Mod-01 Lec-15 Tire Models – Magic Formula

Mod-01 Lec-15 Tire Models – Magic Formula

so in this class we are going to talk about
what are called tire models very short introduction to tire models we go over and look at what
happens in the combined cornering and slip a simple model and we will wind up may be
in the next class about tire models which are used in this combined slip now first of
all what are tire models this is a very important word today in vehicle dynamics because all people who use or interested in
vehicle dynamics they use one software package or the other be it adams or or whatever it
is so they are interested in the way the tire behaves in other words that becomes an input
into the software so they have to give to the software how for example slip versus longitudinal
force varies how alpha to lateral force varies and alpha and moment varies and so on these
graphs are to be input as we can give it as the lookup table which
some of the software accept it as lookup tables are through a formula which is called as a
model so a formula which links this slip for example if i write f it can be fx fy fz
whatever it is as a function of let us say that its longitudinal force fx as a function
of kappa or sigma or slip then it can be a normal force fz which is acting so that camber
angle if it has an effect for example in fy it has an effect so in fy it has camber angle
will have an effect and so on so in the other words this tire model is nothing but a mathematical
equation there are a number of ways in which these
mathematical equations are arrived the easiest of them is to do a curve fitting which we
would call as empirical equations for example i can determine these curves
using experiments for example it can be kappa versus fx and then fit an equation to this
curve the foremost tire model used extensively by people is what is called as the magic formula
tire model you use ti or ty whatever you want but the
magic formula tire model is one of the most important models used today by vehicle dynamics
community there are other models we will see that very short introduction to other tire
models later let us look at magic formula model and let us look at how for longitudinal
force magic formula model is used you will get a feel of it then we will look at the
combined slip first of all what is magic formula you might
be wondering what is magic formula that is the first question a bit of a history this
is one of the greatest contribution of professor pacejka who is a foreigner in the and he has
a very interesting book which i have already pointed it out and he with volvo company mr
bakker of volvo company they together in around 1987 they put up this formula initially called
pacejka tire model and sometime later it was called as magic formula tire model and so
on 87 to 89 there are two 2 or 3 papers in sae
which actually formed the basis or laid the foundation later in 1993 this model was adopted
by michelin tire company and they came up with a modified version of it and they called
this as magic formula model using an empirical technique after this this tire model has undergone
so many changes there are so many versions up to 2006 version we are not going to look at every version
you know it is a huge topic i refer to the book of pacejka on this we will only go back
to the original paper and just look at the philosophy of how this was done that is what
we are going to do before we go further what is this magic formula why is it called magic
formula magic formula is because this formula which i am going to write down now can be
used for all the three curves in other words the frame or the form of the
equation is the same which i can write down this as y=d sin c do
not worry about how you know why this is so complex that we will see it in a minute how
this came about bx-e*bx-arctan bx this has undergone lot of changes let us first stick
to this now this formula this fundamental form with some modifications we will see that
is used for all the 3 cases if i substitute now x for kappa slip in this
form then i would get fx if i now substitute this for alpha here slip then i would get
fy the same way i will get if i substitute for x the slip angle then i would get m and
so on so these 3 cases are the ones which are these are the 3 cases which are important
there are other thing there like turn slip let us now concentrate on these 3 so you will get you have the same form does
not mean that you will have the same values these vales of d c b and e what they are we
will see in a minute how did they come here why is that this so complex we will see that
but these values or these are values which vary with of course fz the normal force and
they take different forms so in a very simple term you can say that i put down a formula
and i have 3 different curves for example i have a curve like that for moment
and then this curve is symmetric with respect to this and then i have another curve which
may be like that for fy so this can be fy suppose like kappa or alpha this can be the
fx curve this can be m curve and so on the reference which i am going to follow in this
class is from the 1987 very first paper i said 870421 sae paper 870421 is a paper which
i am going to follow for this work of course there are after that there are a
number of papers which i have come about but this paper this is a very fundamental paper
and this actually gives the basis for obtaining the ship so that is what we are going to follow
fundamentally we have 4 of them 4 factors d c b and e these are the 4 things that are
there i would say 4 constants they depend upon fz these are the 4 things now the whole
thing started with a very simple equation suppose now i want to represent this equation
form equation form is very easy or important if i want to do a multi body dynamics analysis
we will talk about that a bit later but just have that in mind in other words equations
are important in order that i combine the dynamics of the vehicle with the tire road
interaction if i have to combine them then equations are important remember when we talked about longitudinal
we have to talk about lateral dynamics a bit later talked about longitudinal dynamics we
were interested in the traction force loop fundamentally a physical to mav road now we
just wrote that as f if i now have to go to the next step then that force f has to be
replaced by this curve where did that force come from from the tire so how did that force
developed we saw all the mechanics if i now want to go back to that equation
then i have to put down this so in other words the force developed would now be a function
of my kappa so in that equation go back to the equation in that equation you have a traction
force that traction force will be replaced by an equation which combines kappa and fx
that is why mathematical form of this equation becomes important i can find out in fact if you want me to get
this kind of acceleration whatever you see you have seen lot of advertisements where
they would claim that 0 to 60 6 seconds or 0 to 80 7 seconds whatever it is then i need
that kind of traction force and whether i can develop that force depends upon this equation
i have maximum force that can be developed say for example this would of course depend
upon the road and so on fz and so on of course depend upon the friction characteristics
so this becomes very important where am i going to develop this in other words it also
tells me whether i am going to have the wheel spinning locking and all those things so in
other words that equation has more meaning when i now combine that with the interaction
with the road you would definitely know suppose to make the point very easy to understand we have been lot of questions before as to
what is this formula why am i using it and so on very experienced user of sometime this
would be nothing for a new person who get in this is going to be difficult so suppose i have a curve like this so what
is the maximum force i know this is the maximum force and that is the kappa fx was this kappa
kappa is a slip we have defined 2 slips remember theoretical slip called sigma the practical
slip called kappa so that is what i am plotting here kappa yes i have plotted before sigma
so kappa is the slip percentage slip if you want to call it we have already defined that remember v-omega and so on go back and look
at that if you have questions now the point is this if i now want to develop a force suppose
you are saying that i have a vehicle it has this kind of rolling resistance this kind
of tire and if this is the kind of aerodynamic forces that are acting and this is the mass
of the vehicle and all other things and then now you say that i want to have this kind
of acceleration i want to have 0 to 60 6 seconds so you get an acceleration first of all that
acceleration would result in a requirement for a force and that has to be realisable
whether it is realisable or not this graph would say suppose i require a force like that
obviously that force will not be realised so that would be a problem while plotting
this curve all the other parameters and their effects will be depending on one point that
is a good question in other words this parameter is this curve
a constant what are the other parameters that is what i wrote first suppose i change fz
this curve would change this becomes very important for example when i break or when
i remember we had that redistribution when i break or when i accelerate then fz changes
so that becomes an important question so can i have only one curve is does it not affect
is this curve not affected by fz even if you say that mu is independent of pressure we know this is f and that would depend upon
mu and so it should depend upon fz and so on precisely so this curve cannot be one curve
so it has to depend upon fz now then there can be a series of curves so function of fz for example if you look at cornering you go
to see later may be from the next topic that there is my cornering forces fy becomes important
and its effect on the vehicle dynamics we will see it now here again there is going
to be a load transfer due to the role and again there is going to be a change of this
forces with respect to fy and so on so when i now calculate it for example when i do a
calculation i have to take into account this kind of transfer of load in other words if i have to calculate fx or
fy then i should have an equation which would now give me a new fy or a new fx because of
the load transfer i would shift from one curve to the other curve when you have fz so in
other words this equation d c b e and so on should be a function of fz and that is what
we are going to see hold on to your questions let me finish this then we will look at your
questions i know that is why i am going bit slow people
who know this experienced users bear with me i will develop this slowly now let us now
get into this equation how did you get this equation it looks very complex some arc tan
sin this that b c d e that is the genius of these people who worked in it so what they
did initially was to put see whether this whole thing can be given by a very simple
formula now i am going to modify this please wait
for a minute so they wanted to know whether they can just put sin of bx say d sin bx they
first wanted to this this did not work so actually because let us get back to my all
the 3 curves and we will see why not going to be easy to work one curve looks like this
looks very much like a sin picture the other curve looks like this and the third curve
let us say that it looks like that accordingly you have kappa and alpha that
you know already fx fy so this was not working the sine curve is not working does not work
so i had to now accommodate now that what is there in the sin or in other words i would
adjust what is there inside the brackets in the side so if you now look at this curve
for example this looks as if it is an elongated sine how do i adjust that and i want some function
which is elongated in the x direction so what is the function which is elongated arctan
is a function which is elongated in other words if you now plot an arctan curve
extensively used in so many applications in mathematics x versus y the curve looks something
like this it asymptotically converges to a value what is
that value pi/2 now that is the first thing so this asymptotically converges to a value
of pi/2 the same thing here it is –pi/2 and so on so that is an elongated curve so
i will now replace if i replace what is there instead of x i would replace that value so that i would now put this as y=d sin arctan
bx let me introduce one more term here then i would elongate it but the amount of elongation
i have to know control here it is not very much elongated here it is elongated more and
here it is elongated less and so on this looks like a sinusoid with a lesser elongation larger
elongation very short elongation so i have to now adjust this elongation how do i adjust
that elongation by multiplying it with a constant called c
so what am i doing i am looking at this graph and looking at mathematical expressions which
can actually model this graph and adjusting this equation that is all i am doing obviously
right now you can immediately tell me what should be the maximum value of y this is purely
a graph this is i am just fitting that so what is the maximum value of y=d obviously so this should be the value of d so c in fact
gives me that kind of the shape how elongated it is how sharp it is and so on because it
sort of has an ability to shrink the frequency if i can call it do not get confused with
that word frequency of this sine now i am not still happy with it look at these two
graphs there are some peculiarity as to how actually it rises up apart from the slope
there it has a curve i now introduce into this equation or change
this form of the equation in order to introduce that variation let me call that as the factor
called e so that i will now write down this expression
to be slightly variate and write down this equation to be y=d sin arctan b instead of
x i am going to introduce that phi a quantity called phi and phi=1-e*x+e/b*arctan bx in
other words what i have done is i have introduced this e which actually changes the curvature
at this place now without confusing c and d let me call the c as a shape factor because
this gives the shape and e as a curvature factor now substitute that into this expression for
phi rewrite this expression what do you get you get an expression which is written there
in other words what i do is to use the property of arctan use the property of sine and use
the property of combination and then get a shape which can be now adjusted i have now
4 values which can be adjusted let me give some names to it let me call b as a stiffness factor we see
the stiffness factor is not actually the stiffness c as the shape factor and d is the peak force
this we had already seen d sine of something the maximum sine value=1 so peak force d
and e is the curvature factor why is it called curvature factor because if i now vary the
value of e suppose this is the curve which is e=0 then
depending upon the value with if whether e is between 0-1 or e=- there is a curvature
here would vary sir when we are using this magic formula the structure of the formula
remains the same so our real job would be in relating all the coefficients as a function
of slips say kappa gamma that is what you would feel to your part yes so i am first establishing this formula
which by tweeking this values i can get whatever be the shape now i have to introduce some
further modifications is this clear any questions now bcd are going to take care of all the
other factors that contribute to fx rate including fa energy yes i am coming to that how is that
fz introduced there are a lot of things that are introduced in other words i understand
the question first fy already i have told you that fy is
affected by plastier i have told you that fy is affected by conicity i have told you
that fy is affected by camber now the question is what happened to all that now what do i
do what can i do look at the question carefully now what is the question that i have to introduce
a force when slip angle=0 so in other words it simply means that let us take fy it simply
means that this curve should not pass through 0 origin fantastic so it has to pass through something like that
it is an exaggerated one it would not be so high but just so in other words i have to
shift y conicity the other way so i have to shift x and so on so i am going to introduce
a horizontal and a vertical shift in order to take care of the conicity and plastier
so i am going to shift that curve so let me call that as sv and sh this is only for fy
when we talk about conicity and plastier we are looking at the values of this formula is unique so you can adjust this
sv and sh you can adjust this formula for sv and sh when i said fy obviously we are
talking about conicity and plastier let me redraw that graph very cleanly let me call that as y let me call that as
x let us introduce this x and y and let me call that as sv let me call that as sh and
let me say that this equation is actually that is the curve and the whole curve as well
as the x axis i am shifting so that i would write x to x+sh and y to be y of x+sv so i
have introduced two factors camber will wait for a minute with our friend here is in a
hurry what happens to fz so what i am going to do is i am going to
introduce all these factors as a function of fz before that there is one question what
is actually the slope the slope of this curve is very important the initial slope of the
curve so all safe drivers drive in this linear range what is the slope of this curve look
at that what is the slope dy/dx differentiate it dy/dx b cos whatever is inside then put
arctan as 1/1+x square and so on and then put x=0 so you will get that=bcd so dy/dx at x=0
would not become bcd so bcd is the initial slope of the curve again do not get confused
i am not getting this curve from these equations i have got already this curve for example
i can get this from an analytical formula i can get this from finite element and more
importantly i can get this curve from experiments so i have got the curve with me i am only
fitting an equation my next job is to find out how i am going
to express these factors bcd and so on you just said that what i expressed bcd as a function
of fz but fz itself depends on these factors alpha no no fz is a normal force it varies
depending upon how or what is the way you do a maneuver or how severe is your cornering
how they would manifest in terms of the slip angles absolutely so that is why i am now
expressing this in other words as i told you i have a number
of graphs here and i want to express all these graphs in terms of one equation fi and fx
are also depends on each other right no no we are not like that is a good question so
we are right now looking at independent cornering this equation now that is what i said right
in the beginning we are now looking at cornering separately and breaking separately i have
not come yet to combine cornering and breaking we will develop first a simple mathematical
model in order to say that what it is or in order to enumerate what are the things that
act when i have combined cornering and breaking and then we will indicate how it is done but
even today most people they do not do combined cornering and breaking and all this formulas
that i have used even today are only for a decoupled longitudinal force and a decoupled
cornering forces why are we interested only in kappa and slip
angle because they are the only parameters of course these are the parameters ultimately
i am finding out why are we interested ultimately we are finding out what is kappa what is a
slip and so on so what am i doing i am going to write this
as quadratic equation a1fz square + a2fz how can you say bc are independent of the x they
might also be dependent on this no no bcd when i say b*c*d that is the initial slope
that is all i am saying it is the initial slope dy/dx at x=0 is bcd they do not depend
upon x y z that is the initial slope now these are the parameters let me reiterate these
are the parameters which gives the slope of the curve that is the most important point this curve
depends upon fz i have repeatedly said this and i want to write that so d=a1fz square
+ a2fz and e=a6fz square+a7fz+a8 bcd the initial curve is a3 fz square+a4 fz/e power a5fz and
c these are initial curves than later models have changed to c let us stick to this first
model c=13 for the site force and=165 for breaking and acceleration and are symmetric
=24 for self aligning time so now what essentially i have done is i have
shifted the owners of this curve from bcd and so on to a1 a2 a3 a4 a5 a6 a7 a8 and so
on so in other words i have now a set of parameters a1 a2 a3 which would capture these curves
all these a1 to a8 are determined from experiments they are all determined from experiments these
are only coefficients that you would feed yes these are the coefficients if you want the type of coefficients are how
it would be so you can say that for example fya1 would be -221 please note that it is
not necessarily positive it may be positive or negative a2 will be 1011 this is in terms
of kilonewtons this is the tendency for many people in the tire industry even today to
use pound force then these coefficients would be accordingly adjusted for a9 we will see
how this comes then we will have properties for a9 a10 and so on so a9 and a10 and other things which we are
going to see now in a minute we will then give the values take care of the camber aspects
for fy we said that the camber gives you a camber thrust in other words that gives you
an fy so that would again be a shift factor and that is the shift factor is given by due
to camber horizontal shift factor for the camber is
given by a9 gamma and gamma is the camber angle
and the delta b change in stiffness that is also obtained as delta b gamma again there
are number of parameters i will not confuse you i can again put on a9 a10 11 12 there
is another 13 which again factor for e and so on let us not worry about it so i do not
confuse you let me summarize what all we said in other words what we said is that we have
an experimental curve and the experimental curve has to be input into my say for example
into my mathematical model of the whole vehicle because i am interested there to determine
the forces i want to know whether my tire would develop that forces if it develops the
force what would be the slip angle or what would be the slip at which this forces will
be developed where do i sit actually in that curve and so on and that i have to meaningfully take into
account what is the change in the fz values or the normal forces that are acting due to
the dynamics so in order to take into account all that i have an experimental curve and
then these experimental curve is fitted by means of an equation which has so many values
a1 to a13 so this is not this is for one tire if i have another tire these values will be
different in fact i do not have time but if you are interested we will discuss it later as to how to fit this it is very important
that i get some unique parameters for this formula so fitting this becomes very important
that i had a student who worked on this and so there is a way of fitting this so what
do you get i get a curve then how do i get from this i get a curve not one curve i have
to change fz i will get a series of curves i have to change gamma i will get a series
of curves with all these curves now i have to fit a1 to a13 there are special softwares available for
it and it is quite an amount of research tool it is an optimization problem that can be
used in order to fit a1 to a13 definition de dcd changes will show effects no no definitions
do not change they are all the form of this thing the values of a1 to a12 would change
camber the same thing this is shift and i call this as delta sh this is the shift the
same way i am going to shift that is the shift why you are not considering coefficient of
friction yes coefficient of friction automatically comes in because i am not interested in mu
that is exactly what i said right in the beginning of the class if i now replace fx for example
by a normalized fx/fz curve people tend to call this as a mu curve here i am plotting
fx directly so it is also important that brings out an important topic as to what is the role
of friction how do people determine this graph there are two ways in which they determine
this curve one is by doing an experiment in the road by a tractor trailer for example
tno in netherlands they have a facility to determine this by taking the tire to the road
and finding out this value but many many people what they do is to do an indoor test they
do an indoor test and find out they have a machine on which they have a surface on which
this tire is mounted and they have facility to measure these forces
and the slip slip angle and so on and they determine this in the laboratory there your
question of surface becomes important because the question is how do you characterize mu
so the surface becomes important so people have various surfaces if they have to do this
in the laboratory this is a big topic and these surfaces are actually do have a roughness
factor which would simulate or hold good even if you have to go or this equations would
hold good even if you have to do it in the road next class we will see this they mount a tire
on a spindle and they have a flat surface which is the surface may be moving and then
they measure forces slip and so on in the laboratory scale there are many tire companies
who have this kind of facility in order to determine fx fy and other factors how these
effects with mu with the coefficient no because it would depend on fx to a certain extent
i would say that this is a good question that is why people in those days they used
to do what they used to do is just normalize this curve and then plot fx by fz and then
have the kappa value and then have this curve and they say that this normalized curve is
fine but then this brings out a very important topic on friction itself the friction coefficient
itself can this curve be like this so there being a lot of work as i told you before to
how actually friction works between the road and the tires big topic by itself so this depends upon as i said not only the
contact pressure but what is called as v you know slip and so on if you want to model in
a very detailed fashion the tire behaviour then you have to go into that friction models
when all the systems that we have are based on this we try to get the slip so that the
loss of force is done now if the mu in the road and the test conditions are totally different
right absolutely that is exactly what he was asking what happens when there is a friction coefficient
is different so you have to be careful the question is if i do it indoor would not i
get a different curve than if i do it outdoor in the road because the road characteristics
are different absolutely there will be a difference between the two whether you do it indoor or
whether you do it outdoor and test in fact there is a paper in tire sense and technology
by continental tires which actually bring out and tell you what are all the differences
on friction coefficients and so on so there will be but the theory is such that
the differences are not very high it also depends upon the enveloping characteristics
of the tire so we would say that there would be a difference but for all practical purposes
people use the data which is generated on this kind of rough paper or sand paper or
whatever you want to call it that is the type of thing that is used in order to get this
state any other questions where are the characteristics of tires incorporated
in bcd yes i am not looking at the tire design here tire characteristics i am looking at
the final result in other words the question is how does the tire characteristics affect
this for example if i change the compound or if i change the side wall profile if i
change anything in the belt angles where is it reflected here that is the question it
is not reflected here this is for a tire a given tire you are doing an experiment but there will be some formula that link a1
and all those absolutely go and look at the paper which we published that is a very interesting
question in fact i and my student we published a paper on how to link very good question
i am very happy because it gives importance to the paper which we had written so how a1
for example is affected by design in the next class i will give you the reference of the
paper where we have linked it is a very important question because ultimately
the tire manufacturer wants to know how he has to change the design in order that he
would vary a1 and so on so there is a link between this and the characteristics in fact
you will be interested to know that these are all affected by inflation pressure as
a role to play as well and lot of design parameters we will postpone that till the next class

Only registered users can comment.

  1. To the student who asked the question " What exactly is Kappa here " ?
    Answer : Kappa is the SLIP RATIO that generates the longitudinal force for the tire and alpha is the SLIP ANGLE that generates the lateral grip for the tire.

  2. Can anyone explain what the B coefficient means and how it is that we get it from tire data? Much appreciated

  3. are the longitudinal and lateral force found using magic formula acting on each wheel. How to find the total longitudinal and lateral force then?

  4. The magic formula is quite similar to chemical kinetics file in combustion modeling. Do we have online file like them that could be downloaded ?.

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