Tire Sideslip Angle vs. Side Force



You could project measurements back to a point between the rear wheels, but you wouldn't. What you would do is project them to the CG location.
Regarding whether one can consider the tires rigidly attached to the road, that of course is not true since the simplest modeling, beyond Ackerman steering, is to regard the tires as a compliance (the cornering coefficient of the tire).
The other compliances are there, and are relevant, but are not first order effects. It's mostly all about the tires and their non-linear response, and because of that it's also about weight transfer from inside to outside during a turn and front to rear during acceleration. As far as independent rear suspension goes, the whole point is to keep the tires on the ground, not to maintain an orientation with the car, so I'm not sure what you saying there.
Regarding the prior comment about the front and rear of the car moving in different directions, that's not wrong, but a more useful view of the situation is to view the vehicle as a rigid body, and to then consider the translation of the vehicle mass and rotation of the vehicle inertia about the CG point. Combine this with the idea of the tires being a compliance and you've arrived at a simple vehicle dynamics model, from which you can determine the vehicle slip angle under discussion here.
dave y.
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Considering:
* http://www.ajc.org.tw/pages/abstract/2.3PD/1FR-00-2.PDF
Page 5, Fig. 1: Speed 30 m/s, adhesion coefficients = 1
Example solution via feedforward control:
Process transfer function
Step response of v2 and yaw rate v1 for selected car speed
1.570587 v1'' + 1.262046 v1' + v1 = v2
* http://home.arcor.de/janch/_control/20100623-yaw-control /
Page 1: With feedforward control improved steering behavior Page 2: Without feedforward control (PI-controller ~ similar to driver control)
Note: t = 10 could be e.g. 1 s
Other parameter values for different speeds could be adapted automatically.
JCH
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wrote:

I didn't delve into the paper, but it looked good from a distance! I note the modeling assumptions on page 2:
The roll motion is negligible. The longitudinal acceleration xu is small. Tire slip angles of the left and the right wheels are the same. Tire longitudinal and lateral forces are represented by the linearized tire model.
which are typical of the "bicycle model", a representation of the car as a two wheel vehicle.
dave y.
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wrote:

And relating the results to Tim's original questions, the responses are dynamic. During simulation, the transients are not proportional to the unit-step change in steering angle - the simulated acceleration responses (body lateral and yaw accelerations) have damped overshoot. During the road trial, the steering angles have lots of high-frequency content, very different from the angles expected from the trajectory which is a series of various circular arcs (i.e. regions of constant curvature) each of several seconds duration.
For steady-state vehicle behaviour, after all transients have died down, there may well be strict proportionality, but that is likely to be of little interest to the control engineer. The simulation results suggest that the transients may last several seconds at the higher speeds.
Depending on the nature of the vehicle and its handling, transient behaviour may be significant. A racing car, a high-bodied truck, and a wheelchair will operate in different handling regimes, and have different responses, and Tim's response to his customer will need to vary accordingly.
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All that is described in * http://www.ajc.org.tw/pages/abstract/2.3PD/1FR-00-2.PDF
I just demonstrated one single case for speed 30 m/s and adhesion 1. * http://home.arcor.de/janch/_control/20100623-yaw-control /
Transient behavior changes with speed v and adhesion : A2(v,)*v1'' + A1(v,)*v1 + 1 = v2 (F1)
The coefficients cover a range in question and can be found via 'high accurate process identification and 3D regression methods'. Even appropriate filter (F3) coefficients should be taken into account.
That solves non-linearity problems.
JCH
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wrote:

Sorry, I think I missed it. What non-linearity problem did you just solve?
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I 'explained' how to do it:

I presented just an EXAMPLE for clarification:
A2(30,1)*v1'' + A1(30,1)*v1' + 1 = v2
* http://home.arcor.de/janch/_control/20100623-yaw-control /

JCH
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wrote:

Are you aware there is already an existing body of work regarding tire models, from simple linear models, to complex finite element models, and that an empirical model, using curve fitting methods to find the coefficients, is called the 'magic formula' model and is very frequently used in practice (see Pacejka, MF-Tire, etc)? For example, a simplified version I've used for longitudinal acceleration studies is
mu=A*sin(B*atan(C*slip))
where A,B,C are constants used to fit the formula to actual tire data. There are other similar versions you use for lateral handling.
This formula will produce the the typical shape of the tire mu-slip curve, with a linear low slip region, the peak adhesion region, and then the negative slope region during sliding out to 100% slip. I fail to see how your formula could possibly create a typical tire curve shape. Perhaps you could explain?
dave y.
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That is much different from that I have modeled. I modeled a similar dynamic system as EXAMPLE with appropriate feed-forward control.

But for modeling of e.g. mu=A*sin(B*atan(C*slip)) I would prefer a polynomial:
EXAMPLE DATA
Just for explanation
A = 1 B = 2 C = 3
slip >= 0
c( 0) = -0,00325266 c( 1) = 7,30646 c( 2) = -17,25949 c( 3) = 10,97288 c( 4) = 14,18502 c( 5) = -31,86388 c( 6) = 26,89163 c( 7) = -12,57325 c( 8) = 3,416591 c( 9) = -0,5060488 c(10) = 0,03165701
slip < 0
c( 0) = 0,003292161 c( 1) = 7,311109 c( 2) = 17,3065 c( 3) = 11,15444 c( 4) = -13,82579 c( 5) = -31,45185 c( 6) = -26,60178 c( 7) = -12,44633 c( 8) = -3,382853 c( 9) = -0,5010668 c(10) = -0,03134352
It is easier to implement polynomials in programs.
* http://home.arcor.de/janch/_control/20100630-magic%20formula /
JCH
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wrote:

Guess I'm back to the same question then. I was just wondering what non-linearity you were modeling, without having to reverse engineer your formula to understand what you did. When you stated that "Transient behavior changes with speed v and adhesion ", what physical principal are you trying to capture? Or are you just trying to curve fit some experimental data without any underlying physical basis to the form of the formula?
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I just use and fit reliable measured data. The data include all of underlying physical basis. (Pacejka's magic formular does the same.)
Measured process transfer function data let me find the ODEs for different v and . Least-square approximation methods are applied for finding Ai(v,)coefficients. Ai(v,) is non-linear and appropriate for any v and .
EXAMPLE (for just v0, =1)
* http://home.arcor.de/janch/_control/20100702-ode-approximation /
It's a step to the real world. Generally, measured values are also the necessary basis for proving derived physics. I use them witout physics.
JCH
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wrote:

From my experience, one should always try to base models upon 'first principles', rather than, say for example, curve fitting polynomials to the data. The polynomial curve fit is useless for extrapolation, and dubious for interpolation at higher orders. And what kind of understanding are you developing about the process by just doing curve fits blindly to the data?
In the case of the magic formula, the actual relationship of mu to slip is fairly well known, is supported by a little engineering theory (e.g. the tire brush model), and the magic formula is simply a convenient and compact representation of the curve shape.
And in the case of the whole vehicle, models already exist, both geometric and parametric, from simple to complex, and commercial products are available, so I just don't understand what you're doing or why.
dave y.
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'first principles' are referred to as axioms or postulates. In tradition axiom is a proposition that is 'not proved or demonstrated' but considered to be self-evident. Therefore, its truth is taken for granted.
Don't do that!
Just in case you think 'first princible' is pure physics you have to try finding M, C, D, K and F:
The dynamic equation can be represented as non-linear
M q'' + C(q, q') + D q' + K q = F delta (1)
* http://www.ajc.org.tw/pages/abstract/2.3PD/1FR-00-2.PDF
II. ANALYSES OF LINEARIZED VEHICLE DYNAMICS

I studied process engineering, but worked for years as control engineer. As control engineer I need an accurate process transfer function. Otherwise trial-and-error methods are left for tuning. And that seems to be the topic.
See for extrapolation:
* http://home.arcor.de/janch/_control/20100702-ode-approximation /
There is nothing fitted blindly. Just integrate this ODE for e.g. x = 40 and you have an extrapolated value of y = 9.998.
Conclusion:
Looking at Fig. 13 I think the control quality could be much increased using TRUE process transfer functions.
Note: Measured values are solutions of 'real' physics. Use them.
JCH
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wrote:

I have to reject your idea of 'first principles' are you define it. As applied to modeling, I mean analyzing the system and writing the basic equations of motion. The degree of complexity is entirely up to you, but when you get done, the form of the model represents the actual physics going on. There will be lots of coefficients to be determined, so you can go curve fit happy as you please, but you're doing it to a meaningful model, and the coefficients represent a meaningful parameter. And in the process of doing the model, you can introduce non-linearities in a meaningful way. For example, even in the simplest vehicle dynamics model, the vehicle longitudinal speed will enter the model as a non-linearity. By using 'first principle' models, you can see the effects of understeer gradient and how you go from understeer to oversteer as the vehicle speed increases. However if you just throw in vehicle speed as a variable of a generic equation and do some curve fitting to find equation coefficients, what have you learned?
dave y.
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Dave, there is no point in arguing with JCH. We, at least I, know you are right and JCH is wrong. You are kicking the defenseless. The problem is that the rookies and students probably don't know the difference.
Peter Nachtwey
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