The first simulation concerned the electric motor that actuates the device; the simulation included:

• Phase 1: the electric motor is turned on until the vehicle reaches the desired backward velocity.
• Phase 2: the electric motor is turned off until the vehicle stops

The simulation was carried out under both lubricated and dry conditions. The graph in Fig. 6 shows the angular velocity of the simulacrum (see Fig. 1), which is, of course, directly related to the vehicle speed.
ω₀ is the simulacrum’s angular velocity that corresponds to the desired backward speed of the vehicle. The relevant quantity to be computed was t0, which measured the system’s responsiveness: its computed value was deemed satisfactory.
Lubrication was discovered to be irrelevant in terms of vehicle backward speed. However, it proved to be necessary because the rotating and the translating bodies remain engaged after the vehicle stops, because the spring’s forces are insufficient to overcome the friction forces at the mating surfaces. Fig. 7 shows the system’s configuration after the vehicle had stopped, without lubrication. Lubrication turned out to allow the rotating bodies to resume to their initial position after the vehicle stopped.

Fig. 6 - The vehicle’s backward speed.

Fig. 7 - The system’s final stationary position subsequent to the vehicle’s backward motion

The simulation was then carried out moving the vehicle forwards. This was simulated by applying a ramped angular velocity to the simulacrum. The translating body’s motion along the rear-axle’s axis had to be triggered at a low enough vehicle speed to guarantee the electric motor’s survival. Fig. 8 shows both the translating body’s axial position and the vehicle speed time histories.
ω₁ was deemed to be low enough to avoid failure of the electric motor. Lubrication turned out to have a significant effect on this working condition, as shown in Fig. 9. As was expected, the lubricated systems was more responsive.

Fig. 8 - The disengaging process and the time history of the translating body’s position

Fig. 9 - The disengaging process and the effect of lubrication.

Due to the system’s topology, the disengaging bodies experience the same angular velocity as the rear-wheel axle. So, they can reach the angular velocity that corresponds to the vehicle’s maximum speed. A fixed element method (FEM) analysis was conducted to verify whether those bodies could withstand the maximum speed condition. Fig. 10 shows the computed stress distribution.
The FEM analysis determined that the safety factor corresponding to the maximum stress was high enough to conform to the Piaggio standard. At the end of the disengaging process, when the translating body reaches its final position, the disengaging bodies are affected by the same impact condition that the translating body undergoes. Fig. 11 shows the system’s initial and final positions.

Fig. 12 shows the stress distribution in a disengaging body when the translating body reaches its final position. The maximum stress here was found to be at the same position as in the high-speed, stationary condition (see Fig. 10). The maximum value, however, was revealed to be 10 times smaller in the stationary condition.

Fig. 10 - The disengaging bodies’ structural performance and stress distribution under highspeed conditions.

Fig. 11 - The disengaging bodies’ structural performance and the initial and final positions of the transient condition.

Fig. 12 - The disengaging bodies’ structural performance and the stress distribution in the disengaging bodies during the transient condition.

Fig. 13 - The rotating bodies’ structural performance and the modal superposition results.

The rotating bodies undergo impact conditions when they engage with the translating body. The resulting stress distribution was first evaluated using a modal superposition approach directly inside the multibody model. Using this method, stress levels were obtained which were above the yield limit, due to the method’s inability to consider the material’s plasticity.

Therefore, the multibody model was used to compute the impacting bodies’ velocities, which were subsequently fed to an explicit FEM model as the initial conditions. This latter model was then used to compute the stress state, using proper material data, to take into account the plasticity phenomena induced by high strain rates. Fig. 14 shows the stress distribution at time of impact.
The maximum stress was found to be below the maximum limit allowable by the Piaggio standard.

Fig. 14 - The rotating bodies’ structural performance and the stress distribution computed with an explicit FEM model.