Engine and reaction torques and inertia

Hi all,

I have a simple vehicle simulator going for a game which I'm trying to
move towards a more accurate model, taking into account torques and
inertia down the drivetrain and reactions back from the road. I've
been reading the archives and found some very helpful discussions from
quite a while back, but I have a few lingering doubts which I was
hoping someone could help me with.

The area I'm having trouble with is working out the resulting angular
acceleration of the drivetrain components once I have the drive and
reaction torques. For now I am assuming no differential (which
actually fits with the game - go-carting!). Assuming the clutch is
fully engaged and no braking, what I have at the moment is the
following:

engine torque = throttle * torqueCurve(engine rotation speed)
torque at drive axle = engine torque * gearRatio *
transmissionEfficiency * finalDriveRatio
wheel drive torque = 0.5 * torque at drive axle

For the wheels and tires, I'm not using full-blown Pacejka yet but I
do have a Pacejka-like curve defined which maps slip ratio to
longitudinal force. I'm only worried about straight-line behaviour for
now

friction force = slipRatioCurve(wheel slip ratio) * wheel load
reaction torque = -friction force * wheel radius

So I end up with a drive torque and a reaction torque at the wheel,
but I'm not sure how to apply/combine these to get the angular
acceleration. Do I need to simply add the torques and use the
resulting torque to accelerate the drivetrain, where the effective
inertia from the wheels' point of view is:

inertia of drive wheels + inertia of driveshaft * finalDriveRatio^2 +
gear inertia * (gearRatio + finalDriveRatio)^2 + engineInertia *
(gearRatio + finalDriveRatio)^2

So angular acceleration = (driveTorque + reactionTorque) / effective
inertia (where reactionTorque opposes driveTorque)

Does this sound right? Am I even close?

Thanks,
James

> engine torque = throttle * torqueCurve(engine rotation speed)
> torque at drive axle = engine torque * gearRatio *
> transmissionEfficiency * finalDriveRatio

> wheel drive torque = 0.5 * torque at drive axle

You're assuming each wheel get's equal torque?

> do have a Pacejka-like curve defined which maps slip ratio to
> longitudinal force.

> drive torque + reaction torque

When the reaction torque is less than the drive torque, the tires
are spinning and accelerating. The net torque, drive - reaction,
divided by the angular inertia of the tires, drivetrain, and
engine determines the rate of angular acceleration. Normally
the drive torque falls off at partial throttle once the engine
reaches some rate of rotation (rpm), or the engine hits the
rev-limiter.

From what I understand, Pacejka doesn't deal well with static to
dynamic or dynamic to static friction situations, as these require
seperate curves (once slipping, the reduced reaction force on the
tires from slipping has to be lowered further still before the
transition from dynamic to static friction occurs). The other
issue with Pacejka is slow speeds, or spinning the tires forwards
when stopped or moving backwards, or simply stopped on a banked
or angled (hill) section of track.

Thanks for your response Jeff!

> You're assuming each wheel get's equal torque?

In the absence of a differential (see my original post) this must
happen mustn't it?

> When the reaction torque is less than the drive torque, the tires
> are spinning and accelerating. The net torque, drive - reaction,
> divided by the angular inertia of the tires, drivetrain, and
> engine determines the rate of angular acceleration. Normally
> the drive torque falls off at partial throttle once the engine
> reaches some rate of rotation (rpm), or the engine hits the
> rev-limiter.

I have the drive torque falling off according to a pretty standard
curve so that part is fine. When I try this though, what I seem to
find is that when accelerating the reaction torque very quickly
exceeds the drive torque, so the kart drives off backwards. If say
after one timestep the engine has accelerated the wheels to produce a
0.01 slip ratio which on my curve comes out at about 0.1 Fx/Fn, so
with say 400N load on the wheel this is a 40N longitudinal force per
wheel. If I use this to generate the reaction torque (ie multiply by
wheel radius) this seems to exceed the drive torque pretty easily. My
engine data are based on some torque curves for a go-kart engine I
found on a manufacturer website so I would have thought they were
reasonable. Are my timesteps simply too large? I'm running it at 30Hz
at the moment (there's no suspension so I thought this would do for
now).

> From what I understand, Pacejka doesn't deal well with static to
> dynamic or dynamic to static friction situations, as these require
> seperate curves (once slipping, the reduced reaction force on the
> tires from slipping has to be lowered further still before the
> transition from dynamic to static friction occurs). The other
> issue with Pacejka is slow speeds, or spinning the tires forwards
> when stopped or moving backwards, or simply stopped on a banked
> or angled (hill) section of track.

I'm not using Pacejka, but I have a curve relating slip ratio to
longitudinal force which is shaped similarly to Pacejka (fairly
standard I think, rises more or less linearly to a peak at 0.1 slip,
then falls off with increasing slip). I have some low speed handling
based on SAE950311 in my slip ratio calculation, but the issue seems
to be that the slip ratio very quickly reaches a value approaching my
optimal slip ratio and therefore a large longitudinal force is
generated, which generates a large reaction torque on the wheels
making them spin backwards, which generates an even larger
longitudinal force backwards. I've obviously got something wrong but
I'm not sure what it is :/

Thanks,
James

> after one timestep the engine has accelerated the wheels to produce a
> 0.01 slip ratio which on my curve comes out at about 0.1 Fx/Fn, so
> with say 400N load on the wheel this is a 40N longitudinal force per
> wheel. If I use this to generate the reaction torque (ie multiply by
> wheel radius) this seems to exceed the drive torque pretty easily.

Which sounds like a problem. The torque from the engine is multiplied
(minus losses) to the driven tires, and the resultant driven tire
torque divided by the radius is the force the tire tries applies to
the pavement, except during the dynamic period where the tire's slip
ratio is changing. If this force exceeds the maximum grip, then
the tires slip and accelerate.

You could keep adjusting the slip ratio until the Pacejka calculated
force agrees with the engine generated force for each step. Sort
of like predictor + corrector numerical intergration.

http://en.wikipedia.org/wiki/Predictor-corrector

http://en.wikipedia.org/wiki/Numeric...tial_equations

After re-doing my clutch model which left something to be desired, I
retried my initial ideas and it worked much better. However, I've come
to the conclusion that I really need to run the simulation at a higher
frequency; 30Hz doesn't seem to be enough for the slip ratio
calculations to be stable. I've had some correspondence by email that
also suggested this. A predictor-corrector might help to some degree
as this effectively artificially creates a higher simulation frequency
but I still think a higher simulation rate overall is necessary.
Unfortunately in the environment I'm working in I can't easily adjust
the timestep so I'll need to make some changes to try it out. Thanks
very much for your help though!

On Dec 1 2008, 10:16*am, jlevick > wrote:

> So I end up with a drive torque and a reaction torque at the wheel,
> but I'm not sure how to apply/combine these to get the angular
> acceleration. Do I need to simply add the torques and use the
> resulting torque to accelerate the drivetrain, where the effective
> inertia from the wheels' point of view is:
>
> inertia of drive wheels + inertia of driveshaft * finalDriveRatio^2 +
> gear inertia * (gearRatio + finalDriveRatio)^2 + engineInertia *
> (gearRatio + finalDriveRatio)^2
>
> So angular acceleration = (driveTorque + reactionTorque) / effective
> inertia (where reactionTorque opposes driveTorque)
>
> Does this sound right? Am I even close?
>
> Thanks,
> James

Yes, that's right. Just add the torques together and divide by the
effective inertia.

On Dec 2 2008, 8:12*am, jlevick > wrote:
> Thanks for your response Jeff!
>
> > You're assuming each wheel get's equal torque?
>
> In the absence of a differential (see my original post) this must
> happen mustn't it?
>

If you're simulating a go-kart with a solid rear axle, then you can
just treat the whole axle as a single rotating "thing" with its own
effective inertia including whatever gearing you wanted with the other
shafts and so forth that are attached to it. Then assign both tires
to this same angular velocity and feed it to your tire model for the
next iteration. For the front tires of course you'd just use their
own inertias and have independent angular velocities as you're
undoubtedly already doing.

Things are nice and simple when there aren't any diffs in the mix. :-)


> > When the reaction torque is less than the drive torque, the tires
> > are spinning and accelerating. The net torque, drive - reaction,
> > divided by the angular inertia of the tires, drivetrain, and
> > engine determines the rate of angular acceleration. Normally
> > the drive torque falls off at partial throttle once the engine
> > reaches some rate of rotation (rpm), or the engine hits the
> > rev-limiter.
>
> I have the drive torque falling off according to a pretty standard
> curve so that part is fine. When I try this though, what I seem to
> find is that when accelerating the reaction torque very quickly
> exceeds the drive torque, so the kart drives off backwards. If say
> after one timestep the engine has accelerated the wheels to produce a
> 0.01 slip ratio which on my curve comes out at about 0.1 Fx/Fn, so
> with say 400N load on the wheel this is a 40N longitudinal force per
> wheel. If I use this to generate the reaction torque (ie multiply by
> wheel radius) this seems to exceed the drive torque pretty easily. My
> engine data are based on some torque curves for a go-kart engine I
> found on a manufacturer website so I would have thought they were
> reasonable. Are my timesteps simply too large? I'm running it at 30Hz
> at the moment (there's no suspension so I thought this would do for
> now).
>

Oh yes, 30Hz is way too large of a time step for a car simulation,
especially a kart drivetrain. I run my simulations at 250 or 333Hz.
With higher frequencies you'll still have oscillations, but they get
smaller with increasing frequency. I run my drivetrain and tire
simulation at a higher frequency (just an inner loop of a few cycles
depending on what works well, and only if it's really needed). If I
recall correctly, that SAE paper you mentioned covers the relaxation
approach. If you implement that or something similar and tune it
well, it should reduce the oscillations.

Several simulators seem quite happy with 100Hz so long as you run the
drivetrain in a loop at a higher frequency, mine included, although it
seems to feel nicer at 250+. For the drivetrain you might try
300-500Hz or so.

> > From what I understand, Pacejka doesn't deal well with static to
> > dynamic or dynamic to static friction situations, as these require
> > seperate curves (once slipping, the reduced reaction force on the
> > tires from slipping has to be lowered further still before the
> > transition from dynamic to static friction occurs). The other
> > issue with Pacejka is slow speeds, or spinning the tires forwards
> > when stopped or moving backwards, or simply stopped on a banked
> > or angled (hill) section of track.
>
> I'm not using Pacejka, but I have a curve relating slip ratio to
> longitudinal force which is shaped similarly to Pacejka (fairly
> standard I think, rises more or less linearly to a peak at 0.1 slip,
> then falls off with increasing slip). I have some low speed handling
> based on SAE950311 in my slip ratio calculation, but the issue seems
> to be that the slip ratio very quickly reaches a value approaching my
> optimal slip ratio and therefore a large longitudinal force is
> generated, which generates a large reaction torque on the wheels
> making them spin backwards, which generates an even larger
> longitudinal force backwards. I've obviously got something wrong but
> I'm not sure what it is :/
>
> Thanks,
> James

Matt Jessick used to call this the "bang bang" problem. Wait until
you try to get your clutch to lock :-)

Anyway, just crank up the frequency and it will get better. Starting
from rest will require a relaxation approach of some kind. A quick
and easy way to get around this is to switch to a different slip ratio
calculation at low speed. One based on the velocity difference
instead of the ratio. Then, at some low speed, you switch to the
correct slip ratio calculation. Better yet is to do both calculations
and then use a weighted average so it transitions somewhat smoothly.
It's not perfect, but if you play with it enough most people won't
ever notice the model switching unless they're really looking for it.
I wrote the physics engine for Virtual RC Racing. You think kart
tires are hard to deal with? Try inertias of almost 0... :-P

teTodd Wasson wrote:
> On Dec 2 2008, 8:12 am, jlevick > wrote:
>> Thanks for your response Jeff!
>>
>>> You're assuming each wheel get's equal torque?
>> In the absence of a differential (see my original post) this must
>> happen mustn't it?
>>
>
> If you're simulating a go-kart with a solid rear axle, then you can
> just treat the whole axle as a single rotating "thing" with its own
> effective inertia including whatever gearing you wanted with the other
> shafts and so forth that are attached to it. Then assign both tires
> to this same angular velocity and feed it to your tire model for the
> next iteration. For the front tires of course you'd just use their
> own inertias and have independent angular velocities as you're
> undoubtedly already doing.
>
> Things are nice and simple when there aren't any diffs in the mix. :-)
>
>
>>> When the reaction torque is less than the drive torque, the tires
>>> are spinning and accelerating. The net torque, drive - reaction,
>>> divided by the angular inertia of the tires, drivetrain, and
>>> engine determines the rate of angular acceleration. Normally
>>> the drive torque falls off at partial throttle once the engine
>>> reaches some rate of rotation (rpm), or the engine hits the
>>> rev-limiter.
>> I have the drive torque falling off according to a pretty standard
>> curve so that part is fine. When I try this though, what I seem to
>> find is that when accelerating the reaction torque very quickly
>> exceeds the drive torque, so the kart drives off backwards. If say
>> after one timestep the engine has accelerated the wheels to produce a
>> 0.01 slip ratio which on my curve comes out at about 0.1 Fx/Fn, so
>> with say 400N load on the wheel this is a 40N longitudinal force per
>> wheel. If I use this to generate the reaction torque (ie multiply by
>> wheel radius) this seems to exceed the drive torque pretty easily. My
>> engine data are based on some torque curves for a go-kart engine I
>> found on a manufacturer website so I would have thought they were
>> reasonable. Are my timesteps simply too large? I'm running it at 30Hz
>> at the moment (there's no suspension so I thought this would do for
>> now).
>>
>
> Oh yes, 30Hz is way too large of a time step for a car simulation,
> especially a kart drivetrain. I run my simulations at 250 or 333Hz.
> With higher frequencies you'll still have oscillations, but they get
> smaller with increasing frequency. I run my drivetrain and tire
> simulation at a higher frequency (just an inner loop of a few cycles
> depending on what works well, and only if it's really needed). If I
> recall correctly, that SAE paper you mentioned covers the relaxation
> approach. If you implement that or something similar and tune it
> well, it should reduce the oscillations.
>
> Several simulators seem quite happy with 100Hz so long as you run the
> drivetrain in a loop at a higher frequency, mine included, although it
> seems to feel nicer at 250+. For the drivetrain you might try
> 300-500Hz or so.
>
>>> From what I understand, Pacejka doesn't deal well with static to
>>> dynamic or dynamic to static friction situations, as these require
>>> seperate curves (once slipping, the reduced reaction force on the
>>> tires from slipping has to be lowered further still before the
>>> transition from dynamic to static friction occurs). The other
>>> issue with Pacejka is slow speeds, or spinning the tires forwards
>>> when stopped or moving backwards, or simply stopped on a banked
>>> or angled (hill) section of track.
>> I'm not using Pacejka, but I have a curve relating slip ratio to
>> longitudinal force which is shaped similarly to Pacejka (fairly
>> standard I think, rises more or less linearly to a peak at 0.1 slip,
>> then falls off with increasing slip). I have some low speed handling
>> based on SAE950311 in my slip ratio calculation, but the issue seems
>> to be that the slip ratio very quickly reaches a value approaching my
>> optimal slip ratio and therefore a large longitudinal force is
>> generated, which generates a large reaction torque on the wheels
>> making them spin backwards, which generates an even larger
>> longitudinal force backwards. I've obviously got something wrong but
>> I'm not sure what it is :/
>>
>> Thanks,
>> James
>
> Matt Jessick used to call this the "bang bang" problem. Wait until
> you try to get your clutch to lock :-)
>
> Anyway, just crank up the frequency and it will get better. Starting
> from rest will require a relaxation approach of some kind. A quick
> and easy way to get around this is to switch to a different slip ratio
> calculation at low speed. One based on the velocity difference
> instead of the ratio. Then, at some low speed, you switch to the
> correct slip ratio calculation. Better yet is to do both calculations
> and then use a weighted average so it transitions somewhat smoothly.
> It's not perfect, but if you play with it enough most people won't
> ever notice the model switching unless they're really looking for it.
> I wrote the physics engine for Virtual RC Racing. You think kart
> tires are hard to deal with? Try inertias of almost 0... :-P
> test