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

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.

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

What is the name of the paper mentioned at 48:15?

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

svp comment calculer Fxi ou bien c'est quoi la formule mathmatique de Fxi Fxfr Fxfl ….

I'd love to make my own racing game..but I'm such a noob. I don't even know how to code.

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