Force Based Roll Centre VS Kinematics Roll Centre (The Same Point?!)

[Yunlong Xu]

Hi Guys. I've seen a lot of dicussions of force based roll centre and kinematic roll centre on line with people talking about the difference and pros/cons of each. I've been thiking about these two roll centres a lot. Recently I just realised that I can actually prove that the SAE definition of Roll centre (force based) --- "The point in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll", is exactly the same location as the kinematics roll centre! And now I'm really confused. Does this mean that all those discussions on the difference between these are just pointless or did I just make a ridiculous mistake?

I will upload the photo of the proof on papaer ASAP


Alright, here's the proof that I've got. Please tell me if you find anything wrong.

First let's start with the kinematic roll centre. simple concept and easily found in the figure below

http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/1.jpg

Now consider a vehicle is in roll equilibrium originally(no change in roll angle). Now we need to find a point on the sprung mass where we can put an extra force without causing further roll motion.(by SAE force based roll centre definition)

No roll === No Geometry Change from the original situation

So we can superimpose equilibrium solutions of forces in the system!!!! (equilibrium force solutions 1+ equilibrium force solutions 2 = new force solutions that makes the system stay equilibrium)

Now let's consider:
a new equilibrium solution = equilibrium solution with the extra force on spurng mass - original equilibrium solution
So effectively what we are now considering are the changes of the forces in the system members due to that extra force on the sprung mass that causes no further roll motion from SAE definition. And we want to find the location of that force.

Also, no change in geometry == no change in spring length === no change of force in push/pull rod! (very important!)

Now let's consider a free body diagram of one side of the unsprung mass.

http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/2.jpg

Since there is no force in the push/pull rod in this solution(we are now considering the change of forces!) The unsprung mass is only subjected to three forces T1,T2 and F1. We can find out the directin of F1 just by taking moment about the intersection of T1 and T1 (A-ARM forces). Also the resultant of T1 and T2 must equal in magnitude and opposite in direction to F1 for the unsprung mass to be in equilibrium. The force,F1', on the sprung mass from the unsprung mass must be equal and opposite to the resultant of the A-ARM forces according to Newton's 3rd Law. And hence, F1' = -resultant = F1

Similarly we can find the force from the unsprung mass to the sprung mass on the other side. F2'= F1

Now put all these togeter we get a new Free Body Diagram of the srpung mass. It is also only subjected to three forces F1',F2' and the extra force F. Again for a rigid body with three forces acting on it, the forces must intersect at one point, so we know that F must act through the intersectin of F1' and F2'. And that intersection is the point O, which is the Force Based Roll Centre.
http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/3.jpg

Comparing this with the first diagram, it can be seen that the force based roll centre is the same as the kinematic roll centre.

[Yunlong Xu]

Hi Guys. I've seen a lot of dicussions of force based roll centre and kinematic roll centre on line with people talking about the difference and pros/cons of each. I've been thiking about these two roll centres a lot. Recently I just realised that I can actually prove that the SAE definition of Roll centre (force based) --- "The point in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll", is exactly the same location as the kinematics roll centre! And now I'm really confused. Does this mean that all those discussions on the difference between these are just pointless or did I just make a ridiculous mistake?

I will upload the photo of the proof on papaer ASAP


Alright, here's the proof that I've got. Please tell me if you find anything wrong.

First let's start with the kinematic roll centre. simple concept and easily found in the figure below

http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/1.jpg

Now consider a vehicle is in roll equilibrium originally(no change in roll angle). Now we need to find a point on the sprung mass where we can put an extra force without causing further roll motion.(by SAE force based roll centre definition)

No roll === No Geometry Change from the original situation

So we can superimpose equilibrium solutions of forces in the system!!!! (equilibrium force solutions 1+ equilibrium force solutions 2 = new force solutions that makes the system stay equilibrium)

Now let's consider:
a new equilibrium solution = equilibrium solution with the extra force on spurng mass - original equilibrium solution
So effectively what we are now considering are the changes of the forces in the system members due to that extra force on the sprung mass that causes no further roll motion from SAE definition. And we want to find the location of that force.

Also, no change in geometry == no change in spring length === no change of force in push/pull rod! (very important!)

Now let's consider a free body diagram of one side of the unsprung mass.

http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/2.jpg

Since there is no force in the push/pull rod in this solution(we are now considering the change of forces!) The unsprung mass is only subjected to three forces T1,T2 and F1. We can find out the directin of F1 just by taking moment about the intersection of T1 and T1 (A-ARM forces). Also the resultant of T1 and T2 must equal in magnitude and opposite in direction to F1 for the unsprung mass to be in equilibrium. The force,F1', on the sprung mass from the unsprung mass must be equal and opposite to the resultant of the A-ARM forces according to Newton's 3rd Law. And hence, F1' = -resultant = F1

Similarly we can find the force from the unsprung mass to the sprung mass on the other side. F2'= F1

Now put all these togeter we get a new Free Body Diagram of the srpung mass. It is also only subjected to three forces F1',F2' and the extra force F. Again for a rigid body with three forces acting on it, the forces must intersect at one point, so we know that F must act through the intersectin of F1' and F2'. And that intersection is the point O, which is the Force Based Roll Centre.
http://i863.photobucket.com/albums/ab197/yunlongxu/FSAE%20FORUM/3.jpg

Comparing this with the first diagram, it can be seen that the force based roll centre is the same as the kinematic roll centre.

[exFSAE]

Your picture links are broken.. still broken. I suggest using photobucket.

Also, resize them.

[Pennyman]

I keep asking myself why the roll center is important in the first place...

Why do YOU think it's important?

[Yunlong Xu]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
Your picture links are broken.. still broken. I suggest using photobucket.

Also, resize them. </div></BLOCKQUOTE>

Cheers, will do ASAP.

[Yunlong Xu]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Pennyman:
I keep asking myself why the roll center is important in the first place...

Why do YOU think it's important? </div></BLOCKQUOTE>

To be honest, I don't like the concept of roll centre at all since I don't think they describe what the car is actually doing!

But the roll motion is really important, you need it for quite a LOT of things, such as calculating weight distributions on wheels, camber change, etc... The concept of roll centre (I think) provides a shortcut by its own simplification and approximations and is a good way for looking at roll motions and has been used for many years.

But as I said I don't like it. So in fact I've been trying to workout a way of finding the geometry change of the car, expecially when it's subjected to a lateral force. But to be honest I've had very little success since it's actually quite complicated. So before I have any progress, I guess I'll have to stick to the roll centre model even though I don't trust it too much...

It seems that you know some way of doing it without using the concept of roll centre. Can you give me some hints on how you did it?

many thanks

[BillCobb]

The roll center(s) are important because they define a roll axis who's side view inclination angle is especially important to cars having rear weight bias.

In the profession, a roll center defined by linkage points may be an OK first shot for a suspension layout, but as you get closer to the reality check of first hardware, you can no longer ignore the geometric distortion of compliant members and attachment points. This includes steering gears. Even if you use unobtainium or infinitum materials, the K&C findings will always beat you up. Even on a K&C machine, a kinematic roll center obtained by simulating suspension roll with up and down suspension pad movements will produce disappointing results compared to those produced during a lateral force test. In the lateral force test, side forces applied at the tire patches cause axle, crossmember, track bar (if you have one), wheel bearing and strut bending moments (did I mention tire lateral and vertiacl strains?). Many or most of these are nonlinear, some are bi-nonlinear (is that a new word?). In fact, the best way to produce correct results is to bootstrap the K&C inputs by providing simultaneous FY, MZ and DZ controls and then measuring the steer and camber angles that result.

BTW, with akinematic test, the RC is assumed to be at the centerline of the chassis and the height is calculated from intersections. In a sideforce test, the height and the lateral location are calculated. If you have an asymmetric component set or a Panhard bar, the center of the car ain't where the roll center is laterally.

With the importance of your tire lateral load distribution, having force based reaction points and the simulation to make use of them gets you a car "that's good right off the hauler" as they say. Needless to say, tire MX adds another highly nonlinear input to the overall roll moment distribution. Since these cars are pretty stiff in roll anyway, producing a car that's 'tighty loose' as Terry Satchell would say, is pretty easy. This happens because other managed compliances are introduced to try to offset sideslip stiffness losses as the tire softens. Unfortunately, the tire no longer "listens' to slip angle or camber modifiers at the limit of control. It only hears from forces and moment. The only forces hanging around are FX and FZ, so a Force based suspension management analysis is the only one that can do the job in racing circles. (road courses or ovals, too, actually).

Howzz Zhat?

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by BillCobb:
With the importance of your tire lateral load distribution, having force based reaction points and the simulation to make use of them gets you a car "that's good right off the hauler" as they say. </div></BLOCKQUOTE>

True, though in this series there are so many things most teams don't account for.. even the better ones.. such as floppy frames, floppy wheels, floppy uprights, poor equations of motion.. that anything past the most rudimentary simulation and predictive tools is in most cases overkill and a waste of time.

EVERY computer simulation or roll center calc or tire data plot you make.. is an abstraction of reality. Gotta know how accurate you can hope for and what's worth spending time on.

Also really need to define what you're trying to get OUT of your sim or analysis FIRST, to determine what's important.. and if force-based, kinematic-based, or any roll center approximation is relevant at all.

[Pennyman]

This topic always seems to address compliance as a factor, which I don't deny.

Claude had a good description at his seminar. I probably don't do it justice, but it went something like:

"If you have very stiff spring on the outside, and very soft spring on the inside, wouldn't you agree that the unloaded side of the car is going to lift more than the loaded side is going to compress? So the "force based" roll center is skewed somewhere towards the side with the stiffer spring."

I would think this idea can be extrapolated to motion ratio linearity, tire dynamics, and compliance. Suspension link kinematics in this case only affect instant center movement, which in turn can tell you how much tire force is transmitted to the springs and dampers (elastic WT).

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Pennyman:
This topic always seems to address compliance as a factor, which I don't deny.

Claude had a good description at his seminar. I probably don't do it justice, but it went something like:

"If you have very stiff spring on the outside, and very soft spring on the inside, wouldn't you agree that the unloaded side of the car is going to lift more than the loaded side is going to compress? So the "force based" roll center is skewed somewhere towards the side with the stiffer spring."

I would think this idea can be extrapolated to motion ratio linearity, tire dynamics, and compliance. Suspension link kinematics in this case only affect instant center movement, which in turn can tell you how much tire force is transmitted to the springs and dampers (elastic WT). </div></BLOCKQUOTE>

The compliance hits everything. Compliance in the wheels and upright will mean your slip and inclination angles won't be what you think.. compliance in the frame will mean your roll stiffness split isn't what you think.. and significant manufacturing variation in frame welding and even spring winding won't help either.

It's for that reason that I'm really skeptical of going for anything beyond the basics.

How critical is it then, to use force-based RC's in place of kinematic approximation? How important is it to dive into RC's at all, until you can make sure you're building a rigid car to the real design specs?

[scott_rfr]

Yes compliance does have a big effect on slip angles, inclination angles and just about everything else. We can sit here all day and say this simulation or this design approach is not really worth it because of compliance or this or that. However you do need to start some place and build from there. Yes the simulation program that I wrote is rather basic compared to a full adams model and inputting K&C data, but it is a start in the right direction. Just because I did not account for wheel compliance does it make the outputs useless? No, you just need to take them with a grain of salt. What we have done has proved beneficial on track and in reducing lap times. But by starting somewhere it showed where we needed to go; we finally started testing frame rigidity and input that to the model, now were discussing ways to test wheel compliance if we can't get on a K&C. The point is having a starting point is a good way to get help from experienced professionals in getting the details down.

Ok, enough thread jacking. There is a great paper from Cranfield about force based roll center that was sponsored by OptimumG. Might be worth your time looking into it. Im starting one of our Jr members with that paper and taking our OptimumK outputs and writing a Matlab program to calculate fbrc.

Scott
Rutgers FSAE

[Zac]

I wouldn't say that high level analysis isn't worth anything. Being able to build up a simulation or other analysis tool from first principles is a critical part of being an engineer. But another critical part of being an engineer is following up on the assumptions you made in your analysis. Scott, I think you're on the right track with this.

A simple model will get you a half-decent design direction fairly quickly. Will it correlate 100% to what you see on track? Most likely not, however it does give you a starting point. If we're talking about a lap sim, maybe the next step is to incorporate some global suspension compliances and see what the sensitivity is. How much rear toe compliance can you live with vs. how much you actually have is probably a pretty important question to answer.

But I wouldn't be overly concerned with Wheel Compliance. Yes, the typical wheels used by a lot of FSAE teams are kind of floppy, but I think the total contribution to camber and toe compliance is small compared to the tire and several of the student designed suspension bits. Based on what I've seen during K&C testing and just walking around the paddock, a lot of teams seem to be struggling with overall suspension packaging and upright design in particular.

Then again, it doesn't matter what you do in your analysis and design work if you show up to the track with a junk alignment.

[Yunlong Xu]

Thanks for the replies guys. Sorry that I didn't get to read it in details early (I was away for 10 days after posting this and then found my laptop broken...)

But I don't think anyone has given me the answer to the first question from the post: Is the force based roll centre the same point as the kinematic roll centre? If not, what went wrong in my proof in the first post?

[cmeissen]

The way you determined the Force Based Roll Center is incorrect. The FBRC is determined from vertical force on the tire from the geometric load transfer(load transfer going throught the suspension links) and the lateral force generated by the tire.

For more information I would refer to John Dixon's "Tires, Suspension, and Handling" or his IMechE paper on the subject.

[Yunlong Xu]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by cmeissen:
The way you determined the Force Based Roll Center is incorrect. The FBRC is determined from vertical force on the tire from the geometric load transfer(load transfer going throught the suspension links) and the lateral force generated by the tire.

For more information I would refer to John Dixon's "Tires, Suspension, and Handling" or his IMechE paper on the subject. </div></BLOCKQUOTE>

Thanks cmeissen,

So does this mean the SAE definition of roll centre --"The point in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll", despite having forces in this definition, is in fact NOT the Force Based Roll Centre. (and is the same point as the kinematic roll centre as illustrated in the pictures I uploaded?)

[exFSAE]

Who gives a shit what the SAE definition is.

[Yunlong Xu]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
Who gives a shit what the SAE definition is. </div></BLOCKQUOTE>

Any particular reason for that?

[exFSAE]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Yunlong Xu:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by exFSAE:
Who gives a shit what the SAE definition is. </div></BLOCKQUOTE>

Any particular reason for that? </div></BLOCKQUOTE>

That's actually what I'm trying to ask you. The "roll center" is an imaginary point. Depending on your approach there are a variety of ways to define and approximate it. Why be particularly caught up on how SAE defines it? Gotta figure out what you're trying to do with it first.

Personally, I don't care for the SAE definition. When I think RC's, the word "roll" doesn't come into play at all. I think of it as a way to describe the lateral/vertical force coupling of both tires on an axle, simultaneously.

[Bazanaius]

I don't think a discussion about what the application of the various RC definitions are used for is what is being asked for. If the guy is asking for clarification about what the SAE definition means, tell him - there are reasons for asking other than knocking out a car at the end: genuine interest, historical background, interesting academic discussion... etc. AFAICS this is more of a theoretical discussion about the various definitions and types.

[cmeissen]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Yunlong Xu:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by cmeissen:
The way you determined the Force Based Roll Center is incorrect. The FBRC is determined from vertical force on the tire from the geometric load transfer(load transfer going throught the suspension links) and the lateral force generated by the tire.

For more information I would refer to John Dixon's "Tires, Suspension, and Handling" or his IMechE paper on the subject. </div></BLOCKQUOTE>

Thanks cmeissen,

So does this mean the SAE definition of roll centre --"The point in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll", despite having forces in this definition, is in fact NOT the Force Based Roll Centre. (and is the same point as the kinematic roll centre as illustrated in the pictures I uploaded?) </div></BLOCKQUOTE>

No the definition given by the SAE corresponds to the FBRC. The kinematic roll center concept is based on the Kennedy-Aronhold theorem which only applies to rigid bodies. Thus the kinematic roll center has many assumptions that make it not comply with the SAE definition(i.e it negates the lateral force at the contact patch, assumes the tire is rigid, assumes the tire is pin-fixed to the ground, etc.

Now even though the FBRC agrees with the SAE definition, the car still does not roll about it or the kinematic roll center/axis.

[Yunlong Xu]

Hi Chris, I'm a bit confused now.

As you said, the SAE definition is the FBRC, and hence can't be the same point as the kinematic RC and there was definately something wrong in my derivation. But where did I do wrong? I think I followed strictly what was said on the SAE definition.(ie, I found the point "in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll")

Do you agree with me that when a lateral force is applied to the point I found, it would not cause suspension roll? If not, please state where in the mechanics I've done wrong.

Thanks

[ben]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Yunlong Xu:
Hi Chris, I'm a bit confused now.

As you said, the SAE definition is the FBRC, and hence can't be the same point as the kinematic RC and there was definately something wrong in my derivation. But where did I do wrong? I think I followed strictly what was said on the SAE definition.(ie, I found the point "in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll")

Do you agree with me that when a lateral force is applied to the point I found, it would not cause suspension roll? If not, please state where in the mechanics I've done wrong.

Thanks </div></BLOCKQUOTE>

Chris already explained where the mechanics is wrong:

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">The kinematic roll center concept is based on the Kennedy-Aronhold theorem which only applies to rigid bodies. Thus the kinematic roll center has many assumptions that make it not comply with the SAE definition(i.e it negates the lateral force at the contact patch, assumes the tire is rigid, assumes the tire is pin-fixed to the ground, etc. </div></BLOCKQUOTE>

Ben

[Yunlong Xu]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by ben:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Yunlong Xu:
Hi Chris, I'm a bit confused now.

As you said, the SAE definition is the FBRC, and hence can't be the same point as the kinematic RC and there was definately something wrong in my derivation. But where did I do wrong? I think I followed strictly what was said on the SAE definition.(ie, I found the point "in the transverse vertical plane through any pair of wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll")

Do you agree with me that when a lateral force is applied to the point I found, it would not cause suspension roll? If not, please state where in the mechanics I've done wrong.

Thanks </div></BLOCKQUOTE>

Chris already explained where the mechanics is wrong:

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">The kinematic roll center concept is based on the Kennedy-Aronhold theorem which only applies to rigid bodies. Thus the kinematic roll center has many assumptions that make it not comply with the SAE definition(i.e it negates the lateral force at the contact patch, assumes the tire is rigid, assumes the tire is pin-fixed to the ground, etc. </div></BLOCKQUOTE>

Ben </div></BLOCKQUOTE>

Now I'm more confused...

I didn't use anything related to the kinematics roll centre theory in my derivation of the locatin of th SAE definition of roll centre,(it was purely based on force and moment, no kinematcis was considered) but it just turned out to be the same point as the kinematics roll centre.

So considering what Chris has talked about, I now have the following guesses of what went wrong:

1) The SAE definition doesn't assume a rigid chassis, while mine derivation does.

2) The SAE definition doesn't assume a rigid tyre, while mine derivation does.

Can you confirm if the above guesses are correct?

[Shashi]

I think there is another important factor the definition discounts. The plane that is supposed to have the roll center on it does not always remain transverse with wheel steer due to axle tramp. So I guess it is all a 3D story.

I think I saw this in the OptimumG folder. The FBRC is the intersection of the resultant force acting on the pressure point in the contact patch. Now I think that in a dynamic condition, it is not necessary for the point of intersection of the 2 patches be in the axle plane. Finding that would require knowing exactly what the tire experiences. Could someone point out whether all I said brings me onto the right track?