# Model Updating of a Freight Wagon Based on Dynamic Tests under Different Loading Scenarios
## 5 Calibration
### Sensitivity Analysis  
The sensitivity analysis allowed for the selection of the numerical parameters that most influence the modal responses (five natural frequencies and five modal configurations) and so should be included in the subsequent optimization phase. In this study, 10 numerical parameters were selected and stochastically sampled by the Latin hypercube method. The samples generated based on uniform distributions within the limits of each parameter are given in Table 2.  
The sensitivity analysis results are based on Spearman correlation coefficients (\(e\)) that express the monotonic correlation between two vectors of samples, \(x\) and \(y\), based on their rank-order vectors, \(R(x_{i})\) and \(R(y_{i})\), by the expression:  
\[r_{xy}^{S}=\frac{\sum_{i-1}^{n}\left(R(x_{i})-\overline{R}(x)\right)-\left(R(y _{i})-\overline{R}(y)\right)}{\sqrt{\sum_{i-1}^{n}\left(R(x_{i})-\overline{R}( x)\right)^{2}\sum_{i-1}^{n}\left(R(y_{i})-\overline{R}(y)\right)^{2}}} \tag{1}\]  
in which \(\overline{R}(x)\) and \(\overline{R}(y)\) are the average values of the rank-order vectors \(R(x_{i})\) and \(R(y_{i})\), respectively, and \(n\) is the number of samples of each vector.  
Figure 12 shows the results of the global sensitivity analysis based on a Spearman linear correlation matrix based on 750 samples obtained with the Latin hypercube method. The samples with a MAC value lower than 0.50 have been excluded from the total amount of samples. Additionally, the samples with a correlation coefficient within the range \(-\)0.30 to +0.30 were removed. This strategy allows us to properly identify only the parameters with a significant amount of correlation with the modal responses. Numerical parameters presenting Spearman correlation coefficients, in relation to the modal responses, higher than 0.50 were considered for the further optimization stage [58; 59].  
The results of the sensitivity analysis allowed us to identify specific numerical parameters that have an important influence on the modal responses, both on the unloaded and loaded models, particularly the side bearers' vertical stiffness (\(e\)) and the stanchion masses (\(M_{t}\)). Curiously, there are other numerical parameters, namely the vertical stiffness of the primary suspensions (\(K_{1}\) to \(K_{4}\)), that reveal a high sensitivity with the responses in the loaded model, particularly with the MAC values. The additional mass of the braking system (\(M_{c}\)) and the steel equivalent density (\(D\)) proved to have no influence, or negligible influence, on the modal responses, and, consequently, were not included in the optimization phase. Thus, from the 10 numerical parameters evaluated, 8 of them proved to have an influence on the modal responses, particularly on the side bearers' vertical stiffness (\(K_{s}\)), the stanchion masses (\(M_{t}\)), the vertical stiffness of the suspensions (\(K_{1}\) to \(K_{4}\)), the steel deformability modulus (\(E_{S}\)) and the masses of the back stop and front stop (\(M_{e}\)).  
The stiffness parameters of each suspension showed enough sensitivity to be properly estimated in the optimization stage. This is particularly relevant, since in other studies (e.g., [12; 31; 33]), the reliable suspension stiffness estimation sometimes required the grouping of the suspension parameters, considering two sets or even four sets springs, since the individual stiffnesses per set of springs presented low correlation values.  
Also, the mass parameters (\(D,M_{c},M_{e},M_{t}\)) present a negligible sensitivity to the modal configurations, probably due to the symmetry/near symmetry of the vibration modes, which means that a variation in the values of these symmetrically positioned masses will not change its symmetrical configuration.