# Experimental and Numerical Dynamic Identification of an Aerostatic Electro-Spindle
## 5 Comparison between Experimental and Numerical Results  
The first critical speed of the spindle was experimentally measured by analyzing the amplitude of the synchronous whirl at different rotational speeds. It was found to be 72 krpm and 94 krpm at gauge supply pressure \(p_{s}\) = 0.5 and 0.7 MPa, respectively. If compared with the theoretical frequencies resulting from the simplified model of Section 4.5, the difference is less than 6%.  
The experimental stiffness on the nose at different pressures is lower than the theoretical value calculated near the rotor-centered condition (between 27% and 39% with respect to the theoretical value). Considering that the 30 N external load imposed on the nose involves non-negligible eccentricity ratios on bearings (especially on the front bearing and with low pressure), a non-linear estimation of the stiffness is preferred (see Section 4.4). If compared with the stiffness obtained in this case, the experimental stiffness is more in accordance with the prediction (between 52% and 86% with respect to the theoretical value).  
The experimental damped frequencies and the damping factors evaluated on the shaft vertical vibration at \(z=z_{L}\) are compared with the numerical frequencies in Figure 16. The experimental damped natural frequencies are in good accordance with the numerical frequencies, as they fall in the range defined on the basis of the tolerance considered on the front (\(h_{f}\) = 17.5 \(\pm\) 1.5 \(\upmu\)m) and rear (\(h_{r}\) = 21.5 \(\pm\) 1.5 \(\upmu\)m) clearances. Conversely, the computed experimental damping factors are higher than the numerical simulations. The single-DOF methods were found to be less accurate than LSCEM to capture the modal parameters of experimental signals, especially the damping factors.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(p_{s}\) (MPa) & \(h_{f}\)/\(h_{r}\) & Signal & \(\omega_{d,1}\) (krpm) & \(\zeta_{1}\) (krpm) & \(\varphi_{1}\) (kdeg) & \(A_{1}\) (krpm) & \(\omega_{d,2}\) (krpm) & \(\zeta_{2}\) (kdeg) & \(\varphi_{2}\) (kdeg) & \(A_{2}\) (kz) \\ \hline
0.3 & 16/20 & \(y_{L}\) & 80.51 & 0.1675 & \(-\)19.07 & 0.0737 & n.a. & n.a. & n.a. & n.a. \\  & \(y_{0}\) & 58.28 & 0.3427 & \(-\)54.14 & 0.9738 & n.a. & n.a. & n.a. & n.a. \\
0.3 & 19/23 & \(y_{L}\) & 70.00 & 0.1556 & \(-\)18.80 & 0.1224 & n.a. & n.a. & n.a. & n.a. \\  & \(y_{0}\) & 50.00 & 0.2044 & \(-\)70.50 & 1.115 & n.a. & n.a. & n.a. & n.a. \\
0.5 & 16/20 & \(y_{L}\) & 100.6 & 0.0804 & \(-\)21.40 & 0.0618 & n.a. & n.a. & n.a. & n.a. \\  & \(y_{0}\) & 74.82 & 0.1394 & \(-\)76.58 & 0.7109 & n.a. & n.a. & n.a. & n.a. \\
0.5 & 19/23 & \(y_{L}\) & 86.97 & 0.0620 & \(-\)21.69 & 0.0710 & 66.22 & 0.0667 & 100.1 & 0.0377 \\  & \(y_{0}\) & 65.86 & 0.0629 & 82.46 & 0.7587 & n.a. & n.a. & n.a. & n.a. \\  & \(y_{L}\) & 113.3 & 0.0365 & 21.8 & 0.029 & 87.83 & 0.0604 & \(-\)50.2 & 0.0135 \\
0.7 & 16/20 & \(y_{0}\) & 87.69 & 0.0652 & 115.17 & 0.3245 & n.a. & n.a. & n.a. \\  & \(y_{L}\) & 96.09 & 0.0356 & \(-\)23.30 & 0.0496 & 75.80 & 0.0193 & 79.62 & 0.0218 \\  & \(y_{0}\) & 75.81 & 0.0166 & \(-\)85.73 & 0.6339 & n.a. & n.a. & n.a. & n.a. \\
0.9 & 16/20 & \(y_{L}\) & 122.3 & 0.0124 & 4.855 & 0.0251 & 97.46 & 0.0281 & \(-\)14.7 & 0.0104 \\  & \(y_{0}\) & 97.30 & 0.0265 & 157.25 & 0.3423 & n.a. & n.a. & n.a. & n.a. \\
0.9 & 19/23 & \(y_{L}\) & 102.5 & unst. & unst. & unst. & 82.03 & unst. & unst. & unst. \\  & \(y_{0}\) & 82.03 & unst. & unst. & unst. & unst. & unst. & unst. \\ \hline \hline \end{tabular}
\end{table}
Table 10: Modal parameters resulting from LSCEM.