# Thread: Plotting Cn-Cy and Cn-Ay Graphs.

1. Originally Posted by Claude Rouelle
Tim, I am going to look stupid but I have to ask: what is a caster wheel? Do you have picture or a simple sketch?
A caster wheel ('roulette pivotante' in french accoring to leo.org) in its simplest form is a shopping trolley wheel. The wheel will, in steady state, point along the velocity vector of the mounting location w.r.t. ground. The transient slip angle can be found using the angle and rate measured by the trailing arm and the vehicle velocity as measured by the rotating wheels.

I drew up a design for this while I was still in university but never had the time to make it. The concept can be found here:
https://1drv.ms/b/s!AioEiFs0jfZSgVc5hbwAOWexfqFI

The design was made to have the CG of the trailing arm assy along the vertical pivot axis but the counter balance the Doug mentioned were to be included in the green plate in front of the pivot if any corrections were required.

I've seen a similar concept used in the cottage industry too where its more accurate at low speeds than the GPS based systems which cost 50k+. I know my old uni trialled a similar concept last year. I will let them elaborate if they want.

You still need a gyro though to transform the slip angle from the measurement point to the CG and front and rear axles for cornering compliance analyses.

I doubt the idea can be any good though because my track record is nothing special. My FSAE team always came last and every shool I went to in my life was demolished shortly after I left.

In terms of definitions the slip angle is, in my opinion, unambiguously defined in Milliken, Guiggiani, Pacejka, ISO 8855 and SAE J670e. Clear as a dogs donger in my opinion - I don't get what all the fuss is about.

2. Tim,

Quick question, what is the assembly with #11 pointing at it? Seems like the vertical (slip angle) pivot must be inside the blue cylinder?

Yours looks much more robust than the one I made ages ago, I had the extra constraint of very low vertical height, to fit under the CG of a production car without any big holes in the floor. Too bad that yours wasn't tried, looks like it should have worked OK to me.

3. Originally Posted by mech5496
There was a post in here somewhere, I believe by Bill, on using optical mouse sensors to build a relatively simple and cheap, vision-based slip angle sensor. Shame I cannot find it anymore.
I saved the paper that post was centered around, unfortunately it's above the 19.5 KB (!?) limit for PDF attachments. If anyone is interested google "Development of a Low Cost Slip-Angle Sensor" and it should come up for you in the first couple hits - published 2013 by Nathan Tarlinton of University of Wollongong.

Taking that approach is a nice middle ground between the low-tech castor wheel (and ultra low-tech wire) and a black box high end sensor package. It's arguably going to teach a lot more about modern instrumentation in the process too, as long as the right steps are taken for proper data acquisition.

4. Originally Posted by DougMilliken
Tim,

Quick question, what is the assembly with #11 pointing at it? Seems like the vertical (slip angle) pivot must be inside the blue cylinder?

Yours looks much more robust than the one I made ages ago, I had the extra constraint of very low vertical height, to fit under the CG of a production car without any big holes in the floor. Too bad that yours wasn't tried, looks like it should have worked OK to me.
That assembly is a steel tube which houses the potentiometer (blue cylinder) and 2 bearings for the vertical spindle. The tube is threaded on the outside surface for mounting to the vehicle.

I didn't worry about height as I assumed it would be mounted off the front or rear of the vehicle and then the measurement transformed to the CG using the yawrate.

The only thing missing that I feel is important is the velocity measurement using the wheel speeds. With such small wheels I don't know how accurate the velocity reading would be. Actually there isn't any real reason that the wheels have to be that small.

5. Harry,
I believe I linked to the article you are referring to: http://journals.sfu.ca/vte-j/index.p...article/view/6.

Erik,
I think we're on much the same page here, though we come at the matter with different tacts. Perhaps I am too subtle and that is a failing of mine? I would say that in general, motorsport VD relies far too heavily on models utilising implicit (behavior generation through gradient extrapolation and iteration), Eulerian (fixed particle, flowing field) schemes without necessarily understanding the limitations of the strategies (or perhaps not understanding that the scheme is inherent in the calculations).

The way I read your critique of moment diagrams (save for critiques of both personalities and typographic renderings) leads me to surmise you are suggesting modelling - through time - the motion of a rigid body through space (using tyres for et al) and subsequently measuring the body path through space to determine parameters including yaw rates, slips etc. This could be described at an explicit (current behaviours integrated updated through time over small time steps), Lagrangian (moving particle, fixed frame ) model. This akin to the (bicycle?) model you presented in thread a while back, and is something that I advocate for relatively strongly. When people speak of 'transient' models, this is more often what they are discussing (n.b. transient, non-linear behaviours can be, in general modelled using implicit techniques and are often more efficiently modelled in an implicit time scheme).

Using a well-prescribed test to examine the performance limits allow for much greater insight into the vehicle behaviour; a chirp steer test (constant velocity with increasing steer frequency) is great for looking at normal and yaw acceleration limits. The below is a four-wheel, suspended chassis and suspended wheel model, with absolute bodge vehicle parameters.

Once you have the explicit, Lagrangian vehicle model ready, you can feed in whatever test characteristics you want. This framework allows you to conduct other tests with relative ease (e.g. understeer, constant radius, etc.) to characterise a vehicle concept readily.

6. Tim,

Sorry, just some small corrections.

A caster wheel ... will, in steady state, point along the velocity vector of the mounting location w.r.t. ground.
Actually, it points along the velocity-vector of the Car-Body-reference-frame, at the castor's WHEELPRINT-"POINT", wrt Ground.

So the "Slip-Angle" is measured at a point in the CB's reference-frame that moves around as the castor (or caster) swings left and right. So if the castor has a long "trail", and the corner radius is very tight, then there can be significant differences between the SAs at the castor's "mounting location pivot-point", and at its "wheelprint-point".

However, with a short "trail" of say a few centimetres, and even with the typically tight FS/FSAE corner radii, I doubt this "error" would be a problem. Anyway, it is an error that can easily be subtracted by some post-processing of the signal.
~o0o~

You still need a gyro though to transform the slip angle from the measurement point to the CG and front and rear axles for cornering compliance analyses.
Or you can fit TWO castor-wheels, one at front-of-car, and one at rear. The two measured velocity-vectors, of front and rear of Car-Body-frame, wrt Ground, then give a complete description of the motion of the Car-Body wrt Ground, at least in Flatland.

But, in fact, it is even easier. You just need the angle and rolling velocity of one castor, and then ONLY THE ANGLE of the second castor. Kinematic's "Axiom of Rigidity" then lets you find the velocity of all other points on the Car (ie. you assume the Car-body is rigid, so it stays the same shape, always...).
~o0o~

Nevertheless, I DO LIKE the idea of a castor-wheel. My L-shaped wire has left scratches all over my driveway! (Grrr ... hopefully should buff out...)

Z

7. Rory,

The way I read your critique of moment diagrams ... leads me to surmise you are suggesting modelling - through time - the motion of a rigid body through space ... and subsequently measuring the body path through space to determine parameters including yaw rates, slips etc. This could be described as [being like the] explicit ... Lagrangian ... model you presented in [another] thread a while back...
Yes, that is the way I generally prefer to do things. Three points below to explain the "why":

1. These "mathematical models" are tools. And as with all tools, different people have their different preferences. One person will use a laser-guided, diamond-tipped, boring-tool, while another prefers his sledge-hammer. Ultimately, the only thing that matters is that the job gets done. That is, the job is finished properly, correctly, everthing works as it should, and so on.

But, of course, some tools will get the job done more quickly, or easily, or cheaply, or, most importantly, with less chance of cock-ups.
~o0o~

2. With these mathematical models, I see a great advantage in choosing a model, or a "map", that, to human eyes and mind, looks as close as possible to the "terrain" that you are trying to navigate, or investigate.

So the rather abstract graphs of a YMD/MMM, or the left two images in your last post, might be able to be read by a VD-specialist with long training in such things. But any anomolous car behaviour appearing in that map might NOT be so obvious to the less trained eye. Or, indeed, not even to the trained eye that only has time for a quick glance.

Your third, rightmost, graph (maybe with equal X, Y scales) gives a more obvious and compelling indication of anything that might be wrong VD-wise. Especially if it is animated (ie. moving pictures) and it includes tyre-screeching sounds, possibly followed by the car rolling over and bursting into flames!

That is, I find the model that is easy to understand, and hence with less chance of my missing something important (= less chance of cocking-up), as the preferable one.
~o0o~

3. This next point is a huge subject, and was much debated throughout the 1600s and 1700s ++, so only the briefest comments here.

The old-fashioned Geometric way of solving problems, epitomised in Euclid's "Elements", involved symbolically representing the "real world" with ink lines on flat sheets of paper. (In fact, originally by using a stick to draw lines in smooth sand.) This is a SYMBOLIC ABSTRACTION of the real world, albeit one that is NOT TOO FAR removed from reality. A Chinaman, Masai, or Eskimo will all be able to grasp the connections between "map" and "terrain" quite easily.

This all changed when Rene Descartes added the short appendix "La Geometrie" to a much longer book he was writing on philosophy (~1637?). Here Rene put forward a new method of solving, or proving, Euclidean problems/propositions. I recall he ended an example in that appendix with something like "See, I have solved it without drawing a single figure!". That method is now commonly known as "analytic geometry", and like Rene it still uses "a, b, c"s for the knowns, "x, y, z"s for the unknowns, "Cartesian" axes, and so on.

As noted, when Rene's book was published there was much controversy over this new "cogitatio caeca" way of doing things. Along with "blind thinking", it was also described as "mechanical" or "imaginationless" thinking. These names were given because there was virtually NO connection between the symbolism, namely the abc's and xyz's, and the actual physical problem. It was perhaps AN ABSTRACTION TOO FAR. And once the problem was cast into its alphabet-soup form (<- err, my term), it became nothing more than a mechanical process of stirring the soup (ie. manipulating the equations) until the desired result appeared. With never any hint of how far from reality you might be drifting.

To stress this again, throughout the process of stirring the soup, or manipulating the equations, there is NO VISUAL CONNECTION between the "map" (<- the algebra) and the "terrain" (<- the real problem). Hence the description of a "blind" or "imaginationless" approach. Contrast this with, say, a geometrical solution to Free-Body-Diagrams. Here, each step of the process involves "seeing" something that looks very like the "real forces" acting on the "real body", albeit with the forces combined or decomposed in many different ways.

Anyway, Newton was initially besotted by this "New Maths" as a young man. But when he got older, and wiser, he wrote his "Principia" in a style that was remarkably similar to Euclid's Elements. (One day I will have to rant about the "piece of string" that lies at the core of the Elements, and why the ancient Geometers were so highly regarded...)

More recently the Western Education system has settled for the Cartesian approach, unfortunately exclusively, with never a mention of Euclid! I speculate that this has been driven mostly by the ease of typesetting "equations", and the apparent difficulty of presenting "figures" or "sketches". This Forum is a good example, with NO sketching facility! Grrrrr...
~o0o~

End of rant. Will probably hit character limit soon.

But remember, just because everyone else is doing it, surely don't mean it's the best way!

So YMD/MMM? Or visual graphics of a car fish-tailing around a bend?

I prefer the second, with sound-effects!

Z

(PS. I'll give an example of "visual" versus "blind" thinking tomorrow.)

8. At 5x speed, a chirp steer test conducted at 50 km/h over 20 sec. Red-line connects vehicles CoM to CoM centre of curvature (obviously this tends to +- infinity as the normal acceleration tends to zero), blue-line indicates vehicle heading.

9. Rory,

Yes! The first step to understanding VD, IMO, is being able to watch a car, either real or modelled, do its thing. As such, your above type of "map" should be one of the first outputs of the modelling process.

(And it seems that .gifs delivered via "imgur" work much better than the google/picassa rubbish that I had to jump through so many hoops to get working, many years ago. More grrr...)

Also, as you said earlier, "Once you have the explicit, Lagrangian vehicle model ready, you can feed in whatever test characteristics you want [which] allows you to conduct other tests with relative ease (e.g. understeer, constant radius, etc.) to characterise a vehicle concept...

It also makes it very easy to pull any amount of performance data, or "DAQ", out of the model, because all the data is being generated by the computer as the tests are run (as you also noted). So all the Betas, Thetas, Yaw-rates ("r"s), Tyre-forces, Tyre-SAs, +++, are all already in the box, and you can plot them any which way you want.
~~~~~o0o~~~~~

Back to alphabet-soup, or "blind" versus "visual" thinking.

Here is a screen-shot (hopefully!) of Figure 1 of Claude's second article, RCE2.pdf. At the top of the image is a visual depiction of a car, that, while certainly being a "symbolic abstraction" of reality, is also something that most people can recognize as a top-view of a car, together with some superimposed arrows. At the bottom of the image is the "super"-symbolic, alphabet-soup representation of those "closer-to-reality"-symbolic arrows.

Right at the start of the article Claude writes, "...there are 12 causes for the yaw moment: four tyre lateral forces Fy, four tyre longitudinal forces Fx; and four tyre self-alignment moments Mz.". So it is reasonable to interpret most of those arrows drawn on the visual image as the four tyres' Fx, Fy, and Mz forces.

Now, does anyone see something that looks not quite right, either in the image, or in the soup?

Can anyone see the dog's proverbials?

For me, it took just a quick glance around the visual depiction of the car and its arrows, before I was thinking "Why the hell does he have the front-wheel-yellow-arrows, presumably the "Fx"s, aligned with the chassis-frame, and not in the wheel-frame where they should be, namely perpendicular to the blue-axial-arrows, presumably the "Fy"s???". This ease of "seeing things" in such visual maps, is, of course, their huge advantage.

Anyway, I then trudged through the alphabet-soup version of the same thing, a slower and more painful process, and found many errors. Short list:
1. Both front-wheel-Fx COS- and SIN-delta components missing, given that the Fxs should be cranked around to be perpendicular to the Fys.
2. Both front-wheel-Fy SIN-delta components missing. (They do NOT cancel out, because Fyrf is much bigger than Fylf.)
3. A "+" near the right-side of the bowl of soup should be a "-".

And as a lesser note it would be helpful to have the arrows on the visual map suitably labelled. And to read in the text something like "... aero-forces ignored...".

Z

10. ## Pay Attention Rory, this one's for you !

A Message to Garcia: a widely distributed essay written by Elbert Hubbard in 1899, expressing the value of individual initiative and conscientiousness in work. Hubbard, from my nearby hometown of East Aurora N.Y.

ATTN: I'm trying to promote a .pdf file format for relevant presentation material on this thread's topic, but it's too big to attach. Who can I send it to to load ?