Dynamic Simulation - Step-Steer - Problems

[CWA]

Hey guys, I'm getting my brain twisted a bit by this one.

So I've got a basic Excel "step-steer" simulation on the go. To quickly describe it, time increments go down the page (0.01s).. specified initial conditions are in the form of Beta (0) and yaw rate (0). The simulation is 5 seconds in duration, step steer occurs at exactly 1 second (does not ramp). On this first line, front slip angle equals the new steer angle (also as a function of the initial yaw rate (0) and initial Beta (0)), creating maximum yaw moment (N) (using a, b and L for wheelbase), hence maximum yaw acceleration (r-dot) (using yaw inertia (Izz)), and Beta-dot comes from "(F/mV)-r" (where F= total lateral force, m=mass and V=forward velocity) to end the line. For each following line, I then integrate the Beta-dot and r-dot of the previous lines to give my new r and Beta values, then repeat as described. Sorry if the explanation is a bit brief, but for those who have done similar things before I'm sure you can recognise my process. I have used the 2DOF single-track model equations from RCVD (Chapter 5), using a single linear cornering stiffness coefficient for each axle.

Anyway, my responses seem to be on the right track, in terms of their general trend and also how they vary with certain parameter changes. Slip angles ramp up and then level off into steady-state at believable values after around 1 second, and, for example, when weight distribution is "moved forward" at a modest forward speed, there is slight overshoot of the responses, highlighting the "underdamped yaw-sideslip response" typical of this kind of car architecture. For these reasons, I feel my model is on the right lines, and doing what I want it to do.

There are a few problems though. And currently I'm not sure if they are due to the limitations of this modelling technique, or an error in my modelling. The problems are as listed:

1. The final slip angles (and Beta) outputs do not match the steady-state solutions for my chosen steer angle and forward velocity, according to the same RCVD equations. To explain, the step-steer simulation for V=70m/s and delta=0.2rad may have front and rear slip angles settle at alphaf=Xrad and alphaR=Yrad. But then if I use the RCVD equations to find the steady-state front and rear slip angle solutions for V=70m/s and delta=0.2rad, I will get alphaF=1.2Xrad and alphaR=1.2Yrad, or something along those lines. This seems to somehow come down to the relationship between radius of turn, wheelbase, F&R slip angles, and "ackerman steer" angle. But this is as far as I can get. My question regarding this problem is; shouldn't the slip angle solutions shown by both techniques be the same? If anyone could perhaps point out what I might have neglected hence causing this error (if there is one), that would be a lovely bonus!

The other two problems are less important, but puzzle me none the less:

2. I am currently trying to emulate some response graphs that have been provided to me by someone (who is currently unavailable to defend them!), with Ay, r, Beta-dot and delta all plotted against time for a similar step-steer simulation. My responses look vaguely similar for all except Beta-dot; mine is negative but the plot shows it being positive. Given my rather "high" forward speeds, and if I tell you that my steady-state Beta is negative (indicating that the vehicle has pasts it's "characteristic speed", which is intuitive for anything above modest forward speeds), shouldn't it make sense that Beta-dot also has to be negative? I am confused as to why the response plot I have been given would show Beta-dot as positive if the Beta value is also bound to be negative. To clarify, there is no overshoot or oscillation of Beta.

3. Upon initial steer application, front lateral force is positive (into the turn) and rear lateral force is negative (out of the turn). I understand this is due to how the lateral velocity builds up in relation to the yaw rate build-up? This relationship may be expressed by use of a "yaw dynamic index", taking mass, inertia, radius of gyration and wheelbase into account. I have also read that this behaviour is indicative of vehicles with a yaw dynamic index (let's say DI) value of greater than unity (DI>1). Vehicles with DI<1 will have rear lateral force build in the same direction as front lateral force. I'm unsure on the behaviour of vehicles with DI=1. Suffice it to say that altering Izz and hence my DI value does not change this behaviour in my simulation; rear lateral force always begins by pointing in the "outside of the turn" direction before reversing and acting in the same direction as the front. My question with regards to this is; is this a limitation of my modelling technique, or have I built the model incorrectly? Should varying the DI be reflected in the behaviour of this simple model?

Sorry for the painfully long post, if anyone has managed to read it and can offer me some guidance I'd very much appreciate it.

Cheers,

Chris

[Claude Rouelle]

Chris,

Send me an email at engineering@optimumg.com and I will send you just a few slides of the OptimumG advanced workshop which could put you on the right track

[CWA]

Thanks Claude! You have mail

[Z]

3. Upon initial steer application, front lateral force is positive (into the turn) and rear lateral force is negative (out of the turn). I understand this is due to how the lateral velocity builds up in relation to the yaw rate build-up? This relationship may be expressed by use of a "yaw dynamic index", taking mass, inertia, radius of gyration and wheelbase into account. I have also read that this behaviour is indicative of vehicles with a yaw dynamic index (let's say DI) value of greater than unity (DI>1). Vehicles with DI<1 will have rear lateral force build in the same direction as front lateral force. I'm unsure on the behaviour of vehicles with DI=1. Suffice it to say that altering Izz and hence my DI value does not change this behaviour in my simulation; rear lateral force always begins by pointing in the "outside of the turn" direction before reversing and acting in the same direction as the front. My question with regards to this is; is this a limitation of my modelling technique, or have I built the model incorrectly? Should varying the DI be reflected in the behaviour of this simple model?
Chris,

Not sure about your first questions, mainly because it is hard to know which are your "positive" directions, etc. These puzzles are always easier to solve when all the info is there.....

But for Q3, a simple bicycle model + CG somewhere in middle + Yaw Radius of Gyration, should most definitely give the results you are expecting.

Basically, if the mass-distribution "dumb-bell" is shorter than the wheelbase (eg. like a cannon-ball), then inwards road-to-front-tyre-force causes an outwards MOVEMENT at the rear-wheel (car pivots about cannon-ball), which causes inwards road-to-rear-tyre-force.

If "dumb-bell" is longer than wheelbase, then inwards road-to-front-tyre-force causes rear-wheel to also move inwards (car pivots in plan-view about a point behind the rear-wheel), hence outwards road-to-rear-tyre-force.

If "dumb-bell" is exactly same length as wheelbase (so one half-mass is at front-wheel, and other half-mass is at rear-wheel), then inwards road-to-front-tyre-force only moves front half-mass inwards (car pivots about the rear-wheel and its half-mass), hence no force at rear-tyre, until beta angle builds up...

So, check your equations???

Z

(PS. Also try giving extreme values to Yaw Radius of Gyration, say from 0.1m to 10m, and see what happens.)

[BillCobb]

Your Beta dot term in incorrect.

Beta dot = Ay - U*R where U is fwd velocity (or V to You) and R is yaw velocity. In your own equation, if the sideslip effect were zero, would you believe Beta dot = r ????

Watch your signs and don't forget to subtract the Ackerman steer term in the the slip angle formulation column.

Stay tuna-d

[BillCobb]

Speed = 100 ! Vehicle speed (kph)
sr = 10 ! Overall Steering Ratio (deglee per deg C)
WF = 310. ! killer grams
WR = 290.
L = 1374 ! Wheelbase (m&m)
Step Steer = 5 ! deglees
IZS = 398.225 !SPRUNG YAWn INERTIA
IZU = 152.1 ! UNSPRUNG YAW INERTIA
CAF = 506.7 ! Fig Newtons per deglee. (tars)
CAR = 711.

So what is the understeer & front & rear cornering compliances (deg/g) of this FSAE-like car ?
Limited to 10000 chars so its been truncated.
TIME STEER U R BETA SLIPFT SLIPRR
0.000 0.0000 27.7778 0.0000 0.0000 0.0000 0.0000
0.010 5.0000 27.7778 0.0031 0.0002 -0.4871 0.0042
0.020 5.0000 27.7778 0.0090 0.0004 -0.4662 0.0082
0.030 5.0000 27.7778 0.0148 0.0005 -0.4497 0.0084
0.040 5.0000 27.7778 0.0203 0.0006 -0.4372 0.0053
0.050 5.0000 27.7778 0.0256 0.0006 -0.4282 -0.0008
0.060 5.0000 27.7778 0.0308 0.0006 -0.4223 -0.0095
0.070 5.0000 27.7778 0.0356 0.0006 -0.4194 -0.0204
0.080 5.0000 27.7778 0.0403 0.0005 -0.4189 -0.0331
0.090 5.0000 27.7778 0.0447 0.0003 -0.4207 -0.0473
0.100 5.0000 27.7778 0.0488 0.0002 -0.4245 -0.0628
0.110 5.0000 27.7778 0.0527 0.0000 -0.4299 -0.0794
0.120 5.0000 27.7778 0.0564 -0.0002 -0.4368 -0.0967
0.130 5.0000 27.7778 0.0599 -0.0005 -0.4449 -0.1146
0.140 5.0000 27.7778 0.0631 -0.0007 -0.4541 -0.1329
0.150 5.0000 27.7778 0.0661 -0.0010 -0.4641 -0.1514
0.160 5.0000 27.7778 0.0689 -0.0012 -0.4747 -0.1700
0.170 5.0000 27.7778 0.0715 -0.0015 -0.4860 -0.1885
0.180 5.0000 27.7778 0.0739 -0.0017 -0.4976 -0.2069
0.190 5.0000 27.7778 0.0761 -0.0020 -0.5094 -0.2250
0.200 5.0000 27.7778 0.0781 -0.0022 -0.5215 -0.2427
0.210 5.0000 27.7778 0.0799 -0.0025 -0.5336 -0.2600
0.220 5.0000 27.7778 0.0816 -0.0027 -0.5456 -0.2768
0.230 5.0000 27.7778 0.0831 -0.0030 -0.5576 -0.2931
0.240 5.0000 27.7778 0.0845 -0.0032 -0.5694 -0.3088
0.250 5.0000 27.7778 0.0857 -0.0035 -0.5810 -0.3238
0.260 5.0000 27.7778 0.0868 -0.0037 -0.5923 -0.3383
0.270 5.0000 27.7778 0.0878 -0.0039 -0.6033 -0.3520
0.280 5.0000 27.7778 0.0886 -0.0041 -0.6139 -0.3651
0.290 5.0000 27.7778 0.0894 -0.0043 -0.6241 -0.3775
0.300 5.0000 27.7778 0.0901 -0.0045 -0.6340 -0.3892
0.310 5.0000 27.7778 0.0906 -0.0047 -0.6434 -0.4002
0.320 5.0000 27.7778 0.0911 -0.0048 -0.6524 -0.4106
0.330 5.0000 27.7778 0.0915 -0.0050 -0.6610 -0.4204
0.340 5.0000 27.7778 0.0919 -0.0051 -0.6691 -0.4294
0.350 5.0000 27.7778 0.0921 -0.0053 -0.6768 -0.4379
0.360 5.0000 27.7778 0.0923 -0.0054 -0.6841 -0.4457
0.370 5.0000 27.7778 0.0925 -0.0055 -0.6909 -0.4530
0.380 5.0000 27.7778 0.0926 -0.0057 -0.6973 -0.4597
0.390 5.0000 27.7778 0.0927 -0.0058 -0.7032 -0.4659
0.400 5.0000 27.7778 0.0927 -0.0059 -0.7088 -0.4716
0.410 5.0000 27.7778 0.0927 -0.0060 -0.7139 -0.4767
0.420 5.0000 27.7778 0.0927 -0.0060 -0.7187 -0.4814
0.430 5.0000 27.7778 0.0926 -0.0061 -0.7231 -0.4857
0.440 5.0000 27.7778 0.0926 -0.0062 -0.7272 -0.4895
0.450 5.0000 27.7778 0.0925 -0.0062 -0.7309 -0.4930
0.460 5.0000 27.7778 0.0924 -0.0063 -0.7343 -0.4961
0.470 5.0000 27.7778 0.0922 -0.0063 -0.7374 -0.4988
0.480 5.0000 27.7778 0.0921 -0.0064 -0.7402 -0.5012
0.490 5.0000 27.7778 0.0919 -0.0064 -0.7428 -0.5033
0.500 5.0000 27.7778 0.0918 -0.0065 -0.7450 -0.5052
0.510 5.0000 27.7778 0.0916 -0.0065 -0.7471 -0.5068
0.520 5.0000 27.7778 0.0915 -0.0065 -0.7489 -0.5081
0.530 5.0000 27.7778 0.0913 -0.0066 -0.7505 -0.5093
0.540 5.0000 27.7778 0.0911 -0.0066 -0.7519 -0.5102
0.550 5.0000 27.7778 0.0910 -0.0066 -0.7531 -0.5110
0.560 5.0000 27.7778 0.0908 -0.0066 -0.7542 -0.5115
0.570 5.0000 27.7778 0.0907 -0.0066 -0.7551 -0.5120
0.580 5.0000 27.7778 0.0905 -0.0066 -0.7558 -0.5123
0.590 5.0000 27.7778 0.0904 -0.0066 -0.7565 -0.5125
0.600 5.0000 27.7778 0.0902 -0.0066 -0.7570 -0.5126
0.610 5.0000 27.7778 0.0901 -0.0066 -0.7574 -0.5126
0.620 5.0000 27.7778 0.0899 -0.0066 -0.7577 -0.5125
0.630 5.0000 27.7778 0.0898 -0.0066 -0.7579 -0.5123
0.640 5.0000 27.7778 0.0897 -0.0066 -0.7580 -0.5121
0.650 5.0000 27.7778 0.0895 -0.0066 -0.7581 -0.5118
0.660 5.0000 27.7778 0.0894 -0.0066 -0.7581 -0.5115
0.670 5.0000 27.7778 0.0893 -0.0066 -0.7580 -0.5112
0.680 5.0000 27.7778 0.0892 -0.0066 -0.7579 -0.5108
0.690 5.0000 27.7778 0.0891 -0.0066 -0.7578 -0.5103
0.700 5.0000 27.7778 0.0890 -0.0066 -0.7576 -0.5099
0.710 5.0000 27.7778 0.0889 -0.0066 -0.7574 -0.5095
0.720 5.0000 27.7778 0.0889 -0.0066 -0.7572 -0.5090
0.730 5.0000 27.7778 0.0888 -0.0066 -0.7569 -0.5085
0.740 5.0000 27.7778 0.0887 -0.0066 -0.7567 -0.5081
0.750 5.0000 27.7778 0.0887 -0.0066 -0.7564 -0.5076
0.760 5.0000 27.7778 0.0886 -0.0066 -0.7561 -0.5072
0.770 5.0000 27.7778 0.0885 -0.0066 -0.7558 -0.5067
0.780 5.0000 27.7778 0.0885 -0.0066 -0.7555 -0.5063
0.790 5.0000 27.7778 0.0884 -0.0066 -0.7552 -0.5058
0.800 5.0000 27.7778 0.0884 -0.0066 -0.7549 -0.5054
0.810 5.0000 27.7778 0.0884 -0.0066 -0.7546 -0.5050
0.820 5.0000 27.7778 0.0883 -0.0066 -0.7543 -0.5046
0.830 5.0000 27.7778 0.0883 -0.0065 -0.7540 -0.5042
0.840 5.0000 27.7778 0.0883 -0.0065 -0.7537 -0.5039
0.850 5.0000 27.7778 0.0882 -0.0065 -0.7535 -0.5035
0.860 5.0000 27.7778 0.0882 -0.0065 -0.7532 -0.5032
0.870 5.0000 27.7778 0.0882 -0.0065 -0.7529 -0.5029
0.880 5.0000 27.7778 0.0882 -0.0065 -0.7527 -0.5026
0.890 5.0000 27.7778 0.0882 -0.0065 -0.7525 -0.5023
0.900 5.0000 27.7778 0.0882 -0.0065 -0.7522 -0.5021
0.910 5.0000 27.7778 0.0882 -0.0065 -0.7520 -0.5018
0.920 5.0000 27.7778 0.0881 -0.0065 -0.7518 -0.5016
0.930 5.0000 27.7778 0.0881 -0.0065 -0.7516 -0.5014
0.940 5.0000 27.7778 0.0881 -0.0065 -0.7514 -0.5012
0.950 5.0000 27.7778 0.0881 -0.0065 -0.7513 -0.5010
0.960 5.0000 27.7778 0.0881 -0.0065 -0.7511 -0.5009
0.970 5.0000 27.7778 0.0881 -0.0065 -0.7510 -0.5007
0.980 5.0000 27.7778 0.0881 -0.0065 -0.7508 -0.5006
0.990 5.0000 27.7778 0.0881 -0.0065 -0.7507 -0.5004
1.000 5.0000 27.7778 0.0881 -0.0065 -0.7506 -0.5003
1.010 5.0000 27.7778 0.0881 -0.0065 -0.7505 -0.5002
1.020 5.0000 27.7778 0.0881 -0.0065 -0.7504 -0.5001
1.030 5.0000 27.7778 0.0881 -0.0065 -0.7503 -0.5000
1.040 5.0000 27.7778 0.0881 -0.0065 -0.7502 -0.5000
1.050 5.0000 27.7778 0.0881 -0.0065 -0.7501 -0.4999
1.060 5.0000 27.7778 0.0881 -0.0065 -0.7501 -0.4998
1.070 5.0000 27.7778 0.0881 -0.0065 -0.7500 -0.4998
1.080 5.0000 27.7778 0.0881 -0.0065 -0.7499 -0.4997
1.090 5.0000 27.7778 0.0882 -0.0065 -0.7499 -0.4997
1.100 5.0000 27.7778 0.0882 -0.0065 -0.7498 -0.4997
1.110 5.0000 27.7778 0.0882 -0.0065 -0.7498 -0.4996

[CWA]

Z, thanks for explaining the behaviour; I like the dumb-bell analogy, it helped alot. Having experimented with extreme values of Izz, RoG, DI, I can now actually see what you have described is happening. The behaviour just wasn't as obvious as I had first thought it might be, perhaps because of the simplicity of this analogy?

Thanks Bill Cobb. I really appreciate the numbers you've posted, I'll be running through them soon. One initial observation I've made is that you have separate sprung and unsprung Izz values but have not separated the sprung and unsprung mass values. Could you explain the reason for this? As I have one full car mass value (split into two axle loads like yourself) and one full car inertia value.

Anyway, starting at the top of your post, all I've managed to do as of yet is check my equation for Beta-dot. It comes from RCVD equation 5.2 (Page 146). Derivation as per the book is:

ay = (V*r) + v-dot (V=forward speed, r=yaw velocity, v-dot=lateral acceleration? (unsure of this one though))
ay = (V*r) + (V.Beta-dot) *
ay = V*(r + Beta-dot) (5.2)

I then rearranged for Beta-dot myself, as follows:

ay/V = r + Beta-dot
(ay/V)-r = Beta-dot
Beta-dot = (ay/V)-r

Substituting "Fy=m*ay", or "ay = Fy/m":

Beta-dot = ((Fy/m)/V)-r
Beta-dot = (Fy/(m*V))-r

I have also assumed that, in the second line of the Milliken equation's I have included (*), the substitution they have made relies on the following, which makes sense to me:

meaning: v-dot = V.Beta-dot
meaning: Beta-dot = v-dot/V
meaning: Beta = v/V (v=lateral velocity, assuming V stays constant)

If you could point out the error I've made, I'd be very grateful. I'll bet it comes down to my sloppy algebra, but currently the rearrangements I've made make sense to me..

"In your own equation, if the sideslip effect were zero, would you believe Beta dot = r ????"

I'm afraid I can't answer this at the moment, this is what I'm going to try and get my head around next. Or maybe last, after I've been through your numbers!

[BillCobb]

I gave you a trapdoor to fall thru and you didn't get caught by it, so I believe you are trying to get your head into this. So the Great Karnak will provide you with the answer.

Here's your sim. Sorta lika Toyota Camry:

WF=800 kg
WR=450 kg
L=2644 mm
SPEED=100 kph
SS=1.00 Steering gain of the car (g/100deg/swa)
SWA=10 The steady state steer value deg
DF= 6 deg/g front cornering compliance Its the total front axle sideslip stiffness= a tire + the chassis rubber, bending, kinematics, camber and other crap
DR= 3 deg/g rear cornering compliance Same deal

Some redefined constants to go from industry design units to college simulation units:

WB=L/1000.
U=SPEED/3.6
A=WB*WR/(WF+WR)
B=WB*WF/(WF+WR)
ACKPG=57.3*9.8*WB/(U*U) The ackerman gradient is the vehicles inherent stability gain just because it has a wheelbase.
BETAPG=DR-(ACKPG*B/WB) This is the vehicle sideslip gain (don't need no stinking sim to get that...)
IZZ=(WF+WR)*A*B (Well ya gotta have something for this so make shit up)
K=DF-DR Its the understeer kiddies !
SR=100./((ACKPG+K)*SS) What steer ratio does it take to give you a gain of 1.00 given a wheelbase and an understeer
CAF=WF*9.8/DF Fake tires produced from cornering compliance and NO chassis compliances.
CAR=WR*9.8/DR Fake ones are just as good as real ones if they are the right size and shape...

So this is the stuff to run thru you integration loop:

STEER = SWA* whatever the step function is (pure step, ramp, have her sign, etc
DELTAF = STEER/SR the actual front wheel steer angle
ALPHAF = BETA + A*R/U - DELTAF the front slip angle
ALPHAR = BETA - B*R/U the rear slip angle
FYF = -CAF*ALPHAF let the force be with you [You knew that tire stiffnesses are actual negative, right ???]
FYR = -CAR*ALPHAR FYI its the REAR force
RD = 57.3*(A*FYF - B*FYR) /IZZ yaw acceleration
BETAD = 57.3*(FYF+FYR)/(WF+WR)/U - R See! I was just testing you Keep your units in engineering form, not science spew.
R = INTEGRAL(RD) Integrate your schools Don't forget there is an initial condition, even if its zero.
BETA = INTEGRAL(BETAD) Same to sidestep the issue
AYG = U*(R+BETAD)/57.3/9.8 We all want lateral acceleration in Gs


Cycle thru all that and you will have a winning entry. I'd suggest a Visual Basic Excel function to pop the integration for you.
Post a plot. Doing this in Simulink is a better way to, Student version will do this and much more complicated (3dof + power steering) systems.

Good luck. Thanks or trying.

[CWA]

Well I'm so confused right now, that I'm not sure if the trap door was the Beta-dot equation comment, or your use of two-inertia values?!

Anyway, I've attached a plot of my slip angle responses (left) against your's (Bill Cobb - right). I seem to be out be at least one order of magnitude, and then some..

1. BC my initial question is, how come your front slip angle does not equal your steer angle on the very initial steer input step, at around 0.01s? I have included steer angle in my plot to reflect this, where front slip angle equals steer angle on initial steer input (if I were to include steer angle on your plot BC, it would make the slip angle responses rather unreadable). This outcome is confusing because:

Where; alphaf = Beta + ((a*r)/V) - delta
(alphaf=front slip angle, Beta=body slip angle, a=front axle distance fron CG, r=yaw rate, V=forward speed, delta=steer angle)

On the initial steer input step (0.01s), where Beta=0 and r=0,
alphaf = -delta

But where your steer angle is 5 degrees, your immediate slip angle is only -0.5 degrees
?

2. The trends of the curves look similar, except my slip angles do not seem to reach a steady-state like yours BC (I will next try extending simulation duration to see if my slip angles do flatten off..). My first overshoot peaks also seems to occur much after yours do (my ~0.9s to your ~0.6s). This could be down to different inertia values (?), I ended up using 600kg.m^2 for my single inertia value, but I am still confused and unsure about how you included inertia BC?

Thanks for the further info in your last post too, I'll start having a plug at that to see what I can make of it!

[Marshall Grice]

On the initial steer input step (0.01s), where Beta=0 and r=0,
alphaf = -delta

But where your steer angle is 5 degrees, your immediate slip angle is only -0.5 degrees
?

looks like his steer angle is of the steering wheel and your steer angle is the angle of the front tires. so his alphaf does in fact equal -delta at the instant of step steer input, it's just much smaller than yours.

[CWA]

Blimey, I believe you're right! Steer Ratio = 10 is too much of a coincidence for you not to be right! *blush*

Let me run it all through again! Thanks!

[dynatune]

Maybe some suggestions that could indicate the area of interest for improvement. Typically a step steer response is being applied to a linear system or if the system is non-linear the test is performed in the range where the system should react linear. For a bicycle model there are various methods available to create a linear step steer response. Most important one of these is the Laplace Transformation that provides very accurate results. Taking the Laplace Transformed System responses as a valid reference should be a very good indicator where your excel integrated model shows difficulties. If for instance in your excel model dampers are linear and all your equations for the bicycle model are linear you might have a problem in your tire model and so on. Try to eliminate errors by elimination of possibilities.

Cheers,

Dynatune, www.dynatune-xl.com

[BillCobb]

Typical use of a step steer test is not limited to model validation. The purpose is to run increments of steer angle from say 5 degrees almost up to the limit of lateral acceleration or until the vehicle has insufficient engine power to maintain the constant speed requirement. High gain vehicles may have to start out at 2.5 degrees. The test procedure is not for amateurs. When you are testing a very high performance car (I didn't say high lateral capability car), the response times can be so short that the driver's input signature can be a factor in the result. For that reason, evaluation of the steer response time is necessary to make sure the input is fast enough to beat the vehicle's response.

From this data, each steer segment is analyzed to produce the steady state responses for yaw velocity, roll angle, lateral acceleration and sideslip angle. Also computed are the response times, defined as the time from 50% of steer angle input to 90% of the steady state value of the signal being analyzed. This is all done from the standpoint of real world routine tests that use steering wheel stops and sideslip transducers, etc. Sometimes dozens of these tests are run per week for various manufacturer and competitive cars. Usually there are multiple payload load conditions AND an evaluation of the vehicle at GVW load on its compact or high pressure restricted use spare tire.

Then, functions are evaluated to analyze the linear AND non-linear results, usually from the basis of derivatives. Thus, the understeer gradient, roll compliance gradient, sideslip gradient, steering gain and cornering compliances are computed, usually at 1 or 2 specified lateral acceleration levels. Same for response times. The method shows characteristics typical for each class of vehicle at the payload specified. For example response times at 0.3 g are of interest to car folks and the rates of change of response times with increasing g-level are of interest to truck types. Spare tire analysis is done because there are almost always specifications on the vehicle remaining understeering up to some g level.

The test is also of interest to trailering types and the additional signals for tow angle and trailer responses should be expected.

Testing a full blown race car (or racecar, or rice cah) at operational speeds takes a bit of real estate or a couple of crossing airport runways. Test results often show some undignified 'engineering' snafus and departures from what the old school know-it-alls said would be found. Torque wheels are a special feature of these tests because the results are used to refine the tire data models that are collected from flat belt machinery. Also, drive torque distribution can be observed (say it ain't so) and is often found to be contrary to popular belief.

Obviously the tests are run in both turn directions and for reasons I hope you are able to think through.

But, its NOT just a simple quick step. That's what goes on in "Dancing With The Stars".

There is an ISO procedure for this test. Obviously its also a great companion for simulation since things like tires, roll bars, inertias, damping and the signs of things can get in the way of reality when you are inebriated with the exuberance of your model's ability to actually reach a steady state instead of the usual growing pain and torture of blowing up the computer.

You should discuss the constant radius test next. Certainly the most efficient, practical and high value road test (IMHO). Anybody want to spew on that one ??

[dynatune]

If I am correct the "initial" question in this thread was focused on "starting with a simple 2DOF Bicycle Model" and in that spirit the answer must be seen of using Laplace Transformation.

In my dark past I have been responsible for several years for all handling related issues (testing & simulation) for a major OEM in Europe and developing these test procedures/correlations was one of my missions. I believe we were in those days also one of the first to put an F1 car through full objective testing (with slick tires !) and running full vehicle steering robots to eliminate all the driver problems that are being mentioned. We also ran differential GPS to let a robot car drive itself all ISO procedures. We were in those days also probably one of the first to establish correlations with high DOF multi-body-simulation models. The "problem" we encountered in real life testing was many time the non-repeatability of results of high-g testing or said differently "to high saturation levels of the tires " testing. On a car with standard Mue of 1 doing a step steer to 0.8/0.9g sometimes worked and sometimes not because the car spun out with corresponding lack of repeatability of data. The tire wear was always an issue because after a few runs the tires degraded/overheated, the ambient conditions were too important for high-g testing (humidity, track temp). Same thing with high-g (high tire saturation) frequency response test, nice to have a look at but not robust enough for 100% certainty. The so-called "confidence level" of the testprocedure (which includes preparation of the car by various mechanics, executing the tests on various days, analyzing the test results with various post-processing approaches) was not high enough and made all dynamic high-g testing itself to be treated with proper care. That's is why many OEM's use the step steer and frequency response test mostly in the linear range of the tire (which can go up to 0.7 Mue) to check the correlation of their models with reality and tune the car from there on virtually (which works as we can seen nowadays quite well with all the multi-link kinematics & compliances).

The constant radius test is indeed one of the most efficient test, but it seems that this test is nowadays superseded by the constant velocity full lateral acceleration sweep because the tire wear is less and due to the lack of traction forces (which are in the constant radius test since it starts from 0 kph) less sensitive to "error states".

Cheers,
Dynatune, www.dynatune-xl.com