moment diagram with weight transfer [Archive]

OptimumG

07-23-2013, 04:13 PM

Brian, I believe we spoke about this a little a bit at FSAE-Nebraska.

The typical process of solving a given combination of beta (body slip angle) and delta (steering angle) for a fixed vehicle speed involves an iterative approach. A each point of the diagram you'll hold beta, delta and speed constant, and then iteratively solve for the lateral acceleration and the yaw moment acting at the vehicle cg.

As you mention, after the first iteration, the input lateral acceleration you started with and the sum of the lateral forces from the tyres will not match. This is when you would reiterate over the problem, using the calculated sum of the tyre forces as the input for the next step. After a few iterations there should be a very small difference between the two values, at which point you can call it converged and move on to the next point.

There are various numerical optimisation algorithms that can be implemented to reduce the number of iterations required, and even though the maths (or the name of whatever function you are calling) will change depending on your chosen method, the general process won't.

(As well as updating the weight transfer calculations within each iteration, be sure to update tyre slip angles as the yaw rate will change with each new lateral acceleration, as well as updating the rest of vehicle kinematics (camber, bump/roll steer etc).

I believe I have some slides on hand from the OptimumG seminars which cover this in further detail - if you are interested and have not yet been to a seminar, I can put them up here as well as some further examples.

--
Pete Ringwood
Research & Development Engineer

OptimumG
8801 E. Hampden Avenue, Suite 210
Denver, Colorado 80231 USA
+1 303 752 1562

pete.ringwood@optimumg.com
www.optimumg.com (http://www.optimumg.com) www.optimumg.com (http://www.optimumg.com)

Tiago_Campagnolo

07-24-2013, 06:05 PM

Pete,

I am very interested about this subject as well as further examples that you may have.

If possible, could you please share them with us?

The Moment Method is valuable and interesting tool and I want to create the diagram of our car too.

Thank you in advance.

Tiago Campagnolo

Claude Rouelle

07-24-2013, 11:40 PM

Iago

Race Car Vehicle Dynamics book from Doug Milliken is your #1 source.

The OptimumG seminar is another one.

slicktop

07-25-2013, 10:57 AM

Pete,
Yes, we did speak a bit about this subject at Lincoln. Thanks for the response - it was exactly the piece of the puzzle I was missing.

Viv

03-19-2015, 09:54 PM

Claude Rouelle

03-19-2015, 10:17 PM

Viv,

Do you have the isoline CG slip angle Beta = 0 and the isoline of the steering angle Delta= 0 crossing the origin of the coordinates (Yaw Moment = 0 and Lateral Acceleration = 0) ?

DougMilliken

03-19-2015, 10:33 PM

...I am wondering if it is truly possible to attain zero (or close to zero) yaw moment at max lat g ...
Short answer is yes, this is possible, I've seen it many times. But perhaps it is not possible with your combination of CG location and choice of tires (as modeled) along with any other effects included in your model.

How far are you off? What is the max lateral g available at yaw moment=0?

03-20-2015, 12:44 AM

... I am wondering if it is truly possible to attain zero (or close to zero) yaw moment at max lat g

Viv,

I think you are asking the wrong question. How about:

Q1. In order to win, does an FSAE car really need to be "balanced" at the limit (ie. have zero Yaw-couple at Max-Lat-G)?
Q2. Why?
Q3. If not, then how much unbalance is acceptable, and which way is better (US or OS)?
Q4. Again, if balance does NOT have to be "optimised", then what performance factors should you be focusing on?

And ... a whole lot of other important questions... But, for now, I would answer:

A1. No.
A2. Because bigger fish to fry (see A4).
A3. Production cars usually aim for considerable limit US, but in FSAE conditions quite a lot of OS can work well, although either US or OS are OK.
A4. Aim for MAXIMUM Max-Lat-G, and high enough Yaw-couple (with low enough car Yaw-inertia) to make the driver dizzy.

Note that "balancing" the handling of a car is often done by REDUCING the lateral grip capability of the "strong" end of the car, to bring it into balance with the weak end. This can make sense on 200+ mph Indy oval type racing. Makes little sense in FSAE conditions with virtually NO "steady-state" corners.

Rather than cripple the strong end of the car, keep trying to strengthen the weak end. Meanwhile, drive to suit...

Z

Tim.Wright

03-20-2015, 04:12 AM

Answer to question 1, 2, 3 & 4: What's the point of having a large Max Ay if the driver can never use it?

In my experience, the driver starts complaining of instability and general driving difficulties BEFORE you reach the point of "mathematical" instabilities. I.e. a change of sign of static margin or stability factor. In other words, you NEED a stable balance at the limit otherwise the driver (particularly amateur drivers) will never find it.

FSAE needs high maneuverability but not instability. There are ways to achieve this...

Tim.Wright

03-20-2015, 12:53 PM

Hi everyone. My name is Vivek and I am the suspension lead from the University of Toronto team. I attended Mr. Claude's seminar in Indianapolis where I learnt further details about the yaw moment diagram and have also read chapter 8 from Mr. Milliken's textbook. I have since spent hours programming excel to create yaw moment diagrams for constant radius turns. It uses pacejka models fitted to TTC tire data, ride/roll camber and toe, ackermann steering, and weight transfer. I am however having problems getting the zero yaw moment at max lateral g even after adjusting various vehicle parameters. Currently the vehicle understeers at max lat g.

Pardon my ignorance if this question is trivial but I am wondering if it is truly possible to attain zero (or close to zero) yaw moment at max lat g outside the theoretical world with all these effects accounted for or is there something I am doing wrong with my program. One reason I believe I might be getting these results are because of slight asymmetry in tire data lateral forces (left turn vs right turn). Also the yaw moment at max lat g currently sits at around -500 Nm at lat g of approx 2. I also have little intuition as to how big an effect such a yaw moment would I have (which I suppose really depends on the yaw inertia of the vehicle) and if it is worth spending further hours on the issue considering this is not even the full picture considering compliance, tire aligning moments, etc.

Have you done a bit of a sanity check on the model to see if its working correct? I.e. replaced the pacejka model with a linear one and checked that the results match up with the basic bicycle model equations of motion?

If thats ok the next step I'd try would be to see if the US/OS moment changes in the correct way and direction when you change the CG longitudinal position and the lateral load transfer distribution?

I have seen that when you include non-linear tyre models into a static vehicle model that is close to neutral steer at the limit you can run into some apparently strange numerical behaviours as both axles saturate at the same time. It has even happened to me that the model may not even have a single unique solution in some conditions.

One question, what is the yaw moment digram that you speak of? Is this the MMM?

Viv

03-20-2015, 04:23 PM

Thanks for your replies everyone. I will definitely heed your advice especially do take a few steps back and do sanity checks like the one Mr.Claude suggested and Mr. Tim Wright. My person preference from having driven the car last year is to have a bit of oversteer actually because for drivers like me (not experienced) oversteer maintains steering control (as stated in Milliken). If the car were perfectly balanced and if driver did not realize they are about to breakaway, there would be little opportunity to correct as all wheels breakaway at same time. (At least this is what I understand).

Hi Mr.Milliken,
I am not too far off. Max Lat G at 0 yaw moment is about 1.9 g and the max available is about 2.1.

Hi Claude,

Currently neither of them passes through the origin (a quick sanity check I should have done before). This is odd... The car is generating lateral force at 0 body slip and steering. The isoline at Beta = 0 crosses the yaw moment axis at about -500 Nm and the isoline for gamma = 0 at about -650 Nm.

I shall posted further progress.

Claude Rouelle

03-20-2015, 06:41 PM

Viv,

1. "Currently neither of them passes through the origin (a quick sanity check I should have done before). This is odd... "

That is the #1 mistake made by 99 % of the students who create a Yaw moment Vs lateral acceleration for the first time.

The issue is that you forget to switch from the tire coordinate system to the chassis coordinate system.

A beginning of explanation: Imagine a tire that you put on the tire testing machine. Let's say that you test that tire at 0 slip angle and 0 camber. You notice that that tire has a slight Fy. That is very possible because no tire is absolutely symmetrical. Let's say that that tire has an Fy pulling towards the left. Put that tire on the LF and on the RF of the car and you have 2 Fy pulling to the left, that means some lateral acceleration and some yaw moment without even a steering or a CG slip angle. You see were we are going....

2. If you yaw moment show you a positive number at maximum lateral G that means that the car over-steers ....on the simulation. There are tons of things that you assume in your simulation that are not necessarily real.
- How real is your tire model?
- Are the tire forces and moments that you measured on the tire testing machine for a given input of Fz, slip angle, camber, pressure the same that you will have on the track? And I am not even speaking about tire temperature....
- Does you model take into account compliance? Because your chassis, your suspension elements your uprights etc... are bending, twisting etc... maybe the camber and the slip angle you thought you have is not the one you effectively have.

Can you car effectively take 1.9 G in the skip pad? If the recorded data do not match the simulation. If not then AT LEAST one of them is wrong. on

Does it mean then that this kind of simulations not useful? No. but you need to remember that YOU WORK IN DELTA. THERE ARE TOO MANY PARAMETERS TO BE STOP ON IN ABSOLUTE VALUE. You yaw moment could be wrong by what you are interested in is the DELTA of yaw moment per % of weight distribution or per degree of front camber or rear toe. Then your simulation will be useful.

3. Tim Wright is right.... Before you get to maximum lateral G at the apex you need to enter the corner. Is the car drive-able?

Imagine a tire A that has +/- 10 % of his peak grip between 4 and 6 degree of slip angle. Imagine now a tire B that had twice the grip but only between 4.8 and 5.2 degree. That tire could be beating the lap record on a simulation software but in reality no driver would be able to drive it. The Yaw Moan Vs Lateral Acceleration has several useful indicators.
- The maximum G
- The maximum G at yaw moment = 0
- The control at the corner entry = variation of the yaw moment per degree of steering angle when the CG slip angle beta = 0
- The control at Apex = variation of the yaw moment per degree of steering angle when the CG slip angle beta is the one you have at maximum lateral G. Food for thought: Remember that if all 4 tires are at the ideal slip angle you have no reserve of front grip: if you steer more or less you lose front grip. Mario Andretti said " if everything is under control that means you are not going fast enough" :)
- The stability at the corner entry = variation of the yaw moment per degree of CG slip angle Beta when the steering angle delta is = 0
- The stability control at Apex = variation of the yaw moment per degree of CG slip angle Beta when the steering angle delta is the one you have at maximum lateral G. In other words does your car still has some stability has maximum lateral G? you car

4. Most of the drivers are not very sensitive to the grip but are very sensitive to the balance (yaw moment) the control and the stability. We have worked with professional drivers who can feel the 100 NM or even 50 NM of yaw moment difference but do not feel a difference between 3.0 and 3.1 of apex maximum lateral G.

5. A good advice. Start a yaw moment Vs. Lateral G exercise with a 2 wheel model and constant cornering stiffness, then with "real' non linear tires then and then only with a 4 wheel model and weight transfer. Simple then complicated not he other way around

Good luck,

Keep us posted.

DougMilliken

03-20-2015, 08:11 PM

... Start a yaw moment Vs. Lateral G exercise with a 2 wheel model and constant cornering stiffness,...
This model is available as part of the RCVD Program Suite, on the CD that is packaged with "RCVD: Problems, Answers and Experiments". Of course you will learn more if you write your own, but comparing with ours may be useful as you debug yours.
http://www.millikenresearch.com/rcvdps.html
At the bottom of the webpage are instructions to "register" this older software for recent versions of Windows.

03-20-2015, 10:18 PM

... What's the point of having a large Max Ay if the driver can never use it?
...
... you NEED a stable balance at the limit otherwise the driver (particularly amateur drivers) will never find it.

Tim,

I sense too much time spent in the book world (or Matlab++?), rather than the real world.

Consider two cars:
1. Tim-car - PERFECTLY BALANCED at the limit, which is at Max-Ay = 1.0 G (*).
2. Z-car - AWFULLY UN-BALANCED at the limit, which is at Max-Ay = 2.0 G. Car is only "balanced" (ie. Tz = 0) at Ay = 1.5 G.

Many Tim-cars are out there on the Enduro track, and thanks to their fantastically driver-friendly "balance", they are all cornering right on their Ay = 1.0 G limit,

Now a Z-car goes out on track. Because of its awful balance it cannot get anywhere near its Steady-State limit (although it does have mind-snappingly fast turn-in, which is working a treat through the slaloms). Nevertheless, the Z-driver plods along well below the car's balanced SS limit, and ONLY manages about Ay = 1.2 G through the corners (ie. only driving at "8 tenths"). Said Z-car proceeds to lap the Tim-cars lap after lap after lap... (Do the sim of 1.0 G vs 1.2 G!)

Moral - To win FSAE you DO NOT have to drive your car at its absolute limit. You ONLY need to drive faster than the other (student designed and built! :)) cars.

Z

(* Tim-team achieved its perfect balance by crippling the grippy end of the car, to bring it into balance with the weak end. Z-team just kept chasing more grip at whichever end was the weakest.)

Tim.Wright

03-21-2015, 06:54 AM

If you think you are able to pull an extra 0.5g of steady trim by reducing stability at the limit its quite clear which one of us has never engineered a car in his life...

NickFavazzo

03-21-2015, 08:56 AM

He was just proving a point...

JT A.

03-21-2015, 10:06 AM

He was just proving a point...

He wasn't proving anything. It was an entirely made up scenario, grossly exaggerated to make his point look better. Ironically, he prefaced this made up scenario by telling Tim he needed to spend more time in the "real world".

MCoach

03-21-2015, 03:28 PM

Most papers that focus on designing a lap sim or dynamics model of a vehicle usually end with a conclusion along the lines of:

"In conclusion (cliche) this model accurate models and predicts how our perfectly set-up car can handle around XXX track and correlates well to the car when we have a good day. However, this tool is still not useful for making set up decisions because it puts the car into highly unstable conditions that a driver would normally not run in. A normal driver would have hit the wall with words coming out of his mouth faster than the revs out of his engine. The simulation would drift gracefully within 1mm of every single cone, maximizing the necessary acceleration and yaw moment possible. Future work includes designing a driver control model to make the tool realistic in the limitations that are allowed to occur during operation. For example, a barrel roll is not considered a standard obstacle avoidance technique in ground vehicle motorsport."

DougMilliken

03-21-2015, 04:00 PM

... For example, a barrel roll is not considered a standard obstacle avoidance technique in ground vehicle motorsport."
That's called the Spiral Jump, not meant for motorsports, but a good way to validate a dynamic vehicle model -- see RCVD pages 458-460 <grin>.

Back on topic, perhaps you aren't looking deep enough in the literature? Try our 1994 paper -- "MRA Moment Method — A Comprehensive Tool for Race Car Development, SAE #942538". This is included on the CD that comes packaged in "RCVD: Problems, Answers and Experiments". The second half of the paper describes correlation, done in cooperation with Lotus F1.

BillCobb

03-21-2015, 06:46 PM

The "Astro Spiral Jump" (seen in "Man With the Golden Gun" was perfected using a version of DK4, a Fortran program I worked on as a Co-Op at GM Proving Grounds. This program originated with Ray McHenry at Cornell Aero Labs (now Calspan) for use in helping the Chevrolet Racing circuit teams. It ran from PUNCHED CARDS boys and girls. Back then, aero effects and wings linked to suspensions were the big thing

A fellow we hired from Calspan, Dennis Kunkle, had pulled out the DK4 transient solution leaving the fast steady state summary for use in calculating the first Moment Methods. I can still remember Calspan's Roy Rice's singing the praises of the Moment Method at a GM Vehicle Dynamics Department meeting chaired by Tom Bundorf (another name Blast from the Past: Father of the "Cornering Compliance Concept"). Later, DK4I (the IRS version) was used for development of the mid-70's Corvettes.

I still have the DK4 manuals in my "from GM to be burned" pile, unless my ex-wife already took care of that. The Astro Spiral filming in Thailand only took ONE film shot ! Behold the Power of Simulation.

MCoach

03-21-2015, 08:42 PM

I have read a lot about that particular maneuver for the Bond movie and referenced it for that exact reason. Also, the post was getting too serious and needed to rush it to the emergency room for a healthy injection of fun back into the conversation.
I don't think I have read that particular paper, but will check it out. The reason I say most is that most are published by those who are proud enough to get something working the first time rather than the top tier giants whom are focusing on things such as the influence of brake cooling on the tire tread and tire degradation. (http://scarbsf1.com/blog1/2013/09/06/mercedes-cross-cut-wheel-rims/)

BillCobb

03-21-2015, 09:25 PM

Loose is Fast.

03-21-2015, 11:35 PM

Tim,

If you think you are able to pull an extra 0.5g of steady trim by reducing stability at the limit its quite clear which one of us has never engineered a car in his life...

Of course it is possible! Think it through.

Or else, can you give a well reasoned argument why it is IMPOSSIBLE? (I note that I am still waiting for you to justify your claim that increasing R% reduces response time so much that it is not worth doing.)
~o0o~

JT A.,

... an entirely made up scenario, grossly exaggerated to make his point look better...

Where is the exaggeration?

There are countless cars (of all types) that are very well balanced with Max Ay around ~1.0 G. Putting wider and stickier racing-slicks on such cars can easily push Max Ay up to ~2 G, albeit with unbalanced (= non-neutral) limit behaviour. Such a car with balanced trim around 1.5 G is quite normal.
~o0o~

Claude and Pat keep harping on about how new members of this Forum should always introduce themselves.

I would rather see ALL members being expected to ALWAYS give an engineering justification to support whatever claims they wish to make.

EGs 1 & 2. Tim's claims above.

EG 3. Still waiting for Claude to explain why "Parallel Axes Theorem" must be used when calculating a Car-Body's dynamic motion about its "Pitch/Roll/Yaw axes".

And many more...

Please, a bit more engineering analysis, a bit less "I'm right, you're wrong, so there!".

Z

DougMilliken

03-21-2015, 11:50 PM

The "Astro Spiral Jump" (seen in "Man With the Golden Gun" was perfected using a version of DK4, ...
Hi Bill, sorry but http://xkcd.com/386/

Ray McHenry used a different model at CAL/Calspan for the spiral jump development -- HVOSM. The "Highway Vehicle Object Simulation Model" was initially funded by Bureau of Public Roads, part of the USA government. He and his son Brian (a high school classmate of mine) tell the story best on their own website:
http://www.mchenrysoftware.com/mhvosm1.htm#The%20Astro-Spiral%20Jump

Last year I helped arrange for Ray to give the second William Milliken Invited Lecture at a big ASME conference. He did a great job of telling the story of the spiral jump design and the initial tests run behind CAL/Calspan:
http://www.asmeconferences.org/IDETC2014/Keynotes.cfm

There are also some amusing stories from the very first public jump in the Houston Astrodome (several years before the James Bond version was filmed). CAL engineers went down from Buffalo to help with the production, as detailed in "Equations of Motion", pages 561-563.

-- Doug

JT A.

03-21-2015, 11:57 PM

Well then i'm going to make a car with EVEN WIDER and EVEN STICKIER tires than your car, and it's going to pull 6 g's unbalanced, but still a quite respectable 4 g's with balanced limit trim. And it's going to lap your car 3 times per every one of your laps. This is entirely supported by facts and engineering analysis, and not exaggerated at all.

You people and your silly little matlab simulations need to spend more time in the real world if you want to have any chance at keeping up my & Z's cars.

BillCobb

03-22-2015, 12:07 PM

Ah, you're right. It all comes back to me now. Thanks for the references.

Tim.Wright

03-23-2015, 06:23 AM

Or else, can you give a well reasoned argument why it is IMPOSSIBLE? (I note that I am still waiting for you to justify your claim that increasing R% reduces response time so much that it is not worth doing.)

You are the one putting forward an opinion - the onus is on you to prove it.

Please do so with something tangible. Maybe some real tyre data or some measurements to back up your point. At the very LEAST some feedback from a test driver on a car where you have tried such a setup (and laptimes to prove it was faster). In fact I'd be interested to know if you have ever even tried this setup on a real car or not?

My data on the R% argument is still a work in progress. Unfortunately my work in the cottage industry eats up most of my time...

PS: which reminds me - too much R% will give you linear range OS/instability. If you then setup the car to be neutral also on the limit you will have a car which is unstable everywhere (linear and limit). I assumed one of your tactics with a car with high R% would be to have a forward biased LLTD to make the car controllable at the limit but it seems not.

I hope the Z racing team are good at picking gravel out of the radiators...

03-23-2015, 10:40 PM

You are the one putting forward an opinion - the onus is on you to prove it.

Tim,

You are also putting forth an opinion, so should you not also prove it?

Anyway, the "proof" of my earlier example is in all the data-logging of all the racecars that have ever run. Some of those cars can rightly claim to be "well balanced" at the limit, with that limit close to Ay = 1 G. Other cars, lightweight with sticky racing slicks, have tyre Peak-Mu around 2.0, but not always at both axles at the same time. The "unbalanced" limit handling of my example follows. Check the data.

Oh, and BTW, do you accept that perhaps the easiest way to adjust a car's "setup", and the way that can make the biggest difference, is to change the tyres? Changing wheel widths also helps...
~o0o~

I look forward to your "proof" that Porsche 911s will never win a race. :)

Z

Tim.Wright

03-24-2015, 04:31 AM

I nearly lost an eye in all that handwaving

Viv

03-26-2015, 08:45 PM

Thanks everyone for the discussion and advice.

Hi Claude. Took your advice and began a bicycle model simulation with linear tires. Quick question...Should I be including slip angles resulting from yaw velocity?

Hi Doug. I will look into the book you mentioned with the CD. I should have bought that as a package when I bought RCVD :(

MCoach

03-26-2015, 09:12 PM

Z, I want to know your source for 2G lateral tires...

03-26-2015, 10:56 PM

Z, I want to know your source for 2G lateral tires...

MCoach,

Hoosier, Goodyear, Dunlop, Avon, Michelin, and many others.

Try Google, Ebay, TTC, your local tyre store..., for more leads.

Z

MCoach

03-27-2015, 12:56 AM

I'd go pick up some Maxxis tires if they were made in the right size.

03-27-2015, 11:16 PM

I'd go pick up some Maxxis tires if they were made in the right size.

MCoach,

These "Razr TTs" (http://www.maxxis.com/catalog/tire-191-109-razr-tt) would look great on my high R% brown-go-kart! And love the name...

6" wide fronts and 10" rears (with 11" section width!) sounds just right. Two compounds to choose from, including "ultra-high performance soft". Lots of suitable 10" ATV wheels readily available. Just a matter of checking cost and availability. I reckon the bloke down the road who has the blackboard out front advertising "cheap tyres" could source them. And TTC-data-be-damned. Just bolt them on and go a-testing! :)

Z

BillCobb

03-28-2015, 01:22 PM

Except NowDays, computer simulation and corresponding lab testing of chassis (K&C), tire (high speed Flat-Trac) and steering systems (ZF) can produce a car that's faster than an Old School driver can control. All the major teams that I've worked with are now faced with driver problems. (As your Boy Marcus Ambrosia sadly can attest). Some can adapt, others can't and a few are naturals with high powered, high sideslip and marginally out of control motor vehicles. Start watching for fighter pilots entering the auto racing venues which have closed loop control dynamics.

Here's all of what you will be able to hand off to your next of kin....

Silente

04-02-2015, 09:01 AM

hi guys,

first of all thanks a lot to everybody contributing to this thread. Really interesting.

I built up myself a tool in excel to create the Yaw Moment Diagram of a vehicle and would like to share here the results and open to contribution, since there are since a couple of points i am not sure about.

The vehicle model used a pacejka tire data set, for now only depicting lateral forces (i would like to add self aligning torque and longitudinal forces somewhen). It doesn't consider suspension geometry, for the time being (camber gain in bump and roll, static camber is set to 0, not toe or bump steer) and neither suspension compliance.
It anyway considers weight transfer, cg height, track width and lateral weight transfer distribution as an input, together with aerodynamic downforce and downforce distribution.

Although being a simplified vehicle model, it anyway already shows something interesting and matches pretty good to the logged data (i am using the scale coefficients in the pacejka tire model to achieve realistic performances).

I performed the simulation ranging with Beta and Delta between -12 and 12. I understand it is probably a too big range, but i wanted to be sure i was catching "everything", even with a different (more FSAE style) car model, in case i would need it.

I read with great interest the comments from Claude, in particular, and tried to extract the metrics he mentioned.
In particular, i did it for the same car running in three different conditions:

1) high speed cornering
2) low speed cornering, same setup as 1)
3) low speed cornering, TLLTD switched from 50 to 60% at the front.

The car in question is a high downforce prototype, with a pretty high weight (around 1000kg with the driver).

I added some pictures referring to the above scenarios. I will try to add the missing one (results for the high speed cornering simulation) soon, unfortunately only 5 files at a time can be attached.

What i find very interesting is looking how both control and stability are changing in the two low speed cornering simulations, beside all the other metrics more related to the terminal behavior of the model and to its maximum performance.
I would be interested first of all to hear your opinions to understand if i am looking to the right metrics and it would be great if Claude (or somebody with more experience about this kind of studies) could share some thoughts to understand if the results are anyway close to something realistic or maybe tell us which order of magnitude to expect for each metrics.

Thanks!

Silente

04-02-2015, 09:02 AM

*missing picture

DougMilliken

04-02-2015, 10:16 AM

... What i find very interesting is looking how both control and stability are changing in the two low speed cornering simulations, beside all the other metrics more related to the terminal behavior of the model and to its maximum performance.

I would be interested first of all to hear your opinions to understand if i am looking to the right metrics
Nice start. Now that you've played with a few different car changes, I suggest re-reading Chapter 8 to review some other possible metrics.

One cautionary note -- model results are never any better than the tire model.

Claude Rouelle

04-03-2015, 04:36 PM

Silente,

From what we see from the picture everything seems fine but unless I "play" with the software myself I can't see if everything is working well.

I notice that you have a car pretty neutral at low speed and understeering at high speed although with the high Y axis scaling it is hard to guess the real numbers. The contrary would not have been good for an amateur driver

I just encourage you to make a few sanity checks. For example
- Is the sum of the tire lateral force in the chassis coordinate equal to the car mass * lateral acceleration?
- is the sum of the Fz equal to the mass * 1 G of vertical acceleration + (0.5 * air destiny * Frontal area * Coefficient of lift * speed square)?
- Does the vertical load front / rear distribution remains the same with no downforce or taking into account the aerobalance if you have downforce?
- Is each tire lateral force decreasing once you pass the peak slip angle for a given tire at a given vertical load? Is the tire lateral Mu evolving in a usual / logical way compared to your tire model?
- Etc...

I think it is a good idea that for a first version you did not include the full kinematics but simply use camber Vs roll and heave ratio.

Including, the compliance, the longitudinal forces (and therefore your differential law) would be good but I would encourage to first make sure your software give you something USABLE the day you and/or your team members will want to develop your car

For example it would be good to develop a tool that gives car design and setup parameters sensitivity on performance. For example
- Weight distribution
- Springs, and ARB stiffness (and their motion ratio)
- Front and rear track width (all other parameters remaining the same: usually if you change the track the spring and ARB motion ratio will change)
- Initial toe and camber
- Dynamics camber / heave ratio and or dynamic camber / roll ratio
- Aero-balance
- Front and rear roll centers altitude
Influence on
- Each tire use: are you on the left or the right of the peak slip angle? in other words are you under or over using the tire?
- Grip
- Balance at maximum lateral G
- Control at the entry and the apex
- Stability at the entry and at the apex

It is also useful to do a yaw moment Vs lateral acceleration diagram for a given radius of a given corner and different speed and it that case try to find the ideal beta and delta combination which will give you the best speed and the yaw moment = 0 at the apex.

Remember that in any case this simulation will not be perfect (certainly not without compliance) but it will be useful if you work in variation and not in absolute value
If you are bale to overlay this with you data acquisition; you need your speed, a lateral accelerometer and a gyro

Claude Rouelle

04-03-2015, 05:00 PM

Silente,

Just noticed

Most of the time people show the dN/dDelta as positive and the dN/dBeta as negative (Hopefully stability is an anti yaw moment unless you go at very high beta angle) but I understand it depends on your initial convention sign and if it is a left hand or right hand corner. No big deal.

Claude

04-03-2015, 09:29 PM

Silente,

Hmmm....., well ... it all looks wrong to me!

But I cannot read the labels on the diagrams, I have no idea what units you are using, no idea what your tyre curves look like, no idea which way is which, and a whole lot more unknowns.

Specifically, if your three diagrams are supposed to be similar to Milliken's Chapter 8 MMM diagrams, then are not one set of "construction lines" supposed to slope up-to-right (ie. the effect of steering the front-wheels), and the other set supposed to slope DOWN-to-right (ie. effect of the tail "stepping out")? Maybe you have adopted a different sign convention? Right now, I have no idea what that might be?

Anyway, I think you will get better feedback if you present the following:

1. Sketch of plan-view of car showing your definition of positive X and Y directions, positive rotation (for direction of the Couple, or "turning force", acting on the car), your assumed position of the centre of the corner that the car is driving around, and clear descriptions of all other important parameters such as Beta, Delta, etc., including their DIMENSIONS.

2. Simplified depiction of the F&R tyre slip-angle curves (with DIMENSIONS!), perhaps as in Figure 8.11 of RCVD.

3. A single MMM diagram, perhaps like your low-speed example, but with much clearer annotation of the axes and what the lines on the diagram represent. Values of things like "dN/dDelta @ Beta = 0" can be hand-written in a corner of the diagram, with an arrow pointing to which part of the diagram that value refers to.

(To repeat, at the moment it looks like steering your front wheels rightward causes the car to turn the same direction as steering the rear wheels rightward??? That's one odd car!)

Z

DougMilliken

04-04-2015, 12:34 AM

Z & Claude, I agree that definitions are in order -- good homework for Silente<grin>.

But, I think all is well because FSAE cars normally operate below tangent speed? Everything presented in RCVD is above tangent speed.

Claude Rouelle

04-07-2015, 08:19 PM

Silente,

I sent you a Private Message

Claude

Viv

04-26-2015, 10:58 PM

I wish to quickly confirm a few things regarding corner entry analysis. So following Claude's seminar slides, the thing to look at is control and stability at corner entry i.e. change in yaw moment with steering at body slip =0 (control) and body slip at steering = 0 (stability). Is it fair to say that the weight transfer that has occurred for this is Non-suspended mass weight transfer + geometric weight transfer or should zero weight transfer be assumed? I hope this wasn't an unreasonable question.

Thank you for you help.

Claude Rouelle

04-27-2015, 03:02 PM

Viv,

2 perspectives (going the same way) keeping in mind that the Yaw Moment Vs Lateral Acceleration method is not a transient simulation

1. If you create a steering angle and the beta remains zero you create a slip angle at each front tire therefore you create a tire front tire lateral force therefore some lateral acceleration at the CG therefore you create weight transfer.

2. In your Yaw Moment Vs Lateral Acceleration graph when you follow the isoline beta = 0 and you go from delta = 0 to delta = 1 you create on Yaw Moment that you can observe the Y axis, correct? But you also create a lateral acceleration that you can see on the X axis, correct too? So if you have lateral acceleration unless you CG is on the ground, you will have weight transfer

".... following Claude's seminar slides, the thing to look at is control and stability at corner entry..." ....Yes that and many other things!

04-27-2015, 10:32 PM

...keeping in mind that the Yaw Moment Vs Lateral Acceleration method is not a transient simulation.

Claude,

This has me baffled. Why not?

Or, if not a "transient simulation", then what is it?

Or, given that Viv wants to do a "corner entry analysis", then how should he go about solving this necessarily "transient" problem?

Or, to get to the core of all this, what exactly is your definition of "transient", in VD terms?

Z

DougMilliken

04-28-2015, 08:43 AM

This has me baffled. Why not?
Or, if not a "transient simulation", then what is it?
I invite you to read the opening sections of RCVD Chapter 8 including the large footnote on the first page...

Bill Milliken (my father) applied the techniques of aircraft static stability and control analysis (developed well before WWII) to automobiles. Aero engineers have whole books on this topic, sometimes also called "quasi-static" analysis. A quick google turned up this online course, lecture 2 gives a very simple overview.
http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-333-aircraft-stability-and-control-fall-2004/lecture-notes/

04-28-2015, 10:57 PM

Doug,

Having just skimmed through RCVD Chapter 8, I note the following:

1. Chapter 8 is called "Force-Moment Analysis". I take this to mean that it is an analysis of the "actions" exerted onto the car-body from the road (although "Plan-View Force-Couple...", or "...Force-Torque..." would be better, IMO :)). But half-way through the chapter the diagrams start being labelled with X-axis = "g"s and Y-axis = "lb.ft". Why change only the X-axis from the cause (ie. force) to its effect (ie. acceleration), and not the Y-axis?

2. From the introductory sections it is clear that MMM is a direct application of D'Alembert's Principle. I take it that this is primarily to allow actual physical testing of real cars (or real models), and DAP conveniently gives the testers all the time they want to take whatever measurements they want. That is, it turns a "transient" problem into a "quasi-static" one.

3. There is a confusing section on page 294,
"8.2 Constrained Testing.
... a study of the steady-state forces and moments on the vehicle. These forces and moments are not only those associated with equilibrium (balanced) conditions, but those "unbalanced" forces and moments which are available for linearly and angularly accelerating the vehicle in maneuvers." (Emphasis is in original text.)
I have written (some years ago) "Huh!!!???" in the margin next to this section.

4. Much later (p308) you have a section "Steady State Defined", which, as best as I can interpret it, means that you consider just about any manoeuvre to be SS, just as long as it is only considered for an instant. (BTW, Euclid usefully always put his Definitions on page one of his books.) RCVD's Index contains no entry for "Transient", at least not under the "T"s, so hard to know what you mean by it.

5. My dictionary defines "transient (adj.) 1. for a short time only; temporary or transitory...". Personally, I consider all existence to be transient (or as per the I Ching, "the only constant is change itself..."). Conversely, SS is just an occasionally useful abstraction that dumbs-down a problem enough to make its solution a bit easier. As such, I reckon that whenever a problem is described as SS, the great many simplifying ASSUMPTIONS being made should be very clearly spelled out. Preferably right at the beginning of that Chapter... :)
~~~o0o~~~

Summing up the above, it seems to me that MMM is, indeed, a study of "transient" behaviour.

In its physical, experimental, form it uses a simplified version of DAP to make the measuring more convenient, so it artificially appears to be SS. But in its theoretical, analytic, form it represents (with many simplifying assumptions *) a real car accelerating, laterally and in yaw, on a real racetrack (or else it represents the forces that would cause such accelerations). Each point on the diagram represents the car at one transient instant in time.

Comments and criticisms of above interpretation most welcome! :)

Z

(* PS. On p296 is "... Thus simulations of constant [X, Y, and Yaw] acceleration can be had on a constrained tester." To which I have added "But not gyroscopic effects (due to rotations)...".)

DougMilliken

04-29-2015, 02:48 AM

Z -- I didn't write that chapter, but I'll give these a try.

1. ... half-way through the chapter the diagrams start being labelled with X-axis = "g"s and Y-axis = "lb.ft". Why change only the X-axis from the cause (ie. force) to its effect (ie. acceleration), and not the Y-axis?
This could be a simple matter of practicality -- RCVD took eight years to produce and at the beginning of that period we didn't have a lot of diagrams that we could use publicly (other customers didn't want their results made public). The Chaparral set had been done earlier and were available. Later in the chapter the Y-axis is in Yaw moment coefficient, Cn.

2. ...primarily to allow actual physical testing of real cars (or real models), ...
Similar to traditional wind tunnel testing, but with tire forces as well as aero forces.

3. There is a confusing section on page 294, "8.2 Constrained Testing. ...
What part of the statement bothers you? Note that the car (or model) is constrained, same as a model in a wind tunnel attached to a sting or other balance..

4. Much later (p308) you have a section [I]"Steady State Defined", ... means that you consider just about any manoeuvre to be SS, just as long as it is only considered for an instant.
More recently we have started to use the word "snapshot" -- does this help?

5. ... RCVD's Index contains no entry for "Transient", at least not under the "T"s,
It's under "Stability and control, transient". We did not produce the Index, SAE had that done. Over the years I've added entries and this would be a good one to cross-reference -- "Transient, see under..." It might make the next printing.

The "Comments Requested!" page in the back (either before or after the Index, it has been moved around) has resulted in numerous corrections over the years. SAE lets us make changes as long as they don't change the pagination.

... SS is just an occasionally useful abstraction that dumbs-down a problem enough to make its solution a bit easier.
Exactly, very important before there were powerful personal computers. It also allows presentation of the maneuvering range (envelope) of a car in one diagram. Much easier to think about than a stack of transient responses to a whole raft of different maneuvers.

... whenever a problem is described as SS, the great many simplifying ASSUMPTIONS being made should be very clearly spelled out.
It sounds like you want some things added to Section 8.2? Can you be specific?
* Many of the assumptions have been described in earlier chapters (but we've probably missed some).
* While Section 8.2 could mention gyroscopic effects, the model in Figure 8.1 is held fixed, seems pretty clear to me that gyro effects are not being considered for that case of a straight path.
* We weren't thinking about autocross and specifically FSAE (low speeds, small radius turns, etc) when the book was written--that would have changed the focus in many parts of the book.

04-29-2015, 11:56 PM

Doug,

First some brief answers to questions you asked above (more general comments below).
~o0o~

What part of the [Section 8.2, p294] statement bothers you? Note that the car (or model) is constrained,...

My confusion comes from that section initially discussing the "steady-state forces and moments" acting on the "constrained" vehicle, but then adding that those are also the "...'unbalanced' forces and moments which are available for linearly and angularly accelerating the vehicle". Having read it several times now, I think it would be clearer if it said something like;

"... the steady-state linear-forces and couples acting on the constrained vehicle represent the forces and couples which linearly and angularly accelerate the real vehicle..."
~o0o~

More recently we have started to use the word "snapshot" -- does this help?

Yes! I think "...different points on these MMM diagrams are each a brief 'snapshot' of a potentially transient manoeuvre..." is MUCH better than pretending that the diagrams refer to some sort of fuzzily defined "Steady-State".

I have now found in RCVD that "SS" is quite explicitly defined in the "...Transient..." Chapter 6, on p232, as "steady-state turning was defined as no time variations in the motion variables, r and beta (or v)". However, that quote is actually referring (via the "was defined") to the previous Chapter 5, section 5.9, where SS is much LESS clearly described, with NO use of the word "defined". (More on this below.)
~o0o~

It sounds like you want some things added to Section 8.2? Can you be specific?

Hmmm.., thinking "...snowflake's chance in hell here...", ... but see below... :)
~o0o~

While Section 8.2 could mention gyroscopic effects, the model in Figure 8.1 is held fixed,...

The two "real world" tests shown in Figure 8.2 have the real car undergoing real Yaw motion. So these real cars will feel the gyroscopic effects of their rotating wheels, engines, etc., as changes to their Fz tyre loads, etc. So, any theoretical analysis of these tests that is attempting to be "realistic" should also model these gyroscopic effects.

Or, at the very least, it should give a clear ESTIMATE OF THE ERROR of ignoring such. (Yep, I am going to keep on about "error analysis" for a while yet... :))
~o0o~

More general comments about RCVD...

Over the years I have had quite a few PMs from people struggling to grasp some parts of RCVD, particularly the equations. And, frankly, even though the "Mechanics" in question are quite simple, I too have struggled trying to "reverse engineer" said equations.

Most of the problems seem to be due to the many, many different symbols used, which are not always consistently used, and not always clearly explained. Often several of the earlier chapters have to be read COMPLETELY through in order to discover the many simplifying assumptions, approximations, etc. that are used in the later equations. Even then, some "simplifications" seem to switch on and off at random...

As an example, in Chapter 8, CN seems to mostly mean "N/Wl". But ... looking at the "List of Symbols" on the inside of the front cover (of my edition) I find "CN Yawing Moment Coefficient = N/W". And what are the units, or "dimensions", of these many symbols? Specifically, can I rest assured that "Weight = W = lb = pound = force", always? And is "m = mass" always, and is that always measured in, err..., ummm ..... "slugs"???

So, ... (back to feeling like a snowflake in hell...), maybe the whole book could be heavily pruned, reformatted, and generally tidied-up? With a clear and consistent list of Definitions and Assumptions at the beginning of each of the technical Chapters. And, err..., most importantly ..... METRICATED!!!??? (Aaaarrghhh!!!! What will the SAE say about that!!!)
~o0o~

As a specific example of above, the Chapter 8 MMM diagrams currently have the X-axis as the non-dimensional (and thus rather arbitrary) acceleration in "g"s. But the Y-axis is either in "N = lb.ft", or the non-dimensional, but rather meaningless, "CN = N/Wl"s. As a step in the direction suggested above, these X-Y axes could be relabelled in "Newtons" and "Newton-metres" (ie. the forces acting on the car), or else in "metres/second-squared" and "radians/second-squared" (ie. the resulting linear and angular accelerations of the car, when it is UN-constrained).
~o0o~

Finally, does anyone else think that these MMM diagrams are anything other than maps of the potential TRANSIENT performance of the modelled car?

Or, putting it slightly differently, what is the point of modelling a racecar's "Steady-State" performance, when such SS is taken literally to mean "with nothing much changing, at all"?

Claude???

Z

Edward M. Kasprzak

04-30-2015, 09:21 AM

Finally, does anyone else think that these MMM diagrams are anything other than maps of the potential TRANSIENT performance of the modelled car?

I think MMM diagrams are useful for understanding potential transient performance plus stability, control, trim, sensitivities, balance, limit behavior, on-center behavior and responses to changes in vehicle configuration. I probably missed a few others, too. It certainly isn't the only tool I use but I wouldn't want it missing from my toolbox.

Tim.Wright

04-30-2015, 12:48 PM

Or, putting it slightly differently, what is the point of modelling a racecar's "Steady-State" performance, when such SS is taken literally to mean "with nothing much changing, at all"?

I like looking at steady state responses because the DERIVATIVES of all of the curves (swa vs ay, beta vs ay, Fy vs alpha) tell you a metric shit-ton about the stability and the general character of the car - so they are quite useful numbers to know. Once you fit a pacejka curve to your measured/simulated Fy vs slip measurements of the front and rear axle you are then able to calculate and express all the parameters from RCVD as a function of the lateral acceleration. I.e. stability factor, static margin, yaw natural frequecies, yaw damping, elastic steer angle. You are able to watch how these values evolve as you travel from the on-centre region, through the linear range and then at the limit.

Finally, does anyone else think that these MMM diagrams are anything other than maps of the potential TRANSIENT performance of the modelled car?

I think its a pretty good summary. The potential transients captured in the MMM allow the calculation of the aforementioned metric shit-ton of parameters also in non-steady state conditions.

Consider also that transient is nothing but the passage towards a steady state condition. If you good understand the steady state - you are alerady halfway towards understanding the transient.

Flight909

04-30-2015, 02:10 PM

MMM are constrained steady-state simulation so the situation that is evaluating is not steady-state, once the constraints are "removed" the situation will change (for most cases). By investigating the yaw moment for a given constrained situation will give you an indication of how the vehicule will react when the constrained is removed. Will the rotation increase or decrease as a example. So in such way you can say it is possible to used for transient analysis.

Pennyman

04-30-2015, 08:27 PM

Finally, does anyone else think that these MMM diagrams are anything other than maps of the potential TRANSIENT performance of the modelled car?

Isn't transient performance related how quickly the car changes direction? Isn't that just the rate at which the car moves through the different states of the diagram? How would one extract that information from the diagram itself?

DougMilliken

04-30-2015, 10:48 PM

... I think it would be clearer if it said something like;
"... the steady-state linear-forces and couples acting on the constrained vehicle represent the forces and couples which linearly and angularly accelerate the real vehicle..."
That might be a little better for some people, but the first couple of posts below yours shows that others have been able to work it out...

Yes! I think "...different points on these MMM diagrams are each a brief 'snapshot' of a potentially transient manoeuvre..." is MUCH better than pretending that the diagrams refer to some sort of fuzzily defined "Steady-State".
We keep trying new ways to describe them to customers. Anyone that has taken a course in aircraft "statics" seems to get it right away.

The two "real world" tests shown in Figure 8.2 have the real car undergoing real Yaw motion. So these real cars will feel the gyroscopic effects of their rotating wheels, engines, etc., as changes to their Fz tyre loads, etc. So, any theoretical analysis of these tests that is attempting to be "realistic" should also model these gyroscopic effects.
I believe that our software (which I haven't used in a number of years) does include wheel gyro effects, but not engine. We didn't intend for the book to replace the detailed manual that comes with the software, so this chapter hits major points. Bill/Dad was trying to "teach", not to write a cookbook for someone to write analysis software (which would have been competitive with our own product!) Also, remember that we were not thinking about autocross or FSAE where turn radii are small, and yaw rates are very high.

... As an example, in Chapter 8, CN seems to mostly mean "N/Wl". But ... looking at the "List of Symbols" on the inside of the front cover (of my edition) I find "CN Yawing Moment Coefficient = N/W".
So how long have you known about this error? I just checked back to our final draft (1994) which correctly includes the lower case "L", in script font. SAE made an error in typesetting that we didn't catch, although we did catch hundreds of similar errors in a month where we did nothing else but proof the typeset version. If you told us sooner it would have been fixed by now.

And what are the units, or "dimensions", of these many symbols? Specifically, can I rest assured that "Weight = W = lb = pound = force", always? And is "m = mass" always, and is that always measured in, err..., ummm ..... "slugs"???
Isn't that part of the fun<grin>? Seriously, we tried to coordinate things like this between material generated over many years and by several authors, and obviously we weren't completely successful. As a rough guess, we spent hundreds of hours to get it to the current state.

..... METRICATED!!!??? (Aaaarrghhh!!!! What will the SAE say about that!!!)
Believe it or not, when we submitted the final manuscript (in person, at SAE near Pittsburgh), they asked to have it redone in ISO units. Note that ISO doesn't allow "degrees", all angles are in radians... Since nearly all the developments that we discuss were originally done in US/UK* in "customary units" we weren't about to convert. If nothing else, switching to an unfamiliar set of units opens up another set of errors, not to mention the work involved in re-doing many plots from old research projects to put sensible scales on them.
(* I believe that similar work was also done in German, but we didn't have access to very much of it.)

So, if you want "metric", just which version of metric do you want? I doubt that the Aussie conventions will be convenient for others using their own regional versions of "metric".

...the Chapter 8 MMM diagrams currently have the X-axis as the non-dimensional (and thus rather arbitrary) acceleration in "g"s. But the Y-axis is either in "N = lb.ft", or the non-dimensional, but rather meaningless, "CN = N/Wl"s. As a step in the direction suggested above, these X-Y axes could be relabelled in "Newtons" and "Newton-metres" (ie. the forces acting on the car), or else in "metres/second-squared" and "radians/second-squared" (ie. the resulting linear and angular accelerations of the car, when it is UN-constrained).
We have worked with a wide variety of vehicles, of different sizes. Non-dimensionalizing to CN is actually a nice way to think about it--once you get a feel for it.

Claude Rouelle

05-01-2015, 01:00 AM

Some directly or indirectly related comments

1. Doug Milliken, Edward Kasprzak, Eric Erik Zapletal, Tim Wright, Bill Cobb, Silente…. A good debate of a lot of experts or experienced guys (or old farts, whatever we want to think of ourselves) but no students…. We do not know if the readers of this thread are learning something, understand or not agree or not, and without their reactions we won't not know…. If it is only a “expert debate” that we defeat the purpose of this forum. Students; are you there?

2. Transient simulation supposes to have additional input like damper and inertia. But we also need the road profile (bumps, curbs – good luck!), and most important a transient (and ideally thermal) tire model that goes way beyond the simple notion of the tire relaxation length (or time). Oops several other additional layers of complexity. Passed a certain level, complexity goes against usefulness. I concede that level is higher every year one of the reason for that being the increased speed and the amount of data that new computers can deal with. That being said....
- Even for passenger cars simple simulation of ISO tests or for race car lap time simulation without dampers (as, if I can express it this way, more a succession of “pictures” more than a movie) are quite useful .
- My personal experience is that a 4 or 7 post rig test will tell me more on the transient simulation than most software
- If there is one layer of useful complexity that you want to add in any basic steady state simulation it will be the compliance way before any transient inputs (as noticed in other recent threads).

3. With the risk to explain what some of the readers of thsi forum already know.... In simple terms transient can be viewed as nothing else than the car behavior from one steady state to another. Example: a driver is on a skip pad of a radius of 100 m and drives as fast as the tires, the car and his skills allow him to do. You now ask him to go as quickly as possible without understeer or oversteer to another skip pad of a radius of 50 m. V1 = r1*R1 and V2 = r2*R2, r being the yaw velocity. In the 2 steady state circles r1 and r2 (yaw velocities) are different but are constant so Yaw Acceleration 1 and 2 are is zero. Yaw moment being yaw inertia * yaw acceleration, the yaw moment is (well should be) zero if the car is well balanced on these 2 circles. But from one circle to another there will be a delta speed, a delta time (that you try to minimize) and a delta r. Delta r / delta time means a yaw acceleration, therefore a yaw moment. If you have too much of it you have oversteer, if you have too little of it you have understeer. If you understand this, nothing prevent you to make more iterations and calculate the yaw month needed between the 100 m to 99 m radius, 99 m to 98 m radius and soon on.
So the goal is to get the yaw mot you want when you need it.

Well but there are 13 causes for the yaw moment :
- 4 tire Fx * by their effective ½ track
- 2 tire Fy front * a (a being the distance from the front axle to the car CG)
- 2 tire Fy rear * b (b being the distance from the car CG to the rear axle)
- 4 tire Mz
- Possibly an aerodynamic Mz

Let’s not forget to put every forces Fy and Fx in the car coordinates (not the tire coordinates) system so the 2 front wheel steering angles will have to be taken into account

Each car design or setup parameters (spring or ARB stiffness, weight distribution, aero downforce and downforce distribution, even each of the tire Pacejka coefficients, you name it.... ) will have an influence (sometimes big sometimes small) on some or all of these 13 Yaw Moment causes and therefore the car yaw moment POTENTIAL (yes IMO too that is the right term) to get from one state to another. The goal will be to study the influence that each of these parameters will have on the pure lateral grip or combined lateral + longitudinal grip trying to get the best of each (that means grip, that means speed) AND the Yaw Moment you want when you want (that means balance).

4. The yaw moment Vs Lateral acceleration method is the best tool I ever have used in race car engineering and vehicle dynamics simulation. Kudo to the Millikens and the guys before them who initiated and explained that tool. I will never be perfect but it is always useful to evaluate if not the exact at least the relative influence (remember we work in Delta comparison) of practically all cars design and setup parameters on target such as grip, balance, control and stability. If you overlay the results of this car simulation analysis method with good guided drivers debriefing (where drivers are demanded to qualify and quantify the car behavior in terms of …guess what…. grip, balance, control and stability) in different parts of a given circuits (or even different parts of a given corner of a given circuits) and good onboard acquired data analysis you have THE tool that makes winning cars. We use it with great success in GT, single seater cars, prototypes. AS a design judge I KNOW that the best teams in FS/ FSAE know how to use it.

5. Very honestly I never have included the wheel gyroscopic effect. Maybe I should. Maybe I will. One thing I know is that the Yaw Moment Vs Lateral acceleration method allows us to decide what is the car design and setup parameter(s) we want to modify to reach given goal(s) of grip, balance control and stability. It helps us to make decision that are qualitatively right 9 times of 10 and 6 times of 10 pretty close to our goal quantitatively. With it you win and you know WHY. Perfect? No. Useful? You bet!

DougMilliken

05-01-2015, 01:17 AM

...but no students. ...
Just filling dead air while the students get ready for Michigan competition in 2 weeks...

njgedr

05-01-2015, 11:39 AM

Hello Experts!

My name is Noah Gedrimas and I am the suspension co-lead for Oakland University in Michigan.

Your discussion about yaw moment vs. lateral acceleration is interesting, and more or less how I have always thought about balancing a car from a quasi-static force balance point of view. It makes sense that the yaw angular velocity and lateral acceleration must line up to go around a turn without any over or understeer, assuming steady state around the turn.

From a theory point of view it is certainly elegant, and a good way to explain oversteer and understeer mathematically. However, I was hoping you could give examples of how and when you used it.

What I am thinking, albeit without ever looking at gyro data is:

1. Plot data of yaw acceleration and lateral acceleration on the same graph (gyro and accelerometer)
2. Compare the curves to see the delay between lateral acceleration and yaw velocity.
3. Look at just the yaw acceleration graph
I am expecting I would see the yaw acceleration go above zero then back to zero (the transient stage, corner entry). Stay around zero for some period of time, then drop below zero for corner exit.

4. Adjust the car to shorten the delay between yaw accel and lateral accel? Minimize yaw accel variation (I expect it will oscillate a bit even in steady state section of turn). Something else maybe?

5. Retest and check differences

This sounds very valuable and I want to find the answers on my own through testing, but for sake of discussion and maybe getting some more ignorant voices involved can you discuss some specific instances with us?

In short: A quick example of a test day from your professional careers would be very interesting: e.g. Measured this, driver said that, simulation said this, we determined Roll stiffness was too low, adjusted whatever and it showed up in the data as this.

For me it would really help bridge the gap from the paper or calculator to the track. I am not looking for a simple "if this than that" (though that section of RCVD is excellent). Instead, at this stage of my life I am more interested in the application experiences rather than just the theoretical understanding... the theoretical stuff on this forum is very humbling and impressive to read, but it is too involved for most teams. I expect the experts developed much if not all of their theoretical understanding at least partially based on their application experiences.

As for student involvement on the forum, I think the chassis discussions often swing too far into the abstract side which is tough for students to follow, let alone contribute too. The experts have dozens of years experience and literally wrote the books. I would love to see more discussion around the application side (I am sure there is some great stories and lessons learned from forum members) as it is easier to follow and participate in.

Regards, Noah

Pennyman

05-01-2015, 01:08 PM

Hey Noah, do yourself a favor and get yourself Jorge Segers' book (SAE #R-408)

Claude Rouelle

05-01-2015, 02:23 PM

Noah,

If you want to know more about the theory and also the tips and tricks (especially how to link simulation. data acquisition and driver feedback) on how to use the Yaw Moment Vs lateral acceleration method both in simulation and onboard data analysis, you may want to come to the seminar organized in East Lansing at the Michigan Sate University, right after the FSAE competition. Out of 4 intensive days one complete afternoon is dedicated to this tool

See point 8 of http://www.fsae.com/forums/showthread.php?12067-Conception-Simulation-Testing-Developing-of-a-FSAE-car.-OptimumG-seminar

The yaw moment Vs lateral acceleration diagram is a quasi steady state simulation tool that show the POTENTIAL of the yaw moment and lateral acceleration at
- a given steering angle
- a given car CG slip angle
- a given speed

While if you plot the derivative of your gyro signal multiplied by your car yaw inertia Vs your lateral acceleration you have the real thing.

You will see that the quasi steady state simulation (prediction) and the reality (data analysis) are not the same but if your simulation has been well written you will see also that every time you change something on the car (for example the rear toe) the yaw acceleration and the yaw moment change on your data qualitatively in the same direction and quantitatively about the same percentage that they change on your simulation.

I also want to insist that the biggest tools of the Yaw Moment Vs lateral acceleration method are the quantification of each car design and setup parameters on the CONTROL and the STABILITY. That is the hidden part of the iceberg that most students do not see and do not use.

About a dozens of FSAE / FS teams - manly in Europe I have to say- have been able to create their own Yaw Moment Vs lateral acceleration graph AND USE IT. FSAE / FS STUDENT TEAMS; not "experts"!!! .....

And guess what.... they are often the most competitive teams. Cause or consequence? Maybe they had the time and the money and the necessary education and "culture" to start creating and efficiently using this method? Or maybe that had the curiosity and intelligence to create this tool BEFORE they design their car and that is what made them creating a good team or a good car? Or maybe a bit of both? I don't know. But in any case they had the courage to at least try something unusual in their FSAE / FS world but that doesn't come from the moon: that method has been used with success in the passenger car and the professional race car industry for years and is one of the centers of the RCDV book.

Also, you wrote "Plot data of yaw acceleration and lateral acceleration on the same graph (gyro (Note: not only gyro but also derivative of the gyro) and accelerometer) and compare the curves to see the delay between lateral acceleration and yaw velocity" Not a bad idea but you will have to put your steering angle and your speed in there. After all the yaw moment and the lateral celebration occurs if you first do something to change the steady state nd that "something" is turning the steering wheel. The delay you mention will not be the same even for the same steering input if you are 10, 20, 40, 60 or 80 or 100 km/h.So te speed has to be taken into account.

Let's dream; beyond the use of the Yaw Moment Vs lateral acceleration method, as a design judge I wish to see one day a FSAE / FS team showing me the plot of the steering, the gyro, the derivative of the gyro, and the speed for a given so called "step steer" input at different speed. The team will have to do that test on a straight line, not on a circuit. The display of
- the lateral acceleration response time (LART)
- the yaw velocity rising time (YRT)
- the yaw velocity settling time
- and the yaw velocity damping (YVD)
for different car setup change will tell you more about the car transient behavior than any test on a circuit. In fact it would prepare you to have a better car on the circuit because you will understand and know what car parameters change what transient performance criteria and by how much.

These testing methods are also presented in the OptimumG seminar

Ideally it would also be nice and USEFUL to also make in a large straight line part of a circuit different sine steering sweeps at different speeds and different frequencies; not easy though to make identical and repetitive steering sine waves at a given amplitude and a given frequency; that is why such passenger car tests are made with a steering robot.

Claude Rouelle

05-01-2015, 10:05 PM

Doug,

There are much more students reading this forum than "just" the ones from the Michigan FSAE participating teams. Michigan is not the center of the world... Not mine anyway....

DougMilliken

05-02-2015, 01:06 PM

Hi Claude,

But I think Michigan is still the largest of the competitions? Most cars, most students?

I agree, it would be interesting to see the results of a step steer test -- the test procedure was originally developed by General Motors (primarily), then transferred to the SAE Vehicle Dynamics Committee, and is now an ISO Standard.
ISO 7401:2011 Road vehicles -- Lateral transient response test methods -- Open-loop test methods
http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=54144

I don't have the SAE version number handy, but from memory it's easier to follow in some respects, pre-dates some of the complexity added more recently. If anyone is interested I can dig it out.

-- Doug

BillCobb

05-02-2015, 10:54 PM

For Noah's benefit:

The difference between yaw velocity and lateral acceleration signals you are contemplating (actually speed times yaw velocity and lateral acceleration) is actually something named the "Zero Sideslip Algorithm" most commonly employed in 'rear steer' controls (more correctly, that is, 4 wheel steer). The integrated speed yaw velocity product difference from roll corrected lateral acceleration is the vehicle's body sideslip angle. As used in performance car testing, the constant radius test is a very good method to evaluate your car's setup using this sideslip function because it will reveal the car's "tangent speed", which is the speed at which the body sideslip is zero (i.e. the speed yaw velocity product is the same as the true lateral acceleration). The goal being to get this speed as high as possible. In case you are wondering, you don't need expensive or accurate sideslip transducers or equipment to get this metric. A VERY small trailer hooked to the underside of the car acts as a good measure of sideslip because the tow angle (being zero) is the sideslip angle. In the late 50's and 60's we actually used a small solid tire trailer about 20 inches long for this purpose.

A better road test than the step test (IMHO) is the frequency response test. This is a constant speed test for linear range properties using a chirp steering wheel signal. A sound version of the required constant amplitude chirp is easily made with Matlab and can be played on a cd player into the driver's headphones. (This technique also works well to guide a sine steer test for a smooth perfect steer waveform input). The FR test will hand you the steering gain, response time (from bandwidth), yaw peak to steady state ratio (then converted to yaw damping if you insist), understeer and front and rear cornering compliance. Your zero sideslip goal is revealed by a function which is the difference in phase between yaw velocity and lateral acceleration (sound familiar ?). An FR test is easily done on a straight section of road and does not need a lot of real estate. This is NOT a max lateral performance test, but if your car sucks in the linear range, it's gonna REALLY suck when all the marbles are in play.

Based on known weights and tire properties, I'd estimate the transient properties of FSAE cars are not all that terrific. Low or no understeer, relatively low tire stiffnesses, short wheel base and lots of steering and camber compliance with a very low steer ratio makes them really a challenge to drive. (Let's call it truck like for now). Things don't get much better even if you could get the yaw inertia to zero.

Once again, I'd beg some clever group to bring stuff to an event and sell such a test to get results for posting. Pay extra to hide bad results. Even Shark Tank would bite on this proposal because it's sure to be a money maker....

Cleanup in aisle 3....

05-03-2015, 11:13 PM

From Tim:

... Consider also that transient is nothing but the passage towards a steady state condition. If you have good understanding of the steady state - you are already halfway towards understanding the transient. (My emphasis and slight editing.)

From Claude:

... In simple terms transient can be viewed as nothing else than the car behavior from one steady state to another.

This thread is moving on from the issue of transient vs steady-state, but I feel obliged to have one more go at this...

!!!!! DEFINITIONS !!!!!
=================
As per an earlier post, my dictionary defines transient as "(adj.) 1. for a short time only; temporary or transitory...", which suits me. And, of course, reality is simply made up of a long string of these many transient "dt"s. (Or is it? *)

But what is the definition of steady state as used by Tim or Claude above, or anyone else here???

It seems to me that steady state is just a name that is applied to any arbitrarily dumbed-down model of reality that the speaker happens to be using at the time. The dumbing-down involves, often unrealistically, holding certain variables fixed so as to ease the understanding. Anything that makes said model difficult to understand, such as variables that change with time, is ignored, or else lumped under "transient" behaviour.

As a typical example, consider the commonly understood "steady state cornering". This refers to a car at constant speed, with constant CG-side-slip-angle, on a constant radius, perfectly flat, skid-pad. And a whole bunch of other rarely mentioned and unrealistic simplifying assumptions, such as constant fuel-load, constant tyre-temp, NO tyre-wear, etc.

Anyway, although this car is said to be in "steady state", it is still undergoing MANY CHANGES. Its position (X, Y coordinates) is changing. Its direction (Yaw angle, or heading) is changing. And it is most definitely accelerating and so subject to inertial (= centrifugal) forces. It is most definitely in "transient" circumstances! But for some reason, some people say it is only when the speed changes, or the path-radius changes, or the dampers move, that the car is undergoing a brief and fleeting "transient". After which things go back to that much more comfortable steady state.

So, I would first like to suggest that these VD discussions would be much easier to follow, if, whenever people discuss their particular version of "steady state", they start by CLEARLY DEFINING their particular choice of ASSUMED NON-CHANGING VARIABLES that make the indisputably transient behaviour qualify as steady state.

Secondly, and contrary to Tim's quote above, I reckon once you have a good understanding of transient behaviour, then understanding steady state becomes a doddle! There is nothing else to learn, really!

After all, the whole point of your schools teaching you Calculus, with all those "dt"s of things, is so you can understand transients.

(* Too much to say here... Perhaps try googling Anaximander and the I Ching for one view, and Parmenides and Zeno for the other. All from ~2500+ years ago...)
~~~o0o~~~

Doug,

So how long have you known about this [N/W vs N/Wl] error?

Almost every second page of my copy of RCVD has a "???", or "!!!", or "X!" (= "wrong"), or other less printable comments, in the margins. I am not sure how I am supposed to know which parts of RCVD are typos, and which are as you intended them to be?

Believe it or not, when we submitted the final manuscript (in person, at SAE near Pittsburgh), they asked to have it redone in ISO units.

Great! So, when I get more PMs asking for help in deciphering RCVD equations, then I will let them know that all these equations will be made crystal-clear in the new, SAE-approved, soon-to-be-released, METRICATED version of RCVD! (Yes? :))
~~~o0o~~~

STEP-STEER TEST.
===============
From Noah:

1. Plot data of yaw acceleration and lateral acceleration on the same graph (gyro and accelerometer)
2. Compare the curves to see the delay between lateral acceleration and yaw velocity.

From Claude:

...so called "step steer"...
The display of
- the lateral acceleration response time (LART)
- the yaw velocity rising time (YRT)
...
... will tell you more about the car transient behavior than any test on a circuit.
... because you will ... know what car parameters change what transient performance criteria...
(My emphasis, and "step-steer" also discussed by Doug and Bill.)

I hope that anyone contemplating doing this test, or even just studying the data from other peoples' tests, appreciates this following important point.

The Lateral-Acceleration-Response-Time is hugely dependent on the longitudinal position of the car AT WHICH the Lateral Acceleration (Ay) is measured!!!

That is,
1. Mount your Lat-G-sensor on the nose of the car and the LART will appear to be lightning quick.
2. But ... mount the Lat-G-sensor on the tail of the VERY SAME CAR (in the VERY SAME TEST!) and the LART will be as slow as a wet weekend.

In fact, in case-1 above the transient Ay can exceed the steady-state Ay, and more so the further forward the G-sensor is mounted, whereas in case-2 the transient Ay can start off in the opposite direction to SS Ay, and more so the further rearward the sensor is mounted.

So,
1. CLEARLY DEFINE THINGS.
2. UNDERSTAND TRANSIENTS.

Z

(PS. Noah, welcome to Wonderland! As in, ... down the rabbit-hole -> Mad Hatter's tea party, etc... :))

njgedr

05-04-2015, 09:42 AM

Thanks for the help understanding! The dedication of the inevitably very busy experts to participate on this forum, shows just how passionate they are about VD! I have gotten less help out of some professors I paid very good money to teach me. So thank you much! Testing is a very interesting topic, and at least for me, makes theory very clear in my professional career. I think it is time to really apply it to FSAE.

This year we focused on building a good static VD model and eliminating compliances, which while difficult to quantify was an issue in the past. We also improved dimensional accuracy of points. So, this years platform is our best bet for getting good, usable and consistent data. For Data acquisition, I am thinking at minimum: steering angle, 3 axis accelerometer (at COG), 3 axis gyro, and wheel speeds. I really like the trailer under the car idea to measure side slip too. I also want to to use a governor or throttle stop to try to get more repeatable data, has anyone tried this before?

I know Claude mentioned that other FSAE teams have been successful implementing many testing methods, but at OU it takes just about everything we have to get a reliable car on the track. So any test plan needs to take into account timing and be well documented for future students. I expect most if not all teams are in the same boat. Hopefully, we will have a chance this summer to get some good data and useful processed results. I hope to report back on at least some findings before or at the Lincoln competition.

I feel I should mention, this was a great and much needed motivator, because now I REALLY want to see some plots of yaw vs. lateral with all your suggestions added in of course... (and find the dream job in VD haha). I expect figuring those plots out will help more than anything else in understanding what is happening and motivating the new guys and girls on the team to take an at least partially scientific approach to FSAE.

Z,

The VD rabbit hole keeps getting encouragingly narrower just to open up again every other morning on this forum. The fake and crumbling walls are particularly frustrating. I suppose that is the case with learning anything worth learning.

Best Regards, Noah

BillCobb

05-04-2015, 01:54 PM

Don't worry about the location for mounting your transducer package. Mount it where it is solid and the structure is good and isolated from motor firing frequencies. It is common practice to use math transforms to 'correct' signals for position back to the cg.

Most of the engineering tests that you should run have constant speed requirements. Therefore a throttle stop is not recommended because speed will scrub off. You need to train your driver operator on these requirements as well as how to run the instrumentation, warm up the tires, stabilize the gyros, etc.

Even the best data will or may have some puzzling signal content. I recommend you prepare for this by choosing the appropriate data model equations and practice fitting them to your data channel segments. These generally are the solution form of realistic handling model differential equations.

Therefore the metrics you acquire from the summary data processing software relate directly to the gain, steady state, bandwidth, damping etc. parameters from your 'engineering' education. Here are some examples for Frequency Response, Step, Constant radius tests of a hypothetical FSAE car. Same deal for sine steer, max lat, etc. tests. The point is to avoid the panic of needing results by practicing the data crunching beforehand. Otherwise you are doomed to a future on a team or in some company where they stash you in a basement to do your job.

DougMilliken

05-04-2015, 04:35 PM

... Almost every second page of my copy of RCVD has a "???", ...
I'm going to guess you are still using a first printing, 1995 version? There have been many small changes over the years, probably approaching two hundred, as we patched our errors and those introduced by SAE typesetting. It might sound like a lot, but the text is about a million characters, so the error rate isn't completely terrible! Error reports started to come in once it was used as a text. I remember the 4th printing was much improved over the first three, this would be about 1997.

Any chance you would be willing to share your markup in some way? I'm always looking to improve it. Send me an email or PM if you want to set up something.

I'll let you work on the metric version...first step is to figure out how much of an advance you want from the publisher<grin>.

Claude Rouelle

05-04-2015, 10:45 PM

Bill,

" The integrated speed yaw velocity product difference from roll corrected lateral acceleration is the vehicle's body sideslip angle"

Again a great way to make all us us, the students and the so-called "experts", to think deeper and wider. I knew what you meant, but I did not know it simply because I never looked at it from your perspective.

But.... if you allow me

1. I checked the units and this doesn't add up. I guess you meant the CG side slip lateral speed, CG Vy, correct?

2. If I follow you, if we would integrate the signal from a longitudinal accelerometer we would measure the longitudinal speed. Integration answer is something plus a constant. What is that constant? In my opinion here is too much noise to make this correct. If you method was working way would people design manufacture and sell slip angle senors and why would other people buy them?

3. Your last thumbnails are excellent and thank you for that! But for most of the students, I need to mention that they do not mean anything: the first time those graphs are seen, uneducated and inexperienced people (like me to some extend) do not know what is a "good" car frequency response curve and a "bad" car frequency response curve unless a) you compare those graphs for car A and car B where something in the design and setup has been changed (and of course you know that change) b) your deduction are corroborated by driver feedback and other data measurements C) there graphs are used as input in other simulation. To be very humble I myself never went that far in VD but because of people like you who open intellectual horizons I am looking for it. Thank you.

Claude

Tim.Wright

05-05-2015, 03:05 AM

Claude,

There are 2 main components to the yawrate:
1. A kinematic portion coming from the trajectory and velocity (LatAcc/Vel)
2. An "elastic" component coming from the side slip velocity (BetaP) which is the effect of the car rotating about its own trajectory/velocity vector (due to different front and rear sideslip velocities)

The full equation being:
Yawrate = LatAcc/Vel - BetaP

With BetaP having a negative sign due to the SAE convention used in Milliken.

What Bill is referring to (I believe) is solving the above formula for BetaP and then integrating it (to get Beta):
INTEGRAL(LatAcc/Vel - Yawrate).dt

Quite a powerful construct as it allows you to calculate sideslip velocities and also sideslip (for short maneuvers) without needing a slip sensor.

Silente

05-05-2015, 09:31 AM

Hi everybody,

my apologies for not coming back earlier after all your comment.

i revised my work on the YMD basing on your suggestions and comments, also because i actually found something wrong with the sign convention, as Z said.

The problem, is that typical Pacejka data sets produce Positive Forces for positive slip angles. Anyway, the assumption presented in RCVD is that for a right corner, we would have positive forces with negative slip angles.

I think i now solved the issue and the results should be now consistent. Anyway i would love to read again some of your comments.

Again, the vehicle is a high downforce car, with around 1000kg mass and 50% TLLTD at the front. According to the new results, in a very fast corner, the car would be slightly oversteering in these conditions.
Increasing the TLLTD to 60%, effectively produce the graph to change as shown in RCVD and suggested by Claude, with the right "apex" moving down, showing final understeer.

Attached, the diagram and the results from the 50% case.

Claude Rouelle

05-05-2015, 10:38 AM

Tim, Bill, Doug,

Thank you for confirming what I already know with INTEGRAL(LatAcc/Vel - Yawrate).dt. In fact this explanation has been in our seminar for many years. It is right mathematically but I am skeptical on the application. I made such test many times and I have compared the result of this equation with the slip angle sensor data and the difference was huge and not logical. I am ready to reconsider, maybe I did not use the right filter for the lateral accelerometer and the gyro signals (which is noisy in a race car). Maybe there is something else that I need to learn.

One of the things we did was a post test sanity check the other way around: With the slip angle senors data that we were lucky to use (slip angle sensor of which we took the Vx) and the gyro, we should be able to find the lateral acceleration, correct? Yes I know that would be a very expensive way to measure lateral acceleration but for the sake of the sanity check we did so. The calculated lateral G at the corner apex was 0.25 G different that the one measured (calculated 1.07, measured 0.82). If the 2 numbers are not the same at least one of them is wrong.

I have good connection and I would say even friends with the people who manufacture slip angle senors so it can be said that my opinion is necessarily biased. The fact is that I keep wondering why, if such calculation would be so easy and correct, engineers would keep manufacturing, developing and buying (quite expensive) slip angle senors and the automotive and racing industry buying and using them.

PS: Several FSAE / FS teams started to use slip angle senors (that they did not pay for: they had good collaboration contract established between their university and several sensor manufacturers) so this conversation is in fact very relevant.

BillCobb

05-05-2015, 11:07 AM

Claude: Yes lateral velocity, but you know the fwd velocity so you have the tangent. And, the procedural benefits are well established. Virtually all tests begin and end with a zero AY section before you start a steer input. Not a zero steer angle for obvious reasons. These are real cars with real tires and real suspension installations by humans. A data segment (step steer or FR) also ends with a zero portion. Thus the integration constant you worry about ought to be and IS zero. Yes the noise may give you some accumulated drift error, but the gifted students know how to remove it. You want flies with that ?

The practical application in FR is that the integration is done in the frequency domain. Simply dividing by frequency hands you the integrated complex channel. BTW, this action can also give you roll angle from roll velocity (from a cheap gyro). Yeah, there maybe some low frequency garbage that gets accumulated, but the fitting function fills in the missing information because the fitting function IS the expected response. No need to filter, either.

The identification of good, bad, and ugly cars comes from fingerprinting them. You set up a family of car conditions, you test, and you stare at the metrics. When everbody wakes up, they ALL know what it takes to start producing the best car for that driver. Pure Systems Engineering from Dr. Science.

Note that this methodology hands you some applicable metrics. You probably say the two words "yaw damping" as much as you say "Bon Appetit!". Well, in this example (from the thumbnails) the fitted transfer function directly gives you the "yaw damping" zeta as 0.913 and the natural frequency as 6.376 rad/sec. (from the Matlab function DAMP). As long as you are at it, the sideslip damping is 0.245 with a natural frequency of 16.667 rad/sec. and heck, the lateral acceleration zeta is 0.815 and 9.902 rad/sec. As you and others might imagine, the calculation of yaw damping 'improvements' or degradations is nearly impossible given the realities of testing unless the changes are major and almost ridiculous. Race engineers usually like improvement ratings in believeable values, not the 4th or 5th decimal place. Cutting to the chaise lounge, I'm hinting that lateral acceleration damping ratio seems more practical to score. And this manifests itself as lateral acceleration response time: a good, great, terrific metric. Finally, CHANGES in LART make a darling stability factor. Lateral acceleration kinda jives with lateral position,eh?. Most people will agree that this is a favorable thing to have under control.

Didn't forget you, Tim: You got. However I usually splain to the naïve that the total side-force they are feeling is due to them spinning as well as from being pushed aside. The faster you go, the less spinning you ought to be doing. Attached is the missing thumbnail. Forum allows only 5.

Goost

05-05-2015, 12:00 PM

Claude,

This is very interesting, I don't have access to slip sensors so if true this is disconcerting; would you mind elaborating to make sure this is correct?

'Of which we took the vx'

Did you only use the longitudinal component of the slip speed?

I think it depends on your coordinates:

(ay_sensed) = d/dt(vy) - (vx)*(r) [the second term can be whatever combo you want to measure - (vx)^2/(R)=(r)^2*(R) = (vx)*(r)]

where

(ay_sensed) - accelerometer = "test-mass force sensor" = actual acceleration (body coordinates)
(vy) - lateral velocity (body coordinates)
(vx) - longitudinal velocity (body coordinates)
(R) - 'turn radius' [not practical in general]
(r) - yaw rate (inertial coordinates)

Attached a derivation, including corrections for sensor locations. If there are errors in this I'd appreciate hearing about them.

Claude Rouelle

05-05-2015, 12:16 PM

Goost,

Yes the Vx of the slip angle sensor that was mounted on the car longitudinal axis. As well as the lateral accelerometer.

Claude

Claude Rouelle

05-05-2015, 12:20 PM

Bill (or others)

I really would like an answer to this question and if you have an opinion I and other readers of this post will sure appreciate your perspective: "Why, if such calculation would be so easy and correct, engineers would keep manufacturing, developing and selling (quite expensive) slip angle senors and the automotive and racing industry buying and using them" I think it is a fair and interesting question

Claude

BillCobb

05-05-2015, 04:29 PM

Couple of notions on your question, Claude.

A typical Datron has more options and signals than just sideslip, etc. So it has use if you also need forward velocity and lateral velocity on surfaces where '5th wheels' won't work, are in the way, can't keep up with a wildly serpentining car, and even can't keep up with the maneuver. Can't use the car's own speedometer because different wheels and tires and gears, and errors screw things up. (Ever notice that German car's speedos are hugely too fast ? "Oh, I drive my BMW at 85 all the time". OK pal, then why when I was following you did my GPS read 68 ? (The answer has to do with German law).

These things are now specified as part of stability control validation procedures. Just like robot drivers for rollover certification. And you get a 1% discount for buying more than 10 at a time. And stopping distance (FMVSS 105 etc.) and snow and ice testing for tire traction.

What new school race team would question the need for all the wizz-bang equipment anyways? You do order the custom steering wheel torque transducers, right? or do you deduce steering torque from tie-rod loads ? "Oops we put a different steering gear and ratio in the car and forgot to inform you. Sorry about that ..."

And of course, what teams know what to do with such a signal and how to integrate it into the way they do business ? In my case, we don't want sideslip as much as measuring the rear cornering compliance, because that's a vehicle design specification. You mount it somewhere at the rear centerline and transform the signal back to the rear axle. Keep it out of the exhaust stream because it will interfere with your act. The rear cornering compliance tells you whether the car is loose or tight because of a front related issue or a rear one, so you now have a vector 'why' instead of a scalar explanation.

Since the phone technology is now so advanced, I'll bet that there is already an APP for your phone to broadcastify these data channels. Heck we were sneaking the data outbound on the extra flyback lines of the in-car TV cameras 15 years ago. Mental telepathy is probably "just around the corner" so to speak. (Hey that's a pretty good one, eh ? )

Claude Rouelle

05-05-2015, 06:19 PM

Bill,

You did not answer my question.You know that I respect you so, in the interest of all the readers of this forum, I will respectfully ask again:"Why, if such calculation would be so easy and correct, engineers would keep manufacturing, developing and selling (quite expensive) slip angle senors and the automotive and racing industry engineers buying and using them"

BTW the new generation of slip angle sensors are way, way smaller and lighter than the one on the picture you posted.

Claude

BillCobb

05-05-2015, 07:15 PM

Geez Claude, that picture is almost 20 years old. I have only one idea why special sensors are still being sold to race teams, that being the complexity of bank angle mingled with Ay. Why did you buy one ?

Do you realize that almost all brand new vehicles have such a derived sideslip angle algorithm in their stability controls to pass the Sine With Dwell Test ?

Claude Rouelle

05-05-2015, 08:03 PM

We never bought slip angle sensors. I use them because of our collaboration work with slip angle sensors manufacturers (we beta test for them), car and tire manufacturers and race teams.

"Do you realize that almost all brand new vehicles have such a derived sideslip angle algorithm in their stability controls to pass the Sine With Dwell Test ?" Oh I do Bill. And of course we cannot install slip angle sensor on each passenger car (well we do not know maybe the price will dramatically go down ...) But these algorithms has been validated in testing with many sensors one of them being a slip angle sensor, isn't it?

My question is simple (and I have to admit there was a little trap in my question to you): For having using them I do believe in slip angle sensors but I am ready to accept we do not need them (even for testing and developing a new car) but so far nobody has been explaining to me how and why.

I was hoping you could.

I keep thinking that if there was another way around somebody would have come to a solution and save a lot of money because these little babies are bloody accurate but not cheap....

BillCobb

05-05-2015, 08:59 PM

FYI:

search for "ROBUSTNESS OF SIDE SLIP ESTIMATION AND CONTROL ALGORITHMS
FOR VEHICLE CHASSIS CONTROL
Aleksander Hac
Edward Bedner
Delphi Corporation
United States of America
Paper Number 07-0353"

JT A.

05-06-2015, 09:51 AM

Admittedly I only gave that paper a quick skim, but what I took from it is this-

If you want to measure 20 degree slides with +/- 3 degrees accuracy for stability control systems, deriving sideslip from gyro, wheelspeed, and accelerometer is close enough.

If you're working with race cars where +/- 3 degrees of sideslip covers their entire operating range, and you want to validate your tire force & moment data,you probably want slip angle data that's accurate down to 1/10 of a degree. The calculated sideslip angles just don't have that precision. Also depending on the series, you may only be allowed to test on sanctioned test days at a racetrack. Using a big blank skidpad to do short, controlled step steer or other maneuvers isn't really an option. Over the course of multiple laps, I imagine the integration error from a calculated sideslip angle would become pretty messy.

I also suspect the racing industry has a case of "keeping up with the Jones's" to some degree. Team X has slip angle sensors on their car at every test so we have to do it too or we'll fall behind. Regardless of what value they're actually getting from the data.

Claude Rouelle

05-06-2015, 10:54 AM

The well defended and defined opinion of a guy with solid arguments who have "been there done that" will convince me more than any paper (which I have not been able to determine the date at which it was written).

JT A is right; +/-3 degrees is not accurate enough, IMO even for a passenger car.

Same thing for slip ratio targeted for TC and ESP and torque vectoring control loop definition.

We have compared simple straight line longitudinal speed from the slip angle sensors, wheel speed senor, the GPS and the differences are ugly and ridiculous. I know which one I will trust the most.

Again; if it was that easy to calculate slip angle and slip ratio why would slip angle senors manufacturers spend big amount of time and money to develop these sensors and why would the automotive passenger car and race cars industries buy and use them?

Back to the beginning of this post and just as a reminder; we discussed this because of the need of VALIDATION of the yaw moment Vs lateral acceleration diagram. When you see what the change of one simple degree of CG slip angle Beta does on the lateral acceleration and the yaw moment, for on and off center maneuvers, we cannot kid ourselves with +/- x degrees of accuracy

Kevin Hayward

05-07-2015, 05:27 AM

The paper will be 2006 or later, based on reference dates. Given the topic is close to that presented in reference 2 I would suggest within 2006-2008.

The accuracy at around 10 degrees slip is shown to be much more accurate than +-3 degrees. The accuracy is quite close at the beginning of the maneuver. The potential error sources are documented, most of which could be accounted for when testing. I would also agree with Bill that it is reasonable for FSAE teams to attempt controlled maneuvers within the region of speeds we expect. We are not out racing, and a flat carpark eliminates a lot of problems.

I wouldn't be quick to discount papers as quickly. They are peer reviewed and often written by the exact people that had "been there and done that". I put a little less faith in what I am told as opinions even with solid arguments, especially if they meet one or both of the two following conditions:

1) They are a competitor

2) They are at a race track

While I have learnt a heck of a lot from people meeting both of these conditions, the ratio of bullshit to diamonds is about 1000:1 (if I am less than generous with the manure).

As with any advice or info (paper or person) I would be interested in validating it.

Claude,

I think the main reason that slip angle sensors are valuable is that it is easy to account for banking angles, and the calculations become trivial and accurate. It doesn't mean that FSAE teams should be doing the same. If a manufacturer has 10 researchers on staff at $100k a year it makes little sense to waste their time playing around. A $20k sensor looks pretty cheap. For fsae teams, especially those far away from manufacturers and suppliers (i.e. borrowing not a reasonable option) $20k is a lot of money when a few simple techniques can get you most of the way there.

Is that a good enough reason for why people buy them, and FSAE students (and any lower end racing team) should pay attention to the ideas Bill is presenting?

Kev

BillCobb

05-07-2015, 10:12 PM

You see, I'm a supporter of the notion that a team ought to present the state of their car's handling in terms of the cornering compliances. The optimum recipe for them at each axle will deliver a car that does very well in all driving events. If done poorly, you can't fix it with a fatter roll bar, more spring, or a better driver. It will show that you have designed the handling state of the car instead of discovering it.

You don't need a sideslip transducer to get a handle on it. A steer angle transducer is helpful, some knowledge of what constant speeds you will run at, a lateral accelerometer, a yaw rate gyro and a data acquisition system with a good slew rate. And a properly executed procedure.

I know for a fact that there are professional teams having expensive state of the art sideslip sensors and they run many laps with them. But, show me how that information gives them a better car. Let's see it. Yes there are big advantages. But you should not freeze out a team from designing and validation a good car because they can not obtain such a device. Because, if your cornering compliance recipe is 6 front and 1 rear or 1 front and 1 rear or 1 front and 6 rear, or heaven forbid 6 front and 6 rear, your car will suck. So bad it would make a better (your private entertainment product here) than a vehicle. Excuse the theory in the report. I know that welding is more fun and glamorous, but the pay is low and won't fill up your IRA like good theory does.

Sorry for the multiple thread entries. The PDF is too big and we are limited to 5 pictures...

BillCobb

05-07-2015, 10:15 PM

Second verse, same as the first.

05-09-2015, 06:33 AM

I have been away for a few days, but would like to address some stuff from a few pages back. Haven't had time to read Bill's latest paper above, or his previously linked paper, so some of the below may be covered in those papers...
~~~o0o~~~

From Bill (04 May, bottom p7):

Don't worry about the location for mounting your [acceleration-G-sensor] transducer package. ... It is common practice to use math transforms to 'correct' signals for position back to the cg.

Bill,

There are two problems here.

1. Nowadays very few people know how to do the "math transforms" correctly. Goost's equations have been up for the better part of a week, and no one has yet pointed out his errors (see below). And what happens when the next lot of students download the transforms from the All-Knowing-Interweb, ... and get Goost's equations?

2. More importantly, even if the transforms are done correctly, they still give Lateral-Acceleration AT THE CG. Nowadays there are too many, ahem, "VD experts", who interpret this variable KINEMATIC result as a Dynamic influence that makes forward-CG cars have faster turn-in, or "LART", than rear-CG cars. Very misleading! (More below.)
~~~o0o~~~

From Goost (05 May, mid-p8):

Attached a derivation, including corrections for sensor locations. If there are errors in this I'd appreciate hearing about them.

Goost (Austin),

Nice graphics! But I think your alphabet-soup is a little bit off...

Some nit-picking first (ie. ...old-man whingeing and moaning... :)).

Regarding "communication of information", you have a lot of alphabet-soup in there, but perhaps not enough clear indication of what all those wiggly-little-noodles mean. So, in your first diagram you could have combined the top-left and right corners to give a more concise, but also more complete, account of the various reference-frames you are using (ie. inertial/global-frame = N-E, path-frame = t-n, and car-body-frame = X-Y). For example, your "eta" path-coordinates are useful when studying point-mass-dynamics, but are somewhat redundant for the rigid-body-dynamics here. But ... too many little details here to be able to fully cover how to do this better (maybe next time)...

IMPORTANTLY, whenever specifying any Position, Velocity, or Acceleration (ie. any KINEMATIC quantity), you should, IMO, always specify the TWO bodies/reference-frames involved. You have this in very top-right corner, with "Psi = Theta (Et wrt N)", but unfortunately this very important "with respect to" is missing from all the other Velocities and Accelerations.

Also I am not sure why you have, for example, all three of Psi, Theta, and Omega, when you could have just Psi, Psi-dot, and Psi-double-dot? Also, not too important, but I have got used to Thetas being positional angles, Omegas ("W"s) being angular velocities, "R"s being RADII/radial-position vectors (so why "r" for an angular velocity!?), etc. So, frankly, to have any hope of checking your work I had to translate it all back into my preferred hieroglyphics...

And my bottom-line conclusion is that ... your bottom-lines are WRONG! Namely, your third diagram equations for Apx,y have too many terms, and your bottom-most Ax,y equations have one wrong +/- in them. Close, but no cigar! Also not sure about your second diagram, but that might be a misunderstanding of "? wrt ???". Hopefully, my next post might help you more...

Z

(PS. Should add that your Diagram-3 "Notes..." are OK. 1. Yes, Gyro can be mounted anywhere (that is safe). 2. Any two linear-accelerometers, widely spaced and aligned along a common line, can give you Yaw-velocity and Yaw-acceleration (and Yaw-angle when integrated). So no point having that Gyro. Also, six single-axis-G-sensors, suitably separated and directed, will tell you just about everything. 3. Non-slipping wheels revs should be able to give you Yaw-Rate, but I don't think with much accuracy when most needed (?).)

05-09-2015, 06:44 AM

PLANAR KINEMATICS of RIGID-BODIES and the "ACCELERATION POLE".
================================================== =========
To keep this relevant to Goost's diagrams, and to save some time, I will keep this in "Flatland" for now.

I am sure everyone here is familiar with the concept of the "Instant Centre" (IC) of 2-D Kinematics. This is the point where you can push a thumb-tack through a piece of cardboard that represents the "moving Body", down into the table that represents the "Ground frame" (but NOT Nana's best dining-table!!!), and the subsequent SMALL motion of the Modelled-Body-wrt-Ground represents the possible INSTANTANEOUS motion of the real linkage/sliding-car/whatever.

In the olden-days (~1700s) this IC was also called the "Velocity Pole" for the relative motion, because it was the point in both reference-frames where the Velocity of "this-frame-wrt-that-frame" was zero, at that instant of the motion. So like a "maypole" that children dance around, or the thumb-tack above. Considering, as usual, the Ground-frame "fixed", it follows quite obviously (ie. I won't bother proving it), that ALL points of the moving Body have Velocity-vectors (wrt Ground) that are perpendicular to radii drawn from this "Velocity-Pole/IC", with magnitudes directly proportional to their distances from the V-IC.

So all the Velocity-vectors taken together form a swirling pattern of arrows that are tangent to circles centred on the V-IC, and grow ever longer the further away they are from the V-IC. THIS IS ALWAYS THE CASE! (Err, in Flatland, and with some "degenerate" cases, such as V-IC infinitely far away, so all V-vectors parallel and equal...) Simple, eh! :)

But what about the Acceleration-vectors of all these same points (ie. the accelerations of all the Body-Points-wrt-Ground...)? What pattern do they form? Is there ALWAYS a common pattern? Is there an Acceleration-Pole, or A-IC as we might henceforth call it? If so, then is it always at the same place as the V-IC? Or only sometimes? Or never? And, most pertinently, assuming there is such a thing, how do we find the A-IC???
~~~o0o~~~

Only time now for some quick comments on above questions, taken in no particular order:

1. All this was common knowledge to "small boys" hundreds of years ago.

2. Yes, there IS ALWAYS an A-IC (though, as with the V-IC, it might be a long way away).

3. In general, the V-IC and A-IC are NOT at the same point. In fact, they only coincide in rare-ish cases.

4. The pattern of Acceleration-vectors is similar to that of the V-vectors, in that they swirl around the A-IC, and have increasing magnitude the further away they are. But, in general, all the A-vectors have an INWARD component, TOWARDS the A-IC. And all the A-vectors make EQUAL ANGLES to radii from the A-IC out to the relevant Body-Point. To repeat this, all the angles between the radii and A-vectors are the same, and are NEVER greater than a right-angle.

5. If given the A-vectors for TWO separate points (ie. you need both X&Y components of both vectors), then it is quite easy to find the A-IC, and thus also the pattern for ALL the A-vectors.

6. Likewise, if given the A-vector for ONE point (X&Y components), together with the W and W-dot vectors (see below) for the whole Body, then it is easy to find the A-IC and the whole pattern. Note that W and W-dot are "free-vectors", so they are the same for all points of the Body.

7. So, per Goost's diagrams, I get in vector terms,
(Edit: To stress again, the velocity and acceleration vectors below are of the Moving-Body WRT Inertial-Ground-Frame.)
Ap = Ac + W-dot x Rcp - W^2.Rcp,
where,
Bold = a vector, with +/- indicating vector (parallelogram) addition,
Ap, Ac = linear-Acceleration-vector at general point P of the Body (wrt Ground), or at the origin of coordinates C (metres/second-squared),
W (Omega) = Rotational-Velocity-vector of the Body rotating in its plane ("r" in all Goost's diagrams, and radians/second),
W-dot = Rotational-Acceleration-vector, (radians/second-squared).
Rcp = radius, or "displacement", vector, from C -> to P (metres).
Note:
Second-term-RHS = "rotational or circumferential"-Acceleration of P AROUND C.
Third-term-RHS = "centripetal"(= "centre-seeking")-Acceleration of P TOWARDS C.

In Cartesian-components (which, given poor Rene's X-Y-axes have been turned upside-down in Goost's diagrams, is still correct because of backward (CW) rotation),
Apx = Acx - W-dot.Yp - W^2.Xp,
Apy = Acy + W-dot.Xp - W^2.Yp.

8. And, of course, everything is a bit more interesting in the 3-D world...

9. And I'm sure I missed a lot...
~~~o0o~~~

A longer rant. :)

You are on the school-bus. The driver is maintaining a constant speed and a straight path. So the Velocity-Pole/IC is a long way sideways (say, at "infinity", if a really straight path). The Acceleration-Pole/IC does NOT YET EXIST, because all accelerations are zero! The driver then initiates a right-turn, which we might consider to be a "step-steer".

Immediately a swirling field of Acceleration-vectors (... of Bus-wrt-Ground) springs into life! The A-IC is "born" near the bus-centreline, close to the rear-axle (could be slightly fore or aft depending on Yaw-MoI). Initially, because of no (or very little) Yaw-velocity "W", the A-vectors are all tangent to circles surrounding the A-IC. If "stiff" front-tyres, and large step-steer, then these A-vectors can be quite large.

So, the bus-driver, sitting furthest forward and thus the longest distance from the A-IC, feels the largest rightward Lateral-Acceleration Ay of any position on the bus (well, front-numberplate has more). The cool-kids, sitting in the rear-seat, some distance behind the A-IC, initially feel a LEFTWARD Ay. This cool-kids' Ay-vector subsequently rotates forward (CW in plan-view) and then ALMOST reaches full-rightward, but never quite gets completely rightward...

During this "transient" phase the Velocity-IC is moving leftward (initially, at N! x lightspeed!) towards the bus from far-right. Simultaneously, the Acceleration-IC is moving (much more slowly) from near the bus's differential, rightward and towards the V-IC.

Eventually, should that most unlikely occurance of a "Steady-State" ever being reached, the two ICs, V and A, will meet at the same point called the "corner-centre". But Buckley's chance of them both staying there for long. Yep, the V-IC and A-IC are more like "ships in the night"...

Ahh... transients! Well worth studying. :)
~~~o0o~~~

Final note. It is often said that a racecar driver can better sense the car's "handling" foibles when they sit near the rear of the car. Consider the bus-driver and cool-kids above, and ask, if the bus is being driven particulary vigorously, then who will feel the rear-axle "stepping-out" first?

Does this suggest that an FS/FSAE driver's head (where they keep their personal G-sensors) should be as far forward, or as far rearward, of the car as possible?

Z

Goost

05-09-2015, 03:30 PM

1. Nowadays very few people know how to do the "math transforms" correctly. Goost's equations have been up for the better part of a week, and no one has yet pointed out his errors (see below). And what happens when the next lot of students download the transforms from the All-Knowing-Interweb, ... and get Goost's equations?

[...]

Also I am not sure why you have, for example, all three of Psi, Theta, and Omega, when you could have just Psi, Psi-dot, and Psi-double-dot? Also, not too important, but I have got used to Thetas being positional angles, Omegas ("W"s) being angular velocities, "R"s being RADII/radial-position vectors (so why "r" for an angular velocity!?), etc. So, frankly, to have any hope of checking your work I had to translate it all back into my preferred hieroglyphics...

And my bottom-line conclusion is that ... your bottom-lines are WRONG! Namely, your third diagram equations for Apx,y have too many terms, and your bottom-most Ax,y equations have one wrong +/- in them. Close, but no cigar! Also not sure about your second diagram, but that might be a misunderstanding of "? wrt ???". Hopefully, my next post might help you more...

[...]

7. So, per Goost's diagrams, I get in vector terms,
Ap = Ac + W-dot x Rcp - W^2.Rcp,
where,
Bold = a vector, with +/- indicating vector (parallelogram) addition,
Ap,c = linear-Acceleration-vector at general point P of the Body, or at the origin of coordinates (or CG) C (metres/second-squared),
W (Omega) = angular-Velocity-vector of the Body rotating in its plane ("r" in all Goost's diagrams, and radians/second),
W-dot = angular-Acceleration-vector, (radians/second-squared).
Rcp = radius-, or "position-", vector from C to P (metres),
second-term-RHS = "circumferential"-acceleration of P AROUND C,
third-term-RHS = "centripetal"(= "centre-seeking")-acceleration of P TOWARDS C.

In Cartesian-component terms (which, given poor Rene's X-Y-axes have been turned upside-down in Goost's diagrams, is still correct because of backward (CW) rotation),
Apx = Acx - W-dot.Yp - W^2.Xp,
Apy = Acy + W-dot.Xp - W^2.Yp.

8. And, of course, everything is a bit more interesting in the 3-D world...

9. And I'm sure I missed a lot...

Dang you write some long posts... Wish I had time for that.

"No one has pointed out his errors"

Thanks for reviewing this, seems other people here are contributing more than criticizing, but good to have a bit of both.

" I am not sure why you have, for example, all three of Psi, Theta, and Omega"

Yea, so I use these notes when I cover navigation filtering. So if you mean my definitions at top right, I'm just clarifying the notation for the 'classical notation' guys like yourself who seem to dislike using psi instead of theta.
If you mean the fact I track three coordinates systems (the ENU, path, and body coordinates) that's related to the coordinate systems that measurements are made in.

1)
GPS position is in ENU/NED (from LLA, which you may be familiar with)
Gyroscope readings are inertial of course, which is pretty much equivalent to being here for 2d case

2)
GPS velocity and heading (or 'course', but not 'attitude', which you seem familiar with), describe the Path coordinate system

3)
Acceleration, slip angle, steering angle, etc etc are reported in body fixed coordinates.

Oh, so 'r' is pretty common for yaw rate and so is making right-hand turns positive (you said you read Milliken pretty thoroughly?)
Comes from the aerospace guys I think, Doug could explain better.

So fair that it's cumbersome, however it's important to see which states and sensor errors are 'observable' from measurements (may not be a classical term for this one?)
Explains why you can't get slip angle from a GPS antenna (but can from two), why it's useful to use tires/GPS as a measurement update when estimating gyroscope bias, etc.

moving on...

(I honestly skimmed for a bit here, you criticize instant centers of roll and pitch but like the concept for yaw?
Seems much of that is hard to measure.)

"And my bottom-line conclusion is that ... your bottom-lines are WRONG!
Namely, your third diagram equations for Apx,y have too many terms,
and your bottom-most Ax,y equations have one wrong +/- in them.
Close, but no cigar! Also not sure about your second diagram, but that might be a misunderstanding of "? wrt ???".
Hopefully, my next post might help you more..."

"Ap = Ac + W-dot x Rcp - W^2.Rcp"
"Apx = Acx - W-dot.Yp - W^2.Xp,
Apy = Acy + W-dot.Xp - W^2.Yp."

Ah, finally the 'bottom lines are WRONG!" is addressed here? So back into my notation:

apx = acx - dr*Y - r^2*X = acx - dr*Y - r^2*X = (?) - dr*Y - r^2*X
apy = acy + dr*X - r^2*Y = acy + dr*X - r^2*Y = (?) + dr*X - r^2*Y

where I wrote:

apx = (dvx - vy*r) - dr*Y + r^2*X
apy = (dvy + vx*r) + dr*X - r^2*Y

So, seems we have come down to disagreement over the sign of (+/- r^2*X)?

I think from the way I have drawn my coordinate system and diagram (at least p3?) by my 'right hand turn'/'SAE'/'aerospace' notation, my equation is correct? can you check?

[edit: looking at this again, the sign SHOULD be negative. Good catch Z, thanks]

~~~

"Alphabet soup"
Thank you for noting this is messy, I will probably clean it up before teaching with it again.
Could definitely be cleaned significantly using matrix notation, but I find undergrads don't trust/follow matrix math.

What else is wrong? Or maybe I still misunderstand? I do appreciate the critique!

~~~

The initial reason I brought this to the table is really still over the term

ay=(dvy + vx*r) .

Claude,

if this seems wrong, let me know. If correct, using only the longitudinal slip speed to calibrate an accelerometer will show error for events in which slip is changing.

Could you possibly test
accelerometer_error_1 = (ay_accelerometer) - (Vx_slip_sense)*(gyro_yaw_rate)
vs
accelerometer_error_2 = (ay_accelerometer) - (Vx_slip_sense)*(gyro_yaw_rate) - d/dt(Vy_slip_sense)

If you are willing to share any data with these four signals, I could check and report my findings?

Tim.Wright

05-09-2015, 06:09 PM

Erik,

What you say about the position of the CG influencing the lat acc response is true and it is the reason that I prefer to use the driver H-point as the output for Ay and sideslip when I'm trying to objectively measure driver feedback.

But to quantify the handling response of the car you MUST be looking at the CG response as it's the only point on the car where it is valid to calculate the vehicle responses Ay = Fy/m and YawAcc = Izz/Mz. Moreover, THIS is the point that needs to be accelerated laterally in order to enter a curved trajectory (i.e. go around a corner).

Motorsport is a closed loop game - so trajectory control is the primary thing to get right in a race-car (subjectively referred to as "precise" if your trajectory control is good and "shit" if it isn't). If you have a diminished or lagging response in lateral acceleration at the CG w.r.t. steering angle, it means that the car's ability to follow a curved trajectory is reduced and delayed.

This is bad subjectively (the driver will describe the car as sluggish because he needs to anticipate this delay with his steering wheel input) and objectively (delays like this can induce instablities in the squishy closed loop control system installed between the seat and the steering wheel). Also, the lower the velocity, the longer are these delays in terms of time.

To put it another way - think of it in terms of the frequency response of the vehicle. For a road car, your lateral acceleration response at the CG will start to drop off at around 1.5 - 2.0 Hz until you reach 3-4Hz where its basically non-existant. This means your car's ability to change its trajectory curvature (e.g. to get around a slalom) is diminished. The car is still rotating about its velocity vector (i.e. there is slip velocity and you can feel it) but the car is going straight.

This is easy to see yourself. Give a 30deg sinusoidal input to your car at 80kmh firstly at 0.2Hz and see how many lanes of traffic you cross. Then do the same at 4Hz and observe the car just goes pretty much straight. It still has a response in yaw but there is no lateral acceleration therefore no lateral movement. In short, if you don't have Ay at the CG, you aren't turning.

Therefore, the closer you move your CG to the rear axle, the less curvature response you have at high frequencies which, in my opinion, is the complete opposite of what you want in FSAE.

The goal for a highly maneouverable vehicle should be to maximise the lateral acceleration bandwidth (Ay drop off frequency measured in Hz) as much as possible. One way of doing this is to move the CG forward. The gotcha here is that increasing the AY bandwidth will generally also make the vehicle more underdamped in yaw - so in the end its a compromise.

All of this is possible to see in the frequency response maneuver that Bill mentioned. If you setup a dynamic bicycle model you will see how you can really influence the handling "DNA" of the car by managing the front and the rear cornering stiffnes'.

DougMilliken

05-09-2015, 07:41 PM

...To keep this relevant to Goost's diagrams, ...
Olley developed a qualitative graphical presentation, see "Chassis Design", SAE R-206, Figures 4.11 through 4.15 (or further), starting page 233. Might help some readers visualize for some simple cases?

- - - -

SAE definitions -- these are from Vehicle Dynamics Terminology, J670e, Issued 1952-07, Revised 1976-07. There is version "J670f" now which is much larger than "e".

10.4.1.6 Roll Velocity (p)—The angular velocity about the x-axis.
10.4.1.7 Pitch Velocity (q)—The angular velocity about the y-axis.
10.4.1.8 Yaw Velocity (r)—The angular velocity about the z-axis.

We typically used SAE definitions in RCVD, and J670 is Reference 1 in the back. Capsule history -- CAL aero engineers (often working on contract to General Motors in the 1950's) adapted their aeronautical terminology. It went through several versions in the SAE vehicle dynamics committee with many contributions and I think I remember hearing about some turf wars. One common exception in RCVD is because Bill/Dad hated the sign convention for slip angle (which he pointedly blamed on a specific individual!) -- he often used "steer angle" instead to put the plot of tire Fy vs [slip|steer] angle in the first quadrant.

Goost

05-09-2015, 11:36 PM

Erik,

Looking at it again, I think you were correct - should be

apx= ... - (r^2*X)

Good catch, thanks!

05-11-2015, 12:24 AM

Doug,

I had a quick look at Olley's book. Figure 4.14 confirms what I was getting at above (ie. re: school-bus transient Ays, etc.). I will re-read that section in more detail later... But I note that by Figure 4.24 I have in the margin "X! P should be here!", because the figure is WRONG (P should be outside, and to right of, circle :)).

Regarding aero-coordinates and Yaw-Velocity = "r". I genuinely think that all this is down to different industries trying to develop their own in-house "secret language", mainly so they can appear "smarter" to outsiders. So rather than just adopting Rene's boring old axes with X-to-right, Y-up, rotation-ACW, someone decided they should turn everything upside-down, and pick a whole new set of squiggly-noodles to represent the same-old-stuff... (Incidentally, teenagers also develop their own secret languages, probably for the same reasons.)

As an example, I have been recently reading a rather technical book that starts Chapter 1 with, "...velocity will be denoted by a vector u, or by Cartesian components (u, v, w), or by tensor components (u1, u2, u3), or by..., or by..., according to circumstances."!!! And then every type of "derivative" notation is also thrown in, such as all the different types of "d"?s, and the multiple-dash notation, and also multiple-sub/superscript notations. Seemingly AT RANDOM. With almost NO figures anywhere in the very long book.

IMO this "up-sized alphabet-soup" approach works great if the author wants to look clever. But, frankly, it is useless for imparting knowledge and understanding.
~~~o0o~~~

Austin,

...you criticize instant centers of roll and pitch but like the concept for yaw?

In general I do NOT like 2-D approaches for any VD stuff that clearly needs 3-D analysis, especially when said 3-D approach is quite simple.

Here I decided to stay with a 2-D Yaw-plane approach, just to save time with the explanation, and hopefully to be able to explain some of the simpler and more obvious concepts, more easily. This stuff in 2-D is really very simple. But I guess it is only able to be understood when explained with a (longish?) series of sketches showing how the different vectors develop over time... Something that is very easy to do on a (2-D!) blackboard, or even with a stick-in-the-sand, but harder in words alone. Nigh-on impossible with alphabet-soup alone! :)

The initial reason I brought this to the table is really still over the term ay=(dvy + vx*r).

That expression is notional correct, but ONLY if the various vectors apply to the appropriate reference-frames. Given my interpretation of your terms and their reference-frames, I think you need a "d(Beta)/dt" in there somewhere, as per Tim's post middle-page-8.

Methinks some VERY CLEAR diagrams showing these vectors and their reference-frames are needed here...
~~~o0o~~~

More coming...

Z

05-11-2015, 12:27 AM

Tim,

Firstly, when I wrote "Nowadays there are too many, ahem, "VD experts", who [get this wrong]...", I was NOT specifically referring to anyone here (such as yourself). Rather, over the years I have come across this mistaken notion in countless VD papers, books, etc. Having said that, I reckon your whole post above is WRONG, wrong wrong... :)

Perhaps the best way for you to see this is to model the school-bus I described above. As a "baseline", assume the tyres have infinite Cornering-stiffness (ie. as in wheels-on-rails *). This means you have to "ramp" the steering up from zero to steady-state, because a step-steer will give you unrealistic infinite accelerations. And rather than modelling with Laplace-transforms, I suggest you just time-step in discrete "dt"s.

IMPORTANTLY, produce lots of graphic diagrams (ie. one at each time-step) showing the Velocity and Acceleration vectors of lots of different points on the moving Bus (with all these vectors ... wrt-Ground!).

A number of things should become blindingly obvious.
1. The Acceleration-Pole/IC of the Bus always starts near the rear-axle. That is, regardless of how quickly the front of the Bus accelerates sideways into the corner, there is always very little Ay at the rear of the Bus. This is a simple KINEMATIC TRUTH.

2.
What you say about the position of the CG influencing the lat acc response is true...
NO (I didn't say that)! The above Kinematic-Truth is entirely independent of the Bus's "Dynamics". It matters squat where the CG is, or what are the forces acting on the tyres. To see this, you should be able to make multiple Dynamic models, each with different F/R CG position, and each with appropriately adjusted F/R "tyre-stiffnesses", such that the Bus's path in each Dynamic model EXACTLY matches the baseline Kinematic model above (albeit with different steer-speeds). It follows that the Dynamic model's "CG-Ay responsiveness" has got NOTHING to do with "Dynamics", but rather is the same Kinematic effect as in the baseline model.

3.
... to quantify the handling response of the car you MUST be looking at the CG response ... THIS is the point that needs to be accelerated laterally in order to enter a curved trajectory (i.e. go around a corner).
Again, NO! The Bus enters the corner because the driver steers the front wheels. This action then causes the road-to-front-wheel-forces to accelerate the FRONT-END of the Bus sideways... And ... because the Bus is notionally a "rigid-body", the CG, wherever it might be, HAS NO CHOICE but to follow the front-end of the Bus.

4.
...If you have a diminished or lagging response in lateral acceleration at the CG w.r.t. steering angle, it means that the car's ability to follow a curved trajectory is reduced and delayed.
Again wrong. By loose analogy, that is like saying that because a rearward-CG always crosses the finish-line AFTER a forward-CG ... it follows that a rearward-CG will inevitably give a slower laptime! True premiss, but nonsensical conclusion.

5.
... think of it in terms of the frequency response of the vehicle...
... Give a 30deg sinusoidal input to your car at 80kmh firstly at 0.2Hz and see how many lanes of traffic you cross. Then do the same at 4Hz and observe the car just goes pretty much straight...
This is a blindingly obvious Kinematic effect. Fit infinitely stiff tyres (ie. wheels-on-rails) and regardless of where you put the CG, the effect (ie. car trajectory) is the same. At 4 Hz the front-wheels simply do not have much time to go very far sideways (ie. distance = ~ 0.1 sec x V m/s). This has nothing to do with Dynamics.

6.
...the closer you move your CG to the rear axle, the less curvature response you have at high frequencies
...The goal ... should be to maximise the lateral acceleration bandwidth...
...All of this is possible to see in the frequency response maneuver that Bill mentioned.
Again wrong, because you are again measuring Kinematic effects (ie. of the F/R measurement of Ay)... and then interpreting them as Dynamic effects that are supposedly CAUSED by CG position. WRONG!

I could go on, but I think to understand this we really need lots of graphical images of all the Velocity and Acceleration vectors, throughout all the different stages of the transient corner-entry phase (ie. up til notional "steady-state-cornering").

If anyone cares to produce these, then please post them here! :)

Z

(* PS. Perhaps best way to think about the "baseline" model is as a massless car (it can be a "bicycle-model") on a RAIL-TRACK. You specify car wheelbase and speed, and the shape of the rail-track (maybe a straight-section merging with constant-radius-corner). The resulting V and A vectors are then PURELY Kinematic, because NO Forces or Masses. The car simply follows the track at a constant "speed" (but varying Vs and As!).)

jpusb

05-14-2015, 06:52 AM

This thread went crazy long and deviated. I stopped reading the long posts at page 6 so forgive me if this has already been discussed. That said, I would really love for all you people much more experienced at this to read this post and share your thoughts. One of the things I find amazing of FSAE is the fact that you can talk to such people directly.

Going back to the moment diagrams. I can't even picture why would people try things like these in Excel, all the best for you guys, MATLAB for me.

Now to topic. This year I developed a 4 wheel model with TTC data and the MRA tire model that comes with it. I am working on this very slowly (or best said, really fast each time I sit down on it, which is once every two months or so) because it is only in my free time and I am doing it out of pure curiosity.

At the moment, the model accounts for lateral weight transfer (iterations and all, just like talked in the first page), down force (if I switch it on, it is off for the moment and to set fundamentals), longitudinal forces (the rear wheels "push" the scrub produced by the fronts, and the aero-drag, if the downforce switch is on, here I went lazy with an open-diff), but no camber changes and kinematics whatsoever (to keep things "simple"). Jacking is non-existent, steering is parallel (out of laziness and simplicity, at the moment). Also (important to say, I think), the yaw rate is for B = constant in time, so slip angles are purely geometrical and "locked" to the instant turning radius, just as the yaw rate.

I calculated and plotted moment diagrams for both constant radius conditions, and constant speed conditions. They are very different, I learned a lot from the experience, 1000 thanks to your father, Doug, for such a mind-blowing chapter on this in RCVD. I wish I had my current MATLAB skills back in the day when I took the OptimumG seminar with you, Claude, in 2007 at Oakland University, but I still learned a lot and that got me initiated into this vehicle dynamics world. I even extrapolated the learnings of these diagrams to the downhill mountain bike dynamics and my personal riding and bike setup, mind blowing really. In the process (and still) I got confronted to many questions and doubts. Some, I answered my self later on in the process, some remained, and some other also appeared when reading this thread.

1. First question, Claude, you say to Viv that there must be [YM = 0, Ay = 0] at [B = 0, delta = 0], and that this not being the case was the first mistake students made. This should happen in constant velocity diagrams but not in constant radius diagrams, or am I wrong? Viv said his diagram was constant radius, I think.

2. In my diagrams (I will try to show one later, if I get the time to upload the image), there is a very weird behavior in the untrimmed max-g corner of the diagram. Where the peak of the diagram gets (visually) folded up on itself, just like the wrapping plastic (or paper) on a hard candy. I don't see this in any of the diagrams shown in RCVD, nor I see it in the diagrams uploaded in this thread or found in the web. The untrimmed max-g condition occurs at very different B and delta than the trimed max-g (higher B and very low delta for the untrimmed case). What is your opinion on what could be causing this? Is this physically possible or is it an artifact of my model/geometry or tire model? I bet this is an artifact, but maybe you guys have seen this tons of times. It seems like the rear end has too much grip, which will be, to some extend, fixed when I put the camber vs. roll corrections in there, but I don't think camber will change the diagram that much (at least for Hoosier 7.0 tires on 7 inch rims).

3. Do you guys think it is useful to use this diagram (to find performance) without accounting for the diff? I see many of these studies done with open diff, or (worse, better, does not matter?) with no FXs at all, and by playing with the diff a bit you get to see big changes in the shape of the diagrams, specially when FXs get bigger because the rear wheel push through high front slip angles and high drag with aero. The change in the shape of the diagram changes, of course, the metrics defined previously in the thread. I would agree that the diagram teaches you a lot (even if you make it out of a very simple model), but one thing is to use it to learn and another is to use it as a tool for setup or performance characteristics. In other word, the question is, would you believe the response to input deltas you get from the diagram (ARB, spring, weight dist, track, blablabla) without properly including the diff?

4. In my case, the vehicle understeers at trimmed max. Ay condition. However, if you think about this candy-wrapping paper zone explained in question (2), if I change setup parameters to re-shape the diagram and move this max. untrimmed Ay zone in the Y axis, I can make this zone intersect with the CN = 0 horizontal line. Here, things get really funky because this condition may produce up to three different TRIMMED MAXIMUM Ay possible.

5. Claude (and any other design judge reading this thread), what will you look for in a design presentation brave enough to get into these diagrams? The metrics, of course, and the proof or show that the student really understands what these are for, but from that, what else?

I know number (4) may get too complicated to understand without the diagram, I will try to upload it later.

I will surely come up with more questions later as I get back into this, because I read this thread today but haven't worked on this for months, so I am a bit off.

JP

jpusb

05-14-2015, 12:12 PM

Well, during lunch I took a quick look at RCVD chapter 8, and Figure 8.7 seems to have both its max-g ends also curled or candy-wrapping-paper-like, similar (in concept but still, not in shape) to what I tried to describe in my question (2), but very small. However, there is no zoom on this area that helps enlighten my doubts.

JT A.

05-14-2015, 02:09 PM

The regions where the diagram "folds over on itself" happen to be at high steering angles and high sideslip angles. What are the tire slip angles at those states? What is the slope of the SA-Fy curves at those slip angles? That should help make sense of what you're seeing.

DougMilliken

05-14-2015, 02:18 PM

JT A. gave a good hint.

...1000 thanks to your father, Doug, for such a mind-blowing chapter on this in RCVD.
I'd pass it along, but you are a few years too late -- http://blog.hemmings.com/index.php/2012/08/27/william-f-milliken-1911-2012/

jpusb

05-14-2015, 03:05 PM

I understand what you mean because that was my first thought and it is easy to understand/visualize when saturating just one end of the car. For instance, leaving beta = 0 and increasing steering, it is obvious that the diagram will fold up on itself due to front saturation (as shown at the very beginning of that chapter in RCVD).

However, what is intriguing me is how this fold occurs (to the right, or to the left, or over the same constant-beta line?) and why does it behave like that (the more Ay, the more it folds to the right for a constant beta). I think this is the product of some geometry and angles, but it also may be the combination of increasing scrub (longitudinal drag-force component of Fy) with increasing slip angles, which also produce a negative contribution to CN, I don't know. It may also be as simple as the Fy vs SA curves not peaking for high load (hence, high load transfer) cases. Even more curious is how/why is this effect amplified when the two saturated regions (front and rear) meet, at the untrimmed max-g end of the diagram (which is what actually forms this candy-wrapping paper shape).

To finally shed some light, I took some minutes to prepare the diagrams to be presentable for upload here, I removed the numbers because that is not the point anyway. I will only show the constant velocity diagram but the candy feature appears on all my diagrams (constant velocity and constant radius, including infinite which is the simplest case). Below I put links to the two figures, the first is the complete diagram (the axes on this diagram are symetrical so CN = 0 and Ay = 0 is right in the middle) and the second one is a detailed view of the max-g area. Blue lines represent constant steer situations and grayscale lines represent constant chassis slip. The darker the line, the larger its corresponding absolute value, so the brightest blue and gray lines are for zero steering delta and chassis slip respectively.

Complete diagram https://dl.dropboxusercontent.com/u/22545629/CN-AY%20Diagrams/CNvsAlat%20-%20V11.png

Detailed view https://dl.dropboxusercontent.com/u/22545629/CN-AY%20Diagrams/CNvsAlat_zoom%20-%20V11.png

What do you guys think?

JP

Silente

05-26-2015, 05:49 AM

After the discussion has deviated a bit, i would like to redirect it a bit on the original topic, the yaw moment diagram and its use.

I would love to have the feedback of more expert people about the post i published on page 8:

http://www.fsae.com/forums/showthread.php?6982-moment-diagram-with-weight-transfer/page8

As i said, i revised the tool mainly about the sign conventions and, although the results don't change dramatically in comparison with the old one, it looks now more correct to me.

The study refers to a very high downforce car, about 1000 kg, cornering at high speed (so aero loads are here important).

I would love to have your feedback about the results i am showing there (both the plot and the numbers) and it would be great to have some hints at what you would actually look, to identify car behaviors and tendencies (max lat acceleration, max lat acceleration in trimmed conditions, stability, control, ...?). Could you eventually advice on "realistic numbers for these metrics"?

@ jpusb,

my study doesn't show the candy effect that you i can see in your pictures and i am not really sure what it could depend of. But could it simply be something related to plotting? The lines are going back on themselves at a certain points, maybe your plotting tool draw somehow simply a "bigger" curve.

Flight909

05-26-2015, 10:24 AM

my study doesn't show the candy effect that you i can see in your pictures and i am not really sure what it could depend of. But could it simply be something related to plotting? The lines are going back on themselves at a certain points, maybe your plotting tool draw somehow simply a "bigger" curve.

I think we has to thinking about the tyres, because the limit of the diagram will deppend on the tyre limit. Which is the shape of your tire curve at the limit? If one axle has a big drop-of and the other not, you can for example maybe see this candy effect? I think the quality of the tire model will be very important for this diagrams, maybe not so much for central behaviour (were cornering stiffness will be a mayor factor), but absolutely for limit.

DougMilliken

05-27-2015, 12:08 PM

... drop-of and the other not, you can for example maybe see this candy effect?...
I think Roy Rice (CAL/Calspan, RCVD Fig 13.11, p.464-67) started calling this "fold over" of the tire data, when plotted on one of the MMM-style force-moment plots. It does have sort-of a 3D appearance.

BillCobb

05-27-2015, 05:59 PM

That's why we make 'carpet plots' of tire data, pressure effort curves for steering valves, and the value of money.

Add a third independent variable (maybe camber, pressure or figmosity for instance) to make it a real 3D curve. This just means you are missing something...

jpusb

05-28-2015, 04:57 AM

@ Silente. I don't think it is a plotting feature at all. I can trace this points to actual calculated "car states", and you can see it clearly in the zoomed in figure, where the lines show the trend quite smoothly. What tire model are you using for making your diagram? The boundary in your diagram is so strongly defined! I know you said Pacejka but, could you post some of your Fy vs Sa curves for a sweep of Fz tire curves? (you could plot them without numbers just as I did, I would just like to see their shape). At the moment I am using the TTC round 2, Hoosier 20.5x7.0 front and rear. When I have time, maybe in months haha, I will migrate to TTC 3 which is the latest I have around. Like you, I am also ignoring kinematic curves (motion ratio and stuff are just constants), camber, and so on. I also read you are not considering MZs and FXs. I am considering both, but MZs are really really small compared to the torque the FYs put in. For FXs, in my model, the rear wheels just push the longitudinal component of the front FYs (I call this scrub, is this term correct?), with an open diff though, and no traction circle so they do not compromise their FY capacity for giving thrust. However, I did check this and the thrust they need to give just to react front scrub is so small that they would not loose FY at all.

@ Doug. I don't clearly understand what you say Roy Rice defined as "fold over". I would have guessed the "fold over" would be what is shown in the top-left area of the zoomed in figure I posted, where the diagram folds over on itself due to front tire saturation, as you know. However, the candy feature is produced in the zone combining the two limits and it is different. Is this area what Roy Rice called "fold over"?. If so, damn! I thought maybe someday someone would use the "candy feature" term in a legendary book such as RCVD, you heard it here first! Just in case :D

@ Bill. I took a quick look at the plot you posted and I can't work out the X axis. However, I am sleepy and just took a very quick look, I will look into it with patience and organize my thoughts before rushing to ask you silly questions! Thanks for the input though.

I think this forum has long needed a thread such as this one.

DougMilliken

05-28-2015, 09:14 AM

It's been a long time, Roy Rice died before RCVD was written. I don't remember exactly how he defined things, but my best guess is that this was all called fold over, on the front, rear or both. Saturating the color for larger values seems to work well, a nice touch. One thing to try is to limit the delta-beta range which should reduce (or eliminate) the fold over. Then you can look for other effects at the Ay limit.

Silente

05-28-2015, 09:32 AM

Jpusb,

here a picture showing the slip curve of a front tire i am using, with vetical loads varying between 2000 and 6000N.

What is the point exactly in simulating the longitudinal forces if you don't consider how they are influencing the lateral ones? Just to have the weight transfer effect and see what it brings?

Mz are a good point which i want to include as well, as soon as possible.

Again, i would be interested to share about the metrics to look at and their values. I guess there is no universal rule and the sign convention depends influence also the signs of the results, but it would be nonetheless interesting to discuss about how to "read" and use dN/dDelta or dN/dBeta (as suggested by Claude to measure stability and control) both close to the origin and to the maximum lateral acceleration.

What i see is that both this metrics are, in my case, positive and very big in magnitude close to the origin; they then become negative at max lateral acceleration, at least in the base case scenario with TLLTD = 50% at the front.
Moving the TLLTD more at the front (60%), makes the dN/dBeta to further reduce (bigger in magnitude, but always negative) at max Ay (although the max Ay is now smaller) while the dN/dDelta see an increase (still being negative, but now smaller in magnitude).
Moving the TLLTD more at the rear (40%), makes the dN/dBeta to become positive at max Ay (although the max Ay is now smaller) while the dN/dDelta see adecrease (still being negative, but now bigger in magnitude, so "more negative").

jpusb

05-28-2015, 10:18 AM

@ Silente. Thank you for uploading the curves. Our tires differ quite a bit in how they respond to load. Your tires peak at smaller slip angles as the load is increased, mine are the opposite way. Maybe that is why we get so different shapes, I don't know. Important question, since both our diagrams are at constant velocity, do you correct the turn radius (thus, slip angles) for each Ay calculated?

Regarding FXs. The way I see it, there are various important aspects of including the FXs. The longitudinal component of the FY on the outside front tire is, of course, greater than the same force on the inside front, this introduces a counter-steering moment, which I think is important. Obviously, this effect increases with increasing slip angles are, which is the case near the boundaries we are discussing here. I did include the traction circle but I just turn it off to simplify things (runs faster too). Last, when I turn the aerodynamics switch on (which is off in the diagram posted above), drag becomes important and it must be pushed by the rear wheels, so maaaybe here the traction circle will be necessary, but I don't know, since drag comes with downforce of course. In a near or distant future, all this FX stuff will have a nice spice up once I include a model for the LSD diff and the FXs of the two rear wheels stop being equal, introducing an oversteering moment that could reshape the diagram quite a bit, I think.

Regarding MZs, although they are small they may alter the shape of the diagram if your car is very balanced (say, the max. untrimmed lat. g is at CN = 0), but I never checked their influence since I always included them (it is too easy to include them really). Furthermore, when say, front tires, are saturated, their contribution to CN is also saturated so MZs will also alter the shape of the boundary of the diagram I think.

Of course, if I had a lot of time I would answer many of my questions myself just by turning on/off things like weight transfer, MZs, FXs, considering or not FY-scrub, etc and analyzing more. I would also be considerably more motivated to continue the development of this model and all if I was in a FSAE team with moderate data acquisition capabilities so to validate, understand, and further develop the car. But I don't have either free time nor FSAE car with data at the moment, so I properly speculate away.

I would love to read the input of the rest of the known or unknown authorities on this subject, including Z's.

Silente

05-29-2015, 02:16 AM

@jpusb,

yes, i correct the corner radius for each output point of the diagram. Actually, since it is a constant velocity simulation, calculating the Fy gives automatically an Ay, so the radius is defined automatically as an output. From there i also then derive the slip angles.

By the way, here below the results i got for the three different TLLTD scenarios. I will add another message with the diagrams corresponding to the same scenario.

Silente

05-29-2015, 02:17 AM

second part, diagrams.

06-01-2015, 01:06 AM

REGARDING "YAW-MOMENT vs LATERAL-ACCELERATION" DIAGRAMS.
================================================== =========
(Or "YMDs", or "MMMDs", or "M3Ds"?)

Asked by Silente (p11);

I would love to have your feedback about the results I am showing...
... some hints at what you would actually look, to identify car behaviors and tendencies (max lat acceleration, max lat acceleration in trimmed conditions, stability, control, ...?).
... advice on "realistic numbers for these metrics"?
and,

Again, I would be interested to share about the metrics to look at and their values...
... interesting to discuss about how to "read" and use dN/dDelta or dN/dBeta (as suggested by Claude to measure stability and control) both close to the origin and to the maximum lateral acceleration.

Asked by Jpusb (p11).

I would love to read the input of the rest of the known or unknown authorities on this subject, including Z's.

I, too, would be interested in hearing more about above issues!
~~~o0o~~~

However, and this is probably because of my current NON-EXISTENT experience in using such diagrams, I am becoming skeptical of their use in gaining a deeper understanding of the relevant VD. Especially so, to the challenge of winning FS/FSAE competitions.

Here are some reasons why.

1. Both Silente and JP have produced these diagrams themselves, yet are having difficulty extracting "meaning" from them. What I find especially puzzling here, is that S and J have at their fingertips ALL THE INFORMATION necessary to interpret their diagrams, namely all the input stuff (eg. the tyre curves, etc.), yet the process of manipulating that data into the form of the diagrams seems to have resulted in a LOSS OF UNDERSTANDING.

Why? (I have my suspicions, but would like to hear other peoples' opinions.)

2. The presentation of the diagrams, especially JP's close-up of his right-corner "candy wrapper", suggests a very UNREALISTIC degree of precision. I would prefer to see the lines painted with a 12 inch wide paint-roller, to better represent the vagaries of real tyre-road grip. That way, as long as the roller-painted corner is somewhere near the horizontal-axis, the reader can interpret it as "close to balanced ... depending on conditions...". (Ie., "Any measurement is MEANINGLESS, without knowledge of its uncertainty...".)

3. To win FS/FSAE, I would NOT bother chasing "perfect-balance/neutral-handling". Instead, as I said way back on page 2, IMO much more important is to make your diagram AS BIG AS POSSIBLE!

Yep, "Small but perfectly formed..." can work well on the dating scene, but in FS/FSAE I reckon you really need SIZE. :) If you can make your car's diagram twice as big as the opposition's (ie. 2 x as high and 2 x as wide), then you blow said opposition away even while driving comfortably INSIDE the diagram, and well away from any "unstable limit edges".

Q. And how do you increase the size of your diagram?
A. Stickier tyres, and/or MORE AERO!!!
~~~o0o~~~

Despite above criticisms, I would still like to hear how the experts "read" such diagrams. What features are better or worse, etc...? Especially for FS/FSAE conditions?

Z

(PS. Jpusb, back when I worked in the packaging industry we called your corner feature a "butterfly wrap". I recall renovating some beautiful machines, built about 100 years ago, that would butterfly-wrap "candy" (= "lollies" here) using an almost entirely "mechanical" mechanism. Just a single 3-phase electric motor in the base that drove a big camshaft, then multiple push/pullrods, levers, linkages, etc. They were almost completely silent, and would run all day and night ... forever. The more modern versions had lots of VERY NOISY (!) pneumatics and electric-stepper-motors, and kept breaking down!)

theTTshark

06-01-2015, 08:16 AM

Despite above criticisms, I would still like to hear how the experts "read" such diagrams. What features are better or worse, etc...? Especially for FS/FSAE conditions?

Basically if you want to be able to put a handling metric on a vehicle setup you need to at least calculate certain portions of the MMM Diagram. Knowing targets of where the balance of the handling, driver control of the vehicle, or stability of the vehicle should be saves you time, money, and gains you respect by not wasting other people's time.

Essentially at the beginning the only thing you can really tell is if you are in the ballpark of having a balanced car, and roughly how many Gs you're going to be able to pull at a certain speed. This is because you have no testing data to correlate them with. Now if you have data from an older car and you did your job in diligent testing preparation and note taking you can model that vehicle and essentially see where your driver likes to live. But let's say the car is completely different and you can't. You make a couple of diagrams at several speeds and for a couple of easy setup changes that you plan on performing to get a snapshot of how the balance, control, and stability changes through the various corners you might encounter and you store that information in your testing folder. Then you go testing. You ask the driver okay how did the car feel in terms of balance, driver control, and vehicle stability. You make one of those pre-planned setup changes, do more laps, and repeat the feedback process. After enough setup changes you will eventually learn the delta that your driver works in. This delta takes into account the simplifications in your model as well as driver preference, but it begins to allow you to make incredibly smart setup decisions and saves you time while testing. At the University of Kansas we used these, a Total Lateral Load Transfer distribution spreadsheet, a quarter car model, and damper travel estimations in a variety of conditions to estimate how we should initially setup vehicles and how we should make changes. In 2014 JT and myself never even changed the springs through our whole competition season (Michigan and Lincoln). We were able to dedicate our testing to more important concepts and because of that we gained more respect with our team, peers, and ultimately got us both jobs. Eventually you don't even worry looking at the whole diagram or even any of the diagram. You focus on your balance metric and your control and stability derivative at the limit. From there you create MMMs for entry then for exit, and you start piecing the puzzle together of what the hell the driver wants.

Is it always right? No. Is anything always right? Absolutely not. Does it narrow your setup options down? Yes and thank god it does. I can't speak about my professional life on here very much, but Claude and Optimum G have successfully used this process to win races in a wide variety of series. Do all race teams use it? No. Do successful race teams use it? In some way, shape, or form they are or are working towards using it. There are simply too many benefits to not use it.

MCoach

06-01-2015, 03:03 PM

Yep, "Small but perfectly formed..." can work well on the dating scene, but in FS/FSAE I reckon you really need SIZE. :) If you can make your car's diagram twice as big as the opposition's (ie. 2 x as high and 2 x as wide), then you blow said opposition away even while driving comfortably INSIDE the diagram, and well away from any "unstable limit edges".

Q. And how do you increase the size of your diagram?
A. Stickier tyres, and/or MORE AERO!!!

There are options that are stickier than the current "standard" Hoosier offerings, however ever they have all demonstrated to show their ugly side after lap 1. Mostly because these tires are typically Avon (or some competitor) hillclimb tires and not meant to be heat cycled for more than a minute or so.
The simple answer would be to just get a larger tire, but then what is found is that the tire has too much mass to heat up in an FSAE envirmonment. Dirt track tires? Similar problem, soft, capable, but too much mass with too little contact area. Not a simple solution here.

2x high and wide? I'd settle for 1.1 times bigger. However, this only nets you potential of the vehicle. With increasing the boundary that the vehicle operates, it inherently means the driver needs to be capable of using that area predictably. If the car has "little quarks" here and there, then who would want a car that is 5% faster but ready to bite their head off and can't even use it. Of course, the answer to this was answered many years ago by pilots who now zoom around in statically unstable aircraft that will do circles around their stable brethren from yesteryear. But, without the computers that aid them, the planes handle like monsters and several have crashed in the development stages. Some, like the B-2 have crashed due to simple things like moisture on sensors, showing their nasty side.

Trent gave a pretty good summary of it all, and Chris Patton's paper from a few years ago explores such modelling a little more deeply for quasi-static simulation.

jpusb

06-03-2015, 09:37 PM

Don't get me wrong, I never said I did not understand the diagram or how to read it. Neither did I say that I did not understand the realities/limitations/uncertainties of modeling tires (especially at the limit), complianceless car, etc vs. the real situation. I work doing much more complicated experiments all day which are later correlated to very complex CFD models other people do, so I have a very good idea of this. This does not mean, however, that I don't get curious for how and why my idealized model is behaving like that. The zoomed in candy feature I just posted is just one particular doubt that came up, purely out of curiosity. I just wanted to know if other models (don't care about reality at the time of this doubt) show this. I haven't had time to work on it since I posted them, and when I posted them I just prepared the figures but I haven't worked on the model for 6 months or so.
Sadly, I am not trying to win FSAE anymore, I am just doing this just because I am curious. So I don't have a car to spend my time testing instead of spending it in this model. However, being an experimentalist myself, I see many, many uses for this model and the diagram, most of them already discussed in different ways here, during design events, and in RCVD. Is it the secret of life? Of course not, but it is just another tool in your toolbox. The more tools you have (both from modeling and from testing) the more educated your guess will be when the time comes. This is a tool I would have loved to have in my FSAE suspension guy/driver days.
However, the fact that I understand how and why my diagram is built this way does not mean that I am an expert in the matter (so I am not expected to know what is "usual" and what not), that I can solve the effects of all variables instantly in my head, and that I have seen 200 of these for very different cars. At the same time, with so many variables in the middle, so many angles and coordinates, and especially so little time to go deeply through this, it is normal that I am not entirely sure of if this A-feature is produced by that B-variable, etc. Therefore, I just asked for expert's opinion on how the diagram looks (in a general sense, besides the cool colors I put ;)) and if the candy feature was found usually in other cars' diagrams.

BillCobb

06-24-2015, 12:22 AM

I'm disappointed that the multiple requests for information on the direct use of MMM analysis and diagrams from experts has resulted in the threadbare patient dying on the operating table. As the only result, we are left with impressionistic works of art and without any consistent or meaningful engineering values or metrics. Is that the requiem for it ?

What I wish to see are diagrams produced from an elementary introductory and exploratory vehicle architecture, followed by a kitchen sink diagram for the same vehicle from the advanced hardware stage of the design, with test data obtained from a track, overlayed in a best of the best achievable correlation scenario. If you as a manufacturer's representative were hauled into court by plaintiff's legal team demanding these engineering documents, and all you had was candy wrappers, I believe a judge would park you in a jail cell for a day or so to contemplate your contempt. And, I have seen this done in a Federal Court in Washington, D.C..

Now what say you 'experts'. There is no jury, only jurists.

Otherwise, this thread belongs in an art museum to be contemplated for its passion, and personal subjective interpretations of the vigor of the lines and colors in dramatic repose.

In the applicable words or Claude Monet: (Do I need to remind you that he was an artist?)
"For me, a landscape does not exist in its own right, since its appearance changes at every moment; but the surrounding atmosphere brings it to life - the light and the air which vary continually. For me, it is only the surrounding atmosphere which gives subjects their true value."

His work is a Moment diagram, too, eh? and even with aero effects... !!

rwstevens59

09-27-2015, 01:26 PM

Attached (I hope) are a few photos of a very crude example of the kinematics I believe Z was trying to describe on page 10 of this thread.

After reading his post I had a look in the attic and in a very short time produced this 'model' to play with to more fully understand what Z was trying to describe.

Worked for me.
763764765766

The object of the last photo using two different radii is that in the world of oval track skewing of the solid rear axle wrt the car centerline is a common practice and I just wanted to have a look at what 'crabbing' down the track might look like.

Again, Sorry for the toys.

Ralph

09-30-2015, 08:33 PM

Ralph,

I cannot make out the writing on the arrows ... so I am not sure what they represent? Are they "the forces, from-car-to-ground, at centre of F&R-axles"?
~o0o~

Anyway, I have drawn five sketches of the Transient Cornering Kinematics/Dynamics of "The School Bus", and am just finishing writing up the words. Should have these up by next week.

Interestingly, having gone through this "planar" example in detail now, I am wondering why I bothered. It all seems so obvious! And so simple! Nevertheless, many long posts coming soon...

Z

rwstevens59

10-01-2015, 08:30 AM

Z,

The arrows simply represent the individual coordinate systems of Earth (shown in all pictures on the tile floor grid), body front truck pivot, body rear truck pivot, and an assumed body CG in the middle of the flat car body. There are also small arrows on the front an rear couplers indicating front and rear truck 'steer' relative to the body pivot coordinate systems or Earth system (track).

Being a kinematic problem I was not showing any forces, just directions and displacements to get a feel for how front truck steer on a body with a long wheelbase moves on corner entry in very small steps. Far removed due to the rear truck pivot point, but somewhat similar to your school bus.

Kinematic-ally as I studied this more I realized with a fixed path radii (the track) and pivots front and rear, that if you use equivalent links, as the front truck enters the corner this reduces to nothing more than a crank and slider 2D kinematics problem.

Far removed from a car with pneumatic tires I know, but I think it illustrates your point on how a corner is initiated.

Ralph

BillCobb

10-01-2015, 01:35 PM

rwstevens59

10-01-2015, 02:43 PM

The problem with this analogy is that nowhere in traditional vehicle dynamics straight line or cornering equations are displacement constraints (other than where is the ground). Vehicles initiate a turn because of a moment imbalance and continue to yaw and sideslip until this moment (whether from a front or rear steer angle or a wind gust or a tire induced imbalanced, etc.) is nulled. Depending in your point of view, the final resting trim can either be favorable or unfavorable relative to walls, trees, lakes, rivers or snow banks.

As an added observation, may I point out that REAL train's couplers are body mounted, not truck mounted (as your model trains are assembled). There is a huge advantage to this when negotiated turns at speeds well above a posted speed limits set by bank angles. Yes, train wheel flanges are a displacement constraint and the resulting cornering g levels from flanges plus coupler moments can be almost unbelievable. GM has an instrumented automobile carrier car that often rides along in coast to coast deliveries. The dynamic forces read were eye openers and resulted in major changes to batteries, wheel bearings, and vehicle tie down methods and systems. They are often higher than those experienced during abusive driving! GM uses the results to coax rail owners to repair portions of the route or avoid some portions altogether thank you.

I knew you would respond, correct on every count by the way, BillCobb. I got to thinking about Z's posts on the long wheelbase bus, then the kinematics of a bicycle model on rails that he suggested and that is when I decided, well heck, to get myself started thinking in the right direction about the simple kinematics I'll start playing with my toy trains.

It did not take long to realize that this was folly when compared to a pneumatic tired vehicle, but it was a good exercise none the less. However, I was fully aware of how far away from road vehicle dynamics I had strayed with this crude example.

It does reenforce the 'whose doing what to whom' approach that must be taken in any kinematics problem. Those old words 'With Respect To' can not be glossed over.

What can I say, had a kitchen floor that looks like graph paper, had track, had a long wheelbase flat car, cut some paper coordinate systems and play around on a rainy Saturday morning.

Thanks for the description of, and the data related to, real railway cars which I was unaware of and find very interesting.

Ralph

murpia

10-01-2015, 04:18 PM

BillCobb

10-01-2015, 08:51 PM

Instead, get out your biggest belt sander and turn it upside down. Then take a soft tired model car and put a string on it for restraint. Yes, Rembrandt, the car needs to fit on the sander belt and the restraint string has to be properly located (exam question). Then cut 'er loose. You can push on a front spindle with a straw to generate some initial side force. Now watch for the moment reaction and corresponding sideslip activity. You should be able to immediately see that the rear wheels go the wrong way first, then catch up to the rest of the motion.

For best results use a very fine paper (This is not Darlington), and some All Season tires. Getting a team together with a buzz on, a pizza and a strobe light will make for a REALLY cool evening science project. Video on demand.

Extra credit if the roll DOF is excitable.

10-01-2015, 09:25 PM

... generate some initial [front] side force. Now watch for the moment reaction and corresponding sideslip activity. You should be able to immediately see that the rear wheels go the wrong way first, then catch up to the rest of the motion.
Bill,

I will cover this below (next week?) under "The School Bus... (2) Starts Entering the Corner.".

(Spoiler alert: The "Sports Bus" behaves as per above quote. The "Slow Coach" does not, so is slower...)

Z

DougMilliken

10-03-2015, 02:21 PM

Instead, get out your biggest belt sander and turn it upside down. ...
Once you have the belt sander and model car, you can also constrain the model--see RCVD Figure 8.1. This was done c.1970, the model car has a wheelbase of about a foot (I still have this model). The "belt sander" was a proof-of-concept scale model of the belt installation for the Calspan tire tester, TIRF (where TTC data is measured). We didn't have a good photo, so worked with our artist to create the line drawing.

10-06-2015, 11:50 PM

Please make yourselves comfortable. Eight longish posts follow, covering YMDs (aka MMMs), the Acceleration Pole, and this whole business of transient cornering.
~o0o~

Better Yaw-Moment Diagrams.
========================

First I would like to clarify some of the criticisms I made earlier regarding the general use of YMDs.

The main problem I see with these types of diagrams, especially when they are presented in the manner given by Silente and Jpusb on this thread, is that too much of the useful information that could/should be displayed is NOT THERE. In short, a rainbow of pretty colours, even when drawn with computer precision, does NOT maketh good communication of information.

At the very least, these diagrams should include ALL the information necessary for their construction. So for these YMDs, the Velocity of Car, Radius of Corner, Details of Car (eg. is it a "bicycle model", or does it have width and weight-transfer), Details of Tyres, UNITS of X,Y-Axes!, etc., etc., should all appear somewhere in the diagram.

And this applies even if such diagrams are only for your own personal use. This is because in a few years time you might be working on a YMD problem and you remember that you solved a similar problem a few years back. So you get out your old YMDs (or open that file on your i-3000-whatsit...). And then ... you spend the next week trying to figure out what units you used for the axes!
~o0o~

More importantly, these STATIC diagrams might have been the bee's knees back in 1950, but much more can be done today.

Interestingly, Doug Milliken (an M in MMM) gave a very useful link on the "Pitch Axis" thread.
"Here is an interesting website that takes a novel (to me) approach to computing and displaying dynamic systems, http://worrydream.com/KillMath/ note the animations attached to the text. This sub-page uses a very simple car model as an example, http://worrydream.com/LadderOfAbstraction/ ."

I reckon those links give a really good approach to understanding these problems.

For example, you might start with the YMD in one corner of your computer screen. In another corner you have a plan-view of the car showing its major dimensions, corner-radius, etc., and also a display of each tyre's "Friction Circle" potential (as in Figure 8.25 of RCVD, or the "FRC" diagram on front cover). Elsewhere you have all the other important information.

The YMD/MMM has a cross-hair on it that can be moved about to highlight any given point on the diagram. As this is done the tyre-force arrows on the Friction Circle Diagram automatically update to show how much of their Friction Circle potential the car is actually using. And a whole bunch of other useful stuff can also be updated, such as all four tyre-slip angles, total force vector acting on car (as a single Force-vector acting along some LoA, or Force + Couple drawn at CG), and so on.

If you are not interested in certain information, then you simply ignore it.

But the more information that you have immediately available, then the more useful is this whole process to solving your given problem.

The following posts give a hint of some of the things I would like to see alongside any YMD I might draw...

Z

10-06-2015, 11:52 PM

Sketch 1/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ==============

These posts and sketches all relate to my earlier comments on the "Acceleration Pole" (http://www.fsae.com/forums/showthread.php?6982-moment-diagram-with-weight-transfer&p=123640&viewfull=1#post123640), and the fact that all this YMD stuff, and, in fact, all of Vehicle Dynamics generally, is about UNDERSTANDING "TRANSIENTS".

So get the idea out of your heads that you only have to understand the Steady-State stuff, "...because "transients" are just the brief and unimportant "joining-up" bits". Wrong!

EVERYTHING IS TRANSIENT!!!

Learn transients, and all that boring SS stuff becomes a doddle... :)
~o0o~

PRELIMINARIES - If I was doing this as a proper paid job, say, like Claude does, then I would start by listing ALL DEFINITIONS and ASSUMPTIONS (aka Axioms) needed to make sense of what follows. And what follows would be very rigorously built-up using ONLY those initial Definitions and Assumptions. But these posts are just interweb-waffle, so I will skip much of that rigour.

However, I will stress here that what follows is taken from a small corner of the idealised world known as "Mechanics". This world exists inside the similarly imaginary universe of "Applied Mathematics". So, the "Kinematics" discussed below deals only with ideal bodies that are perfectly rigid, perfectly smooth, perfectly massless, perfectly without forces, and moving in perfect Euclidian space. And later on there is also a pinch of perfect "Dynamics" thrown in, and a few other unrealistic things, such as all motions are only in perfectly parallel planes.

Some practical Engineers might say "But..., real racecars are not perfectly rigid, and you have to know all of their real compliances to understand them. So all that 'idealisation' makes the theoretical approach useless!".

Well, the fact is that any useful engineering analysis has NO CHOICE but to build from the above very solid, albeit idealised, foundations. To take just one example, in order to measure the compliances of your floppy racecar, YOU MUST FIRST ASSUME some sort of Euclidian reference-frame that the floppy bits move "with respect to". The assumed motion of the floppy-bits wrt some reference-frame, namely the "Kinematics" of the situation, is a necessary prerequisite for measuring the compliances. If no clearly postulated Kinematic foundation, then no measurements!

Anyway, enough of this deep and meaningful stuff. The following simple idealised descriptions are a good starting point for understanding how your cars corner. To ease this understanding, in this and the next four posts I cover the SPECIFIC case of the aforementioned School Bus going around a corner. In the last post below I cover GENERAL Kinematic Accelerations, albeit still in the idealised Planar world (= Flatland).

All this is done in a very brief and non-rigorous way, but hopefully good enough to give you a taste of how transient cornering works. For those of you interested in the more general case of fully 3-D Accelerations, well, some hints in the last post. Or you might ask Claude at one of his many seminars?
~o0o~

STRAIGHT AND STEADY - This first sketch shows the School Bus travelling (almost) straight and steady in a Northerly direction. The main thing of interest shown here is the "Velocity-vectors", with respect to "Inertial Space", of selected points on the bus.

Note that most of this and the following sketches has been worked out with real numbers, so is "to scale". For example, this particular bus has a wheelbase of 5 metres, width of 2.5 metres, CG slightly aft of mid-wheelbase (55R%), and the Velocity-vectors here represent a speed of 15 metres/second (= 54 kph, ~34 mph). (I have these numbers written down on numerous scraps of paper, and over too many weeks now, so I hope they all still add up...).

https://lh3.googleusercontent.com/-vTHZ8tlVEBw/VhSQSDH1U-I/AAAAAAAAAS0/v2rdzkgBzH8/s800-Ic42/AccDgm1.jpg

Because all the Velocity-vectors of different points on the bus are aligned North-South, it follows that the "n-lines" (ie. lines of NO motion) of all those points are aligned East-West. IF all the Velocity-vectors are perfectly N-S, then all the n-lines will also be perfectly parallel E-W, and so all the n-lines will "intersect at infinity".

But we assume that the Vs are NOT quite perfectly parallel, and the n-lines actually intersect at a Velocity Pole some distance out to the right (East) of the bus, perhaps just out past Pluto. So the bus is not actually travelling in a perfectly straight line, but is in fact cornering, albeit around a very large radius corner. This gives the bus a very small Rotational Velocity W (= "Omega", and more commonly called "Angular Velocity"). In fact, it is an extremely small W, because W = V/R, and while V = 15, R is extremely large.

Likewise the slow change in direction of all the Velocity-vectors amounts to "centripetal" (= "centre-seeking") accelerations of all the points toward the right. But these accelerations are also extremely small, because A.centripetal = V^2/R = 225/a-very-big-number. So I have not bothered drawing any Acceleration-vectors in this sketch. No pencil is sharp enough!

More coming...

Z

10-06-2015, 11:58 PM

Sketch 2a/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ==============

ENTERING THE CORNER - Now things start to get interesting. The school kids are shouting louder than usual, which wakes the bus driver from his slumber, and he suddenly realises he must turn into a side-street. So he yanks hard-right on the steering-wheel, performing what Vehicle Dynamicists call a "step-steer". This puts a "steer-angle" (or a "slip-angle", as some VDs call it) between the direction of motion of the two front-wheelprints (= the V-vectors at the front-wheelprint centres) and the centreplanes of the wheels, sometimes called the wheels' "heading-angles" and shown on the sketch as "centrelines".

After a very short time known as the "tyre relaxation length" (= distance tyre rolls before its sidewalls distort and "tense" up) the two front-wheels develop axial-forces Flf and Frf, in the direction of their axles. These are often referred to as the tyre's "Fy" forces, but note that they do NOT point in the bus's Y-coordinate direction. That is, they are NOT PURELY SIDEWAYS to the bus, but also point a bit backwards.

The two front-wheel axial-forces intersect some distance to the right of the bus and add together as a resultant force Ft ("t" for total, because "r" for resultant gets confused with "r" for rear). We also assume that all the upward ground-to-wheel forces cancel out all the downward gravity-forces, and there are no other forces acting on the bus, such as from aero, etc. So Ftotal is indeed the total external force acting on the School Bus as a "free-body".

The big question is, what does this assumed sum total of all forces acting on the School Bus, Ftotal, do to the bus?

The answer is easily found with some old-school geometry. The force Ft is slid along its Line-of-Action (LoA) to the point where the bus's CG is perpendicular to the LoA. Then, using the construction shown in the sketch, a semi-circle is drawn on the line through Ftotal and CG, such that the semi-circle also passes a distance "K" from the CG. ("K" = Radius-of-Gyration of the bus in Yaw. See below for more details...)

BINGO! This Planar Dynamics problem is now fully solved. :)

https://lh3.googleusercontent.com/-OiSxsGTFRMo/VhSQVBXrBrI/AAAAAAAAATM/0foBLcACn-8/s800-Ic42/AccDgm2.jpg

The above method tells us that the Acceleration Pole (AP) of the bus has now moved to the point M2, just in front of, and slightly to the left of the differential. We also know that all points on the bus are accelerating clockwise about the AP, with the magnitude of the A-vectors increasing in direct proportion to their distance from the AP, and with all their directions PERPENDICULAR to radii drawn from the AP.

We can even calculate that the front numberplate of the bus is accelerating rightward at about 28 m/s/s, or ~2.8 G, the front-wheels are accelerating rightward at ~20 m/s/s (~2 G), and the bus's CG is accelerating rightward at just under 7 m/s/s (~0.7 G). All this comes from a slightly optimistic front-tyre Mu = 1.5, and the other assumed dimensions.

Importantly, in the "Low-K Sports Bus" (main drawing at left of sketch), we see that the cool kids sitting in the rear seats are accelerating LEFTWARD! Likewise, the rear-wheels of this bus are also accelerating leftward. This is good, because it means the rear-wheels quickly develop the sort of slip-angles that produce a rightward axial-force, which is the direction the driver wants to go.

By contrast, in the smaller "High-K Slow Coach" sketch (bottom-centre), and at this very early phase of corner entry, the Acceleration Pole is some distance BEHIND the bus. So ALL points on the bus are accelerating RIGHTWARD. This is bad, because the rear-wheels initially move rightward while they still point due North, so they develop tyre-axial-forces that push the rear of the bus LEFTWARD, and out of the corner. This is NOT the direction the driver wants to go. It follows that High-K (= large-Yaw-Inertia) does NOT maketh a sporty bus.

Also worth noting that at this very early phase of "transient corner entry", the bus is still travelling essentially straight-ahead. (I gave it a very slight deviation from North at this early time-step.) And because all the V-vectors are still pointing close to due North, the Velocity Pole is still a long, long, way away to the right.
~o0o~

More coming...

Z

10-06-2015, 11:59 PM

Sketch 2b/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ===============

EQUIVALENT MASS SYSTEMS.
=========================
The old-fashioned Planar Dynamic Analysis done above is useful for solving a large range of problems. These problems just have to be close to "planar", where the Body can be "real 3-D", but all its motions must be in parallel planes. A similar approach also works for more general 3-D problems, and is very useful to understanding them! See Centre-of-Percussion, Compound-Pendulum, +++, for examples. The following is just a brief introduction to all this.

The gist of the method is that the real body, whatever its "true" nature, is first assumed to be a "continuum mass distribution". That is, the School Bus is assumed to be made of idealised stuff that has Inertial Mass properties (ie. it resists acceleration per N's Laws), but this stuff can be indefinitely subdivided, and may also be of variable density. Various Integral Calculus methods are then used to find things like the Total-Mass, Centre-of-Mass, and Moments-of-Inertia. Check your school books, or ask your teachers.

The next step is to replace the original complicated shaped body with a much simpler body, but one that has EXACTLY THE SAME Dynamic properties as the original body. This new, simpler, body then allows much easier calculations. Fortunately, Nature has been incredibly kind to us, making this process very easy!

All we need do is make sure that the first three "Mass-Moments-of-Inertia" of the differently shaped bodies are equal. No higher order MoIs are needed! Each "MoI" = Sum-over-all-i's-of-(Mi x Ri^N), where Mi is the elemental mass, Ri is the position vector to Mi in some reference frame, and each Ri is raised to the power N, which is the "order" of the MoI.

So, the first three MoIs are;
N = 0, the "zeroth MoI", which gives the Total-Mass of the body (note that R^0 always = 1),
N = 1, the "first MoI", which gives the CG position (or more correctly, the CoM) of the body, and
N = 2, the "second MoI", which gives the Second-Mass-Moment-of-Inertia of the body.

https://lh3.googleusercontent.com/-OiSxsGTFRMo/VhSQVBXrBrI/AAAAAAAAATM/0foBLcACn-8/s800-Ic42/AccDgm2.jpg

One "simpler body" that can be used to represent the original body is a circular ring, as shown at left of sketch. This ring has the same Total-Mass Mt as the School Bus, the CGs of ring and bus are coincident, and the radius K of the ring gives an equal 2nd-MoI-in-Yaw (= Mt x K^2) to that of the bus. Unfortunately, this ring does NOT simplify any calculations, because it is still a distributed mass, so it is still subject to Euler's Rigid Body Equations, just like the original School Bus. (Some deep-and-meaningful stuff here, but moving on...)

So an even "simpler body" is to reduce the bus down to just TWO POINT-MASSES connected by a "massless rod". Yippee! Now we can use Newton's Laws of Motion directly, which is significantly simpler than using Euler's RBEs. There are two infinities of ways of choosing our two point-masses, which is good because it gives us a lot of freedom in solving the problem. Very briefly, (in this planar case) we choose the position (= X,Y coords) of one point-mass in any way we want, and then the size of both point-masses and the position of the second point-mass are automatically determined. This is shown, rather non-rigourously, at bottom-right of the sketch.

One simple choice of the two point-masses is to have two masses, each Mt/2, at each end of a diameter of the circular ring of radius K. This gives a "dumb-bell" of length 2 x K that accurately represents the whole bus. Note that for these planar problems the orientation of the dumb-bell is irrelevant. It can be aligned along the centreline of the bus, or it can be lateral to the bus, or at any other angle, but the mid-point of the "massless rod" connecting the two half-masses must be at the bus's CG.

Here, a more useful choice is to have the first point-mass M1 sitting on the LoA of Ft, which lies a distance "a" from the CG. The location of the second point-mass M2, at a distance "b" on the other side of the CG, is then determined by the semi-circle method, namely "K^2 = a.b". Use Pythagoras to verify the little drawing at bottom-right of sketch. The sizes of M1 and M2 are found from the "0th & 1st MoIs", namely M1 = Mt x b/(a+b), and M2 = Mt x a/(a+b).

The last step of this Dynamic Analysis is to note that the force Ft can only act on M1 but NOT on M2 (<- important!), because, although the "massless rod" connecting M1 and M2 is capable of carrying forces (perfectly well!), at this instant there is NO COMPONENT of the force Ft directed along the massless rod. Therefore, while Ft accelerates M1 to the right, the second mass M2 just sits there minding its own business, as per NI (aka Galileo's Law of Inertia).

Therefore, at this instant, M2 has no acceleration whatsoever, and is thus the Kinematic Acceleration Pole of the School Bus.

Interestingly, note that the Sports Bus has both Ft and M1 in front of, and OUTSIDE, its K-ring, which puts M2, and thus the Acceleration Pole, INSIDE the K-ring. However, the Slow Coach has Ft and M1 inside its K-ring, putting M2 and the AP outside the K-ring, and thus a long way rearward. This greatly affects the transient performance of the two buses, as noted in previous post.

Of course, all the above is only the case for the very first instant of this Dynamic problem, because as each "dT" time-step passes the AP moves to a new location, and other things happen...

So, where does the Acceleration Pole move to next?

More coming...

Z

10-07-2015, 12:00 AM

Sketch 3/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== =============

"PURE" CORNERING - This sketch shows the fabled and fleeting condition sometimes referred to as "Steady-State" cornering. Here the Velocity Pole has moved quite close to the bus, now only 15 metres to the right of the bus's CG. By an incredible coincidence the Acceleration Pole is at this same point!

Note that in this sketch the Velocity-vector at the bus's CG is parallel to the centreline of the bus. This is an unlikely coincidence, and is NOT always the case in pure cornering. In fact, this "tangent condition" is an exceptional case, and I should have drawn it more generally with Vcg NOT parallel to the centreline. But I was probably thinking of other stuff...

More importantly, when a bus on conventional tyres is in this "SS" cornering, the Corner Centre, namely the Velocity Pole, is always forward of the rear-axle-line. This is a direct consequence of the "slip-angles" that are generated whenever the flexible carcasses of pneumatic-tyres are subject to the cornering forces shown as Flr and Frr. So also necessary for SS is that the rear-wheels generate forwards driving forces. So the rear-wheels, and hence also the engine, must constantly expend power (= Force-vector dot-product Velocity-vector) to overcome the "slip-angle-drag" of pneumatic-tyres.

Interestingly, if the bus is rolling on "perfectly frictionless rails", then all wheels have zero slip-angle, the Velocity Pole lies on a rightward extension of the rear-axle-line, the Kinetic Energy of the bus is conserved during cornering (ie. it is equal to when it travels straight-ahead), but the Velocity at the CG is slightly less during cornering than the 15 m/s of straight-ahead travel. The KE seemingly "lost" due to the slower speed of the CG during cornering is made up by the extra rotational KE of the whole bus. In a "conservative" system (ie. bus on frictionless rails), when the bus exits the corner the rotational KE is transferred back to linear KE, and the CG gets back to its original 15 m/s (draw the FBDs!).

https://lh3.googleusercontent.com/-2V4P6R8uLAI/VhSQTBh6TsI/AAAAAAAAAS8/Hs3_7y3phD8/s800-Ic42/AccDgm3.jpg

Unlike in Sketch-2, here the Acceleration-vectors of all points on the bus are quite large. (Edit: This is because the front-wheel axial-forces have increased slightly, but more importantly the rear-wheels are now also contributing quite large forces...) The A-vectors are slightly larger on the outer (left) side of the bus, and slightly smaller on the inner (right) side. If a point of the bus were at the Corner Centre (= VP, and = AP here), then it would have zero acceleration, which is, of course, the definition of the Acceleration Pole.

An important feature of this sketch is that all the A-vectors point DIRECTLY at the Velocity Pole. This is a direct consequence of the resultant (= sum total) of all tyre-forces (ie. Flf + Frf + Flr + Frr = Ftotal) also pointing directly at the VP. Thus, with the A-vectors perpendicular to the V-vectors at all points, the magnitudes of all V-vectors must remain constant, although their directions do change. This change of direction of the V-vectors is at a constant rate, and amounts to a constant clockwise Rotational Velocity W of the whole bus (which, incidentally, is 1 radian/second here).
~o0o~

[Some Tech Notes:]
1. At each point on the bus, the A-vector merely represents the small change "dV" in the Velocity-vector, over a small period of time "dT". Here all the A-vectors (= dV/dTs) are drawn at a scale that makes them big enough to see. So, if the sketch represents the situation at a given time Tn, then all the velocities at the next time-step Tn+1 (= Tn + dT) are found by vectorially adding a small dV-vector (= An x dT) to each current V-vector. That is, Vn+1 = Vn + An x dT. This is only approximately correct for biggish dT, but more exactly so as dT -> 0. In this sketch all the dVs are perpendicular to their Vs, so the dVs cannot change the Vs' magnitudes, but they can change their directions.

2. The sketch is drawn mainly to show the Kinematics of the situation. In this case you can think of the bus as being "perfectly massless, energy-less", and so on. However, if you want to think of the Dynamics of the bus, then relabel all the V-vectors as "P-vectors", which represent the MOMENTUM of each small but massive particle of the bus (ie. Pi = Mi x Vi). Now the A-vectors, suitably scaled down by multiplying them with dT, represent the CHANGES in Momentum "dPi" of those massive particles over a period of time dT. Those Momentum changes are CAUSED by the nett Force-vector Fi acting on each particle over that time-step, so the magnitude of each dPi is directly proportional to its impressed Force Fi, and directions of Fi and dPi are the same, per NII.

3. The sometimes confusing thing about working with "continuum mass distributions" and Euler's Rigid Body Equations, is that each and every "small particle" Mi of the bus has a great many Forces acting on it, but most of these Forces come from the other particles of the bus. So, because of NIII, the vast majority of the Forces in the problem cancel each other out. Thus Euler's RBEs allow us to just consider the much smaller number of "external Forces" that act on the "whole body", while we ignore the vast majority of "internal Forces" that are in play. We then simply stir the alphabet-soup of RBEs and the right answer pops out, even though we have ignored the majority of the problem. Much more that can be said here, but hopefully the above gets you thinking...
[End Tech Notes]
~o0o~

Lastly, the constantness of the Rotational Velocity W of the bus here implies that the Rotational Acceleration "W-dot" is zero (aka "Angular Acceleration", and nowadays often denoted on web as "Alpha", but I will stick with W-dot for now...). This lack of Rotational Acceleration of the bus is why some people call this "Steady-State", even though there is obviously an abundance of A-vectors.

The extreme unlikeliness of this sort of (high-acceleration!) SS is seen by noting that it only occurs while Ftotal points directly (= "perfectly/exactly"...) at the VP. If any one of the four tyre-forces varies the least little bit, either in magnitude or direction, then this condition is lost. As soon as a single tyre-force varies, the LoA of Ftotal misses VP, the AP moves away from VP, W-dot becomes non-zero, all the Vs start changing in magnitude as well as direction, and it becomes somewhat pointless to talk about "Steady-State".

More coming...

Z

10-07-2015, 12:02 AM

Sketch 4/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ==============

EXITING THE CORNER - Eventually the driver decides he has spent enough time cornering, so he starts unwinding the steering-wheel. The sketch shows the situation when the heading-angles of the front-wheel centreplanes are aligned (perfectly!) with their Velocity-vectors. So now there are NO forces acting on the front-wheels, but the rear-wheel forces (from-ground->to-bus) are the same as in Sketch-3. These rear-wheel forces now amount to the sum total force, Ftotal, acting on the bus.

What does Ftotal do to the bus now?

In the blink of an eye, the Acceleration Pole jumps from the Corner Centre (= VP) to a point near the middle of the front-half of the bus. Wow, that AP sure can get around! As a result of this movement of the AP, itself directly caused by the changing pattern of forces acting on the bus, the front of the bus is now accelerating LEFTWARD (fast!). Meanwhile, the rearmost parts of the bus are still accelerating rightward towards the CC, but now even FASTER than when the bus was in "pure cornering"!

Of course, the Velocity-vectors of all points on the bus remain almost the same as in Sketch-3. There are only very tiny changes to the V-vectors, because only very small time dT has elapsed since Sketch-3.

https://lh3.googleusercontent.com/-bwiUGROFjr0/VhSQUEYH2AI/AAAAAAAAATE/6M0oIixXixk/s800-Ic42/AccDgm4.jpg

The exact pattern of the the A-vectors can be found by using similar geometric methods and Equivalent Mass Systems as in Sketch-2. That is, we replace the distributed mass of the whole bus with two appropriately chosen point masses, M1 and M2. Here, both these masses lie very close to the "K-ring", so they are both very close in mass to Mt/2. In fact, M1 is very slightly larger than Mt/2, because "a" is a bit less than "b", but this is just a coincidence here. But now there is an extra twist to this method.

Because M1 and M2 are part of a single rigid body (ie. they are connected by a "rigid but massless rod"), and because that body is ALREADY ROTATING with finite Rotational Velocity W (wrt Inertial Space!), each mass necessarily feels a force acting on it towards the other mass. These forces acting on each mass, coming from the other mass and transmitted by the "massless rod", are also necessarily equal in magnitude. So each mass has a component of acceleration towards the other mass that is inversely in proportion to the size of its own mass (ie. smaller mass M2 accelerates towards M1 slightly faster than M1 towards M2).

Taking the above into account, while using a similar geometric method as in Sketch-2, gives the magnitude and direction of all the A-vectors, and so also the location of the Acceleration Pole. As seen, all the the A-vectors are now swirling ANTI-CLOCKWISE around the AP, with larger magnitudes when further away from AP. Also, all A-vectors now have a radially inward component, towards the AP. So all A-vectors make an angle LESS THAN 90 DEGREES with radii drawn from the AP. Furthermore, as shown by "((" on the A-vectors near the front of bus, ALL these angles are EQUAL. Here all the angles between A-vectors and radii are ~77 degrees.

It is probably not of much benefit to spell out the above geometric method in detail here. The same results can be obtained with the more conventional Euler's Equations methods that are usually taught these days, and which are very briefly outlined in next post.

However, it is well worth closely studying the overall pattern of A-vectors here, and the fact that some features of this pattern are ALWAYS PRESENT. These general features covered next.

More coming...

Z

10-07-2015, 12:04 AM

Sketch 5/5. More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ==============

GENERAL PLANAR ACCELERATION DIAGRAMS - This sketch shows the essential Kinematic features that can be found in ALL general planar problems. Across the top of the sketch are shown the three "elemental" types of planar acceleration, and at top-right the general result when they are combined. (But note that the leftmost "Linear" acceleration can be thought of as a special case of either the second "Rotational", or third "Centripetal", acceleration, but taken a long (= "infinite") way away from their centres.)

The A-vectors in those four topmost sketches are all drawn to the same scale. So vectorially adding the Linear+Rotational+Centripetal A-vectors at any given point on the rectangular Body gives the resultant A-vector shown on the rightmost ("GAD") Body. For example, at the very right of the sketch is shown a little vector triangle of L+R+C vectors that sum to the zero magnitude Acceleration Pole of the GAD Body. Practise doing this and you see that summing L+R moves the AP "sideways" (to the Ls), and L+C moves the AP "forwards" (in direction of Ls).

The most general features of ALL Planar Acceleration Diagrams are:
1. They ALWAYS have an AP (ie. point where Acceleration = 0), with all the other A-vectors swirling around the AP in the same sense (ie. all ClockWise, or all Anti-CW). However, the AP may be a long way away, maybe infinitely far away...
2. The magnitudes of A-vectors ALWAYS increase in direct proportion to distance from AP (so double distance = double magnitude).
3. The directions of A-vectors ALWAYS make the SAME angle with the radii from the AP.

The above means that Acceleration Diagrams are "linearly superposable", which means that vectorially adding together any two ADs simply gives another AD with the same general features as above. Yet another feature of this "linearity" is that all Body points lying on a straight line ALWAYS have all the arrowheads of their A-vectors also lying on a straight line (see top-right corner of sketch, or the vector-tips of any group of points lying on a straight-line). Too easy! :)
~o0o~

We can also put on our Dynamic hats and ask: What sorts of forces systems must act on the massive Body (ie. the rectangle) to cause the three different types of basic acceleration? The answers are fairly obvious, so no rigourous proof given here.
1. A pure Linear-Force acting on a LoA that passes (perfectly/exactly!) through the CG of the Body causes purely Linear acceleration.
2. A pure Couple acting ANYWHERE on the Body causes purely Rotational acceleration, with the AP at the CG.
3. A pre-existing rotation of the Body (wrt Inertial Space), with NO EXTERNAL FORCES WHATSOEVER acting on the Body, causes purely Centripetal acceleration, again with AP at CG. Interestingly, in this case all the little massive particles that make up the Body have forces INTERNAL to the Body acting on them, and thus causing their accelerations (or more correctly, causing their Momentum-changes), but all these internal forces cancel-out in a FBD of the whole Body.

Quite obviously, a "general force system" acting on the Body causes the General Acceleration Diagram. In Planar Dynamics a "general force system" can be seen either as a single Linear-Force on a LoA that does NOT NECESSARILY pass through the CG, or else as a Linear-Force on LoA that DOES pass through the CG, PLUS a Couple acting on the Body as a whole.
(Notes:
1. A pure Couple acting on Body = Infinitesimally-small-Force on LoA-infinitely-far-from-CG.
2. In 3-D problems it is generally NOT possible to have only a single Linear-Force acting on the Body with NO Couple. Search Forum for "Force Screw".)

https://lh3.googleusercontent.com/-PoxlTir09mo/VhSQVzWvMTI/AAAAAAAAATU/J9RkQRi70pk/s800-Ic42/AccDgm5.jpg

The lower two-thirds of the sketch shows how to determine all the A-vectors of the Body, namely the whole Acceleration Diagram for this planar problem, when given only a small amount of initial information. Sufficient information is to know the magnitudes and directions (or "X,Y" components) of two A-vectors at two separate points of the Body, or else to know one A-vector plus the magnitudes of W and W-dot. (Note that in these planar problems the directions of both W and W-dot vectors are always assumed to be perpendicular to "the plane".)

[Notational Note: Having checked this stuff on the web, it seems that a de-facto standard nowadays is to subscript the RoTational Acceleration component with "T" for "tangential", and the CeNtripetal component with "N" for "normal". This notation is derived from the "path coordinates of a moving POINT", but I reckon this can be confusing. For example, here the "tangential" component is, in fact, "normal" to the line joining the two Body-points (1) and (2)! I tried to find some better alternatives, but eventually settled on the above emboldened mnemonics. Ahh..., alphabet-soup! Proceed with caution.]

Anyway, here is how to determine your car's Planar Acceleration Diagrams. You mount two "two-axis G-sensors" on two widely separated points of the car. Perhaps mount one G-sensor to Body-point-(1) near the car's tail, and another G-sensor at point-(2) near the nose, as seen in X-Y plan-view. At any instant in time, these sensors give you the A-vectors A1 and A2 at those points, and so also the X&Y-components of each vector, as shown in the sketch.

You also know the Radius-vector R21 (= "radius of (2), wrt (1)"), and more specifically its magnitude |R|, because you measured this distance with a tape-measure. Then, from the alphabet-soup scattered around the sketch, you find W, W-dot, the location of the Acceleration Pole, and the whole Acceleration Diagram. Some hints to ease this process:

1. The "vector-polygon" at bottom-right of sketch shows how A2 = A1 + At + An. The T and N components, "of (2), wrt (1)" are found using the equations at left of sketch. Incidentally, the angle the "At+An" vector makes with the Radius-vector is the same as the angles that ALL A-vectors make with radii drawn from the Acceleration Pole.

2. W-dot (Yaw-Acceleration) is found very easily, and at any instant is "the difference of y-component-accelerations of the two points, divided by distance between the two points". See "At =..." equation at left of sketch.

3. W (Yaw-Rate) is a bit trickier, because while its magnitude is as easy to find as W-dot's, the "square-rooting" operation (or more correctly, the "double vector-cross-product") destroys the directional information. See "An =..." equation. Put simply, a given speed of ClockWise rotation produces EXACTLY THE SAME Centripetal Accelerations as the same speed Anti-ClockWise rotation.

So, you cannot know the direction of W from a single instantaneous reading of the two G-sensors (ie. CW or ACW?). But as long as you track W-dot from a known condition, such as "car stationary at start-line", then the direction of W should be obvious. Of course, you can also determine W by integrating the W-dot readings over time. But that process gradually accumulates errors, so the "instantaneous" measurement of |W| is preferable because more accurate.

4. In principle, 6 x 1-axis-G-sensors, suitably mounted at different locations around the car, can give a complete 3-D motion history of the car. But this requires integration of the W-dots, so can lead to drift errors. Mounting 3 x 3-axis-sensors at three widely separated points of the car gives enough redundancy to obtain the 3-D components of |W| directly, so more accurate.

5. A greater separation of the sensors, namely a bigger |R|, gives more accuracy. So maybe mount the sensors at nose, tail, and top-of-MRH. Keep in mind that the sensors are mounted to a floppy-frame, and not to a perfectly-rigid-body, so filter out the many vibrations before drawing wildly wrong conclusions about maximum Yaw-rates, etc.
~o0o~

Last little 3-D Dynamic note:
The conventional use of Euler's Rigid Body Equations to solve 3-D Dynamic problems requires the "general force system" to be seen as (or converted to) a Linear-Force-on-LoA-through-CG, plus a Couple-acting-on-Body. The Eulerian version of Newton-II then becomes:
1. Linear-Force F causes change of Linear Momentum P of the Body (ie. F -> P-dot, with P = Mtotal x Vcg, which is much like the dynamics of a Newtonian particle).
2. Couple T causes change of Rotational Momentum L of the Body (ie. T -> L-dot, with L = Body-Inertia-Tensor x W, and L also called "Angular Momentum" or "Moment of Momentum").

The interesting thing here is that, in general, the 3-D Body-Inertia-Tensor makes the Body's Rotational Momentum L-vector point in a different direction to its Rotational Velocity W-vector. And since, in general, the Force F-vector and Couple T-vector acting on the Body also point in different directions, the resulting 3-D pattern of A-vectors gets a bit more interesting than shown here.
~o0o~

More coming...

Z

10-07-2015, 12:08 AM

Last of ... More PLANAR KINEMATICS and ACCELERATION POLES, with a Dash of DYNAMICS...
================================================== ============

CONCLUDING REMARKS - Having reread all the above posts now, it all seems so simple and obvious to me that I wonder if this level of detail was really necessary. On the other hand, skimming through the earlier pages of this thread, and also through many VD textbooks, seems to suggest that a lot of the above is NOT so obvious?

So, since I am rather worded-out now, I guess I will leave it to the rest of you to determine how obvious all this "transient" stuff is. Are the answers to these following questions obvious, or not?

1. Is LART (= Lateral Acceleration Response Time), all by itself, really the most important metric for a transiently sporty car?

2. What about YART (= Yaw Acceleration Response Time)!?

3. Does a forward-biased weight-distribution really maketh a sportier car (as claimed by others, earlier), and if so, then WHY!?

4. Is the generally faster LART of high-F% cars merely by-product of the "LA" being measured closer to the front-wheels, which is where, on standard "front-steered" cars, this whole transient cornering process is initiated?

5. Will a high-R% car, such as a Porker, EVER win a race, and why?

6. What general correlations can be expected between changes to the F:R% of a car, and the consequent changes to its LART and YART, if "all other things are kept equal"?

7. How big an influence to transient cornering is the car's "K", namely its "Yaw-Radius-of-Gyration" (or "Yaw-MoI")?

8. Considering the Sports Bus and Slow Coach of Sketch-2, what can be expected of their respective time-histories of F&R tyre-forces as they move from Sketch-2 to Sketch-3, and are these bus's names thus appropriate?

9. Considering the Bus-CG, the Velocity Pole, and the Acceleration Pole, which of these can move most quickly, which moves most slowly, and does this explain why some quite third-rate "LapSims" give remarkably accurate results? (<- :D)

And probably many more deep-and-meaningful VD questions that could be asked...

But I reckon all answers are very straightforward, and easily found in the above sketches. I hope you students now also see all this a bit more clearly.

As always, comments, criticisms, or questions most welcome. :)
~o0o~

Most importantly, keep in mind that ALL IS TRANSIENT, so do not waste too much time thinking about that boring Steady-State stuff...

Z

maxay1

10-08-2015, 07:01 AM

I guess the information above is common knowledge to everyone but myself, given the lack of response, but thank you Z for taking the time to put that together.
The transient behavior of vehicles having different yaw DI's is very interesting to me, something I've been looking at for some time. As much as I appreciate the
sketches and accompanying explanation, your questions have really made me think, and test my understanding. I hope this topic continues as it should.
Adding the roll d.o.f, as Bill mentioned a couple of posts back, to discuss its effects on the relation between the contact patch and the wheel centerline,
will be interesting as well. More as soon as I have time, I just wanted to say thank you.

Wil

BillCobb

10-08-2015, 02:03 PM

I await with tongue biting anticipation, the responses and additional words of wisdom to Z from the gurus, professors, mechanics, charletons, blowhards, snake-oil salesman, lawyers and politicians who could reply but have headed for the hills fearing the tribulation.

The answers to Z's questions are pretty straightup, but require a little bit of Deus ex machina to show cause.

What say you Caesars ? Acta est fabula, plaudite !

rwstevens59

10-08-2015, 03:37 PM

Z,

First, thank you for all of your work. It is as per usual clear, concise and most important a nicely compact piece of educational material.

Second, my question.

Outside of the realm of FSAE (therefore somewhat off topic) I am usually confronted with a completed race car. I did not build it, there are usually no drawings.

Equipment available in the shop is reduced to a set of wheel scales.

So...total mass, easy. Center of mass X & Y location in a chosen frame wrt ground easy. Center of mass Z location wrt ground somewhat rough using the old tilt the car method due to the center of mass being fairly close to wheel hub height.

The big problem, for me, has always been coming up with K short of building a very large trifilar pendulum. :D

Is there a way to reasonably estimate the second moment of mass about the Z axis?

I know that in the Milliken Olley book, Olley estimates K based on car dimensions of his day, probably with some laboratory pendulum data to back it up, but that would be a poor estimate for the car configuration I am looking at.

Thanks,

Ralph

DougMilliken

10-08-2015, 07:49 PM

The big problem, for me, has always been coming up with K short of building a very large trifilar pendulum. :D

In the aircraft world this is part of "weight and balance". The practitioners even have a professional society, http://www.sawe.org/

When the car comes apart, make sure that every part is weighed and entered into a spreadsheet with its X, Y & Z location, or weigh spare parts if the car isn't coming apart anytime soon. With enough parts in the spreadsheet you can make a pretty good estimate of the inertias and also the CG height.

rwstevens59

10-08-2015, 08:16 PM

DougMilliken,

Thank you for the link, I never knew that.

I am very familiar with the spreadsheet method you propose and it works well with attention to detail in setting up a measurement reference frame and taking care with your measurements.

I guess my enemy is time. Time not available during tear down. Time when parts are free for measurement. And an amateur teams view of spending the time to do something they may not see any benefit in doing.

I'll find the right mix of time, people and interest someday.

I have only done a really thorough job of a weight analysis once many many years ago back in my school days when we were building an SAE high mileage vehicle. No Excel back then. A paper spreadsheet and hand calculation.

Thanks again.

maxay1

10-08-2015, 10:20 PM

For those who need a 'real' VD book to convince them of the validity and necessity of this information, velocity and acceleration fields, and the inflection circle, are covered in Guiggiani's The Science of Vehicle Dynamics. Or you can just look in a classical mechanics text.
I'd been learning (or more accurately, trying to learn) about the effects of rotating a rigid body about a non-principal inertial axis, such as would very likely be the case when a vehicle's roll axis is not coincident with it's inertial axis (the inertia tensor has has non-zero products of inertia).
There may be a brief, but important, yaw response due to roll acceleration that your driver may react to. Add to that a vehicle that is underdamped in yaw, where the initial yaw rate overshoots the steady-state value, and you may have a driver that
raves about the 'turn-in', but complains of mid-corner understeer.....

BillCobb

10-08-2015, 10:46 PM

For weight analysis, sure do the summation of point masses in a spreadsheet. Inertia, rig the tri-filer pendulum. Engine and transmission assembly is not a point mass and it alone could account for 1/2 the total inertia of the whole car. Plus, it's inertia ellipsoid's axes probably won't line up nicely with the car coordinates you choose. There are some other ways to get these values, including methods for obtaining unsprung inertias.

10-10-2015, 12:13 AM

Ralph,

The big problem, for me, has always been coming up with K short of building a very large trifilar pendulum.
Is there a way to reasonably estimate the second moment of mass about the Z axis?

1. The rough calculation by summing contributions from component masses (ie. ~Sum-all-i's(MoI.i + Mi x Ri^2) is the quickest, easiest way, albeit probably at +/-10% error (?). But 90% of an answer is better than no answer. Doing this is also good from a "design" viewpoint, in that doing the calcs many times over makes it clear how big the influence is from masses at the extremities of the car (because of R-squared).

2. Here is my first post on an earlier thread (http://www.fsae.com/forums/showthread.php?4459-Avarage-Cg-of-an-FSAE-Vehicle&p=85043&viewfull=1#post85043) discussing other methods. There is a simple sketch on the next page (p3) showing the "Compound Pendulum" method of finding the car's Yaw-Inertia (the appropriate equation is a few posts before that sketch). But this method is probably better suited to smaller FS/FSAE cars, and maybe not your full-sized ones? Although years ago I did measure some off-road-buggys' CG-Heights and Yaw-Inertias by "swinging" them from a tree branch! (As noted, windy days = !!! :( !!! )

3. Another method is becoming commonplace nowadays, probably because even easier. You build a platform a bit like a playground merry-go-round, or a "turntable". The car sits on the turntable with car-CG directly above the vertical axis. The turntable has a spring between itself and the floor that provides the rotational "restoring force" (measured as torque/rotation-angle, say N.m/radian). You then set this rotational spring-mass system oscillating and do the fairly simple maths to get Yaw-Inertia of the car alone (ie. subtract the turntable's MoI).

I reckon said turntable could be as simple as a front-wheel+axle-bearings+upright assembly. The (steel) wheel sits on its side on the floor, with the upright uppermost. Bolt a very simple frame (angle-iron or welded-RHS...) to the upright. Gently lower the car onto this frame, with car-CG directly above axle (an offset CG will tilt the whole assembly, so adjust car position until it sits level). The "restoring force" spring might just be an appropriately sized steel coil-spring working in both tension and compression, and mounted between car-nose and heavy-object-sitting-on-floor. You give the turntable+car a small rotation from its zero-position, let go, and measure time for ten rotational oscillations. Divide by 10 for period "T", and do the maths...
~~~o0o~~~

Wil,

I'd been learning (or more accurately, trying to learn) about the effects of rotating a rigid body about a non-principal inertial axis, ... (the inertia tensor has ... non-zero products of inertia).

Yes, despite the extreme simplicity of Newton's Laws (ie. little more than F -> P-dot), once you glue a few of N's particles together into a single "rigid body", and then get that body rotating, it all get very interesting, very quickly!

Here is a short thread where I discuss Principal-Axes... (http://www.fsae.com/forums/showthread.php?7498-Skewing-roll-axis-and-its-effects-on-handling) about half-way down the first page, and clarify again at top of 2nd page. (Wow, ten years ago...). Note that many different issues are discussed on that thread, and not always very clearly because of the difficulty of trying to explain in words what really needs a lot of pictures.

The gist of what I was getting at in above thread, is that even in the very simple "Steady-State" cornering shown in Sketch-3 above, WITH the "tangent condition" (ie. the "centrifugal force" at CG is exactly perpendicular to car-centreline, so it is purely "lateral" to the car), it is possible for the car to have a Dynamic LONGITUDINAL-load-transfer (ie. F:R "axle-weights" are different to when car stationary on scales)! This happens whenever the car's principal-axes are not "square" with its rotation W-vector (eg. longitudinal principal-axis NOT horizontal).

The dumb-bell analogy I gave is not completely accurate, but it gives a reasonably clear explanation of the effects. For a completely accurate way of turning a 3-D Eulerian problem into a Newtonian problem of "point-masses connected by massless-rods", you only need a slightly more complicated "dumb-bell". :)

Z

rwstevens59

10-10-2015, 04:14 PM

Ralph,
1. The rough calculation by summing contributions from component masses (ie. ~Sum-all-i's(MoI.i + Mi x Ri^2) is the quickest, easiest way, albeit probably at +/-10% error (?). But 90% of an answer is better than no answer. Doing this is also good from a "design" viewpoint, in that doing the calcs many times over makes it clear how big the influence is from masses at the extremities of the car (because of R-squared).

Z,

I will take a 10% error over having nothing at all.

After all, in almost all cases I am interested in the relative size of some effect, not calculating an exact number for the purposes of chassis tuning.

The dumb-bell analogy I gave is not completely accurate, but it gives a reasonably clear explanation of the effects. For a completely accurate way of turning a 3-D Eulerian problem into a Newtonian problem of "point-masses connected by massless-rods", you only need a slightly more complicated "dumb-bell". :)

Hmm...a more sophisticated dumb-bell...thinking. The first thought that comes to mind is that the dumb-bell must be viewed in both side and plan with the angle of the principle axis in side view being accounted for in the plan view. But that is probably me just being simple minded again.

Off topic but...

Check this link to a Dover book on 'The Variational Principles of Mechanics' by Cornelius Lanczos. Click the Google preview button and go to his section on de'Alembert. I would be interested in your thoughts.

http://store.doverpublications.com/0486650677.html

Also one of my old texts 'Analysis and Design of Mechanisms' by Deane Lent circa 1970 covers the construction of your General Planer Acceleration Diagram quite well for draftsman working out linkage accelerations on a drawing board. Very basic book, think it lists at about five U.S. dollars these days, but I use it quite often as a reference in solving graphic vector problems.

Lastly, a question.

When the kids on the bus applied an input to the driver to cause him to initiate his 'step steer' input how did you estimate:

1. The amount of steer angle input?
2. The associated V to Tire centerline angle (i.e. 'slip angle')?
3. The forces generated at the front tires for your chosen steer angle? (Remember a post way back where you mentioned to someone generating representative curves for tire data)

In closing, the sound of the crickets on this thread is deafening. :)

Ralph

rwstevens59

10-10-2015, 04:40 PM

Z,

It is probably not of much benefit to spell out the above geometric method in detail here. The same results can be obtained with the more conventional Euler's Equations methods that are usually taught these days, and which are very briefly outlined in next post.

However, it is well worth closely studying the overall pattern of A-vectors here, and the fact that some features of this pattern are ALWAYS PRESENT. These general features covered next.

I, for one, would find it informative I'm sure. But, as my 'Z binder' is turning into a textbook size, I realize one can only ask for a finite amount of 'free' education. :(

Ralph

10-13-2015, 10:47 PM

Ralph,

Hmm...a more sophisticated dumb-bell...thinking.

One helpful way of understanding 3-D Dynamic problems is to reduce the complicated 3-D body down to three "rigidly connected" dumb-bells that intersect at 90 degrees to each other at their mid-points (which are at the body's CG), and with each point-mass at the ends of the dumb-bells = Mt/6 (because six point-masses). Each dumb-bell is aligned with a Principal-Axis, and their three lengths are adjusted to match the three respective MoIs of the "real body" about its Principal-Axes. (Note that "MoI.X" = MoI-of-BOTH-Y-AND-Z-dumb-bells about the X-axis = (Mt/3) x (Ry^2 + Rz^2), where Ry is half-length of Y-dumb-bell, etc.).

It is quite fascinating that the dynamic behaviour of ANY body, no matter how complicated its mass-distribution, can ALWAYS be modelled with such a simple "three dumb-bell" shape (assuming reasonable rigidity). For example, consider that the most complicated piece of "modern art" (= pile of steel junk welded together and dumped in town-square) ALWAYS has three mutually perpendicular axes within it, about which it can be spun with perfect dynamic balance (ie. no wobbling). Amazing!

(Edit: It is amazing that there is a SINGLE axis about which any odd-shaped body can be spun perfectly smoothly. More amazing that there are always three such axes (at least), and they are always exactly perpendicular to each other!)

In Sketch-2 I could also have found the AP by considering the bus's mass-distribution to be a simpler, symmetric, dumb-bell with equal masses Mt/2 at each end of a massless-rod 2 x K long. I would then have had to decompose the total force Ft into two separate forces, F1 and F2, both parallel to the original Ft, and acting at the two point-masses. Then work out each mass's acceleration (ie. proportional to its force), then draw a line through the tips of the two A-vectors to find the position of zero acceleration, namely the AP.

In the case of the Sports Bus, the Ft force lies "outside", and in front of, the dumb-bell, giving a rightward F1 on the forward dumb-bell half-mass, and leftward F2 on the rearward half-mass. So the AP lies "inside" the symmetrical dumb-bell, or "K-ring". In the case of the Slow Coach the Ft force lies between the two half-masses, so both forces and accelerations are rightward (but greater on forward half-mass), so the AP lies outside and behind the dumb-bell or K-ring.

More on this below...
~o0o~

Check this link to a Dover book on 'The Variational Principles of Mechanics' by Cornelius Lanczos. Click the Google preview button and go to his section on de'Alembert. I would be interested in your thoughts.

Well, I was disappointed that he starts that section with "... the fundamental Newtonian Law of Motion ... mA = F"!!!

I will keep stressing that the gist of N's LoMs is "... an impressed force causes a change in the quantity of motion...", with Newton defining "quantity of motion" as "the velocity and quantity of matter conjointly." (ie. what we call "Momentum" nowadays). So "F causes P-dot". Thinking in terms of momentum is much more helpful in solving all these problems, especially when things start rotating!

Otherwise that section of the book seems to confirm that d'Alembert's Principle works just fine, everywhere. Without exception!
~o0o~

Lastly, a question.

When the kids on the bus applied an input to the driver to cause him to initiate his 'step steer' input how did you estimate:

1. The amount of steer angle input?
2. The associated V to Tire centerline angle (i.e. 'slip angle')?
3. The forces generated at the front tires for your chosen steer angle?

The overriding concern when doing these sketches is not so much "solving the given problem", but more a question of how to fit-in all the necessary information on one sheet of paper, preferably without overcrowding in one part of the sketch and large blank spaces elsewhere.

So at the very first "rough draft" stages I was trying to find a reasonably realistic sized bus, travelling at a realistic speed, and cornering around a realistic corner radius, preferably with the bus AND corner centre visible on the same page (as in Sketch-3 and 4). And I also wanted the various vectors to be at a scale where the smaller ones are visible, but the larger ones are not shooting off the edge of the page. And where something important like the AP is not in an extremely crowded part of the sketch...

So, in Sketch-2 the front-steer/slip-angles and the amount of positive-Ackermann, etc., was chosen simply to put the "Flf + Frf = Ft" vector parallelogram in an un-crowded part of the page! The steer/slip-angles of ~15 degrees are on the high side, but they make clear that the front-wheel forces are hardly ever purely sideways to the bus, and this then puts M1 and M2 (= the AP) off the bus's centreline, and so on.

The perhaps unrealistically large tyre-forces I used, based on tyre-road Mu = ~1.5, were chosen so I could get a large enough Yaw-Rate W to give visibly large enough Centripetal accelerations in Sketch-4. As it turned out, the A-vector of mass M1 is just long enough to be visible, at about 3 mm long on the A4 sheet of paper (it appears as a short arrowhead "Am1" at the bottom of M1).

But most importantly here, the position of the AP in Sketch-2 is independent of the magnitude of the force Ft acting on the bus. It depends only on the relative positions of the LoA of Ft and the CG, and on the Yaw Radius of Gyration K. I give the standard derivation of this below.
~o0o~

...to spell out the above geometric method [used in Sketch-4] in detail here.

Very briefly, the point-mass M2 is put on the LoA of the total EXTERNAL force Ft, with the bus's CG perpendicular to the LoA at M2. But M2 also has an INTERNAL force acting on it, coming from M1 via the "massless rod", with this force due to the bus's already pre-existing Yaw-Velocity W. So M2's A-vector "Am2" is NOT parallel to Ftotal, but has a small centripetal component towards CG (ie. toward top of sketch). The size of this centripetal component is worked out using the known W (= 1 rad/sec) and distance between M2 and CG (ie. A.centripetal = W^2 x R). Or it can be worked out using known W, and distance between M1 and M2, and relative masses of M1 and M2.

KEY STEP! -> In bottom-right of Sketch-5, the angle the "At+An" vector makes with the R-vector (ie. the angle shown as "((" ), is the same as the angle ALL A-vectors make with radii drawn from the AP. In vector terms "At+An" = A2 - A1. So, back in Sketch-4, add MINUS-Am1 to Am2, which moves the tip of Am2 up about 3 mm... Now the angle between this new vector "Am2 - Am1" and the line joining M2 and M1 is the angle between ALL A-vectors and radii from the AP.

I did the above geometric construction of the Am1 and Am2 vectors at a larger scale, on the back of a bit of cornflakes-packet cardboard. The angle came out to about 76.6 degrees (my drawing-board protractor gives that level of accuracy). I used scissors to cut out a wedge of the cardboard with that included angle. Then simply a matter of placing one edge of this wedge along Am2 and drawing a pencil-line at ~77 degrees to Am2. This pencil line must pass through the AP. Then another line at 77 degrees to Am2 (Am2 necessarily has direction M1-M2)...

BINGO! The AP is at the intersection of these two lines. Then use the cardboard wedge to draw the directions of all the other A-vectors, all at 77 degrees to radii from the AP. Then measure out magnitudes of all A-vectors by direct proportion to, say, the scale of the Am2 vector (which is also easily done with a triangle of cardboard). DONE! :)

(Note: Just a little extra thinking required to make sure that all the "77 degrees" are going the right way from the radii, but this is fairly easy...)
~o0o~

(Bit more coming... 10k char limit!)

Z

10-13-2015, 10:49 PM

CENTRE OF PERCUSSION & CENTRE OF OSCILLATION - <- Google these terms and you find essentially the same problem as in Sketch-2.

For the School Bus the Centre of Percussion is the point M2, namely the point on the LoA of the force Ft acting on the (non-rotating) body, with the body's CG being perpendicular to the LoA at this point. We take it as given here that the distance from CG to Ft's LoA (or M2) is "A".

We seek the Centre of Oscillation (this term coming from pendulum theory, and = AP here), namely the point about which the whole Body (= bus) starts to rotate when first acted on by the force. We take it as given that the Body has total mass = Mt, Radius of Gyration = K, and lives in Flatland.

Step 1. We first convert the single linear force Ft into an "equivalent force system" of another force Ft', of same magnitude and direction as Ft but acting on a LoA through the CG, together with a couple T = Ft' x A acting anywhere on the Body.

Step 2. By Euler's Rigid Body Equations (student-special "Flatland-lite" version), force Ft' causes rate-of-change of Linear Momentum (Mt x V) of the Body, and couple T causes rate-of-change of Rotational Momentum (Mt x K^2 x W) of Body. Or, assuming Mt and K are unchanging, and taking "=>" to mean "causes",
Ft' => Mt x V-dot, and
T (= Ft' x A) => Mt x K^2 x W-dot.
In words:
2a. Force Ft' Linearly Accelerates every point of the whole bus, in the direction of Ft', at rate A = V-dot = Ft'/Mt (like top-left of Sketch-5).
2b. Couple T Rotationally Accelerates every point of the whole bus, in Clockwise direction about CG, at rate W-dot = (Ft' x A)/(Mt x K^2) (like top-second-from-left of Sketch-5).

Step 3. The acceleration of any point of the bus (wrt absolute space...) is simply the vector addition of the above two patterns. Since, in rotating motion V-dot = W-dot x R (where R is the radius vector, from-centre to-given-point, and this much dumbed down here...), the pattern of 2b above gives at any point a linear acceleration of V-dot = R x (Ft' x A)/(Mt x K^2).

We seek the point where the acceleration is zero (ie. the AP), which is the point where the two vectors in 2a and 2b are equal and opposite. So, we seek Ro such that,
Ft'/Mt = -Ro x (Ft' x A)/(Mt x K^2).

The force Ft' and the total mass Mt cancel out (so easy!), leaving K^2 = -Ro x A. And writing -Ro as the more usual "B" gives,
K^2 = A x B... ALWAYS!

Or..., the position "B" of the Acceleration Pole is dependant ONLY on the distance "A" from CG to the acting force (= ~distance from CG to front-axle of the bus), and on the Radius of Gyration "K" of the bus.

Or..., the position of the AP is NOT influenced by how hard the bus driver pulls on the steering-wheel, because the bus ALWAYS starts rotating about the same point near the rear-axle, regardless of the size of the front-wheel forces. (Err..., not considering changing live loads, or boisterous school kids!).

Z

12-04-2015, 11:42 PM

I received the following PM recently. I reply here so that other students with similar questions can see it.

Re: Turn Radius and Yaw Velocity.

... School Bus Example, I have a question, if you could help me:

In order to calculate the Turn Radius and the Yaw Velocity: should I use the kinematic approach, conecting the perpendicular line of the slip angles and measuring this distance(turn radius), or should I use a dynamic approach, using LatAcceleration = LatForce/mass, and Turn Radius = Velocity/Lateral Acceleration?

Because I think I can find a turn radius in both cases, but which one should I use when? I make this question for MMM reasons(modeling).

Thank you very much, it will help me a lot!

This problem is indeed a deep one. Because, sadly, the problem boils down to a complete lack of foundational education. The only way I can think to (start to) resolve this problem is to restress some of the things I wrote earlier. So...
~o0o~

Classical Mechanics (CM) is a sub-field of APPLIED MATHEMATICS!

It is NOT a part of Engineering, or of Physics, or of any other "Sciences". However, it can be used to considerable benefit by Engineers, Scientists, and so on.

CM deals either with idealised "point masses" (Newton), or idealised "continuum mass distributions" (Euler). A mountain of empirical evidence suggests that "reality" is NOTHING like "point masses" or "continuums". In fact, the overwhelming evidence is that at its core "reality" is downright weird. The word "spooky" is most often used to describe this level of reality (see any textbook on Quantum Mechanics).

Nevertheless, for any system bigger than a small bunch of "atoms" (whatever they are, particles or waves?), Classical Mechanics gives results that are extraordinarily accurate, far more accurate than most any Engineer or Scientist can ever measure. This accuracy, plus CM's core simplicity, is what makes the "CM-model" so useful to everyday usage by Engineers and Scientists.

But even more important than this practical utility is that CM is a Mathematically rigorous "Axiomatic-Deductive" approach to thinking about the given subject (namely, how does reality work?). (Note here that "A-D" is the modern name for this approach, but, for historic reasons, it is perhaps better described as "Geometric". This "Geometric" approach is about a lot more than just triangles! Google Plato or the Pythagoreans, for a start.)

Perhaps even more important than the "Axioms" and "Deductions" just mentioned, is that this whole "clear thinking" approach STARTS with very clear DEFINITIONS of all the relevant terms. Only after these Definitions are given, follow the very clear statements of what will be ASSUMED (ie. the "Axioms", or Euclid's "Postulates and Common-Notions"). And then follow, in a clear, incontrovertible, step-by-step way, the Deductions of lots of useful factoids (which, to repeat, must be derived ONLY from the Definitions and Axioms!).

The gist of what I am getting at here, is that Classical Mechanics, when it is taught properly, is always, and entirely, CRYSTAL CLEAR!

The whole edifice of CM is complete and consistent. Every question has its answer. There are NO "unknowables". Even the relatively modern development (~1900, Poincare) of Chaotic Dynamics gives exactly predictable "clockwork" results. (The apparent "unpredictability" of chaotic-systems, such as the weather, is merely a by-product of the imprecision of real measurements.)

The really disappointing thing is that this crystal clear "Geometric" approach to thinking has long been abandoned.

Nowadays students are just fed a random selection of the factoids (= "Deductions" above), usually in the form of poorly explained equations (= alphabet-soup), which they are expected to believe and understand even though they are given no supporting or foundational explanations of how these factoids work, or why they must be true.

And, even worse, there then follows a gradual process whereby the miscellaneous factoids are corrupted and perverted (ie. errors creep in), until the whole subject is reduced to gibberish (eg. you are told to "mind your migrating RCs", "use the parallel-axis theorem", etc.!).
~~~o0o~~~

Anyway, enough ranting, and some brief clarifications for above student.

"In order to calculate the Turn Radius and the Yaw Velocity: should I ..."

1. "... use the kinematic approach, conecting the perpendicular line of the slip angles and measuring this distance(turn radius),"

This only works if you ASSUME the car is a rigid body moving wrt a ground-frame (a reasonable assumption). If so, then the "turn-centre" (or IC, or Velocity-Pole) is found directly from its DEFINITION (= ~place with zero relative velocity). And then, of course, the "turn radius" depends entirely on which point of the body you are considering!

2. "... or should I use a dynamic approach, using LatAcceleration = LatForce/mass,"

This equation has NO direct connection with a car going around a corner! It could be a body (or point-mass?) accelerating laterally, WITH NO ROTATION AT ALL.

3. "... and Turn Radius = Velocity/Lateral Acceleration?"

Well, Velocity-squared! But this equation is based on the ASSUMPTION of a point moving with tangential-Velocity along a path with a given Radius-of-curvature. Why bother "calculating" the Radius ... when it is given to you?

Also, you will never find a Yaw Velocity from this equation because it applies to a point, and a point has no "front", no "rear", no means to establish "direction", so it cannot "rotate". (See Euclid, Book 1, Page 1, Definition 1, "A point is that which has no part.".)

Lastly,

"... I make this question for MMM reasons(modeling)."

I have never done any MMM (or YMD) modelling, and doubt I ever will.

But I understand Claude teaches this method. I note he has been very silent on this subject recently, on this and other related threads.

So, Claude, in return for the student's money, do you not feel an obligation to teach this subject clearly? And, preferably, in accordance with CM?

Z

Charles Kaneb

12-06-2015, 10:19 PM

Z,

Got an even easier way to do this with the schoolyard turntable. Drive the car onto the turntable, attach a spring scale to the unused holes in the license plate, measure the static friction of the assembly (moment required to start turning it) and the dynamic friction of the assembly (moment required to keep turning it at a constant rate) march around with as close as you can to perpendicular with a constant force on the scale, measure angular acceleration of the turntable. Then drive the car off the turntable and put the same moment on the empty turntable and measure its angular acceleration. For an even more accurate measurement stack weights at the center of the turntable to represent the weight on the bearings with a much lower Izz...

Given that most cars won't break when you hop an FIA curb I think a car's hub and wheel bearings would be adequate here.

-CPK

Flight909

12-06-2015, 11:42 PM

I have never done any MMM (or YMD) modelling, and doubt I ever will.

Maybe you should tried it, its classical mechanics.

All the questions you discuss have to do with the tire force and calculations of slip quantities.

ggscott

12-11-2015, 05:08 PM

Well I might be the wrong person to do this since I am measuring my cumulative exposure to vehicle dynamics in months rather than years but, nonetheless, I'll take the bait and answer the questions posed to the best of my understanding.

--------------------------------------------------------------

"1. Is LART (= Lateral Acceleration Response Time), all by itself, really the most important metric for a transiently sporty car?

2. What about YART (= Yaw Acceleration Response Time)!?

3. Does a forward-biased weight-distribution really maketh a sportier car (as claimed by others, earlier), and if so, then WHY!?

4. Is the generally faster LART of high-F% cars merely by-product of the "LA" being measured closer to the front-wheels, which is where, on standard "front-steered" cars, this whole transient cornering process is initiated?

5. Will a high-R% car, such as a Porker, EVER win a race, and why?

6. What general correlations can be expected between changes to the F:R% of a car, and the consequent changes to its LART and YART, if "all other things are kept equal"?

7. How big an influence to transient cornering is the car's "K", namely its "Yaw-Radius-of-Gyration" (or "Yaw-MoI")?

8. Considering the Sports Bus and Slow Coach of Sketch-2, what can be expected of their respective time-histories of F&R tyre-forces as they move from Sketch-2 to Sketch-3, and are these bus's names thus appropriate?"

---------------------------------------------------------------------------

1) No. Clearly a helpful measure but it is dependent on other factors.

2) At least as important. A fast YART will lead to faster development of slip angles in the rear and a quicker LART.

3) Not necessarily. An obvious factor in this that was brought up was location of measurement for LART is critical. An important correlation is that would be that if the COG is more forward the reading at the COG will read higher. Perhaps a more universal way to compare between vehicles (or different COG locations)would be to measure LART (directly or with transforms) at the centrepoint of the wheelbase. That being said for analysis of a single car knowing the LART at the COG is likely more helpful.

4) As in 3.

5) Obviously they can. One distinct advantage in cornering is the potential for higher instantaneous angular acceleration about the COG. As the COG moves backwards the moment arm from the front tires to the COG becomes longer. With tires that have a constant coefficient of friction this results in a 50/50 distribution being optimal. With real tires, that have a decreasing coefficient of friction with higher normal force, the point of maximum moment and therefore maximum anglular acceleration in the event of a step steer moves backwards. The attached graph shows this for a 1N car with a coefficient of friction of 1 at 0 load.

Rear weight percentage has a couple other significant benefits outside of cornering. Overall braking should be improved as it more evenly distributes normal forces to the tires, which with all else equal should improve overall traction. In cars with enough torque to cause wheel spin acceleration will also be significantly improved due to higher overall traction force on the drive tires.

6) As in 5. I would expect higher YART with a slight increase in rear weight percentage (optimized for whatever the normal-force to coefficient of friction response your tires have). More angular acceleration in step-steer should cause quicker slip-angle creation in the rear of the tires and higher LART.

7) Obviously huge. Lower K means higher angular accelerations, faster YART, and therefore faster LART. Faster response overall means faster generation of cornering forces.

8) I am not sure I am interpreting this question the correct way. Assuming the bus’s mass is the same in sport mode or slow coach mode, the actual forces at any given geometric point in a turn will be the same. Looking at the two modes over the same timeframe we would see the slow coach develop slip angle in the rear much slower, and from steady state to corner exit the reverse would be true. The sport bus should “snap” out of the turn much quick due to the lower energy required to change the angular momentum and the faster develop of slip angle change.

--------------------------------------------------------

Other comments:
Much of what was shown is obvious from rigid body mechanics, and likely very apparent to most people reading this, but I at least really appreciated the concise explanation of a geometric approach. It is also helpful in thinking about the effects of F/R weight distribution

From my perspective it seems like the geometric approach Z showed has more commonalities than differences from the MMM. Both allow analysis of balance at varying time and a way to visualize the handling of a car at any given moment in a turn. For any students who have not read the second SAE paper on MMM (SAE 800847) I would highly recommend it. I found it gave a bit more context than RCVD on the development of MMM that was helpful with applying the technique.

A slight rear weight bias might be something to seriously consider for FSAE. We are running a hybrid configuration that has some other complicating requirements but I justified a rear weight bias based on a couple factors:
1) Most critically was to maximize rear traction in braking. In conjunction with larger tires a rear weight bias will allow us to maximise regen. Obviously not a factor for many teams. Might be something to consider for teams running really small brakes. It is much easier to fit big differential brakes than big front brakes.
2) Maximize acceleration. The hybrid powertrain provides significantly more torque than the tires can handle. A better option may have been to downsize the engine for lower weight (already a 250cc). This will likely be very similar to any high power FSAE team.
3) Encourage neutral handling with larger rear tires. For the reasons above 8” tires are used on the rear with smaller tires up front. More rear weight means less understeer from the higher rear grip.
4) The final argument (which I will not take credit for using in my design) would be the one discussed above with regards to improving YART by a slight shift of the COG backwards. While the graph I provided demonstrates this effect the actual results will obviously vary significantly with varying tire properties. From a first look it would appear the advantage is likely to only be realized with about a 3-5% shift rearward (Unless your tires have a huge drop in friction with load).

---------------------------------------------
A couple notes on the plot. It was assume that K remained the same regardless as to CG position. With respect to Z's pictures this would be the same as keeping the dumbell rod the same length but shifting it rearward.

The formula for moment about COG was:

M = (1-F%) * Wheelbase * (F% * Weight)*Cf

Where Cf=1 for the baseline and Cf=(1 - .05*Front Weight) for the "real" tire model.
892

-Geoff

BillCobb

12-11-2015, 06:34 PM

What is the basis for your comments about respnse times, responsiveness, sporty-ness, etc.? From testing and data processing of the results, simulation studies, or SWAGs ? The answers, based on testing, experience, statistics and validated by simulation may surprise you (and, I'm pretty sure, quite a few others. That's why they have not chimed in).

MCoach

12-11-2015, 08:56 PM

"?

7. How big an influence to transient cornering is the car's "K", namely its "Yaw-Radius-of-Gyration" (or "Yaw-MoI")?

7) Obviously huge. Lower K means higher angular accelerations, faster YART, and therefore faster LART. Faster response overall means faster generation of cornering forces.

-Geoff

In terms of making it around the track, it might not make a lick of difference if you increased yaw inertia by 10x. In terms of stability and the magnitude of the impact, that's case dependent.

DougMilliken

12-12-2015, 10:46 AM

... [I]"1. Is LART (= Lateral Acceleration Response Time), all by itself, really the most important metric for a transiently sporty car?

2. What about YART (= Yaw Acceleration Response Time)!? ...
I suggest making a distinction between a "sporty car" and a "racy car", with parallels to linear range response characteristics ("linear range" of the tires) and limit characteristics (like max acceleration, yaw balance).

For example, work was done by VW in the 1970's which showed that minimizing the product of response time (tau) and vehicle slip angle (beta) was a good thing for passenger cars -- ordinary drivers liked this and performed better in various tasks. Here is one paper:
http://papers.sae.org/741105/ "Driving Simulator Studies: The Influence of Vehicle Parameters on Safety in Critical Situations"

There might be an earlier VW paper also, not sure, if so it will be referenced in 741105 (my copy isn't handy just now). VW used symbol "TB" for "Tau x Beta".

If you think about linear range response time and vehicle slip angle (measured at CG), one straightforward way to minimize these is with a front heavy car -- which may be sporty in street driving, but not very racy when pushed hard.

Note that the original Golf/Rabbit was introduced in 1974 and the architecture was reversed from the VW Beetle. I believe vehicle dynamics research at VW ramped up at that time, including publications.

BillCobb

12-12-2015, 05:01 PM

"Response Time" all by itself is a devious evaluation metric because it's easy to game it up by adding a lot of understeer to the vehicle. This lowers the damping in the state variables, which quickens up the response. A pair of metrics formed from the cornering compliances makes a good start for evaluations of handling. Simply put, the gain of the car is defined by the difference between the front (DF) and rear (DR) cornering compliances (assuming that the scalar values are positive) while the lateral stiffness is related to the sum of them and the damping is proportional to the sum (DF+DR) over the product (DF*DR). The goal of scoring a 'good' car is then to try to get DR as low as possible using 'nice' suspension properties and high tire cornering stiffnesses. The penalty function has packaging, durability, cost of materials and parts, and poor ride quality as constraints.

As far as which response time(s) to use, consider a FSAE type car (217kg, 1775 mm wb) on some relatively big tires running at 80 kph. With a steer input sufficient to get some heat into the tires, simulated responses for lateral acceleration (Ay), yawvelocity (R), yaw acceleration (Rdot) and sideslip (Beta) can easily be produced using a home grown simulation with nonlinear capabilities. I've chosen to use a step like steer input most likely to be achieved by a talented driver. After turning the crank, I normalized the data channels in question by their steady state values (except for Rdot, which used the peak value). The reference input time is when 50% of the steer angle is reached. The following figure shows the character of all the signals as is labeled with the time to reach 90% of the steady state (or peak for Rdot) level.

Note that Human Factors studies done over and over and over again using rate tables, sliding seats, tilt tables and othe human rated contraptions have shown that response time under about .25 seconds are probably not detectable. Its just 'there' without a sensed delay. That throws out a couple of candidates. Because sideslip depends on a good transducer that may not have been available or affordable back in the day, that leaves Ay response time as the chosen measure of bandwidth.

Even in this example, the nonlinear tire representation was made from tests of a real tire which had some residual force and moment characteristics, hence the effect of this vehicle crabbing down the hiway with zero steer angle applied. Under actual test conditions, this complicates the response time metric formation.

Knowledgeable readers may already realize that the Ay reponse times of this 'vehicle' are pretty high and comparable to a fully loaded pickup truck's. Anybody wanna venture a guess why ????

BillCobb

12-13-2015, 05:04 PM

Statements like this presume that an architecture starts by collecting up a bunch or stuff and building a car. Ain't so. A car starts as a human factors experiment to determine what the consmer desireables are: steering gain, effort levels, response time, max lat, ride/vs. handling fraction, wet skid, fuel economy, number of seats, trunk size, base, uplevel, sporty, trailer towing, cost to build, global market, etc. All these are relative to a cross section of competitive vehicle metrics. Once these are established, weights, powertrain and powertrain features (FWD, RWD, AWD) and wheelbase are flowed down into chassis and tire hardware requirements based in part on tire reserve load, spare tire yes or no, to deliver the chosen ride and handling metrics.

So, the car doesn't start life as RWD and transgender to FWD. Its very possible to produce a FWD with pretty much the same steady state gain and response time properties, also including roll. However comma, the stuff to do this can be troublesome. First of all, if the tire construction is the same for both types, the FWD will have more understeer just from the tire cornering stiffness difference resulting from the drive type switch. Understeer increases the natural frequency of the handling state variables, so you will see a quicker response time. The reduction of the rear weight will lower the rear cornering compliance, and you get an improvement in LART(shorter) from that. The higher relative front weight increases the Mz tire output, so the tierod loads are higher. This loads the steering gear housing mounts (more compliance understeer) and the convoluted steering shaft (end of column to the gear pinion) sees more load too. Then you add the FW drive torque (more tierod load) and the process snowballs. BTW: a test of FWD cars with driveshafts removed and the vehicle pushed up to speed and coasting through handling tests sure tells a lot.

But, it's a product development decision to accept all of this. Even if the two cars (FWD and RWD) achieve identical gains and response times, they will 'feel' different because the rearranged cornering compliance recipes necessary to meet these common goals will produce different yaw velocity and sideslip (hence lateral acceleration) overshoot/damping/peak to steady state ratio of the cars. And we ain't mentioned steering effort and effort gradient disparities, either.

So seating position is not technically the cause. However, a headlevel accelerometer on the seat headrest sure tells a lot about all the interactions, including seat tuning. A lousy seat in a good car equals a lousy car, just as a good seat in a lousy car is a lousy car.

12-19-2015, 10:55 PM

I have been away at Oz-comp recently, so apologies for slow response times :) to above posts. Some quick comments:
~o0o~

Geoff,

A slight rear weight bias might be something to seriously consider for FSAE. We are running a hybrid configuration that has some other complicating requirements but I justified a rear weight bias based on a couple factors:
1) ... rear weight bias will allow us to maximise regen...
2) Maximize acceleration. The hybrid powertrain provides significantly more torque than the tires can handle...
3) Encourage neutral handling with larger rear tires. For the reasons above 8” tires are used on the rear with smaller tires up front....
4) ... improving YART by a slight shift of the COG backwards.

Yes. Your reasoning is similar to mine in this "Why More R%" (http://www.fsae.com/forums/showthread.php?11906-2014-FSAE-Australasia&p=122428&viewfull=1#post122428) post in the "2014 Aus-comp" thread (two consecutive posts, Competitions section of Forum). Also worth noting that More-R% = Less-wheelspin-out-of-corners = Less-energy-wasted = Better-Efficiency..., so less batteries for H/E-cars, etc.

... the graph I provided ... From a first look it would appear the advantage is likely to only be realized with about a 3-5% shift rearward...

Your parabolic curves of "Moment-About-CG vs F%" are correct. But IMO they show that even with 30%F:70%R you still have an acceptably high moment-about-CG from the front-tyres (ie. the "Moment" is still near the top of the hill).

If you then also add additional moment from different Fx's from the now well-loaded rear-tyres (ie. from "torque-vectoring", or "brake-steer"), then you get frighteningly fast steering response. In fact, I suggest this "extra" is something that should be thoroughly tested before being introduced. Perhaps add some years down the track (it can be awesomely powerful!).
~~~o0o~~~

MCoach,

... it might not make a lick of difference if you increased yaw inertia by 10x...

I might misunderstand you, but IMO just doubling the yaw-inertia of an FSAE car would give it zero chance of winning a typical AutoX or Enduro. Likewise it would have reduced prospects in Skid-Pad (driver would have to slow on second CW lap in order to transition to ACW laps). But no problems in Acceleration, in fact beneficial.
~~~o0o~~~

Bill,

... consider a FSAE type car ...
... Ay, R, Rdot and Beta ... can easily be produced using a home grown simulation with nonlinear capabilities ... step steer...
...(nonlinear tire representation = vehicle crabbing down the hiway)...
...
Knowledgeable readers may already realize that the Ay reponse times of this 'vehicle' are pretty high and comparable to a fully loaded pickup truck's. Anybody wanna venture a guess why ????

Response Times.JPG

Hmmm... Well, yes, the response times are fast (Ay and Yaw...).

But more interesting is the little step ~40% up the Ay curve. This occurs at the same time that Sideslip is most negative (and a lot negative, given it must first overcome the "crabbing" from the tyres).

So, my initial guess is that this car has a lot of Sports Bus in it, and is no Slow Coach in terms of its position of Acceleration Pole for the first ~0.2 seconds after step-steer? But, as always, having knowledge of all the little details makes these puzzles a lot easier to solve...

A lousy seat in a good car equals a lousy car, just as a good seat in a lousy car is a lousy car.

This is a universal truism. Basically, there are a lot more ways of cocking things up, than of getting them right. Entropy!

Z

MCoach

12-25-2015, 06:39 PM

Yaw inertia has been shown to have a small correlation in steady state maneuvers and a strong correlation in transient maneuvers in developing slip angles and stability. However, most of this can be overcome by the driver. Inertia on FSAE vehicles is very small compared to most vehicles. There is usually little to no overhang on any side of the tires. I will contend there may be up to a few seconds difference with 10x more yaw inertia.

Rather than reckon, we can always do math and simulate.

12-25-2015, 09:59 PM

MCoach,

... on FSAE vehicles ... I will contend there may be up to a few seconds difference with 10x more yaw inertia.

I have spent many hours driving vehicles with Yaw-radii-of-gyration roughly equal to their wheelbase. This puts their Yaw-MoI at roughly 4x that of comparably sized "normal" vehicles (ie. which have K = ~WB/2). Such vehicles take "forever" to turn into a corner, and just as long to exit it.

An FSAE car with 10x the typical Yaw-MoI will be MUCH, MUCH SLOWER around a typical AutoX lap than "a few seconds difference"!
~o0o~

Rather than reckon, we can always do math and simulate.

My dictionary has, "Reckon - vb. 1. to calculate, or ascertain by calculating; compute. ... [derived from ... "to count"...]". This is my intended meaning whenever I use the word.

Please "do math and simulate" (or "reckon"!) the relative laptimes of these low and high Yaw-MoI FSAE cars. It is an easy problem to solve, and one for which I have already "done the maths" on several other threads.
~o0o~

Better yet, someone please do some REAL TESTS on your current or old FSAE car.

Again, easy to do. Just two heavish masses, either both mounted close to CG for Lo-Yaw-MoI, or else both mounted equally long distances from CG (say, one mass long way in front of car, and other mass long way behind) for Hi-Yaw-MoI. (This way you can keep Total-Mass and CG-Position constant.) Make life easy for yourself by aiming first for only a "4x" Hi-MoI figure (ie. do the "reckoning" to see how long a beam you will need, so you can mount the two masses far enough away from the CG to push Hi-Yaw-MoI up to 4x its Lo figure!)

A video of this Hi-Yaw-MoI car trying to negotiate an autocross track would be a scream. And very educational, too. :)

Z

nowhere fast

12-25-2015, 10:35 PM

Inertia on FSAE vehicles is very small compared to most vehicles. There is usually little to no overhang on any side of the tires. I will contend there may be up to a few seconds difference with 10x more yaw inertia.

Rather than reckon, we can always do math and simulate.

MCoach,
The comparatively low yaw inertia of FSAE cars is accompanied by short wheelbases, meaning that the yaw moments generated by FSAE cars may also be comparatively low. It is perhaps better to think about the ratio between the radius of gyration in yaw (k) and the wheelbase, or the k^2/(ab) metric where a and b are the distances to the centre of mass from the front and rear axles respectively. A reduction in overhang will tend to improve these ratios.

I posted a simplified simulation of the influence of yaw inertia through a slalom in this thread:
http://www.fsae.com/forums/showthread.php?1389-Any-way-to-objectively-choose-engine/page17

One point to note is that it is essential to allow the driving line to change in order to exploit the benefit of reduced yaw inertia. Here is a comparison of the lines taken through a slalom by a car with high yaw inertia (green) and low yaw inertia (blue), the x’s represent where the car switches from constant yaw acceleration to constant lateral acceleration in this simplified simulation.
http://farm8.staticflickr.com/7009/6471717763_422cc6a83d_b.jpg (http://www.flickr.com/photos/67284757@N06/6471717763/)

09-24-2017, 08:11 PM

0/5. STEP-STEER-SIMULATION - INTRODUCTORY WAFFLE.
=================================================
Well, there is more action at the local graveyard than there has been on this Forum for the last year or so. Certainly, the last few months have been dead quiet... :(

Nevertheless, I have received a PM from one of the few FS-ers still active here, and he was interested in the "time-stepping" approach to modelling VD step-steer tests that I mentioned on this, and other related, threads back at the end of 2015, start of 2016.

So, following this rather long-winded post are five more posts covering this issue. Yes, I know that many of you will think it too much work to read through all those words. But on the up-side, there are many colourful images!
~~~o0o~~~

As introduction to the posts that follow, I should give a brief explanation of how to "read" the images, and how they were produced. The most important point to note is that all the images below were generated by an exceedingly simple computer program. (I wrote said program over a period of a few weeks back in early 2016, but, as noted, no interest back then, so it was put on ice.)

I give more details of this program later, but briefly for now, the program knows naught about Vehicle Dynamics. So it knows nothing of vehicle-path-curvatures, beta-angles, understeer-gains, +++. Nor does it know about Laplace Transforms and their attendant Dirac Delta functions and Heaviside Step functions, or how such mathematical methods are used to analyse the oscillating spring-mass-damper system that is the whole car during cornering manoeuvres.

Nevertheless, the results are quite rewarding, being "fully transient analysis". This ability to model "transients" is, in fact, an inevitable consequence of the program being founded on Newton's Second. Again, more details later, but the core "Mechanics Engine" (the gamers usually call it a "Physics Engine") does nothing more than some very simple NII calculations during succesive short time periods. See near the top-right of each image for the length of this adjustable "Step dT". Note that this right-side area is used for all user input to the program, all done with mouse-clicks on the "+", "-", "Clear", and "Run" buttons.

However, since this program is only a simple first step to VD-modelling, its universe is dumbed-down to "Flatland". Furthermore, the "car" is only a "bicycle", with only one front-wheel, with the wheelprint shown as a blue-dot, and one rear-wheel, shown as a red-dot. The car is treated as a single rigid body, with given WheelBase and Mass distribution properties, shown in right-side "Body Specs". Extending this for more realism, such as to a 3-D world and 4-wheels, is straightforward and is covered later.

The Flatland simplification means that at any instant the car has a "current dynamic state" that consists of the six parameters of its CG's North position, East position, and Heading angle, and similarly a Northerly velocity, Easterly velocity, and Yaw-velocity. The starting values of these six parameters are shown under "Start Specs". Whenever the user clicks the "Run" button, the car is "shot out of a cannon" at its starting velocity, from its starting position. The car has no "drive" to any of its wheels, so it relies entirely on its initial momentum to complete its manoeuvre.

The major, left-side, area of each screen shows a plan-view of the "Track Area" where the car does its thing. The car is first drawn at its start position, and is then repeatedly "ReDrawn" at an adjustable interval of time (every 0.3 or 0.5 seconds in below images). For simplicity, the car is drawn as a forward pointing triangle 2 x WB long, and 1 x WB wide at its base. The car's centreline is also drawn to help see its direction more clearly, and the front-wheel-angle is indicated by a black line (1 x WB long) pointing forwards from the front-wheel's blue-dot.

The car's Moment-of-Inertia in Yaw is set via its "Yaw Radius of Gyration" (at "Body Specs - Radius Gyr"). This is depicted as a circular ring with radius equal to RadGyr, and with this ring centred on the CG position, which is shown as a black-dot, and also adjustable wrt the wheels via "Rear Fraction". So the Flatland mass-distribution properties of the car are the same as if the Total Mass of the car is distributed along that ring. Of course, in Flatland, and as covered in earlier posts, the dynamic behaviour of the car would also be the same if the Total Mass were divided between the two ends of a dumbell. The circular ring is used here because of its better visibility.

The car has quite realistic "tyre-models". Yep, better than Pacejka! Each tyre is specified with only two parameters, namely its "Peak Mu" (= peak "Coefficient of Friction"), and the "Peak Angle" at which the tyre becomes "saturated", and after which it can give no more (see "Tyre Specs"). In more detail, the wheel's Axial-Force/Slip-Angle curve rises like a parabolic arch from zero SA, and then levels-off as it reaches its "Pk-Angle". This is broadly similar to real tyre-curves, which Bill Cobb has described as "softening springs".

More importantly, at Slip-Angles greater than Pk-Angle the wheel's Axial-Force remains constant. Until, of course (!), SA gets close to +/- 180 degrees, when the Axial-Force drops back to zero, which Pacejka does NOT do. I doubt that including Pacejka's other ~57 secret-herbs-n-spices (= parameters) would make much difference to the modelling. There are many other more important factors to add in first! The program code also includes a bi-linear tyre-model that has a straight line from SA=0 up to its Pk-Angle, with a horizontal line after peak. Both parabolic and bi-linear tyre-models amount to ~1 line of code each. Using a bi-linear-Pk-Ang that is about 0.75 x parabolic-Pk-Ang gives almost identical car behaviour in both cases. For low-G matching use bi-linear-Pk-Ang = 0.5 x parabolic-Pk-Ang.

To help the user better understand the car's dynamic behaviour, the F&R-wheels' Axial-Forces (aka tyre-"Fy" forces) are drawn during each Run. Each individual "ReDraw" of the car shows a blue "force-vector" pushing on the front-wheel, and a red "force-vector" pushing on the rear-wheel. These forces are the ones "acting from-ground, to-wheel", so think of the vector "arrow-heads" being at the wheels. During the times that the car is not being ReDrawn, only the tips of the tails of these vectors are drawn. This results in a full trace being left of the F&R Axial-Forces during the manoeuvre. Similar traces are left of the F&R wheelprint-paths. Note that whenever a force is at its maximum (ie. tyre is "saturated") the force trace leaves a thicker and darker line.

The adjustable "Step-Steer" parameters appear near bottom-right-side. The Step-Steer begins a certain "Start Time" after the car is shot out of its cannon (typically 0.3 or 0.5 seconds in below images, to coincide with the first car ReDraw). The front-wheel steer-angle then ramps up linearly from zero degrees to "Front Angle" over the "Ramp Time" period.

Finally, because it is oh-so-easy to add in extra little things to these simulations, this very simple program was also given the ability to steer its rear-wheel, in addition to its front-wheel. The "Step-Steer - Rear Fraction" parameter steers the rear-wheel in a fixed proportion to the front-wheel (+ve RF gives same steering direction, -ve RF steers in opposite direction). No results for such tinkering are given here, but testing such ideas with simple computer simulations surely can save a lot of wasted time with the real thing.
~~~o0o~~~

Enough waffle, and onto the pretty pics!

Z

(Edit: Pics may not appear!!!???
Seems that it is now NOT possible to put images in-line with these posts (at a readable size!)?
Idiocracy is here!)

09-24-2017, 08:12 PM

(Note: Introductory post on previous page!)

1/5. STEP-STEER-SIMULATION - EFFECTS OF "RADIUS OF GYRATION".
================================================== ==
It has been said many times that a small "Yaw Radius of Gyration" (or "Yaw MoI", or simply "Iz") is good for FS racing conditions, whereas a large "Iz" is bad. But how much extra "goodness" or "badness" do you get from given changes to Iz?

The two images below give some indication of the performance effects of changes to Iz.
~~~o0o~~~

As I have covered elsewhere, the practical maximum limit for any reasonable vehicle's RadGyr is about equal to its WheelBase. This gives an equivalent ring-mass with diameter equal to twice the WheelBase, or similarly, a "dumbbell" of length twice the WheelBase. At the other extreme, the almost impossible to reach lower limit of Iz is a ring-mass diameter, or dumbbell-length, of half the WheelBase (ie. RadGyr = WB/4).

So the image below shows two cars, IDENTICAL IN ALL RESPECTS, except that the car at bottom of image has the long dumbbell mass-distribution of length = 2 x WB = 3.2 metres, while the top car is a very short dumbbell, or "cannonball", of length = WB/2 = 0.8 metres. Note that the parameters at right-side of image describe the bottom car, which was the last "Run". Both cars have the same Mass (typical of a lightweight FS-car with jockey-driver), same WheelBase, R%, Tyres, and both are shot out of their cannon at 20 m/s, or 72 kph. Both Step-Steers start at the second drawing of each car (ie. at 0.3 s x 20 m/s = 6 m East).

https://photos.app.goo.gl/zG4MBSmZFIcoZshd2
http://www.fsae.com/forums/attachment.php?attachmentid=1245&stc=1
(SSRadGyr.png)

Interesting points are:
1. The Lo-Iz car, or "agile cannonball", turns very comfortably inside the Hi-Iz, "sluggish dumbbell", car. A zoomed-out shot shows the Lo-Iz car completing a U-turn entirely inside the path of the Hi-Iz car. Clearly, there is a huge difference in "agility" here, albeit from two cars at the extreme ends of the practical mass-distribution spectrum.

2. The reason for the sluggishness of the Hi-Iz car is seen right at the start of its step-steer. The rear-wheel starts pushing the car the wrong way! The driver wants to turn left, but the rear-wheel spends the first 4 metres, or 0.2 seconds, pushing the rear of the car to the right. This is as covered in earlier posts about the School-Bus. By comparison, over that same time-period the Lo-Iz car's rear-wheel is pushing the car hard to the left. In fact, at the end of the 0.2 seconds the Lo-Iz car's rear-force is nearly at its peak.

3. Note that both car's rear-wheel force-curves start off asymptotically to their wheel-paths. This is because the rear-wheel can only generate significant Axial-Force after the WHOLE CAR has "steered" relative to its direction of travel. This inevitably takes a lot longer for the whole car to do than the much quicker changes in steer-angle that are possible with the front-wheel alone.

4. The third and fourth snapshots of each car (ie. between 10 and 20 m East) show how much more quickly the Lo-Iz car develops its body-slip-angle compared with the Hi-Iz car (hint: compare the centrelines of the cars with their wheel-paths). Note also how the Lo-Iz car has both its wheels well and truly at their peak-force at this stage, while the Hi-Iz car has its wheels reaching their peak-force much later (peak-forces are shown as thicker, darker traces).

5. Compare the lengths of the force-vectors of the two cars at their peaks to see that they are of the same magnitude. That is, same mass cars, same tyre-Mus, so SAME FORCES acting on the cars. But, again, the big difference is that the sluggish Hi-Iz car takes almost forever to get its tyres up to their peaks.
~~~o0o~~~

This next image shows a zoomed-out shot comparing step-steers of three otherwise identical cars but with different RadGyrs. The leftmost car has RadGyr = 0.4 m (ie. very agile), middle car has RadGyr = 0.8 m (ie. typical of many FS-cars), and rightmost car has RadGyr = 1.2 m (ie. very sluggish, but not quite as extreme as the one above).

Note that all three cars' step-steers start immediately at the first drawing of the cars, to better fit it all in. Also, the cars are now travelling at 30 m/s (= 108 kph) at the beginning of the manoeuvres, and I have fitted "stronger" tyres to all cars' rear-wheels (Rear-Pk-Mu bumped up to 1.7).

https://photos.app.goo.gl/TuvdvqB00xp06qwk2
http://www.fsae.com/forums/attachment.php?attachmentid=1246&stc=1
(SSFishTail.png)

Main points of interest are:
1. Each car shows the characteristic "fish-tailing" of an underdamped spring-mass-damper system when it is excited by an "impulse" (ie. the "step" force at front-wheel). The wobbles in the red rear-force traces indicate this fish-tailing. Note that these oscillations are there despite this program knowing absolutely zilch about how such oscillating spring-mass-damper systems should work, which is a good sign! (Oh, and the reason I increased speed is to reduce "yaw-damping", to give clearer oscillations. And I stiffened the rear-tyres to stop the cars spinning-out at this higher speed. Plenty of that coming later...)

2. The fish-tailing kicks in progressively later as the car's Iz increases. This is again the result of the rear-forces building much more slowly on the Hi-Iz cars. The Lo-Iz car (left) has its rear-force hitting peak about 0.2 seconds after the step-steer is initiated, while the Hi-Iz car (right) takes about 0.6 seconds for its rear-force to hit peak. Again, this is a direct consequence of the slower Yaw-acceleration of the long dumbbell cars. Or, put another way, a higher mass (= Iz) with a given spring-rate (= tyre Mu), gives a lower frequency of oscillation, or longer time period.

3. The tighter overall path curvatures of the middle and right cars (especially the middle one) compared with the leftmost Lo-Iz car can be explained by looking closely at their wheel-path traces. The Lo-Iz car has its red rear-wheel-path outside its blue front-wheel-path for only the short period from about 0.2 seconds to 1 second (ie. between first and third car images). That is, it is only "Over-Steering" for less than a second. On the other hand, the Mid- and Hi-Iz cars are "OSing" for about twice as long, which tightens their paths. Furthermore, this longer period of initial OS scrubs off more of the initial velocity of the Mid and Hi-Iz cars, which further tightens their line.

Z

09-24-2017, 08:13 PM

2/5. STEP-STEER-SIMULATION - SMALL CHANGES TO F/R "PEAK MU".
================================================== ========
The first image below shows the same car performing the same step-steer manoeuvre, but at three different initial velocities. The top run is at 20 m/s (72 kph), the middle run at 30 m/s (108 kph), and the bottom run at 40 m/s (144 kph).

The most important car parameters worth noting here are the Tyre-Specs, which are identical front and rear. The car has a moderate amount of rear weight bias, with R% = 60%, but since the tyre MUs are the same, the wheel Axial-Forces are always in the same proportion to inertial forces at the wheelprints, such as the centrifugal forces during cornering.

So this car can be said to be "perfectly balanced". It has zero "static margin", so it has "neutral stability", so "neutral handling", and so on. But as harped on before, this program knows nothing of such VD esoterica. Nevertheless, the car's performance is as the VD experts would predict.

https://photos.app.goo.gl/9jyGTN0aoSOwl0Xk2
http://www.fsae.com/forums/attachment.php?attachmentid=1247&stc=1
(SSRMu160.png)

At the slowest speed of 20 m/s (top) the car develops quite a high body-slip-angle after the step-steer. But this high sideslip scrubs off a lot of speed and the car slows enough to finish the manoeuvre rolling forwards, albeit at only about 7 m/s .

At the medium speed of 30 m/s the car gets well and truly sideways, loses almost all its speed, and ends up rolling very slowly backwards.

The fastest run of 40 m/s has the car doing a full 180 degree spin and then finishing the manoeuvre rolling backwards at ~8 m/s or ~30 kph.

The key lesson here is that regardless of how "well balanced" a car is, the faster you go, the easier it is to lose it. And probably something about "critical speeds", and such, if you want to call yourself a "Vehicle Dynamicist".
~~~o0o~~~

Next image shows almost exactly the same car again, but this time with its Rear-Tyre Peak-Mu REDUCED a smidge to 1.58 (from 1.6 previously). Same three runs, at 20 m/s top, 30 m/s middle, and 40 m/s bottom. The image is zoomed-out a bit to fit the complete runs in.

https://photos.app.goo.gl/ANp2xpLy6j8UyGQJ3
http://www.fsae.com/forums/attachment.php?attachmentid=1248&stc=1
(SSRMu158.png)

The obvious thing to notice is that in each run this car develops significantly more body-slip-angle than in the previous image. It only just manages to complete its slowest run going forward, but in the two faster runs it spins out completely.

The obvious lesson is that fitting "weaker" tyres at the rear will NOT help you finish the race going forwards.
~~~o0o~~~

So, let's go the other way and fit "stronger" tyres at the rear. Here everything is identical to before, but Rear-Peak-Mu is now bumped-up to 1.62.

https://photos.app.goo.gl/IJV5j1C4NbtUcbHm1
http://www.fsae.com/forums/attachment.php?attachmentid=1249&stc=1
(SSRMu162.png)

Happy days!!! What a difference that tiny 0.02 makes!

Now the car NEVER spins-out so far that it is going backwards. Well, to be fair, it does get a lot sideways at the higher speed, especially the bottom 40 m/s run. And ... any run above ~45 m/s, or 100 MPH, does have it going backward.

But the key point here is that quite small changes to the "size of your rudder" can make a big difference to ease of driving near the limit of grip.

After all, which of these three cars would you prefer to throw into a high speed corner?
~~~o0o~~~

On the basis that, for stability, the "stronger" tyres should be at the back of the car, it follows that the second car above, with F-Mu = 1.6 and R-Mu = 1.58, should be stable when driving in REVERSE!

Well, the top two runs in the next image show such runs of this car, at initial velocity of 20 m/s, 72 kph, at top-left, and 40 m/s, 144 kph, at top-right. (Note that the Step-Steer angle is now -10 deg to give the same "left-turn" appearance on screen.)

As for stability, it is quite obvious that in the slower run the car very quickly spins 180 degrees. And at the higher speed it manages to spin 180 degrees Anti-ClockWise as before, and then manages another 180 degree CW spin, before finally rolling backwards as expected.

The main problem with driving in reverse, or equivalently, steering-with-rear-wheels, is that the rear-wheel starts off the manoeuvre by pushing the rear of the car the "wrong" way, namely out of the turn. (Edit: Look at blue force-traces starting at the second car snapshots of all runs.) In effect, the rear-wheel gives the car a big OverSteer kick right at the beginning of the manoeuvre, from which it can be difficult to recover.

https://photos.app.goo.gl/LPWBO5sP3EAPaDLR2
http://www.fsae.com/forums/attachment.php?attachmentid=1250&stc=1
(SSReverse.png)

At the bottom of the image are two similar, but different, runs to the first two, at 20 m/s bottom-left, and 40 m/s bottom-right. But this time the car has much greater stability built into its tyres. Now F-Mu = 1.7, and R-Mu = 1.5, which, of course, is stable only in reverse!

Now, despite the OS-kick from the rear-wheel, the car stabilizes itself in both runs and then completes the manoeuvre as requested by the driver. The stabilization is seen clearly in the bottom-right run, where the car spends much time sliding sideways with body-slip-angle around 90 degrees, but then straightens itself up to finish in the correct, err..., reverse direction. In fact, these tyres give so much stability (in reverse!) that the car corrects itself even at 90+ m/s (after which the zoom-out function is not big enough to show it all...).

Biggest lesson to learn from this last image is that it takes very special tyres, or a very good driver, to drive fast in reverse.

Z

09-24-2017, 08:15 PM

3/5. STEP-STEER-SIMULATION - BIG CHANGES TO F/R "CORNERING STIFFNESS".
================================================== ================
Production cars are very rarely driven anywhere near the "Peaks" of their tyre-force curves. It follows that much OEM study of VD is focussed on the lower parts of these curves, where the curves are just starting to rise up from zero Axial-Force (Fy) and Slip-Angle. The slopes of the curves in this area are variously called the tyre's Cornering-Coefficients/Compliances/Stiffnesses, or some such (clarifications to terminology are welcome!).

The "stronger" tyre is then considered to be the one with the steeper curve, and it is that tyre/wheel that should be fitted to the rear of the car for stability. And that is what has been fitted to the car in the image below. Same car, same step-steer, but five runs at five different initial speeds.

The important parameters to look at here are the Tyre Specs (which are same for all runs).
1. The Front-tyre is quite "floppy" with a large Peak-Angle = 15 degrees.
This means its Axial-Force rises rather leisurely with increasing Slip-Angle, and it only reaches its peak-force at SA = 15 degrees. But when it eventually gets to this high SA its Peak-Mu = 1.6.

2. The Rear-tyre is much stiffer, with a very small Peak-Angle = 5 degrees.
So its Axial-Force rises like a cliff face, but when it gets to top-of-cliff its Peak-Mu = 1.5, which is just under that of the front-tyre.

So, considering each run in turn:

Top-Left - Here the car has initial velocity of 5 m/s or 18 kph. This is slow enough that "cornering forces" are very low, and there is very little resultant slip-angle from either tyre. So the car corners close to its "Ackermann radius". (Quick check: Front-wheel steer-angle = 10 degrees = a slope of ~1/6. So drawing a triangle with two long sides aligned with F&R axle-lines, and WB = 1.6 m on third, short, side, gives radius of corner = ~ 6 x 1.6 = ~9.6 metres, which matches near perfectly the ~19 metre diameter path of the car in the image!)
~o0o~

Top-Middle - Now the speed has been bumped up to 10 m/s, or 36 kph, so the cornering forces are increasing, and so too the slip-angles at F&R wheels. Because the front-wheel is fitted with the "floppy" tyre, it naturally "slips", or "crabs", more than the rear-wheel, and the car's path opens out. The car is beginning to show an "UNDER-steering" behaviour (at least in the more common, less precise VD-ish, sense of this word).
~o0o~

Top-Right - Speed has doubled again, to 20 m/s, or 72 kph. And, yep, this is definitely an understeering car. No doubt about it! The faster it goes, the more its nose wants to "push" straight ahead. (In fact, this run only tightens its line towards the end of the run because the high "slip-angle-drag" of the front-tyre has slowed the car significantly.)
~o0o~

https://photos.app.goo.gl/xqCUVvZhPCdmf73n2
http://www.fsae.com/forums/attachment.php?attachmentid=1251&stc=1
(SSDiffCfCr.png)

Bottom-Left - Only a slight increase in speed now, to 25 m/s, or 90 kph. And, ... hmmm. Err, ... so what happened? Why the spin?

Well, the problem is quite clear from the red, rear-wheel, force-trace. Barely 0.3 seconds after the start of the step-steer (ie. just after third car snapshot) the rear-tyre reaches its peak-force, after which it is "saturated". This means the rear-wheel can no longer keep its end of the car tightly controlled, and the rear of the car slides progressively further out from the desired path. This increases the slip-angle of the whole car, which necessarily also increases the slip-angle of the front-wheel. And so the front-wheel is also fully sliding less than 0.3 seconds later.

Now the car's stability is controlled entirely by the behaviour of its tyres at or above their "Peak-Slip-Angles". The front-tyre is clearly the stronger one here, and the rear weaker, so the car switches from UNDER-steering to OVER-steering (in the common sense of those words).
~o0o~

Bottom-Right - Speed is increased slightly again, to 30 m/s, or 108 kph. No surprises here, and at any higher speeds the results will be the same. She's going to spin!
~o0o~

Many lessons possible here.

Perhaps the most important one is that just because you have done extensive testing at low and medium "Gees", and those tests show your car to be incredibly well-behaved and easy to drive, that does NOT mean that your car will be the same "at the limit". And to do well in FS/FSAE you should be getting close to the limit.

Z

09-24-2017, 08:17 PM

4/5. STEP-STEER-SIMULATION - FIDELITY OF THE MODEL.
=================================================
As noted right at the start of this series of posts, these simulations are based on a VERY SIMPLE model of a car. But the real car, in its full "reality", is much more complicated. So there should be times when this model suggests one type of behaviour, while the real car will exhibit a quite different type of behaviour. In fact, based on the extreme simplicity of this model, we might expect it to get many, and maybe even most, of its predictions wrong.

Well, NO, not at all.

As long as the parameters being modelled are reasonably accurate reflections of the real ones, and they are the DOMINANT parameters in the behaviour being studied, and no other significant parameters are being overlooked, then the modelled results WILL match the real behaviours with reasonable fidelity. As always, the results depend on the details.

The image below shows an example of a possible failing of this current model, or not. Two runs are shown, both with exactly the same car parameters (ie. same mass-distribution, same tyre-specs, etc.), but the top run has a step-steer angle of 10 degrees (same as most others in earlier posts), while the bottom run has its step-steer at the larger angle of 20 degrees.

The car has the slightly weaker Peak-Mu = 1.58 tyres fitted to its rear-wheels (fronts have Mu = 1.6), which makes it an Over-steerer. Both runs are at initital velocity = 30 m/s, or 108 kph. This is at a high enough speed for the car to spin out, as seen in the top run. In fact, this top run is the same as the one shown in the second image of post 2/5 (albeit with shorter Run Time).

So, why is it that in the bottom run, with the only difference being the higher front-wheel step-steer-angle, the car manages to complete the manoeuvre without spinning out?

https://photos.app.goo.gl/wazApLhjtqqQ0qYi1
http://www.fsae.com/forums/attachment.php?attachmentid=1252&stc=1
(SSDiffStrAngs.png)

The reason a car "spins out" is that the Yaw-couple exerted on the car by the combined forces of all its wheels during cornering is such as to cause the spin. Typically, and simplistically in this "bicycle model", if the front-wheel pushes the front of the car harder towards the corner-centre than the rear-wheel, then a spin ensues. In the calculation of this behaviour, both the magnitudes of the F&R "cornering" forces, and the distances of these forces from the car's CG, must be taken into account.

In both runs above the magnitudes of the F&R-wheel forces are at their peaks (for most of the runs), so are the same. Same car mass and distribution, so same vertical-load-per-wheel, and same Peak-Mus, so same horizontal forces. BUT (!) the bottom run has its front-wheel Axial-Force tilted at a greater angle to the car-centreline than in the top run. And it is this change in geometry that changes the resulting Yaw-couple.

An easy way to see this is to decompose the front-wheel Axial-Force into its CAR-COORDINATE X (front<->rear) and Y (sideways) components. Specifically, Car-coord-Fy = Wheel-coord-Faxial x COS(Steer-Angle). So, the bottom run with its 20 degree steer-angle has a SMALLER Y-component of force (in car-coords!) than the top run with its only 10 degree steer-angle. And this is why I always refer to a wheel's "AXIAL" force (ie. "in the direction of its axle"), and not its "Fy" force.

So the bottom run has a LESSER oversteering couple acting on the car, so NO SPIN. Increasing the step-steer-angle to 30 degrees lessens the front-wheel's "Fy"-force (in car-coords!) even more, and the car's path opens out even further.
~o0o~

The above effect is a very real thing. It is noticeable on real cars (and tractors! :)), most obviously when driving in slippery conditions such as wet clay. Cranking the steering-wheel around to give large front-wheel-steer-angles often just gives a large "braking" effect from the large car-coord-Fx component of the Axial-force, and very little "cornering" effect. But reducing the front-wheel-steer-angle to just 5 or 10 degrees can suddenly give much more "cornering" effect.

Note, however, that the above paragraph refers to a constant velocity condition where the braking effect from the steered front-wheel is being countered by an equal thrust from the driven rear-wheels. What happens if the braking effect is NOT countered? Well, it depends on the details.

A later version of this program moves the model into "Flatland-Plus". It includes the effects of a CG that is ABOVE ground level. This naturally results in "longitudinal load transfers" whenever there are car-coordinate longitudinal forces at either wheelprint. Modelling the two runs in the image above with the car CGs at ground level gives exactly the same results as shown. But including a typical above-ground CG-Height results in quite different behaviours. Both cars spin, and they spin earlier and harder.

In fact, the high front-steer-angle car of the bottom run spins even more readily than the top run, because the "braking" force from the front-wheel (together with inertia from the above-ground CG), throws extra load onto the front-wheel, with this load coming off the rear-wheel. This is a "first order" effect and allows significantly MORE destabilising peak-Axial-Force from the front-wheel, and significantly LESS stabilising force from the rear-wheel. Spinning follows post haste.

Anyway, if you can build your FS-car with its CG within a few millimetres of the ground, then probably NO need to model the effects of CG above ground. On the other hand, if your CG is a foot above ground, and given the relatively short wheelbases of these cars, then I think it advisable to include the effects of CG-Height.
~~~o0o~~~

FINALLY, PROOF THAT EVEN OLD-FARTS LIKE TO GOOF-OFF OCCASIONALLY!
================================================== =============
The "Mechanics Engine" of this program has to keep track of the car's Yaw-velocity, and this variable has to be initialised at the beginning of each run. So why not shoot the car out of its cannon with some initial "spin"?

The image below shows two such runs, both of identical cars, both given a ClockWise initial spin of just under 1 revolution per second, but with the top run having Easterly velocity 20 m/s, 72 kph, while the bottom run is at 30 m/s, 108 kph. The step-steer starts at the second car snapshot of each run.

https://photos.app.goo.gl/ojyUtWVA9wZ66byx2
http://www.fsae.com/forums/attachment.php?attachmentid=1244&stc=1
(SSSpinning.png)

What lessons to learn?

Well, a specific one might be that once you are in such a spin there is precious little hope of exiting it with a single step-steer. Believe me, I tried! Maybe possible with multiple corrections during the spinning? But that would mean being able to see through all the vomit on the inside of your visor/windows.

The more general lesson is that despite the extreme simplicity of this model, it nevertheless gives a surprisingly broad range of quite realistic predictions of car behaviour. And exploring some of these more bizarre behaviours can be quite enjoyable! :)

Z

09-24-2017, 08:18 PM

5a/5. STEP-STEER-SIMULATION - WHERE TO FROM HERE?
===============================================
The last post showed that a very simple computer program, intended inititially to model only the simple "Step-Steer" manoeuvre, can nevertheless model quite complicated car behaviours.

So, can this program be extended to form a more general foundation for an FS-team's decision making processes? And if so, how?

Firstly, it is important to restress here that the biggest factor in the success of any FS-team is how effectively they utilise their scarce resources (see Geoff's/Big Bird's "Reasoning..." thread!). The best "compass" a team can use, to guide itself in the right direction of resource use, is the "points simulator". The single largest component of such a points simulator is the "dynamic simulator" that estimates a given car-concept's performance on track (the other significant part being the "cost estimator", which estimates both real and Cost-Event dollars, as well as "time" cost).

Most mid-level teams' dynamic simulators that I have seen model the car as a "POINT-MASS". This point-mass either accelerates or brakes in a straight line, or corners at a fixed radius and speed. These straight and curved paths are then stitched together to represent the car's performance around a given FS/autocross-style lap. Claude's "Optimum-Lap" is one such simulator, and given that it is available for FREE, there is absolutely no excuse for teams not to be working at this level of dynamic simulation, at the very least.

However, such point-mass dynamic simulators have significant weaknesses:

1. The point-mass has no resistance to Yaw-acceleration, so such "transients" are NOT modelled. This is a significant deficiency in FS conditions.

2. The COMPUTER DRIVES THE CAR. So considerable time must be spent developing the "AI" that drives the car, and this necessarily takes away time from creating a higher fidelity model of the car itself.

3. The easiest "driver-AI" to develop is that which drives the car always at the car's absolute limit. This can give misleading results (see next).

4. Since the computer is driving the car, the user, namely the FS-student, has NO idea how easy said car would be to drive. Or, indeed, if it is the type of car that is all but impossible for a real person to drive "at the limit".

5. The "output" of the computer driven lap is usually a single number, namely the "laptime". This is a very "narrow-bandwidth" result, which does not make apparent why the car was so fast (or so slow), nor how things could be improved. If, instead, the student has to follow the car around the lap, metre by metre in slow-motion, and with a clear picture of, say, the forces acting on the car, then the student has more time to gain a deeper understanding of why the car is behaving as it is. Thus they are in a better position to improve the car.
~o0o~

Turning the Step-Steer Simulator of this series of posts into a "Car Dynamic Simulator" that overcomes the abovelisted deficiences is relatively straightforward. The "extras" needed are:

1. A "track"! The easiest way to "create a track" is to place a large number of orange-dots on the main track-area on the left-side of screen. Just like real FS-tracks (ie. with their "orange-cones")! A conventional road with edges/curbs is NOT necessary. Different "track-maps" (= lists of N/E-coordinates of orange-dots/cones) should be able to be editted, saved and reloaded. It is useful to see a zoomed-out view of the whole track-area, roughly 300 m square, together with a closer view of the car and forces acting on it, roughly 10 m square. Different runs of different cars running on said "tracks" should be able to be saved, reloaded and replayed, both in normal time, and also in slo-mo, with rewind, etc. It is helpful to show two different cars running at the same time on a given track, to compare their strengths and weaknesses.

2. The car needs some means of "driving" it. So add some simple Accelerator, Brake, and Steering controls. I just use mouse-clicks on graphics buttons for these controls. So "Acc" and "Brk" can be set anywhere from 0-100%, and "Steering" set anywhere from full-lock L<->R. There is then a "DriverStep" button that advances the simulation by an adjustable 0.1 -> 1 seconds. The driver uses big timesteps when going down boring straights, and smaller timesteps when negotiating the twisty bits. (Note that the "Mechanics Engine" timesteps are much smaller, typically 0.001 seconds.) Changes to driver inputs (Acc/Brk/Str) are linearly ramped from-old -> to-new setting, over the DriverStep period. A "BackStep" button is also useful, to allow the driver to have multiple attempts at a given section, perhaps Braking earlier before a corner, etc., without having to rerun the whole lap.

3. The above driver-controls have to control something, so Engine/Transmission and Brake system models must be added. Again, this is straightforward, if perhaps a bit tedious. The general ease of extending the model means that things such as variable-torque-split-All-Wheel-Drive are easy to do. Similarly, the All-Wheel-Brakes should have adjustable bias, and as noted, All-Wheel-Steer is also easy to do. The "tyre-model" for combined Acc/Brk/Cornering-forces needs some thought, and many (realistic!) variations are possible. I suggest keeping all wheel-forces inside a "friction circle" to begin with.

4. It is well worth adding the aero parameters of CxA, CyA, and CzA (ie. drag-, side-, and down-forces). The Lines-of-Action of these forces should be carefully thought about, and included as adjustable parameters. The three aero-couples can also be added, but these are so fickle in reality that trying to model them may be a waste of (scarce!) time. Air density can be taken as a constant, or else used as a global multiplier of all the aero-forces. Adding a steady wind blowing over the track is also easy to do, quite educational, and relevant to the real problem at hand.

5. Just being able to drive, "video-game style", the simple bicycle-model car of the above posts around a track is hugely educational. It makes obvious that a seemingly fast car might also be almost impossible to actually drive fast, whereas another car that is nominally slower might be effortless to drive fast. Adding the aero-parameters to the simple bicycle-model takes the usefulness of the simulation another big step ahead. After that, CG-Height also makes a noticeable difference. Going to four wheels separated by Track-widths, especially when static-toe-angles and "Ackermann geometry" are added, makes a big difference to perceived "driver feel". Differential choice then also becomes a major performance factor. And the list goes on. Adding toe-compliance according to Fx/y forces is easy, and again, very educational (eg. see what braking or dropped-throttle induced rear-toe-out does to your ease-of-driving!).

Anyway, the above type of "Driver-In-The-Loop" dynamic simulator allows quick and cheap exploration of many different car concepts, both in terms of outright lap-time performance, but also DRIVEABILITY of the concept. This allows a clever team to best utilise their scarce resources to build a winning car.
~o0o~

(More coming... Character limit!)

Z

09-24-2017, 08:19 PM

5b/5. STEP-STEER-SIMULATION - LAST BIT...
=====================================
Finally, I better briefly spell out just how simple this Step-Steer program is, at its core.

The "inner-loop" of the program simply repeats Newton's Second, over and over again. Very simple, very boring, and very much what computers are good at.

This is a three-step process, essentially as follows:

===============================================
STEP 1 - AT BEGINNING OF INNER-LOOP TIME-STEP (= "dT").
===============================================
* CALCULATE AND SUM ALL THE FORCES
that will act on the car throughout this TimeStep,
based on the car's "Current State".

The "Current State" of the Car consists of its:
Driver-Settings (= Wheel-Steer-Angles, Throttle/Brake-settings, etc.), and the car's
Dynamic-State (= the car's Position and Momentum wrt Ground/Wind, etc.).

================================
STEP 2 - THROUGHOUT THIS TIME-STEP.
================================
* DO "NEWTON'S SECOND"!

Namely, dP <= Force * dT, both Linear and Rotational. In detail:
dPx = Fx * dT,
dPy = Fy * dT,
dMoMz = Tz * dT.

(MoMz = "Moment of Momentum about z-axis" = "Angular Momentum in Yaw".)
(Tz = Couple/Torque about z-axis.)

This is TOO EASY!!!

=============================
STEP 3 - AT END OF THIS TIME-STEP.
=============================
* UPDATE CAR'S POSITION & MOMENTUM
to their NEW values at the end of this dT
(which gives the Dynamic-State of Car at beginning of next dT).

Note: ASSUME CONSTANT MASS (ie. no fuel lost or bug-splats gained,
and no change-of-shape-of-car, so no "MoI" change,
and all this is in FLATLAND, so no "Euler" or gyroscopics, ... yet!).
So changes to Momentum P (= m.V) are only from changes to Velocity.

So,
dVx = dPx/CarMass,
dVy = dPy/CarMass,
dOmega = dMoMz/(CarMass*RadGyr^2).

(Omega = "W" = Yaw-(rotational)-velocity.)

So NEW Velocity at end of TimeStep given by,
NewVx = OldVx + dVx,
NewVy = OldVy + dVy,
NewOmega = OldOmega + dOmega.

And NEW Position at end of TimeStep given by (via simple "centre-differencing"),
NewX = OldX + (OldVx + dVx/2) * dT,
NewY = OldY + (OldVy + dVy/2) * dT,
NewTheta = OldTheta + (OldOmega + dOmega/2) * dT.

(Theta = "heading"-direction/angle.)

Again, TOO EASY!
==========================================
TIME-STEP dT IS FINISHED, SO LOOP BACK TO STEP 1.
==========================================
~~~o0o~~~

All the rest is just the "look and feel" of the program!

Of course, this "look and feel" is very important to get right, in order to make the whole thing a useful tool. So, you should spend much time polishing these many aspects of the program, such as making the inputting of the many parameters as quick and painless as possible, the "driving" of the car should be easy and smooth, and the many possible "metrics" that can be derived from the car's performance should be available in visually easy to read ways, and so on.
~o0o~

Anyway, it is now 2017, and the computer hardware and software technology needed for this type of program was readily available 30 years ago. In fact, I wrote this Step-Steer program in a freeware version of Basic that was on my computer from ~10+ years ago, when one of my kids started to learn programming. Much older technology would work just as well.

So there is really no excuse for modern FS-teams to NOT be using such "Driver-In-Loop" simulators to help them set their overall car design directions.

So, please, NO MORE Facebook posts showing the new turbocharger fitted to your already 60+ kW engine, in your "super-lightweight" CF-car, which for some reason has LESS THAN 50% REAR-WEIGHT (!!!!!), and a caption that reads...
"Now let's see what this BAD-BOY can do!"

Groooaaannnn!!! It will do NOTHING well, other than turn its rear-tyres into blue-smoke, you fools! Obvious, once you drive it in a simple DIL-simulator.

Z

maxay1

09-24-2017, 09:03 PM

Z,

I'm just beginning to read through this, but I want to thank you for taking the time to post this. Very much appreciated.

Wil

BillCobb

09-24-2017, 09:08 PM

Well done, Z! The KISS principle personified. (KIZZ ?) You maybe should mention that its usually better to make the car simpler than to make the computer program more complicated.

Here's my KISS and tell process. (maybe should been call Newton's Engine). Was first run on a 1965 IBM 1620 in Fortran from punched cards at Hutch-Tech High School in Buffalo N.Y. (I went to South Park, but hitch hiked by bus to the smart zone). I resisted the urge to write it in MACHINE LANGUAGE (a decimal , not hexadecimal era machine).
The only extra complexity is the reality of spaghetti steering shaft stiffness. Yes the Pacejka tire can be overlooked if you know how to twiddle the 4 coefficient that matter (as I have done for a program validation exercise).
Now we have Matlab to blame for the appearance pf pretty wallpaper suitable GUI's, colors, graphs and large values of 2. Yes the rear steering car is not much different that a front steer job, but you know that just about all cars have "rear steer" via roll steer, and by design. I would ding you for the mass spring damper analogy, it sorta works, but Ay gets an extra kick from the wrong way first sideslip awakening and that doesn't happen with my yo-yo unless you touch it off with a hammer blow.
see attached. Yes, these simple codes can objectify 90% of the cars that people like because they are fun to drive1253.

enf = 0;
u = SPEED/3.6;
R = 0; %initial condition for yaw rate
BETA = 0; % initial condition for sideslip
AYG = 0;
WHL_LIFT = [0 0];

WF2 = WF/2;
WR2 = WR/2;

dwf = 0;
dwr = 0;

dt =.01;
n=0;
for t= 0:dt:2.5; % 2.5 seconds ought to do it.
n=n+1;
time(n)= t;
dwf = AYG * LTF;
dwr = AYG * LTR;
wlf = WF2 + dwf;
wrf = WF2 - dwf;
if(wrf <=0) wrf = 0;
WHL_LIFT(1) =1;
end
wlr = WR2 + dwr;
wrr = WR2 - dwr;
if(wrr <=0)
wrr=0;WHL_LIFT(2) =1;
end

ALPHAF = BETA + A*R/u - delta + enf;
ALPHAR = BETA - B*R/u;

ALPHALF= ALPHAF-TOEIN/2;
ALPHARF= ALPHAF+TOEIN/2;

fylf = Pacejka4_Model(FY,[ALPHALF,-9.806*wlf]); %Nonlinear tire FY representation
fyrf = Pacejka4_Model(FY,[ALPHARF,-9.806*wrf]); %
fylr = Pacejka4_Model(FY,[ALPHAR, -9.806*wlr]); %
fyrr = Pacejka4_Model(FY,[ALPHAR, -9.806*wrr]); %

nlf = Pacejka4_Model(MZ,[ALPHALF,-9.806*wlf]); % Nonlinear tire MZ representation
nrf = Pacejka4_Model(MZ,[ALPHARF,-9.806*wrf]); %
nlr = Pacejka4_Model(MZ,[ALPHAR, -9.806*wlr]); %
nrr = Pacejka4_Model(MZ,[ALPHAR, -9.806*wrr]); %

fyf = fylf + fyrf;
fyr = fylr + fyrr;

nf = nlf + nrf;
enf = -sign(nf).*ENFB*ENFC/100.*log(abs(nf)./ENFC+1)/2;

rd = (180/pi)*(A*fyf - B*fyr +nlf +nrf +nlr +nrr) /IZZ;
betad = (180/pi)*(fyf + fyr)/(WF + WR)/u - R;
R = R + rd*dt;
BETA = BETA + betad*dt;
AYG = u*(R + betad)/(180/pi)/9.806;
ayss(n) = AYG;
end
DELTA = delta; % cheezy workaround for global pass.
RADIUS = (180/pi)*u/R;
Radius_Error = TARGET - RADIUS; % return a turn radius error value.

Vishnu Sanjay

09-27-2017, 08:54 AM

Thought there would be a flood of responses by now….

First of all, Z, thanks a ton for taking the time to prepare and post this. It links back to many interesting issues that have been discussed on these boards, and the ‘aerial view’ is very helpful at putting together VD pieces.

I’m slightly confused about what conclusions to draw from the last picture of the inertia sensitivity analysis. The first picture shows the super low IZ car turning within the turn radius of its super high IZ counterpart, but in the next pic (cars with IZZ= 0.4 kg m^2, 0.8 kg m^2 and 1.2 kg m^2), the x displacements are about 25m, 25 m and 30 m respectively, while the y displacements are around 72, 65 and 68 m respectively, in the same time period. IF these were all that were needed to represent these cars (neglecting the effects that you have mentioned having neglected), wouldn’t Car 2 be ‘more agile’ than Car 1? The response times that the driver perceives would be different (assuming driver sits at same position along wheelbase for both cases), would anything else enter the picture in favour of Car 1? At the very least it seems to indicate that the advantages you mentioned after the first picture drop off or yield diminishing returns with further decrease in IZZ. This is somewhat counterintuitive, and was one result that surprised me. I’ll have to look into this further.

Same goes for the bit about load transfers and longitudinal force effects. I find it difficult to know anything about the relative effects of these unless they are quantified and simulated. That’s pretty much why these tools are necessary; else it’s all just speculation.

The other examples are great and most of them match with what I’ve got when running my sims. The biggest difference is with the way the results are presented, the aerial view allows a clearer visualisation of what’s happening. I think I’m going to try and figure out how to write these into my MATLAB code.

Regarding the ‘stronger’ tires at the rear, this is the same as having higher rear cornering stiffness than front. (Lower rear cornering compliance than front), and is applicable with the other compliance and kinematic factors that contribute at each end. With same tires front and rear and a rear weight bias, in the linear regime, the rear does maintain balance because of increased cornering stiffness due to increase normal load, but at the limit, where the rear tires starts to ‘soften’, the car spins. I think Bill has mentioned, on this very thread if I am not mistaken, how important the cornering stiffness derivatives are.

Wish more people here would find this interesting. I for one would love to see what results you have found in ‘Flatland plus’ and the ‘Car Dynamic Sim’…

Other threads that relate to what has been revived here (in case anybody’s interested)

http://www.fsae.com/forums/showthread.php?12263-Bicycle-model

http://worrydream.com/LadderOfAbstraction/ (for what I think is the rationale behind Flatland-> Flatland plus -> Car Dynamic Sim, originally shared by Doug Milliken)

http://www.fsae.com/forums/showthread.php?12211-The-DYNAMICS-Part-of-Vehicle-Dynamics

http://www.fsae.com/forums/showthread.php?12213-Coming-Soon-to-a-Theater-Near-You...&

Should do for starters…

I’m preparing a summary of my current understanding of and a few questions regarding handling parameters related to driver feel….so as to be able to figure out what is 'acceptable to an FS driver' behaviour in a CDS type sim.

P.S I’m currently working on a transient sim (that applies step, ramp and frequency inputs to the car model) using the equations that Bill has detailed above for the bicycle model (no roll DoF, but load transfer, compliance included). Going to try and add your visualisation techniques so I can see clearly if anything funny is happening. Your roadmap of order to add other complexities (weight transfer->steering geometry->diff->other stuff) in is also pretty much what I’m taking. Wish I didn’t have to remind a few people on my old team to figure this stuff out before bickering about an ‘essential for improvement’ 100kW motor….

09-30-2017, 10:59 PM

Firstly, for those few people reading this, Google/Picassa has decided they do NOT want me to put appropriately sized images (ie. big enough so you can read the details) in-line with the text. It used to be possible, albeit difficult, but now seems IMPOSSIBLE. Ahh..., "progress"!!!

Anyway, very small images come up on my screen, but only after I login! When I click on the "google...photo" link I can see the images at reasonable scale, but maybe that's just for me, because they are my images? So, not sure if you people can see them full size?

Grrrrr...
~~~o0o~~~

Bill,

... Was first run on a 1965 IBM 1620 in Fortran from punched cards ...

I have fond memories of sitting at an industrial strength, hammertone grey, punch-card machine, writing my first programs back in the early 1970s. Beautiful bit of machinery, very similar in size and appearance to an anti-aircraft cannon, but about TWICE AS LOUD. Most enjoyable!

Ahh, yes..., back when things just worked. And I probably got my first dose of industrial deafness from that machine. :)
~~~o0o~~~

Vishnu,

I’m slightly confused about what conclusions to draw from the last picture of the inertia sensitivity analysis...
...(cars with IZZ= 0.4 kg m^2, 0.8 kg m^2 and 1.2 kg m^2)...
...wouldn’t Car 2 be ‘more agile’ than Car 1?
... would anything else enter the picture in favour of Car 1?
... seems ... somewhat counterintuitive...

As first clarification, note that:
"Moment of Inertia in Yaw" = "Iz" = Car-Mass x RadGyr^2.

So for the lightweight car examples given earlier, with Car-Mass = 200 kg (= TOTAL mass including driver), and with the "RadGyr"s given:
Lo-Iz (= "agile cannonball") = 200 kg x 0.4 m x 0.4 m = 32 kg.m^2.
Mid-Iz (= "typical FS-car") = 200 x 0.8 x 0.8 = 128 kg.m^2.
Hi-Iz (= "sluggish dumbbell") = 200 x 1.2 x 1.2 = 288 kg.m^2.

Worth noting that I have seen a student spend a whole year doing a thesis to measure this number, and they ended up with a number that was ORDERS of magnitude away from the above numbers!

So, in short, when using metres and kilograms the "Iz" number should be of similar order of magnitude to the "total-mass" number, but preferably about half its size. So ~100 kg.m^2 for a small FS-car, and ~200 kg.m^2 for a heavyweight.
~o0o~

Secondly, in the second image of my earlier post 1/5, it is important to note that the "transient" stuff, namely the cars' behaviour related to their Izs, is mostly over early in their runs. The reason I used a zoomed-out image was to show that the Hi-Iz car had a much longer "transient" phase than the Lo-Iz car (ie. Hi-Iz was still fish-tailing much later in its run).

Where the cars ended up on the track, at the "end" of their runs, is mostly governed by other "steady-state" performance factors. Specifically, all these cars are "Under-Steerers" (and quite strongly so, given their F&R-Mus), so "the faster they corner, the larger their cornering radius". The cars' Ackermann, or low-speed, turning radius is under 10 m (see calcs given earlier for the 10 deg step-steer). In that second image all the cars are clearly US-ing, and following paths of much larger radius than 10 m.

The reason the Mid-Iz car has the tightest overall path is simply that it loses the most speed (ie. momentum) shortly after its step-steer. So it finishes the manoeuvre at a slower speed, and hence at a tighter radius.
~o0o~

Here is a close-up of exactly the same three cars, with same step-steers, but this time with their runs overlaid in the image. So all start their step-steer at the same place, at E,N (= X,Y) = 0,10.

https://photos.app.goo.gl/YndIWQfXFKTFIa7O2
http://www.fsae.com/forums/attachment.php?attachmentid=1257&stc=1
(SS3RadGyrsZin.png)

The image is a bit messy, which is why I didn't display it this way in earlier post, but it makes clear that the Lo-Iz car is the most "agile". At Time = 1.5 seconds (ie. fourth car image) the Mid-Iz car is already seen to be lagging behind because of its lost speed/momentum due to excessive side-slip shortly after its step-steer. Try to follow its wheel-paths to see this side-slip.
~o0o~

Here is a zoomed-out shot of exactly the same runs as above. Now you can see that the Lo-Iz car has maintained more of its speed/momentum, compared with the other two cars, which have "scrubbed-off" a bit of their speed. Note that the car "ReDraws" are always at the same instants in time, every 0.5 seconds.

https://photos.app.goo.gl/EJVO27HU1lRkGrtf1
http://www.fsae.com/forums/attachment.php?attachmentid=1258&stc=1
(SS3RadGyrsZout.png)

Because the Lo-Iz car is going a little faster after its "transient" phase is over, and because of its US-ing tyres, it naturally travels on a slightly straighter path than the other cars. If the Lo-Iz driver wants to tighten their radius, then a light tap on the brakes will do the trick. Especially so, when CG-above-ground is factored in!
~o0o~

This next image shows two DIFFERENT cars to above. Now,
1. Both cars have EQUAL F&R-Mu tyres, so have nominally "neutral" handling.
2. The Lo-Iz car has RadGyr = 0.6 m, which is a practically achievable number for a tightly packaged FS-car.
3. The Hi-Iz car has RadGyr = 1.0 m, which is also realistically possible, say if a team increases its R% to 60% by pushing their heavy engine as far rearward as practical, while still keeping the driver's knees above the front-axle, as is typical with most FS-cars nowadays.
4. Both cars' step-steers are now at 20 degrees. So this is representative of a sharpish U-turn, but not quite at the minimum radius hairpin.
5. The two overlaid runs at top-left of image are done at an initial velocity of 10 m/s, 36 kph, and the runs at lower-right are at 15 m/s, 54 kph. All step-steers start immediately at the first drawing of the cars, then take 0.1 s to ramp-up to the final 20 deg.

https://photos.app.goo.gl/VbhEEUYSeCcYwgno2
http://www.fsae.com/forums/attachment.php?attachmentid=1259&stc=1
(SS2RadGyrsAtVel10and15.png)

Again, a bit messy, but which of these realistic-Iz cars would you rather be driving, assuming you wanted to win the race?

(Will reply to your latest PM soon.)
~~~o0o~~~

Lastly, all the usual VD-metrics can be pulled from this Step-Steer program. All the raw data is there, at a convenient 1 kHz sampling rate in above images, so drawing LART, YART, Side-slip, Vel/Acc-pole, energy-loss, +++, curves is straightforward.

As suggested before, the "helpfulness" of such metrics is largely related to how good is the "look and feel" of their presentation. Being able to overlay the curves for different cars can make comparison easier. Or it might be too messy? The end result depends on the many little details. I suggest a lot of trail and error to help find what works best.

(And BTW, the "look and feel" of putting images into these posts is now an abomination. The downhill slide is getting ever steeper!!!)

Z