## 4 Dynamic Tests
### Determination of the Material LVE Properties from Dynamic Tests
#### 4.3.1 First Method: Optimization of the 2S2P1D Model Constants to Match Experimental FRFs  
Among the 10 constants of the 2S2P1D model, only four constants (E\({}_{0}\), k, \(\delta\) and \(\tau_{\mathrm{E}}\)) have a significant influence for the considered range of frequencies involved during dynamic tests and need to be optimized. The complex Poisson's ratio has a very small influence on the calculation of the FRFs below 20 kHz. [PERSON]'s ratio cannot be back-calculated with this procedure; however, it is necessary to assume values for constants \(\
u_{00}\), \(\
u_{0}\) and \(\tau_{\
u}\) to back-calculate the complex modulus at each temperature. For each tested temperature, the vector X of the four constants (E\({}_{0}\), k, \(\delta\) and \(\tau_{\mathrm{E}}\)) to be identified was optimized iteratively so that the calculated FRFs match the experimental FRFs. The values of the six other constants were fixed to classical values for AM: E\({}_{00}\) = 100 MPa, \(\
u_{0}\) = 0.19, \(\
u_{00}\) = 0.45, h = 0.53, \(\beta\) = 250 and \(\tau_{\
u}\) = 31.6\(\tau_{\mathrm{E}}\). Only the values of the experimental FRFs at frequencies around the resonance frequencies are used as input in the optimization according to previous studies [19; 20; 21] that showed  
Figure 6: Finite element mesh and boundary conditions used for the FEM calculation of the FRFS.  
their meaningful importance. Ten frequencies were selected along each resonance peak and the error function to minimize was defined as follow:  
\[\text{Error}=\sum_{j=1}^{N\text{peaks}}\sum_{i=1}^{10}\left(\left|\frac{\left| \text{H}_{Exp_{ji}}\right|-\left|\text{H}_{\text{C}_{ji}}\right|}{\left|\text{H} _{Exp_{ji}}\right|}\right|\right) \tag{11}\]  
where \(\text{H}_{\text{Exp}}\) is the experimental FRF, \(\text{H}_{\text{C}}\) is the calculated FRF, \(\text{N}_{\text{peaks}}\) is the number of resonance peaks, \(\text{j}\) is the index of the peak and \(\text{i}\) is the index of the frequencies. The number of peaks considered for the optimization at each temperature corresponds to the number of peaks that are visible below 20 kHz (maximum considered frequency with our experimental device). This number is given in Table 2 for each temperature.  
The optimization was performed in MATLAB with the \"fminsearch\" algorithm and the optimization was stopped when the error and the parameter tolerance of 1% is reached (e.g., when the variation of the error and of all the values of the four constants to be identified is less than 1% between two iterations of the algorithm). The final vector \(\text{X}_{\text{f}}\) of the four constants (E\({}_{0}\), k, \(\delta\) and \(\tau_{\text{E}}\)) was then used with the three fixed constants related to the complex modulus (E\({}_{00}\), h, and \(\beta\)) to back-calculate the complex modulus at the resonance frequencies of the peaks used as input for the optimization at the considered temperature. The optimization procedure to identify the four 2S2P1D model constants (E\({}_{0}\), k, \(\delta\) and \(\tau_{\text{E}}\)) at each temperature is explained in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & \(\mathbf{-20}\) & \(\mathbf{0}\) & \(\mathbf{15}\) & \(\mathbf{35}\) & \(\mathbf{50}\) \\ \hline Number of peaks & \(1\) & \(1\) & \(1\) & \(1\) & \(2\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of peaks considered for the optimization of the 2S2P1D model constants at each temperature.  
Figure 7: Method 1: Principle of the optimization procedure to identify the four 2S2P1D constants (E\({}_{0}\), k, \(\delta\) and \(\tau_{\text{E}}\)) at each temperature. \(\text{X}_{\text{f}}\) is the final vector of the four 2S2P1D constants to identify.