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Vehicle DYNAMICS 201 (Testing Your Sim Model or Your Car)
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.
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Thank You for that Suggestion.
But that didn't work. So, let's try another work-around...
Take a look and the snip: