# Experimental and Numerical Dynamic Identification of an Aerostatic Electro-Spindle
## 6 Conclusions  
This paper describes the experimental and numerical investigation of an electro-spindle supported by aerostatic bearings. The numerical model was validated against literature results in the case of plain aerodynamic bearings and against static and dynamic experimental tests carried out on the electro-spindle at null rotational speed. Both single-DOF methods and a multi-DOF method were proposed to estimate the damped natural frequencies and the damping factors. The accuracy of the model results with respect to the experimental data was discussed. In particular, the damping factor evaluated with LSCEM was in good agreement with the experimental data, while the single-DOF methods overestimated it. Future investigations will regard the dynamic behavior of the spindle at different rotational speeds in order to study its unbalance response and stability. Figure 16: Comparison of experimental and numerical damped frequencies (**a**) and damping factors (**b**) on rear plane. **Author Contributions:** Conceptualization, F.C., L.L., A.T., T.R. and V.V. ; methodology, F.C. and L.L. ; software, F.C. and L.L. ; writing--review and editing, F.C., L.L. and A.T.; supervision, V.V. ; project administration, T.R. ; funding acquisition, T.R. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by Carbonech Company. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**  
\(A_{r\,ij}\)\(r^{th}\) modal costant  
\(C\)  
Bearing radial clearance  
\(D\)  
Journal bearing diameter  
\(dt\)  
Time step  
\(d_{s}\)  
Supply holes diameter  
\(e\)  
Static rotor unbalance  
\(F_{c}\)  
Bearing reaction force  
\(F_{ext}\)  
External force applied on rotor  
\(f_{s}\)  
Sampling frequency  
\(g_{in}\)  
Input flow per unit surface through the supply orifices  
\(h_{i,j}\)  
Impulse-response function (IRF)  
\(h_{f}\)  
Radial-film thickness for the front journal bearing  
\(h_{r}\)  
Radial-film thickness for the rear journal bearing  
\(I\)  
Transversal inertia moment of rotor  
\(I_{p}\)  
Polar inertia moment of rotor  
\(k\)  
Radial stiffness of journal bearings  
\(k_{f}\)  
Radial stiffness of the front journal bearing  
\(k_{nose}\)  
Radial stiffness of the spindle evaluated on the nose  
\(k_{r}\)  
Radial stiffness of the rear journal bearing  
\(k_{\theta_{f}}\)  
Tilting stiffness of front journal bearing with respect to its center  
\(k_{\theta_{r}}\)  
Tilting stiffness of rear journal bearing with respect to its center  
\(L\)  
Journal bearing length  
\(l_{f}\)  
\(l_{r}\)  
Axial distance between the rotor center of mass and the centers of the front  
and rear journal bearings  
\(M_{c}\)  
Reaction moment in bearings  
\(m_{r}\)  
Mass of rotor  
\(N\)  
Order of the least squared complex exponential (LSCE) fitting  
\(p\)  
Absolute pressure  
\(p_{a}\)  
Ambient pressure  
\(p_{s}\)  
Supply absolute pressure  
\(r\)  
Journal bearing radius  
\(s_{r}\)  
\(r^{th}\) pole of the system  
\(x\)  
Generic position measured along the \(x\)-axis  
\(x_{G}\)  
Center-of-mass position measured along the \(x\)-axis  
\(x_{nose}\)  
Spindle nose position measured along the \(x\)-axis  
\(x_{1}\)  
Position measured along the \(x\)-axis in the front plane  
\(x_{2}\)  
Position measured along the \(x\)-axis in the rear plane  
\(y\)  
Generic position measured along the \(y\)-axis  
\(y_{G}\)  
Center-of-mass position measured along the \(y\)-axis  
\(y_{nose}\)  
Spindle nose position measured along the \(y\)-axis  
\(y_{rec}\)  
Signal reconstructed by means of the identified modal parameters  
\(y_{1}\)  
Position measured along the \(y\)-axis in the front plane  
\(y_{2}\)  
Position measured along the \(y\)-axis in the rear plane  
\(z\)  
Generic position measured along the \(z\)-axis  
\(z_{f}\)  
Front-bearing center axial coordinate  
\(z_{G}\)  
Rotor center-of-mass axial coordinate