Bicycle model

[MaiUnaGioia]

Hi guys, I'm making some simulations with matlab to understand vehicle dynamics using a simple bicycle model.
It requires to know some vehicle parameters.

One of these is cornering stiffness.
I have calculated it using this formula from Pacejka's tyre model (96) using data from TTC:



Since it is a 2 dof model, with no pitch motion, I used a static weight for Fz (no load transfer).

I assumed the same value for front (Cf) and rear (Cr) cornering stiffness.
I found a lot of data in which there are different values for Cf and Cr.
Here is an example:



I don't know by what criteria I can state to have different value for the front and the rear.

I would like to know your opinion.
Are my assumptions reasonable? Have you any suggestions?

I also tried to compare linear tyre model with Pacejka's one.
I made a Matlab script but linear model seems to be not tangent to Pacejka's curve for lateral force.





Here is the script I'm using
https://www.dropbox.com/s/1quqpgym2s...ar_fsae.m?dl=0

[BillCobb]

Step on him...

[Flight909]

I would said you can try to set this following coeffiicents to zero: PHY1,PHY2,PHY3,PVY1,PVY2,PVY3,PVY4

If looking at the equation you see they will shift the curve up & down and left&right.

[BillCobb]

So:
According to your data, if I were to assume that your tire properties are PER WHEEL, then you can easily compute this vehicle's oversteer and understeer traits without simulation at all.

I get -0.55 deg/g or -.0095 rad sec^2/m for condition 1, and 1.84 deg/g or .00327 rad sec^2/m for condition 2. No need for any 'simulation' here. It's just algebra-I

If your 'tire stiffnesses' are for each axle, then the results are pretty close to yours.

1) This is your first post, so please introduce yourself so we know about the audience we are replying to (Name, school, student year, degree being pursued, Team name, birthday, Mother's middle name, etc.) Otherwise you will receive scorn, insults, comedy, sarcasm and equally puzzling replies. And that's just from me !

2) Always show plots with all axes labeled with parameter names, units, a legend, maybe some gridlines, really cool colors, thick lines and the source of the data. You're using Matlab, use Matlab's features to explain your case and methods.

3) I take it your coefficients are not from the TTC data library. I've never seen so many zeros in a tire cornering stiffness!

4) Just out of curiosity, how would you propose to get such different Cf and Cr stiffnesses from tires that are operating at the same wheel loads ?

5) Try to keep values for your input design parameters in units of the trade (N/deg, deg/g, etc.) all those fractional bits are sleep inducing and awkward to type into my phone's calculator.

6) Ask you question(s) again with a goal in mind. Different "WHAT" for front and rear (Masses, lengths, tire stiffnesses, dimensions ) ?

Then we can help you in small, understandable steps. You certainly are brave to get this far !

BTW: Your "Vehicle" as described as 'understeering' has 90% lateral acceleration response times of about 0.33 sec. and a yaw damping value (zeta) of 0.823. Values even the most simple of simulations ought to compute from the parameters you have provided here.

[MaiUnaGioia]

I would said you can try to set this following coeffiicents to zero: PHY1,PHY2,PHY3,PVY1,PVY2,PVY3,PVY4

If looking at the equation you see they will shift the curve up & down and left&right.

Thanks.

So:
According to your data, if I were to assume that your tire properties are PER WHEEL, then you can easily compute this vehicle's oversteer and understeer traits without simulation at all.

I get -0.55 deg/g or -.0095 rad sec^2/m for condition 1, and 1.84 deg/g or .00327 rad sec^2/m for condition 2. No need for any 'simulation' here. It's just algebra-I

I know that I can say if the vehicle is oversteering or understeering simply using understeer coefficient.
I would make some simulation to see vehicle response to arbitrary control and disturbance inputs.
For example I would use different inputs for steering (step, ramp, sine).

Then I would like to compare results from these simulations with the ones from a more complex model, for example an Adams model with more d.o.f.
Vehicle data I posted are generic, found it in a book.

These are vehicle parameters I'm using for simulation:

m = 315.5; % kg vehicle mass
a = 0.78; % m distance from front tire to C.G.
b = 0.83; % m distance from rear tire to C.G.
Ca0 = 27537.5; % N/rad front cornering stiffness (single tyre)
Cp0 = 27537.5; % N/rad front cornering stiffness (single tyre)
Iz=34.42; % kg*m^2 moment of inertia about z axis

If your 'tire stiffnesses' are for each axle, then the results are pretty close to yours.

I took the cornering stiffness from Ky_alpha formula and then I multiplied it for 2 so I have axle stiffness.

1) This is your first post, so please introduce yourself so we know about the audience we are replying to (Name, school, student year, degree being pursued, Team name, birthday, Mother's middle name, etc.) Otherwise you will receive scorn, insults, comedy, sarcasm and equally puzzling replies. And that's just from me !

I'm sorry. When I registered I didn't find a section for introducing myself.
I'm a student. I'm part of formula student electric team at Sapienza uni.

3) I take it your coefficients are not from the TTC data library. I've never seen so many zeros in a tire cornering stiffness!

I'm using old data from a report of Stackpole Engineering Services.
Data are relative to Avon 7.2/20-13.

4) Just out of curiosity, how would you propose to get such different Cf and Cr stiffnesses from tires that are operating at the same wheel loads ?
6) Ask you question(s) again with a goal in mind. Different "WHAT" for front and rear (Masses, lengths, tire stiffnesses, dimensions ) ?

I'm assuming the same load on every wheel since in the 2 d.o.f model there is no load trasfer. Indeed I'm using the same value for front and rear cornering stiffness.
I asked this question to find out if there was some other parameter to consider, in addition to the normal load on the wheels, that maybe I was forgetting.
Maybe the error is to consider the same load on every wheel.
According to vehicle parameters maybe it's better using 797,8 N (front single tyre normal load) and 749,7 (rear single tyre normal load).
Then use this different values for calculation of cornering stiffness.
I should get Cf=32496,9 N/rad and Cr=33137,5 N/rad

[Z]

These are vehicle parameters I'm using for simulation:

m = 315.5 kg vehicle mass
a = 0.78 m distance from front tire to C.G.
b = 0.83 m distance from rear tire to C.G.
...
Iz = 34.42 kg*m^2 moment of inertia about z axis
...
I'm part of formula student electric team at Sapienza uni.

MaiUnaGioia,

"Sapienza", eh?

The "University of Wisdom"???

Yet ... so many mistakes...
~o0o~

The Yaw-Inertia (Iz) number is quite important for these simple bicycle model simulations. You say your Iz = 34.42 kg.m^2, which implies you think you have accuracy of 4 significant digits?

By my reckoning, the lowest practical mass of a tyre/wheel/axle/upright "corner-assembly" in FS/FSAE is around 10 kgs (taken as a round number of only 1 significant digit). The distance of this mass (again taken for simplicity as a point-mass) from the whole car's CG is around 1 metre (ie. your a, b = ~ 0.8 m, together with half-track = 0.6 m, gives radius = 1.0 m, thanks to Pythagorean 3/4/5 triangle).

So, your Iz, for the FOUR CORNERS ONLY, is AT LEAST Iz = 4 x 10 kg x 1 m^2 = 40 kg.m^2.

(Gee, I hope you are NOT factoring in some of Claude's "negative masses", that he claims you get by using pushrods-and-rockers!?)

Anyway, your four-significant-digit number has NONE of its digits close to correct. In fact, the whole number is close to an order of magnitude wrong.

Not wise.

Wiser is to try to get your order of magnitude right first, then maybe try for an accurate first digit.
~~~o0o~~~

Some More Comments About Pacejka's "Magic Formula".
=========================================
Back around Xmas-2015-New-Year-2016, Bill Cobb posted some "Step-Steer" simulations on various other threads. These used a simple four-parameter Pacejka MF, similar to the OP's above, together with Matlab code to solve the Laplace transforms of the step-steer. Spurred by the results of these simulations I had a go at doing similar, but with a "time-stepping" Classical Mechanics approach.

And given how easy all the coding was, I got a lot of very interesting results! I might post some of those results on the appropriate threads at a later date. But right now I am busy with other things, and I would like to dress-up the outputs a bit more so they are easier to follow (ie. better labels on graphs, more graphs, +++, all of which takes time...).

Anyway, one result that quickly became obvious is that in some areas of tyre behaviour, the Pacejka Magic Formula (and anything similar) is completely WRONG!

And, no, I am not referring to the "divide by zero velocity" error that gets many mentions on the first page of Google hits on the subject. Nope, this is actually a much bigger error, although I have yet to see any mention of it. This error is actually very easy to fix in the simulation code. But, in order to fix it, you must first be aware of it. So it should be clearly spelled out on page-one of any discussion of PMFs. But, I guess, this is just another example of the education system going down the drain.

Anyway, gold star to the first student to point out this fatal flaw. (It is "fatal" because it completely messes up a good simulation ... at certain times.)

Hints: This is a BIG-PICTURE thing. Forget about all the little details ... like using four parameters, or four-significant-digits when you are not even close on the first one. I ended up with very realistic simulations (albeit only of "bicycle model" cars) using only two parameters to specify the shape of each of the F&R tyre curves. But I did have to add that bit of big-picture code to make it work realistically.

This "feature" of tyre curves SHOULD be obvious, and it should be spelled out whenever tyre curves are discussed...

Z

[Tim.Wright]

For steady state stuff I've always found the pacejka 4 parameter model to represent experimental/simulation data extremely well.

I'd be interested to find out what you have 'discovered'.

[Z]

Tim,

Hints:
* Assume the car is a tad oversteery. Or a lot oversteery...
* What happens to such a car when given a large-ish step-steer while travelling at significant speed?
* Also assume the fussy old-fart likes to see a reasonably realistic trajectory of the car after the step-steer, all the way until it rolls to a stop (note that "slip-angle drag" quickly drains the kinetic energy of the car, and I am assuming a "coasting" car during the test, with no "driving" forces).
* Perhaps too big a hint. => Under what circumstances does a OS car become a US car?

But when you get the answer, can you wait a while to see if any of the students can figure it out...?

Z

[BillCobb]

There are actually several flaws in the Pacejka theory of tires, but you must farhten before you can go fig Newton. Old Hans lets a good one fly and 3:36

https://www.youtube.com/watch?v=DNFzYEih72g

[Danschwind]

"Perhaps too big a hint. => Under what circumstances does a OS car become a US car?"


When it hits the wall.