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The previous section has shown that the optimized method provides a more accurate prediction of the stresses and displacements in the main chassis beam. The maximum stress and displacement values pre- dicted by the optimized method are nearly ten percent lower than the values acquired with the conventional

method. However, the resultant stresses and nodal displacements obtained with the optimized method could be too optimistic, which could lead to an underestimation of the stress or strain levels in the main chassis beam. The loadcases of the optimized method are separately applied to the model, and hence the finite element analysis consists of distinct static structural analyses. On the other hand, a transient analysis also considers the deformation of the main chassis beam caused by previous loadcases. The deformation of the chassis beam and accumulated stress levels in the material are therefore taken into account by a transient analysis when a new loadcase is applied. From this perspective, a transient analysis could be regarded as a more realistic method for modelling the deformation of the main chassis beam. A comparison between the results from the optimized method and the transient analysis should reveal if too optimistic results are obtained by the optimized method.

The loadcases for the optimized and conventional method were obtained by creating two different envelopes around the load data, which was acquired from the kinematic model with a sample time of 0.25 seconds. These loadcases are representative for the entire load data set and all passenger occupancy configurations. It therefore sufficed to consider only a limited number of loadcases during the separate static analyses that were performed. On the other hand, a transient analysis is not characterized by a relatively small number of loadcases, since all loads should be evaluated in the correct sequence. Additionally, the loads induced on the main chassis beam should be evaluated by means of a separate transient analysis for each passenger occupancy configuration. The load data set contains 401 loadcases for each passenger occupancy configura- tion, and therefore 1604 loadcases should be evaluated in total by the transient analysis. This total number of loadcases is orders of magnitude larger than the number of loadcases that had to be evaluated in case of the conventional or optimized method.

Load Type Configuration Minimum RPD Average RPD (> 5) Maximum RPD

Fx,total 1 0.13 % 56.09 % 200.00 % 2 0.36 % 58.85 % 200.00 % 3 0.20 % 59.32 % 200.00 % 4 0.03 % 61.86 % 200.00 % Fy,front 1 0.02 % 4.50 % 31.56 % 2 0.15 % 4.52 % 30.40 % 3 0.29 % 4.57 % 29.20 % 4 0.44 % 4.72 % 28.56 % Fz,front 1 0.00 % 16.93 % 200.00 % 2 0.01 % 9.11 % 200.00 % 3 0.08 % 11.38 % 200.00 % 4 0.04 % 14.48 % 200.00 % Fy,rear 1 0.01 % 1.59 % 19.21 % 2 0.00 % 1.59 % 19.26 % 3 0.01 % 1.65 % 19.31 % 4 0.00 % 1.70 % 18.71 % Fz,rear 1 0.01 % 11.79 % 200.00 % 2 0.00 % 5.83 % 200.00 % 3 0.01 % 8.57 % 200.00 % 4 0.00 % 9.45 % 200.00 % Tx,rear 1 0.00 % 8.58 % 200.00 % 2 0.00 % 0.78 % 35.50 % 3 0.00 % 3.25 % 200.00 % 4 0.00 % 0.87 % 155.45 %

Table 3.4: Relative percentage difference between the reaction forces obtained by the kinematic model in Simulink and a transient finite element analysis in Ansys

Except for the nature of the analysis, the finite element model used for the transient analysis is completely identical to the one previously used for the conventional and optimized methods. The loads and constraints are applied in a similar way as described in the previous chapters. Nevertheless, the reaction forces from the kinematic model and Ansys are compared to each other to verify if the transient analysis is conducted correctly. Table 3.4 shows the relative percentage difference between the reaction forces for all four passenger occupancy configurations. The RPD values for the sum of forces inx-direction are again substantial, and even twice the magnitude of the RPD values acquired for the optimized method. This significant difference could be caused by the application of loads on a deformed beam during a transient analysis. The kinematic model on the other hand always comprises an undeformed chassis beam, and the deformation of the beam during the transient finite element analysis could therefore affect the reaction force values. A comparison between the RPD values of both analyses also reveals significant differences for some other load types. Aside from the sum of forces inx-direction, also the RPD values forFz,front,Fz,rear, and Tx,rear significantly

deviate from the RPD values acquired with the optimized method. The different reaction forces lead to RPD values that are multiple times larger for the transient finite element analysis. The forces applied in they-direction are an exception to this general pattern, since the RPD values resulting from the transient analysis are comparable or even slightly lower than the RPD values shown in Table 3.2 for the optimized method. Furthermore, the extent to which the RPD values from the transient analysis deviate from the values of the optimized method also differs for each passenger occupancy configuration. The magnitude of the main chassis beam deformation apparently differs for each passenger occupancy configuration, which consequently leads to significantly different RPD values for each configuration. However, it cannot be stated with absolute certainty that the effect of the beam deformation fully accounts for the large difference in RPD values between both methods. Therefore, no clear explanation can be formulated that fully eluci- dates the origin of the increased RPD values for the transient analysis. Aside from the RPD values for the sum of forces inx-direction, the magnitudes of the RPD values for the other load types are in general acceptable though. The finite element model used for the transient analysis is therefore approved, despite the apparent increase in RPD values with respect to the optimized method. The acceptable RPD values for the majority of the load types namely outweigh the large RPD values for the sum of forces inx-direction. The Von Mises stresses and nodal displacements resulting from the transient analyses are presented in respectively Table 3.5 and 3.6. The largest values for σmax and dmax in these tables are observed for the

first passenger occupancy configuration. This configuration is characterized by four passengers and thus the largest mass of the rotating part, which explains the relatively large stress and displacement values for this configuration. However, the mass of the rotating part is not the only parameter that contributes to the stress and displacement levels, as can be deducted from a comparison between the second and third config- uration. The second passenger occupancy configuration includes three passengers, while two passengers are seated on one side of the gondola in case of the third configuration. The relatively large mass imbalance associated with the third passenger configuration leads to larger stresses and displacements than the second configuration, despite the lower total mass. The increased rotational velocity of the gondola due to the mass imbalance apparently leads to larger stress and displacement values. This observation indicates that the rotation of the gondola can significantly affect the stress and strain levels in the main chassis beam. Hence, it was correct to consider all passenger occupancy configurations during the conventional, optimized, and transient analysis.

σmax [MPa]

Configuration 1 Configuration 2 Configuration 3 Configuration 4

Minimum 3.53 3.47 2.80 3.03

Average 8.89 8.15 7.50 6.85

Maximum 27.23 24.80 25.45 18.27

dmax [mm]

Configuration 1 Configuration 2 Configuration 3 Configuration 4

Minimum 0.032 0.033 0.029 0.028

Average 0.088 0.083 0.076 0.070

Maximum 0.217 0.197 0.199 0.160

Table 3.6: The maximum nodal displacements determined with the transient analysis

A comparison between the results obtained with the transient analysis and the optimized method shows that

σmax and dmax are practically similar for both analyses. Hence, the optimized method with 39 loadcases

predicts similar maximum stress and displacement values as a transient analysis with 1604 loadcases. The evaluation of the 39 loadcases of the optimized method takes approximately fifteen minutes, while the total duration of calculating 1604 loadcases exceeds eight hours. If the prediction of the maximum stress and displacement is main objective of the analysis, it is advisory to use the optimized method instead of the conventional or transient analysis. The stress and displacement values obtained with the optimized method namely have a negligible error with respect to a transient analysis.

For both the transient analysis and the optimized method, a loadcase that leads to a maximum stress or strain level can be related to a certain passenger occupancy configuration and a specific moment in time. Given a maximum stress or displacement, it is therefore possible to determine the corresponding passenger occupancy configuration and the location of the vehicle along the track lay-out with both methods. Only the transient analysis enables a graphical plot of the stress or displacement level as a function of time though, as depicted in Figure 3.6 for the first passenger occupancy configuration. Appendix N presents the stress and displacement levels as a function of time for the other three configurations. It can be deducted from these graphs that aside from the maximum stress or displacement value, the passenger occupancy configuration also affects the development of the stress or displacement as a function of time. In general, the stress and displacement profiles are to a large extent similar. However, a comparison between these graphs also indi- cates that different passenger occupancy configurations can significantly impact the maximum stress and displacement levels at specific sections along the track lay-out. This again underscores the significance of loads induced on the main chassis beam by spinning gondolas.

(a) Maximum Von Mises stress (b) Maximum nodal displacement

Figure 3.6: The maximum Von Mises stress and nodal displacement as a function of time for the first passenger occupancy configuration

The deformation of the main chassis beam is shown alongside the kinematic model in Figure 3.7. The plot of the kinematic model namely provides an indication for the location of the vehicle along the track lay-out. The corresponding deformation plots depict the eight largest maximum Von Mises stress values found for the first passenger configuration. Hence, these maximum values correspond with the eight largest peaks in the stress profile depicted in Figure 3.6a. Please note that the deformation of the main chassis beam in these plots has been exaggerated by means of a scale factor.

(a)t= 21 s (b)t= 24 s

(c)t = 37 s (d)t= 43 s

(e)t = 54 s (f) t= 56 s

(g)t= 70 s (h)t= 73 s