Problem with yaw moment diagram

[prox]

Hi everyone,

my name is Alexander and I write from Lund Formula Student in Sweden. I'm having some difficulties producing yaw-moment-diagrams with constant corner radii. As can be seen in the picture below, the diagram with infinite corner radius looks reasonable(?) (I'm using the SAE-axes convention), however, the one with a constant corner radius (and therefore varying yaw-rate which affect slip-angles) seems incorrect to me. For example, when beta=0 and delta=0 the yaw-rate should be zero(?) and therefore all slip-angles should be zero as well (which also means that all isolines should be shifted vertically compared to the current diagram). Therefore, it seems to me that the yaw-rate converges to an incorrect value in my iteration loop. When there is no yaw-rate (infinite corner radius), the slip-angles and lateral acceleration converges to reasonable values. Any hints as to what I'm missing? How do you correctly calculate the slip-angles iteratively when they depend on vehicle speed and yaw-rate? I'm using a four-wheel bicycle model with parallell steer and equal slip-angles left to right on each axle.

Some car parameters and the beta/delta sweep range differ between the diagrams, so don't compare absolute values. The solid lines are the delta isolines and the dashed lines are the beta isolines.

Thanks in advance!

[DougMilliken]

Ref. RCVD p.310-311 - It looks like you are having problems with the third case in Table 8.4 -- which we call CN-CY, finite radius, road load. Increasing speed on a constant radius increases CY (or AY).

Perhaps plotting yaw moment against speed (instead of lat. acc.) will help you sort out your thinking? For example, beta = 0 will happen at the tangent speed...

[prox]

Thank you for responding, Doug. Currently I'm calculating the speed as a function of lat. acc. (which in turn is a function of the lateral tyre forces and corner radius). I'm unsure on how to calculate the tangent speed correctly as I my model features weight transfer, aero and non-linear tyre characteristics (should I use yaw moment and force derivatives from the YMD?). Are you saying that the speed should equal the tangent speed along a beta isoline? If that is the case I guess that a corner radius of 10m would give far too high lat. acc.. Perhaps the beta/delta sweep range should vary with the corner radius in a Cn-Cy diagram? I have successfully produced Cn-Ay diagrams with the script that I am using but I might have missed some of the differences compared to Cn-Cy. I guess I have some reading to do!

It should also be said that what I want to achieve is a snapshot of the vehicle performance envelope given a certain corner radius (to be used in a lapsim). Any advice on what the best approach would be?

[BillCobb]

I wold be interested in the details (parameters) used in your model:

front and rear weights
wheelbase
tire brand, size, pressure, rim-width (assuming TTC data is available). [don't need the data only the description and test Round number].
overall steer ratio
roll gradient specification or roll center heights & total or sprung cg height.
Anticipated lateral load transfer distribution fraction front
And any anticipated K&C parameters (front & rear roll steer & camber, deflection steers, front Mz steer, etc)

enough for me to put together a transient response model for further investigations. Then we can compare this to what you have going on now.!

You can P.M. me if you wish or just reply here!

[prox]

Thank you, Bill. I will reply here so that others can learn something as well. I've attached a constant speed, varying radius diagram for you to look at.

Total mass: 275kg
Front weight distribution: 47.5%
Wheelbase: 1590mm
Tyres (front and rear): Hoosier 18x7.5-10 R25B, ~85psi, 8" rim, TTC round 6
Steer ratio: 2.7 (deg steering wheel/deg delta)
Front RCh: 50mm
Rear RCh: 80mm
Sprung CGh: ~320mm
FTTLTD: 45% (what I've used as a baseline when simulating, might vary a few percent after tuning in reality).
Unfortunately no K&C parameters as of yet (kinematically there is very little front bumpsteer and none in the rear).

The Pacejka89 formulation is used in the simulation below (zero camber, zero toe). Given the weight distribution and FTLLTD I'd be inclined to believe that the limit behaviour should show more oversteer rather than neutral/understeer. But perhaps this is the case since a lot of Fy is "lost" to steering angle. However, I'm sure that our Drexler LSD will add quite a bit of understeering moment on throttle (which is not included in the simulation).

[BillCobb]

I always take a pragmatic approach to working these situations, so, here goes:

I believe in using analysis and synthesis techniques that have Face Validity. My evaluations, testing and problem identification and fixes
are based on a philosophy of "Test what you Simulate and Simulate what you can Test for". That being said, with just a few (your) parameters,
I read your car as a classic case of being 'too academic'. Yes, you have weights & geometry and even a nonlinear tire representation
(and probably only FY inclusion). ISO test procedures (Step, Constant Radius, Frequency Response, and Constant Steer) indicate that you have a VERY high gain car that is
just about neutral steering and would be quite a handful to drive especially in closed loop. There is a wee bit of net rigid body MZ, But, so what?
This will NOT mimic the car you build ! I can pretty much guarantee it.

Adding the truth gets you a more driveable car, though. There is no way you will escape the deflection steer (FY, MZ and MX loading)
characteristics inherent in all cars, even Racecars. Roll steer and roll camber won't amount to much because these cars don't roll very much.
A soggy steering system will add some understeer, but only at low g levels because they eventually tighten up and MZ drops off as you
increase FY. I walked around in the caster shop a bit and this will help some but will increase your steering effort (which will be at nightmare
levels anyways because of your 2.7:1 steer ratio). Try playing around with negative caster in the rear ! This adds a teeny tiny bit of
deflection understeer that can cancel out the deflection oversteer you likely actually have.

I suggest you add the extra complexity to you model by digging deeper into the sources of compliance and manage them. Lateral force based
understeer is the only friendly one in the tent. Any camber change characteristics are sort of a lost cause because (ask your tire dealer).
Your limited slip will strap the car down but only under power so that's a source of confusion for the driver.

First suggestion is to increase the ratio. Quite a few YouTubers showing the unmanageable and unruly driver behaviors that result from such
a high gain car (g's per 100 degrees SWA) Your car is a the limit of control at 14 degrees steer and only 60 kph). I did not adjust your roll
stiffness distribution because that's your job AND because it takes quite a bit of change to accomplish much because these tires are very load
sensitive (They don't give up a lot with added load). Next might be to look at tire pressure and wheel rim splits. But, hopefully, you get
the idea and the recommendation(s). Finish the car early and run some ISO validation tests. You may have to explain these details to a few
of the judges because for so many, a stop watch is the only instrumentation they are familiar with.

IMHO, you should also study the effect of roll center heights (i.e. roll axis height because the low numbers increase the spring mass roll inertia.
This can make a car feel lazy, especially if your yaw velocity peak frequency goes above your roll peak frequency (as it will with speed). Any
roll coupling K&C parameters that accompany this convolution will be troublesome for any driver especially if you ever go really fast into a corner.

I include some snapshots of my form of analysis, in just a few minutes and a little more data, you would be done. I've condensed your car's traits
down to front and rear cornering compliance factors just because it's a freebee for the way I do business. Otherwise, you will have to go to
the YawMoment Depot and figure out how you are going to fix this car some other way. Enjoy. Keep in mind that 'optimizing the car' does not
generally make it easier to drive over a 'long' period of time. A knife edge car gets 'dull' very quickly and that makes adapting to the variability a
loosing battle. Any questions? Just ask ! My hockey season is
over but the hay is starting to grow. I'll be at Michigan maybe a day or so. Look me up !Alex1.jpgAlex2.jpgAlex3.jpgAlex4.JPGAlex5.jpg

[BillCobb]

One more Beauty Shot. We are limited to only 5 at a time...Alex6.jpg

[DougMilliken]

Elsewhere in this forum, Bill explains his methods in more detail. Unfortunately the FSAE.com Search seems to be broken (at least for me). The first few hits from this Google search point to some old-but-good posts by Bill (copy the whole line and paste in the Google search box):

site:http://www.fsae.com/forums bundorf

R. T. "Tom" Bundorf's work is discussed in some detail, along with references to SAE papers. Bundorf is also referenced several places in RCVD, check under his name in the Index.

I'm with Bill--to make simulation work relevant to your car, you need to add the important compliances. The lack of K&C inputs to your model was obvious from your first infinite radius MMM diagram (2nd diagram in your initial post). The give-away is that delta (steering)=0 across the range of lateral acceleration (below the tire peaks).

Further to my earlier post--the discussion of tangent speed in RCVD isn't indexed directly (this choice was made by the indexer, we didn't generate the Index). Look under "S":
+ Stability and control, steady-state
+ + speeds, significant (on p.879)

[BillCobb]

This one's still my favorite: Measured understeer of my boat. (The fix for it is a 5 blade Mercury Hi-Five ported prop. K=slightly negative).

All you need is forward speed and yaw velocity with a fixed steer angle and slow acceleration. This was done with a VBOX, but there are probably a few phone apps that could do this on the fly.C-Speed Test.jpg

[prox]

Bill and Doug, thank you very much for replying! I will take a look at the sections that you mention, Doug. Regarding compliance values, we have measured the hub-to-hub stiffness of our frame and done tension/compression tests of the toe- and steering links so far (not near enough, but it's a start). I will add compliance effects to the simulation when everything else is working properly.

I always take a pragmatic approach to working these situations, so, here goes:
I believe in using analysis and synthesis techniques that have Face Validity. My evaluations, testing and problem identification and fixes
are based on a philosophy of "Test what you Simulate and Simulate what you can Test for". That being said, with just a few (your) parameters,
I read your car as a classic case of being 'too academic'. Yes, you have weights & geometry and even a nonlinear tire representation
(and probably only FY inclusion). ISO test procedures (Step, Constant Radius, Frequency Response, and Constant Steer) indicate that you have a VERY high gain car that is
just about neutral steering and would be quite a handful to drive especially in closed loop. There is a wee bit of net rigid body MZ, But, so what?
This will NOT mimic the car you build ! I can pretty much guarantee it.

Tyre Mz is included in the simulation. We plan to conduct tests in order to validate our simulations as well as we can. We ran a constant radius test with last year's car and swept the FTLLTD (we run air springs), our yaw-moment diagrams predicted the steady state limit behaviour quite well when accounting for Mz due to our LSD (at least the trends are correct). We tried to produce the understeer gradient as well. Regarding steering sensitivity, is there any paper that links this metric to actual driver experience/performance (I'm sure this has been investigated a lot for passenger cars, but for racecars)? What is the trade off between increased steering "resolution" and less arm movement? Fortunately, our drivers have a lot of karting experience and will be used to the "quick" steering ratio and high steering sensitivity. Regardless, I agree that the car should be as easy to control as possible, and therefore we should look into this even if our drivers can adapt.

Adding the truth gets you a more driveable car, though. There is no way you will escape the deflection steer (FY, MZ and MX loading)
characteristics inherent in all cars, even Racecars. Roll steer and roll camber won't amount to much because these cars don't roll very much.
A soggy steering system will add some understeer, but only at low g levels because they eventually tighten up and MZ drops off as you
increase FY. I walked around in the caster shop a bit and this will help some but will increase your steering effort (which will be at nightmare
levels anyways because of your 2.7:1 steer ratio). Try playing around with negative caster in the rear ! This adds a teeny tiny bit of
deflection understeer that can cancel out the deflection oversteer you likely actually have.

We have thought about compliance quite a lot and tried to minimize it (or making it work for us) by taking the right decisions in the design phase. In order to quantify the effects of compliance, I have swept static toe-values in the simulation and looked at the derivatives of the beta- and delta isolines. The low steer ratio is to keep the steering wheel angle ~90 deg when maneuvering around the tightest hairpins (the cockpit is quite small, experience has taught us that this is quite important). The steering effort is manageable for a fit driver due to the mechanical trail only being 10mm (we have some caster offset), steering self alignment is not an issue.

First suggestion is to increase the ratio. Quite a few YouTubers showing the unmanageable and unruly driver behaviors that result from such
a high gain car (g's per 100 degrees SWA) Your car is a the limit of control at 14 degrees steer and only 60 kph). I did not adjust your roll
stiffness distribution because that's your job AND because it takes quite a bit of change to accomplish much because these tires are very load
sensitive (They don't give up a lot with added load). Next might be to look at tire pressure and wheel rim splits. But, hopefully, you get
the idea and the recommendation(s). Finish the car early and run some ISO validation tests. You may have to explain these details to a few
of the judges because for so many, a stop watch is the only instrumentation they are familiar with.

Good suggestions, thank you.

IMHO, you should also study the effect of roll center heights (i.e. roll axis height because the low numbers increase the spring mass roll inertia.
This can make a car feel lazy, especially if your yaw velocity peak frequency goes above your roll peak frequency (as it will with speed). Any
roll coupling K&C parameters that accompany this convolution will be troublesome for any driver especially if you ever go really fast into a corner.

This would be interesting to do, I will look into it. I have certainly experienced what you mention in a passenger car. The "low" roll center heights is mainly to increase the elastic weight transfer sensitivity to spring/ARB-setup changes.

I include some snapshots of my form of analysis, in just a few minutes and a little more data, you would be done. I've condensed your car's traits
down to front and rear cornering compliance factors just because it's a freebee for the way I do business. Otherwise, you will have to go to
the YawMoment Depot and figure out how you are going to fix this car some other way. Enjoy. Keep in mind that 'optimizing the car' does not
generally make it easier to drive over a 'long' period of time. A knife edge car gets 'dull' very quickly and that makes adapting to the variability a
loosing battle. Any questions? Just ask ! My hockey season is
over but the hay is starting to grow. I'll be at Michigan maybe a day or so. Look me up !

Thanks for helping out! I would look you up if I wasn't in Sweden!