## 2 Expanding Mandel Design
### Collet Chuck Deflection Due to Centrifugal Force  
The variation in the clamping force, \(F_{s3}\), due to centrifugal force \(F_{cm}\) on each of the collet jaws, and on the transmission system, is calculated by Equation (18), where \(F_{se}\) is the preset static clamping force before the collet chuck has been rotated, calculated using Equations (5), (13) and (15); \"+\" indicates external clamping and \"-\" indicates internal clamping. \[F_{s3}=F_{se}\pm F_{cm} \tag{18}\]  
As shown in Figure 12, the centrifugal force \(F_{i}\) of the element \"\(i\)\" is computed as a function of the rotational speed \(n\), the mass \(m_{i}\) and the radius of the mass center \(r_{ci}\), where \(i=cn\), \(pt\) or \(pn\) (which represent transmission system, piston, and collet, respectively), by Equation (19). \[F_{i}=m_{i}r_{i}\left(\frac{\pi n}{30}\right)^{2} \tag{19}\]  
Due to the different positions of the components, the centrifugal forces cause different radial stiffness and thus, a radial deflection at the clamping position. As such, the total radial stiffness of the collet chuck for each centrifugal force, \(k_{r,Tj}\), is given by Equations (20)-(22). \[k_{r,T1}=\frac{F_{pn}}{\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+ \varepsilon_{4}} \tag{20}\]  
\[k_{r,T2}=\frac{F_{cn}}{\varepsilon_{1}+\varepsilon_{2}} \tag{21}\]  
\[k_{r,T3}=\frac{F_{pt}}{\varepsilon_{1}+\varepsilon_{2}} \tag{22}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Deflection** & **Equation** \\ \hline \(\varepsilon_{1}\) & \(\frac{F_{s}}{k_{r,s}}\) \\ \(\gamma_{1}\) & \(\frac{F_{s}\left(\frac{d_{p}}{2}+d_{am}\right)}{k_{r,s}}\) \\ \(\varepsilon_{2}\) & \(\gamma_{1}\left(\frac{d_{p}}{2}+d_{am}\right)\) \\ \(\varepsilon_{3}\) & \(\frac{F_{s}}{k_{r}}\) \\ \(\gamma_{2}\) & \(\fracThe different amounts of deflection at each clamping position caused by centrifugal forces can be determined by the equations summarized in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Deflection** & **Equation Collet** & **Equation Transmission** & **Equation Piston** \\ \hline \(\varepsilon_{1}\) & \(\frac{F_{m}}{k_{em}}\) & \(\frac{F_{m}}{k_{em}}\) & \(\frac{F_{m}}{k_{em}t}\) \\ \(\gamma_{1}\) & \(\frac{F_{m}\left(d_{p}+d_{m}\right)}{k_{f,m}}\) & \(\frac{F_{m}\left(d_{p}+d_{m}\right)}{k_{f,m}}\) & \(\frac{F_{m}}{k_{f,m}t}\) \\ \(\varepsilon_{2}\) & \(\gamma_{1}\left(\frac{d_{p}}{2}+d_{am}\right)\) & \(\gamma_{1}\left(\frac{d_{p}}{2}+d_{am}\right)\) & \(\gamma_{1}\left(\frac{d_{p}}{2}+d_{am}\right)\) \\ \(\varepsilon_{3}\) & \(\frac{F_{m}}{k_{em}}\) & \(-\) & \(-\) \\ \(\gamma_{2}\) & \(\frac{F_{m}d_{am}}{k_{f,m}}\) & \(-\) & \(-\) \\ \(\varepsilon_{4}\) & \(\gamma_{2}d_{am}\) & \(-\) & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Deflections of collet chuck due to centrifugal forces. Figure 12: Centrifugal forces and distances at clamping position. Substituting the equations summarized in Table 3 into Equations (20)-(22) yields Equations (23)-(25). \[k_{r,T1}=\frac{1}{\frac{1}{k_{r,s}}+\frac{d_{p}^{2}+d_{pn}d_{pn}+2d_{pn}d_{an}+2d_ {p}d_{an}}{2k_{f,s}}+\frac{1}{k_{r,p}}+\frac{d_{pn}d_{an}}{k_{f,p}}} \tag{23}\]  
\[k_{r,T2}=\frac{1}{\frac{1}{k_{r,s}}+\frac{d_{p}^{2}+d_{an}d_{p}+2d_{cn}d_{an}+2d_ {pn}d_{an}}{2k_{f,s}}} \tag{24}\]  
\[k_{r,T3}=\frac{1}{\frac{1}{k_{r,gt}}+\frac{d_{p}^{2}+2d_{an}^{2}}{k_{f,gt}}+ \frac{3d_{an}d_{p}}{2k_{f,s}}} \tag{25}\]  
By assuming the loss of clamping forces \(\Delta F_{s}\) and the centrifugal forces \(F_{i}\) are always equal on the collet side and on the workpiece, we have Equation (26). \[\frac{\Delta F_{s}}{k_{w}}=\frac{F_{pn}}{k_{r,T1}}+\frac{F_{\alpha}}{k_{r,T2}} +\frac{F_{pt}}{k_{r,T3}}+\frac{F_{se}}{k_{r,T}} \tag{26}\]  
Figure 13 shows the diagram used to determine the variance in the clamping force depending on the stiffness of the collet chuck or expanding mandrel and the workpiece. In Equation (26) and in Figure 13, \(k_{w}\) is the radial workpiece stiffness.