# Viscoelastic Dampers for Rotors: Modeling and Validation at Component and System Level
## 2 Viscoelastic Model Identification
### Experimental Characterization and Identification by Genetic Algorithm
The GA is implemented exploiting a population of 60 individuals in 60 generations according to the following procedure:  
1. Computation of \(K_{E}\) from Equation (7) at steady-state (\(s=0\)). Random generation of the initial population of the parameters \(K_{1}\) and \(C_{1}\). * 3 Computation of the fitness value for each chromosome by means of \(F_{obj}\). * 4 Selection process with the NGS, TS (with 100 rounds) and RWS procedure to select the best set of individuals. * 5 Application of the elitist selection method to avoid the loss of best chromosomes during the creation of the new population. Elitism allows for saving a priori best individuals before the production of the next generation. * 6 Execution of the Arithmetic Crossover Procedure to generate new individuals. * 7 Execution of the Uniform Mutation Method to perturb the population with an occasional random alteration to avoid sub-optimal convergence of the solution. * 8 Going back to step 3 of this procedure and evaluation of the new population fitness value according to \(F_{obj}\). The termination of the iteration occurs at the production of the 60 th generation. One occurrence of the best population of the optimized parameters is reported in Figure 6 for the two dampers. The three selection procedures are compared (NGS in solid line, TS in dashed line, and RWS in dotted line) showing the convergence to the same final values with a good level of robustness of the optimization method. NGS is adopted because of the slightly lower execution time. However, the other two methods can be adopted without the loss of performance. The identified parameters are  
\[ \begin{array}{l}K_{1_{\text{SES}}}=\ 1.1034\times 10^{5},\quad C_{1_{\text{SES}}} =\ 8.2923\times 10^{2},\\ K_{1_{\text{CS}}}=\ 8.9207\times 10^{5},\quad\quad C_{1_{\text{CS}}}=\ 1.1615\times 10^{3}. \end{array}  \tag{11}\]  
The comparison between \(H_{\text{N}UM}\) and \(H_{EXP}\) is shown in Figure 7. The noise measured at frequencies higher than 1 kHz is due to bad measurement but it can be neglected since it is out of the range of interest. Indeed, the nominal speed of the pump where the dampers are supposed to be installed is 48,000 rpm. The values of magnitude and phase at frequency higher than 1 kHz have not been taken into account in the GA identification. The frequency response of the damper (\(G_{D}\)) is then obtained as  
\[G_{D}=k_{E}+\frac{k_{1}s}{s+\frac{k_{1}}{C_{1}}}=\frac{F_{ACT}}{x}-ms^{2}- \frac{sK_{ACT}}{s+\frac{K_{ACT}}{C_{ACT}}}. \tag{12}\]  
In the final part of the identification, the dynamic of the damper is obtained from Equation (12) to be usable for rotordynamic models. Two contributes are identified in the frequency range of operations: in-phase or storage stiffness (\(K_{P}\)) and in-quadrature or loss stiffness (\(K_{Q}\)), obtained as follows:  
\[K_{P}=Re(G_{D}), \tag{13}\]  
\[K_{Q}=Im(G_{D}). \tag{14}\]  
Figure 8 reports the trends of \(K_{P}\) and \(K_{Q}\) of the SES (a) and CS (b) dampers of the pump. It is possible to obtain equivalent viscous damping (\(C_{eq}\)) of the support with  
\[C_{eq}=\frac{1}{\omega}K_{Q}\, \tag{15}\]  
where \(\omega\) is the actuator excitation frequency in rad/s. The elastomeric support provides a low value of stiffness and damping at low frequencies. The maximum contribution of damping is recorded in the central area where the stiffness is rising. At high frequencies, the stiffness converges to a maximum value, while the damping is steadily decreasing. In the systems that are equipped with viscoelastic supports, the goal is to make the transition area of the damper correspond with the natural frequencies of the structure in order to exploit the support where its damping action reaches its maximum. This objective is pursued by properly designing the  
Figure 8: Dynamic behavior of the damper usable in rotordynamic models. In-phase stiffness \(K_{P}\) (solid line). In-quadrature stiffness \(K_{Q}\) (dash-dotted line). (**a**) SES support of the turbo-molecular pump. (**b**) CS support of the turbo-molecular pump.