Paolo Pinto

Aeronautical Engineer and Computer Programmer

# The Pacejka Equation

Since the ‘70s Dr. Pacejka has been developing several tyre behaviour models which led to the “Magic Formula” : a function simulating with relative simplicity and good approximation the main tyre characteristics.

The Magic Formula is a transcendent function :

Y(x) = D sen(C arctan(Bx – E [ Bx – arctan(Bx) ] )

where  B,C,D,E are coefficients. relevan

The x, y variables can be associated from time to time to different parameters ; for example :

x = slip angle   ,  y = Fy             if studying the tyre’s ability to provide centripetal force

x = slip ratio    ,  y = Fx             for the tractive force

It is also possibile to take into account  camber, ply steer and conicity, with slight modifications..

The function actually has the near-magic property of being useful for simulating many different tyre phenomena just by changing the coefficient and the meanings of   x, y.

# Function study

The Y(x) function is anti-symmetric ; it always goes through the axis’ origin and it always has there a null second derivative.

In tyre models  y’(0) >0 is always desiderable ; it is possibile to demonstrate that this implies B,C being of the same sign.

For the  same reason it must always be  y’’’(0) < 0 ; this implies

E > -(1 + C2/2)

The curve always shows an horizontal asymptote for x tending to infinity.

The asymptote’s value is :

D sen(Cp/2)                               if E <1

D sen (C arctan (p/2))                if E =1

-D sen (Cp/2)                             if  E >1

As a consequence, the need to have y(x)>0 for x tending to infinity leads to coefficients :

E<1  ; 0 <C <2

It must be said that several other different couples of values for  E, C could suit this need.

## The B parameter

B is called Stiffness Factor

It controls the slope of the curve at the origin ; in practical models it must always be B>0

It must be pointed out that B also exerts a strong influence on the relative minimum and maximum position

## The C parameter

C is called Shape Factor

The possible presence of a relative maximum (a “peak” in NON mathematically correct terms) on the right of the zero depends on C being >1 (provided the previously stated conditions are met) .

This maximum happens at  am ; a position which may be obtained from :

B(1-E) am + E arctan(Bam) = tan (p/(2C))                    (*)

The bigger the value of C, the more pronounced the maximum.

Typical values for C are 1.3 for the centripetal force simulation and 1.6 for the tractive force simulation

## The D parameter

D is also called Peak Value

It constitutes a superior limit to the function’s values, since the factor : .

sen(C arctan(Bx – E [ Bx – arctan(Bx) ] )

cannot obviously exceed 1 .

## The E parameter

E is also called Curvature Factor; it is usually set at a value less than zero. As the absolute value of  E goes down the curve assumes a flatter shape

### PHISICAL APPLICATION

Not many data are available in technical literature about the real tyres’ characteristics, expecially about race tyres.

Although this is understandable for the latest developements, this desolating scarcity is the same even for 20 years old designs.

When data are available they are often only partial : for example in  the Lateral Force vs Slip Angle plots stop at a = 6°,  and only 4 different values of vertical load are considered.

This makes the Magic formula useful to those wanting to create a race vehicle simulation., since it allows to extend the tyre behaviour to the whole range of slip angles and vertical loads, once that the parameters are set to match a limited quantity of experimental data.

Obviously it will be impossible to have a perfect match ; one must also consider that all of the analytical formulas (and many numeric solutions too) used in Engineering are useful only to give a first approximation, to the Experimental Engineers delight.

Another application of the curves is the simulation of tyre for which no data is available at all, by making use of the ones of similar tyres.

For example, if wanting to simulate a tyre similar to a given one, but  a softer compound, it will often be enough to raise the value of D.

This because (AS A RULE OF THUMB…) , B,C,E depend on the carcass’ construction, while D depends on the compound.

If  it is desired a tyre able to give the pilot a very clear warning of the limit’s approaching (as in videgames, where a lack of the acceleration input ought to be compensated by an enhanced visual input), the parameters of choice are C and E, since they rule on the adherence’s fall after the maximum.

A wider (and hence stiffer) tyre can be simulated by an increase of  B, which will increase the initial slope. In this case it will also be useful a decrease in C, to diminish the BCD parameter on which the peak adherence slip angle depends.

The Pacejka curves are here used to plot the various a,m diagrams , i.e.  slip angle a versus  adherence coefficient  m, at varying vertical  loads  FZ

It must be said that this kind of approach is not very common in literature, since usually the centripetal force is plotted, instead of the adherence coefficient.

The two values are anyway linked by a proportionality factor, thereby legitimating this change.

It won’t be unuseful to remember that “adherence factor” is not an exceedingly formally correct term , since it would be more orthodox to refer to a “normalized force” Fy/Fz

The easy understanding of  m will anyway led to its use in these pages.

#### D coefficient variability under vertical load

In starting the calculations for a complete tyre, the first value to be fixed should be  D, wich takes into account the adherence factor.

For a Formula One tyre, a  value of 1.8 under a vertical load of 600 kg will be in the ballpark

It is very important to make D dependent on the vertical load Fz , since the maximum adherence value depends on it.

A good formula, presented (in different ways)  in  and  ,  will be  :

m = a1 Fz + a2                              (**)

where a1<0 ; a2>0

It ought to be pointed out that the nonetheless interesting paper   suggests (Chapter 24) positive values for both these coefficients, a statement not congruent with experimental data.

As an alternative it can be used the formula shown in   :

m = m0 / (1+mtFz)                            (***)

Of course the right coefficient must be found; referring to the experimental values given (in a different form) for a Goodyear front F1 tyre in , after a brief comparative analysis  it is observed that (**) provides a better match, at least for this kind of tyre.

Plausibile values for the coefficients are :

a1 = -00138

a2 = 1.988

where  FZ is measured in Kg

To all practical effects D is the adherence factor under a zero vertical load; the ample variability of D under load (it can easily vary between 1.8 and 1.2) helps understanding how far are the tyre’s adherence mechanics from the simple Couloumbian model

#### Cornering stiffness variability under load

Cornering stiffness is normally meant as dFy/da ; in this paper it will be meant as dm/da .

The value dFy/da  increases with the vertical load; yet dm/da decreases with Fz.

Slip angle stiffness depends on thje vertical  load by the formula  :

BCD = a3 sen (2 arctan(FZ/a4))

By all practical means this leads to intervene on  B and C , since  D is chosen on different grounds.

Yet it is important that the value of D be the one appropriate for the given load .

Unfortunately, it is not possible to use equation (*) to close the algebraic system, since this would introduce the new variable E.

The best approach would of course be a numerical one, which finds the best B,C,E triplet ; yet many numerical solution benefit from reasonably accurate starting values.

Now, some hints are given for C, at least : as previously stated its value ought to be comprised beetween 0 and 2 ; Ref.  quotes   1.3  as a good value (although it is not stated which kind of tyre was examined to determine this)

So it will be, at least for a first evaluation

C = 1.3

The previous formula can therefore be rewritten as :

B = (a3 sen (2 arctan(FZ/a4))) / (C (a1 Fz + a2))

The values to research are  a3 and a4 , since all other parameters are set.

Once again it is possible to refer to the experimental values

The a4 parameter controls the diagram bending , while  a3 is an intensity term.

Reasonable starting values can be :

a3 = 1.37

a4 = 120

where vertical load is expressed in Kg.

The maximum’s position

As a check it is possibile to observe what happens to am when the load increases ; it should increase too.

Let’s remember its value is given by :

B(1-E) am + E arctan(Bam) = tan (p/(2C))                    (*)

A PRACTICAL EXAMPLE

Here is a plot of Pacejka’s curves for a tyre of the following characteristics , under different vertical loads:

 C 1,35 E -0,4 a1 -0,00138 a2 1,988 a3 1,37 a4 120

##### Adherence coefficient vs Slip Angle for a front F1 tyre ##### Glossary

Fx : force acting along the road in a direction orthogonal to the wheel axis

Fy : force acting along the road  in the wheel’s axis’ direction

FZ : Vertical load on the tyre

am = slip angle of maximum adherence

a = slip angle

Bibliografia :

 M. Guiggiani : Dinamica del veicolo ; Città Studi Edizioni

 M. Milliken, D.Milliken : Race Car Dynamics ; SAE

 Brian Beckman ; The Phisics of Racing ; Online Document

 G.Rimondi, P.Gavardi ;  A new interpolative model of the mechanical characteristics of the tyre as an input to handling models ; Rivista ATA 6/7/1991