# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 2 Vibration Simulation of the Milling Process
### Dynamic Performance of Workpiece and Stability Prediction of Cutting System  
The dynamic performance of the workpiece is often obtained by the analytical method and FEM. FEM is very convenient and common, but its results are limited to the mesh resolution. The analytical method is often calculated with [PERSON]'s theorem of motion and the Lagrange method. Considering the semi-conical shell workpiece is a kind of the thin-wall shell structure, the calculation of the dynamic performance based on the theory of shells can be adopted. The spectro-geometric-Ritz method (SGM) proposed by [PERSON] et al. [31] is an effective method to conduct transverse vibration analysis of plates and shells with arbitrary elastic supports. With the trigonometric series expansion and Rayleigh-Ritz technique, their method could be generally applied to a wide scope of the boundary conditions, which can be used in this paper. With the use of SGM, the semi-conical shell workpiece under consideration is regarded as a kind of rotating shell with longitude, latitude and uniform thickness, as shown in Figure 6. The middle plane of the shell where an orthogonal coordinate O\({}_{\text{m}}\)-X\({}_{\text{m}}\)Y\({}_{\text{m}}\)Z\({}_{\text{m}}\) is situated is taken as the reference plane. The X\({}_{\text{m}}\)-direction represents the displacement in the longitude direction, which ranges from 0 to \(L\), the Y\({}_{\text{m}}\)-direction represents the angle between the normal of a point on the latitude, and the rotation axis, which ranges from 0 to \(\Phi\) and the Z\({}_{\text{m}}\)-direction represents the position in the thickness direction, which ranges from-\(h\)/2 to \(h\)/2. In the calculation process, the vibration displacement of the semi-conical shell in the X\({}_{\text{m}}\), Y\({}_{\text{m}}\) and Z\({}_{\text{m}}\) directions are represented by \(u\), \(v\) and \(w\), respectively and their corresponding admissible displacement functions are proposed in a trigonometric series form as shown in Equation (2). \[\left\{ \begin{array}{l}u(x_{\text{m}},y_{\text{m}})=\sum\limits_{m=-2}^{ \infty}\sum\limits_{n=-2}^{\infty}U_{m,n}\varphi_{m}(x_{\text{m}})\varphi_{n}( y_{\text{m}})\\ v(x_{\text{m}},y_{\text{m}})=\sum\limits_{m=-2}^{\infty}\sum\limits_{n=-2}^{ \infty}V_{m,n}\varphi_{m}(x_{\text{m}})\varphi_{n}(y_{\text{m}})\\ w(x_{\text{m}},y_{\text{m}})=\sum\limits_{m=-4}^{\infty}\sum\limits_{n=-4}^{ \infty}W_{m,n}\varphi_{m}(x_{\text{m}})\varphi_{n}(y_{\text{m}})\end{array} \right. \tag{2}\]  
where \(U_{m,n}\), \(V_{m,n}\) and \(W_{m,n}\) are the expansion coefficients for the semi-conical shell, and they can be solved by the Rayleigh-Ritz procedure. The coefficients \(m\) and \(n\) are limited by the truncated coefficients \(M\) and \(N\), which are set as 15 to achieve a good balance between calculation efficiency and accuracy in the calculation process. \(x_{\text{m}}\) and \(y_{\text{m}}\) represent the position coordinates in the X\({}_{\text{m}}\) and Y\({}_{\text{m}}\) directions of the O\({}_{\text{m}}\)-X\({}_{\text{m}}\)Y\({}_{\text{m}}\)Z\({}_{\text{m}}\) coordinate system. \(\varphi_{\text{m}}(x_{\text{m}})\) and \(\varphi_{\text{n}}(y_{\text{m}})\) are trigonometric functions which can be given as:  
\[\left\{ \begin{array}{l}\varphi_{m}(x_{\text{m}})=\sin\lambda_{am}x_{\text{m }},\,m<0\\ \varphi_{m}(x_{\text{m}})=\cos\lambda_{am}x_{\text{m}},\,m\geq 0\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}\varphi_{n}(y_{\text{m}})=\sin\lambda_{bn}y_{\text{m }},\,n<0\\ \varphi_{n}(y_{\text{m}})=\cos\lambda_{bn}y_{\text{m}},\,n\geq 0\end{array} \right. \tag{4}\]  
where \(\lambda_{am}=m\pi/L\) and \(\lambda_{bn}=n\pi/\Phi\). The boundary conditions of the semi-conical workpiece are set based on the practical condition shown in Figure 2a. The workpiece is fixed on three separate fixtures with HY-194 epoxy adhesive, and all three fixtures are mounted on the worktable of the machine-center with T-shaped bolts, which will affect the stiffness distribution of the workpiece.