Filippo Ponchia – filippo.ponchia@gmail.com – Degree in Mechanical Engineering

## Introduction

A continuously variable transmission (CVT) is a transmission that can change steplessly through an infinite number of effective gear ratios between maximum and minimum values. The project aim is to analyze the functionality of a friction CVT. The particularity of this specific type of CVT is that the torque is transmitted by friction forces.

The main technical problem which the CVT is intended to resolve is that of providing a traction system for vehicles whitout a discrete trasmission ratio gear change system. By doing this the internal combustion engine can work for most of the time with the maximum efficiency.

## Objectives

The purposes of our work are the following:

- compare different contact models applied on the toroidal CVT;
- verify the correct working of the toroidal CVT;
- verify that the transmission is homokinetic;
- dynamic analysis.

## Mechanism description

The mechanism is very simple and it’s essentially composed by three main elements: two conical shafts (one for the input and one for the output) and a circular friction element called friction sphere (in this case are present two of them for a better force balance). The last one has the function of transmit the torque between the input and the output. The friction sphere can rotate around his axis on a carrier. Also the friction sphere’s carrier can rotate around a pin, by doing this the contact points on the two shafts change their position and so do the transmission ratio. While the contact point moves on the shaft surface, the radius from the shaft rotation axis to that contact point also change due to the shaft particular shape. Here below there is a video that explains the mechanism working.

## The Modelling Problem

The first step was to create the essential geometry for the assembly of the CVT system. Concerning the parts of the model which allow to trasmit the motion through the friction forces (shaft and friction sphere) there has been created two geometries: one functional which is shell type (“surface thickness) and another which is classic CAD type for the presentation.

Because of the contact models chosen there has been needed the shell geometry. In fact some of these models reqired that the affected bodies had to be built in a certain way. Moreover the use of this geometry permitted to reduce the calculation time of the CAD contact model also having reduced the number of the required cells for the meshing of the involved bodies too.

After having assembled the mechanism with the required joints, we carried on implementing three different kinds of contact models. These models are:

- CAD contact;
- Sphere to revolved surface contact;
- Analytic contact model.

For this contact models we adopted these main parameters:

- spring coefficient 1e+006 Nm
- damping coefficient 1000 kgs
- friction coefficient 0,5

Between the friction sphere and his carrier we added a force of 1000 N so that we could obtein a contact pressure between the shaft and the friction sphere itself.

## Contact Models Comparison

LMS Virtual Lab provides a several number of methods for modelling the contact and the relative forces between two bodies.

For the study of this mechanical object we have adopted three different contact models to apply between the two shafts and the friction spheres. Two of them are supplied from Virtual Lab while the third has been developed by us through the application of some analytic expressions.

### CAD Contact model

The first model that we have tried is called “CAD contact” by the software, we first chose it because it is probably the easiest model to implement since it works with every geometry. At the expense of its simplicity, we have found several limitations on analysis results. With this kind of contact model solid geometry associated with each body is tessellated with user-specified parameters. The first parameters we used to define the tessellation gave rise to very disappointing results (heavy oscillations of the parameters analysed even if they should be theoretically constant). We have so tried a thicker tessellation to avoid these problems in order to better fit the tessellation with real geometry. However the thickening of the tessellation has the negative consequence of requiring more computing resources and also time. Below is reported a plot of input and output torque for a configuration with the friction sphere in the neutral position (transmission ratio of 1:1).

For the constant input torque of 20 Nm we can observe that the output presents some oscillations around a value of about 12 Nm. Beyond the oscillations this contact model notices a heavy loss of torque trough the mechanism (a value approximately of 40%). This loss seems to be really too high for being realistic. An explanation of this high value could be done by the observation of the the contact point’s displacement calculated by the model among the y axis.

This value should be 0, but the model assignes an oscillation of about 1,2 mm around it. This is due to the tessellation of the geometry and cause an extra rolling torque much more high than what it should be.

Our conclusion is that this contact model is not suitable with this kind of problem and in general with every sort of not planar geometry.

### Sphere to Revolved Surface Contact Model

In the second model we tried to simulate the contact forces using the sphere-to-revolved-surface-contact.

Initially we tried to simulate the friction sphere with a single contact model’s sphere with the same radius of the real one. This model didn’t work because the software requires that the trajectory of the center of the sphere collides with the revolved surface. If it does not happen the model doesn’t create any contact forces, even if the sphere and the surface touch each other.

To avoid this problem we decided to change the problem modelling adopting two fictitious spheres, one in contact with the input shaft and the other with the output shaft. In this way we have been able to simulate the interaction between the friction sphere and the shafts.

We have created a joint between the two sphere in order to make them rotate with the same spin speed. Doing so the tangential speed of the two spheres on the contact points are exactly the same, like if there was a single friction sphere to rotate.

Unlike the CAD model the evolution of the analyzed parameters shows much more steadiness in time, without the previous observed oscillations. Moreover we can point out that the loss of torque reflects badly at null values (the contact model, in fact, doesn’t include the resistance at the rolling between the two superficies). As regards the power, on the other hand, we notice a loss that is identified around the 3,5%; this is due to the presence of a certain sliding speed between the friction sphere and the two shafts due to the quite-coulumbian friction adopted by the software (threshold speed 0,01 m/s).

### Analytic Model

The third model has been personally developed by us by using the only application of forces calculated through analytic formulas in order to simulate the contact between shafts and friction sphere. Specifically, for every contact point three forces have been applied: a normal force at the surface of contact and two of them at the tangent plane to the surface.

For the two tangent forces, therefore simulating the friction force, a friction formula of the quite-coulumbian kind has been adopted:

**V _{st }/ (|V_{st}| + V_{0}) * μ * F_{n}**

where:

**V**: is the sliding speed_{st }**V**: is the threshold speed (more smaller it is, much the model approaches to a perfectly coulumbian one)_{0}**μ**: is friction coefficient**F**: is the normal force_{n }

The contact point where all the forces have been applied has been beforehand chosen and supposed to be at the half of the friction sphere’s contact surface.

Even in this case the loss of couple turns out to be null since the created model does not consider the losses for rolling friction. As regards the powers we can point out that a loss in the amount of 11% with a speed threshold of 0.01 m/s, while 1,3% with a speed threshold of 0,001 m/s. These remarkable gaps are due to a different sliding speed caused by the variation of the parameter V_{0}.

## Proof of concept

We have therefore accomplished a simulation in which we have verified the CVT operation, that is to say the variation of the transmission report at changing the position of the friction sphere’s carrier. This simulation has been made for both sphere and analytic models. The CAD model has been rejected for the poor quality of results.

We notice that with a constant rotation speed of the input shaft, at the variation of the friction sphere’s position, we obtain a speed in variable output. In particular, with a law of linear variation of the tilt angle of the friction sphere’s carrier, we can notice that this does not happen for the evolution of the output shaft’s rotation speed. It can be considered trivial, since the variation of the distance of the contact point of the shaft’s rotation axis does not vary in linear fashion with the friction sphere’s angle.

Between the two chosen models (the analytical and the sphere one) we can appreciate how at a macroscopic level the evolution of the angular velocity of the shaft output is the same. However there are some small differencies probably caused by the different sliding velocity calculated from the friction model and the fact that the analytical model does not consider the penetration of the friction sphere caused by the load. Below in the table you can find the relations of the transmissions concerning the two different models.

## Homokinetic Test

A mechanism can be defined homokinetic if in a constant input rotation speed even the output results the same. We have therefore executed several tests with different friction sphere’s positions to verify that the CVT system in analysis have effectively that propriety.

Below is reported the graph with the results from which we deduce that the given mechanism is effectively homokinetic.

## Dynamic Analysis

We did a dynamic analysis of the CVT system using as an input a costant torque while as an output an inertia and a resistant torque with linear law on the rotation speed.

Concerning the transmission of the torque we obteined the results showed in the graph below.

As we can see the functionality of the CVT system is confirmed in these case too. Because of the fact that the evolution of the transmission ratio is not linear, the exiting output does not follow a linear law with a variation of the tilt angle of the friction sphere’s carrier. Furthermore we can appreciate how the results obtained by the two calculation models are fundamentally the same.

Regarding the friction forces operating in the contact points we can notice (see graphic below) a discontinuity in the sphere model at 1 second of the simulation time. We do not understand why yet. Probably it is a bug of the software considering that the evolution of the torque do not show any discontinuity. However hereafter the two simulations give the same results.

Concerning the normal contact forces the two models give the same results because both depends on the only push force operating on the friction sphere and on the geometry itself.

An interesting aspect of this CVT system is the fact that the analysis show that it is necessary a considerable torque in order to guarantee the variation of the angle of the friction sphere’s carrier. This torque derives by the presence of a friction force on a normal plane to the rotation axis of the friction sphere’s carrier and it is generated by the sliding of the friction sphere to the shaft. As we can notice from the graphic below, the values of this torque are rather high. This fact surely invalidates in a negative way the efficiency of the CVT mechanism.

We also analysed the system submitting it to an input of a torque which raises linearly in time, so that we could verify the maximum transmissible torque. As we can notice both of the models agree attesting that the maximum transmissible torque has a value of about 58 Nm.

Below there are the angular velocities of the two shaft in order to do a comparison between the two models.

## Conclusions

In this work we analysed the working of a CVT friction transmission system. Apart from having verified the effective working of the mechanism and having done some dynamic analysis, we focused in particular on the comparison of different contact models offered by the software LMS Virtual Lab.

Concerning the contact models we tried three of them:

- CAD contact;
- sphere to revolved surface contact;
- analytic model, implemented by us.

The first of these three models had been unsuitable regarding the study we were doing.

From the analysis we did the mechanism showed a continual variation of the transmission ratio to the change of the position of the friction sphere’carrier, but with a non linear evolution. Moreover it has been verified that the mechanism is homokinetic.

The analysis also allowed us to underline a negative aspect intrinsic of this kind of CVT system. In other words the fact that this system requires a considerable torque in order to permit to vary the position of the friction sphere’s carrier and so the transmission ratio.

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