Issue 30

G. Belingardi et alii, Frattura ed Integrità Strutturale, 30 (2014) 469-477; DOI: 10.3221/IGF-ESIS.30.57 474 Theoretical values of the above quoted forces may be calculated by the analytical combination of tangential, radial and axial components; these equilibrium equations are the following: x t r F F sin φ F cosφ     =157.18 [N] (1) y t r F F cosφ F sin φ      =-198.68 [N] (2) x a F F   =-134.69 [N] (3) where t F , r F , a F are respectively tangential, radial, axial forces and φ =15° is the misalignment angle of gears 3 and 4. From the analysis of Fig. 3, it may be observed that numerical averaged forces well matches with the analytical ones. Fig. 4 shows the Internal Dynamic Factor (K v ) trend of gear 4 during 6 second of the simulation. To calculate K v values, the following equation [7] has been used: static multibody F dynamic multibody Total v static static static static F F F F K 1 1 F F F F        (4) where static F is the static theoretical force of the meshing gears obtained by means of above quoted equations for gears 3 and 4, while dynamic F is the force trend obtained by means of the RecurDyn Simulation menus the Static one. Figure 4 : Dynamic Factor of Gear 4 for Rigid simulation. By analysing the data reported in Fig. 4, it is possible also to calculate the mean value of the dynamic factor (Eq.(4)), that for gear 4 is equal to 1.125. This dynamic factor is very important to be determined already in the design phase because the fatigue strength of gears depends on it, as indicated in ISO Standard [7]. Thanks to multibody simulations, however, it is possible to calculate a well defined value of K v as in experimental tests, even if a high computation time is required for the simulation of the complete model of the transmission (38 to 47 hours). The contact force components versus time of Gear 4 obtained by the Rigid-Flexible simulation are shown in Fig. 5; the forces directions are referred to the ground system of the model. Averaged values are: force in x direction F x =157.2 N, force in y direction F y =-198.69 N force in z direction F z =-134.79 N. As it can be seen by Fig. 5, the trend of the forces in the three directions, respect to the rigid simulation (Fig. 3), is quite different, but the average values are practically the same. The difference is probably due to the flexibility of the teeth, as the Rigid-Flexible simulation results do not present the peaks emphasized in Fig. 3. Finally, the trend of K v factor (Fig. 6) is proposed: the average value for Gear 4 is 1.13, similar to that obtained by the rigid simulation. The trends of forces and of K v factor of the Rigid-Flexible simulation is slightly more accurate and do not present peaks respect to the rigid one, but a very high computational time has been required for this calculation (384 hours respect to a mean of 41 hours of the full rigid simulation).