# Viscoelastic Dampers for Rotors: Modeling and Validation at Component and System Level
## 2 Viscoelastic Model Identification
### Experimental Characterization and Identification by Genetic Algorithm  
The proposed procedure is tested on the two different dampers that have been installed in the compressor side (CS) and in the shaft end side (SES) of the turbo-molecular pump that is used for the experimental validation at the system level, as reported in the final section of this paper. The transfer function \(H\) (Equation (7)) is measured experimentally (\(H_{EXP}\)) with a sweep test from 0 to 1.5 kHz, and is then compared to the numerical equivalent (\(H_{NUM}\)) to identify the values of \(K_{E}\), \(K_{1}\), and \(C_{1}\). The identification is performed automatically on the DSP (Digital Signal Processor) by means of a Genetic Algorithm (GA), aiming to minimize the difference between the numerical (\(H_{NUM}\)) and the experimental (\(H_{EXP}\)) results. GAs are search and optimization techniques that are inspired by two biological principles, namely the process of natural selection and the mechanics of natural genetics. They consist of a mechanism of selective reproduction acting on a set of potential solutions, called chromosomes, whose correctness is measured by means of a fitness function. The collection of potential solutions, called population, is manipulated to perform a search based on the survival principle of the fittest. The chromosomes are the encoded representations of all the parameters of the solution. Each chromosome is compared to the others in the population and evaluated with a fitness rating that indicates how successful the solution is. The first population is created randomly while the following are generated to encode better solutions using genetic operators or evolution operators, such as elitism, crossover, and mutation. Elitism allows for saving the best chromosomes before reproduction, crossover, and mutation, enabling the new reproduction by either merging the chromosomes in the current population or by modifying some of them. Several algorithms of crossover and mutation are present in the literature, in the present case the Arithmetic Crossover Procedure and the Uniform Mutation Method are adopted. The selection of the best individuals for each generation can be performed with different methods, in this case, three of them are tested and compared: Normalized Geometric Selection (NGS), Tournament Selection (TS), and Roulette Wheel Selection (RWS). The creation of new generations stops when a fitness target is reached or after a certain number of reproductions [11; 12; 13]. In the proposed GA, each chromosome is encoded as a vector of real numbers of the same length as the solution vector. The parameters to be optimized are \(K_{1}\) and \(C_{1}\), therefore the chromosome \(j\) of generation \(i\) is represented as  
\[C_{j}^{i}\ =\ \Big{[}c_{j1}^{i},c_{j2}^{i}\Big{]}, \tag{8}\]  
where \(c_{j1}\ =\ K_{1}\) and \(c_{j2}\ =\ C_{1}\). The upper and lower bounds of the two parameters for both the characterized dampers are  
\[ \begin{array}{lcl}c_{j1,min}&=&1\ \times\ 10^{5}\ \mathrm{N/m}\,,\\ c_{j1,max}&=&4\ \times\ 10^{6}\ \mathrm{N/m}\,,\\ c_{j2,min}&=&100\ \mathrm{Ns/m}\,,\\ c_{j2,max}&=&400\ \mathrm{Ns/m}\,.\end{array} . \tag{9}\]  
The fitness function \(F_{obj}\) is established to provide a performance measure of the generated chromosomes. The difference between analytical (\(H_{NUM}\)) and experimental (\(H_{EXP}\)) results is considered as the error \(e\) to be minimized by means of Mean Square Error (MSE) computation. Both magnitude (\(Mag\)) and phase (\(Pha\)) are considered according to the fitness function  
\[F_{obj}=1/\left(W_{m}\frac{\sum_{j=1}^{a}\left(Mag\left(H_{NUM_{d}}\right)-Mag \left(H_{EXP,d}\right)\right)^{2}}{n}+W_{p}\frac{\sum_{j=1}^{a}\left(Pha\left( H_{NUM_{d}}\right)-Pha\left(H_{EXP,d}\right)\right)^{2}}{n}\right) \tag{10}\]  
where \(n\) is the size of the frequency vector, and \(W_{m}\) and \(W_{p}\) are the weighting coefficients for magnitude and phase MSE, respectively, both set to 0.5 in this case.