Paolo PintoAeronautical Engineer and Computer ProgrammerThe Pacejka EquationSince 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) ] )whereB,C,D,E are coefficients. relevanThe x, y variables can be associated from time to time to different parameters ; for example :x = slip angle,y = Fyif studying the tyres ability to provide centripetal forcex = slip ratio,y = Fxfor the tractive forceIt is also possibile to take into accountcamber, 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 ofx, y.Function studyThe Y(x) function is anti-symmetric ; it always goes through the axis origin and it always has there a null second derivative.In tyre modelsy(0) >0 is always desiderable ; it is possibile to demonstrate that this implies B,C being of the same sign.For thesame reason it must always bey(0) < 0 ; this impliesE > -(1 + C2/2)The curve always shows an horizontal asymptote for x tending to infinity.The asymptotes value is :D sen(C/2)if E 1As a consequence, the need to have y(x)>0 for x tending to infinity leads to coefficients :E