Yaw Rate

[scott_rfr]

I have a question about how I am solving for yaw rate in my simulation. The way I have things set up right now is it's an open loop simulation, input steer angle and forward speed, and the rest is just a simple bicycle model with load transfer. My issue is that I solve for yaw acceleration then I was integrating this to get yaw rate to use for my tire model and slip angle calculation. However I realized this was not going to work as in a steady state condition my yaw acceleration would be zero and integrating zero would just give me zero.

Does anyone have any hints on how I may want to go about this?

Scott
Rutgers SAE

[scott_rfr]

I have a question about how I am solving for yaw rate in my simulation. The way I have things set up right now is it's an open loop simulation, input steer angle and forward speed, and the rest is just a simple bicycle model with load transfer. My issue is that I solve for yaw acceleration then I was integrating this to get yaw rate to use for my tire model and slip angle calculation. However I realized this was not going to work as in a steady state condition my yaw acceleration would be zero and integrating zero would just give me zero.

Does anyone have any hints on how I may want to go about this?

Scott
Rutgers SAE

[BilletB]

Integrating zero in this cause doesn't give you zero. It gives you zero + constant. You said you know your forward speed and steering angle. Calculate your yaw rate. That is your constant.
-B

[exFSAE]

Are you doing a steady state, or transient sim?

If the former there's no need for integration of any sort.

For a given speed, and input steer angle...

Have it iterate on beta (chassis sideslip angle) until it finds a solution where yaw moment = 0. Done.

[scott_rfr]

It's steady state in that I pulled the dampers out for the load transfer calculation. I didn't want to have it iterate on beta to find a solution where Cn=0. One of the reason for this simulation is to evaluate how certain aspects affect the Cn.

Scott

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by scott_rfr:
It's steady state in that I pulled the dampers out for the load transfer calculation. I didn't want to have it iterate on beta to find a solution where Cn=0. One of the reason for this simulation is to evaluate how certain aspects affect the Cn.

Scott </div></BLOCKQUOTE>

Ok.. so is it steady state, or is it transient? What are you trying to evaluate in the end? If you're not including the dampers, might as well just make it a steady state simulation. That will tell you a lot by itself.

Also not sure it's worth doing dynamic stuff if you don't have a transient element to the tire model.

On turn in, initially you will have a non-zero yaw rate acceleration, so you will integrate to a non-zero yaw velocity.

Alternatively you could write your own Milliken Moment Method code to show you a variety of trimmed and non-trimmed (CN != 0) responses as a function of steering and Beta.

I'm honestly not entirely sure how you "solve" for Beta in a transient simulation. Turn in... you've got some unbalanced moment, thus a yaw acceleration, you can start to integrate to have a yaw velocity, but if you integrate from there you get heading angle rather than sideslip.

Somehow you'd have to come up with an expression for Beta_dot I think, probably in the chassis co-ordinate system.

Probably in RCVD but I'd rather have this soup than look it up

[J.R.]

Cn must equal zero to be steady state (MMM)

The model that I made was not MMM, but it took forces from all four tyres and used their F&M to calculate a yaw acceleration, derived this to get yaw rate. Think about your yaw rate, and the distance the axles are from the CG and you should be able to figure out your beta value.

[exFSAE]

Ah, but CN doesn't have to be 0 for MMM http://fsae.com/groupee_common/emoticons/icon_smile.gif

Which is what is nice about it. Captures some transient "zones" while being a steady-state "sim."

[J.R.]

True, but he said he wanted a steady state sim.

[scott_rfr]

Still not sure how to calculate yaw rate. Here is my logic in all of this maybe there is a problem in there.

First step I am going straight, no slip angle or beta. Then I start a sweep of steer angle, take first point at say 1 deg of steer angle, so I now have 1 deg of front slip angle? This creates a yaw moment which gives me a yaw acceleration, yaw rate and beta. Next step I recalculate slip angles based off this yaw rate, beta and new steer angle.

Scott

[Zac]

I think the best way is to do this is a numeric integration. what you need to integrate should be apparent from your differential equations. something along the lines of a simulink block diagram is probably the easiest way to actually get this into your computer.

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by scott_rfr:
Still not sure how to calculate yaw rate. Here is my logic in all of this maybe there is a problem in there.

First step I am going straight, no slip angle or beta. Then I start a sweep of steer angle, take first point at say 1 deg of steer angle, so I now have 1 deg of front slip angle? This creates a yaw moment which gives me a yaw acceleration, yaw rate and beta. Next step I recalculate slip angles based off this yaw rate, beta and new steer angle.

Scott </div></BLOCKQUOTE>

To go around a steady state circle, for a given lateral acceleration and speed, there is only one solution for yaw rate. YawRate * Radius = ForwardVelocity.

Front Slip Angle is a function of Beta, YawRate, and Steering. Rear Slip Angle is a function of Beta and YawRate. They're pretty simple and in RCVD.

A limit sim may not be the best idea since the raw tire data won't work for it, and I don't trust when its fudged. Might be best to...

1. Pick a speed
2. Pick a turn radius (or lateral G level)
3. That will define your load transfer, so you can instantly set the loads on the 4 corners.
4. Let the computer iterate on steering and Beta to find the trimmed, steady-state solution.

From there you can figure out your under- or over-steer (difference in front and rear axle slip angle)

[J.R.]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
From there you can figure out your under- or over-steer (difference in front and rear axle slip angle) </div></BLOCKQUOTE>

Difference in lateral forces produced by the front and rear tracks, which are a f(SA,IA,MZ)

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by J.R.:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
From there you can figure out your under- or over-steer (difference in front and rear axle slip angle) </div></BLOCKQUOTE>

Difference in lateral forces produced by the front and rear tracks, which are a f(SA,IA,MZ) </div></BLOCKQUOTE>

Hm? Not sure I follow. Guess I'll elaborate on my first post...


<LI>Set all your 'macro' vehicle parameters (wheelbase, track, corner loads, TLLTD, roll stiffness, a simple polynomial camber curve, compliances if you want...)
<LI>Pick a speed
<LI>Pick a turn radius (or lateral G level)
<LI>With the above set, this should instantly define your lateral load transfer, corner loads, corner cambers, and yaw rate
<LI>You have two unknowns (steering angle and beta/sideslip angle), and two equations. Sum of forces has to equal your lateral cornering level, and sum of moments = 0 for steady state.
<LI>Write a quick script to iterate on Steer and Beta until the above two constraints are met
<LI>You know have a trimmed, steady state cornering solution


Since this is steady state, the front and rear axles are experiencing the same lateral acceleration. The best bet for evaluating balance at this point is a simple difference in front and rear slip angle. Front slip angle higher = understeer, rear slip angle higher = oversteer.

From here you can tweak your TLLTD, your corner weights, kinematics, etc to get the balance level you want.

You can also have it drop out a term for slip angle drag, and play with different tire sets and vehicle setups to minimize that.

IMO, you will want some understeer in steady state cornering... since front load transfer under braking will tend to "free" the car up and you probably don't want the car to spin out on corner entry.. just be nice and neutral.

[J.R.]

But what if one end has more weight on it than the other? Then a simple slip angle measurement tells you nothing. Or what if your rear axle is solid and doesn't produce wheel camber?

Over/ Under steer is a linear range term. Race cars operate at the limit. Limit Over/ Understeer are not really applicable terms. That being said, I understand what you mean by them. Understeer = plow, Oversteer = spin, Netural = drift. So, its more insightful to look at the moments that the front and rear tracks are causing about the CG. If the front (destabalizing) moment is greater, you will spin, if the rear (stabilizing) is greater you will plow.

Also, the steady state modeling works great for skidpad, or NASCAR turns. This year, we designed for skidpad, since I didn't know how to make a dynamic analysis. This is where you have to make a decision, if your team's at the point where they can do dynamic analysis, definitely go for it, if not design for skidpad and you'll get close enough.

All this being said, if you have the time, go through and derive the dynamic equations yourself, you will learn why most of the general rules of thumb work, which lets you use them to your advantage.

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by J.R.:
But what if one end has more weight on it than the other? Then a simple slip angle measurement tells you nothing. Or what if your rear axle is solid and doesn't produce wheel camber?

Over/ Under steer is a linear range term. Race cars operate at the limit. Limit Over/ Understeer are not really applicable terms. That being said, I understand what you mean by them. Understeer = plow, Oversteer = spin, Netural = drift. So, its more insightful to look at the moments that the front and rear tracks are causing about the CG. If the front (destabalizing) moment is greater, you will spin, if the rear (stabilizing) is greater you will plow.

Also, the steady state modeling works great for skidpad, or NASCAR turns. This year, we designed for skidpad, since I didn't know how to make a dynamic analysis. This is where you have to make a decision, if your team's at the point where they can do dynamic analysis, definitely go for it, if not design for skidpad and you'll get close enough.

All this being said, if you have the time, go through and derive the dynamic equations yourself, you will learn why most of the general rules of thumb work, which lets you use them to your advantage. </div></BLOCKQUOTE>

Hm? You may want to review your vehicle dynamics notes.

More weight on one axle or another is irrelevant once you calculate front and rear axle slip angles. The difference between the two is all that matters. In the above description given that you have to include all the corner loads in your vehicle model, this all takes care of itself! Furthermore if your rear axle is solid and doesn't produce wheel camber, then you have a pretty simple rear camber curve for your sim!

Furthermore, understeer/oversteer is not a purely linear range term. Understeer gradient generally is. Understeer is a quantity that can be evaluated at any combination of lateral and longitudinal acceleration. For free-rolling cornering simulation for example you can plot understeer versus lateral acceleration and see how your balance evolves as you initially start cornering, up through the linear and transitional range, and to the limit where you'll hit an asymptote.

Race cars operate at the limit some of the time. Not all of the time. Most FSAE cars probably operate at the true limit of combined traction a small percentage of the time, as a function both of driver and how courses have become running at speedways. What happens in the linear range does help set up a lot of what happens at the limit of traction. The two are GENERALLY not mutually exclusive. A racecar's steady-state response and balance is the fundamental building block and first step in both design and track tuning. If you don't have that foundation set, you are hosed.

You can take the above simulation a step further though and include longitudinal effects if you've got that available in your tire model, in which case you can start to do a 2-d map of vehicle balance as a function of Lateral G and Longitudinal G. That tells you a LOT for your handling, and is appropriate for much more than "NASCAR turns" ([sic].. most NASCAR corners are fairly dynamic, including brake-in-turn and are traction limited on throttle.. in some ways they are orders of magnitude more dynamic than FSAE)

[J.R.]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
Hm? You may want to review your vehicle dynamics notes.

Furthermore, understeer/oversteer is not a purely linear range term. Understeer gradient generally is. Understeer is a quantity that can be evaluated at any combination of lateral and longitudinal acceleration. For free-rolling cornering simulation for example you can plot understeer versus lateral acceleration and see how your balance evolves as you initially start cornering, up through the linear and transitional range, and to the limit where you'll hit an asymptote. </div></BLOCKQUOTE>

Opening my RCVD textbook to review.... P. 313... 'The concept of under/oversteer is associated with the linear range of tire operation ...', later in the paragraph, 'Lateral tire forces is approximately proportional to slip angle (at constant load)'

So, you are correct, if you don't include load transfer in your model, and your tyres have the same CS F&R, then the slip angle is the only thing that matters.

However, we have load transfers, and tyres are completely non-linear, and objects are moved by forces and moments. Newtons laws.

Fixing this, edits in bold.

Think about this situation, bilinear tyres. You have different tyres front and rear, your front ones with a lesser CS, and your rear with a greater, say 50lb/deg and 100 lb/deg. REAR tyres peak at 200lbf, Front at 300lbf. 50/50 weight distribution, no LLT. So, you go around a corner and turn the wheel.

Steady state cornering, at the limit, your front and rear tyres must balance themselves, producing equal moments, in this case matching lateral forces (given 50/50 weight distribution). The rear axle slip angle is 2deg, and the front is 4 deg. In your definition, this would be understeer. However, the Front tyres can go up to an axle slip angle of 6deg before being saturated, leaving a DESTABILIZING moment available, and thus being an OVERSTEER vehicle.

By the way, this situation is comparable to having 50/50 lltd, and more weight on the nose, for a four wheel vehicle. Perhaps Radials rear, bias front for your CS values, or perhaps operating the tyres near their load limit where cornering stiffness will regress with increased load.

[jrickert]

Wouldn't be the only contradiction in RCVD. I don't really like that book, but havn't found one better. Its full of partially formed and poorly related thoughts.

[Zac]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Steady state cornering, at the limit, your front and rear tyres must balance themselves, producing equal moments, in this case matching lateral forces (given 50/50 weight distribution). The rear axle slip angle is 2deg, and the front is 4deg. In your definition, this would be oversteer. However, the rear tyres can go up to an axle slip angle of 3 deg before being saturated, leaving a restoring moment available, and thus being an understeer vehicle. </div></BLOCKQUOTE>

higher slip angle on the front axle =/= oversteer

[J.R.]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Zac:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Steady state cornering, at the limit, your front and rear tyres must balance themselves, producing equal moments, in this case matching lateral forces (given 50/50 weight distribution). The rear axle slip angle is 2deg, and the front is 4deg. In your definition, this would be oversteer. However, the rear tyres can go up to an axle slip angle of 3 deg before being saturated, leaving a restoring moment available, and thus being an understeer vehicle. </div></BLOCKQUOTE>

higher slip angle on the front axle =/= oversteer </div></BLOCKQUOTE>

Right, sorry, wrote that while eating dinner, see corrected version.

@Cricket, in a book that long, there are bound to be a few mistakes, but I don't think this is one of them, anyone know for sure the SAE definition of Over/understeer? Usually when I find something in there that seems contradictory, its because I didn't understand it the first time through, but that very well could just be me. What do you find contradictory/ poorly related?

[Zac]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Right, sorry, wrote that while eating dinner, see corrected version. </div></BLOCKQUOTE>

I'm still not sure what you're trying to argue. That you need to look at more than slip angle difference?

in your example you're neglecting that once the rear axle force "clips" that the slip angle in the rear is going to increase dramatically. the vehicle you're describing will plow up until the point the rear breaks loose. it both understeers and oversteers depending on what the driver is asking it to do.

[J.R.]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Zac:

I'm still not sure what you're trying to argue. That you need to look at more than slip angle difference? </div></BLOCKQUOTE>

Exactly. You need to look at moment difference.

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">
in your example you're neglecting that once the rear axle force "clips" that the slip angle in the rear is going to increase dramatically. the vehicle you're describing will plow up until the point the rear breaks loose. it both understeers and oversteers depending on what the driver is asking it to do. </div></BLOCKQUOTE>

What exactly is going to cause this sudden increase in slip angle? I don't know if I understand what you mean by clips. Slip angles are related by geometry, and if you have the vehicle operating at a certain condition, unless your beta or delta change, there is no reason for the alpha to change. If I understand you correctly, then your telling me that at steady state cornering, ie situation described above, your going to get a larger rear slip angle without having a yaw moment move the car, or the steering angle alter the front moment, to increase your rear slip angle?

[Zac]

What I'm talking about is that if you slowly and steadily increase the steering input from 0 to full lock (i.e. a ramp input) the vehicle will be understeer at lower levels of lateral acceleration, but once the rear saturates the balance will move to oversteer. in this case the rear slip angle goes to infinity(car spins) since the rear tires can't generate any additional yaw moment to balance the front.

and by "clips" I mean the force saturates

[jrickert]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">
@Cricket, in a book that long, there are bound to be a few mistakes, but I don't think this is one of them, anyone know for sure the SAE definition of Over/understeer? Usually when I find something in there that seems contradictory, its because I didn't understand it the first time through, but that very well could just be me. What do you find contradictory/ poorly related? </div></BLOCKQUOTE>
Its a good book don't get me wrong. When i wrote the comment i had just finished mulling over a few chapters. It frusterates me how it spends lengthy amounts of time on some topics while grazing others that are equally as important. It seems a bit disjointed at times. It talks about particular concepts without trying to relate them most of the time. IE: This is what happens in a steady state corner in a perfect situation but heaven forbid we throw in braking or acceleration. I wish it went more into alternate ways of analyzing lateral/vertical force relationships. IE: It talks at great length about side view geometry in terms using anti coefficients to relate longitudinal forces to vertical forces, but makes almost no mention that the same analysis can be applied to front view geometry.

[exFSAE]

Still don't quite think you're quite getting the point I'm trying to make, JR.

Remember, we're talking steady-state simulation. By definition, the yaw moments (from Fy, Fx, Mz, aero..) are balanced at that point, so looking at yaw moments doesn't tell you anything. They do at the absolute limit of traction, but at that point your simulation isn't steady state, and even then that doesn't really tell you much about the vehicle handling. That only gives you ONE data point, at max lat G's.

What you want to come up with is how the handling feels through the entire range of lateral (and eventually longitudinal) G's. Something like the following (shamelessly borrowed from MRA's site).

http://www.millikenresearch.com/MMM820.gif

http://www.millikenresearch.com/MMM823.gif

Though in the latter I'd plot just "understeer" instead of the gradient.. not quite sure how Doug has that calculated. In any event this shows you how the car balance feels starting in the linear range, through transitional, and you can see pretty clearly what the limit response is from the asymptotes.

Anyway. Understeer [angle] isn't just a binary 1 or 0 attribute you apply to a car. It's a value you can calculate through the whole range of handling.

To your (and Zac's) earlier point, if you use imaginary tires then yes, any kind of linear to limit range handling is possible http://fsae.com/groupee_common/emoticons/icon_smile.gif Generally you will find on real cars, with real tires, with real setups, the kind of behavior in the above MRA plots. Ie, what happens at low G levels sets up what happens at the limit.

[Jersey Tom]

Interesting posts.

As far as I'm aware, (alpha_f - alpha_r) is indeed one perfectly acceptable metric for balance and works equally well for racecars as anything else. I use pretty much that exact thing on my F1000 car (shameless plug.. I'd include the web address again but I'm afraid I'll get autoflagged as a spambot!)

Certainly on a 'real' racecar you'd want a good understand of how the thing is going to react both steady-state and under all sorts of dynamic inputs. Steady-state is certainly a good place to start, and a legitimate, accurate 'skidpad' simulation is probably beyond what a lot of FSAE schools do.

[scott_rfr]

After some advice from some people I finally got a steady state sim running. It a Moment Method Diagram, and I have it linked into my kinematics program as well. Granted it dosent tell you much about the transient but, wow I have picked a lot up. Next step is adding Ax and a combined slip/slip tire model....

Back to some of the arguments that are being made. I dont agree that you have to look at the moment difference, but I also don't agree that you can just look at slip angles. When you have different tires on the front and rear of the car you may or may not need different slip angles to have zero yaw moment.

You really need to look at: delta(alphaf)/delta(Ay_f)=delta(alphar)/delta(Ay_r)

This is in both Milliken and presented at Claude's lecture. It also works out this way in my sim to.

Scott
Rutgers SAE

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by scott_rfr:
You really need to look at: delta(alphaf)/delta(Ay_f)=delta(alphar)/delta(Ay_r)

Scott
Rutgers SAE </div></BLOCKQUOTE>

Certainly. But for steady state, it does simplify a bit.

For all practical purposes, Ay_f = Ay_r for the car to be trimmed (minus very small changes in corner radius). Likewise, assuming you're not running an oval track setup, at 0 Ay your alphaf and alphar should both start at 0. Equation then juts becomes alphaf = alphar (at multiple lateral accelerations)

So in my opinion, it boils down to keeping alphaf and alphar close to each other through your cornering range, though I again believe in a little bit of understeer (alphaf grows a little faster than alphar) to make up for how much the car will free up on the brakes.