Pacejka magic formula, longitudinal force and slip ratio

Hello,

I am trying to code my own car simulator, and have some difficulties
to understand how slip ratio is to be interpreted and how I should
interpret it to generate proper wheel angular velocities. I've read
this carefully:
http://groups.google.com/group/rec.a...3b90f 0f630e3
but there are still some contradictions I don't grasp.

From what I understand, in real life, slip ratio is the proportion of
contact patch that is sliding (usually the aft part as the wheel tread
is lifted as the wheel rolls) over the part that adheres on the ground
(fore area of the patch, pressed into adherence by the tread being
pushed down on the ground as the wheel rolls). In other words, in my
wheel model, as long as slip ratio is not 100%, at contact patch wheel
tangential velocity (radius x angular velocity x 2PI) more or less
matches ground velocity. If slip ratio is 100%, then the wheel slides,
and the generated force is constant: whatever the velocity delta, the
resulting force is the same, and wheel angular velocity change depends
on the resultant of all torques, and the wheel inertia (plus
connexions to the drive train).

But Pacejka magic formula takes as input a totally different slip
ratio: it is the ratio of the two velocities I mentioned previously,
measured at the previous simulation step. Using these formulae,
typical road tyre generates maximum force at slip ratios between 10
and 20%, and doesn't generate anything significant when slip ratio is
0. This means that for the tyre to generate maximum force the totality
of the contact patch slides, because I can't really well see how 80%
of the patch could be in static friction state when I have a
tangential velocity of 1.2m.s-1 over a ground that travels at 1m.s-1.
And I see that even less when the tyre has a tangential velocity of
60m.s-1 with ground at 50m.s-1 (same slip ratio, but 10m.s-1 of
relative velocity!).

This means that when I want to render the wheel angular position,
using the velocity ratio definition, it constantly slides over the
ground, which is hardly acceptable.

So, how can I use Pacejka magic formula, slip ratio as it is said to
be defined, and still have realistic wheel angular position
rendering ?

Thanks for any insights on that matter,


Benoit Germain

I can't provide the math that describes this, but you can find it in
Wong's 'Theory of Ground Vehicles'. Wong refers to "Juliens theory"
on slip deformation, and provides the equations.

I can describe the mechanical/physical reason that the circumferential
velocity of the wheel/tire doesn't match the ground speed of the
vehicle during longitudinal acceleration and braking. The reason is
that as a tractive force is applied to the tire, the tire tread
immediately in front of the contact patch is compressed. As these
compressed tread elements enter the contact patch the distance that
the tire travels when subject to driving torque is less than a free-
rolling wheel.

I suppose that as tractive force increases the compression of the
tread increases which increases the slip ratio. At low slip ratios
adhesion remains constant through the contact patch. The relationship
between drive torque and tractive effort remains relatively linear
until approaching peak slip ratio. Then the tractive efficiency tails
off as more and more of the back side of the contact patch starts to
slide relative to the road.

Pat Dotson
UltraForce Simulations LLC
http://www.ultraforcesim.com


On Mar 18, 5:39*am, wrote:
> Hello,
>
> I am trying to code my own car simulator, and have some difficulties
> to understand how slip ratio is to be interpreted and how I should
> interpret it to generate proper wheel angular velocities. I've read
> this carefully:http://groups.google.com/group/rec.a...e/browse_frm/t...
> but there are still some contradictions I don't grasp.
>
> From what I understand, in real life, slip ratio is the proportion of
> contact patch that is sliding (usually the aft part as the wheel tread
> is lifted as the wheel rolls) over the part that adheres on the ground
> (fore area of the patch, pressed into adherence by the tread being
> pushed down on the ground as the wheel rolls). In other words, in my
> wheel model, as long as slip ratio is not 100%, at contact patch wheel
> tangential velocity (radius x angular velocity x 2PI) more or less
> matches ground velocity. If slip ratio is 100%, then the wheel slides,
> and the generated force is constant: whatever the velocity delta, the
> resulting force is the same, and wheel angular velocity change depends
> on the resultant of all torques, and the wheel inertia (plus
> connexions to the drive train).
>
> But Pacejka magic formula takes as input a totally different slip
> ratio: it is the ratio of the two velocities I mentioned previously,
> measured at the previous simulation step. Using these formulae,
> typical road tyre generates maximum force at slip ratios between 10
> and 20%, and doesn't generate anything significant when slip ratio is
> 0. This means that for the tyre to generate maximum force the totality
> of the contact patch slides, because I can't really well see how 80%
> of the patch could be in static friction state when I have a
> tangential velocity of 1.2m.s-1 over a ground that travels at 1m.s-1.
> And I see that even less when the tyre has a tangential velocity of
> 60m.s-1 with ground at 50m.s-1 (same slip ratio, but 10m.s-1 of
> relative velocity!).
>
> This means that when I want to render the wheel angular position,
> using the velocity ratio definition, it constantly slides over the
> ground, which is hardly acceptable.
>
> So, how can I use Pacejka magic formula, slip ratio as it is said to
> be defined, and still have realistic wheel angular position
> rendering ?
>
> Thanks for any insights on that matter,
>
> Benoit Germain

On 18 mar, 21:42, wrote:

> The reason is that as a tractive force is applied to the tire, the tire tread
> immediately in front of the contact patch is compressed. As these
> compressed tread elements enter the contact patch the distance that
> the tire travels when subject to driving torque is less than a free-
> rolling wheel.

Thanks for you reply.
Let's see if I get this right:

Because of engine induced torque, a particular tread point compresses
tangentially in front of the patch. This compressed point enters the
patch, is kept compressed because of friction, and then is released
when the tread leaves the ground.In wheel hub reference frame, this
point travels a circle of unloaded radius sligthly deformed toward
wheel hub when traversing the contact patch and its neighbourhood.

As a result, because of tangential compression and relaxation,
tangential velocity decreases in the compression phase, is more or
less constant while the point is on the ground, then increases again
when it starts sliding as load diminishes because the wheel rolls. I
suppose that if we measure tangential distance between the tread point
rest position and its position in fully compressed state it should be
on the order of a few millimeters, which is at the very most 1% of the
wheel's circumference. If a tread point moves more than that, I would
expect the sidewalls to deform considerably to the point of generating
significant ripples, and I don't think it happens in real life even
with a powerful vehicle. I suspect the wheel would spin before it can
happen.

But more important for me, during one wheel revolution the considered
tread point travels less than the unloaded circumference, but more
than the loaded circumference, the exact amount being closer to the
former (this is my gut feeling here). In the mean time, the vehicle
has moved forward by that exact same distance, since there is always a
part of the contact patch that does not slide. Am I correct, or is
there something I miss here ?

> Then the tractive efficiency tails
> off as more and more of the back side of the contact patch starts to
> slide relative to the road.

Now back to my original question: Assume that peak performance is
indeed reached at 15% slip ratio on real life everyday tyres on an
everyday road, and we have a vehicle in steady state (constant
velocity) at that slip ratio. In practice, it is constantly
accelerating (bad choice of words, but I don't know how to state it
better) to oppose frictions, but as you said, the major part of the
contact patch *does not slide*.

Now, in my simulation I don't do actual contact patch simulation, but
get the generated force from the Pacejka magic formulas. Because of
this, slip ratio is not a consequence of patch simulation and the
various forces acting on the tyre, but a cause of those forces. I
therefore have to measure slip ratio, then feed it to the formula,
that yields a force. According to what I have seen so far, slip ratio
in that context is the ratio of vehicle forward travel velocity and
tread point tangential velocity, computed as a fraction of the
unloaded velocity (this fraction being of the order of 99%, or even
100% if the loaded radius is neglected). And here comes my mind
dislocation:

On one side during a full wheel revolution, the distance travelled by
the vehicle should be near equivalent to the wheel's circumference
(unloaded or not, as stated above) since the major part of the contact
patch is not sliding.

On the other side, for Pacejka formula computed force to oppose
friction and be able to maintain current velocity, I should feed 15%
of slip ratio, and therefore measure 15% of velocity difference.
Replace 15% with 3% everywhere in the above text, and it still would
be visible and not acceptable, as they represent respectively 53° and
10° of excess angular travel each time the vehicle move forward the
equivalent of a wheel circumference.

Which means that I should render my vehicle with wheels rotating that
way, and therefore show them spinning much faster than they should
compared to the vehicle velocity. Because in real life, if I try to
move forward a wheel the equivalent of its circumference while I
rotate it by one revolution plus 53 degrees, I just can't see how I
can do this while having the majority of the contact patch in static
friction with the ground at the same time.

So, I'm afraid I still don't understand the true reason why Pacejka
magic formulas take such a velocity difference as input, and how it
can mix well with vehicle rendering.

Benoit.

I think you understand the basics of the explanation that I gave. The
tread is compressed circumferentially prior to entering the contact
patch. That means that the portion of the contact patch that is in
constant adhesion to the road is shorter circumferentially than it is
at any other point in the rotation of the tire. Slip is the ratio of
the difference in length between compressed and uncompressed sections
of tire tread. You have to integrate(?) that difference over a
complete rotation of the tire to realize the potential magnitude of
the difference.

The main thing to keep in mind is that, even though it's called
'slip', the tire is really not necessarily sliding at all. At slip
ratios less than 100% some part of the tire is in constant contact
with the road.

I hadn't thought it through enough to consider the magnitude of
angular difference. It is amazing that at 15% slip ratio the wheel is
really turning an extra 53° every time the vehicle travels a linear
distance equaling the circumference of the tire. Per my understanding
of slip ratio this is exactly what is happening!

And to consider that the tire does that while in constant contact with
the road is equally amazing.

It might help to imagine taking a cross section of the tire, laying it
down on a flat surface, and then applying a weight to the tread
portion of the cross section. This would be akin to tangential force
applied to a tire via drive torque. Imagine a 1" thick cross section
of tire laying on a table. Place a 1000 pound weight on the tread
part of the tire section. How much will the rubber tread be
compressed? It will probably be significant, and I imagine you've
gone well past 15% compression. This is essentially what is happening
with regard to slip ratio.

> Now back to my original question: Assume that peak performance is
> indeed reached at 15% slip ratio on real life everyday tyres on an
> everyday road, and we have a vehicle in steady state (constant
> velocity) at that slip ratio.

If you assume steady state (constant velocity) then there should be no
appreciable slip ratio. It takes very little throttle input to
maintain a constant speed, which means there is very little force
working to compress the tire tread at the contact patch. You would
have to have an unreasonable amount of drag on the car to reach 15%
slip at constant velocity.

Do I get to beta test?

Pat Dotson
UltraForce Simulations LLC
http://www.ultraforcesim.com



On Mar 20, 6:00*am, Maximilien > wrote:
> On 18 mar, 21:42, wrote:
>
> > The reason is that as a tractive force is applied to the tire, the tire tread
> > immediately in front of the contact patch is compressed. *As these
> > compressed tread elements enter the contact patch the distance that
> > the tire travels when subject to driving torque is less than a free-
> > rolling wheel.
>
> Thanks for you reply.
> Let's see if I get this right:
>
> Because of engine induced torque, a particular tread point compresses
> tangentially in front of the patch. This compressed point enters the
> patch, is kept compressed because of friction, and then is released
> when the tread leaves the ground.In wheel hub reference frame, this
> point travels a circle of unloaded radius sligthly deformed toward
> wheel hub when traversing the contact patch and its neighbourhood.
>
> As a result, because of tangential compression and relaxation,
> tangential velocity decreases in the compression phase, is more or
> less constant while the point is on the ground, then increases again
> when it starts sliding as load diminishes because the wheel rolls. I
> suppose that if we measure tangential distance between the tread point
> rest position and its position in fully compressed state it should be
> on the order of a few millimeters, which is at the very most 1% of the
> wheel's circumference. If a tread point moves more than that, I would
> expect the sidewalls to deform considerably to the point of generating
> significant ripples, and I don't think it happens in real life even
> with a powerful vehicle. I suspect the wheel would spin before it can
> happen.
>
> But more important for me, during one wheel revolution the considered
> tread point travels less than the unloaded circumference, but more
> than the loaded circumference, the exact amount being closer to the
> former (this is my gut feeling here). In the mean time, the vehicle
> has moved forward by that exact same distance, since there is always a
> part of the contact patch that does not slide. Am I correct, or is
> there something I miss here ?
>
> > Then the tractive efficiency tails
> > off as more and more of the back side of the contact patch starts to
> > slide relative to the road.
>
> Now back to my original question: Assume that peak performance is
> indeed reached at 15% slip ratio on real life everyday tyres on an
> everyday road, and we have a vehicle in steady state (constant
> velocity) at that slip ratio. In practice, it is constantly
> accelerating (bad choice of words, but I don't know how to state it
> better) to oppose frictions, but as you said, the major part of the
> contact patch *does not slide*.
>
> Now, in my simulation I don't do actual contact patch simulation, but
> get the generated force from the Pacejka magic formulas. Because of
> this, slip ratio is not a consequence of patch simulation and the
> various forces acting on the tyre, but a cause of those forces. I
> therefore have to measure slip ratio, then feed it to the formula,
> that yields a force. According to what I have seen so far, slip ratio
> in that context is the ratio of vehicle forward travel velocity and
> tread point tangential velocity, computed as a fraction of the
> unloaded velocity (this fraction being of the order of 99%, or even
> 100% if the loaded radius is neglected). And here comes my mind
> dislocation:
>
> On one side during a full wheel revolution, the distance travelled by
> the vehicle should be near equivalent to the wheel's circumference
> (unloaded or not, as stated above) since the major part of the contact
> patch is not sliding.
>
> On the other side, for Pacejka formula computed force to oppose
> friction and be able to maintain current velocity, I should feed 15%
> of slip ratio, and therefore measure 15% of velocity difference.
> Replace 15% with 3% everywhere in the above text, and it still would
> be visible and not acceptable, as they represent respectively 53° and
> 10° of excess angular travel each time the vehicle move forward the
> equivalent of a wheel circumference.
>
> Which means that I should render my vehicle with wheels rotating that
> way, and therefore show them spinning much faster than they should
> compared to the vehicle velocity. Because in real life, if I try to
> move forward a wheel the equivalent of its circumference while I
> rotate it by one revolution plus 53 degrees, I just can't see how I
> can do this while having the majority of the contact patch in static
> friction with the ground at the same time.
>
> So, I'm afraid I still don't understand the true reason why Pacejka
> magic formulas take such a velocity difference as input, and how it
> can mix well with vehicle rendering.
>
> Benoit.

On 20 mar, 14:50, wrote:
> Slip is the ratio of
> the difference in length between compressed and uncompressed sections
> of tire tread. *You have to integrate(?) that difference over a
> complete rotation of the tire to realize the potential magnitude of
> the difference.

OK, I understand what you mean.

> I hadn't thought it through enough to consider the magnitude of
> angular difference. *It is amazing that at 15% slip ratio the wheel is
> really turning an extra 53° every time the vehicle travels a linear
> distance equaling the circumference of the tire. *Per my understanding
> of slip ratio this is exactly what is happening!

In fact, I didn't reach the figure that way, but it happens that 53°
is actually 15% of 360°, so that's consistent.

> And to consider that the tire does that while in constant contact with
> the road is equally amazing.

Yes, to me as well, hence my dubiousness. But it looks like I have to
make myself a reason and accept is as reality.

> If you assume steady state (constant velocity) then there should be no
> appreciable slip ratio. *It takes very little throttle input to
> maintain a constant speed, which means there is very little force
> working to compress the tire tread at the contact patch. *You would
> have to have an unreasonable amount of drag on the car to reach 15%
> slip at constant velocity.

Yeah, for example 350km/h as in a racing game. Aerodynamics drag is
quite enourmous, and tractive force should compress the tread notably.
I'd love to see a high frequency video capture of a F1 racing car
wheel at that velocity to see if angular velocity is perceivably
higher than ground velocity... Not that it would be visible when
rendering of course, because the wheel spins so fast we would use a
different material to render the spinning rim, or else we would just
get a horrible stroboscopic effect.

> Do I get to beta test? *

Given the stage of the project, I fear you'd in for a few years wait.


Benoit.

On Mar 20, 12:59*pm, Maximilien > wrote:
> On 20 mar, 14:50, wrote:
>
> > And to consider that the tire does that while in constant contact with
> > the road is equally amazing.
>
> Yes, to me as well, hence my dubiousness. But it looks like I have to
> make myself a reason and accept is as reality.

Wong's book describes exactly the idea that I relayed to you. I trust
that source so I assume its accurate.

I'll confess that I've never looked that closely at the concept of
slip ratio before. My only real racing experience is with relatively
underpowered karts, so I've never seen a need to apply the concept.
All I knew was the basic idea that its the difference between angular
wheel speed and linear travel. I never knew the mechanics behind it.
This has been a good learning experience.


> Yeah, for example 350km/h as in a racing game. Aerodynamics drag is
> quite enourmous, and tractive force should compress the tread notably.

I'd never considered that before, but you are absolutely right. At
terminal velocity with 100% throttle all available engine torque is
working through the tires to overcome aero drag. Can you put a number
to the drag force in terms of pounds or newtons?

I think, though, that available torque at the tires is going to be
greatly reduced by gearing, such that in 7th gear there may not be as
much torque on the wheels as one might imagine.


> > Do I get to beta test? *
> Given the stage of the project, I fear you'd in for a few years wait.

It was really only a joke. I noticed your email domain after a couple
of replies and responses. I had previously thought you were doing
this for fun.

Hope this discussion helped in some way.

Pat Dotson

> This means that when I want to render the wheel angular position,
> using the velocity ratio definition, it constantly slides over the
> ground, which is hardly acceptable.
>

Benoit/Maximilien:

Man, I think it's time to get a grip/smell the coffee/stop missing the
boat here.

All mixed metaphors aside, please be advised that what you deem
"hardly acceptable" is the de facto standard in the automotive
engineering world. Many, if not most, custom tire models are
proprietary enhancements to the Pacejka tire model.

>in real life, slip ratio is the proportion of >contact patch that is
>sliding (usually the aft part as the wheel tread is lifted as the
>wheel rolls) over the part that adheres on the ground

Semantics. In my "real life," the slip ratio is as defined by the
Pacejka model.

It is almost amusing that you're bemoaning what you consider an
unsatisfactory formulation utilized by the rest of the automotive
engineering world because you can't get your mind around the concept.

But in short, you are getting tire design issues and vehicle dynamics
issues confused, as well as missing a few key vehicle dynamics
considerations. I'll cover both levels.

The mechanics of how a given tire generates forces (what is going on
in the contact patch) is studied in the world of tire design. The net
force(s) a tire DOES generate that act upon a vehicle are studied in
the field of vehicle dynamics. Your perceived problem is one of level.
Consider the following comparison:

Materials develop their strength based on molecular slip; i.e., under
load a material deforms based on whether atoms/molecules are slipping
past one another, or if the space between them is merely stretched.
Applying your point of view to the strength of materials, the only
valid view is some explicit measure of molecular slip and
displacement. Net properties such as yield strength, ultimate
strength, percent elongation, and fatigue properties would be "hardly
acceptable."

What causes slip and distortion in the contact patch? The only thing
that CAN cause it is a difference in the states of motion of the
contact patch and the axle. Which means what? It means that tire
properties that are described as a function of net motions such as
slip ratio and slip angle are both mathematically and philosophically
valid.

>it constantly slides over the ground, which is hardly acceptable.

Maybe you haven't thought about this, but if the contact patch sliding
over the ground is truly "hardly acceptable" to you, then you should
have already been having heartburn over lateral force generation and
the concept of slip angle.

I can tell you for a certainty that tire testing done for the purposes
of characterizing a tire's handling characteristics for vehicle
dynamics purposes, concerns itself with net slip ratio, net slip
angle, camber, radius, load, etc. - all net properties, and you get
six net forces and moment in your data.

>So, how can I use Pacejka magic formula, slip ratio as it is said to be
>defined, and still have realistic wheel angular position rendering ?

If your tire performance data is right, the wheel speed that comes
from a Pacejka model will be right, because the force generated is
related to the speed of the axle and the speed of the contact patch.
Advanced Pacejka models are speed-sensitive, and can take into account
the speed-related concerns you have.

Contact-patch modeling for force generation is done largely in FEA, as
I recall. You can incorporate that into a full-vehicle simulation if
you like, but go raise a family or something to while away the time
once you hit the "solve" button, because it will move with glacial
speed.

If someone (with an understanding of tires and data for a given tire)
tells you a given tire generates maximum force at around 15% slip (the
definition of slip you cite in conjunction with the Pacejka model is
not limited solely to Pacejka formulations, BTW), please be advised
that said tire was MEASURED to perform that way on a tire test
machine, therefore any any wheelspeed "mismatch" per se, IS a REAL
attribute of that tire, if the tire test was carried out properly and
the data processed properly.

You cite the case of F1 cars at 350 kph. It's a good case. I've got a
news flash though: An F1 car at 350 kph is NOWHERE NEAR delivering
maximum longitudinal force to the tire. Notice I said longitudinal
force, and not power. Q: When is a car capable of deliviering maximum
longitudinal force to the tires? A: Regardles of the power band of the
engine, THE CAR DELIVERS MAXIMUM FORCE TO THE DRIVEN WHEELS WITH THE
TRANSMISSION FIRMLY IN FIRST GEAR!!! I hope I don't need to repeat
that. Think about it, the principle is simple: Torque muliplication.
F1 tires, as well as NASCAR tires, as well as street tires, have to
have longitudinal force capacity to permit the car to get its power/
torque to the road WHEN THE CAR IS IN LOW GEAR. Delivering the power
(longitudinal force, really) in top gear becomes a piece of cake.

So, in reality, The F1 car at 350 mph is nowhere near pushing the
tires to their longitudinal force limits, nor is a Nextel/Spring Cup
car at 205 mph doing so, either. In top gear, those cars likely have
integer multiples of longitudinal force capability in the tires above
and beyond what the motor and gear combination can generate.

I really hope this helps, and that it doesn't come across as merely a
good b*tch-slapping.

Pat recommended Wong's "Theory of Ground Vehicles" to you. I'd like to
recommend a couple as well. I have Wong's book, but I typically do not
like it. I suggest Gillepsie's "Fundamentals of Vehicle Dynamics". I
don't know if it covers Julien's theory, but where it and Wong do
cover the same material, I like Gillespie's treatment much better. I
would also heartily recommend Milliken & Milliken's "Race Car Vehicle
Dynamics" for tire data treatment and much, much more.

speedmd

On 20 mar, 22:50, speedmd > wrote:

> All mixed metaphors aside, please be advised that what you deem
> "hardly acceptable" is the de facto standard in the automotive
> engineering world. Many, if not most, custom tire models are
> proprietary enhancements to the Pacejka tire model.

This is exactly why I came here in the first place. I am by no way a
physicist, and my own reality sensors (eyes, namely), told me that
when a vehicle moves, the wheels rotate so that vehicle velocity and
tread tangential velocity match with no perceptible difference.
Because of this, figures saying that maximum force occurs with a 15%
difference or more seemed strange. It happens that I got things wrong
and mixed transitional and steady states (more on that below)

> Semantics. In my "real life," the slip ratio is as defined by the
> Pacejka model.
>
> It is almost amusing that you're bemoaning what you consider an
> unsatisfactory formulation utilized by the rest of the automotive
> engineering world because you can't get your mind around the concept.

I am just bemoaning that fact that I don't understand it, nothing
more. I know very well that all this doesn't come out of thin air. I
just wanted to know where I was wrong in my "understanding" of the
tyre dynamics.

> Maybe you haven't thought about this, but if the contact patch sliding
> over the ground is truly "hardly acceptable" to you, then you should
> have already been having heartburn over lateral force generation and
> the concept of slip angle.

I wasn't clear enough. The only unacceptable thing (to me) here is not
the fact that it happens, but the visual discrepancy between what I
see in real life, as described above, and what I think I will have to
render. But it seems I got the englightenment I needed (see below).

Regarding slip angle, I had no preconception, true or false (which I
admit were what caused my misunderstanding of longitudinal force
generation). And just visually speaking, a few degrees difference
between car heading and wheel heading aren't as visible as in the
longitudinal case, so it simply didn't bother me as much.

> Contact-patch modeling for force generation is done largely in FEA, as
> I recall. You can incorporate that into a full-vehicle simulation if
> you like, but go raise a family or something to while away the time
> once you hit the "solve" button, because it will move with glacial
> speed.

Well, the family is already being raised, so I'd have to find
something else. For lack of other ideas, I guess I'll take the easy
route and forget contact patch modeling :-)

> You cite the case of F1 cars at 350 kph. It's a good case. I've got a
> news flash though: An F1 car at 350 kph is NOWHERE NEAR delivering
> maximum longitudinal force to the tire.
> ...
> So, in reality, The F1 car at 350 mph is nowhere near pushing the
> tires to their longitudinal force limits, nor is a Nextel/Spring Cup
> car at 205 mph doing so, either. In top gear, those cars likely have
> integer multiples of longitudinal force capability in the tires above
> and beyond what the motor and gear combination can generate.

Here is exactly what I've needed to know: slip ratio is very low even
at high velocities, which was not self-evident with my background.
Just out of curiosity, if we assume that the car has the necessary
drive train capacity, at what velocity would the tyre peak ratio be
reached ?
Anyway, if I had understood/known this in the first place, then all
would have been much clearer.

So I assume that if a skilled driver accelerates just short of
starting wheelspin, and I can somehow watch the wheel rotation
compared to ground movement, I'll actually observe this 15%
difference, but it will be much less when he reaches a constant
velocity however high it might be.

> I really hope this helps, and that it doesn't come across as merely a
> good b*tch-slapping.

It does help a lot, and it was worth the understanding I gained in the
process :-)

> Pat recommended Wong's "Theory of Ground Vehicles" to you. I'd like to
> recommend a couple as well. I have Wong's book, but I typically do not
> like it. I suggest Gillepsie's "Fundamentals of Vehicle Dynamics". I
> don't know if it covers Julien's theory, but where it and Wong do
> cover the same material, I like Gillespie's treatment much better. I
> would also heartily recommend Milliken & Milliken's "Race Car Vehicle
> Dynamics" for tire data treatment and much, much more.

I am already aware of Gillespie's and Milliken&Milliken's books, but
haven't had time to get them yet. But once I do I'll have to read them
too.

>
> speedmd

Thanks to you and Pat for the explanations, they were really
necessary. Maybe I should have started by explaining that I can't
really work on any given subject and apply whatever models it without
having the necessary understanding. If I don't have it, I can't see
the difference between what I do wrong because I didn't do it right
(and know this is the case), or because I don't understand what I am
doing (and therefore can't tell if I do it right or not, which gets on
my nerves). I was just seeking that understanding.


Benoit.

I've observed something in everyday driving that has always puzzled
me, and it may have its roots in this discussion...

Have you ever noticed that when you apply throttle in a car the engine
RPM will increase immediately, and any increase in velocity will lag
behind RPM increase. Then, when you release the throttle, RPM will
drop with no corresponding drop in velocity. The point being that in
real life the relationship between engine RPM and velocity is not
entirely linear.

I always figured maybe it was just mechanical 'slop', or something
else in the drive train that causes this observable disconnect between
engine RPM and vehicle speed.

Slip ratio seems to explain what I've observed. Maybe Manuel can
confirm this?

Pat Dotson

> You cite the case of F1 cars at 350 kph. It's a good case. I've got a
> news flash though: An F1 car at 350 kph is NOWHERE NEAR delivering
> maximum longitudinal force to the tire. Notice I said longitudinal
> force, and not power. Q: When is a car capable of deliviering maximum
> longitudinal force to the tires? A: Regardles of the power band of the
> engine, THE CAR DELIVERS MAXIMUM FORCE TO THE DRIVEN WHEELS WITH THE
> TRANSMISSION FIRMLY IN FIRST GEAR!!!

Not with a high downforce car like a F1 car. With 800hp, 1300 lb
car, and downforce up to 2 g's. Even though first gear on F1 cars
redline around 100mph, there's downforce isn't high enough and the
car still has enough power to spin the tires in 2nd gear until
enough downforce has been achieved. 1g of downforce occurs around
120mph (depending on wing settings), and my guess is that maximum
slip ratio occurs about that speed.

Link below to actual downforce numbers for 2001 F1 cars which had
the more powerful 3 liter V10s. The left side list the tracks,
the car setups vary from track to track. Minimum weight of a 2001
car plus driver, oil, basically everything but fuel is 605kg. This
is a French web site. In the table Cx = Cd (coefficient of drag),
Eff = efficiently (overall lift to drag ratio), V = velocity,
the speed at which all the other parameters were measured or
calculated, A = aerodynamic downforce (in kg).

http://www.one-pablo.com/technique/tablaero.gif

Regarding slip angle, the Indy Racing League cars have tires with
the stiffest sidewalls, and their "optimal" slip angle is only
about 2 degrees. Formula 1 cars are closer to 3 degrees. Typical
radial tires are around 3 to 5 degrees, and bias ply racing slicks
are even higher. Going back to the old days of pre-downforce
Formula 1 cars, 1967, the bias ply racing tires reached "optimal"
grip at over 10 degrees slip angle.