# Proof of concept

For demonstrating the right working of Edyson CVT some simulations have been done on the virtual model.

In a first preliminary analysis an input constant torque is given to input shaft making rotate all pinion satellites in the verse permitted by free wheel mechanisms. In this case the disk does not rotate, the power transmission and efficiency are null: all input power is dissipated in mechanical losses.

In the virtual model realized, the CVT works properly only with counterclockwise input torques, so a clockwise input torque brings to free rotation of the input shaft.

Edyson CVT free rotation due to clockwise input torque

A different behavior is obtained giving an opposite torque to the input shaft imposing a rotation of at least one pinion satellite in the direction locked by the free wheel mechanisms. In this case the disk rotates and there is a power transmission between input and output.

Fixing the slide position so that the input shaft and the output shaft result aligned, all the four pinions satellites are always blocked to rotate about themselves, and the output shaft rotates with the same velocity of the input shaft.

Edyson CVT working with counterclockwise input torque and null slide displacement

Next analysis wants to demonstrate how the CVT works; the slide moves varying input-output shafts offset from 0 to 25 mm with discontinuous ramps. The input torque is set constant and equal to 100 Nm and the damping coefficient at the output shaft is equal to 3 m2 Kg/(rad s).

The figure A represents, at bottom, the value of the displacement of the slide versus time adopted in this analysis and, at top, the transmission ratio. The transmission ratio is defined dividing the input shaft rotational velocity by the output shaft rotational velocity: The mechanism is a revolution multiplier, so this quantity is always lower than one.

According with previous considerations the transmission ratio is equal to 1 when the slide displacement is zero; when the slide displacement assumes values greater than zero there is an offset between the input shaft axis and the output shaft axis and the transmission ratio is lower than one, as it’s possible to see from the previous figure, and decreases as the slide displacement increases.

The virtual model of the gearbox shows that Edyson CVT works correctly as expected by description: the user can choose a certain slide displacement and can vary it continuously to reach the desired mean transmission ratio. But it is possible to observe that the transmission ratio does not rest constant even if the slide displacement does not vary (fastened input and output shafts offset); this is due to the variation of the distance between the axis of the single satellite gear locked and the axis of the output shaft during the rotation. Minimum transmission ratio happens at the horizontal lateral wheel position, represented in figure B, and maximum transmission ratio happens at its insertion/disinsertion position represented in figure C.

In conclusion if a constant rotational velocity or a constant input torque is imposed to the input shaft, the same entity related to the output shaft is not constant even for a constant displacement of the slide and so the mechanism is not omocinetic. Furthermore oscillations rise when the slide displacement increases and consequently for the lowest transmission ratio the rotational velocity of the output shaft has the highest variations.

Before analyzing the oscillations frequency it’s necessary a description of the behavior of the system imposing an entering torque to the input shaft; in this case the output torque decreases when the displacement of the slide increases. The velocity of the input shaft is not constant, but it oscillates between two limits, and the same consideration is correct also for the output shaft. Analytically: In this particular model the resistance torque applied to the output shaft is proportional to the output shaft rotational velocity with a constant damping coefficient c, but this is an assumption we made to build the model; the real gearbox may works in different ways and so every consideration related to obtained results is valid for this working mode of the mechanism.

From the last relationship, assuming a constant damping coefficient c and about constant efficiency η, because the input torque doesn’t vary during the analysis, if the displacement of the slide increases, and so the transmission ratio τ decreases, then the rotational velocity of the output shaft and the output torque decreases. But the system is a multiplier, so the output shaft rotational velocity is higher than the input shaft one; increasing input-output shaft offset the output shaft rotational velocity decreases and so the input shaft rotational velocity decreases to lower values than those relative to the output shaft.

The next video illustrates the behavior of Edyson CVT when applying constant input torque.

Edyson CVT working with constant input torque

When the slide reaches high displacements it’s possible to observe from simulation results that the frequency of the oscillations of the transmission ratio decreases, coherently with previous considerations.

Referring to the previous case, where input-output shafts offset varies with the ramp of figure A, input and output torques are plotted in the next figure:

The input torque is imposed constant; the output torque mean value varies with the variation of the slide displacement. When the slide is in the initial position the output shaft is aligned with the input shaft; the output torque does not oscillate and it’s equal to the input torque, coherently with a unitary transmission ratio value. The output torque decreases when there is an increment in the input-output shafts offset and it oscillates with similar trend of the transmission ratio from a minimum to a maximum value; oscillations amplitude are higher when the slide displacement is higher.

The following graphs confirm what described above. The first graph compares transmission ratios reached with various displacement of the slide.

Coherently with previous considerations the mean transmission ratio assumes lower values for bigger displacement of the slide; furthermore the frequency of the oscillations of the output torque is lower for bigger displacement of the slide, like described before. Even the amplitude of this oscillation is bigger for higher slide displacements.

The second graph compares the output torques reached in different cases.

This graph also confirms previous considerations, valid for the model under analysis: with higher slide displacements the output shaft rotates slower and the output torque is lower than a case with less displacement.

The Edyson CVT works like expected. However, as predicted, the mechanism is not omocinetic, so it is not indicated for applications where constant output shaft velocity is required applying a constant input velocity.