# Feed Rate Variation Strategy for Semi-Conical Shell Workpiece in Ball Head End Milling Process
## 2 Vibration Simulation of the Milling Process
### Vibration Response Analysis  
The machining vibration of the workpiece is often divided into free vibration, forced vibration and self-excited vibration. Free vibration can be ignored because it is easily dissipated with the oscillation, friction and damping of the workpiece. The self-excited vibration, also known as chatter, is the disgusting result of the dynamic instability of the whole machining system. It is necessary to conduct a stability estimation after the selection of the machining parameters, as mentioned above. The variation of the spindle speed can induce chatter, but the feed rate variation can hardly do that. Because the variation of the spindle speed is the variation of excitation frequency, the chatter will occur if the frequency is matched with a natural frequency of the workpiece, cutter or even machine tool [47]. However, the feed rate variation affects the value of the cutting force, which hardly conducts dynamic instability for a stable process. The forced vibration is mainly influenced by the amplitude and frequency of the cutting force, which will last for the whole milling process. Consequently, the vibration prediction is regarded as the calculation of forced vibration response using modal analysis. The \(i\) th modal milling vibration of the semi-conical shell workpiece in the thickness direction can be modeled as a continuous system as indicated partially in Figure 11. It should be pointed out that the \(\mathbf{F}^{\mathrm{m}}(\omega)\) is the modal force which reflects the excitation component of each mode provided by the external force. The equation of motion of the milling system can be expressed as:  
\[[\mathbf{m}]\big{|}\big{\{}\mathbf{z}_{\mathrm{m}}\big{\}}+[\mathbf{c}] \big{\{}\dot{\mathbf{z}}_{\mathrm{m}}\big{\}}+[\mathbf{k}]\big{\{}\mathbf{z}_ {\mathrm{m}}\big{\}}=\{\mathbf{F}\} \tag{16}\]  
where \(\mathbf{k}\) and \(\mathbf{m}\) denote the stiffness and mass matrix as mentioned above. Variables \(\mathbf{c}\) and \(\mathbf{F}\) denote the matrix of damping and the column vector of exciting force, respectively. It should be pointed out that the \(\mathbf{z}_{\mathrm{m}}\) in Equation (16) is the vibration response vector in the thickness direction of the \(\mathrm{O}_{\mathrm{m}}\)-\(\mathrm{X}_{\mathrm{m}}\mathrm{Y}_{\mathrm{m}}\mathrm{Z}_{\mathrm{m}}\) coordinate. By assuming the cutting force and vibration response are harmonic motions, they can be expressed by \(\{\mathbf{F}\}=\{F(\omega)\}\mathrm{e}^{\mathrm{j}\omega t}\) and \(\{\mathbf{z}_{\mathrm{m}}\}=\{\mathbf{z}_{\mathrm{m}}(\omega)\}\mathrm{e}^{ \mathrm{j}\omega t}\). Equation (16) can then be rewritten as:  
\[\Big{(}-\omega^{2}[\mathbf{m}]+\mathrm{j}\omega[\mathbf{c}]+[\mathbf{k}] \Big{)}\{z_{\mathrm{m}}(\omega)\}=\{\mathbf{F}(\omega)\} \tag{17}\]  
Based on the theory of the modal analysis, the expansion theorem is used, and the displacements of the masses are expressed as a linear combination of the normal modes of the system, as shown in Equation (18). This linear transformation uncouples the equations of motion so that we obtain a set of \(n\) uncoupled differential equations of second-order. \[ \begin{array}{lcl}z_{\mathrm{m}}(\omega)&=&\delta_{1}(\omega)\{\mathbf{E}_ {1}\}+\delta_{2}(\omega)\{\mathbf{E}_{2}\}+\ldots+\delta_{mnum}(\omega)\{ \mathbf{E}_{mnum}\}\\ &=&\sum\limits_{i=1}^{mnum}\frac{[\mathbf{E}_{i}]^{T}[\mathbf{F}(\omega)][ \mathbf{E}_{i}]}{-\omega^{2}M_{i}+\mathrm{j}\omega C_{i}+K_{i}}\end{array}  \tag{18}\]  
Figure 11: Schematic diagram of vibration response in the milling process in the thickness direction. where \(\delta_{1}(\omega)\), \(\delta_{2}(\omega)\dots\)\(\delta_{mum}(\omega)\) are frequency-dependent generalized coordinates.