166 G. Saletti et al. Table 1 System parameters. I (Kg/m2) r (m) C(Ns/m) z cr φ kc (N/m) 0.0158 0.0705 4038.2 50 1.75 20° 2.6e8 Where the parameters considered are collected in Table 1. To validate the methodology proposed, the reference results are obtained by direct time integration of the system expressed in equation 14. The solutions are obtained implementing as a MATLAB algorithm the Newmark method with β =0.5 andγ =0.25[14]. The simulation is carried out until steady state is reached. Non-linear forced responses Different analyses are performed for multiple levels of constant torques (T) applied, namely 100 Nm, 200 Nm and 300 Nm. The number of harmonics considered in the MHBM solution is H= 14. Since the system has only one degree of freedom (DOF), it is possible to simulate it using a large number of harmonics, although good results could be obtained by retaining fewer harmonics. It was also intention of the authors to properly capture the contact force behavior, considering that the unrealistic step gap function induces instant changes in the contact status leading to steep gradients of the contact stiffness during the rotation of the gears. In Figure 5 the non-linear forced responses of the system are presented; the jumps phenomena are highlighted by black arrows. The HBM simulations are performed implementing a continuation method, however, it was hard to obtain the unstable branches of the response. The simulations for both DTI and HBM are carried out starting from the lowest frequency of the range (forward, “forw”) and from the highest as well (backward, “back”). The different simulation methods are in excellent agreement with each other. All the peaks are predicted accurately both in terms of amplitude and frequency. The jump phenomena typical for this kind of systems are predicted accurately by the HBM simulations. A slight difference seems to be present around the jump of the first super-harmonic (around 1250 Hz), in that region the continuation method struggled to obtain a converged solution due to the presence of an unstable branch of the response. The apparent difference is in fact due to the missed convergence of the numerical solution. In Figure 6 a comparison of the non-linear forced responses is presented. It is worth noting that the different responses seem to be scaling linearly with the constant torque applied. This effect is not present in more accurate simulations or in the experimental results available in the literature [6, 15]. Typically, the increase of the constant torque level tends to “linearize” the system, resulting in less softened peaks. In fact, high levels of constant torque counters the complete separation of the teeth at excitation frequencies close to resonance. The presented system does not completely capture this behavior as a consequence of the simplified contact modelling. More realistic contact modelling will be introduced in future work, to both improve convergence of the solution method and increase accuracy of the results obtained with respect to experimental results available in the literature. 500 1000 1500 2000 2500 3000 3500 Excitation frequency (Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RMS of the Transmission Error (m) 10-5 Non-linear forced response DTI back (100Nm) DTI forw (100Nm) HBM back (100Nm) HBM forw (100Nm) 500 1000 1500 2000 2500 3000 3500 Excitation frequency (Hz) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 RMS of the Transmission Error (m) 10-5 Non-linear forced response DTI back (200Nm) DTI forw (200Nm) HBM back (200Nm) HBM forw (200Nm) 500 1000 1500 2000 2500 3000 3500 Excitation frequency (Hz) 0 0.5 1 1.5 2 2.5 3 RMS of the Transmission Error (m) 10-5 Non-linear forced response DTI back (300Nm) DTI forw (300Nm) HBM back (300Nm) HBM forw (300Nm) Fig. 5 Non-linear forced responses of the system simulated with DTI and HBM with different static torques applied
RkJQdWJsaXNoZXIy MTMzNzEzMQ==