# Tire Modelling

Josef Bertucco, josef.bertucco@studenti.unipd.it

# Introduction

This project analyses how Adams View studies tire behaviours studying tyres from FSAE. The project begins from “Basic Tyre” project developed during lecture on tire studies. ## Objectives

1. verify condition of free/pure rolling of the tire
2. apply as input slip ratio and slip angles to the tire and verify that outputs are coherent comparing with self-developed MATLAB scripts and with Tire Fitting Tool of Adams Car
3. compare results with Adams Car and self-developed MATLAB script
4. verify limit of frequency of motor applied on the tire

# The modelling problem

Starting from the model “Basic Tyre” some changes are applied in order to control a better axis of rotation, inspired by the MTS machine, a common system machine used to test tire in indoor conditions.

## Tire

Starting from the tire, a tire part has been adding with the Tire tool of Adams View. First of all an FSAE .tir file is chosen

Following set-up is performed

• Right tire according to file .tir
• Mass set in order to have Fz=800 N
• Inertia arbitrary (according with thin disk thin disk Izz=Iyy+Ixx)
• Location at the origin, z > unloaded radius. This permit to use this model for different tires having different radius
• Orientation: normal to the ground, parallel to x-axis (Euler angles 0,pi/2,0)

Road is chosen in order to maximize forces and torque of the tire, thus no road interaction reduction with tire is consider. For that reason, mu friction coefficient is set to 1.

Regarding the geometry, road is attached to the ground

## Support for tire

### Wheel speed cylinder

Once defined the wheel centre position, a first cylinder has been adding to the system, with a revolute joint between the wheel centre tire and the base of this part.

### Vertical part

this part helps the system to adapt itself to many different tires with different radius, thanks to a translational joint that add a DOF on the vertical axis. Obviously, this system is acceptable only if a flat road is used, because no vertical response will be performed by the system, except for the weight of the tire (set inside the tire definition)

### Horizontal part

This cylinder is connected to the vertical part with a fixed joint.

### Z_cyl_top

This component is connected to the horizontal part with a revolute joint which axis is parallel to absolute z-axis. On this joint a revolute motion is applied in order to control yaw-angle of the tire

### Machine to the ground

This part connects the system to the ground, with a fixed joint between this one and “z-cyl-top” part, and a cylindrical joint with the ground; the cylindrical joint has same coordinate in terms of x with the centre of the tire, and its axle is parallel with x-axis; this permits to apply two motors on this joint:

1. Traslational motion (Vcx_mot) that control the translational velocity (parametrize with a design variable Vcx_DV) on x-axis. This will be useful to control slip ratio of the tire on slip ratio simulations
2. Revolute motion (Roll_mot) that control the roll-angle of the tire, assuming that roll axis of the tire pass through the centre of contact patch C

# Simulations and results

## Free/Pure rolling conditions

The aim of this first simulation is verify how Adams View define slip ratio, thus conditions of free and pure rolling are replicated.

### Free rolling condition

To simulate free rolling condition, wheel speed motion is deactivated in order to guarantee the degree of freedom on y-axis of rotation, as free rolling definition says:

“free rolling condition when the tire is rolling without the application of a driving or braking torque” Instead, to guarantee the moving of the system, a velocity on translational motion “vcx_mot” is applied. The velocity is parametrized with the Design Variable “Vcx_DV” set to 5 m/s in order to avoid too much time for reaching the steady-state condition. In the image below is possible see that Grubler equations returns 2 DOF, becuse the wheel speed motor is deactivated (the other DOF is the traslational joint on z-axis)

### Simulation set-up Most important condition to impose in the simulation condition is “start at equilibrium”, this avoid bumping condition in the first few seconds of simulation. Uncheck “update graphics display” reduce a lot the time of computation, But first simulation is suggested a render to verify that the motion of the system is correct

### Free rolling condition-results

As imagine obtained from Postprocessor shows, on the top-left rolling resistance torque is plotted and is non-zero as expected, on the top-right the longitudinal force, non-zero as well; in the bottom-left the vertical force Fz to verify that the system used to apply vertical force on the system is correct; on the bottom-right the longitudinal slip ratio, near to zero but because of low-speed, this values is considered reasonable compared also with Fx obtained.

A fast check of Fx(k) is done via MATLAB, calling the function “MF52_long_combined.m”, a fuction containg the Pacejka 2002 formulas that has the following input parameter:

• Alpha, set to zero because of pure longitudinal condition
• K, slip ratio input, set equal to slip ratio measured on Adams View simulation
• Fz, vertical force, set equal to input choose in the simulation
• Gamma, inclination angle, set to zero according with simulation condition
• tireData, is a struct file containing the parametres of .tir file

the output is according with Adams simulation, see result below

### Pure rolling condition

To simulate pure rolling condition, wheel speed motion is activated in order to apply the torque necessary on the tire, keeping activated the translational motion: The angular velocity applied on wheel motor is obtained from the formula of slip ratio: Imposing zero the slip ratio, according with definition of pure rolling

pure rolling condition when a driving torque is applied to the wheel that overcomes the rolling resistance moment

For that reason, on the function builder of the wheel speed motion the following formula is applied:

`-Vcx_DV/rolling.r`

Where

• Vcx_DV is the DV of velocity applied on the translational motion
• r is a function measure time dependent obtain from the function measure of the tire. This passage is necessary because only function measure time dependent can be usable in the function builder

### Pure rolling condition-results

The result is overlapped with free rolling condition (red=pure rolling, blue=free rolling), in order to verify the nullify of longitudinal force and slip ratio. As expected, Fx and slip ratio both tend to zero (respectively 10-9 and 10-14, assume equivalent to zero). Regarding Rolling resistance torque, it shows a non-zero result. The reason of that result is inside the Pacejka formulas: Assuming neglectable the parameter QSY3 and QSY4, is clear that QSY1 give a My non-zero also when no longitudinal force is applied on the tire. In order to verify that another simulation is performed setting to zero the QSY1: As expected, the rolling moment now tend to zero.

## Slip Ratio

Second part of the project consists on import a slip ratio input (.txt file) and reply that valueas in the system.

### Create file

A small MATLAB script is created to generate txt files. Is important to set a number of values equal to the number of the steps in order to avoid errors on the fitting of the import (a CUBSPL will be used to fit the spline and use it in the function definition of the wheel speed motor). This txt files contain also a column with the time indication (independent variable) that will be set during the importation in Adams, for that reason is important impose in the .txt files the same time of the duration of the simulation. ### Import

As shown in the figures below, two different import has done: the first one imports the data as a spline, to use it to define the function of the wheel speed motor; the latter is imported as measure to overlap it in the postprocessor and verify the quality with the measure of the slip ratio simulated ### Slip Ratio – results

As shown in the figure, the simulation overlaps perfectly the slip ratio vector

## Slip Angle

Similarly to the slip ratio simulation, the slip angle simulation verify the input. Differently from previous case, the yaw_angle_mot will be used. In particular the spline input will be used as it is because this motor is displacement-control so the angle input given by the spline is the angle to give as input. The wheel speed motor will be deactivated to simulate the free rolling condition.

### Slip Angle – results

As shown in the figure, the simulation overlaps perfectly the slip angle vector

## Comparison with MATLAB and Adams Car

### Slip Ratio

First comparison is on slip ratio and longitudinal forces. To compare the result, an Adams View is initially performed; the slip ratio output of the simulation is used as input in Adams Car and MATLAB script.

To compare results with MATLAB, a function has been used to evaluate forces on tire according to Magic Formula 5.2 (developed during Formula SAE project). In this function, the Pacejka formulas are implemented, so giving an input of slip ratio, inclination angle, vertical force, slip angle, an output of forces is returned. #### Slip Ratio – Results As shown in the table, MATLAB results are coherent with Adams View results. On the contrary, Adams Car give an output of Fx hugely different from the others. That’s why the file .tir used is based on MF52 (Magic Formula 5.2) and Adams Car, that use pac2002, need to fit the file .tir in order to extrapolate parameters according to pac2002.

In order to solve this incongruence with the other results, the value “PROPERTY_FILE_FORMAT” of the file .tir, from “MF_05” has changed to “PAC2002”; in this way no fitting has done in Adams Car. This solution is possible because the pac2002 version and MF 5.2 are quite similar, on pure longitudinal and pure lateral.

The last row of the table represents the Adams Car result with the modify inside the file .tir. As a result, the longitudinal force is not identical to the other result, but is comparable ( 5% of difference with other results ).

### Slip Angle Similarly to slip ratio result, Adams View, Car and MATLAB are comparable between them. In particular in this case Adams Car results and MATLAB are identical, instead the Adams view is a bit different. That’s why during simulation, the slip ratio is non-zero. Different simulation has been performed, to check the difference in term of lateral force (consider that combined condition of slip ratio and slip angle change the value of lateral/longitudinal force) The conclusion of this analysis is that all cases doesn’t return a pure lateral condition. The only way to obtain slip ratio equal to zero is consider that the tire when yaw-angle is non-zero the Vcx of the tire is not the velocity set on the system, but that velocity multiply for the cosine of the slip-angle, so the function of the wheel speed motor must be correct multiplying for the cosine of the slip_angle.

## Verify limit of frequency

Last study of this project focus on define the limit of frequency of motions. To study that, the input signal of slip ratio used on wheel speed motor changes the frequency, setting the number of period of slip ratio in MATLAB script that create the vector to import in Adams. It’s important verifying that the shape of the input signal is good (sine function is given); if not, the number of steps must be increased till the shape of input is acceptable. Many simulations are performed increasing the frequency of the signal:

First simulation: frequency=33.3Hz Second simulation: frequency=100Hz Third simulation: frequency=200Hz Fourth simulation: frequency=400Hz Fifth simulation: frequency=600Hz As the image shown demonstrate, the motor follows perfectly the input signal. It’s not possible increase the frequency of the signal because is necessary increase the number of steps (to guarantee the correct shape of the input) and that request too many PC resources, but the max frequency used is consider a good value for this type of simulations.

# Conclusions

As expected the results obtained is in according with the Theory of Pacejka models. Further analysis could take in account combined conditions of slip angle slip ratio and inclination angle; another interesting study could be on test on different roads, to analyse how forces change when non-flat roads are used.

# References

 H.B. Pacejka, Tire and vehicle dynamics, 3rd ed, Butterworth-Heinemann, 2012

 D.J.N. Limebeer & M. Massaro, Dynamics and optimal control of road vehicles, Oxford University Press, 2018