Vehicle Moment Axes

[Lemon Lime]

Hello,

Introductions :

Marwan Alabassiry , Helwan University , Vehicle Dynamics & Suspension , Cairo , Egypt .


Earlier ; I read about vehicle moment axes , And I was wondering ; What is the moment point where the vehicle chassis rotate about either pitch / yaw ... as for rolling I'm aware of the suspension roll centers and the roll axis ... Because when I read about which subject I found that mostly the point where the vehicle rotate about whether in pitch or yaw is the CoG !!


So my questions simply are :


1) Is it really the CoG where the vehicle rotates about ?! If it is ... Why ?!
By the way , I'm not convinced by such assumption .

2) If I had a vehicle which has a certain percentage of antix ( Anti-dive - Anti-squat ) ... Then I'll be able to derive the SVSA and the side-view instantaneous center .

So I was asking ; Can I derive a center " kinematic pitch center " where the vehicle rotates about in the vertical longitudinal plane of the vehicle just like what we do in the transverse vertical plane to derive the kinematic roll center ?

3) About the yaw ! Will the lateral load transfer be able to effectively produce enough moment to " yaw " the vehicle .. If so ; where is the center where the vehicle will " yaw " about if it isn't the CoG already ?!

4) Last question .. If I had a vehicle model with such configurations :

a- Double wishbone suspension system .
b- In the longitudinal vertical plane ; The two wishbones are parallel to each other and parallel to the ground surface .

... I tried this configuration on a simple old fashion kinematics analysis software called " Wishbone.exe " and the results where that this vehicle has :

* 4.4 % of anti-dive in case of inboard brakes .
* 0.0 % of anti-dive in case of outboard brakes .

The question ; Why does this vehicle model have a 0.0 % anti-dive and why not 100 % anti-dive ?



Sorry about my way of asking ; I really tried hard to understand about which subject but alas " Goolge " couldn't help me through which ! And I thought many times before posting such a discussion , Maybe because I was afraid ; Afraid to get mocked or to being let down by such subject drifting which occurs often when the person asks is a bit off track or not ready enough to tackle one subject .. !!


One last thing ; I stick to the rules of the forums ; I introduced myself , Tried to be polite when asking and I searched about this subject and I found one result which I thought it might help me !! But it didn't and I believe that Pat Clark had a thread discussing the pitch axis and it's effect on load transfer .. But alas again .. Couldn't help much !!

So do me a small favor ; Try to take this forum more serious please specially when posting a reply in this discussion or any other discussion http://fsae.com/groupee_common/emoticons/icon_smile.gif !



Thank you for your precious time and Thanks one more time for your well perceive and understanding .


And Good luck for all the teams at FS Germany .. God be with you all http://fsae.com/groupee_common/emoticons/icon_smile.gif


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[Owen Thomas]

I am by no means an expert, but here goes:

1) Plainly, No for roll and pitch, Yes for yaw.
Roll and pitch motions are kinematically controlled by suspension geometries, yaw motion is not. I believe the key to understanding this is thinking about wheel travel. If you look at the vehicle in front view or side view and apply forces at the tires, you would expect to induce and observe movement up or down of the wheels (relative to the chassis). This is because they are allowed to move, and it is clear that their path is defined by the suspension geometries. Now in the top view, apply the same forces and you will not expect to see any movement (neglecting scrub, compliance, etc.). There is no prescribed movement of the wheels relative to the chassis, and the vehicle acts as a rigid body in this plane. Thus, the movement of the whole body is about the yaw center, which is the center of yaw inertia, which is the center of mass (CoG). Consider that an infinitely stiff kart would roll, pitch, and yaw about the CoG, because it has no kinematic "suspension".

2)You can find a pitch center for each axle (instant center) in side view the same way you do for instant centers in front view. There is a google image result for "pitch center" that shows this quite nicely. I am unsure about finding the vehicle pitch axis, since (unfortunately) I have never had a use for it. I think it's the same as the roll center. Hopefully someone with more know-how can help you here.

3)See 1.

4)See 2. The diagram described shows how to define anti(x). If you have an infinite pitch center, then your line from the contact patch to convergence of your control arm planes is virtually horizontal at ground plane. This is 0% of the height of the CoG, and thus 0% anti.

For the record, it seems like you have the knowledge to figure this stuff out. You might need to just take a step back and look at it again. Try talking to someone about it, even if they know almost nothing on the topic, you will probably have a few lightbulbs go off.

[wil oberlies]

I think the point about which the vehicle yaws is a function of moment of inertia, CG location, and the ratio of tire cornering stiffness (front to rear).
If you can find a graph of SWA, Ay, yaw velocity, and beta (vehicle sideslip angle), look closely at the yaw velocity (psi dot) curve; you might see a brief plateau, while Ay and beta continue to develop. Here, the car is rotating about a point on the rear axle, and this point moves forward. Think too about a vehicle in a spin: the point of rotation is now about a point on the front axle....
You will (obviously, I guess) see a similar 'plateau' in a graph of front and rear lateral force vs time: the front axle curve will have a brief plateau as the rear axle begins to develop lateral force.
I'll see if I can find some graphics that might help.
Wil

[Francis Gagné]

About 3) If you define the instantaneous center of rotation as a point in 2D space where the body rotates around but there is no translation, I will have to disagree with Owen.

If the car yaws around the CoG, then you have some serious oversteer problems! Think of a skidpad, the car will (should) rotate around the center of the skidpad. Thus, the yaw IC will be near the CoG at turn-in when you create an oversteering moment to accelerate the yaw speed.

As for the pitch center IC, you can find it using the same principle as in roll.

[Zac]

I think it's pretty common for people to get mixed up between body fixed coordinates and global coordinates. The axis for roll, pitch, and yaw (and the corresponding translations) are attached to the CG of the body. Don't make this more confusing than it has to be.

[Lemon Lime]

I'll reply to each reply individually ... But before which .. have a look on this pic just clear things up a bit ... Is this picture theoretically right ?!


Thanks one more time

http://s2.postimg.org/a816ywpqx/1081570_10151757187220944_1711816718_n_g.jpg

[Claude Rouelle]

LemonLimne,

Let's say your car is "kinematically" symmetrical. Think just 2D for a while. Your kinematic roll center is in the middle of the car. At least to start with.

But on the left you have 100 N /mm springs and on the right 1000 N/mm spring.

Do you think that with, let's say, one G of lateral acceleration that the car will rotate around the kinematic roll center?

Similarly, if your kinematic pitch center is where your sketch shows (nice software by the way) and you put super stiff spring on the front and super soft on the rear that the car will rotate around that kinematic pitch center under acceleration and braking?

[Claude Rouelle]

Another quick comment about yaw axis.

This is not the answer but a simplified perspective which will make you think.

Some people think the yaw axis is an axis perpendicular to the ground passing though the car CG. Yeah... but the lateral force is applied to the CG, right?

Moment = Force * leverage, correct? So how do you create a yaw moment if you have a force but no leverage. No yaw moment to enter in a corner.... Or Exit.

Food for thoughts....

[Claude Rouelle]

LemonLIme,

More questions and only a few answers. Welcome to the club!

You thought Vehicle Dynamics and Race Car engineering were going to be easy?

If so think about fishing...

And yet....

[Lemon Lime]

Claude Rouelle,

Allow me to start from downwards ...


1) No , I don't think that vehicle dynamics studying would be easy at all , I've been struggling through all my way ... But lets say I'm trying to wake up just to make my dreams come true .. Or that is what I hope !

And by the way ; Great to mention fishing http://fsae.com/groupee_common/emoticons/icon_smile.gif Because I think Fishing and vehicle dynamics have something in common .. It's all about being patient enough till you reach your goals http://fsae.com/groupee_common/emoticons/icon_wink.gif


2) About yaw moment ; If I started by dividing the total lateral force which impacts right at the CoG among the four tires then started working ; I'll find my self ending up with a moment !! I'm not sure of course about which !!


3) Aha ! About 100/1000 stiffness idea ; That was really helpful ... My thoughts ; No , Of course there would be a certain ratio by which the kinematics roll center will be shifted according to the same ratio of the springs stiffness .

And by the way ; This isn't OptimumK ... This was Karam's HFS'13 project analysis made using Susprog3D ... Of course it would be an honor if I can show you mine when I finish it in a couple of weeks .



Earlier this day I was discussing this with Karam " I hope he join in here soon " He himself has a lot to ask about ... Anyway ; Eventually ; I came up with another question ... ** See the picture ** If you can imagine the wishbones from the plane view you'll be able to see a V-shaped diagram with an included angle ... Usually we put the rear one with a larger included ... If we have the L distance larger than the M distance as shown in the picture attached so ; If I am right about my assumption ; This might affect the pitch axis as well and shift it towards the front ?!


http://s16.postimg.org/lk978l0jp/Explain.png


Sorry about your time ... I can see you were typing really fast this time on the keyboard http://fsae.com/groupee_common/emoticons/icon_wink.gif , And I know you're busy about 24 hrs SPA .. By the way Good luck .. I've seen the Z4 pics today on OptimumG facebook page .. They're just gorgeous .. Amazing to behold http://fsae.com/groupee_common/emoticons/icon_smile.gif .. Good luck http://fsae.com/groupee_common/emoticons/icon_smile.gif


Again many many thanks about everything .. And thanks for your time and your reply http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Z]

Marwan,

Firstly, your questions are quite valid, and it is good that you can ask them more clearly than many native English speakers (eg. "vertical longitudinal plane" is a clear, precise way of explaining yourself ...).

Secondly, and more sadly, you are unlikely to find good answers to your questions in the automotive "Vehicle Dynamics" world. I partly explain why this is so in this post regarding the Many Definitions of RCs (http://www.fsae.com/forums/showthread.php?4390-Roll-Center-Migration&p=16320&viewfull=1#post16320).

IMO your best way forward is to simply consider the problem as one of 3-dimensional Mechanics. So a few rigid, massive bodies (1 x big-heavy-ground-body + 4 x wheel-bodies + 1 x car-body) that act on each other via contact forces, and also interact with inertial space when they are accelerated relative to it. The "Jacking Force" (http://www.fsae.com/forums/showthread.php?4063-Jacking-force) thread gives a (simplified 2-D) example of how to do this. Also on that thread is a link to (retired) Professor J. Tatum's "Classical Mechanics" (http://orca.phys.uvic.ca/%7Etatum/classmechs.html) webpages that are probably more useful than anything you will find in the automotive literature (eg. Owen gets it wrong above, then Francis corrects him, but then also gets his second last sentence wrong...).

Much more to say, but maybe later...

Z

(PS. The blue-lines in your first image are the "vertical longitudinal plane" wheelprint n-lines for the motion of the wheel-uprights wrt the car-body. Thus these can be used to determine the LoAs of forces acting between ground<->wheelprint<->body, through the suspension control arms. BUT!, this only applies for outboard-drive/braking. So with inboard-drive/brakes (where the driveshafts are now part of the suspension linkage) the wheelprint n-lines may have different slopes.)

[Karam Atteia]

http://s17.postimg.org/akl8p3ym7/Untitledsde.png (http://postimg.org/image/m9p8d2pkr/full/)
screen grab (http://postimage.org/app.php)


First of all i would thank Marwan for posting about this subject the Moment Axes, where i really have lots of questions.

Second .. I'll try to collect some info.
About the rotation motions of the vehicle, it is around the Roll center in the transverse front view as a roll motion. and around the pitch center in the longitudinal side view. while looking at the top view of the vehicle, around the yaw axis perpendicular to vehicle through the CoG. But, all these info are about kinematics only !!

Back to Mr. Claude's questions about the situation of a vehicle that have non-symmetrical springs. Here i posted a photo that may illustrates my thoughts (Excuse me, if any fatal mistakes)

first i'll think about the anti-roll torque from the right (1000 N/mm) spring and the left (100 N/mm) spring. So, there is a roll torque produced from the Lateral Acceleration around the roll center. This Roll torque will compress the outer 100 N/mm spring and extend the 1000 N/mm spring, but not with the same displacement. Where the outer is compressed a small distance and the inner one extends a larger distance.

I think here we are having a different situation, Kinematically the Roll center migrates to this point when the Chassis rolls. But,here due to the different left/right springs stiffness, As Marwan Says the new center of rotation is shifted towards the stiffer spring. But where is this Roll center?? then these is my thoughts, I think the center of rotation will starts to move towards the stiffer right spring some where in the orange circle in the photo and by increasing the Lateral acceleration at an instant the chassis will starts to take the big orange point on the right spring as a fulcrum to rotate around. where the left inner wheel could be lifted above the ground at this instant taking in consideration the lateral load transfer.

Same for the pitch rotation motion in the side view, having the situation of super stiff/soft front and rear springs. If having an aggressive braking situation, the chassis rotates around a center near the stiffer spring. and at an instant we would have the same situation of the new Fulcrum at the stiffer spring.

.................................................. .................................................

About the Yaw moment produce at the Yaw axis of the vehicle, I think this Yaw moment is produce due to having an uneven weight distribution front/rear and unequal Lateral forces generated by the front and rear axles. M=F*d >> Moment=Lateral Force at the axle*distance from the Cog to axle .. then if having different lateral forces front and rear and uneven weight distribution .. then we will last with a Yaw moment around the Yaw axis through the Cog


Karam

[Claude Rouelle]

Karam, LemonLime,

You are getting more and more in depth and you ask the right questions.

You will have to solve a system of differential equations which include your kinematics, your masses, your inertia, suspension springs and ARB stiffness, suspension damping and tire stiffness (and maybe damping). And still that is simplified with no compliance.

Putting these equations on paper is the first thing to do.

Can you do that?

[Karam Atteia]

Mr. Claude,

Thank you again for your attention and help.

I know you are too busy, but when you are free please give us more info or hints to work on. May be hints about the starting point only, on the subject of solving a system of differential equations including kinematics, masses, inertia, damping.

we'll do our best, Sir.

Karam

[Owen Thomas]

I thought about it some more, and realized I was most certainly wrong regarding the Yaw moment.

In the vehicle axis, the whole car should yaw about the Neutral Steer Point and not the CoG.

Karam, a good place to start is to check out RCVD chapter 5, if you haven't already.

[Lemon Lime]

Claude,

Karam,



Thank you Karam for joining .


After thinking again ... I discovered that I myself have a mistake which I should be working on ... Which is :

The difference between the kinematic roll center and the force-based roll center .



In fact to answer Claude's question in a more precise way ; First let me put it this way :


1) In the transverse vertical plane ( AKA : Front view geometry ) we have symmetric suspension layout " geometry " but an asymmetric springs stiffness .. Which must direct one's attention back to the definition of the force-based roll center .

2) The kinematic roll center will remain unchanged as long as the geometry is symmetric , Unlike the force-based roll center ... Here is a picture to demonstrate one of my thoughts which might be wrong as well !



http://s16.postimg.org/77s45vyjp/image.png



But here it goes ; In the picture you can see that the KRC " Kinematic Roll Center " will remain unchanged even if we " played " with the geometry a bit .. regarding the lengths not the angles ... But for sure the lateral forces distribution among the tire contact patches will certainly be affected .

So what I've concluded for now ; Neither the KRC nor the KRC migration are my number one priorities , And I'll do care more for the force based roll centers .. At least one someone proves it opposite ! " Just a share of thoughts "



And I do believe that we can do pretty much the same regarding the KPC " Kinematic Pitch Center " ... *** See the first picture posted *** , We can get the pitch center geometrically but what do you want to do with it is the question !! Till now I'll do nothing with the kinematic pitch center ... I might benefit from the force-based pitch center in load transfer and springs displacement differences from rear to front according to the applied direction and magnitude of the load transfer .



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**Conclusion ** :

Know why , Before knowing how !



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Again thanks for all who replied and cared and thanks for your time .


Good luck to all of you !

[Claude Rouelle]

Owen,

There will be one moment when the "yaw axis" goes through the car CG: roughly at the apex of the corner when you are for a little moment in steady state and the yaw moment is 0.

On a larger perspective,.....In fact there is only one complete car (or suspended mass) 3D instant axis combining yaw, roll and pitch. But a step by step top view, front view and side view help to later acquire the 3D view concept.

Simple than complicated, not the other way around.

[Claude Rouelle]

LemonLime,

Let me help you to go further about "real" roll centers; it is not only about kinematics and stiffness......

Imagine (still in 2 D) you have a perfectly symmetrical kinematics and suspension a tire with a Muy of 0.25 on the left and 1.75 on the right. Do you think you will have the same lateral and vertical force base roll centers lateral and longitudinal movements with 1 G of lateral acceleration in left hand an din right hand corner?

At OptimumG we use 3 different roll and pitch centers (and axis) definitions
- kinematics
- force based
= "FMA" = Force Motion Axis

Each one is closer of the reality but again it is till far way from the real world: the main reason being compliance which is absent on this simulations. Then there is the driver who has even more unknown "tuning" bottom

These are just comparative reference which helps us to "walk less in the dark"

But at least you go on the track better knowing what you will be testing. There is a time where you stop the intellectual madness and and you declare; "enough of calculations let's go testing"

*******

As far as "knowing why before knowing you" I can't agree more. That is where you see the great race engineers.

At OptimumG we have 3 phrases
- If you don't know why you win, you won't know why you lose
- Win. Know why. Win again.
- Knowledge shared is knowledge squared. That is how we successfully work with our consulting customers.

[Lemon Lime]

Claude,


Thanks for your reply and for your care again .



But excuse me Sir ! I didn't quite understand the " Muy " !? " Third line from top " .. Surely I'll answer your question then .



About solving a system of differential calculations ; I think it is not a difficult job to do ... The problem is why would I want to ! If it is really necessary ; Sure I'll do it .. And I'll put all my thoughts and my efforts in which !



To know whether it's necessary or not I think that this will take such a period ...



For now ; A deep breath and a step backward !

Then starting all over again !



By the way ; I really like this one : Knowledge shared is knowledge squared



Great motto ; No question a great company such as OptimumG and a great engineer just like you will do that http://fsae.com/groupee_common/emoticons/icon_smile.gif



Again ; Many many thanks Claude ... And good luck at 24 hrs. SPA http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Karam Atteia]

Owen,

Do you mean the neutral steer point, the geometric and instantaneous turning center (the intersection of the lines perpendicular to the wheel path). No Offense but Sorry, I don't think so. because lets see if we have a lateral force at the front axle then where should we multiply this lateral force to get the Yaw moment, and if the lateral forces are also perpendicular to the same wheel path .. then M=F*(0)= 0 .

I still convinced with what i wrote above and what you said at the beginning of the discussion. where there would be a car entering a corner really it will rotate around the geometric turning center but the car will still have a yaw motion due to due to having an uneven weight distribution front/rear and unequal Lateral forces generated by the front and rear axles. M=F*d >> Moment=Lateral Force at the axle*distance from the Cog to axle .

and about chapter 5 RCVD. i don't think thats what Claude's mean. As it is all about steady state response, bicycle model, no transients, no kinematics, nor damping.

Karam

[Claude Rouelle]

LemonLimne,

Mu Y = y for lateral and Mu for coefficient of friction

[Claude Rouelle]

LemeonLime,

This weekend I am in paddock where I know at least 2 dozens engineers who are better than me.

The difference is that rarely do share... so they don't square.

http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Lemon Lime]

Claude Rouelle ,



About the question ; No !! I don't think it would make any differences .. Or that's what I can think of right now according to my limited knowledge about which subject !



And you still are the best in my eyes Sir ! http://fsae.com/groupee_common/emoticons/icon_smile.gif



Thanks http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Lemon Lime]

Claude Rouelle,



No wait !! Second thoughts !



Yes it would affect the force-based roll center ...


If the left tire would have a smaller value ( MuY ) and the vehicle is experiencing a left corner then the force-based roll center will be shifted further from the left tire towards the right one ... Or at least that was what I've been able to conclude !


Anyway ; The force-based roll center will certainly shift its lateral position .. But about it's vertical position ; I still have no clue about which .. Sure I'll answer when I organize my thoughts !



Many thanks Claude http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Owen Thomas]

Originally posted by Karam Atteia:
Owen,

Do you mean the neutral steer point, the geometric and instantaneous turning center (the intersection of the lines perpendicular to the wheel path). No Offense but Sorry, I don't think so. because lets see if we have a lateral force at the front axle then where should we multiply this lateral force to get the Yaw moment, and if the lateral forces are also perpendicular to the same wheel path .. then M=F*(0)= 0 .

I still convinced with what i wrote above and what you said at the beginning of the discussion. where there would be a car entering a corner really it will rotate around the geometric turning center but the car will still have a yaw motion due to due to having an uneven weight distribution front/rear and unequal Lateral forces generated by the front and rear axles. M=F*d >> Moment=Lateral Force at the axle*distance from the Cog to axle .

and about chapter 5 RCVD. i don't think thats what Claude's mean. As it is all about steady state response, bicycle model, no transients, no kinematics, nor damping.

Karam
I do not mean the turning center. The neutral steer point is defined in RCVD as the point (in top view) at which a lateral force can be applied and no yaw moment is put on the car. The location of this is related to cornering stiffness. By this definition, and as explained in Chapter 5, the neutral steer point is located at the CG for a neutral steer car. As Claude mentioned, this is for steady state.

The reason I mention chapter 5 is that it is a good place to start - in the world of steady state. The equations laid out involve yaw control, damping, and stability, as well as other things. If you can wrap your head around this stuff it will put you in a good place to begin understanding transients. In theory, anyways. I am still stuck in the laymans world of steady state http://fsae.com/groupee_common/emoticons/icon_razz.gif

[Buckingham]

1) Is it really the CoG where the vehicle rotates about ?! If it is ... Why ?!
By the way , I'm not convinced by such assumption .

You are right to be skeptical!

All motion can be defined as a combination of translation and rotation about any point. We CHOOSE to do the math about a certain point (cg) because sometimes it is makes math easier for us. Sometimes doing so makes us dumber because we accept regurgitated 'practice' as laws of physics.

We can choose to define points on a graph in either cartesian or polar coordinates. On the same token, you could claim that are car driving straight down a highway is not translating at all, rather it is rotating about a point an infinite distance away.

As long as you use a consistent coordinate system, your math will be correct.

Most people use a coordinate system with the origin at the CG, because it is 'intuitive' ... not because it is a law of physics.

End Rant

[Z]

Well, THANK YOU Buckingham for getting it (almost) right!

Certainly a lot better than most of the other codswallop on this thread so far!!!

(BTW. "All motion can be defined as a combination of translation and rotation about <STRIKE>any point</STRIKE> a particular line in 3-D space (= the Motion Screw, or ISA).")
~~~~~~~~~~~~~~~o0o~~~~~~~~~~~~~~~~


Originally posted by Claude Rouelle:
You will have to solve a system of differential equations which include your kinematics, your masses, your inertia, suspension springs and ARB stiffness, suspension damping and tire stiffness (and maybe damping)...

Putting these equations on paper is the first thing to do.

Absolute nonsense!!!

Claude,

Why are you complicating it first, before making it simple?

This notion that all questions will magically be answered when you have the right "equation" (no doubt cut-and-pasted off the web), is why so many students and too many of their teachers (!!!) have NO REAL UNDERSTANDING of Vehicle Dynamics.

Idiocracy, here we come!!!
~~~~~~~~~~~~~~~~o0o~~~~~~~~~~~~~~~~

Marwan (Lemon Lime) and anyone else interested in VD,

You have used the terms "Kinematic Roll Centre", "Force Based RC", and talked about motions of the car in "Roll, Pitch, and Yaw". But it is clear that you DO NOT UNDERSTAND the meaning of these terms. That is why I gave the link (on page 1) to another post on the "Many Definitions of RCs" (http://www.fsae.com/forums/showthread.php?4390-Roll-Center-Migration&p=16320&viewfull=1#post16320). Ie., to stress that you must CLEARLY DEFINE a term before you have any hope of understanding it. Only then will you have a chance to usefully use it.

So, please answer these questions:

Q1. What is your definition of the "Kinematic Roll Centre"?
Give a complete and concise answer that allows anyone using your definition to always find the same KRC.

Q2. What are your definitions of the Kinematic Pitch Centre, and the Kinematic Yaw Centre?

Q3. Do your above definitions apply to both 2-D and 3-D problems?

Q4. What are your definitions of "Force Based" Roll, Pitch, and Yaw Centres?
Explain why these are different to the Kinematic R/P/Y centres. In fact, are they? Why???

Q5. What are your definitions of Roll, Pitch, and Yaw Motions?
Make very clear what your reference frame is in these definitions!!!!!

Q6. Do these apply to both 2-D and 3-D problems?
~~~o0o~~~

For bonus points, answer the following...

Imagine a car negotiating a right hand bend. Not in hot and sandy Egypt, but in the cold and icy North. The car oversteers and spins off the road and onto a large frozen lake. There the car's CG keeps sliding in a straight line, at a steady 10m/s directly North, while the whole car keeps spinning (yawing) at a steady 1 rev/s, clockwise in plan-view. (Think recent James Bond movie...)

There are three points painted onto the car's centreline.
Point "A" is 1 metre in front of the CG.
Point "B" is 1.6m in front of CG.
Point "C" is 3m in front of CG (the car has a long nose).

Q7. At any instant, where is the "Instant Centre" for this essentially horizontal, planar motion of the car's body, WITH RESPECT TO (= wrt) the frozen lake?

Q8. Describe the paths of points A, B, and C, wrt the car body.

Q9. Describe the paths of points A, B, and C, wrt the frozen lake (draw and post these here, if possible).

Q10. What is slightly special about the path of point B, wrt the frozen lake?

Q11. What would be the EXACT position of point B (on car centreline) to make this path EXACTLY special?

For double bonus points:
Q12. What are the fancy sounding mathematical names for the paths of points A, B, and C (wrt frozen lake)?

For triple bonus points, and assuming the frozen lake is an "inertial reference frame", and neglecting aero forces:
Q13. If your body has a mass of 100kg, then what are the major forces acting on your body when you sit at points A, B, or C???
Give magnitude and direction at any instant.

For quadruple bonus points:
Q14. Is it possible to determine all the forces acting on your body in the above question, if you know the exact position of the IC for the car's (and hence also your) motion wrt frozen lake, but you only know this IC position AT THE INSTANT?

For absolutely NO points at all:
Q15. Do you really have to "solve a system of ODEs" before you can answer the above questions?
~~~~~o0o~~~~~

So, Lemon Lime, if you can answer the above, then you should be able to unlock a few locks. If not, then still a lot of keys to find...

Z

(PS. Why did you call this thread "Vehicle Moment Axes"? What is your definition of a "Moment Axis"?? Does this have any connection at all to the age-old subject of Mechanics???)

[Lemon Lime]

Z,



Firstly ; Warm greetings and afterward ...



Thank you for your care and your reply .



I'm just a 19 years old "boy" just messing around with some basics , But if you would allow me I want to tell you Sir something ... A simple tiny thing ... Maybe I would be wrong about it no doubt ... But listening won't be a bad idea either .. right ?! http://fsae.com/groupee_common/emoticons/icon_smile.gif



P.S : I don't mean by my words that I'm lecturing anyone !! Of course no ... Who am I to do so !! .. The following part of the reply may sound so .. But honestly ; I didn't mean it would be so !



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Alright , If you allow me ...



Firstly :
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There are two sorts of positive discussions :



1) There is a discussion which removes a negative idea from the persons involved in the discussion .

2) There is another sort of discussion which adds a positive idea to the persons involved in the discussion .



Both are great ; But personally ; I would rather prefer the first sort of discussions ... You know why Sir ?



Because if the base is sabotaged be too many bad ideas and false basis ; Then by all means don't try to build over which .. Instead I would repair my bad ideas and thoughts , Then I would start gaining positive ideas .



But you seem Sir that you've chosen to follow the second sort of discussion ... Not a bad decision ... But as I said ; I do prefer the first one .. And I think most students prefer the first one as well .



Secondly :
------------------



About being complicated ; Complicity is a matter of perspective ... I might see that solving a system of differential equations is easy .. You Sir might think the opposite ... Not difficult for you Sir ... But you may see it's difficult for a student at my age ... And I respect such a point of view ! ... But your postulations Sir might be wrong ! ... It is a difficult choice I can't deny ... But still a right choice .


Life is about two choices ... The easy choice and the right choice ... Of course we all heard about that .. But allow me to add something to this ... The right choice is always the most difficult choice among the other difficult choices that life offer to us !



P.S : I'm terribly sorry again if the above lines sounded like lecturing ... Frankly ; I didn't mean it to be so .



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About the terms which I've used ... You are right Sir ... I'm definitely would want to understand more and more about which subject ... Which is why I came here ... To find answers to my questions that may illuminate me in a way or another .



About the questions Sir ; I'll answer them as soon as possible ... I guess right now I'll take a deep breath and start from scratch again trying to remove some bad ideas first then building the good ones .



I called the discussion " Vehicle Moment Axes " ---> Yaw axis , Pitch axis and Roll axis ... I wanted to know their exact positions .



Eventually ; Thank you Sir for you contribution and thank you for your well perceive ! http://fsae.com/groupee_common/emoticons/icon_smile.gif

[Z]

Marwan,

I am not sure why you think I am only adding "positive ideas" to the discussion, rather than removing your "negative ideas".

What I am certain of is that the "basis" of your thinking is a deep swamp. Until you drain that swamp (ie. get rid of the rubbish cluttering your mind) you will never have any hope of understanding your problem.

To build a solid foundation (= "base") for your thinking you must start with clear and precise definitions of the concepts you wish to use. Ie., you must give good answers to the first six questions I asked above.

If you do not do this, then your thinking will forever keep sinking back into the swamp that is your current "basis".
~o0o~

The bonus point questions I asked above are a very simple example that might help you make some progress in understanding Vehicle Dynamics.

The answer to the quadruple point question (Q14) is "NO!". Namely, knowing the position of a Yaw Instant Centre, but only AT THE INSTANT, is of (almost) no help whatsoever in understanding the Dynamics of the problem (ie. no help in calculating the dynamic forces on the massive and accelerating bodies). Simply looking at the paths-of-motion of the various points on the car body (CG, A, B, C...) , relative to the position of the Yaw IC, should make this obvious.

So, finding the "Roll, Pitch, or Yaw Axes" (however you choose to define these), is of little help when solving VD problems.

Rather, if you want to solve these VD problems, then I strongly suggest you study the age-old subject of Classical Mechanics first. Note that this is rarely taught in the automotive literature, so you will have to go outside of this field. It is up to you whether you want to take the "easy choice", or the "right choice".
~o0o~

Finally, the term "moment" has many different uses in the technical fields. For example, it can refer to a "moment of area", or "moment of mass", or "moment of motion", or "moment of force", or "moment of momentum", or generally to the "cross product of any two vectors".

So, again, what is the meaning of your thread's name, "Vehicle Moment Axes"?

This whole thread is quite useless so far, because some posts are about motions about some (unspecified?) axis, others are about forces about some (different??) axis, and others are about ... who knows??? http://fsae.com/groupee_common/emoticons/icon_confused.gif

Z

[Lemon Lime]

Z,


Mr.Erik ... Sorry for being very too late ... But I didn't forget about the questions by the way.


Before trying to answer those questions I was trying to get my hands on knowledge first ... I read some pdfs and threads on this forum and other forums as well ... and look what I've found http://www.google.fr/patents/US6702265 ... Well I didn't understand a thing ... But I think that this was too complicated for me .. Anyway the answers !


1- The kinematic roll center in the vertical transversal plane is the intersection of the line joining between the instant center point and the intersection point of the centerline of the contact patch with the centerline of the vehicle's chassis.

2- I have no clue on yaw center ! About pitch center; I would say it is the same explanation as the KRC above but in the vertical longitudinal plane.

3- Yes ... Every line to plane and every point to line.

4- No idea .. I was hoping anyone could help me on that ... I read on other forums that the force based definition of roll center was published on one of the SAE papers ... Which I don't have ! Also I read that the KRC lateral migration is not considered a thing at all by SAE definitions.

5- Ok .. I don't know how exactly to define them scientifically ! I'll try; In 3-D if we have 3 axes X, Y and Z each axis is perpendicular to the other, And if we have a moment about each axis we can name them accordingly. Or, More simply; If we have a small kite with two ropes attached at both ends of the kite ... while airborne pull one rope and release the other to make some pitch motion ... Roll is when a storm comes and make the kite sideways ... Yaw is putting the kite on a Air-hockey table and pull one rope and release the other to make a yaw motion.

6- I guess so !

7- The CoG I guess !

8- Circular !

9- Spiral !

10- mmmm ... I thought maybe point "A" is a bit special .. Because it's 1 meter away from CoG and the Car rotates about it self at 1 rev/sec and has a constant velocity of 10 m/sec.

11- At "A" 1 meter away from the CoG !!

12- A: Parabola ... B: Spiral ... C: Elliptical !

13- The further I sit the more I would feel the lateral forces and the more I feel "dizzy" !

15- Well ... Practically I answered none ... None was correct ... And I understood nothing at all ... But I'm still hopeful ... I wish I can get 25% correct so I can be a politician ! http://www.fsae.com/forums/images/icons/icon6.png


54

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The questions are really wonderfull ... I wish I could really answer them ... I tried too look for a software that enables the user to generate a motion on a certain body (drawing) so I can understand the whole thing better ... But I couldn't find any !


Also I removed that locks and keys quote that I made myself ... Because I found myself a lock of seven HUNDRED keys !!


Cross product of any two vectors ... mainly force vectors !


You're right ... Quiet useless thread ...


Anyway; Can you Mr.Erik recommend me a book to help me understand those basic stuff ... I already took them in school and college ... But you Sir no nothing of the Egyptian education ... Total failure and a mess ... I can't actually believe that those who teach me do have a Phd !! Doesn't make any sense !


Many thanks for contributing in this thread and many thanks for trying to help me !


Wish you a great day !

[Z]

Marwan,

Z - "Q1. What is your definition of the "Kinematic Roll Centre"?
Give a complete and concise answer that would allow anyone using your definition to always find the same KRC."

M - "1- The kinematic roll center in the vertical transversal plane is the intersection of the line joining between the instant center point and the intersection point of the centerline of the contact patch with the centerline of the vehicle's chassis."

Firstly, note that anyone can define their personal KRC in any way they want. BUT!!! it is only a worthwhile definition if it is both unambiguous (anyone and everyone will always find the same "centre"), and it is also useful (see below!).

If, in your "end-view" (= "vertical transversal plane") the suspension is asymmetric, then there will be TWO points on the "[vertical] centreline of the vehicle's chassis" (ie. the 2 x lines from the 2 x ICs to their respective wheelprints will intersect the centreline at two different heights). So this definition does not define ONE centre, but rather TWO centres!!! Not useful...

Do not be upset by this because the SAE definition of RC is no better. It only defines a "height", and rather ambiguously at that (ie. there is no clear specification of lateral force distribution between the two road-to-wheel forces). Also, a few pages ago Claude wrote,
"At OptimumG we use 3 different roll and pitch centers (and axis) definitions
- kinematics
- force based
= "FMA" = Force Motion Axis"
I have yet to see any clear definition of these either. Especially "Force Motion Axis"!!! Huh??? Claude???

A traditional definition of the KRC might be (although I have never seen it explicitly spelled out this way),
"For the two independent suspension linkages at the given end of the car, the Kinematic Roll Centre is the intersection point of the two kinematic n-lines belonging to the two respective wheelprint centres, with those n-lines lying in the vertical plane containing the two wheelprints. See Figure ..."

Of course, this definition relies on an earlier definition of "Kinematic N-Lines". These are (briefly here) the lines of "No motion", or lines "Normal to the path of motion" of any point on any link of a kinematic linkage, wrt a given reference frame (typically another link, say, "the frame"). Making definitions based on earlier, more fundamental, definitions, is standard practice. Read Euclid!

In the common 2-D suspension design textbooks, the line drawn from "wheelprint centre" to "Instant Centre" is an n-line. The intersection point of these two n-lines, in end-view, is the KRC. It does not have to be, and rarely is, on the car vertical centreline.
~~~o0o~~~

Z - "Q2. What are your definitions of the Kinematic Pitch Centre, and the Kinematic Yaw Centre?"

M - "2- I have no clue on yaw center ! About pitch center; I would say it is the same explanation as the KRC above but in the vertical longitudinal plane."

Yes. For KPC similar principles to KRC apply, but in side-view. But note importantly that accelerating and braking n-lines are not always the same (in fact, usually quite different).

I would say that any definition of "KYC" would be meaningless. Of course, I am sure that many people will happily talk about KYCs, invariably without giving any sort of definition!!!
~~~o0o~~~

Z - "Q3. Do your above definitions apply to both 2-D and 3-D problems?"

M - "3- Yes ... Every line to plane and every point to line."

??? Anyway, any 3-D kinematic linkage has lots of n-lines passing through 3-D space, and any 2-D plane taken out of this has some of the n-lines in it, so the concept works well in both 2-D and 3-D (provided you use it correctly!).
~~~o0o~~~

Z - "Q4. What are your definitions of "Force Based" Roll, Pitch, and Yaw Centres?
Explain why these are different to the Kinematic R/P/Y centres. In fact, are they? Why???"

M - "4- No idea .. I was hoping anyone could help me on that ... I read on other forums that the force based definition of roll center was published on one of the SAE papers ... Which I don't have ! Also I read that the KRC lateral migration is not considered a thing at all by SAE definitions."

I have never seen any of these "Force Based..." definitions either (other than nonsensical "circular" definitions like "they are the centres based on the forces...")!

Note that any linkage can only carry (or resist) a force acting on the linkage when that force's Line-of-Action is along an n-line of the linkage (obvious, really). So suggesting that there is another "FB" centre is quite pointless...
~~~o0o~~~

Z - Q5. What are your definitions of Roll, Pitch, and Yaw Motions?
Make very clear what your reference frame is in these definitions!!!!!

M - "5- Ok .. I don't know how exactly to define them scientifically ! I'll try; In 3-D if we have 3 axes X, Y and Z each axis is perpendicular to the other, And if we have a moment about each axis we can name them accordingly. Or, More simply; If we have a small kite with two ropes attached at both ends of the kite ... while airborne pull one rope and release the other to make some pitch motion ... Roll is when a storm comes and make the kite sideways ... Yaw is putting the kite on a Air-hockey table and pull one rope and release the other to make a yaw motion."

Firstly, "Kinematics", roughly speaking, is one-third of the larger subject of "Classical Mechanics" (the other two-thirds being "Statics" and "Dynamics"). Kinematics is about the idealised geometrical MOTIONS of perfectly rigid, frictionless, massless, forceless shapes, wrt each other, or wrt a certain "ground" reference frame.

So thinking about real bodies moving in certain ways because you are pulling on them with strings is acceptable, and may be helpful to your understanding, but it is not necessary. Plenty of time to include things like "forces" when you get to Statics, and "masses" when you get to Dynamics. Much later you can add "friction", and "flex" (= compliance, Modulus of Elasticity+++), etc....

Not enough time for a full explanation here, but it is very important to note that "rotational motions", which can be described with "vectors", are sometimes (not always!) best thought of as "free vectors". Try to find out the difference between "free", "sliding", and "fixed" (or "bound") vectors, and where and when they should be used, or not used...

Roll, Pitch, and Yaw motions could be defined as the "free vector components" of a body's total rotational vector, wrt some ground reference frame (preferably an "inertial" one), with those components taken in the directions of some chosen, mutually orthogonal, set of axes fixed to the body. These body-axes are typically "longitudinal X-axis" for Roll, "lateral Y-axis" for Pitch, and "vertical Z-axis" for Yaw.

Again, VERY IMPORTANTLY, this only covers some of the COMPONENTS of the total motion. For the total motion, wrt ground, you must know other things, such as the details of the ISA!

Read this thread (http://www.fsae.com/forums/showthread.php?10551-Cars-with-wings&p=36671&viewfull=1#post36671) for some background. It is mainly about the Force Screw, but the Motion Screw (= ISA) gets some coverage at bottom of page 4 and on page 5, and they have similarities...
~~~o0o~~~

Z- Q6. Do these apply to both 2-D and 3-D problems?

M - "6- I guess so !

Well, yes, if you do it all correctly...
~~~o0o~~~

I will go through the bonus point questions in a few days. Meanwhile you might want to read about Trochoids. (http://en.wikipedia.org/wiki/Trochoid)

Z

[Ben Coburn]

Regarding Q4, the admittedly somewhat hazy/poorly remembered definition I've seen for the "force based roll center" is "the point (I assume on the same "vertical transversal plane" as above) at which a force applied to the chassis will not produce any body roll". I don't remember for sure, but it may have specified a purely horizontal applied force as well. I think this is the same point as Z's definition of the kinematic roll center above, and can't remember what the difference between this point and the supposed "kinematic roll center" actually was.

(a simple argument for why the two definitions should be the same: any force applied at that point can be considered as two components with one along each of the intersecting n-lines, neither of which should produce any motion by the definition of an n-line.)

[Lemon Lime]

Ben Coburn,


Mmmmm ... About your definition; I don't think Ben ... I think this definition should refer to the "CoG" point in the vertical transversal plane.


Moment of roll = Force vector x Displacement vector ... If displacement = zero ; Then the mentioned body won't rotate ... Taking into considerations that the lateral forces/Centrifugal force acts directly onto the "CoG".


So the force acts at the "CoG" and the body rotate about another point which is the kinematic roll center. If the kinematic roll center lies at the same coordinates in the vertical transversal plane then the moment will equal zero and the body won't roll.


Thanks for your contribution ... Have a nice day !

[Karam Atteia]

hello,

i think these threads would improve your definitions,

http://www.auto-ware.com/ortiz/ChassisNewsletter--October2010.htm

http://www.auto-ware.com/ortiz/ChassisNewsletter--December2004.htm

http://www.auto-ware.com/ortiz/ChassisNewsletter--August2004.htm

GOOD LUCK

[Z]

Aaaarrrghhh!!!!!! Groooaaannn .... mummble ..... vommmitt !!!!!

Why, why, why??? No, no, nooooo!!!

This is NOT THAT HARD!!!!!

All teachers of Engineering, in every school, in all of the world, must immediately sack themselves and publicly declare that, thanks to their efforts, we are already in The Next Dark Ages. Geeeeez, and I used to think there was at least a hundred years left.... :(
~~~o0o~~~

Regarding the above three posts.

Ben's description of a FBRC is similar to the SAE definition (!?) of a RC. That is, it only gives a HEIGHT of the LoA of a horizontal force, and in a rather ambiguous way at that (*). So, NO POINT CALLING IT A "CENTRE"!!! (* In general, if this above ground horizontal-lateral force is reacted mostly by the left wheel, then the resulting behaviour is different to when the reaction is mostly from the right wheel. Typically, in one case the body jacks up, and in the other it jacks down. This different jacking changes the geometry of the linkages, so everything else, like roll angle, also changes.)

Much as I like Mark Ortiz's writings on Vehicle Dynamics (linked by Karam), I am very disappointed by his muddying of the waters on this issue. (He is usually much better. Given those newsletters where written in 2004 and 2010, I hope Mark has since changed his mind on this one...) I can only guess that he was unduly (and unhelpfully) influenced by the SAE definition, since he describes the RC as being like "an imaginary roller in a vertical slot" (= "RIAVS"). Why!!!???

He thus concludes that the "intersection of the force lines" (which is the same as the "intersection of the n-lines", and = the traditional RC) does not give an accurate account of the car body's behaviour. This is simply because a RIAVS does not (CAN NOT!) pass the vertical jacking forces from the wheelprints onto the body. So when Ortiz does his calculations on an RC/RIAVC that is offset sideways from HIS PARTICULAR coordinate origin, his calculated body roll is wrong.

For the record, Ortiz IMPLICITLY assumes that the jacking forces act on the body at the mid-track centreline position of the car. W. Mitchell, and probably Danny Nowlan as well, IMPLICITLY assume the jacking forces act through the CG. Both of these UNSTATED ASSUMPTIONS are a result of the respective authors' choice of coordinate origin (MO = mid-track, WM/DN = CG). I honestly doubt they even know what is going on while they crank the handles of their analytic-equation engines. That is, they are all using Descartes' approach of "cogitatio caeca", which literally means they are "THINKING BLIND"!!!

The errors inherent in the above algebraic approaches, and the sensibility of using the traditional "RC" (= intersection of n-lines) is blindingly obvious when the problem is solved graphically. I gave a start to this type of graphical solution on the "Jacking Force" thread (http://www.fsae.com/forums/showthread.php?4063-Jacking-force&p=17492&viewfull=1#post17492) (bottom of page 4), but it appears that the message is not getting through. Errors made using the graphical approach are very obvious (well, to anyone with at least a little practice). However, the algebraic equations of Descartes' "Analytic Method" are all just jumbled alphabet soup. Implicit errors like those mentioned above are all but invisible!

Who would have thought that it would take only one hundred years of NOT teaching Euclid before we'd all find ourselves in the Engineering version of TNDA!!!???

Marwan, read the "Jacking Forces" thread, learn how to do FBDs CORRECTLY, learn the difference between KRC and "Motion Centre" for Roll (I linked to this earlier), and learn the difference between a "Moment of a Force" and a "Couple of Forces"!

As a final BTW, I recently ordered Den Hartog's "Mechanics" and "Mechanical Vibrations" from Amazon (yet to arrive). I don't need these books, but I got them out of curiosity because they are both quite old (pre-1950?), so they should explain all this stuff properly. They cost US$3.99 each!

Z

(Edit: Geez!!!!! Now all the links to threads/posts on the previous Forum don't work, and they have to be changed!!! This is "progress"???!!! :()

[Z]

Marwan,

Regarding "The Spinning-and-Sliding Car".
==============================
I gave this example to show that "Instant Centres" (and even ISAs), in themselves, are quite useless when solving Vehicle Dynamics problems...


For bonus points, answer the following...

Imagine a car negotiating a right hand bend. Not in hot and sandy Egypt, but in the cold and icy North. The car oversteers and spins off the road and onto a large frozen lake. There the car's CG keeps sliding in a straight line, at a steady 10m/s directly North, while the whole car keeps spinning (yawing) at a steady 1 rev/s, clockwise in plan-view. (Think recent James Bond movie...)

There are three points painted onto the car's centreline.
Point "A" is 1 metre in front of the CG.
Point "B" is 1.6m in front of CG.
Point "C" is 3m in front of CG (the car has a long nose).
~~~~~o0o~~~~~

Z - "Q7. At any instant, where is the "Instant Centre" for this essentially horizontal, planar motion of the car's body, WITH RESPECT TO (= wrt) the frozen lake?"

M- "7- The CoG I guess !"

With respect to the frozen lake, the IC is always 5/Pi metres (~1.59 m) to the East of the car's CG, and thus moving Northwards in a straight line at a steady 10 m/s.

(BTW, the ISA for this 3-D motion is a vertical Screw Axis passing through the above 2-D IC, and with pitch = 0.)
~~~~~o0o~~~~~

Z - "Q8. Describe the paths of points A, B, and C, wrt the car body."

M - "8- Circular !"

The three points are PAINTED ONTO the car body. So, "WRT(!) the car", the points are STATIONARY!
~~~~~o0o~~~~~

Z - "Q9. Describe the paths of points A, B, and C, wrt the frozen lake (draw and post these here, if possible)."

M- "9- Spiral !"

This Wiki page of Trochoids (http://en.wikipedia.org/wiki/Trochoid) shows some of these curves. Trochiods (from Greek "wheel") are the paths traced out by a point on the side of a wheel as it rolls without slipping along a flat road.

In plan-view the car's body is moving like a side-view of a wheel rolling along a road represented by the line traced out by the IC of Q7. (Note that the side-view IC for a wheel rolling on a road is the "wheelprint/contact-patch"!).

So,
Point A traces out an "s" shaped path (like lower curve of Wiki pic at lower right).
Point B, almost on "the circumference of the wheel", traces out a "m" shaped path (middle curve of Wiki pic).
Point C, further out from "the circumference", traces out a path that occasionally loops back on itself (upper curve of Wiki pic).
~~~~~o0o~~~~~

Z - "Q10. What is slightly special about the path of point B, wrt the frozen lake?
Q11. What would be the EXACT position of point B (on car centreline) to make this path EXACTLY special?
For double bonus points:
Q12. What are the fancy sounding mathematical names for the paths of points A, B, and C (wrt frozen lake)?"

M - "10- mmmm ... I thought maybe point "A" is a bit special .. Because it's 1 meter away from CoG and the Car rotates about it self at 1 rev/sec and has a constant velocity of 10 m/sec.
11- At "A" 1 meter away from the CoG !!
12- A: Parabola ... B: Spiral ... C: Elliptical !"

These questions are not particularly important to understanding VD, but you should know that the answers are NOT "Parabola ... Spiral ... Elliptical"!!!

As above, they are "Trochoids" (or, more generally, "Cycloids"), and can be further prefixed with "contracted/curtate/common/extended/prolate" depending on how far from the centre of the wheel are the generating points.

Point B would be slightly more special if it was EXACTLY 5/Pi (~1.59) metres from the car CG, because once a second it would come to a complete stop (= zero velocity, wrt frozen lake).
~~~~~o0o~~~~~

Z - "For triple bonus points, and assuming the frozen lake is an "inertial reference frame", and neglecting aero forces:
Q13. If your body has a mass of 100kg, then what are the major forces acting on your body when you sit at points A, B, or C???
Give magnitude and direction at any instant.

M - "13- The further I sit the more I would feel the lateral forces and the more I feel "dizzy" !"

In all three cases your body always feels the same downward gravitational force (= your weight of ~980 N), and an equal and opposite upward force from the car body.

Your body also feels an inertial force directed away from the car's CG (= centrifugal, or "centre-fleeing" force), and an equal and opposite inward (centripetal = "centre-seeking") force from the car body (or else you fall off the car!). However, the centrifugal/centripetal forces are smaller when sitting at Point A, and greater when sitting at Point C.

The centrifugal force, which is always of constant magnitude, and always directed away from the car's CG, has magnitude;
at Point A = m x R x W^2 = 100 x 1 x (2.Pi)^2 = ~3,948 N,
at Point B = ~6.3 kN,
at Point C = ~11.8 kN = more than a TON.

So, yes, I guess you would fell a little "dizzy".

Note that the paths the three points follow have different curvatures at different places, and the velocities of the points change significantly along the paths (eg. Point B occasionally comes almost to a complete stop, and at other times moves at 20+ m/s). Nevertheless, the magnitudes of the inertial (centrifugal) forces are constant, and their directions change smoothly (ie. steady rotation, wrt frozen lake).
~~~~~o0o~~~~~

Z - "For quadruple bonus points:
Q14. Is it possible to determine all the forces acting on your body in the above question, if you know the exact postion of the IC for the car's (and hence also your) motion wrt frozen lake, but you only know this IC position AT THE INSTANT?"

M - ?

The whole point of the above is to show that there is next to NO CONNECTION between the position of the IC at any instant, and the magnitude and direction of the forces acting on your body (specifically, the centrifugal/centripetal forces).

At one instant the centrifugal force might point directly TO the IC, at another instant it points directly AWAY from the IC, and at any other instant it can point in ANY OTHER (horizontal) direction.

Knowing the path of motion of your body (ie. the above trochoids) is more helpful because the centrifugal force always points to the convex side of the curve (although to get the magnitude and exact direction of the force you must also know the speed along the curve).

Bottom line here is that to solve Dynamic problems you must know both the position of the IC (more strictly, ISA!) at "the instant", and also how this position changes with time.
~~~~~o0o~~~~~

Z- "For absolutely NO points at all:
Q15. Do you really have to "solve a system of ODEs" before you can answer the above questions?"

IMO, you have to be able to answer all the above questions correctly, BEFORE you can even write down the correct "system of ODEs"!

Or put another way, students who cannot answer the above correctly, but are handed a "system of ODEs" on a plate, will never know if their solutions are even close to correct.

But that is the way it is done these days, isn't it!?

Z

[BillCobb]

In working in the professional Vehicle Dynamics field for about 40 years, the only instance of instants I've endorsed were coffee, pudding, and soup. The problem as I see it in using "them" is there is no way to correlate the math to the road test. People have tried film analysis using targets etc., but the resolution didn't resolve anything. Stick to what you can measure and what measurements you can simulated.