## 4 Conclusions
### Nomenclature  
\begin{tabular}{l l} \(a\), \(b\), \(c\) & section length of tool spindle \\ \(\dot{a}\) & axial depth of cut \\ \(a_{\rm n}(t)\), \(b_{\rm m}(t)\) & generalized coordinates \\ \(A(z)\) & cross-sectional area of the shaft \\ \(c\) & tool length \\ \(C_{xx}\), \(C_{xy}\), \(C_{yx}\), \(C_{yy}\) & damping matrix of the bearing using GPEM \\ \(C_{xyj}^{\rm b}\), \(C_{xyj}^{\rm c}\), \(C_{yyj}^{\rm b}\), \(C_{yyj}^{\rm c}\) & damping coefficients of the j-th bearing \\ \(d_{\rm O}\), \(d_{\rm I}\) & outer and inner diameter of the shaft and tool \\ \(\mathrm{d}A\) & area of cutting segment \\ \(\mathrm{d}b\) & differential cutting edge length \\ \(\mathrm{d}F_{\rm u}\,\mathrm{d}F_{\rm r}\), \(\mathrm{d}F_{\rm a}\) & differential cutting forces in tangential, radial, and axial direction \\ \(\mathrm{d}F_{\rm x}\), \(\mathrm{d}F_{\rm y}\), \(\mathrm{d}F_{\rm z}\) & differential milling force in Cartesian coordinate \\ \(\mathrm{d}s\) & differential length of the curved cutting edge segment \\ \(\mathrm{d}z\) & differential height in axial direction \\ \(E(z)\) & elastic modulus of the shaft and tool \\ \(f_{\rm d}\) & feed per tooth (mm) \\ \(f_{\rm n}\) & natural frequency (rad/s) \\ \(f_{1}(\varphi)\), \(f_{2}(\psi_{\rm i})\), \(f_{3}(\psi_{\rm i})\) & coefficients of cutting force \\ \(G\) & gyroscopic matrix of the rotor using GPEM \\ \(h\) & chip thickness \\ \(l_{0}\) & helix angle at meeting boundary between the ball and shank \\ \(I_{\rm D}\), \(I_{\rm D}\) & diametral and polar mass moment of inertia of the shaft \\ \(I_{\rm D\
u}^{\rm d}\), \(I_{\rm F}^{\rm d}\) & diametral and polar mass moment of inertia of the i-th disk \\ \(K_{\rm e}\) & stiffness matrix of the shaft using GPEM \\ \(K_{\rm tc}\), \(K_{\rm cc}\), \(K_{\rm ac}\) & tangential, radial, and axial cutting force coefficients \\ \(K_{\rm tc}\), \(K_{\rm cc}\), \(K_{\rm ac}\) & tangential, radial, and axial edge force coefficients \\ \(K_{\rm xx}\), \(K_{\rm xy}\), \(K_{\rm xr}\), \(K_{\rm xy}\) & stiffness matrix of the bearing using GPEM \\ \(k_{\rm coj}^{\rm b}\), \(K_{\rm xyj}^{\rm b}\), \(K_{\rm yyj}^{\rm b}\), \(K_{\rm yyj}^{\rm c}\) & elastic constant of the j-th bearing \\ \(k\) & angle in a vertical plane between a point on the flute and the \(z\)-axis \\ \(l\) & total length of the tool spindle \\ \(M\) & mass matrix of the rotor using GPEM \\ \(m_{\rm d}^{\rm d}\) & mass of the i-th disk \\ \(m_{\rm k}\) & vibration cycle number of tool during the cutting time from one flute \\ \(N_{\rm d}\) & total number of the disk \\ \(N_{\rm f}\) & total number of the flutes \\ \(N_{\rm b}\) & total number of the bearing \\ \(N_{\rm p}\) & total number of polynomial \\ \(N_{\rm f}\) & number of flute \\ \(N_{\rm cr}\) & critical speed in rpm \\ \(R_{0}\) & ball radius of tool \\ \(R(\varphi)\) & tool radius in \(x\)-\(y\) plane at a point defined by \(\varphi\) \\ \(\dot{t}(\psi_{\rm i},\dot{k})\) & uncut chip thickness normal to cutting edge in milling \\ \(x(\dot{t}_{\rm i}),y(\dot{t}_{\rm i})\) & deflections of tool-tip at the present time \\ \(\alpha\) & rake angle of cutting parameter \\ \(\dot{\beta}\) & friction angle of cutting parameter \\ \(\gamma\) & chip compression ratio of cutting parameter \\ \(\rho(z)\) & density of the shaft and tool \\ \(\theta\) & tool rotation angle, measured from \(y\)-axis (CW) \\ \(\theta_{a}\), \(\theta_{b}\) & the start and exit angles in the milling process \\ \(\varphi\) & lag angle between the tip (z = 0) and a point on the helical flute at height z \\ \(\psi_{\rm i}\) & instantaneous angular immersion in the global coordinate system \\ \(\tau\) & shear stress of cutting parameter \\ \(\Omega\) & spindle speed \\ \end{tabular}