(\theta_{1}\) can be defined as (3). Here, the function \(E(\theta)\) can be defined as (4). The definition of \(R_{a}\) is given by Equation (5). \[R_{a}=\frac{1}{\varphi}\cdot\int_{0}^{\varphi}\left|h(\theta)-h(\theta_{1}) \right|d\theta \tag{2}\]  
\[\theta_{1}=\varphi-\arctan(\frac{\sqrt{4\cdot E(0)^{2}\cdot R^{2}\cdot\varphi^ {2}-\left[\varphi^{2}\cdot({R_{o}}^{2}-R^{2})+E(0)^{2}\right]^{2}}}{\varphi^{2 }\cdot({R_{o}}^{2}-R^{2})+E(0)^{2}}) \tag{3}\]  
\[E(\theta)=R\cdot EllipticE(\sin(\varphi-\theta),\frac{R_{o}}{R})-R_{o}\cdot \sin\varphi \tag{4}\]  
\[R_{a}=\frac{2}{\varphi}\cdot\left[H(\theta_{1})-H(0)-\theta_{1}\cdot h(\theta _{1})\right] \tag{5}\]  
Here, \(H(\theta)\) is the integral of \(h\) and can be expressed by Equation (6). In addition, all of the integral constants were omitted in the modeling process and will be reflected in the final verification result in the form of amplitude multiple \(K\) and translation constant \(C\). Surface roughness is a direct demand indicator that can directly express machining quality. Due to the geometric envelope relationship between the ball end milling tool and the curved surface, the machining surface created by the ball end tool without any wear still remains a residual height of material, which leads to \(R_{a}\) a non-zero initial value. Since the tool is not worn at this situation, the calculation of the theoretical initial value of \(R_{a}\) only needs to substitute the required geometric and kinematic conditions, the milling line spacing \(p=0.15\) mm and the sampling length of \(R_{a}\), which is replaced by the center angle corresponding to the sampling length \(\theta=10^{\circ}\). There is no need to consider the effects of material properties and processing systems. The initial roughness obtained by Substituting the above geometric parameters for Equation (5), and Equation (5) has its initial value of \(R_{a}=0.67507\) um. This value of \(R_{a}\) means the initial value cause only by geometric factor of ball end tool and curve surface without tool wear when \(t=0\).