Racecar Engineering magazine; August 2004.


There is an excellent article in this issue that uses William Mitchell's software to show why the classic racing line around a 180 degree turn of starting wide and making a nice smooth arc down to the apex and then a smooth arc back out is not the fastest way through the corner (even if trailbraking heavily down into the apex or late apexing).

The fastest line is a parabolic line; nearly half of an ellipse. Essentially turning into the corner earlier and driving the car in a straighter arc to the apex. As the car gets nearer to the apex it is heavily braked and after slowing the car is cranked hard around the apex before getting hard on the gas and accelerating back out. The parabolic line has a much slower speed at the apex than the classic line, but its pure cornering phase at the apex is tremendously shorter. The parabolic line carries more speed over a longer period of time before the apex and begins accelerating sooner after the apex. To top it off it also covers less distance than the classic line.

Note that the parabolic line is still different than late apexing a corner. The late apex does allow you to get on the throttle earlier, but its entry speed is still not maintained as long as the parabolic line. Essentially in the car you would start wide and turn in early, aiming the front of the car towards the corner. You can carry more throttle longer because you are not doing as much turning and therefore have more grip for acceleration/braking. Once closer to the apex the car is braked very hard and then trail braked over a short distance while cranking the car very hard around the apex (the speed here at the apex is actually considerably slower than the classic line would produce). The amount of turning done is much tighter than the corner may seem (this also means you may not be on the curb at the apex, you may be a couple feet from it to give room to crank the car around) and so the car will stay to the inside of the track longer just after the apex. Since it already got most of its turning done however it can now use more of it grip to immediately accelerate hard out of the corner. The parabolic line has a much shorter pure cornering phase and much more opportunity to carry throttle longer on the way in and earlier on the way out. Less overall distance is just a bonus.

I can't find any diagrams right now but they are all in the article and it's several pages long. Excellent reading if you can find it at your local magazine stand.

This diagram is not an actual racing pic but illustrates the difference in shape. You can see in the parabolic one the majority of the turning is done right at the apex as opposed to turning it smoothly throughout the corner. Since the parabolic line is straighter on the way in and back out the tires have more grip available for braking/acceleration in these areas.

Parabolic line:

Classic line:

 

Interesting...

Did they have any numbers in terms of actual performance difference on a track?

-=Russ=-

 

I'll have to buy a copy of that magazine but I would bet that one assumption behind that analysis is a relatively high HP to weight ratio.

That is the ideal line when you are traction limited. If you don't have enough HP to break traction in the forward direction, your best corner exit begins to favor that classic line.

As a matter of fact the ideal line changes based on the length of the incoming and outgoing straight. When the straight gets very short (for example an autocross) you begin to prefer the shortest path (the inside line).

Everything is a compromise based on what you have to work with. There is no one universal ideal line.

ed

 

The article completely recognizes this along with the fact that every corner on a racetrack is different and requires a different set of compromises in each one to add up to the fastest lap time.


The entire premise is that the classic line insists you don't have any traction left to brake or accelerate in a forward direction because you are using all of your grip to do nothing but corner over that long arc through the turn. You must begin braking earlier and wait longer to accelerate when using the classic line compared to the parabolic line.

Compared to the classic line the parabolic line gives a faster segment time because is it much faster into and out of the corner than the classic line. Horsepower to weight doesn't matter because you would still be faster through the initial phases of the corner and be able to accelerate sooner and more aggressively at exit using the parabolic line than the classic line. You do give up more speed at the immediate apex with the parabolic line but you more than make up for it at the entry and exit phases of the corner because you do the majority of your turning in that small distance around the apex where you're going "slow". Since the majority of your turning is done at the apex instead of throughout the entire corner you have more grip left to brake and accelerate the car everywhere else in the corner.

The downfall of the classic line is the fact that its apex speed is so high. If you are carrying that much velocity around the apex you don't have much grip left to actually turn the car, so you can turn the car only slightly and thus you must corner over a longer period of time. During that time you are pure cornering and thus can't slow or speed up the car without sliding. You know already that you balance your brake/throttle with the steering wheel. More brake/throttle requires less steering and vise versa; the friction circle of the tires' grip insist this to be so. The classic line does not maximize tire grip through all phases of the corner, only at the apex. Everywhere else it is losing time.

The article is based on mathematical models by using force vectors through the corner. Anyone comfortable reading vector diagrams will be comfortable and understand.

 

Here's the given problem: A 180 degree turn with a short straight leading into and away from the corner, no banking, bumps, sand etc on the surface.

If you were a bystander viewing the track perpendicular to the straightaways you would obviously know right away that the car that can come down the straightaway, turn, and come back out at the highest speed will be faster through the segment. The speed that counts isn't around the corner, it's the speed on the way in and on the way out. By purposely apexing the corner at a slower speed you can do all your turning in a very small portion of the racetrack and therefore not spend as much of your traction budget cornering. You'd spend more of your grip budget and time going fast in a straight line.

These digrams are scans from Erik Zapletal's article in the August 2004 issue of Racecar Engineering. Each of these diagrams shows the force vectors present on the car throughout the corner. The longer the vector, the more force it represents. The "V" vector represents straightline velocity of the car (speed) and the "A" vector represents acceleration force (In physics an acceleration is defined as a change in vector quantity. Therefore driving at a constant speed around a circle is in fact an acceleration since direction is constantly changing. Speeding up and slowing down in a straight line are of course accelerations as well). The parabolic line allows the car to generate greater velocity at a greater acceleration for a longer period of time than does the classic line. This therefore means it can cover the given distance more quickly. The parabolic line allows the car to spend more time at a higher velcoity than the classic line does, even though the classic line does carry more speed at the apex. When viewing the diagrams you can see that the parabolic line has greater vector sums than the classic line when the entire corner is analyzed.

These diagrams are just the tip of the iceberg. The entire article goes much greater into the depths of explanation. You do need to understand vector quantities though.

 

great stuff Damon.

that's exactly how i try to drive turns 1, 3, 11a and 14 at VIR If I enter a turn from a long straight thats my chosen line. Not for beginners though because the rear has to pivot to make the apex also much less room for error in regard to braking. Use the tires aggressively once to get the car pointed then back on the gas rather than a slower longer drift through the corner which is a slower line and actually wears the tires more. Again you better have good car control and know how to brake Bottom line is it's a faster line but also a more dangerous one.

 

Everyone also keep in mind the difference in racing other cars and in just turning fast laps. In a real race lap time doesn't matter; track position does. Many times when you put a pass on a car into a corner you blow by him in the first half of the corner and then have to practically stop the car to get around the apex because you overcooked it at entry. You got through the corner more slowly but you passed someone in doing it. As long as you didn't overcook it so much that he can dive back under you at exit the position is yours. Track position is what counts, not lap time. You'll see this in any race. The guys fighting eachother for position can't keep up with the guy out in the open. The guy in the open doesn't have to proctect himself and can drive purely for speed.

When you're just lapping by yourself or not in direct competition with other cars on the track you're just interested in bragging rights: fastest laps. Racing door to door is different. Think of it as the difference between qualifying and racing.

 

I agree that the parabolic path uses the traction circle to best advantage but you still need the HP to take advantage of the traction circle. If you take the example to the limit of a car with no HP, the fastest way around the track is the way that retains the most speed. Lesser HP cars cannot completely utilize the acceleration half of the traction circle so they need to concentrate on carrying more speed.

An ITS Rx7 that is not prepped to the limit is right on the edge of being considered a low HP racecar (in my opinion - or maybe no racer thinks he has enough HP, I don't have the money to make my engine a "consumable"). The higher speed the corner is, the more your racecar will act like a low HP racecar - the more you will want to favor the classic line.

Going into a corner you would still want to favor the parabolic path (trail brake) but coming out of a corner depends on how much acceleration your car is capable of at that point. You would adjust your path so that you can maximize the usage of your available traction circle.

ed

 

Interesting.

I disagree that HP/weigh doesn't matter, though.

Let's take the ultra-extreme example- a car that cannot accelerate at all; aka, an engine-less car.

In the classic line, the car brakes and begins his turn as he lifts off the brakes. The apex speed is fairly constant (The constant corning section) and he exits with "roughly" the same speed as he entered.

Now, the parabolic line: you enter quickly and brake MUCH more heavily for the apex. As you round the apex, you have no ability to accelerate, so your exit speed is MUCH slower. Sure, you covered a smaller area- but if the straight is sufficiently long, the parabolic corner is going to be slower for our engineless car.

As a result, a parabolic curve is probably only better assuming you have sufficient power to overcome the fact that you're exiting the corner very slowly.

 

ok...


Not necessarily. If you have more speed but have to follow a longer path your net time could in fact be greater.



Depends. All corners don't have the same arc radius therefore you can't make a blanket statement like that. There is certainly a point for every car where the corner becomes so tight you'd want to trade distance for velocity, but you'd rarely find that on most racetracks.


The classic line will make you drive further around the corner and the bigger the corner is the farther you're going to have to drive. The lesser distance advantage of the parabolic line will increase as the corner radius increases. In your example you're carrying more speed around the apex so you don't have to accelerate as much at exit. In doing so you are traveling a greater distance and are in fact spending less time with your foot to the floor. The corner would have to be verym very tight IMO. Shorter distance will beat higher speed the majority of the time.

Once you've trailbraked in you're already committed to how fast you're going to come out. The first phase of the corner dictates how the rest of the corner will be. If it doesn't then the first phase of the corner was taken too slow...

Just being argumentative. Nothing on a racetrack or with a car is ever "perfect" but don't be so quick to figure it doesn't apply just because the car is "slow".

 

I don't think most production closed-wheel chassis can handle suspensions loads on the parabolic race line, thus the "classis" race line is a compromise?
I can't even imagine what kinda loading is on the chassi when it hits the parabolic apex???

You can maybe get away with the parabolic line on a slick-shod, open wheel race car, but no way your FD is going to pull it.

A possible example for the west coast guys is T2 on Laguna Seca.
Instructors will recommend taking the turn as a modified double apex, but when the CART (or whatever they are called now) comes into town every year, they take that particular turn as a single apex.


-Ted

 

If the car is not taking full advantage of the friction circle it is not going as fast as it could be. If the car is in fact power limited and doesn't have gut wrenching acceleration you would merely not drive the car out as wide at corner exit (since you don't have much acceleration potential but have lots of cornering potential) and you'd spend more time going straight down the following straightaway as opposed to tracking across it... You'd still cover less distance through the corner iself as well. The straightway lasts longer on the way in and starts earlier on the way out so to speak.

 

Absolutely you can.

The parabolic line has absolutely nothing to do with type of car, type of suspension, type of tires are type of engine. The parabolic line takes the most advantage possible of the tires in every portion of the corner. As long as the car is on rubber tires it applies...

 

"Out as wide"? If the car has NO power at all (as shown in my example), you have to admit that a a sufficiently long exit straight would favor the classic line. Eventually, he's going to pass you.

So if it's true for a powerless car, what about a 1HP car?
What about a 10HP car?

See where I'm going?

 

Yes. But I believe it to be flawed and not representative of what happens to a powered automobile on the far majority of corners

So you guys have it all figured out already. How long is sufficiently long and how little power is underpowered?

I'm not by any means saying this technique applies to all corners in all situations. The author fully realizes that as well; everything in racing is always more complex than it seems.

However, before everyone poo-poos the article and insists it doesn't apply here, there and everywhere I highly recommend you actually read it first... My comments are by no means a condensed version of the article, they just represent some points to think about. If you do everything the same way forever you'll never have the opportunity to improve on what you are doing right now...

 

Yep you just shift to lower gear if necessary. Its not about power its about carrying as much speed as possible through a given distance from beginning to end. The classic approach sacrfices speed before turn in so you could argue that the higher hp car has the advantage

 

No poo-pooing involved, Damon. Just discussing your article, that's all.

Hell, I even posted it over on Corvetteforum to see what those guys have to say: http://forums.corvetteforum.com/show...p?p=1547876837

 

Damn, I can't quote that well...

Given the same corner and a car that can't accelerate "hard", the parabolic path is shorter and slower. The circular path is faster but longer. Immediately at the end of the corner the parabolic car will probably be ahead but moving more slowly. At some point the faster moving car will pass the slower moving car.

But if the car can accelerate "hard", the parabolic car will already be moving faster than the circular car and it will already be ahead.

The real question is when is a corner going to favor the circular path or the parabolic path. If it's a fast corner your car might only be able to accelerate at .2G. That might not be considered "hard" acceleration. In a slow speed corner you could accelerate at .8G that probably would be considered "hard" acceleration.

I'm not even going to try to analyze what happens when the outgoing straight isn't long enough to reach "terminal velocity". At some point everything shifts to simple shortest path where a circular path around the inside of the corner is quickest. You'd probably only have to worry about that at an autocross but I have two tracks I go to that have slow second gear corners that make me wonder. There was one corner at Shannonville that definately favors the inside line.


As far as blanket statements, you are right. I shouldn't have said that because any car can get "hard" acceleration if it is going slow enough. The question is still what kind of corner you are talking about in relation to your racecar's ability.

I'm just saying that if a car cannot accelerate well, the circular path should be quickest but if the car can accelerate well the parabolic path is better. When does it change from one to the other? I don't know. I tried to write a program to figure it out but I failed.


When I talk about going parabolic into a corner and circular out of the corner I am talking about shape with speed adjusted to blend the two. The shape might be half in and half out but the position would have to be adjusted to fit on the track. I think the fastest line can shift from one to the other at diferent points based on the speed of the corner and ability of the car.

I'm definately going to read the article and see if I can't learn something new. (or prove me wrong)

ed

 

Thats an an interesting premise. I agree that the higher HP cars will be able to use this type of turn more efficiently than the lower HP cars.

I used to race at the "Lone Pine Time Trials" every year. It was at a little over 4000' elevation, and I had a CSP FB with around 150 HP at sea level. At 4000' I lost a lot of power. At one end of the track was a 180 degree turn that was around 60 mph in my car. One of the course workers was timing the cars thru the turn. I used the Classic line, and tried to keep an even speed all the way thru the turn. The course worker later told me that I had the fastest time, and that he could hear my exhaust (no muffler) at a steady level all the way thru the turn. I made the best turn in that sitution. It is possible that the higher HP cars were going faster than me at the exit, but I got there faster.

Another possible problem with the "parabolic line" is that it will be much harder on brakes. If you are on a track, and going to be out there a while, this may not be the way to go.

 

Regardless of the power of the car the circular path will insist every single time that you spend more time slowing the car and travel a longer distance on the way in (and out). The real key to this whole discussion is time; not speed.

When speaking of lap time around a closed circuit the driver who spends the most time during the lap with the gas pedal pressed flat to the floor while traveling the least distance will always be the fastest, regardless of what his apex speeds are. The classic line insists you spend less time on the gas and more time with greater steering angles trying to change direction. Even though the classic line carries a higher speed at the apex and you may in theory pass a parabolic car there if you were beside him just before the apex, the parabolic car is going to win the drag race down the following straight because he can floor the throttle earlier. Even though he was going slower than you at the apex and is behind you right now he will pass you on the straight because he will spend more time on that straight with the throttle buried. The parabolic car goes slower across the apex but not so slow as to take away the advantages of its time spent with the throttle farther open.

The key is getting to the gas earlier in the corner than the other guy; NOT carrying the most speed around the apex. A race course is a pretty defined path since you have to stay on the pavement. If you can drive the course with the throttle on the floor for 60% of time it takes to compete a lap wouldn't you expect to cover a lap distance in less time than if you only held the throttle floored 55% of the time?

 

It's time AND distance AND exit speed. You are braking earlier but you are breaking to a faster cornering speed.


But look at it another way. A parabola is the result of a path where the acceleration vector always points directly back up the straight (for a 180 degree turn). It's easy to see that this situation has a minimum of wasted acceleration and so it should be the fastest.

But what happens if at any point you don't have enough HP to max out your friction circle? You are not using your maximum available longitudinal acceleration. If you cannot use your max available longitudinal acceleration then you need to alter your path so that you do. By going a little faster at midpoint you will need more radial acceleration at the expense of your longitudinal acceleration (which you didn't have enough HP to use anyway)

You might not actually need to go all the way back to the circular path but something between that and the parabolic path. A slightly longer path but at a higher speed.

As far as I can see the parabolic path is only fastest when you have enough HP to keep your acceleration maxed to the edge of your traction circle. But I haven't read the article yet.


ed

 

Yeah, I think this is where the hang-up is.
Late-apex is taught for a "safer" line, and the ability for a vehicle to spend as much time accelerating out from the turn.
If the turn is isolated from the rest of the track, the parabolic line would be quick, as pointed out.
But, realistically, a single turn does not make for a complete "lap".
We're all trying to connect turn to turn, so you need to account for other factors that does affect how quick you get to the next turn.
Of course, wheel-to-wheel racing would just mess up any line.


-Ted

 

If the driver feels he needs horsepower to max out his friction circle he is a stupid driver. If the car has no more acceleration left he should be using the rest of the traction budget to corner at higher speed (not brake as much on the way in) and use a tighter line until he IS using up his traction budget.

How much horsepower or acceleration the car is capable of has absolutely nothing to do with using up the friction circle. You always trade acceleration (or braking) and cornering grip back and forth. If you're not doing much of one you do more of the other, otherwise the driver is a dumb bell.

 

Let's put some numbers to this and prove to you guys that corner exit speed is NOT king. Getting on the power earlier in the corner than the other guy is more important; even if you must give up speed in the corner to do so.

Vf=Vi + a(t)

Vf= final velocity
Vi= initial velocity
a= acceleration
t = time

This is a basic formula for velocity. It says that final velocity is equal to the initial velocity plus acceleration times time. Let's put some numbers in there. I know for a fact that my car can accelerate at .7 g very easily. 1g= 32 ft/sec/sec so .7g= 22.4 ft/sec/sec.

Let's say a "classic" car goes through a corner. He carries 35 mph around the corner and floors the throttle onto the straightaway. Let's then say it takes him 3 seconds to cover the straight before he has to brake, turn, whatever.

In this case:
Vi= 35 mph = 51.33 ft/sec
a= .7g = 22.4 ft/sec/sec
t= 3 sec

therefore Vf= 118.53 ft/sec = 80.816 mph

So the classic car leaves the corner doing 35 mph and after 3 more seconds of acceleration he is doing 80.816 mph.


Let's say a "parabolic" car goes through the same corner. He can only carry 30 mph through the corner because of his tighter turning (that's about 15% slower than the other car did there!) Because he gets his turning done earlier in the corner he can floor the throttle .5 seconds sooner. This gives him 3.5 seconds down the straight before he has to brake, turn, whatever.

In this case:
Vi= 30 mph = 44 ft/sec
a= .7g = 22.4 ft/sec/sec
t= 3.5 sec

therefore Vf= 122.4 ft/sec = 83.45 mph

So the parabolic car leaves the corner doing only 30 mph but gets to accelerate for 3.5 seconds. This leaves him doing 83.45 mph!

So the parabolic car goes through the corner 15% slower (that's a lot! That's going through a 100mph corner at only 85 mph for instance) than the classic car but since the parabolic car gets on the gas earlier (he finished his turning earlier since he was slower in the corner. Since his turning is done earlier he can accelerate earlier) he will pass the classic car down the straightaway Exit speed is over rated! It's how much time you can spend with the throttle to the floor that counts...

Also take note that if the classic driver insists on taking every corner in that fashion the parabolic driver's straightaway will get even longer and he will be able to spend even more time accelerating because he not only gets on the gas sooner, he stays on it longer into the next corner. The parabolic driver will gladly apex every single corner on the track slower than the classic driver.

Exit speed from corners is not the most important thing. The most important thing is to spend as much time as possible with the throttle wide open while keeping an eye on the length of your line.

 

New line seems like the old double apex to me