rec.autos.simulators

I'm adding some variables to my car model, while reading 'Fundamentals
of Vehicle Dynamics' from Thomas Gillespie. Well, actually, I'm
converting the body and wheels to rigid body classes which use
quaternions and inertia tensors for their rotation (so I have faster
functions and gyroscopic precession for free for all the
forces/torques I want to add) (this 'simplified' model will allow
easier inclusion of new forces/torques in a more clean way).

Anyway, in the book, Gillespie follows the engine torque to the
wheels, and does it like this:

Te=point on the engine's torque curve
Tc=Te-IeAe
  where Tc=torque at the clutch, Ie=rotational inertia of the engine,
and Ae=engine rotational acceleration.

Td=Tc-...
Ta=Td-...

(Td=torque output to the driveshaft, Ta=torque on the axles)

So he keeps subtracting torque in every step, based on the
acceleration. Now in simulation, the acceleration is *the question*,
rather than the answer to fill in the equations. It seems I can't use
these formula's like that.
It seems to me that since everything is fixed together, the torque
remains the same, but all inertia of all connected components must be
taken into account, so:

totalInertia=I_engine+I_driveshaft+I_transmission+I_rearaxle(RWD)+...
torque=Te

And then calculate rotational acceleration by torque/totalInertia
(taking into account the transmission's torque multiplier ofcourse).
Right?

The differential will probably be the most problematic part, where I
have to model how it will allow both axle outputs to rotate (which
means the totalInertia might change if the diff locks up more or
less).

This problem looks a lot like what I had once before, namely the
rotation of the wheels; what influence the drivetrain inertia has
together with the wheels on the total change of the engine's rpm. That
was mainly solved by just applying the torque and using the slip ratio
model.
But it seems strange to me what Gillespie writes, namely that torque
*decreases* as it moves from engine to the wheels. At least his
formula's imply so.

Any thoughts? Thanks,

Ruud van Gaal, GPL Rank +53.25
Pencil art    : http://www.racesimcentral.net/
Car simulation: http://www.racesimcentral.net/

When thinking of the engine driving the wheels, the fricitional losses
always cause a drop is power when going from the source (engine) to
the point of interest (wheels).  When under engine braking the
fricitional losses will continue to be dissipative.  In this case the
power under engine braking will be higher at the wheels, smaller at
the engine.

Bob

  I'll probably be needing your help with quaternions :0)  Good luck on the
conversion.  

  It's strange how Gillespie's equations show a decrease in torque.  I don't
have his book, so can't comment on that.

  As far running the rotation stuff through the driveline goes, I use something
similar to what you posted:

  That's how I do it.  Not sure if it's right though :0)

  Right now, I loosely simulate the differential action by assigning the drive
torques from the engine independently to each rear wheel, and letting them
accelerate on their own.  This way, they're free to run at their own speeds.
It seems to work pretty well, as the inside tire spins first in a turn.  By
matching the speeds to either the high or low speed tire, it works just like a
spool/locked rear end, exhibiting push and snap oversteer at high throttle.
Now, I need to know how the torques are supposed to really be applied, rather
than just splitting it 50/50 all the time.

  Strange about Gillespie's stuff.  Again, I haven't seen it so can't comment.

Todd Wasson
---
Performance Simulations
Drag Racing and Top Speed Prediction
Software
http://PerformanceSimulations.Com

The torque doesn't decrease, rathersome of it is expended to accelerate
the rotational inertia of the rotating parts. You have to expend
power to accelerate them up, and  it also
takes more braking to slow them back down.
As the book explains, when accelerating the car, you are also
accelerating the rotating parts of the engine and the
drive train.

If you are accelerating a drag racing car at 3 G's with the
clutch not slipping and the final drive locked, you can
expend a noticible part of the engines torque production
just spinning up all this rotational inertia. This effect is most
noticible in the lowest gear, because when the clutch is
locked, the engine is accelerating very fast because
of the large gear ratios commonly in use.

As Gillespie notes, you can express the overall effect of
this as an "effective mass". This is through combining all the
rotational terms together, and noticing that they act just like
the mass in    Forward_force = mass times acceleration
This becomes
    acceleration = force / (mass + effective_mass_rotating_parts)

Where the effective mass can be several hundred pounds
if you aren't careful.  Drag racers are careful, and do things like
using short aluminum or carbon drive shafts instead of long
steel ones, etc., to reduce this penalty. High tech road racers use
*** wheel materials for similar reasons when allowed and
they can afford it.

The effective_mass of the rotational parts has terms like:

(inertia_wheels +
  inertia_driveshaft * final_drive_ratio**2
  inertia_engine      * overall_gear_ratio**2) / tire_radius**2

given final drive ratios  of 4-5 and overall first gear ratios of 12 or
more
you can see where those gear ratio squared terms can
get out of hand if not controlled.

To answer your question about how to get the accelerations,
"just" ;)  solve the resulting equations for
the unknown accelerations.

It's messy but straightforward for each configuration of the driveline.
(Clutch slipping or not slipping, transmission in gear or out of gear,
differential locked or open = 8 or more cases! ;)

I use the Mathematica program to do this kind of algebra,
otherwise I'd be spending all my time correcting errors ;)

For your case with all nodes locked:

Te = point on the engine's torque curve
Tc = Te - Ie Ae
Tp = Gi Tc - Ip Ap      p = pinion or driveshaft
Aw = (Gfd Tp - FxR)  / Iw         w = drive wheels,
                      FxR is the "Load" torque of the tires forward force

and noting:
Ae = Gi Gfd Aw,
Ap = Gfd Aw

Here is Mathematica script  to solve this:

Ae = Gi Gfd Aw
Ap = Gfd Aw

Tc = Te - Ie Ae
Tp = Gi Tc - Ip Ap

(* then solve the combined equation below for Aw:
  Aw = (Gfd Tp - FxR)  / Iw
*)

Solve[{Aw == (Gfd Tp - FxR)  / Iw}, {Aw}]

The solution is:
Aw = (-FxR + Gfd * Gi * Te) / (Gfd**2 * Gi**2 * Ie + Gfd**2 * Ip + Iw)

and the other accelerations easily derived from that.
(Gillespie gives essentially this solution in Chapter 2.)

Our games also include the transmission input shaft which adds some more
complexity but it is still straight forward but tedious without
Mathematica ;)       Get Mathematica, particularly if you are a student.
They have student packages because it is more ***ive than cigarettes!
;)

- Matt

P.S.   Ruud, please note Very Important Caution interspersed below

...

Yes, strange that mass and inertia can be interchanged like that. Like
E=mC^2. So this acceleration would be the linear acceleration of the
center of mass, right?

...

Yes, I've looked at this big formula multiple times. But I wondered
what would happen to it if one of the tires is off the ground, or
broken off (!), or getting light because of transverse
weight-shifting. It would seems that tire_radius (esp. when one of the
2 tires is broken off) will mislead the calculation, or not?

Or should I in the case of a missing wheel (broken off), remove 1
wheel of the 'inertia_wheels' variable? Sounds ok. But perhaps remove
a power from tire_radius as well?
((inertia_wheels+...)/tire_radius**(number_of_connected_wheels))

Thanks for the help, I've printed it and will reread it until I fully
understand it. :)

Ruud van Gaal, GPL Rank +53.25
Pencil art    : http://www.marketgraph.nl/gallery/
Car simulation: http://www.marketgraph.nl/gallery/racer/

Thanks. I've done a little writing this morning. Now for a little time
off and back on it this evening. I'm basing things on Richard Chaney's
RigidBody class (www.chaney.karoo.net). This uses a linear
position/velocity, and a rotational position/velocity as the state for
a rigid body.
After that, he present formulas and source code to add torques and
forces (applied at body or world points) to the body, then moves on to
integrate, using either Euler or Runge-Kutta 4th order. All the
ingredients to make a new soup of things which will need some
debugging to get right. Fortunately, I have things working nicely as
it is with yaw/pitch/roll, so I just have to end up thereabouts again.
It also does precession. Funny is perhaps that linear/rotational
acceleration is not part of the body's state. That's only used when
applying forces/torques for a moment and in the RK integration.
I'm beginning to understand quats a little better step by step. They
have a funny origin, but are basically just rotations around a 3D
axis. Where the magic thing is that q1*q2 is both rotations
geometrically added, although I wouldn't know how to do this if just
given 2 axes and 2 angles. Nice things! Besides, they normalize very
quickly, because it's just a 4D vector. So rotation matrices won't
have squeezing problems.
I intend to convert the body first. The wheels, being so restricted,
are hardly free moving bodies. Although I wonder how big the effect of
gyroscopic precession is on turning if the wheels are spinning fast.
Perhaps its effect on Mx, the aligning moment, is greater than I think
(which is necessary ofcourse to in the end come up with great force
feedback).

It seems, reading Matt's reply, we can do better. Sure, ofcourse, this
is an endless field of optimizations and improvements! ;-)

...

You mean you take the speed of one of both tires, and apply its
rotational velocity to that of the *entire* axle? (making a solid
axle)?

Ruud van Gaal, GPL Rank +53.25
Pencil art    : http://www.marketgraph.nl/gallery/
Car simulation: http://www.marketgraph.nl/gallery/racer/

  Yikes!  I better get Gillespie's book too, as I must be doing this
incorrectly.  There's been a long, long series of debates in the past (in other
forums) on mechanical efficiency through a drivetrain.  Engineer types usually
insist it's around 95% or better, while drag racing folks insist it's usually
from 75-90% or so.  This changes with engine torque of course, so in my drag
racing predictor, I allow people to input a "torque lost to friction" value.
The results have been excellent in terms of prediction accuracy, but perhaps
I'm merely understating the rotational inertia effect as you pointed out.  Any
thoughts, Matt?  This program will be needing a complete re-write after my 3-D
physics engine is done.

Todd Wasson
---
Performance Simulations
Drag Racing and Top Speed Prediction
Software
http://PerformanceSimulations.Com

  Exactly.  This keeps both rear tires locked to the same speed, and seems to
work just like a spool type differential.  I'll have to study limited slip
diffs a lot more in order to get behavior for those done right.  It seems the
graphs for torque split vs. wheel rpm difference are the key, as there is a
region defined for "locked" behavior in many cases.

  As far as Richard Cheney's stuff goes, it's very nice.  He emailed me that a
couple of weeks ago (I didn't know he had it on a website like that.)  Too bad
I'm not using an OOP language for this.  Still, the entire physics engine runs
at 300Hz main sampling, with 3,000-30,000Hz sampling around the rotational axis
for the tires, in about 1 millisecond on a P400.
  I'll need to try converting vector3b.exe to use quarternions like you
mentioned.  If that works, converting the car sim to using them should only be
mildly difficult (yeah, right!)  
  BTW, what's the name of Gillespie's book ?
Todd Wasson
---
Performance Simulations
Drag Racing and Top Speed Prediction
Software
http://PerformanceSimulations.Com

--
Matthew V. Jessick         Motorsims

Vehicle Dynamics Engineer  (972)910-8866 Ext.125, Fax: (972)910-8216

These equations just properly account for the different rotational
accelerations being experienced by the different rotating parts.
If the overall gear ratio is 12, the engine is accelerating
at 12 times the radians/sec/sec of the wheels.
So just adding the physical rotational inertias doesn't
account for the different accelerations the parts are experiencing.

Inertia * acceleration is larger for the faster elements
so you have to properly account for that when you combine
them to form an effective inertia (which you are going to
multiply by only one acceleration value).

If you want to use one "effective inertia" for the entire
locked together drive train, the terms combine so that
the gear ratio terms are actually squared. I can't think
of a good simple motivation for why it turns out that
way, other than the "cascade" of the equations:
Tc = Te - Ie Ae
Tp = Gi Tc - Ip Ap        p = pinion shaft or drive shaft
... etc.

From just this part of it, with Ae = Gi Gfd Aw
and Ap = Gfd Aw

Tp = Gi (Te - Ie Gi Gfd Aw) - Ip Gfd Aw

so you can start to see how collecting the
inertia effects of several rotating parts geared in series
results in the proper "weighting" of the inertias
in calculating one effective inertia term for the
whole assemblage involves the square of the gear ratios
such as    Gi Gi Ie  above.

The reason it doesn't much matter for drag racing
is that it would be _too much_ if the
inertias weren't purposely made as light as possible.
A lot of the first gear time is also spent with
the clutch slipping so the engine acceleration
isn't involved. (The engine starts out at a high speed
and roughly stays there so it doesn't have to be
accelerated, at least until the clutch engages.)

I happen to be struggling against this effect at the moment
in some work I'm doing because I need to use higher
inertias than I should otherwise in order to avoid
numerical instabilities in the drive train.
Until we fix that next week, I've lost a couple
tenths of a second off my ET because of it.

--
Matthew V. Jessick         Motorsims

Vehicle Dynamics Engineer  (972)910-8866 Ext.125, Fax: (972)910-8216

It's an excellant basic book. His primary field of
expertise is in vibrations I believe, so besides
being good for basic vehicle dynamics, it also
covers vibrations. The appendices include the
SAE J607e recommended practice for vehicle dynamics
terminology and the SAE J6a Ride and Vibration Data Manual
so it is a convenient way to get those as well.
--
Matthew V. Jessick         Motorsims

Vehicle Dynamics Engineer  (972)910-8866 Ext.125, Fax: (972)910-8216

While I haven't measured power losses through drag racing gearboxes
(or differentials), I can understand why "engineer types" don't agree
with efficiencies of 75-90%.  Just look at the heat generation values
implied with low efficiencies like this.  While this might not be a
significant factor in the short time frame of drag racing, think about
for a longer race where steady state values are reached.  Using an
*average* horsepower of 500 bhp = 373 kW, an efficiency of 75% tells
you that the heat rejection is 25% of 373 kW, or 93 kW.  There is no
way that amount of heat is being rejected in a racing gearbox!

  Right!  This is what I don't understand.  In writing SAS (in my link below),
I read lots of debates on this subject and still don't have a clear answer on
what's really going on there.  Patrick Hale, creator of Quarter and Quarter Jr,
the pioneer in this field, appears to be the only one using efficiency values
in the 95+% range in his software.  Since it's listed as "transmission
efficiency", perhaps he's got another efficiency for the rest of the drivetrain
already built into his software.

   For some reason, when I use values that are unrealistic to engineers
(75%-90%), I come up with almost exactly the same outputs as Mr. Hale's
programs do.  I'm hoping that Ruud Van Gaal's book observations and Matt
Jessick's explanations will shed some light on this subject for once :0)  I
suspect that I, along with several other developers of drag racing predictors,
may be miscalculating the torque required to accelerate the drivetrain.  Who
knows?  I'll have to get Gillespie's book and do some studying.  

  On the other hand though, the high performance car and drag race magazines
and racers seem to agree that the driveline efficiency is indeed this poor,
regardless of what anybody says about the incredible heat that "should" be
generated.  The main thing right now that makes me suspect the lower values are
true is that my top speed predictions remain far more accurate with the low
efficiency numbers racers and dyno testers swear by, even though they should be
generating enough heat to do quite a bit of damage, from what others say.  

Todd Wasson
---
Performance Simulations
Drag Racing and Top Speed Prediction
Software
http://PerformanceSimulations.Com

...

That's where my problem still is, I can get to the Aw=... formula, but
FxR is an unknown variable in my head too. :) (making 2 unknown var's,
1 equation, deadlock)

Ok, so R = radius of the wheel.
Fx is then just the last (longitudinal) force that was calculated the
last 'frame' (derived from the slipRatio graph)?

I'll keep this in mind. Thanks for the tip. :)

Ruud van Gaal, GPL Rank +53.25
Pencil art    : http://www.marketgraph.nl/gallery/
Car simulation: http://www.marketgraph.nl/gallery/racer/