# Model Updating of a Freight Wagon Based on Dynamic Tests under Different Loading Scenarios
## 5 Calibration
### Optimization
#### 5.2.2 Loaded Vehicle  
The second optimization process was performed for the loaded vehicle, considering only 5 numerical parameters, particularly the 4 vertical stiffness values of the primary suspensions (\(K_{1}\) to \(K_{4}\)) and the vertical stiffness of the side bearers (\(K_{s}\)), as well as 10 modal responses.  
The values of the remaining numerical parameters associated with the additional masses (\(M_{e}\) and \(M_{t}\)) and steel deformability modulus (\(E_{s}\)) were assumed as deterministic and derived from the optimization of the unloaded vehicle numerical model.  
The proposed sequential optimization strategy allows limiting the extent of the optimization problem by reducing the number of design variables while incorporating the eventual modification of the vertical stiffness of the primary suspensions and side bearers in the optimization results, due to the eventual non-linear behaviour of these elastic components under higher load levels [53].  
Thus, maintaining the same strategy of the unloaded vehicle optimization, Figure 16 represents the ratios of the values of each numerical parameter relative to the bounds indicated in Table 2 for four independent optimization runs, GA1 to GA4.  
Figure 17 shows the variation of the objective function residue during the optimization process. The optimal parameters are referred to the optimization run GA1, which presented the lowest value for the objective function.  
Comparing the results obtained in the two optimization processes, and focusing on the primary suspension parameters, no significant variation on the vertical stiffness values were detected. Additionally, the estimated stiffness values are significantly higher in comparison to the nominal values usually found in the literature, which are typically evaluated considering the vehicle under operation and are in the range between 800 kN/m and 1000 kN/m [20; 22; 23; 53]. As example, Figure 18a,b shows the evolution of the numerical parameter \(K_{1}\) as a function of the number of individuals and for the unloaded and loaded vehicle models, respectively.  
Figure 16: Values of the optimal parameters for the optimization runs GA1–GA4 for the loaded vehicle numerical model.  
These aspects are particularly relevant since as the vehicle was at rest, or almost at resting conditions, and the primary coil suspensions apparently kept their behaviour in an elastic or almost elastic regime, independently of the level of the vertical static loads, in this case the tare load (unloaded configuration) and a regular operation loading (loaded configuration). However, considering the vehicle in a real operation scenario, where, due to a variation on the movements of the suspension components due to the dynamic loading, non-linear behaviour of the suspensions is expected. This non-linear behaviour is due to the friction/slippage contact phenomenon that tends to occur in some of the suspension components, which justifies an overall stiffness reduction [36].  
Regarding the vertical stiffness of the side bearers, an important increase in the vertical stiffness was detected, in comparison to the optimal values obtained with the unloaded configuration. This is probably associated with the pronounced non-linear behaviour of these components, since, under higher vertical forces, the force-displacement curve presents a higher tangent stiffness.  
Figure 17: Residue of the objective function throughout the optimization process for the loaded vehicle numerical model.  
Figure 18: Variation of the optimal values of numerical parameter K1: (**a**) unloaded vehicle; (**b**) loaded vehicle.