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12-08-2015, 11:21 AM
Vehicle Quasi-Statics 101: Testing Your Sim Model And /Or Your Car
This started life as a Word document so it kinda fell apart as it got pasted in here. Oops.

OK, to establish some facts on these cars amongst all the claims, noise, religion and trivial pursuits, I want to submit a few procedures that can be used to MEASURE some significant design parameters of them. These procedures have been used industry wide to evaluate vehicle performance relative to design intent. Believe it or not, vehicles today are designed around a set of constraints, goals and specifications that are flowed down into hardware, as opposed to slapping some metal, plastic and rubber pieces together, hoping for the best and changing parts and adjustment holes until you run out of time.
The first one is a simple but elegant way to measure the understeer of a car. The procedure has been called a Characteristic Speed Test or more simply a "C-Speed" test. That name used to be referenced when 'characteristic speed' was a favorite term. The Characteristic Speed is the speed of an understeering road vehicle at which the steering gain is 1/2 of the initial open loop gain. For the oversteering vehicle, there is a Critical Speed where, you guessed it, another factor with a 2 in it. The reason to know the approximate understeer of any vehicle is to gauge the sensitivity of steering and response bandwidth to variations and deviations in chassis, tire performance and operating speed parameters. Now I’m personally not a fan of using understeer/oversteer (tight or loose) because the metric doesn’t identify which end of the vehicle is at fault. In fact, you can ‘fix’ a vehicle in these conditions by manipulating the wrong end of the car.
This procedure could be, can and should be used on simulations of handling to determine recognized metrics indicating potential and real performance as well as parking lot tests with all the bells and whistles working on a real live car. It’s also just a great way to see what the laugh factor of your simulation is.
All you need is a yaw velocity or correct lateral acceleration signal and a forward speed signal, plus a tape measure to determine the wheelbase. You also need some way to hold the steering wheel fixed during each road test segment. We often used a seat belt pulled tight through the steering wheel rim to do this.
The test utilizes the notion that the decreasing, increasing or unchanging curvature of a vehicle path as speed (hence lateral acceleration) increases with a fixed steering wheel angle is a direct indication of its understeer or oversteer gradient.
In the good old days ('60's), a lunch box with a rate gyro and a fifth wheel were all that was needed to grab a car, head to a small test pad and get a plot on an analog paper sheet plotter to get a PFC measurement. The box had a built in analog multiplier to produce the lateral acceleration and a divider to produce curvature analog signals wired to the plotter. Even before that, a checker-boarded road and a movie camera were used to compute the speeds and yaw rates frame by frame. In the late '90's and 2000's, I used a VBOX to measure the two signals and reproduce values nearly identical to results from step, frequency response, and constant radius tests. Today, who knows what you guys will dream up.
The procedure is as follows. I’m focused on the road test. A simulated test is a no brainer. (At least I can imagine that).
Fix the steering wheel rigidly to the car body somehow and slowly and smoothly increase throttle until you reached a high lateral g level. Best to use a fixed transmission gear for automatic and manual cars for each data segment that can cover the desired range of g's without jerking the vehicle because of shifting.
Run the test as multiple segments covering several different fixed steer angles and both turn directions. Yes, this is how you can measure the understeer of a boat.
With the data piped into Matlab, Excel, Fortran, Basic or a calculator, you can finally get to the end of your search for the Holy Grail of your first New School Vehicle Dynamics report card.
I’ve attached two summary examples, one to show it works and the other to show what you might find out. Sometimes the truth is stranger than fiction.
First example is the use of a full blown nonlinear handling simulation throttled down to only a few linear constants which define a known, installed value of total vehicle understeer. I.e. I put into the simulation parameters that deliver the constant understeer response I want the data processing to deliver. I install specific values, turn the crank and compute the metrics I'm expecting from the simulation data. Yes, it works.
Next, I up the ante with generic FSAE car parameters (171 kg.) and add a few likely nonlinear effects (steering compliance and some nonlinear tire data with some forward TLLTD). You can see some of the effects of steering compliance and TLLTD effects in the plotted results. BTW, I'm not trying to deal out a full blown FSAE car because I don't (want to) know how much geometric and deflection steer you have, but you might want to know it.
Using both quasi-static simulation and road test results I'd bet we'd all want to see some results for a few sims (Danny ???) and some real track data (come on, cough it up). Once you have a tool like this, from simulation and road test, you will know exactly how to apply tuning and balancing changes to get a decent car. It should be obvious to you all that having a decent trustworthy and validated simulation cuts through all the lore and guesswork on the way to production of a really great vehicle.
Anybody got Game ? If some of you are still awake after contemplating this technique, I will post a vehicle DYNAMICS test procedure to make you really look and sound professional. Remember, the Human Factors aspect of vehicle design and specification is just as important as durability, etc.
Here's some data processing code in Matlab:
%% Constant Steer Increasing Speed Test: The C-Speed Test Bill Cobb Vehicle Dynamics 101
% Take out the trash: delete possible residue from previous activities:
clear all
% Get me some SIM or DAC data:
csmp_read % get simulation data
% Discard the startup transient if you have one:
inx = find(TIME > 1); % skip any blips.
URG = URG(inx); % yawrate * speed = ~ay(ss)
YAWVEL = YAWVEL(inx); % deg/sec
SPEED = SPEED(inx)*.277778; % Convert from km/hr to m/sec
L = L(inx); % Don't start sweating, this is a sim artifact to transmit the wheelbase parameter
% Turn the crank:
wb = L/1000; % convert wheelbase to meters
r = YAWVEL*pi/180; % convert yawrate to rad/sec
radius = SPEED./r; % computed turn radius
curvature = r./SPEED; % the math majors term for it
b = spap2(2,4,URG,curvature); % Spline fit. Use polynomial if you don’t have spline toolbox.
db = fnder(b); % derivative function
d = fnval(db,URG); % evaluate the derivative over the range
k = -wb.*d*180/pi; % understeer deg/g over the range
plot(URG,k,'r', 'LineWidth',2);
xlim([0 2.0]); ylim([-2 12])
xlabel('Steady State Lateral Acceleration (g)')
ylabel('Understeer (deg/g)')
supertitle(['Characteristic Speed Test Results']) % this function is from the Matlab Users Group.
title({[title2]},'Interpreter','none') % Use a title passed down from the sim
sidetext(['Bill Cobb zzvyb6@yahoo.com'],'Interpreter','none');
grid on
legend('Installed Understeer in this vehicle is 3.0 deg/g') % comment only for this example
%legend('Understeer in this vehicle is from TLLTD and steer compliance')

12-09-2015, 10:11 PM
To establish even more facts on these cars amongst all the claims, noise, religion and trivial pursuits, I offer another procedure that can be used to MEASURE some significant design parameters that indicate gains, bandwidth, response times and damping , you know TRANSIENT RESPONSE stuff. The ‘dynamics’ in Vehicle Dynamics. This one is a simple but elegant way to measure them. The procedure has been called a Frequency Response Test or more simply an "FR" test. It is a foundation procedure used in most Control Systems Engineering processes to synthesize and analyze all sorts of linear and (Yes) nonlinear electronic, mechanical and fluidic systems. Synthesize means we want to know: what are the properties of the components of a 'plant' such that under closed loop control, it achieves desired response characteristics? That implies we either can produce an appropriate and practical tire, suspension, steering and mass distributed vehicle given some constraints (total mass, wheelbase, etc., and some human factors requirements: max steering effort, max steering wheel rotation, lateral acceleration max and phase preference. These are items pertaining to Performance Engineering.

This procedure could be, can and should be used on: simulations of handling to determine recognized Metrics that indicate potential and real performance and on-road tests with all the bells and whistles working on a real car. For simulations, it helps you pass the Laugh test.
All you need are steer angle, speed, yaw velocity, corrected lateral acceleration and forward speed signals. A sideslip transduce is handy and useful but not absolutely necessary. The test utilizes the notion that the open-loop system (i.e. the car's vehicle dynamics) is compatible with the driver's sensation and strength abilities in closed loop control. An analogy often used in course work on this subject are music systems with speakers, amplifiers and media. All must complement each other to please a listener.

The most common procedure is as follows: After the vehicle is washed and gassed and the tire pressures set, you head down a straight flat road at a constant designated speed (the faster you go, the more exciting it gets). With a fixed throttle or speed, you begin a maneuver referred to as a 'chirp' it's a frequency modulated constant amplitude steering wheel motion. Some drivers use modulated sine wave forms, others can do bang-bang pulse moves. You collect all your motion data hopefully at a reasonably high sample rate 50 - 100 Hz, for a length of time that produces a nice happy power of two number of scans per channel (like 512, 1024, 2048, 4096, you should be smart enough by now to appreciate this sequence). Doing this a number of times to get a half dozen or so 'segments' of this maneuver is typical for on the road tests and not necessary for simulations. This is done to calculate the transfer functions obtained from the data channels with an aura of confidence called 'coherence'. Coherence is simply the R2 correlation coefficient function for each frequency point member of the transfer function. We like high correlation coefficients because it indicates consistency for all the runs (segments) you made and that your car is a 'stationary' system (i.e. minimal time dependent parameters). Yes, yes, yes, tires warm up, shocks heat up and leak, you burn off some gas and your driver sweats quite a bit. But what was your coherence level? High or low? In reality, friction can be a game changer in coherence but that's a learning feature of this analysis. Friction is usually not your friend in high speed vehicle dynamics.

OK, so now we turn a crank on the data and compute some transfer functions that indicate the amplification or attenuation of one data channel by another as well as the phase angle. Phase usually means 'lag' if it’s negative in Vehicle Dynamics and 'Holy Crap' if its positive'. A list of typical metrics calculated for these functions include 'Steering Sensitivity' gain, roll gain, yaw peak to steady state ratio (yaw damping), and lateral acceleration response time to name a few.
From the steering sensitivity you compute the understeer, from the sideslip gain (if you have the signal) or the yaw velocity minus lateral acceleration phase difference) you compute the rear cornering compliance and the sum is then the front cornering compliance.
That’s a LOT of useful information about a car, especially if you have a bad one and want produce a good one.
To guide your test driver for producing the best possible chirp steer input, I've produced a tone signal they can hear on a cd or laptop.
After a few practice runs, they usually get it just right for consistent amplitude, frequency sweep and duration.
But what if your driver is an idiot or your Walmart parking lot test area is too full of texting shoppers? Then, there is an alternative procedure with a little more variability but much less trouble. You put in a finite pulse type steer input with a decent length steer hold-in. The PSD of the input (power at frequency) is very different, but you might get away with such and input. You certainly do in a simulation. The road test is harder because the driver’s attempt at a square wave can be slack.

Here are two examples, one to show it works and the other to show what you might expect. Here I'm using a full blown nonlinear handling simulation throttled down to only a few linear constants which define a known total vehicle. I.e. I put into the simulation a few parameters, turn the crank and the data processing delivers the expected outputs. First, I do the simple primitive model followed by the Dream-On FSAE car.

As some Einstein out there may imagine, the gain and phase plots produced by the data processing can be fit to analytic transfer functions (you know, complex numerator and denominator stuff). Once you have these, you can ask Matlab etc. to compute response times, damping, and natural and damped peak frequencies. THEN, if you can get this far, you won’t be waiting in line very long at the Human Resources hiring tent at a Job Fair. Remember, this is New School, it’s all about designing to specification in order to deliver required performance. Now go and check out the plots… All you need next is the appropriate metrics from these results in terms of Control Systems Engineering performance indicators.
Anybody got Game ?

Matlab code should follow. The Topic Manager thinks some of the Matlab statements are pictures and rejects the coding listing example.

Marshall Grice
12-10-2015, 12:08 PM
for posting the matlab code you might find the following BBCodes to be useful, {code} and {/code} (where you replace the braces {} with brackets [] ). it keeps the software from trying to get too smart with your code.

as an example

insert code here

12-10-2015, 09:16 PM
But that didn't work. So, let's try another work-around...
Take a look and the snip:

12-10-2015, 09:36 PM
Heres the transfer function fitting routines. This code was copied from my GUI function that does the FR data processing, hence references to the 'handles' structure. "Fit" is a Matlab routine that works with a pattern regression formula.

Here are the fit-types:
handles.roll_fit = fittype('(n0/sqrt(16*pi^4*d2^2*x^4+4*pi^2*x^2*(d1^2-2*d2)+1))','coeff',{'n0' 'd2' 'd1'});
handles.yaw_fit = fittype('sqrt(4*pi^2*n1^2*x^2+n0^2)/sqrt(16*pi^4*d2^2*x^4+4*pi^2*x^2*(d1^2-2*d2)+1)','coeff',{'n1' 'n0' 'd2' 'd1'});
handles.ay_fit = fittype('sqrt(16*pi^4*n2^2*x^4+4*pi^2*x^2*(n1^2-2*n0*n2)+n0^2)/sqrt(16*pi^4*d2^2*x^4+4*pi^2*x^2*(d1^2-2*d2)+1)','coeff',{'n2' 'n1' 'n0' 'd2' 'd1'});
and the processing:

% compute the xfer function parameters:
handles.ssg = fit(handles.f,handles.ss_gain,handles.ay_fit,'star t',[.01 .015 handles.ss_gain(1) .01 .1]);
handles.yawvg = fit(handles.f,handles.yawv_gain,handles.yaw_fit,'s tart',[handles.yawv_gain(1) .1 .01 .1]);
handles.betag = fit(handles.f,handles.beta_gain,handles.yaw_fit,'s tart',[.5 handles.beta_gain(1) .005 .05 ]);
handles.rollg = fit(handles.f,handles.roll_gain,handles.roll_fit,' start',[handles.roll_gain(1) .005 .04]);

12-14-2015, 10:22 PM
If I understand correctly, the end result of the analysis returns a series of decoupled linear transfer functions in the s-domain. This is useful because there is a wealth of vocabulary and techniques available to analyse the system once you get a plant model in the form of a transfer function.

The resulting transfer functions are linear. However, I know that the tire behaves in a non-linear fashion. Lets say I want to characterize the limit behaviour of the vehicle and start working in the region where the tire forces start acting very strangely. Could I still apply the same concept where I apply a chirp steer input with a DC offset? Would the resulting transfer functions have any useful meaning?

12-15-2015, 11:01 AM
The technique you describe (step or chirp with dc offset) is the most common form of analyzing such behavior. Some people are disturbed by the use of "Linear Analysis" techniques on nonlinear systems, but a car's nonlinearities are all continuous except for friction. They are simply softening springs. So, testing in math or on the road about a fixed oiperating point (as you describe) still valuable. The downside of this is the amount of roadway needed to accomplish the test in real life. However it's perfect for testing the procedure at a circular or oval track.

If you are clever, fortunate, ambitious or dedicated enough to make use of a vehicle handling simulation with nonlinear capabilities, it will be readily apparent that the nonlinearities won't interefere in your 'fingerprinting' of base car performance at any speed.

12-16-2015, 03:33 PM
Another jewel in the analysis and testing crown of simulations AND actual vehicles is the Constant Speed Step Test. The objective of this analysis is to characterize the response to steering up to the limit of capability by means of a series of incremental step steer inputs. A step (or actually a step-like) steer input is used to excite the yaw, sideslip roll and lateral acceleration responses of the vehicle. Traditionally performed by test drivers with the aid of steering wheel angular limiters, the availability of steering robots in the industry makes their use practical. Acquiring data for 10 - 15 runs in each turn direction pretty much defines the success or failure of the vehicle to live up to its handling design goals.

A lot simpler step steer contraption/mechanism can be employed for FSAE cars. This test tends to require a lot of real estate to get a car up to speed, perform the maneuver, and return for another run. A large parking lot of the crossed runways at a small local airport have been used.
Results consist of cross plots of steady state steer angle, yaw velocity, roll angle sideslip and response times (time to 90% of steady state value) for these response variables. Findings and learnings from these plots indicate desirable (and also undesirable) handling design factors. It is common to combine the step steer application with braking, acceleration and other disturbance inputs to quantize the stability characteristics of vehicles.
This test procedure provides high value to simulation studies of a vehicle design before actual construction begins. Testing of actual beta or prototype builds should provide confirmation of a design’s intended behavior and probability of success. I imagine that Design Judges would be impressed by the presentation of results from any of these test procedures.

Processing of data produced by simulation or actual tests is remarkably easy, although road test signals usually have enough confounding spurious trash content to bewilder and confuse even the most patient engineering technician. For that reason, I am a proponent of using software techniques to fit the raw data to equations which are the actual solution to the expected vehicle dynamics responses. This eliminates all the crapola resulting from road irregularities, tire imbalances and other force and moment offsets. The solution equations (found in any table of Inverse Laplace Transforms) consist of a few constants, an exponential and a cosine function. A few lines of Matlab code (for example) are all it takes to reduce the data, take some key derivatives, and compute gains, frequencies and damping factors over the range of lateral acceleration achieved. In the early 90’s, I used Lotus 1-2-3 and Fortran programs to do this same thing with quite a few more steps (Never did get the hang of Symphony, though).

As a final comment, I note that discussions about ‘vehicle dynamics’ usually mention the ‘neutral’ car, ‘yaw damping’, and a few other code words for ‘I have X spurteze’. Well, here’s where all the dirty laundry gets hung out to dry. Your measured steering gain will show you under or over steer all the way out to the nerve endings of your driver. The response times and slope of response times vs. Ay will show you if you have enough tire on the car and if your TLLTD is set properly, and the presence of any overshoot in your simulated or road tested Ay, Yaw rate or Sideslip data channels will dispel any doubt about how ‘neutral’ your car and how ‘critically damped’ it is. Say it ain't so, Weezer.

Here’s a slice of the data processing code that gets this job done:

First is the generic fitting function. I even fit the steer channel to this to determine the quality of the step and the satisfactory timing of it. The 50% response level is used for time of steer initiation.

function f=cr_fit(x,time)
% step response function solutionn model.
% might want to consider using abs(x(4)) to keep the fit consistent.
% Matlab will try using + and - parameters.
f=x(1) + x(2).*exp(-x(3).*time).*cos(2*pi.*x(4).*time + pi/180*x(5));

Next is the production of coefficients using least squares curve fitting and the generation of the fitting function.

Yaw velocity signal example:
R0=[YAWVEL(end) 10*YAWVEL(end) 2*pi 1. -20] % initial coefficient guesses
R=lsqcurvefit('cr_fit',R0,TIME,YAWVEL,[],[],options) % turn the crank
R_fit=cr_fit(R,TIME); % comparison data
plot(TIME,YAWVEL,TIME,R_fit) % make a comparison plot.

Compute 90 percent response times (industry standard of time to 90 percent of steady state value:

rsp=spline(TIME,R_fit-.9*R_fit); % spline the data shifted down 90 percent
Tau_R=fnzeros(rsp); % compute the zero crossing of this shifted signal.

Second verse, same as the first for Ay, Roll, Sideslip, Net Tierod Load, Wing Force, etc.

AY results are in the figure.

12-18-2015, 03:53 AM
Hey Bill, I'll have a crack at the simulation part of things.

I did the characteristic speed test by starting at 18 km/h with 5 degrees of tire steering, then accelerating up to ~92 km/h at around 0.1g. I added some SAT compliance steer, but I can't say I know exactly how much Fy/SAT compliance steer we have. One day I'll get some proper K&C values...
It seems posting properly sized images inline is far more difficult than this analysis, so all the images are attached as thumbnails.

For the frequency response test I ran at 100 km/h with 3 degrees of steering. My post-processing leaves a little to be desired, but nevertheless they are there. I fit your given transfer functions to the data, but the roll gain transfer function seems to not play nicely with negative gains, so I left it as is.

I hope to have some physical data for all of these tests at some point in the future, until then I guess it's up to you to figure out the simulation's laugh factor.

Also thanks again, your posts are gold when it comes to vehicle dynamics, especially on topics where there is very little published literature.

12-18-2015, 11:08 AM
This analysis is A+++.
There is no Laugh Factor needed for anything presented here. Your Sim results look like what I found from my own based on a few known and some guessed paramters. Your steer ratio is a bit lower than I chose which produces the higher steering gain. Also, "we" generally run Roll by Lateral Acceleration(g) instead of roll by steer, although your results have some eye opening features. And, It's typical to invert the signs of roll and sideslip in road testing so that the data acquisition screen size and resolution shows to the driver signals all going 'up' for right turns. (Just a convention).

As for aligning torque compliance steer, this is always a stiffening spring mechanism. A simple but satisfactory model of the data for simulation purposes is a log function defined by just two parameters. First is the initial slope and second is the loading for 67% of the highest deflection. For production cars, the initial slope ranges from 0.3 to 3.0 deg per 100 Nm and the range factor can be 30 to 300 Nm. The 'best' cars are the 0.3 and 300 jobs: The most linear over the amount of MZ that can be generated. Here' a typical 'good' plot'. A 'bad' results can be soft and change quickly or even worse, be bi-linear, as in loose parts, really bad gear mounting, structural failure, and u-joints buckling.

A typical good use for the C-Speed test is examining the effects or 'workings' of locked, or limited or selective limited slip drive differentiials. You will often see lurches in the understeer curves as the diff messes with curvature, especially when it grabs or lets go at inappropriate times and places. Aero effects are also well suited to studies with this test. A constant radius test is sometimes applied to these studies, but the effects turning on and off cause problems for the driver trying to "stay the course", so to speak.

Now you have the opportunity to pull your Team out of the garage and into the Engineering Office. Now you can engage Team members specializing in Math descriptions of vehicle subsystems as analytic functions that are piped to your simulation. When approved in the Math, your hardware specialists can get to work with quite a bit fewer adjustment holes.

So, if all these cars are so 'neutral' as the bookworm lore goes, why is your YawVelocity by Steer peak to steady state ratio ~1.133 ? (That translates to .819 damping factor? If this was a 'neutral' car, the damping factor would be close to 1.00 . Sup wit dat ?

Great job. Show us a few more results from TLLTD shifts, a locker diff and maybe a comparison of peak YawVelocity gain frequency vs. Roll by Ay frequency and the speed where they might cross over. Then puzzle yourself with why the use of rear roll understeer in some cars at very high speed have oscillatory handling problems.

Now if you could engage Z into designing a practical K&C mechanical device that could be used to obtain compliance steer and camber and geometric steer and camber for one axle at a time, you could form a company selling the data at all the events. I've actually used an old wheeel alignment machine to do this at an assembly plant. Maybe call your stuff MoreZ MeaZurementZ.

I was hoping to have seen a submission by other simulationists (Hoy Goys...), but all I can imagine at this point is their product leaves something yet to be desired, or the focus on track evaluations eliminates the ability to perform SAE or ISO handling tests procedures.

12-18-2015, 11:43 PM
I chucked a couple of shifted TLLTD values through. Anything lower than about 50% TLLTD struggles to put power down at the higher speeds due to aero drag. A locked diff would help to power down, but as with a lot of things its implementation into my sim is on the horizon.

The Roll by Ay plot has been brought to life, but it also doesn't play too nicely with the given transfer function. It looks about right to me, so am I just completely missing something?

With the yaw velocity vs roll/Ay, I'm not too sure exactly how you can get a cross over point (at least with a FSAE car). When running at different speeds the roll by Ay plot doesn't really change much, but yaw velocity gain increases with speed. Unless I'm missing a unit conversion somewhere it seems like these two plots are a hop, skip and a jump away from being able to intersect. A guess at what you're getting at though is if a vehicle has its peak yaw velocity response at the same frequency (and magnitude) as its Roll by Ay, when driving over rough road/side wind gust/under acceleration (assuming torque steer is a player), these two effects will constructively interfere and make the vehicle get a progressively bigger and bigger tank slapper going on.

12-19-2015, 02:31 PM
Sweet ! Roll just looks heavily damped. Wouldn't your car have some peak roll just from tires, alone, though?

On the other question, roll probably doesn't have a lot of speed dependency while yawrate has quite a bit of. On some production cars, there can be a speed where the peak frequencies cross over. Above this speed roll steer signs invert: rear roll understeer behaves like roll oversteer (which ain't a real good feeling at 180 kph). Corvette is a good example. So, some manufacturere adopt a policy in which roll oversteer is designed into the rear "because it feels so good above 180 kph", other run some seriously stiff roll damping to attenuate the problem and others just can't figure it out.

Problem with the rear roll oversteer theory is that your base cars suffer from slow and lazy responses, require extra stiff rear tire stiffness to get the rear cornering compliance back down and your signature cars acquire a reputation of loosing control by "speening out" and crashing.

12-22-2015, 06:23 AM
Yep, my damping values were a bit off. I gave it another go with a few changes and it's looking a bit more believable. As for now, I think it's time for some eggnog.

01-30-2016, 05:09 PM
I managed to get a simulation up and running. It is an 'idealized' model with no compliance effects and very primitive suspension. Regardless, lets give this a go and see what we get.

The test conditions mirror what Geist ran - 100 km/h with 3 degrees of steered angle. For interest sake, I cranked up the chirp frequency up to 6 Hz just to see what happens at higher frequencies. The regression fits to the magnitude, but to double check I plot the phase as well.

Here is what I got.

EDIT: Re-uploaded plots to be reasonable sized

01-31-2016, 08:53 PM
Use the "unwrap" function to remove the jump in phase plots to make them continuous. To include the effects of suspension compliance, incorporate their action as a calculated loss in tire cornering stiffness. Cornering compliances as equivalent tire stiffnesses, makes this analysis easy. It also reveals how the best tires in the world are easily ruined by having suspension compliances that wash them out.

If you apply this technique to real world road testing, try to develop the finite width pulse step steer input technique instead of the chirp. Your driver will appreciate the shorter segment duration. The steer PSD is different, but as long as there is some power in it, your transfer functions will be revealing.

Since you have produced a ~zero frequency gain value, calculate the car's understeer from it.

01-31-2016, 11:34 PM
Just for fun, I applied the same technique to some random data I have laying around. Formula North has some nice slaloms for this purpose. Near constant forward velocity, equal cone spacing and nearly equal amplitude left-to-right. Combined with a driver that likes to wrestle with the steering, I get a response that vaguely represents a pulse step steer if you squint.

From the DAQ data, the forward velocity is 45 km/h with the steering oscillating between 7 degrees-ish of steering. The lowest logging frequency is 20 Hz, which in hindsight isn't ideal. The analysis suffers and only the response below 3 Hz is observed.

The phase plots are questionable but the magnitude plots could be believable. Lets say the steering sensitivity of the car is 0.135 g/deg just inspection of the steering sensitivity plot. Cracking open a trusty copy of Milliken & Milliken, I can find the DC understeer. The final computed value: 1.5 deg/g.

Some quick observations: Both the steering sensitivity and yaw-by-steer DC gains are a lot lower than what the simulation predicts. The yaw-by-steer response is heavily damped and the roll-off occurs much earlier in reality than in simulation. The data is sketchy at best. This is a car that has been accused of "understeering like a pig".

EDIT: And only now do I realize I am comparing two datasets with different operating conditions. Oops.

05-01-2016, 05:05 PM
I'm surprised this thread and its counterpart over in the 'Static Events' section haven't gained more traction, considering Bill is serving it all up on a silver platter. Anyhow, I thought I'd chuck some more plots in the mix to get the ball rolling again.

Building upon the frequency response plots, I've thrown together the lateral acceleration and yaw rate phases for both 60 km/h and 100 km/h. The Mean Phase Difference can then be computed (yaw - lat accel). You can then churn out the 0 Hz gradient of the Ray function to come up with a rear cornering compliance values of 1.94 and 1.71 deg/g for 60 and 100 km/h respectively. It's a bit of an academic exercise considering it's using tire data and a vehicle simulation instead of real on-track stuff, but it's a bit of fun nonetheless. It seems like the forum isn't allowing me to upload images no matter what I do, so here are some links.

Phase plot - 60 km/h - http://i.imgur.com/2xVAu0C.jpg
Phase plot - 100 km/h - http://i.imgur.com/IGLt72U.jpg

Shifting focus slightly, I've temporarily dropped the non-linear tire model for some nice constant cornering compliance values. The result is a carpet plot of vehicle yaw damping and lateral acceleration response time for various rear cornering compliance (DR) and understeer (K) values. The yaw velocity peak to steady state ratio is from the same old frequency response stuff and the lateral acceleration response time is from a step steer input (calculated from 50% steer to 90% steady state lateral acceleration). The results are pretty much what you would expect, being that while you're cooking up a new vehicle and choose the wrong recipe, you (or the driver) are gonna have a bad time.

Carpet plot - http://i.imgur.com/XdGIUCW.jpg

05-02-2016, 04:05 PM
Not really surprized by the null conversation about these synthesis tools. That's because, really, there are more mechanics than engineers involved in this pendeavor, and the engineers appear to be more fluent in CAD than CAE. That essentially makes their cars 'kit' versions, because a design process to meet specifications alludes them. They just hope their cars don't fail in mass, durability, and performance. ("How much does my skin made up of little triangles and squares weigh ?")

What makes this more difficult is the lack of competitive performance analysis (Called "fingerprinting" in the trade.) You measure the BOB and WOW characteristics (Best of the Best and Worst of the Worst) of several or many different versions of a vehicle and weigh the merits of each design feature based on performance. Then you assign values to your 'winning' choice and flow down the key attributes from the metrics into sub-system and component specifications (I.e Rear cornering compliance and aligning moment compliance steer given a tire cornering coefficient or stiffness. Your (Excellant BTW) carpet plot of performance surface features is esentially the beginning of a good vehicle design. Cost, mass, manufacturability, strength and specifications then assign a pin-point on this map.

Yaw velocity peak to steady state ratio at some (high) speed is a desireable objective metric because it is directly obtained from a step or frequency response test data processing program. But, just in case you are stuck with a controls engineer at the helm, it is easily transcribed into 'Yaw damping', yaw dampin', or yaw dampink' by the following function:

function y=po2zeta(po)
y= sqrt((log(po/100)^2)/(pi^2 +(log(po/100)^2)));

so 4.32% overshoot is a dam pink value of 0.707. Sound familiar ? (I thought so...) At Lateral Acceleration Response time is a good counter measure because it help constrain the cornering compliance specifications. You can actualy make it too fast (as in there's too slow, but don't over-do it because once your driver can't sense it, that's short enough).

C'est la vie.

05-03-2016, 08:21 AM
I think there also isn't many posts here because on most teams you usually have 1 to maybe 3, if you're lucky, team members working on vehicle dynamics projects (plus building the car and school work), so for this time of year you're probably not gonna hear much from them. I know when I was in college from January to May I would just occasionally check the forums here. I also think a lot of people don't necessarily want to post on FSAE.com despite reading it because of the high likelihood of being called an idiot, lazy, etc... So you have a lot of people who just view and learn without actually joining the fray.

Hopefully I can play around with this here soon since I have two off weekends this month.

05-06-2016, 02:46 AM
I would love to post something here and contribute a meaningful amount, but I imagine like others; so far I have only been a student of this thread. I have been learning Matlab for a relatively short time and within my team I'm currently (and have been for a few years) the sole student on kinematics/dynamic behaviour.

I enjoy this thread and check up on it regularly, so don't feel your posts are ignored. I for one appreciate the insight given and look forward to more.

05-06-2016, 10:19 PM
Enf is the symbolic name for front aligning torque compliance steer on some planets: 'E' = steer, 'N' is aligning moment, 'F' as in front.

So, put your car on grease plates, lock the steering wheel in the straight ahead position very firmly, take your torque wrench and a precision protractor, measure and record road wheel angle vs. moment for a series of readings at say 5 or 10 Nm increments up to 10 or so readings in both directions. We would normally do both wheels at the same time on a K&C machine, but wtf, do it one wheel at a time.

Using Matlab, fit a log function to the data and scaled so that parameter 'A' is the initial slope, and parameter 'B' is the 63% moment value to the final limiting steer level. Change some parts (wheel bearing, steering gear preload, steering gear mount type, column configuration, etc.) and show us the pltted results and a table pf the 'A' and 'B' coefficients. Yeah, you can do this in Excel, but that's for accountants.

Nice little project to be greatly valued by all on this Forum. Then try it at a few different steer angles. This experiment is usually the beginning of Reality Checking.

05-07-2016, 11:25 PM
Here's an example of a way to parameterize your data for K&C types of tests: Run it through a function.

Here's the function:

function [enf]= enf_fcn(enfb,enfc,nf)

You could speculate on the looks of data from good cars, bad cars, awful cars and great cars, but a simple test will answer that question directly.

Here is some Matlab to demo it:




enfb2 =.5
enfc2 =100

enfb3 =15
enfc3 =5


legend(['ENFB= ' num2str(enfb1) ', ENFC = ' num2str(enfc1)], ...
['ENFB= ' num2str(enfb2) ', ENFC = ' num2str(enfc2)], ...
['ENFB= ' num2str(enfb3) ', ENFC = ' num2str(enfc3)],'Location','SouthEast');
legend boxoff

xlabel('Aligning Moment (Nm)')
ylabel('Aligning Moment Steer Compliance (deg/100Nm per wheel)')
title('Two Parameter Nonlinear Compliance Steer Model')

Make it so ....

05-08-2016, 10:31 PM
Whether simulation or on road testing, Matlab tools require only the simplest scripting to reduce your raw data to Engineering results. You should not be 'programming in Matlab the way you would in BASIC, FORTRAN, or PL/1. The Matlab functions are already written, debugged and waiting for you to select them.

For example:

Step steer processing. You can either paw through the data segments to get steady state averaged values and run backwards through the segment to determine the time for 90% or 67% or 50% of the steady state value to be reached. Whoopie.
Or, you can fit your channels (including steer BTW) to this function using one of several parameter ftting routines (like LSQCURVEFIT):

function f=cr_fit(x,time)
% step response function solutionn model.
% might want to consider using abs(x(4)) to keep the fit consistent.
% Matlab will try using + and - parameters.
f=x(1) + x(2).*exp(-x(3).*time).*cos(2*pi.*x(4).*time + pi/180*x(5));

This equation contains the Systems Engineering traits of your signals (the X values, silly), including the steady state, phase, and damping factors. After a series of runs in a linear or nonlinear mode, you will have run results suitable to process for gains (steering sensitivity, sideslip, roll, yawrate, cornering compliances) and the responcs times, damping, phase angle vs. Ay, or Steer, whatever you pick. It's that simple.

Frequency Response processing. With a simulation, just a single run is needed. I use a 40.96 second chirp at 100 Hz. sample rate with an initial and final zero steer angle to produce a marvelouse FFT array. On the road we use multiple identical steer attempts and use a process to average the real and imaginary components of the FFT so we get a 'Coherence' calculation. Coherence = consistency, confidence level, repeatability, whatever you want to call it. High coherence = high believeability. It's like an R^2 at each frequency point.

Same input channels now processed in pairs.

warning off
YAWACC = gradient(YAWVEL)./gradient(TIME); % Math channel
warning on

nfft = pow2(round(log2(length(STEER)))) ;
[sstxy,f] = tfe(STEER(:,nseg),LATACC(:,nseg),nfft,sps,hamming( nfft),nfft/2); % ayg by steer transfer function
[rolltxy,f] = tfe(LATACC(:,nseg),ROLL(:,nseg),nfft,sps,hamming(n fft),nfft/2); % roll by ay transfer function
[betatxy,f] = tfe(LATACC(:,nseg),SIDSLP(:,nseg),nfft,sps,hamming (nfft),nfft/2); % beta by ay transfer function
[yawvtxy,f] = tfe(STEER(:,nseg),YAWVEL(:,nseg),nfft,sps,hamming( nfft),nfft/2); % yawv by steer transfer function
[yawatxy,f] = tfe(STEER(:,nseg),YAWACC(:,nseg),nfft,sps,hamming( nfft),nfft/2); % yaw accel by steer transfer function
[rollyawvtxy,f] = tfe(YAWVEL(:,nseg),ROLL(:,nseg),nfft,sps,hamming(n fft),nfft/2); % roll by yawvel transfer function
[rollswatxy,f] = tfe(STEER(:,nseg),ROLL(:,nseg),nfft,sps,hamming(nf ft),nfft/2); % roll by steer transfer function

inx=find(f > 4 | f < .10); % The reality of Bandpass


ss_gain = 100.*abs(sstxy); % convert to Bode form for gain
ss_phase = angle(sstxy).*180/pi;
yawv_gain = 100.*abs(yawvtxy);
yawv_phase = angle(yawvtxy).*180/pi;
yawa_gain = 100.*abs(yawatxy);
yawa_phase = angle(yawatxy).*180/pi;
roll_gain = abs(rolltxy);
roll_phase = angle(rolltxy).*180/pi;
beta_gain = abs(betatxy);
beta_phase = angle(betatxy).*180/pi;
rollswa_gain = 100*abs(rollswatxy);
rollswa_phase = angle(rollswatxy).*180/pi;

plot(f,ss_gain),%ylim([0 2])
plot(f,yawv_gain),%ylim([0 30])
plot(f,yawa_gain),%ylim([0 300])


Yes, you need LSQCURVEFIT and TFE tools to do this job professionally. If you didn't know it already, you can buy a killer version of Matlab for home use if you promise you won't show up on Shark Tank to sell your analysis product. If you really need an extra Toolbox, its $50. Your iPhone bill per month is higher than that. Go search for "Hobby version of Matlab"

All the same parameter metrics mentioned in the step response processing are easily produced from your FR results. Now all you need to do is resolve the discrepancy between what you built and what you intended to build in Math or Material. Don't worry about the nonlinearities. The effects of them are actually very disappointing (like miniscule in the Big Picture of things). Keep your max Ay value below 0.4 and you will live to test another day.

08-23-2016, 04:01 PM
OK, so if some forms of testing are too hard, too complicated or too expensive for a student team, try a very simple test from the infant stages of vehicle dynamics with road cars: The "C-Speed" test. No, its NOT a constant speed test, it was originally planned as a "Characteristic Speed" test. An understeering vehicle's characteristic speed is the speed where the steering gain is down by 1/2 . Caused by the development of understeer with increasing lateral acceleration.

All you need is a yawrate gyro to measure yaw velocity and also forward speed. A borrowed VBOX will do it. That's it. You set the steering wheel to some non-zero value, hold it firmly and start driving with a slight forward acceleration. This creates lateral g's. You process the yawrate and speed into curvature and lateral g's and you get a plot and a function of understeer vs. lateral acceleraton. Since a number of us are still awaiting a realistic test of this metric from a team, I thought I's kick up some dust off of this procedure. Heck, I even ran it on my bass boat.

In the first picture, I show a validation test of the process: I take a 'vehicle' model with an installed linear value of understeer. Second pic shows a likely result of runing a FSAE type car with some typical nonlinear tire and steering compliance factors installed. Kinda goofy, ain't it ? Anybody want to speculate on why the curve looks like a polynomial fit to some funky data ? (Its not, its the effects of the two nonlinearities interacting).

SO, isn't this more exciting than studying boring rules, tube frames, throttle thickness, fuel viscosity, restrictor condoms, cost of manufacturing, etc. or looking at employment ads wanting engineers who can tweet and text at over 1000 words per minute while eating hummus and drinking a Starbucks simultaneously ?

Here's the Matlab code: (BTW: You'll need a committee to measure your car's wheelbase and to vote on what the correct value is).

% Take out the trash: % delete possible residue from previous activities
clear all

% Get me some SIM or DAC data:
csmp_read % get simulation data

% Discard the startup transient:
inx = find(TIME > 1);
URG = URG(inx); % ~yawrate * speed = ay(ss)
YAWVEL = YAWVEL(inx); % deg/sec
SPEED = SPEED(inx)*.277778; % Convert to m/sec
L = L(inx); % Don't start sweating, this is a sim artifact

% Turn the crank:
wb = L/1000; % convert wheelbase to meters
r = YAWVEL*pi/180; % convert yawrate to rad/sec
radius = SPEED./r; % computed turn radius
curvature = r./SPEED; % the math majors term for it
b = spap2(2,4,URG,curvature); % Spline fit. Use polynomial if necessary.
db = fnder(b); % derivative function
d = fnval(db,URG); % evaluate the derivative over the range
k = -wb.*d*180/pi; % understeer deg/g over the range

plot(URG,k,'r', 'LineWidth',2);

xlim([0 2.0]); ylim([-2 12])
xlabel('Steady State Lateral Acceleration (g)')
ylabel('Understeer (deg/g)')
supertitle(['Characteristic Speed Test Results'])
sidetext(['Bill Cobb zzvyb6@yahoo.com'],'Interpreter','none');
grid on

legend('Installed Understeer in this vehicle is 3.0 deg/g')
legend('Understeer in this vehicle is from TLLTD and steer compliance')

08-23-2016, 08:46 PM
The heck with a VBOX, Use this app (or the free version) and use your phone to compute your FSAE cars understeer !


Charles Kaneb
11-10-2016, 10:50 AM
BillCobb, if you're ever feeling underappreciated for your work on this thread, I've talked to two FSAE teams in the last week and they are both using some of your ideas on testing.

12-05-2016, 09:22 AM
BillCobb, if you're ever feeling underappreciated for your work on this thread, I've talked to two FSAE teams in the last week and they are both using some of your ideas on testing.

I'd also like to thank you for these contributions, the Lund team will use this.

And I'm planning to test my own car, as soon as winter lets go of its crippling grip.

12-05-2016, 04:29 PM
Thanks to all for your interest. Let's see some results. Don't have to wait for better weather, do some indoor tests of steering compliance and overall steer ratio. The compliance results will guide you when it's correlation time. The ratio tests will tell you how good your as-built steering geometry is. Build a 'side-pull rig' to evaluate calculated vs. actual lateral load transfer distributions with the car stationary. Do this at several steer angles and see how it changes.

Then prep with a simple but elegant simulation of the on-track tests you intend to run. Take the sim data and post-process it to have a good idea what to expect from your test results: signs, order of magnitude, units, data volume, data editor/extractor, toolbox functions (Matlab kinda stuff), GUIs, summary data metrics, metrics database and graphs.

Then the testing will hit the ground running and you will be pleased with your preparations and be able to run a predetermined set of conditions valuable to your team: tire pressures, springs and bars, reinforcement braces, wheel rim effects, tire temps, static caster/camber settings, etc.

'Knowing' what to expect will allow you to prioritize test conditions. A high volume of outdoor testing will also tax your powertrain and result in valuable durability confidence levels.

12-05-2016, 11:54 PM
A indoor lab test whose importance is so very often overlooked, is a test for overall steering ratio. Simple stuff. Measure steering wheel angle, measure one or both road wheel angles, calculate a slope and you are done, right ?

Not so fast. Because of build tolerances being out of spec, because of part simplification(s), and because of non-coaxial geometric consequences, there can be some REALLY bad results obtained. Most often occuring with a large angular difference between steering column and pinion axes, where no (or a bad) attempt at using a double Cardan joint is made, angular variations in wheel angle displacement and velocity become pronounced. There is a lot more to this than just a little steering gain effect.

Here's an example from the history books.

First plot is just the raw data. Looks pretty innocent. Some curvature because of Ackerman geometry design intentions. Now lets look at it in more detail. The next plot shows the higher resolution derivative of the overall steering ratio functions for both road wheels. This vehicle has a problem with so-called 'lumpy steering'. Because the column was not carefully installed, the input and output angle of the steering intermediate shaft which connects the end of the column to the pinion nose are not co-axial and the phase angle on the shaft is no longer correct for the setup. So, a nominal 17:1 steer ratio has high and low values as you turn the wheel. Corresponding to the steering gain effect, will be a steering effort cycling that can be felt even bt casual evaluators. So the wheel is light in the straight ahead zone where the gain is highest. This tends to feel weird and unnatural when you drive the car straight either with both hands on the wheel or with one of your knees pushing on it as you eat your lunch with one hand and swig a gulp of soda held by the other. Its not really severe, though, where it gets bad is when that min/max gets shifted off to one side by more than a few dozen degrees. Then you can sense a tendency for the vehicle to lead or pull in one direction.

Now, for the few geeks who are still with me, there is another tragic consequence. To compute the understeer of a vehicle from step tests, etc. You subtract the so called Ackerman gain (the wheelbase and speed effects) from the meaured steering wheel angle divided by the overal steering ratio. (but, you see, that's just NOT a constant value in this case). If you DO use a single, constant value, the understeer curve you generate will be distorted by the lumpiness effect. This can lead to all sorts of bad judgements: you have a bad tire, a bad alignment, a badly installed part or some other unknown whimsically blamed item.

To correct the understeer calculation, you would need to use the actual left and right wheel steer angles and here is how you do it. Take the nonlinear ratio data fitting function, integrate it out to the steering wheel angle input value and produce the true reference steer angles.

Here is the other singer, those of you who have some engineering education on control theory, would know that the yaw rate by steer angle transfer function has TWO numerater terms: that means a vehicle at speed responds to both steer angle AND steer velocity. Now take a look onece again at the lumpy ratio function. You should recognize that it is possible to have an increasing steer angle but also a negative steer veolocity. Now you could understand why some cars and trucks seem to have a 'deadband' at some steer angles. It's because your steering commands to the pinion are inconsistent, undesireable, unnatural and explainable.

"Welcome to the New World, Captain Rameous."

Over the years, I quickly learned to value this test by a large amount. If the ratio functions looks goofy, then, most likely, the car feels goofy, the test results will look goofy and the results of other tests at multiple speeds, load conditions and tire contstructions will be a total waste of time and energy.

So, spend a little time in the shop if the weather is bad outside and get this information. The presentation value is huge, the judges will be shocked and unable to talk, and a lot of handwaving will be avoided.

Here is the nonlinear ratio function I've used in this example. Keith Adams came up with this formulation. It handles nonlinear (by design) gears, Cardan joint effects, Ackerman steer geometry, etc. Matlab, of course... I use lsqcurvefit to compute the appropriate and fitting coefficients. A GUI to explore all the possible effects is sorta entertaining, actually. The coefficients that model the plotted data are listed on the plot for both road wheels.

function [RATIO] = sr_func(SR,swa)
%VHTRIMS Overall Steer Ratio fitting function
% [RATIO] = sr_func(A0,A1,A2,B,C,D,F,G,swa)

% sample data typical for full blown curve fit:
%A0 = 15.57 Ratio Points
%A1 * 10^3 = -0.4974 ASYMMETRY
%A2 * 10^6 = -9.4283 BOW
%B = -0.56 LUMPINESS AMPLITUDE (Ratio Points)
%F = 550.82 VARIABLE RATIO RANGE (Degrees SWA)

A0 = SR(1);
A1 = SR(2);
A2 = SR(3);
B = SR(4);
C = SR(5);
D = SR(6);
F = SR(7);
G = SR(8);

F(F==0)=realmax; % realsteve could do the job, too.

RATIO = A0 + A1/10^3*(swa) + A2/10^6*(swa).^2 + B*cos(2*(swa- C)/57.3) + D*exp(-(4*(swa-G)/F).^2) ;

Vishnu Sanjay
08-14-2017, 08:03 AM
First of all, I'd like to thank Bill for all the gold on this thread.

I'm Vishnu Sanjay. I used to be the mechanical lead of the FSE team at my University, and graduated a few months ago. I thought I would share some data that we obtained, to show evidence of how important this kind of testing could be to the teams in this competition. It definitely tells you what to concentrate on improving, before expecting a quick car.
This data is representative of the 4ze Racing car, from SRM University, Chennai, that my batch designed and ran (and won with) at our national competition Formula Green in March this year. The car has already ended its competition life cycle, and is not one that the current team will be taking to another competition.
I think the data shows the cars around here need improving.

First, an example of the steer ratio data. The raw data is quite noisy, due to the free play in the system and the the fact that we used our toe alignment jig (thread running longitudinally from two bars on the front and rear bulkhead) with the test conducted at 2 am and with 150+ manual readings recorded).
The fit is a nonlinear fit of the kind that Bill has specified, and it shows lumpiness and the asymmetry in the steering. Yes, the UJ phase was not installed right, and the manufacturing resulted in the rack being a little off centre.
If you are running the test, do it with digital encoders if possible, like we intend to do in the future. The (large amount of) play at the end does drive the residuals of the fit up.


Next, a plot of the aligning moment compliance of the car. Our car is definitely closer to the 'bad' example in Bill's post than it is to the 'good' one. Comes from thinking of only a few of the many 'springs' that get loaded in series when an aligning moment is applied to a tire.
We used ratchet straps and dial gauges to get this data. The method is crude but if you are clever and constrain the car right, then you could get useful data very quickly.
A future test we plan involves strain gauge rosettes on all the springs to find the 'soft' culprits.


Definitely useful information to have before you start your design stage. If I were leading the mechanical team I would have the vehicle dynamics, steering kinematics AND structural engineering guys all being involved when any dynamics are being checked, just so that everyone understands what they need to do in order for the car to perform as simulated.
For the car the readings were taken on, it happened because we didn't think in great detail about this kind of stuff back then. Kind of disappointing thinking you've designed a close to neutral steer, high-g machine and then finding out on test day that you don't get what you want.

What do you guys think? Do you have any suggestions/experience regarding processing data with a lot of noise/play? (Our SR goes off to infinity at the ends because of the play, and we dealt with it by not including data points near the ends, for the fit)

08-28-2017, 05:01 PM
I am disappointed but not surprised at the lack of response, comments or interest in the steering compliance results posted here by Vishnu. (Commendable job, BTW, and very brave to reveal your data and model coefficients). I mean, of course, the form, traits, values and character of the test results.

No response because the hunch I have has the probability of being 100% correct. You see, chances are good that there are no regular members here who have any ideas on what to do with this information. It's simple, really, the books, the forums, the papers and the plethora of software(s) have no "window" to acknowledge, accept, and incorporate this compliance effect. Words like "tire stiffness", "bump steer" (really when you do the bump???) are apparently the only real and necessary factors which influence vehicle dynamics expectations and results. Right ? Gotta have that "load transfer" and "TLLTD" vernacular probably, too.

So, what's the Big Deal about this data and information? Where does it come from, why is it so important (BTW: it's VERY important no matter how many g's you are pulling). what are the manifestations and infestations of it, how do you incorporate it into simulation tools and how may have it contributed to this being a "winning" car ? (They said it was a winner).

Anybody still awake? Can we read your take on this information, how should it get used and evaluated, why is it so important, and what did it do for this car (if anything) ? No further points for absent answers from the Soprano Prima Donnas. Let's discuss vehicle dynamics instead of wheel studs vs. nuts, dental floss for carbon fibre wings or what kind of wax to use for that 4th decimal place aero drag coefficient. Jump into the TTC Forum if you feel its necessary to continue...

08-28-2017, 07:05 PM
I'm with Bill on this. Once you get past the stage of keeping the wheels on the car, then it is time to figure out where they are, relative to the sprung mass. Also, which way they are really pointing (steer and camber) ... under realistic loading conditions. Until you have done some kinematic and compliance testing, you won't have a clue.

For anyone not familiar with the term compliance in this context it is 1/stiffness, the deflection for a given force. For example inches/pound or, more conveniently, "inches/1000 lb" (possibly "M/1000N"). Rotational compliance is in "deg/100 lb-ft" ("deg/100 N-M"). Morse Measurements have some nice tech articles on their site, but you don't need their expensive test time to get some rough data in your own shop.

08-28-2017, 07:41 PM
Thanks, Doug !. It's been so boring here, I went out and cut some hay and baled it.

08-29-2017, 04:35 PM
And thank you both for sparking a very interesting topic! Now where's Z, Geoff, Rob Woods and the rest of the guys? Missed a good old time thread like this. Seems that most of the guys nowadays spend their time on faceplace instead.

Vishnu Sanjay
08-30-2017, 05:39 AM

I was hoping for some contributions as well. When I said my data was from a 'winning' car, I was also being a bit ironical. Somebody like Michael, from Monash, who posted on the thread earlier will probably have values more representative of a winner (They've won 7 of the past 8 Aussie comps, I think?). I was hoping for some feedback from someone like him (although he has mentioned earlier in the thread that he doesn't have any values for these at the moment).

For starters, I thought I'd post a few results from the edited version of a simple sim that you had uploaded on:


I haven't changed much, just the way the inertia is represented (this way it's easier for a team to read the value directly from their CAD package).

The two pics show what happens when your design (or not designed for) compliance is not the same as your manufactured compliance. The 'car' still has the same 'balance' or 'understeer' though (DF-DR=1, in both cases)



The sim is, as you have noted elsewhere, is a simple one of a linear stiffness mechanism. But it does tell you (a lot of) what to expect and what not to leave out.

As you have mentioned, the TTC Forums and data, can build on this.....

You might also get a sense of what the "reasonable loading conditions" that Doug mentioned actually are.

Hopefully this brings in some new inputs.

08-30-2017, 09:10 PM
This is probably when the carpet plots come in handy, and a chirp/pulse/square steer test that was briefly discussed earlier on.

Imagine if you could combine this steady-state force/moment analysis to visualize the evolution of the transients with different initial/operating conditions.

09-05-2017, 10:53 PM
I use the term "ENF" to describe steer ("E") from aligning moments and tierod forces (N) at
the 'F'ront axle not resulting from any steerig wheel movement. Other compliances (inverse of stiffnesses)
recognised, simulated and tested for are attributed to lateral force ("Y"), as in EYF and EYR, camber ("G")
i.e. lateral force (GYF and GYR), and roll steer (EF and ER) and roll camber GR and GR. The force,
moment or displacement induces the movement. The effect of a compliance is a direct slip or camber angle
modifier. These compliances are designed for, managed, accounted for and verified in the specification
of handling performance in virtually all productio vehicles. This is what K&C machines are used for.

In the case of ENF, slip angle is attenuated because the steered wheels yield to the tire induced aligning
moments as well as the effects of loads on tirerods from all sources (caster, caster offset, overturning
moment, etc). The loading of the steering mechanism is usually the biggest contributor to this
characteristic either by design or ignorance. Intermediate shaft cross section, isolators, Cardan joint expansion,
steering gear mounting, rack push-away, T-bar stiffness in hydraulic power assist systems, wheel bearings and
wheel stiffness are all players in this term.

Even if the ENF term could be made to be zero, there is still a set of MZ moments from each tire that act
on the vehicle as a rigid body. They can't be ignored as they are contributors to total vehicle
understeering traits.

ALL vehicles have the "features" of the steering compliance situation revealed here by Vishnu:
it is a stiffening spring. It can get ugly, as you should all know, because tires are softening springs.
Thus the flow of forces and moments becomes a dance to a certain extent that the vehicle processes
to balance all of the forces and moments acting on it (usually input by YOU). Considering that the tires
do not react instantaneously, your can be assured that the computation of all the interactions between
these characteristics stirred together is mostly beyond the technology of most handwaving scientologists.
Thus, they are ignored, and your computer simulated car handling is shown as nothing more than axle weights,
wheelbase and a tire lateral force generation element. Say it ain't so, Rembrandt!

As you would know from being TTC members, tire Mz's go NEGATIVE after a spell and so much of this
disertation has the opposite effect: Your understeering effects become oversteering. When too much is
enough, drivers loose control and you "speeeen out". At low sideforce levels your car is "tight" and
higher up its "loose".

A saving grace to all you FSAE designers, optimizers and wrenches, is that your tires have very
low Mz to FY stiffness ratio (am I allowed to say that ???) and the tires are very load sensitive (Most are
radials) so your rearward weight bias is livable, AND you also don't really go that fast so the effects
of low understeer and some limit oversteer are second fiddle.

BUTT, now the reality: Low tire pressure, + compliance, then higher pressure from heat makes the car
"non-stationary" in engineering terms. i.e. it can be different every time you examine it, just like
photons in the Michaelson-Morley demonstration.

OK, still awake ?

Most of you would be horrified to know that the list of favorite cars from Road & Track etc have gobs of
this ENF stuff because there's no other way to make it safe (understeering) for Grandma when she takes
the Ferrari out for a spin (so to speak). (Me thinks I have posted a scatterplot of the statistics on
here previously). This quickens the response of the car by raising its natural frequency and settles the
steering gain down as you increase speed. This loss of steering gain is often patched with a lower numerical
overall steer ratio, but guess what: she-it happens: the lower gear ratio increases the loads on the
gear mechanism so you can wind up with a car that's actually "slower" (less responsive) because ENF jumps
up when you have steering parts made up from the wrong kind of pasta !!!

OK. So, to recap:

1) I've posted previously a 2 term log type math function that captures the essence of aligning moment/
kingpin compliance. The B term regulates the initial stiffness while the C term indicates the 67% range
of influence. Actual ENF has another useful property. Toyota refers to it as "viscous feeling" which is
a 1st order steer velocity response trait happening because of the kinds of parts used. This affects the hysteresis you
would measure as part of an ENF test. It implies that the steer velocity you apply affects the shape
and width of the hysteresis area. A Simulink model of this makes for cool wall art.

2) Tuning a car for constant understeer ("soft" steering stiffness with "hard" tires, merged with "hard"
steering stiffness with "softening" tires as you corner harder) can produce an exilerating feeling vehicle.
Most uniformed/naive evaluators decry understeer as a blemish on a car, but actually the departure from
linearity is the hated understeer trap. Hence Derivative of Understeer wrt Ay is the actual flaw.

3) The "best" cars feel good. It's a haptic realization that many drivers use. Steering MOMENTS are used
as control inputs, not angles. When all is said and done, the steering wheel moment running through all
the friction, gears and geometry transforms is how you control a car with a closed loop driver and a
path constraint. The bandwidth and transmissibility qualities of steering controls can make a vehicle fun,
safe, easy to drive and last a long time. If you have to wrestle a car to make it behave, the car wins.
ENF is THE major player in this behavior. The car accepts your moment and cancels it.

There you go ! Now let's have some twit bump us off this thread with an inquiry as to whether they
should use psi or kPa for tire pressure settings... Ain't it the truth ?

Mikey Antonakakis
09-28-2017, 02:09 PM
Bill and others, thank you for your contributions in this thread! I have spent a long time away from FSAE, having graduated about 6 years ago. I'm now back in school (MBA), and we have a first-year team getting things going now. I think there is enough (Enf?) info here to keep me and the rest of the team busy for some time... I look forward to learning from it!

02-05-2018, 10:26 PM
I would like to know if anyone at Leeds could resurrect the simulation mentioned in this SAE paper. Since there has been little if any sim/test material much less correlation work presented anywhere, it would be a valued exercise to amend this ancient work with real tire data and run a play with an ISO test procedure (as in Constant Radius, Constant Steer, Constant Speed, or Constant Figmosity).

This paper clearly shows that the simulated test circle radius was the same as the road tested one ( ! ), but the Frequency Response results hint at more interesting findings: The car has more understeer than the simulation predicts. And the reason given for the difference is (believe me) NOT due to lack of a tire relaxation block, as theorized.

Just wondering. The paper is well written and the findings are relevant to everyone here. The "WHY" details are begging to be explored.

If not available, how about some primitive vehicle parameters and the chosen tire: weights, wheelbase, steer ratio, speed(s), Izz, you get the picture. I'd like to take a stab at the analysis. Yes we need the road test results to make any real meaning of it. Without that, all it is is sandbox engineering playtime (IMHO) and perpetuation of the (bad) notion that cars are just barbells with tire 'springs' on them.

PAPER SERIES 2002-01-0567
Lap Time Simulation for Racing Car Design
Blake Siegler and David Crolla
University of Leeds

07-30-2018, 08:45 PM
The computational performance of the NLP puts processing power into perspective.

Speaking of ISO test procedures, whats up with the different understeer gradient tests (constant speed vs constant steer vs constant radius vs constant confusion)? Sure the procedure says you can use either or, but there is a thesis paper on the interwebz that suggests that they aren't always yield the same UG. How would that compare to a side force disturbance response test? (other than the carnage and destruction of a side impact)

07-30-2018, 10:28 PM
A good question about the UG you probably ought to determine for the sake of beating up those design judges who are way out of their element.

Just the facts, Ma'am:

1) Constant speed procedure. Takes a LOT of real estate to get the job done. This includes getting it up to speed, collect some data and then hitting the binders without smoking the tires. The test can produce a complete profile of a vehicles capability: nonlinear steady state gains and transient response characteristics of lateral acceleration yaw velocity and roll. You would normally want to a assess steer removal responses, too, because getting into a corner ain't the same cigar as asking the tires to straighten you out. Transient responses can get interesting, too. For all the hype about rice cars, these ugly machines tend to produce traits that even Grandma would object to if her Honda acted this way. Either your simulation matches the results or it doesn't. You can't hide behind a "test Variation" clause because these cars ought to be super-duper, but many of them obviously aren't. (just look at some recent pictures posted on Faceplant). Testing at and airport using crossed runways is a Best Practises move. Plus you can run at any speed the car will go and maintain the speed during the cornering segment. Some people cheat by going in hot and scrubbing bubbles to get a constant average speed.

2) Constant steer procedure: An easy one as long as the steering is nicely designed and assembled. Otherwise you will have some (usually) puzzling steer position effects. No transient metrics, just steady state gains. As some may recall, I ran this procedure on my Bass boat on Woodland Lake a few years ago to look at propeller blade effects. Still a good test to validate a nonlinear computer simulation. Yes there are/can be speed (aero, tire, tractive force) effects, but the results need to pass a laugh test. All you would really need is yawrate and speed to get you a plot of UG vs. Ay out to the "limit".

3) Constant Radius Test: Known amount of real estate makes it very practical. Requires a high bandwidth driver 'cause its a closed loop test. (that means you need at least a learners permit to get to the test track, much less operate on it. One benefit of this procedure is the determination of the 'Tangent Speed'. I.e. the speed where body sideslip is ZERO. You don't need an expensive sideslip transducer to get a value of ZERO at some speed around the circle (where the vehicles axis is tangent to the test circle). The tangent speed is a great contest and tuning metric. Changes to the front or rear of the car that raises the tangent speed are 'good'. Things that lower it are 'bad'. There are two ways to run the test. One is to slowly increase speed while staying on track. This implies throttle and differential and aero blending but that's the way it goes. The other method uses segments of constant speed runs. Run it in both directions, please. Your simulation will naturally include all these complex interactions for which you will have no data to characterise them.

4) The constant confusion test is a last resort used in the Design Event where you wave your hands and paw through you many notebooks and logs to explain why a naturally occurring oversteering yet stable vehicle is horribly understeering during the autocross. Say it ain't so even though you memorized RCVD in 3 languages.

Don't forget about a Frequency Response test ! When you pipe your iPhone into a Bose, you don't compare the sound at one frequency or loudness level, do you ? All the steady state values and transient response stuff (bandwidth, get it ?) are right there in front of you with a few hand typed Matlab statements. AND, the evil effects of nonlinearities (tires and steering) are way overstated and hyped by the non-believers. (It's Fake Vehicle Dynamics theory).

All these tests are intended to confirm that you have accounted for all the tire and chassis structural modes in the vehicle that you said you have. And what you said you have I'm sure you feel is the bestest, happiest, warmest, amazing, wonderful, awesome, and incredible university-speak that money can buy. But once you produce a test result with a measured UG, the lawyers will be all over you. Just make sure you have the sim that goes along with it.

Make it so...

07-31-2018, 07:20 PM
Lets get excited about frequency response.

So you've got your UG test, but now with the constant speed test we have transient data? Who cares about what the transient test procedure is, as long as there is enough frequency content - time domain, frequency domain, (z-domain? o_O).

No matter how the system gets identified you hope you get correlation in both time and frequency domains. Simulating and parameter estimation in time domain is conceptually closer to watching the real thing. Frequency domain is ???. You have a preference?

08-01-2018, 05:28 PM
The 2 processes are essentially interchangeable. Each one promotes a set of constitutive describing functions based on gain, damping and natural frequency. What makes FSAE cars so special is their VERY low understeer (theoretically) and thus corresponding heavily damped (i.e. sluggish) responses. You can see this is recent videos posted. Actually, the mention and consideration of "damping" (as in 'yaw damping') is almost ridiculous in a FSAE car context because the tires and architecture of them eliminates the traits associated with such metrics. The fact that some cars may have a 'yaw damping' trait means that the car is hurting for some engineering.

All this boils down to real estate (how much room have you got ?) and what kind of theory do you subscribe to. And if you have the background in solutions to differential equations, a time based formulation of a vehicle's step steer transient response contains 2 gain terms, a damping term, a frequency term and a phase term.

Note that the interchangability of Time and Frequency domains is demonstrated by the chirp_test and VHSIM simulations posted on my Google drive.

06-11-2019, 01:48 PM
Hello everyone, the discussion on this thread has been extremely helpful. I was also wondering how useful can estimating TF from just Slalom or DLC data using lsqcurvefit or some nonlinear fitting routine in MATLAB be? I collected data on a non FS vehicle and fitted a TF using not-so-close initial guess for the TF coefficients (num,den) from the bicycle model to match the Ay signal of the VBOX. The results seem to be fairly decent IMO, to check this I fitted the data for a DLC run at a speed close to 65kmph which produced my TF. Further I compared the data for a Slalom at 65kmph with the output of the TF for the steer signal. I am skeptical about the validity of this approach and wanted to discuss it here hoping something useful could come out of it. I understand that the TF won't be as accurate when the tires enter the non-linear regime but can teams use this as a starting point to study metrics?
Edit: A chirp or pulse with enough frequency content can be more accurate (using system ID) but in this case I did not have enough real estate and a trained driver for doing this on a passenger vehicle.
Y axis - right in figure 2 is Speed in kmph
Figure 3 is the USG from this TF, example for metric evaluation

06-12-2019, 10:38 PM
For a first round experiment, this looks very good ! First suggestion would be to improve/reduce speed variation. With constant speed, you eliminate a very important state variable.

Next suggestion would be to reduce the maximum/peak lateral acceleration. No need to push the tires that far into non-linear zone.

Then, forget the DLC. There's really only about 1 frequency in this maneuver.

A pulse steer input (approximating a step steer input) should work pretty well. I have a Matlab Demo program on my Google/FSAE drive which allows you to play with various types of inputs, filters/smoothing and processing techniques.

Try to fit your data to a TF involving the num/den of an equivalent bicycle model. That will reveal quite a bit about the makeup of your car (steady state, damping, and bandwidth/response times.1354

11-16-2019, 07:07 PM
Hey BillCobb, what's the impact of the relaxation length on the quality of fit? This of course means there are four states to track now.

11-18-2019, 03:46 PM
Hey BillCobb, what's the impact of the relaxation length on the quality of fit? This of course means there are four states to track now.

I'm not seeing a huge range of fit quality if the transfer function in Bode Form are used. Yes the resulting damping and frequency parameters that you would obtain will be different.

2 Examples: a Pretty complete vehicle model with nonlinear tire FY, MX, MZ, etc. and nonlinear MZ chassis compliance. These are all measured vehicle parameter values.

First case is No Relaxation followed by Slow Relaxation. 'No' is essentially really fast.

Then I switched over to a 2DOF bicycle model with a tire having only FY generation properties. Pretty much the same results: you can see the relaxation effects progression. However no change in fit quality. shorter response times, less damping, higher damped frequency in all modes (Yaw velocity and sideslip).1355135613571358

03-04-2020, 02:41 PM
Hi Bill,

Thanks for lending us your clever brains. As mentioned before, I think most teams struggle to get real data due to resource (also difficult to convince people as to why that data is useful, too, when instead they could be building a tangible car), and once they do have the data, they don't know what to do with it. A lot of stuff we learn academically is based on rigid models which are very simplified and are used to teach fundamental theory.

I had a question about the different compliance terms that you mentioned. Our team is receiving KnC data for an older car. My understanding is as follows (and hoped you could correct me if I'm wrong): In terms of KnC tests, we have lateral non-trailed toe compliance, lateral trailed compliance, and pure aligning moment toe compliance. All in and out of phase. ENF would be obtained from the pure aligning moment (so no trailed Lateral force, just the pads moving underneath to provide an Mz)? And the lateral force compliance EYF would be pure lateral force applied at TCP (no Mz). Technically, the Lateral trailed then should give us a combination of the two and summaries the total effect, but since we have them as separate tests, we can isolate and decouple them better in our simulation, especially since the lateral force and moments from the tyres aren't synchronised?

The difference between in and out of phase tests will also show the corners (suspension parts) only, and remove the effects of everything in-between such as steering. And then there's the camber compliance side of it too.

03-04-2020, 10:26 PM
Have you seen this presentation I had Pat post on the Facebook FSAE site ?


I have a rear compliance effects deal waiting in the wings, too.

I'm not sure of the question you are asking. Snow over again, I didn't get the drift...

The trailed lateral force (Fy) inputs plus the Mz loads are what the tierods will see. Thus the steering mechanism, its mounts, it's transmissibility up the steering column is the bad actor for the most part. As you add caster, the Fy component blooms.

There's also this piece I posted for Greg Locock (Ford Australia). If you know the tire properties all the cornering compliance allocations are solveable. My own observations (and from others) tells me that the FSAE cars are quite flimsy.

All the major players can be measured for sufficient accuracy on a wheel alignment machine. Put in some forces and moments, write down the toe and camber readings. Get used to the idea that wheel bearings do have a sizable contribution to the the ENF compliance.
Sometimes the wheel, too. It's incorrect that they test tires on wheels made from Unobtainium. The wheel may look pretty but can actually be pretty ugly from a compliance standppoint. It all depends on the reference sensors on the K&C machinery.


I would like to stress that this ought to strike you as common sense because NOTHING in a factory designed suspension or chassis is left to natural causes.

Its ALL designed with a certain amount of elasticity in mind for each of the handling related compliance terms. In ALL cases, the tires by themselves can NOT do the entire job.

If left to the imagination, they will crap all over the car's handling quality.

03-05-2020, 11:10 AM
Have you seen this presentation I had Pat post on the Facebook FSAE site ?

Thanks Bill, those are great slides. In my FSAE lecture I try to cover some of the same ground. After an overview of tire force and moment properties, applying the tire data to a vehicle model requires operating conditions--what is the load on each tire and where are the tires pointing. That is where K&C comes in because it's not at all obvious how a suspension is going to deflect in operation.

I strongly recommend that anyone planning to use K&C data witness a test in person. There is a lot going on during the test and there are many detail differences from one rig to the next--some machines (ABD, MTS, original Chevrolet VHF) are comprehensive, others test piece-wise and may not cover every combination of loads and deflections.

For example, John Ellis* built an early rig which applied horizontal force to move a vertical axis pivot laterally, this pivot supported a pad under the tire contact patch. To apply varying amounts of aligning torque resulting from the applied force, he would move the pivot point (and wheel pad) away from the contact patch center. Of course this meant that he wasn't applying a pure moment (the way it's commonly done now). From memory he developed a clever way to separate the two effects(?) and his machine was much lower cost to build. *https://www.worldcat.org/title/vehicle-dynamics/oclc/655191086

At the Chrysler Proving Ground, they used to have two separate axle testing rigs, one capable of heave & roll for kinematics, the other for compliances.

Before designing their SPMM, ABD used their 4-post ride rig with additional fixturing/actuation to measure partial K&C properties for their customers.

Lotus built their own low cost rig in house, it took multiple setups to apply forces in different directions. They may have sold some copies (not sure).

The history goes back a good ways. Maurice Olley at GM had a clue. Here's a short quote from our book, "Chassis Design" which is based on his notes from the early 1960s,
3.7 Lateral Deflection Steer

In addition to the changes in the pointing of the wheels (relative to the median plane) due to the roll of the car, there are sometimes appreciable changes due to deflections in the suspension mechanism under lateral forces. These have been less thoroughly studied than roll steer effects, partly because they are more difficult to measure accurately.

Probably the only realistic way of measuring them is by instrumentation on the road. However static shop tests give clear indications that lateral deflection steering effects exist. Their importance became apparent with increasing use of rubber bushings.

-- Doug Milliken

03-05-2020, 11:42 AM
When I hired in to GMPG, they had a machine in addition to the Chevy VHF in Warren) loaded by pumping water in and out of tanks to alter loads. I also remember a Lotus rig that Jack Turner (r.i.p.) ordered for the Chevy P.G. garage. We really needed Mz compliance steer. AND, the machine came unequipped to do it. What a farce.

Turns out we gave the machine away because no one wanted it. Used for vehicle storage and wheel alignment. As in: Lotus never considered ENF as a player in handling. WTF... From the statistics of their car measurements, they still don't !

03-05-2020, 12:34 PM

Thanks for the presentation. My original question was, what is the difference between the three different tests (in terms of what it tells us, not how its measured) 1. Lateral toe compliance 0mm offset 2. Lateral toe compliance Xmm offset 3. Mz Compliance.

And more specifically, why would we want to look at the 'pure Mz' or 'pure lateral force' graphs as opposed to 'lateral force offset', which would induce a force + moment simultaneously.

I attached some real KnC data from our previous car for the above tests (in-phase). I'm trying to figure out how to utilize it, and more importantly, understand it.


03-05-2020, 12:53 PM
...what is the difference between the three different tests...

You may need to go back to the rig operator to really understand what was tested? Based on experience, I think I can guess, but that isn't good enough. K&C measurements depend on precise definitions of force/moment application and measurement locations. Best to witness a test and discuss details with the operator.

What I take from your plots: it looks like this car has nearly +/- 0.2 degrees of free-play (slop) at the front wheels, which is way too much. Time for some mechanical detective work to find out what is loose! And fix it. Don't feel bad, I've judged a number of cars at Competition with a similar problem.

03-07-2020, 06:46 PM
The Fy input at "Zero Offset" I presume to mean that the force is applied at the wheel plate directly below the spindle center point at "the Ground"

The measurements pick out the separate steer and camber effects from this applied force (same on both ground patches of the axle). This gives you EYF, GYF, EYR, and GYR. The you apply the Mz moments on the wheel plates and record ENF, GNF, ENR and GNR traces.

The next bridge you must cross is the assumption that superposition of all of these rotations and displacements is valid and in realistic load ranges. In some cases it is not, but there are ways to map the actual functional relationship for a MIMO mechanism.

I have no clue as to why you would use a presumed offset of 115 mm. That's a lot off mm (certainly not any kind of 'pneumatic trail' effect).

On another page, Where are these measurements being done and who is processing the data ? Not any facility I'm used to seeing. Is that Excel output? What's up with the sign convention Fy to the right should be positive, not negative. Also, it looks to me like the steer channel is of the wheel plate [Earth] and not the wheel based on the hysteresis signature.

Plus the FY steer traces are not typical, as in soggy wheel bearings, etc. Sut wit dat ?

If you have the actual raw data available to you, it's time to produce more relevant, zoomed data fitted to data models.

I can help with this if you need a jump start.

Then all you need to do is add up the compliances form each mechanical path (CAF + EYF + GYF + ENF + GNF + EPF + NF) into Front Cornering Compliance and then (CAR + EYR + GYR + ENR + GNR + EPR + NR) into Rear Cornering Compliance and do the integration. . There will need to be a little iteration loop defined to solve for the common slip angle that balances the forces and moments at each slip angle, but, as in Nature, all this works out
Some geeks go with a matrix inversion process to solve this dilemma, but I recommend an analog solution done with a small delta time step. I don't believe Mother Nature utilizes a matrix inversion process in the nonlinear physics of a Newtonian Universe.

03-08-2020, 09:14 PM
Yep, I thought I recognized Morse Measurements results...1369

03-10-2020, 07:55 PM
Yep, I thought I recognized Morse Measurements results...1369

Or is this an Anthony Best result ???1371