# 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
In the calculation process, the spring stiffness of the clamped boundary is set as \(1\times 10^{8}\) N m\({}^{-1}\), and the free boundary is set as 0 N m\({}^{-1}\). It should be pointed out that two straight boundaries located at \(y_{\rm m}=0\) and \(y_{\rm m}=\Phi\) are clamped partially instead of totally, and two semicircle boundaries located at \(x_{\rm m}=0\) and \(x_{\rm m}=L\) are free. The potential energy of the boundary constraints can then be calculated based on the boundary conditions and the theory of shells. The total energy function of the semi-conical shell is taken as the sum of energy contributions from the shell, which is composed of the strain energy \(U\), kinetic energy \(T\) and potential energy of the boundary constraints \(U_{\rm b}\). Thus, the Lagrangian energy function of the semi-conical shell can be expressed as:  
\[Lagn=U+U_{b}-T \tag{5}\]  
Substituting the admissible displacement functions from Equation (2) into Equation (5), with the use of Rayleigh-Ritz method, the Lagrangian expression can be minimized by taking its derivative with respect to these expansion coefficients  
\[\frac{\partial Lagran}{\partial\vartheta}=0,\vartheta=W_{m,n},U_{m,n},V_{m,n} \tag{6}\]  
Considering the truncated coefficients are represented as \(M\) and \(N\), a total of \((M+5)\times(N+5)+2\times(M+3)\times(N+3)\) equations are obtained. By substituting Equation (5) into Equation (6), it can be summed up in a matrix form as follows:  
\[\Big{(}\mathbf{k}-\omega^{2}\mathbf{m}\Big{)}\mathbf{E}=0 \tag{7}\]  
where \(\mathbf{k}\) and \(\mathbf{m}\) denote the stiffness matrix and mass matrix, respectively. Based on the orthogonality of the modal vectors, the modal stiffness and mass of the \(i\) th modal shape can be obtained by:  
\[\{\mathbf{E}_{i}\}^{T}[\mathbf{m}]\Big{\{}\mathbf{E}_{j}\Big{\}}=\left\{ \begin{array}{c}0,i\
eq j\\ M_{i},i=j\end{array} \right. \tag{8}\]  
\[\{\mathbf{E}_{i}\}^{T}[\mathbf{k}]\Big{\{}\mathbf{E}_{j}\Big{\}}=\left\{ \begin{array}{c}0,i\
eq j\\ K_{i},i=j\end{array} \right. \tag{9}\]  
where \(K_{i}\) and \(M_{i}\) denote the modal stiffness and mass of the \(i\) th modal shape, respectively. \(\mathbf{E}_{i}\) denotes the \(i\) th column of the eigenmatrix E. By solving a standard matrix eigenvalue problem, the natural frequencies can be obtained, as shown in Table 3. The natural frequencies obtained with the use of FEM and experimental modal analysis (EMA) are also shown in Table 3 for better comparison. It can be  
Figure 6: Schematic diagram of the semi-conical shell workpiece and coordinate system O\({}_{\rm m}\)-X\({}_{\rm m}\)Y\({}_{\rm m}\)Z\({}_{\rm m}\). observed that the results obtained by SGM are superior to FEM, and they are very close to the results of the EMA. Considering the vibration energy often distributes in the range of low frequency, and the frequency of the cutting force is 120 Hz, which is known as the tooth pass frequency, only the first three modes are considered in the following calculation. For further calculation, the \(i\) th modal shape function in the thickness \(Z_{\text{m}}\)-direction of the semi-conical workpiece is normalized as:  
\[w_{\text{nor},i}(x_{\text{m}},y_{\text{m}})=\frac{1}{\sqrt{M_{i}}}\sum_{m=-4}^ {\infty}\sum_{n=-4}^{\infty}W_{m,n,i}\varphi_{m}(x_{\text{m}})\varphi_{n}(y_{ \text{m}}) \tag{10}\]  
where \(w_{\text{nor},i}\) denotes the \(i\) th normalized modal shape displacement function in the thickness \(Z_{\text{m}}\)-direction of the workpiece. \(W_{m,n,i}\) is the expansion coefficient of the \(i\) th modal shape. Ultimately, the first three modal shapes in the \(Z_{\text{m}}\)-direction can be obtained, as shown in Figure 7.  
From Figure 7, it can be found that with the increase of the modal sequence, the modal shapes are getting more complex. Since the radius on the right side of the workpiece is larger than the left side, the vibration on the right side is more intense and complicated. In the same figure, three chosen contour paths from different heights are plotted for easy understanding and comparison.