Slow going...

For all my adoring fans, I apologize for the slow pace here as of late. Somehow in the past week or so there have been a million and a half things to do at work... and I've had some software "issues" with doing more car design.

So hopefully all that will be squared away and I'll be ready to rock out after break, refreshed on turkey and mashed potatoes. By turkey I mean Guinness, and by mashed potatoes I mean... more Guinness. Yum!

Aligning Torque - It's no joke

Another item I unfortunately didn't understand well during FSAE is the importance of tire self-aligning torque (Mz). It's not just about steering feel, it directly affects your vehicle handling (steering effort and feel are also pretty critical but that's another story). I recently was talking with a friend of mine, a tire data and DAQ guy on a Nascar Sprint Cup team. He wasn't really interested in Mz as "we have power steering," until I pointed out the stuff below and his later comment was "really fucking brilliant." (You know who you are... I had to mention it!)

Let's take a gander at the following force / moment equivalence.


There is a force (Fy) and moment (Mz) generated by each tire during cornering. I can sum all of that up and act as if it's a total force, and total moment, acting at the CG. Then, if I want, I can take that force and moment and resolve it into two different forces acting at the front and rear axle. The effect should be apparent. Mz effectively "robs" the front axle of cornering power and will always generate an understeer moment. On "real" racecars the magnitude of extra Fy needed at the front can be quite significant. I haven't run the numbers on a F1000 or FSAE car yet, but on the latter it might be significant just based on the fact that a short wheelbase really amplifies the effect.

The other important thing to note is how Fy and Mz behave in the tire's linear range and at the limit. The picture I have here isn't the best.. I found it on the interwebs.. but it will have to suffice.



The above plot becomes VERY asymmetric with high camber, but for simplicity's sake...

In the linear range (within a couple degrees of slip angle) both Fy and Mz build up at a constant rate, and the two are related by the "pneumatic trail" of the tire (Mz/Fy). So as the driver is approaching a turn and beginning to "feel out" the entry the effect is a normal buildup of slight understeer. At the tire's peak grip where the footprint is transitioning to full slide, Fy saturates but Mz drops to zero. The picture doesn't really show it well but generally Mz will peak and start to fall off much earlier than Fy does.

My suspicion is that cars shod with tires with high aligning torque will feel good on entry and with initial steer angle, but then at the limit will transition (gradually or snap) to something much more free. Aggressive camber curves probably play into this as well as they have a big effect on peak and slide Mz.

Generally the wider the tire footprint is the less Mz you have to deal with. Wider rim widths on a given tire will tend to make the footprint wide, and higher pressures will tend to shorten the footprint up as well.

The importance of frame rigidity

Chassis (or really hub-to-hub) torsional rigidity is a term thrown around FSAE Design Event tents frequently. When I was doing the FSAE thing I knew it was important and that you didn't want a floppy frame, but really didn't know why. So, and I suspect a lot of folks do this too, I would BS in the design event about how it adds another spring to the system, makes the car hard to tune, etc.. without a real explanation.

Thought about this a bit last night after I woke up from an entirely too long nap, had a Red Bull-Vodka, and watched a bit of Police Academy and Blacula (and its 1973 sequel, Scream Blacula Scream). Here's the real deal, to the best as I can determine.

Two of the biggest and most common tools we have to adjust vehicle balance (understeer vs oversteer) are adjusting spring and anti-roll bar (ARB) rates. Really what we're trying to adjust roll stiffness distribution front to rear, and both the spring and ARB act as roll-resisting springs in parallel (they add) at both ends of the car. Generally we assume the frame to be a rigid member. See picture below.


Kwheel is the wheel rate, since we're technically interested in how much stiffness the spring is providing to the tire/ground and this is NOT equal to the actual spring rate. Anyway, more roll stiffness at the front relative to the rear -> more load transfer at the front relative to the rear -> front tires have less grip than rears -> understeer. The corollary is true, ie stiffer rear -> less rear grip -> oversteer. That's all well and good. The fact we're tuning relative stiffness and not absolute is a key point. You'll often hear the term Total Lateral Load Transfer Distribution (TLLTD) used to describe the ratio of front to rear load transfer in roll.

In reality, particularly in the FSAE class, the chassis ALSO acts as a torsion spring in series to the front and rear suspension. For simplicity's sake I'll assume the sprung mass is a point mass, connected by a front and rear chassis "spring" to their respective suspensions. See picture below.



Springs in series act like resistors in parallel. For example if I have a 10 ft-lb/deg spring in series with another 10 ft-lb/deg spring, the resultant springrate is 5 ft-lb/deg. Oh snap, serious implications. If you have a compliant front end of your frame (in this case from the CG to the front suspension mounts) you're effectively changing your roll stiffness distribution, TLLTD, and vehicle balance! If it's bad enough you can make a massive spring or ARB change on one axle and get almost no change in balance. Additionally it's not really good enough to look at global frame rigidity, it's better to have the stiffness distribution fairly equal between sections.

How stiff is stiff enough? Tough to say, really depends on how load sensitive your tires are and what your suspension rate is. The stiffer your suspension rates are the more rigid your frame has to be. For a relatively softly spring FSAE car a well-designed tube spaceframe chassis is perfectly fine, and you quickly hit a point of diminishing returns trying to go stiffer. For an absurdly stiffly sprung F1 car a carbon tub is mandatory. An order of magnitude higher than the highest axle roll stiffness might be a decent minimum target value.

I'll throw in now, another reason Solidworks' Weldment feature is awesome is you can VERY quickly drop all the geometry into a beam-element FEA study that automatically picks out all your element MOI's etc. And beam-element analysis is wicked fast. Bitchin'!

Now you know. And knowing is half the battle.

Or at least that's my theory on the whole thing.