## 5 Conclusions and Future Work
**Nomenclature**  
\begin{tabular}{l l} \(A_{1}\), \(A_{2}\) & The amplitudes of vibrations \(y_{1}\), \(y_{2}\) [m] \\ \(ae^{\pm t}\) & The envelope of free viscous damped vibration velocity response [m/s] \\ \(b\) & The damping constant [s\({}^{-1}\)] \\ \(C\) & The constant of velocity signal integration [m] \\ \(D_{df}\) & Theoretical dynamic amplification factors of vibrations [ ] \\ \(D_{qf1}\),\(D_{qf2}\) & Dynamic amplification factors of vibrations \(y_{1}\), \(y_{2}\) produced by shafts 1, 2 [ ] \\ \(D_{qfc}\) & Dynamic amplification factor of resultant vibration \(y_{1}+y_{2}\) at average frequency \(f_{c}\) [ ] \\ \(dy_{1}/dt\), \(dy_{2}/dt\) & The derivative of vibration displacements \(y_{1}\), \(y_{2}\) (vibration velocities) [m/s] \\ \(f\) & The frequency of harmonic excitation of the lathe headstock [Hz] \\ \(F_{1}\), \(F_{2}\) & The horizontal projection of the rotary unbalance forces generated by shafts 1 and 2 [N] \\ \(f_{1}\),\(f_{2}\) & The frequency of vibrations \(y_{1}\), \(y_{2}\) [Hz] \\ \(f_{b}\) & The beat frequency [Hz] \\ \(f_{c}\) & The frequency of the resultant vibration \(y_{1}+y_{2}\), or combination frequency [Hz] \\ \(LAS\) & Instantaneous angular speed [rad/s] \\ \(k\) & The stiffness of headstock and lathe foundation [N/m] \\ \(m_{1}\), \(m_{2}\) & Unbalance mass on rotary shafts 1, 2 [Kg] \\ \(n\) & A natural number involved in the definition of the beat period T\({}_{\rm b}\) \\ \(p\) & The natural angular frequency [rad/s] \\ \(p_{1}\) & The angular frequency of damped harmonic vibration [rad/s] \\ \(r_{1}\), \(r_{2}\) & The distance between the center of the unbalance mass and the rotation axis on shafts 1, 2 [m] \\ \(s\) & The addition of vibration displacements s = \(y_{1}+y_{2}\) [m] \\ \(s_{i_{1}}\), \(s_{i+1}\) & Two successive displacement samples of vibration [m] \\ \(s_{i+h}\), \(s_{i+h-1}\) & Two successive displacement samples of vibration [m] \\ \(t\) & Time [s] \\ \(t_{0}\) & The origin of time for the theoretical model of free damped vibration velocity [s] \\ \(t_{zcj}\), \(t_{zcj+1}\) & Two successive zero-crossing moments of the displacement vibration signal involved in \\ \(t_{zcj}\), \(t_{zcj+1}\) & frequency measurement [s] \\ \(\Delta t\) & Sampling interval for a numerically described signal [s] \\ \(T_{1}\), \(T_{2}\) & The periods of vibrations \(y_{1}\), \(y_{2}\) [s] \\ \(T_{b}\) & The beat period, with \(T_{b}=Jf_{b}\) [s] \\ \(T_{c}\) & The period of the resultant vibration \(y_{1}+y_{2}\), with \(T_{c}=1/f_{c}\) [s] \\ \(v\) & The velocity of the resultant vibration in beating [m/s] \\ \(v_{gl}\) & The vibration velocity of the headstock during a free damped response [m/s] \\ \(v_{i}\) & A sample of the vibration velocity [m/s] \\ \(y_{1}\), \(y_{2}\) & The vibration displacement generated by shafts 1, 2 [m] \\ \(y_{1s}\), \(y_{2s}\) & Simulated vibration displacement signals [m] \\ \(\alpha\) & The phase angle at the origin of time \(t_{0}\) for a theoretical model of free damped vibration \\ \(\alpha\) & velocity [rad/s] \\ \(\varepsilon_{f}\) & The error in the frequency measurement [Hz] \\ \(\varepsilon_{j}\), \(\varepsilon_{j+1}\) & The calculus errors for two successive zero-crossing moments [s] \\ \(\gamma\) & The shift of phase between the excitation force and the vibration displacement in the free damped response [rad] \\ \(\theta_{1}\), \(\theta_{2}\) & The instantaneous value of the angle of centrifugal forces to the horizontal direction [rad] \\ \(\varphi_{1}\), \(\varphi_{2}\) & The values of \(\theta_{1}\) and \(\theta_{2}\) at the origin of time, \(t=0\) [rad] \\ \(\omega\) & The angular frequency of harmonic excitation of the lathe headstock [rad/s] \\ \(\omega_{1}\), \(\omega_{2}\) & The instantaneous angular speed of the rotary shafts 1, 2 [rad/s] \\ \(\omega_{1}\), \(\omega_{2}\) & The instantaneous angular speed of the rotor shafts 1, 2 [rad/s] \\ \(\omega_{1