## 2 Modeling Methods
### Application Scenario  
#### 2.2.1 The Back-to-Back Test Rig  
To be able to validate the presented approach, a back-to-back test rig was used as reference (Figure 3). It consisted of 2 parallel axes gearboxes connected by means of two shafts in a closed mechanical loop. The slave- (better called service-) and test-gearboxes had the same gear ratio (17/18) but a different number of teeth (34/36 and 17/18 respectively). The slave reduction had helical gears, while the gears of the test one were spur. A rotating hydraulic actuator was used to preload the system. An e-motor connected to the main shaft supplied the power that was dissipated in terms of losses during operation. While this approach has already proven to be very computationally efficient, by substituting the bearings with equivalent springs, the stiffness of which was calculated with separate simulations, the computational effort could be further reduced, and plenty of simulations could be performed in a reasonable amount of time. In fact, in the past the authors have modeled the same configuration by means of traditional FE software. The time required for the solution of a single time step on 115 GFLOPS hardware was about 30 h. With the present approach, the simulation of the same gearbox performed, in the same amount of time and on the same hardware, about 700 time-steps. The substitution of the bearings with equivalent springs (Figure 4) further doubled the number of time-steps that could be computed in the same amount of time. Figure 2: The matching interface \(\Gamma\). The stiffness of a bearing is a function of the applied load. Considering an applied load of 300 Nm on the long shaft, the stiffnesses of the ball bearings of the test gearbox (left in Figure 4), resulted in 0.77 kN/m (left bearing) and 0.86 kN/m (right bearing). For the 4 supports of the longer shaft of the slave gearbox, the stiffness results, starting from the right side, were 2.46 kN/m, 0.82 kN/m, 1.71 kN/m, and 0.03 kN/m; while the 4 bearings of the shortest shaft showed values of 0.01 kN/m, 2.22 kN/m, 1.11 kN/m, and 3.39 kN/m. The different values of stiffness that similar bearings showed when mounted on different shafts and/or on different axial positions, was related to how the total load transmitted by the 300 Nm is varied. Figure 4: Simplification of the model. Figure 3: (a) Layout of the back-to-back system; (b) numerical model. the mating gears was shared among the supports. This effect was more evident for the slave gearbox, in which the helical gears also promoted an axial loading with opposite directions on the two shafts. While one of two adjacent bearings was always higher-loaded and, consequently, its stiffness was higher, from the extracted values it can be clearly observed that, while in the longer shaft, the force flowed from the gears mainly to the left end of the shaft, in the shorter shaft, it was transmitted to the housing mainly by means of the right bearings. By inverting the rotational speed, for instance, the stiffness values changed. Considering that the dynamic analysis is aimed at extracting the displacements (and after a double integration also the vibrations) of the model at a certain regime, the time discretization (sampling) must be selected accurately. On one hand, the time discretization should be enough dense to avoid aliasing phenomena, while on the other hand, it should be kept as large as possible to limit the computational effort. The factors that affect the time-step to be selected are the rotational speed, the number of teeth, the number of meshing pairs, the number of harmonics to be considered, and the desired number of time-steps for each meshing period. The latter strongly defines the resolution of the analysis. To determine the sampling time, the gear-meshing frequencies (GMFs) of the two pairs must be calculated:  
\[GMF=\frac{z_{min}\cdot u_{min}}{2\pi} \tag{2}\]  
where \(z\) is the number of teeth and n is the rotational speed.