# Viscoelastic Dampers for Rotors: Modeling and Validation at Component and System Level
## 3 Rotordynamic Validation for the Identified Damper Models  
The final part of the study is focused on the validation of the approach at system level. To this end, the two dampers previously characterized are installed in a turbo-molecular pump to perform the experimental characterization. The machine adopted has a pumping capacity of 1.2 L/s, the weight of the rotor is 0.9 kg, and the nominal speed is 48,000 rpm. The whole system is represented by means of an FEM model of the rotor and stator that is connected through the two dampers. The rotordynamic model has been realized by means of a FE (Finit Element) code (Dynrot), representing the rotating and the static parts with Timoshenko beam elements. Figure 9 shows the main nodes of the mesh along the shaft and the stator; (1) represents the stator that in the left-hand side has been considered as an unstructured mass on the left-hand side whose stiffness contribute is null (dotted part); (2) is the rotor whose pumping part is replaced by a mock-up provided with two discs to replicate the inertial behavior of the original bladed impeller; (3); and, (4) are the bearings and viscoelastic supports on the CS and on the SES, respectively. Gyroscopic effects are taken into account in all of the elements and rotating masses. The dynamic equation of the rotating parts in the rotor reference frame is  
\[\mathbf{M}\ddot{q}(t)+(\mathbf{C}+\mathbf{G})\dot{q}(t)+(\mathbf{K}+\mathbf{H} )q(t)=F_{u}, \tag{16}\]  
where \(q(t)\) includes the generalized displacements, \(\mathbf{M}\) is the symmetric mass matrix, \(\mathbf{C}\) and \(\mathbf{G}\) are the symmetric damping and the skew-symmetric gyroscopic matrices, \(\mathbf{K}\) and \(\mathbf{H}\) are the symmetric stiffness and the skew-symmetric circulatory matrices, and \(F_{u}\) is the unbalance force [16]. The experimental tests have been conducted in three steps: test on the shaft in free-free condition to validate its rotordynamic model (A); test in static conditions (B) and in rotation (C) on the assembled pump including rotor, stator and dampers. (A) The rotor model is validated by means of impact tests on the free-free rotor. The transfer functions between the force (input) and the acceleration (output) are measured (Figure 10). The force is provided with an instrumented hammer in correspondence of the higher part of the impeller mock-up, while the acceleration is measured in the lower part of the shaft in correspondence of the CS (a) and SES (b) damper location. Solid and dashed lines report the experimental results and the numerical computations, respectively. Figure 10: Rotor characterization. Transfer functions measured between the force on the higher part of the impeller mock-up (input) and the acceleration in correspondence of the SES (**a**) and CS (**b**) damper locations. Figure 9: Turbo-molecular pump discretization for Finite Element Method (FEM) rotordynamic analysis. (**a**) Complete system. (**b**) Shaft. (1) Stator. (2) Rotor. (3) Bearing and viscoelastic support on the compressor side (CS). (4) Bearing and viscoelastic support on the shaft end side (SES). (5) Rotating part of the electric motor. (6) Impeller mock-up. (B) The assembled pump that is equipped with the two characterized dampers is tested in static condition with impact tests to obtain the transfer functions between the force (input) and the acceleration/displacement (output) (Figure 11). The force is provided with an instrumented hammer in correspondence of the higher part of the stator. The acceleration is measured on the stator (a) and on the rotor (b-d). The relative displacement between the rotor and the stator is obtained with capacitive sensors (e,f). Solid and dashed lines are experimental and numerical results, respectively. The natural frequencies in the range of interest are located at 83 Hz and 472 Hz, where both dampers are in the transition area and can provide damping, as illustrated in Figure 8.  
The proposed modeling and identification methodology can be indifferently applied to rigid and flexible rotors.