Appendix 4 | The Tire Force Models


Introduction

Virtual CRASH 3 comes with three tire force models which the user can choose from. In this write-up we will describe how to select a tire force model, as well as describe the various user input parameters that can be specified.

 

Tire Force Components

The total force on each tire can be decomposed into three components, where each force vector is projected onto the corresponding tire’s reference frame \((\hat{x}^{\prime\prime}, \hat{y}^{\prime\prime}, \hat{z}^{\prime\prime} )\) [1]. The longitudinal axis is given by \(\hat{x}^{\prime\prime}\), whereas the lateral axis is given by \(\hat{y}^{\prime\prime}\). The tire normal force along the \(\hat{z}^{\prime\prime}\) is orthogonal to the road surface at the idealized contact patch.

Tire Force Parameters

The longitudinal tire force is generally defined as a function of various parameters, such as normal force, longitudinal slip, and lateral slip angle. The longitudinal slip can be calculated by using the \(\hat{x}^{\prime\prime}\) projection of the velocity vector component at the wheel center, \(v_{x^{\prime\prime}}\), the wheel’s angular velocity, \( \omega \), and its radius \(r\). During acceleration, the longitudinal slip can be defined by [2]:

$$ S_{Acceleration} = {v_{x^{\prime\prime}}-r \cdot \omega \over r \cdot \omega } \tag{97} $$

and for braking: 

$$ S_{Braking} = {v_{x^{\prime\prime}}-r \cdot \omega \over v_{x^{\prime\prime}} } \tag{98} $$

The lateral tire force is primarily a function of the slip angle, \(\alpha\). This is given by the angle between the velocity vector at the wheel center and \(\hat{x}^{\prime\prime}\):

$$ \alpha = \tan^{-1} \left( {v_{y^{\prime\prime}} \over v_{x^{\prime\prime}}} \right)  \tag{99} $$

Selecting a Model

With your vehicle selected, go to the “axles” menu in the left-side control panel. You can simultaneously select all wheels by using ctrl+left-click to select axle 1 and axle 2. To ensure both right and left tires receive the same mode parameters, make sure that the box next to “symmetry left-right” is checked (see next figure).

Left-click on the box next to “tire model-left” to reveal the tire model pull down menu. Finally, select either (1) constant, (2) linear, or (3) tmeasy [3].

The Linear and Constant Models 

Longitudinal Force

By default, all vehicles in Virtual CRASH 3 are assigned the “constant” tire force model. In Chapter 8 | Vehicle Controls, we show how the longitudinal tire force normalized to normal force is expressed as a fraction of the maximum drag factor at the contact patch \(f_{x^{\prime\prime},j}\). The user can either define the total acceleration or deceleration on the vehicle, from which the \( f_{x^{\prime\prime},j}\) values are derived, or the values of \( f_{x^{\prime\prime},j}\) can be specified by the user explicitly. For the rest of this treatment, we will drop the double primed notation, as it is understood that the tire forces are evaluated in the local tire reference frame.

Lateral Force

The lateral force is defined as a function of slip angle. The characteristic steep linear turn-on leading to plateau shape of the lateral force as a function of slip angle has been the subject of immense research. The Virtual CRASH 3 linear tire force simulation models the shape of the lateral force turn-on curve as a piece-wise continuous function of two parts [4] (see next figure).

First, is the steep turn on line, which extends from 0 to the maximum value of 1.0 at the saturation slip angle value \(\alpha_{max}\). In the region \( \alpha \gt \alpha_{max} \) the tire transitions from the (linear) elastic region to the (sliding) frictional region of the lateral force curve, where the transition region is modeled as vanishingly narrow. This is described by equation (100).

$$ { \tilde{F}_y \over \mu F_z } = f_y = \big\{ 1, \alpha \gt \alpha_{max} ; -{\alpha \over \alpha_{max}}, \alpha \leq \alpha_{max} \big\} \tag{100} $$

The cornering stiffness is obtained by the slope of the initial turn-on, or

$$ C_{\alpha} = {\mu F_{z} \over \alpha_{max}}  $$

Note, in the condition \( \mu  \rightarrow \mu^{\prime} \lt 1.0 \), the value of \( \alpha_{max} \) is adjusted to \( \alpha_{max} \rightarrow \alpha_{max} \cdot {\mu^{\prime} \over 1.0} \) to account for the fact that a lower slip angle is needed to cause saturation in such a case. 

Combined Forces

Because the total tire force at the contact patch should not exceed the maximum limit \( \mu \cdot F_{z} \), the combined lateral and longitudinal forces are limited by the upper bound friction circle condition:

$$ {\sqrt{\tilde{F}_x^2+\tilde{F}_y^2} \over \mu F_z} = \sqrt{f_x^2 + f_y^2} \leq 1 \tag{101} $$

Braking Only

The next figure below illustrates a tire force diagram for a braking only scenario. Here we see the fraction of maximum possible friction is less than 0 indicating a braking force, and is bounded by \( f_{x} \geq -1 \).

Braking in the Elastic Region

For a tire undergoing braking, given by \( f_{x} \), and steering with slip angle in the elastic region (the linear turn on region), a new elastic limit upper bound is defined and given by: 

$$ \tilde{\alpha}_{max} = \alpha_{max}\cdot\sqrt{1-f_x^2}. $$

In this situation, the linear tire force simulation models the lateral tire force by:

$$ {\tilde{F}_y \over \mu F_z} = f_y =- {\alpha \over \alpha_{max}}, \alpha \leq \tilde{\alpha}_{max} \tag{102} $$

This case is illustrated in the next figure below:

Unlocked Wheel Braking in the Frictional Region

For a tire undergoing braking without wheel lock, given by condition \( |f_{x}| \lt |\cos\alpha| \), and steering with slip angle in the frictional (plateau) region, \( \alpha \gt \tilde{\alpha}_{max} \), the magnitude of the lateral tire force is bounded from above such that: 

$$ { \tilde{F}_y \over \mu F_z } = f_y = -sign(\alpha) \cdot \sqrt{1-f_x^2} \tag{103} $$

This case is illustrated in the next figure below:

Locked Wheel Braking in the Frictional Region

For a tire undergoing braking with wheel lock, given by condition \( |f_{x}| \gt |\cos\alpha| \), and steering with slip angle in the frictional region, \( \alpha \gt \tilde{\alpha}_{max} \), the magnitude of the lateral tire force is bounded from above such that: 

$$ { \tilde{F}_y \over \mu F_z } = f_y = -\cos\alpha \tag{104} $$

The magnitude of the longitudinal tire force is also bounded such that:

$$\left| { \tilde{F}_x \over \mu F_z }\right| = |f_x| = |\sin\alpha| \tag{105} $$

This case is illustrated in the next figure below:

Acceleration

Perfect symmetry is assumed in the tire behavior between acceleration and braking, thus the linear tire model works the exact same way for an accelerating wheel as described above, where \( f_{x} \gt 0 \).

 

Modifying the Linear Tire Force Model

By selecting the linear model from the pull-down menu, the linear force model window appears (see next figure).

The linear model has only a single adjustable parameter: “max slip angle.” This is given by \( \alpha_{max} \) in equation (100). This value is defaulted to 10 degrees, a good approximation for the transition between linear to elastic region in lateral force curves observed in experiments across many tires. \( \alpha_{max} \) can be adjusted either using the left panel, or interactively with the control grip.

The Constant Tire Force Model

By default, Virtual CRASH 3 uses the constant tire force model. This is identical to the linear model, but has no user adjustable parameters. In this model, we have \( \alpha_{max} = \) 10 degrees.

 

The tmeasy Model

The tmeasy model is a semi-empirical tire force model that was developed to more accurately model the behavior of tires undergoing simultaneous steering and braking/acceleration maneuvers, but to do so in a way that is practical to implement in a simulation environment [5]. Full and thorough treatments of the tmeasy model can be found in many articles which can be obtained online [6]. Below, we explain how to access and modify the tmeasy tire parameters.

Select tmeasy

Once the tmeasy model is selected from the pull-down menu, the tire parameter window show appear.

You’ll notice interactive control grips which allow you to change the tmeasy model parameters on the fly; you can also use the data fields in the left side panel.

The tmeasy Parameters

The tmeasy model contains 10 adjustable parameters. For convenience, the various parameters and variable names used by Rill et al are shown in the next figure below.

Note the \( F \) and \( dF \) parameters are normalized by the factor \( {1 \over \mu\cdot F_{z}}\), and are therefore dimensionless [7].


Notes:

[1] The Virtual CRASH reference frames are described in Appendix 3 | The Virtual CRASH Coordinate System.

[2] This is one of many possible definitions for longitudinal slip.

[3] Both the linear and tmeasy models are reviewed in SAE 2009-01-0102.

[4] This linear approximation approach has been utilized by other vehicle dynamics simulators as well. See PC-CRASH Technical Manual Version 9.0 for example.

[5] The tmeasy model has also been successfully used by vehicle dynamics simulation packages such as PC-CRASH. See PC-CRASH Technical Manual Version 9.0.

[6] It is highly recommended that the user read “User-Appropriate Tyre-Modelling for Vehicle Dynamics in Standard and Limit Situations” by Hirschberg, Rill, and Weinfurther, Vehicle Systems Dynamics, 2002, Vol. 38, No. 2, pp 103-125. This can be found online at: https://online.tugraz.at/tug_online/voe_main2.getVollText?pDocumentNr=137855&pCurrPk=30050

[7] Note some versions of Virtual CRASH may display units of Newtons for max force and full sliding force parameters, as well as on the vertical axis of the force versus slip graph. These are to be ignored as the parameters are dimensionless. 



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