Line Theory: Perfect Corner 1 & 2

In Perfect Corner 1 and 2, Adam Brouillard has made the most important contribution to racing line theory since the first technical book on the subject, Taruffi’s 1959 The Technique of Motor Racing.

A few months ago I started reading, studying and applying knowledge gained from these two books. They are an exposition of line theory, the theory of what driving lines are most efficient, i.e. the fastest, around a race track. Brouillard takes a physics-based approach.

With Perfect Corner 1 and 2 we can look at all the various permutations of line theory since 1959 and clearly understand what is right about each one and why and what is wrong about each one and why. When a new theory explains all the previous ones, that makes for a powerful theory.

Brouillard teaches that there are only three types of corners in all of road racing and autocross. (He tells me he began as an autocrosser.) The three are the standard corner, the chicane and the double-apex. That’s it.

A standard corner has enough space before and after to stand alone so that the entry and exit can be optimized without regard to the previous or following element. A chicane is defined as two corners in opposite directions that are so close to each other that they must be optimized together. (The autocrosser’s slalom is two or more chicanes end to end.) A double-apex is two corners in the same direction that are so close to each other they must be optimized together.

The books give rules for how to classify each corner you will encounter and rules for how to determine the most efficient line through each type.

My approach to autocross is now forever changed because of Brouillard. What I’ve realized in doing autocross events this year with his concepts and rules in mind is that many autocross courses are more complicated than most road-race tracks. Some autocross courses essentially have no stand-alone elements. Everything is connected in a series of chicanes and double-apexes with only the rare standard corner. So, applying the methods and rules he gives is not easy. Some of it is immediately applicable, some is not. He tells me he’s thinking of writing an autocross book. I hope he does. Soon!

In the meantime, get these books and start a new journey into autocross.

What Trail-Braking Looks Like

TAC3 180

Big Sweeper At TAC/TVR #3

 

One of the difficulties in learning anything is working through the trash. The ‘trash’ is what I call all the myths, supposed common-knowledge and just plain wrong stuff people tell you that can send you down less than optimal learning paths. If you’re someone like me that has to first get it in his head intellectually before the body can do it, you may be particularly susceptible to well-meaning but wrong advice or supposed facts that aren’t so factual. Even correct information delivered at the wrong stage of development can cause learning to go off track. We can’t start at the top. We have to start at the beginning. There’s always a progression.

I guess if you begin with a world-class instructor in a well-developed field (one where effective teaching techniques have been developed over time and are widely known, like music or golf) then the trash problem is minimized. I don’t think autocross is quite there yet, but the data revolution is changing that. If you’re Dad or Mom happens to be a great autocrosser, knows why she’s fast and can teach it, then you’re in the soup. Very few get so lucky.

In my case my Dad was a multi-sport athlete and tremendous competitor who could never understand this nutty autocross thing. He always wanted to come watch the event if I was racing in his city but he never, ever rode with me.  Not once. He just wouldn’t do it and I never understood the reluctance. He would say, “I don’t want to encourage you” and smile as if it was a joke.

Trash example: the purpose of trail-braking is to help get the car turned in a long corner, like the one shown above.

Maybe I heard it wrong (multiple times?) but this is what I remember people would say when discussing track driving and the difference between braking for a corner in a straight line then turning in for a late apex (Slow-in, fast-out) vs. the more advanced (and, OMG! dangerous!?!) technique of trail-braking into a corner. My problem was that I’d believe stuff like that and think it was the real reason for trail-braking when maybe it was just an easy thing to say, or it was being said by someone who didn’t really know why trail-braking was a technique for Saving Time. (Yes, I’m kinda slow like that.) I carried that idea into autocross.

It was in my head and wouldn’t come out without great difficulty, i.e. progress in learning that can replace the simple idea that trail-braking is for rotating the car with a more sophisticated idea.

I’ll tell you The Non-Trash Truth: no one can Save a lot of Time in autocross without trail braking the heck out of any long corner, like the one shown at the top. We see a lot of those in autocross and many tracks have something similar. The feature shown above didn’t even have an apex cone. Just an entry and an exit and you figure out how to get from one to the other as fast as you can.

Trail-braking has little to do with turning the car by putting weight on the nose and freeing up the back tires to slip more. Sure, you can use it for that and may need to, depending on the type and setup of the car, but it’s not the most important reason why you should trail the brakes entering most long turns.

The real reason is because of the physics of tire performance. Unlike me when I started out, tires can do two things at once. They can both brake and turn at the same time, just like they can accelerate the car forward and turn it at the same time, but that doesn’t seem as hard to understand. The two capabilities added together are more powerful than used separately. Proper use of trail-braking allows you to brake later into the corner, thus extending the time spent at a higher speed (extending the length of the previous straight for you track drivers), to take a shorter, elliptical path to the apex, and to take that path at a higher average speed. Those three things sound like they’d Save some serious Time, don’t they?

So, go learn how to trail-brake.

This isn’t a how-to article on trail-braking, but I will show you what it looks like in data. If you’re like me, you need some convincing first so you can really commit to learn it later. Read this article then go read some books on racing. I like Krumm’s Driving On The Edge. He’s a professional racer that figured out how to drive long corners by trail-braking into a double-apex by analyzing data of the same corner driven over and over again various ways by various drivers.

Then, go practice. Where? At the autocross event, of course, where a spin only costs you a little tire rubber.

The data below is for the turn shown at the top of this post. The top trace is speed, the middle trace is how hard the car is turning (lateral force) and the bottom trace is how hard the car is braking (negative) or accelerating forward (positive).

From the point marked ‘Lift’ the LongAcc goes steeply negative. This is hard braking. Notice that just above the LatAcc is turning positive. That means I started turning left at exactly the same time as I was braking. (This is a little unusual, but I was in a bit of a hurry.) And I keep it up.

In the section marked ‘Trail braking’ the negative acceleration is gradually trending up to zero, i.e. I’m gradually coming off the brakes. Meantime, the LatAcc continues to build up to well over 1g. The tires are providing the ability to brake and turn simultaneously. This is the data signature of trail-braking.

TAC#3 180data

TAC/TVR #3 First 180

The other thing to notice is the shape of the path. It’s an almost perfect portion of an ellipse. The physics of the situation dictate that it be this way if you do it correctly.

A Real World Comparison

At the Blytheville Pro-solo a few weeks ago I put my data device into another car and got data for three different drivers: Ryan, Tom and me.

Ryan and Tom were in a BSP Miata on race compound Hoosiers; I was in my BS Corvette on Bridgestone street tires. The course contained an almost perfect, more-than-180 degree sweeper, entered from a slalom just like in TAC/TVR #3, above, marked by an entry cone, a center “apex” cone and an exit cone. Each of us did this corner in his own way. You can see the path differences in the right of the figure and the data on the left. 

BPS180 data

2018 Blytheville Pro-Solo Turnaround (Left Side)

 

Looking at both the LongAcc (longitudinal force) and the LatAcc (lateral force) we see the trail-braking signature in the data. After braking hard, Ryan’s red line only very gradually heads back to zero, that is, he’s staying on the brakes as he turns in more and more, only very gradually releasing the brake pedal, taking best advantage of the tires’ ability to multi-task. This allowed him to maintain the highest entry speed and yet not overshoot.

The major difference as I see it was that Ryan, clearly the highest level driver of us three, did a much faster straight-in approach and a perfect trail-brake entry. His minimum corner speed was 38.7mph. I (green) did a slightly wider approach and a less than perfect trail-brake, attempting to agressively go shallow and accelerate to the apex.  (It’s a big corner, much larger than the corner from TAC/TVR #3, so big and with such a fast and difficult entry that everyone was accelerating to what would normally be called the apex. Effectively, we all double-apexed this monster.) My minimum corner speed was 35.8mph, almost 3mph slower than Ryan, not too surprising given the car/tire difference and the different strategy. Tom (blue) went widest for a classically best entry angle, did not trail-brake, but was able to accelerate to the apex sooner than I and catch back up to me. His minimum speed was 38.3mph, just slightly less than Ryan.

Once at the apex cone all three cars had speeds contained within a 1mph band. From entry to apex cone took a bit more than 4 seconds during which time Tom and I lost 0.25s to Ryan. This can be seen in the bottom trace, where Tom and I (blue and green, respectively) are compared to Ryan, the horizontal red line. The more the blue and green lines are above the red, the more time they’ve lost to red.

For my part, I think trail-braking is what allowed me to match another car to the apex that had greater grip but whose driver didn’t trail-brake. 

Addendum

I’ve become aware that Brouillard claims that the shape of the trail-brake curve is an Euler spiral, not part of an ellipse as I stated above. I’ve now ordered all his books and will study on it. I don’t see how Brouillard can be correct (if this is actually his claim… I read it in Wikipedia) when the radius of curvature of such a spiral varies linearly. That’s the definition of an Euler spiral.

Euler spirals were first used in the railroad industry to transition from a straight to a curve without literally jerking the passengers around. They also reduce loads on the tracks. In autocross, of course, we’re not too worried about a little jerking, which is literally the time derivative of acceleration. Lots of little jerks in autocross.

The trail-braking curve seems a non-linear situation, even if we assume a perfect circle for the tire traction “circle” and a linear release of the brakes, since the radius varies with the square of the velocity. I think the the curve shape is more complicated, more like an ellipse with a non-linear variation of the radius of curvature. My assumption of an ellipse, based on what the path actually looks like in the data, may be an approximation that’s not mathematically correct. So far, I’ve not found a mathematical description of the trail-braking curve geometry. Maybe I’ll find it in Brouillard’s books. If so, I’ll come back and tell you about it.

 

 

Extra Twist?

Someone asked this question in an on-line critique of various run videos from our latest event: Not being the expert you guys are, I enjoy the critiques. What I notice is that I and others will start a turn, hold it for a while and then just as we [get] to the cone we give the wheel an extra twist to get around the cone and on the line we want. Or am I just seeing good technique?

While we all make mistakes, and we all have to make corrections (for instance, the level of grip is not necessarily constant in even a single turning element) Steve Brollier (multi-time national champ) taught in our autocross school last year that we should strive to turn once for each slalom cone, for instance, and once for each offset cone. I think this applies, in general, to all turns.

As someone pointed out in my video (which can be seen here TAC/TVR#3 Run video) at 1:07 in the final turn to the finish I make a preliminary turn and then the “real” turn. As a result, I have to turn sharper, which means slower, and I lost time there.

The “extra twist” being talked about may be a valid technique in certain situations. I’ve always called it taking advantage of the ability to dynamically shock the tires and get a little extra out of them. There aren’t many places where you can use it, however. If you’re doing it at every corner, it’s probably covering up a basic fault. You’re probably cornering too much under the limit over a large portion of the turn and only at (or above) the limit in the final phase. You may be turning too early and too slow, rather than turning later but with greater steering wheel speed.

I remember doing a lot of the “extra twist” technique when I was new. I think it may be caused by the lack of confidence in turning hard at higher speeds. We get comfortable with turning hard a low speeds first, so that’s where we do it. As our level increases we get more comfortable with quickly getting to the cornering limit at higher and higher speeds. Turning the steering wheel as fast as conditions allow reduces the transition time from one turn to the next, or from going straight to turning in, which has a direct effect on the speed you can carry, how late you can brake and ultimately elapsed time on course.

I think the process of “getting fast” is 1) learning how to evaluate the proper line to take, for your particular car and driving style, 2) developing the car control skills necessary to make certain maneuvers and be able drive the line you’ve decided to take, which again is highly dependent upon the type of car, and 3) gradually reducing the number and severity of mistakes, which implies that you have gained the knowledge of what constitutes a mistake. Making multiple inputs in what should be a single, smooth arc is definitely a mistake, but doesn’t by itself mean you won’t be “fast” in relation to someone else just because you’re not perfect. A lot depends upon the magnitude of the “mistake.” It does mean you have room for improvement. (I’m discounting the often-rapid corrections you have to make to keep a car on the limit of adhesion.)

Earlier this year I got to sit in on a video critique session with a group of accomplished autocrossers. One of the top drivers on the national circuit (another multi-time national champion) was watching his own video from the course we’d all run that day. The level and completeness of the critique he gave himself was impressive. “Oh, I got late there,” he says at one point, and I’m looking at it thinking the error was so incredibly slight that I would have never noticed it. Upon first view I would have said it was a flawless run. Only after repeated viewings could I see what he saw.

There are levels and levels.

3 Ways To Take A 180

The course at our last event of the season had an interesting more-than-180 degree sweeping turnaround. Partly by design and partly by not being overly consistent I got data on three different approaches to the corner.

The three afternoon runs are shown in Figure 1, below. Run 1 is green, run 2 is blue and run 3 is red. The day was sunny but cool. The tires definitely got better (warmer) from 1 to 3, which partially determined the changing approaches, but I had already planned to use this corner as a data-analysis opportunity.

figs.001

Figure 1  Turn-around

Here was the plan: Run 1, green, was to be the “momentum maintenance” line. Green would enter the corner beyond cone A faster, via less braking, attempting to carry more speed from the fast section that preceded, even though it would mean a bigger turning arc beyond cone A. Green would continue turning right about half-way down to B to get a good angle when turning back on that cone, again braking as little as possible.

Beyond B Green would take a wider path to get the best possible angle on cone D, not worrying about keeping close to cone C. A good angle on cone D would allow early power application for the following acceleration zone.

Run 2, Blue, was intended to make maximum use of the short acceleration zone between A and B. To do this, the plan was to execute a little more braking on the approach to A and open the steering wheel earlier for the earliest possible application of power. Blue would accelerate for as long as possible and sacrifice some angle on cone B, accepting that the car will go deeper beyond B.

Going deeper beyond B was not necessarily such a bad thing as it would allow a good angle on cone D, just like Green.
Run 3, Red, was planned to be the tightest, minimum distance path from B to D. Red would slow enough before A to allow some acceleration down to B and then get wide enough to cut in at B to cross from B to D by a tight path that was close to cone C.

Cutting a shallow, fast line across from the two cones that form the entrance and exit to a 180 degree turnaround is a strategy I’ve used on larger 180’s at certain national events. It’s only good if the exit cone is not constrained by needing to be backsided. That is, the line must be open beyond that cone so that significant braking for that cone is not required. This was not the case for this particular section.

Which path do you think saved the most time? Or, would you have done something different? Before you decide, I’ll give you some data to think about.

The relative path distances: From A to E the green path is 439 feet, Blue is 428 feet and Red is 425 feet. So, Red was the shortest path, but only by three feet as compared to Blue. As expected, Green was the longest path, 11 feet farther than Blue, 14 feet farther than Red.
Entry speeds: At A, Green is moving 41.6mph, having braked less per the plan. Blue is at 38.1 and Red is at 39.8. There’s no real reason for Blue to be slower than Red. Chalk it up to driver inconsistency and maybe cooler tires.

Maximum speeds between A and B: 41.7mph for Green, i.e. essentially no difference from the entry speed. This makes sense because Green is supposed to be turning smoothly at max lateral g at all times.

Red has managed to accelerate a bit from 39.8 to 42.3mph.

Blue, which had the slowest entry at 38.1mph, gets up to the fastest at 43.9mph, as intended. Of course, if Blue had entered as fast as Red it would have reached an even higher speed.

The path distances and speeds indicate to me that the three runs were driven basically according to plan. The only anomaly is that Red’s entrance speed is a little faster than Blue when it should be, by all rights, a little slower. I was probably driving more aggressively on the warmer times for the Red run. This gives an unfair edge to Red.

So, what actually happened as far as time lost or saved? The Delta-T chart tells the tale in Figure 2, below.

figs.002

Figure 2  Delta-Time

Now, Red was the fastest run overall, so the software has set it as the baseline. It’s a completely flat line. The time difference for the other two runs appear as lines that waver around the red line. When the Green and Blue lines are descending they’re gaining time on Red. When ascending they’re losing time to Red.
From Position 290 to 460 Blue has saved 0.16s over Red. Red has saved 0.08s as compared to Green. But, as you can see, it’s not quite that simple.

Green is catching Red as the data ends and by the time the cars get to the next cone (not shown) Green equals Red. Why? Because the better angle that Green had on cone D allows it to pass D faster and maintain a speed advantage over Red from then on. Blue maintains an advantage over Green, however, having rounded cone D just about as well.

To fully understand what happened we need to separate the story into two parts: before B and after B.

Before B, at Position 360, Blue has pulled a lead of 0.19s on Red. Green initially had a speed advantage on Red and Blue around the cone at B as seen at Position 300, but then loses it due to a longer path and inability to maintain that speed. The relative speeds can be seen in Figure 3, below.

figs.003

Figure 3  Speed

Paradoxically, Green catches Red by Position Index 360. (See Figure 2) It does this by braking less than both Blue and Red as the cone at B is approached. This is the great advantage of the momentum-maintenance mindset: you can really focus on carrying speed and staying off the brakes, which spurs you to take corners faster than you think you can.

Beyond Position 360 we have a give-it-up-to-gain-later situation. Red, the tight path, initially makes up tremendous ground on both Green and Blue, leading Green by 0.29s at Position 410 (and even slightly leading Blue), by virtue of the shorter path, a higher minimum cornering speed (about position 370 in Figure 3) and by a consistently higher speed from 370 to 400.

Red loses everything by cone D, however, having to slow tremendously to get tightly around that cone. Even with slowing a full 6 mph as compared to Blue at D, Red still pushes out a little wider than optimum, pulling a 1.34 lateral g shock at D in an attempt to limit the damage, as seen in the Lateral-g chart in Figure 4, below.

figs.004

Figure 4  Lateral g

Red is royally screwed for the entire next section of the course and both Green and Blue claw back most of the advantage they had previously at Position 360.

Conclusions
1.Maximizing the acceleration and speed from A to B as done by Blue was a good strategy, especially for a relatively powerful car.

2. The big hump in Blue’s delta-T trace centered at Position 410 in Figure 2, indicates that Reds strategy of a flat, fast path across the 180 turn was equal to Blue’s line in the 180 itself. Blue had gained earlier and Blue maintains that lead after cone D, so Red was not the optimum line overall, but might have been if the next feature after cone D had not been as constraining.

3. Given that this feature required more than 180 degrees of turning, ignoring the cone at C and taking a wider path from B to D was necessary to get a proper angle on cone D. This allowed early acceleration from a higher starting speed as the course straightened. Even though Red had warmer tires and was able to accelerate harder than both Green and Blue, it couldn’t catch up beyond cone D. This was a true (and rare) instance of the value of a “late-apex” in the road-race sense thanks to a severely decreasing radius path dumping into an acceleration zone.

How Lowering Affects Your Autocross Car

Part 1- Estimating the Effect on Weight Transfer and Roll 

944-rake.jpg

I’m setting up a 1989 Porsche 944 with the sport suspension (Porsche option code M030) for street-class autocross. This suspension design, MacPherson strut in front with threaded spring perches and semi-trailing arm in the rear with indexable torsion bars, allows me to adjust the ride height of the car, i.e. I can lower it from stock. Figure 1 shows the front suspension, looking from the front.

DSC00303

Figure 1

Adjusting the front couldn’t be easier… just twist the spring perch. It’s right there at the base of the spring in the photograph.

Adjusting the rear is the opposite of easy. It’s so horrifying to contemplate that I really don’t want to talk about it. At least it can be done, which is an advantage this car has over many others.

Right now I’m in the middle of swapping M030 torsion bars for stock and re-indexing them, along with renewing all the bushings. I’m aiming at 1” lower than stock as the base setting. The car has a separate method for adjusting it slightly from the base position apart from indexing the bars. That method is intended mainly for corner balancing.

Before I go crazy with lowering the car I want to estimate how much cornering power I can expect to gain compared to the stock ride height and balance that advantage against certain disadvantages that will occur.

One thing people forget is that being able to quickly estimate an answer is just as important as being able to grind out a precise figure. For one thing, maybe the estimate is enough to tell you you’re going down a blind alley. For another, it allows you to know when your precise answer is obviously wrong because you made an error somewhere. (If I only had a nickel for every time a young engineer believed that his obviously (to me) wrong answer just has to be correct because, well, it just has to be because he spent so much time on it and it was derived exactly from first principles as taught in school. Usually, they come back with “I found a slight mistake… how did you know?”)

If anyone can show me where this information is already available for the 944, I’d sure like to see it! And, if you have better numbers than what I’ll be using, I’d love to have those too.

Lowering a 944 (and just about any other car) does two good things: 1) it creates an increase in the maximum available static negative camber, which is limited to much less than optimum for autocross in most cars as they come from the factory, and, 2) it reduces lateral weight transfer in the corners. Both of these effects tend to increase tire performance at the limit.

Lowering the 944 also has at least one quite bad effect, namely, a reduction in roll stiffness with the MacPherson strut front. I’ll explain why this happens as we go along.

Reducing the roll stiffness means the car rolls over to a greater angle than before in the corners. The extra roll reduces the negative camber of the outside tires (some of which was gained by lowering) just when you need it most, and tends to reduce the transient response of the car by making it take longer to go from full cornering in one direction to full cornering in the other direction. Transient response, while important in all forms of motorsport, is especially critical in autocross.

A race car that can change the springs can counteract this decrease in roll stiffness. In SCCA Street class autocross, I can’t change the springs to anything not offered stock and I can only change one anti-sway bar.

Now, there are things that might be done to limit the roll even in Street class (specially designed bump stops for this particular car) and to increase the transient response (increased low-shaft-speed damping in the shocks) but I’m not going to get into that here. Those are possible mitigation measures that also have their own trade-offs, though I expect to use both to some degree before it’s all over.

There are a lot of opinions floating around out there on the subject of lowering a 944. In general, the advice is to lower the car, but not too much. Some people scream, “Whatever you do don’t let the roll center go under the ground” and point to the angle of the lower control arm in the front suspension, saying it shouldn’t go below horizontal. (The roll center is the imaginary point that the sprung mass of the car rolls around when you enter a corner. Actually, there’s one for the front suspension, one for the rear. The roll axis connects the two.)

There may also be bump steer effects, which make a car hard to drive, tire rubbing possibilities and a concern with over-rotation at the lower ball joint which has the potential to crack the lower control arm.

Figure 2, below, shows the relationships that determine the roll center for a generic MacPherson strut suspension. I’ve added some extra dashed lines to show what happens when the lower control arm angle is changed from slanting up toward the center of the car to slanting down without changing the strut angle. (In reality, the strut angle does change a little, making things even more complicated.) 

roll centers.001

Figure 2

What suspension designers call the roll center was termed in my mechanical drawing class an instantaneous center of rotation, or instant center for short. We learned to derive these for various linkages and mechanisms. The key point being it’s where it is for an instant, not forever. That is, the damn things move around when the suspension articulates. Lowering the 944 is an articulation of the suspension that moves the roll center. See how simple this will be?

Looking at figure 2, this is how you find the instant center called the roll center for a MacPherson strut mechanism:

1- draw a perpendicular from the top of the strut

2- draw a line parallel to the lower control arm

3- from where those two lines intersect draw a line to the tire contact patch

4- the roll center is at the intersection of that last line and the car’s centerline

The roll center is the point about which the sprung mass rotates at the start of cornering. As the car rolls, the roll center is actually going to move, maybe only a little, maybe not so little, but I’m going to have to ignore that.

I’ve also added a circle to mark the center of gravity (CG) of the sprung mass. When cornering, a lateral force acts on the sprung mass at that point. This is the big arrow in the figure. (The opposite reaction, not shown, is at the tire contact patch.)

That force causes the sprung mass to rotate about the roll center. The distance from the CG to the roll center is a moment arm. It’s a lever that’s working on the springs and bars. The longer the lever, the more the effect on roll for the same lateral force.

When the car is lowered the control arm angle changes. I’ve drawn a new one pointing downward. You can see that the new roll center is just about sitting on the ground. It’s quite possible that it descend below the ground, but only if the control arm angle slants even more sharply downward. (It may do this dynamically in a turn.)

With a control arm angle limited to horizontal the roll center can get close to the ground, but, because of the geometry, it can never descend below it. You might want to stare at the figure until that becomes clear.

What I haven’t shown is the new CG point. For the new control arm angle it will have dropped a little bit from where it’s shown. But, the way things work is that the roll center drops more than the CG point, so the length of the lever between them, the moment arm, gets longer as the car is lowered.

I think this is where the rule of thumb to not lower the car so much as to create a control arm angle below horizontal comes from. People somehow got the idea that a below-ground roll center was the kiss of death. Realizing the relationship just explained, they saw a way to prevent it from happening, i.e. don’t let the control arm sink below horizontal and the roll center can never descend below the ground.

I think it’s bunk. I ain’t skeered of below-the-ground roll centers.

Ok, maybe I’m a little scared!

Low roll centers do a really good job of banishing jacking force, one of three components of lateral weight transfer which we’ll talk about later. Jacking force produces loads on the suspension components that tend to jack up the sprung mass. This force gets smaller, however, the closer the roll center is to the ground. It turns negative and jacks down the car if the roll center goes below the ground.

One possible bad effect of transitioning between positive and negative jacking is that the forces in the suspension components reverse direction. Now, a control arm couldn’t care less, but all the joints might, especially if there were any play in them.

All modern performance cars have low roll centers. In fact, jacking force reduction is the primary way lowering this suspension reduces lateral weight transfer. If you aren’t calculating jacking force you’re nowhere, man.

For another thing, the control arm on my car as I’ve been running it is guaranteed to go below horizontal dynamically in bump and make the roll center descend below the ground, at least for a moment. The car seems not to explode.

Take a look at Figure 1 again. Notice that, at full droop, the arm is definitely slanting upward toward the center of the car. But, imagine that we mounted a wheel and dropped it off the jack-stands. The arm is going to be right about horizontal once the spring compresses. The idea that it could be kept above horizontal during it’s normal range of motion doesn’t seem credible to me.

I may be able to take some measurements and model how this actually occurs on my car. (If anyone already has the suspension measurements I’d like to have them.) In any case, I want to put some approximate numbers to what’s happening so I can do a first-order approximation to estimate the total effect on cornering power, as well as other factors, that result from lowering the car. Later, if I can get better numbers, I plan to refine the results.

Weight Transfer Reduces Cornering Power

Lateral weight transfer, by definition, is the transfer of load from the inside tires to the outside tires while cornering. Lateral weight transfer is important because it reduces total tire grip, or more precisely, total cornering power.

The four tires on your car work best when load is distributed evenly among them because, unlike what you were taught in high-school physics class, the load vs. friction (lateral grip) relationship is not linear in the real world where the rubber meets the road, so to speak. Unfortunately, equal load distribution is never the case when we want it to be, like when cornering or braking or accelerating or any other damn thing we are doing while driving the car, ordering your vente-mocha-espresso at the Starbucks drive-thru excepted.

How big is this effect? Carrol Smith, in Tune to Win, worked an example for a light race car and a particular tire performance curve. He found a 6% decrease in cornering power in an 80% load transfer case. Unfortunately, my car is clearly not much like the one he was working with.

Herb Adams, in his book Chassis Engineering, did the calculations for a 3,000lb car with 50/50 front to back weight distribution with 1000lbs of weight transfer while cornering at 1g. That’s 66.7% weight transfer, i.e. 1000lbs out of an original 1,500lbs on one side of the car transferred to the other side. His example has numbers very similar to my street-class Porsche 944 cornering on the sticky (and very expensive) street tires we use in autocross these days.

Herb’s method for calculating the weight transfer is less rigorous than what we’ll do, but what matters to me is that he found that the theoretical cornering power of 1.13g’s for the tire performance curve he used was reduced by weight transfer to 1.05g’s. That’s a 7% decrease for a 66.7% weight transfer. I will boldly make a linear relationship out of this data, namely that the cornering power will decrease 7% divided by 66.7% or about 0.1% for each 1% of weight transfer. This is an easy relationship for my feeble brain to remember.

By the way, the effect of lateral weight transfer is worse the farther the car is from 50/50 front-to-back because the heavy axle is affected more than the light one, making the two axles different in cornering power at different lateral-g levels. This affects the balance at the limit, making the car harder to setup and drive. It’s also why a weight distribution close to 50-50 is very nice to have. The 944 isn’t bad at all in that respect.

The main point for the autocrosser is this: if you reduce the lateral weight transfer the car should corner faster because the tires perform better. At least in theory. Except for the bad things that happen when you reduce weight transfer by lowering the car.

Calculating Weight Transfer

I’ve read several expositions of how to calculate lateral weight transfer, including Herb Adams’s and Carrol Smith’s mentioned previously, but the one I like the best is from Dennis Grant’s Autocross To Win website. He’s got a section entitled Weight Transfer and I highly recommend it.

A quick review: Grant calculates three sources of lateral weight transfer in a cornering automobile that add to produce a total value. He calls them the unsprung weight transfer, WTu, the sprung geometric weight transfer, WTg (also called jacking force) and the sprung elastic weight transfer, WTe. Each one gets it’s own formula that allows us to calculate it’s magnitude. Depending on what you do to the car, these values can change up or down.

In Figure 3, below, I’ve reproduced the formulas as Grant presents them. M is mass, LatA is the lateral acceleration. CGh is center of gravity height. RC means roll center. T is the track width.

WTe

WTg

WTu

  Figure 3

Studying the equations we see that in order to calculate the total lateral weight transfer in a corner we need to know, measure or estimate the values for only seven variables: the unsprung mass (Mu), the sprung mass (Ms), the track width (T), the CG height of the car’s sprung mass, the CG height of the unsprung mass, the height of the roll center (RCh) and the value of the lateral acceleration (LatG) you think (or know) the car can generate. That’s it. You can use these equations roughly for the whole car, or for each end independently. Working each end independently will give insight into the roll distribution, front to rear, which is major when it comes to oversteer/understeer balance.

Now, I can hear Bubba Bratwurst asking, “Hey, it’s the lateral acceleration I want to increase, but I have to know the lateral acceleration in order to calculate the lateral weight transfer that I want to decrease. Isn’t that circular?” Yes, it is. But, it’s only a problem in your head, Bubba. Stick a number in for LatA and have another brat.

It’s also very easy to get the units messed up if you aren’t an engineer, or, even if you are. If you want to keep it simple and all you know (or want to know) about slugs (look it up) is that they’re slimy things that crawl in the dirt then here’s what you do: use the weight on the Earth’s surface (what the bathroom scale says) in pounds for mass, convert all inch dimensions to feet and use g’s for the lateral acceleration, as in 1g lateral, which I’m using here. Then you don’t need any conversion factors… they’re all built in and the answers are pounds of weight (vertical tire load) transferred.

If you want to do it in metric, go on ahead. (Bless your heart.)

Things we don’t need to know: how stiff your springs are, how stiff your anti-roll bars are, or, your shock forces. I expect more than a few of you might be surprised at that.

Don’t stiffer springs reduce roll? Yes. Therefore, weight transfer is reduced also, right? Nope. (Or, at least, not much.) Don’t believe me? Bless your heart.

Since spring rate, for instance, doesn’t appear in any of the three formulas, that means you don’t need to know your spring rates in order to calculate weight transfer. Same for horsepower, barometric pressure, phase of the moon or any other of an infinite number of parameters you care to think about. Not in the formula? Fugetabowdit.

Spring, shock and sway bar roll stiffnesses are important… I’m not saying they aren’t. They primarily affect how much of the total weight transfer goes to which end of the car and how fast it gets there after you make the boneheaded digital steering input that causes the front end to push out toward Saturn. They do not affect the total.

You might think about it like this: place a coil spring of 100lbs/in spring rate on the ground and set a 100lb weight onto it. How much weight does the ground under the spring feel? 100lbs, neglecting the spring weight, right? Now change the spring rate to 200lbs/in and put the 100lb weight back on top. How much weight does the ground feel now? Still 100lbs. The new spring only deflects half as much, but it transmits the exact same force to the ground.

Same thing in a car. The spring rate determines how far the car rolls, but doesn’t affect the value of the force that causes it to roll and it has nothing to do with how much weight is transferred due to the compression/extension of the springs.

Take a kart, for another example. Assume an infinite spring rate, which is almost correct since they don’t have a suspension. You don’t doubt that weight transfer still takes place do you? (See the unsprung weight transfer formula in Figure 3.) A passenger car has almost infinitely soft springs compared to a kart. Weight transfer still takes place, just in a more complicated manner. (All three equations from figure 3.) If you can take a variable in both directions towards infinity and nothing much happens then that variable isn’t important.

So, spring rate doesn’t affect weight transfer. This is what all the books say. Of course, all the books are slightly wrong. (Bless my heart.)

Or rather, they leave a little something out which may or may not be significant. I think the books are slightly wrong because the sprung mass of the car, as it rotates on the springs, is not absolutely constrained from moving laterally with respect to the tire contact patches. The sprung mass rotates about a roll center that’s usually (in a modern race car) very near the ground and therefore significantly below the CG, so the CG very definitely moves left and right as the car corners. This lateral movement of the CG that accompanies the rotation creates a little additional lateral weight transfer, just like the tool box in the trunk we forgot to remove would change the weight distribution during a corner if it slid to one side and how the gasoline in the tank sloshes, which we are also ignoring.

Our three equations don’t take this extra lateral movement into account. Neither did Smith or Adams or any other book on automotive suspensions that I’ve seen. A complete kinematic model of the car, which I imagine is the more usual way for a professional race team to do it these days, could include this effect, along with precise roll center movement.

I’ve never seen any numbers for the lateral CG movement; I have no idea how big it is. I think we can expect that the higher the CG is above the roll center and the softer the springs the more it will be. So, for an old-time race car with both a high CG and a high roll center (I hear they used to design suspensions that way) the effect on weight transfer must be vanishingly small. Such a car doesn’t roll much at all. (But, it will flip!)

For a modern race car, with a relatively low CG, low total roll, and a low roll center, it’s probably also very small. For a narrow and tall passenger car, say a 1970’s Saab 96, maybe not quite so small. In the mean time, like a typical engineer, I will arrogantly neglect any effect I can’t easily calculate!

(Yesterday I read Vivek Goel’s blog post on weight transfer in his blog Beyond Seat Time, which I highly recommend, in which he references a paper that does take the extra lateral movement of the CG into account. The author of the paper does a sensitivity study and concludes that the effect on weight transfer is negligible in all cases. I’m not totally convinced, but Ima-gonna-go with it for now.)

The Numbers

So, let’s estimate some numbers and put it all into a spreadsheet. I haven’t yet weighed or corner-balanced the car, so, for now, I’ll use the spec weight of 2900 lbs, not counting driver.

I have weighed the wheel and tire combination: 39lbs. Double that weight will be my estimate of the unsprung mass. That makes the unsprung mass total 2 x 4 corners x 39 lbs/corner = 312lbs. The sprung mass, Ms, is then 2900 – 312 = 2588lbs.

I know the average track, T, is 57.6”.

The CG height of the unsprung mass, CGhu, is usually taken as the wheel centerline height. For the stock wheels and tires that’s pretty close to 13”.

The CG height of the sprung mass, CGhs, is a little more difficult. I haven’t been able to find an actual measurement. I will guesstimate it at 20”. I’ve seen numbers like that for similar cars. For more modern Porsches, like the Cayman, I’ve seen a number in the 17” to 18” range, so 20” for the 944 seems reasonable. March of progress, you know.

The roll center height is another dubious number with no reliably measured values to be found. Several people are on record as saying the stock roll center is 4” to 6” above the ground, but they never say where or how they got it. I’ll go in the middle with 5”.

Now for the hairiest of the estimates. To calculate the effects of lowering the car we’ve got to know how much the roll center height changes as the sprung weight CG height changes. I have some real data for the double A-arm Corvette, but only hearsay for the strut/semi-trailing arm arrangement in the 944. Various sources (possibly not independent) claim 1” of sprung mass CG height change causes 2” to 3” of roll center height change. My initial guess from looking at the geometry is that the truth is on the high side of that range, so I’ll assume a 3 to 1 ratio to start with. Later, I’ll put in a 2 to 1 ratio to check the sensitivity. (The real change is probably not linear, either.)

So, for each inch of suspension-derived lowering we are going to say the roll center goes down 3”. That means the roll center is getting farther from the CG as the car is lowered. The distance between these two is the roll moment arm, so the arm is getting longer as we lower the car.

The lateral acceleration of the sprung mass during cornering produces the force on the end of the moment arm trying to roll the mass. A longer moment arm means more leverage at the roll center and, since I can’t change the resistance of the springs, the sprung mass portion of the car rolls farther. This is why lowering the car makes it roll more. What a pain!

Spreadsheet Results

The results of putting Dennis Grant’s equations and my values for the stock car into a spreadsheet are shown in the following figures. Figure 4, below, indicates that 1g lateral will produce 969lbs of total lateral weight transfer. So, 66.8% of the weight that was on one side gets moved to the other side. Using our previous tire performance estimate of 0.1% reduction in cornering power per each percent of weight transfer, we can calculate a value for the total cornering power reduction due to weight transfer of 6.7%.

fig4

Figure 4

Please notice that only 70lbs of weight transfer is from the unsprung weight. This is exhibited as a force that tends to pick up and flip the unsprung mass, just like a kart flips. 225lbs is geometric weight transfer (jacking force) which produces internal forces in the suspension components that tend to jack up the sprung mass. The vast majority, 674lbs, is elastic weight transfer, exhibited as roll of the sprung mass reacted by the springs and bars.

By the way, the 66.8% weight transfer is within 0.1% of the example given by Herb Adams. Maybe he was talking about a 944?

The first change I made to the car before I ever autocrossed it was to join the ‘tiny Rivals” club (coined by Burglar on Rennlist in this thread, post 83) which had the effect of lowering everything by 1/2” while not affecting suspension geometry. This slightly reduces weight transfer, as seen in the “Short Tires” column in Figure 5, while not affecting anything negatively. 

fig5

Figure 5

Total weight transfer drops 25lbs (from 969 to 944), most of which is a reduction in geometric weight transfer (WTg) which is directly proportional to the height of the roll center if you look at the equation. Elastic weight transfer (WTe) is unchanged because both the CG and roll center heights drop equally, maintaining the same moment arm distance from one to the other.

Next, I look at what happens if I take advantage of the M030 sport suspension and drop the overall ride height by 1” from stock, which then drops the roll center 3”, given our initial assumption. The results are shown in the “1” Drop” column of Figure 6. 

fig6

Figure 6

Geometric weight transfer (WTg) has plummeted 158lbs (from 225 to 67.4) because the roll center height has moved down from the initial 5.0” above ground to 1.5” above ground. (The first 0.5” from the tiny Rivals, then 3” more from the 1” drop.) But, the elastic weight transfer has increased from 674lbs to 764lbs, a 13.3% increase, thanks to more distance between the CG and the roll center. This means the car will roll 13.3% farther than stock or with tiny Rivals.

Unsprung weight transfer doesn’t change as compared to the short tires value and total weight transfer is reduced 70lbs (969 to 899) from stock. Cornering power reduction due to weight transfer has dropped 0.5% compared to stock, meaning we now have more cornering power.

How about a 2” drop? Most people seem to think this is possible with the sport suspension but probably too much if you can’t make other changes, because of bump steer effects plus a roll center that gets too low. (OMG, it’s below the ground!) I actually don’t yet know how much lowering I can get.

At a 2” drop the weight transfer numbers look even better, as shown in Figure 7. 

3to1

Figure 7

Geometric weight transfer (WTg) is now negative! (Anti-jacking.)

In spite of the anti-jacking making the geometric weight transfer go negative, it’s countered by a 26.7% increase in elastic weight transfer (WTe) as compared to stock, 854lbs vs 674lbs. This means 26.7% more roll than stock. Unsprung weight transfer doesn’t change and we are left with a total weight transfer decrease of 115lbs as compared to stock (854 vs 969). The total reduction in cornering power from weight transfer, as compared to stock, has decreased from 6.7% to 5.9%, meaning we have a net increase in cornering power of (100-6.7/100-5.9)-1 = 0.0086 or .86% (I think I did that right.)

How Much Time Saved?

How significant is a 0.86% increase in cornering power? Let’s put some numbers to it.

Each year at Dixie Tour there’s a 180 degree turnaround that’s about 160 feet in diameter. At 1g a car can negotiate that curve at 34.61mph. At 1.0086g (0.86% greater) that speed increases to 34.75mph. That extra speed saves 0.02s of time around that turn. Not really very much!

Similarly, if we assume that 25% of a 60s course is spent at max lateral-g, then the time saved over the course is 0.064s. While not nothing, and it could easily be the difference between 1st and 3rd, I’m having my doubts that it’s worth a significant decrease in roll stiffness and the loss of camber that would result.

How Sensitive Is this Result?

Remember that the preceding is predicated on a critical assumption, the 3 to 1 ratio of roll center movement to CG movement. What if it’s only a 2 to 1 ratio, as some have said?

For a 2 to 1 ratio we get a new, final chart, figure 8, below. 

2to1

Figure 8

Notice a pattern? The decrease in cornering power reduction is exactly the same as before, going from 6.7% initially to 5.9% now, the same 0.8% improvement.

The big difference is the decrease in roll. The 2 to 1 ratio of movement creates only a 13.3% increase in roll as compared to 26.7% increase for the 3 to 1 ratio. So, the roll stiffness is very sensitive to how much the roll center actually moves w/r/t the CG height, but the weight transfer is not. These estimates show that I’ve got to have better information. I’ve got to nail down the actual ratio or I’m just spinning my wheels, so to speak.

Also, how do we know that we aren’t decreasing grip faster due to camber loss in roll than we are increasing it with reduced weight transfer? I’ll need to experiment and measure.

How does the roll center actually move? How bad is this extra roll and what does it actually do to transient response.? We’ll explore these questions in Part 2.