The research and development in tires is gaining importance these days. Tires play pivotal role in defining the driving dynamics of the car, ride and comfort, fuel economy etc. Tires are the interface between the car and the road and play a significance role in performance of any road going vehicle. Numerous methods have been proposed over the years to understand the mechanism of the tire, the forces generated by the tire and how varying certain parameters affect the performance of the tire.
With the development in FEM simulation, tires are understood in a broader way. Before selecting a particular method of tire modelling, one must give an emphasis on the type of application the vehicle simulation will address. To put this in picture, to study about suspension loading durability, importance must be given to the contact forces generated at the tire-road interface must be determined when the tire strikes an obstacle such as a pothole or a bump (Blundell & Harty, 2015).
To have more insight in to ride and vibration, the model must capture the transmission of forces from the road surface where the inputs are small but has high frequency (Blundell & Harty, 2015). To study the vehicle handling, especially manoeuvring on a flat track, it is necessary to establish the forces and moments occurring at the tire-road interface and resolve these forces to the wheel centre and then in to the vehicle. The Magic Formula tire model developed by Pacejka, (2007) is a semi-empirical tire model, meaning the equations are not physically derived, rather these are mathematical approximations of the curves that were recorded in a test bench. Different versions of Magic Formula has been proposed over the years which addressed different drawbacks of its previous versions. However the Magic Formula tire model is limited to quasi steady state conditions only i.e., pure braking or pure cornering or a combination of these two and not suitable for non-steady state conditions. Also, it does not capture the thermal effects, tire wear, durability and off road usage (Schmid, 2011). The FEM tire model CDTire developed by Fraunhofer ITWM is a software which caters to different applications. In this software one can model the different layers of the tire to capture its behaviour accurately. The drawback of this is the high computation time. To get accurate results, one must look in to the application of the tire model and based on the application, a suitable tire model must be selected for analysis.
Axis system definition
To start with tire modelling, an axis system has to be defined. The figure below shows the axis system of SAE and ISO. The axis system used in Magic Formula model is adapted from SAE with few changes.
018542000SAEAdapted SAE ISO Adapted ISO
Figure 1: Tire axis systems. Source: (Pacejka, 2007)
Where, V = Speed of the wheel centre
?= Side slip angle
?= Camber angle
?= Turn slip velocity
Fx & Fy are Longitudinal and Lateral force respectively
Fz= Normal load
Mz= Self aligning torque
Magic Formula Tire Model
The Magic Formula tire model is one of the commonly used modelling techniques in the automobile industry. This modelling method is semi empirical, meaning the mathematical expressions are an approximation of the experimental data that is obtained on a test bench. The method was developed by Pacejka, (2007). Several versions of this formula has been published over the years for e.g., the model is further extended for combined slip conditions by Pacejka & Bakker, (1992). The magic formula is undergoing continuous development since then. The basis for the formula is the steady state testing of the tire. Tires are tested on a test bench under steady state conditions and tire force and moment curves are obtained as shown in figure 1. The curve looks like a sine function, modified by an arctangent function that tends to a horizontal asymptote.
Figure 2: A graph showing Lateral force measured at different loads. The loads and the slip angle are normalized so that it gets reduced to the same curve
Source: Tyre and vehicle dynamics by Hans B Pacejka pg 157
The general formula for the magic formula is
y=Dsin?[CarctanBx-EBx-arctanBx]Where, B= Stiffness factor
C= Shape factor
D= Peak factor
E= Curvature factor
The magic formula produces a curve which passes through the origin and then leads to a horizontal asymptote. The factors B, C, D and E are factors which has no physical meaning and is used just to fit the curve to the test data. When a tire is manufactured, due to the manufacturing defects, plysteer and conicity can arise. Due to this, there will be an offset of the data points from the origin. To take account of this, there are equations which makes the magic formula curve fit to have an offset from the origin. The equations are:
YX=yx+SVx = X + SHwhere, Y = Output variable, Fx, Fy or MzX = input variable, tan? or ?
Figure 2 shows the side force characteristics, which illustrates the significance of the coefficients used in the formula.
Figure 3: Plot of the magic formula, indicating the significance of the coefficients
Source: Tyre and vehicle dynamics by Hans B Pacejka pg 173
This simple version of Magic formula is applicable for pure slip conditions i.e., pure braking/acceleration or pure cornering. The complex case of combined slip has been addressed in the work of Pacejka & Bakker, (1992). To analyse the combined slip conditions, new slip quantities have been introduced. The complete list of formulas are listed in the appendix section of Pacejka & Bakker, (1992). Although this version addressed the combined slip, there are other parameters which cannot be analysed. The dynamic conditions are not captured, for eg., changes in inflation pressure of the tire or the wear of the tire. The major disadvantage of the Magic Formula tire model is that it cannot capture transient behaviour i.e., time dependant changes. But with other approaches along with magic formula it can be done and these are discussed further.
To address the transient behaviour, PAC2002 version was developed. The low frequency behaviour (up to 15 Hz) is called the transient behaviour. To do this, there are two methods; Stretched string and Contact Mass approach MSC Software. In stretched string approach, the tire belt is modelled as stretched string. The tire belt is supported to the rim with lateral and longitudinal springs as shown in figure 3
Figure 4: Stretched string model
Source: MSC Software
The unique feature in the string model is the separate model for the carcass and the contact patch. The lateral deflection v1 of the string is calculated in terms of the slip angle and relaxation length. With the similar approach, the deflection in the longitudinal direction is also formulated. In both the cases, the relaxation length are defined in terms of vertical load. Using the relaxation length and deflections, the practical slip quantities ? and ? are defined as
?’=u1?x.sinVx ?’=atan?(v1??)Using these practical slip quantities, magic formula equations can be used to calculate the forces and moments using the relation:
Fx=Fx(?’, ?’, Fz)Fy=Fy(?’, ?’, ?, Fz)Mz=Mz(?’, ?’, ?, Fz)M’z=M’z(?’, ?’,Fz)For more details regarding the equations, the reader is advised to refer MSC Software.
In the contact mass model, the tire carcass and the contact patch are modelled separately. In the previous method, to describe the compliance effects of the carcass, relaxation length was used. In this method, the carcass spring is modelled separately. The contact patch is given inertia effects to ensure computational causality (MSC Software). At higher values of slip, this modelling method compensates for the lagged response to slip and load changes. The contact patch can deflect both in longitudinal, lateral and yaw directions with respect to the wheel rim. A set of differential equations are obtained that governs the dynamics of the contact patch of the tire. Equations for the longitudinal, lateral and yaw deflections are determined. The transient slip equations for side slip, turn slip and camber is determined, with this the composite turn slip quantities are calculated. Finally the tire forces are calculated using the turn slip quantities. For more details about the equations, the reader is advised to refer MSC Software (pg 37-39).
The swift model stands for Short Wavelength Intermediate Frequency Tire model. The previous model discussed above i.e., the string model had to be restricted to lower magnitude to make the theory linear. Several approximations were also introduced in Pacejka, chapter 8 (2007), which enabled a simple non-linear extension, but even those approximations limited the model for longer wavelengths (approx. >1.5m). To address the situation in shorter wavelengths (>20 cm) and higher frequencies, the swift model can be used. In the first version of the SWIFT, the development of the model was more towards the responses to variations of longitudinal and side slip. Furthermore, the possibility to analyse the behaviour of the tyre encountering cleats (distinct road irregularities) was also introduced in the model. Further developments of this model include the addition of camber variation and turn slip. The figure below shows the representation of the SWIFT model
Figure 5: Representation of the SWIFT model
Source: Schmeitz et al., (2005)
The five main aspects of the SWIFT model are:
A 6 degree of freedom elastically suspended ring which represents the belt and the sidewalls, also takes in to account the mass and inertia factors. The ring is considered rigid because of the restriction of the frequency to 60 Hz
Introduction of residual stiffness and damping between the rigid ring and the contact patch to ensure total static tyre stiffness in longitudinal, lateral and yaw directions. Also the total tyre compliance is made up of the carcass compliance, residual compliance and the tread compliance
The contact patch model is of brush type and features horizontal tread element compliance and partial sliding. Based on this model, the finite length and width of the foot print are included
To simulate the tyre moving over an uneven road surface, three inputs for the road profile are used: the height of the road plane, the slope of the road plane and the effective rolling radius (which is a result of effective forward road curvature)
The steady state Magic Formula model to describe the nonlinear slip force and moment properties
Fiala Tyre model
The fiala tyre model is one of the methods earlier developed by E. Fiala. This software is well known amongst MSC.ADAMS users. ADAMS stands for Automatic Dynamic Analysis of Mechanical System. The Fiala model is a simple model because it requires only ten parameters. The parameters can be easily obtained and relates to the physical parameters of the tyre. The simplicity of the model makes way for some disadvantages and they are listed below (Blundell & Harty, 2015):
The model cannot be used to analyse the combined slip conditions
The effect of the camber angle on lateral force and aligning moment cannot be modelled
The variation in cornering stiffness at zero slip angle with tyre load is not considered
The conicity and plysteer effects on lateral force and aligning moment is not considered
The list of parameters for the Fiala Tyre model are (Blundell & Harty, 2015):
R1 The unloaded tyre radius
R2 – The tyre carcass radius
Kz Tyre radial stiffness
Cs Longitudinal tyre stiffness
C?- Lateral tyre stiffness due to slip angle
C?- Lateral tyre stiffness due to camber angle
Cr- Rolling resistance moment coefficient
? Radial damping ratio. The ratio of tyre damping to critical damping
?0- Tyre road coefficient of static friction
?1- Tyre road coefficient of sliding friction
The common parameters from the above mentioned parameters, that are used in all the tyre models are R1, R2, Kz and ?, which are used to calculate the vertical load in the tyre. As discussed above in the disadvantages about Fiala tyre model not considering the influence of camber angle in the analysis, the parameter C? is not used. The resultant coefficient of friction ? is calculated as a function of comprehensive slip ratio SL? as shown in figure 6. The comprehensive slip ratio SL? is the resultant of the longitudinal slip coefficient S and the lateral coefficient S?.
SL?= S2+ S?2center381000
Figure 6: Linear tyre to road friction model
Source: Blundell & Harty, 2015
Using the SL?, the instantaneous value of tyre to road friction coefficient ? can be obtained by linear interpolation. Using the value of ?, we can now calculate the critical value of longitudinal slip ratio S*, above which the tyre starts to slide.
S* = |?Fz2Cs|If |S| is less than S*, then the tyre is said to be in elastic deformation, then in that case the longitudinal force is
Fx= -CsS If |S| is greater than S*, then the tyre is said to be sliding and then in that case the longitudinal force is given by
In the lateral case, the critical slip angle ?* is calculated using
If |?| is less than ?* then the tyre is said to be in elastic deformation and in that case,
and the lateral force is Fy=-?Fz1-H3sgn(?).
If |?| is greater than ?* then the tyre is considered to be sliding and in that case the lateral force
Source: Blundell & Harty, 2015
Similarly the equations for the aligning moment are calculated and can be referred in Blundell & Harty, 2015. As one can see that Fiala tyre model uses very few parameters to calculate the forces and moments and it is relatively simple compared to other models. But the simplicity comes with disadvantages as discussed above.
CDTireCDTire is a FEM/physical tire modelling approach as opposed to the semi empirical approach of Magic Formula, developed recently by Fraunhofer ITWM. With the help of tire construction assistant, different layers of tires can be physically modelled with different properties for each layer. The software has different physical models for sidewall, belt and tread for different applications. CDTire can capture dynamic conditions which can help in studying for e.g., changes in tire inflation pressure, changes in the road terrain etc. CDTire gives emphasis on tire dynamics and captures the vibrations in both amplitude and frequency by the interaction with 3D road surfaces. CDTire calculates the spindle forces and moments acting on each wheel in the model and the contact forces developed between road-tire interface during multibody simulation. CDTire can be used for many applications, which include:
To study ride and comfort
To study handling of car on flat and 3D roads
Influence of tire pressure changes on forces developed by the tire
Realtime application like SIL/HIL/MIL
To study active safety system such as ABS, ESP
To study interaction of tire with flexible wheel rim
Harshness analysis when a tire runs over an artificially created obstacle for e.g. cleat
CDTire is a collection of different physical models for different applications. The documentation released by Fraunhofer ITWM gives an insight in to different models:
3D shell based model of sidewall and belt
Separate modelling and parameterization of all functional layers of a modern tire
Includes models for belt, carcass, plies and thread
Brush type contact model
Can handle variations in inflation pressure up to total pressure loss
Accurate in frequency range of up to 150 Hz
Application include from ride comfort to durability analysis
Model to predict to predict the temperature creation and propagation in a tire
Finite volume based description
Auto meshing functionality
Easy to parameterize
Can be coupled with CDTire/3D, CDTire/MF++ and CDTire/Realtime
Temperature enhanced Magic Formula for use with CDTire/Thermal in advanced handling applications
Based on Magic Formula 5.2
Empirical model to predict contact patch shape under various driving conditions
Can be used for modal analysis and imported in to NVH tools
Discrete local contact area excitation
Software toolbox to derive a linear model from CDTire/3D for a rolling tire
Software tool for parameter identification
Standard tire measurements and formats
Automatic execution of test rig simulation scenarios
Cross section construction assistant
To capture real environments, the surface of the road must be captured so that the simulation can be accurate. To do this, CDTire has road models which gives surface positions and more importantly, the friction coefficient of the road surface. The road models can be arbitrarily driven in both translational and rotational directions for more accurate test rig applications. The computation time of CDTire is quite high compared to other methods. FEA approach is used to understand about crash analysis and a Flexible belt Brush-type contact model is used for durability and ride/comfort analysis. A rigid ring empirical model is used for active safety and ride comfort analysis (B?cker & Gallrein, 2012).
The tyre modelling methods discussed above are some of the well-known methods in use. The tyre modelling method depends on the application for which it is used. A semi empirical method like Magic Formula method will give accurate results for steady state analysis and also it is relatively simple to implement. The model gives good results for analysis in flat road conditions. But Magic Formula alone has certain limitations such as not capturing certain dynamic changes such as inflation pressure changes, tire wear and also the model is not suitable for analysis in flexible terrain (off road applications), but adaptation of Magic Formula with certain physical modelling approach can enable dynamic analysis. Two approaches that were discussed in this paper was the SWIFT model and the PAC2002 which captured the transient behaviour of the tyre. Different layers of tire can be modelled such as the tire carcass, belt and sidewall which can capture the dynamics of the tire accurately. Several physical methods have been introduced over the years for describing the behaviour of each layer. One such method that is discussed in this paper is the string model and its adaptation with Magic Formula for forces and moments calculations. CDTire is a physical tire modelling method with different models for different layer of tyre. The vast application of CDTire allows the user to cater for different scenarios such as off road usage, NVH analysis, temperature changes and real time analysis. The computation effort required is high but the vast application range shadows this drawback.