78 J. Koutsoupakis et al. Table 2 Gear geometry related parameters. The corresponding MBD model of the single stage gearbox is shown in Figure 5. Fig. 5 Single stage helical gear transmission system – MBD model. A spring-damper force is used to simulate the stiffness at the bolted joints at the base of the gearbox, while the shafts are connected to the gearbox through bushing forces, used as a simplified means of modeling bearings. While this does not allow for modeling the precise bearing dynamics, it was sufficient for the present work, where gear-related phenomena are of interest. Last, the gears are considered fixed to the shafts and a contacting force is acting between them. A constant velocity profile is used as an input to the model while a torque acts on the outlet shaft, simulation the load at the end of the gearbox. Following the optimization process described in the theoreetical background section, the model was optimized based on the healthy state acceleration measurements of the gearbox. The parameters used in the optimization process were the elastic modulus of the gears, assumed equal for both gears, the contact force model (Table 1), the coefficient of restitution and the nonlinear exponent. The final contact stiffness value was estimated based on the Hertzian stiffness formula Equation (1). The updated values for these parameters are shown in Table 3. It should be noted that the parameters of the bushings used in the shafts as well as the spring-damper component that was used to connect the gearbox to the ground could have also been included in the optimization but were omitted for the sake of speeding up the process. This is due to the fact that population-based methods, such as the CMAES algorithm used in this work, require a large number of instances/population for each iteration, which increases as the number of model parameters increases. For these components, nominal values were derived from literature, based on properties used in similar applications, and were then fine-tuned manually, ensuring the final model yielded realistic results. Table 3 Optimal model and parameters. Figure 6 shows a comparison between the experimental and optimal MBD frequency responses of the single stage drivetrain, measured at the Y axis of accelerometer A. As shown, while both systems display peaks at the orders corresponding
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