Physics Help Forum Wondering About Racing Line
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 Apr 1st 2017, 06:32 AM #1 Junior Member   Join Date: Mar 2017 Posts: 9 Wondering About Racing Line Have only just last night found these forums and appreciate the opportunity to post and ask true experts if they might help, even tho I am a physics idiot and will probably embarrass myself with pre-Newtonian ignorance of what is appropriate here and where if at all my interest could be directed. Just guessing this is the place. Attached are some diagrams created out of my imagination. The last (Aremberg) was posted on a simracing forum in 2010, and the rest were created since then. Note the POV is that of a point moving on a line looking forward to a future point on a line that is not yet there and can be changed, (optimized), and it is not the Cartesian POV of an objective observer. Can’t find anything like this, don’t know what it is or even if it is anything...is it I simply do not know where to look? Just made it all up starting with no idea. I definitely do not know where to look for help or more info. If anyone has any recommendation regarding how I might learn more about the physics/math of this POV perhaps you can help with a reply? Or if you want to tell me I am for sure a physics idiot that's okay too. All these ideas have been thought up while having fun. Lately been wondering whether there is anything to it and if so what. Appreciate any comments, thanks in advance. Best to all, John Attached Thumbnails           Last edited by simracer; Apr 1st 2017 at 06:52 AM.
 Apr 1st 2017, 08:15 AM #2 Physics Team     Join Date: Jun 2010 Location: Naperville, IL USA Posts: 2,225 Hello simracer - as a sometime track junkie myself, I like to think I know a bit about proper racing lines. But I don't understand what you are asking. Are you looking for a a formula for the geometry of a proper racing line? Or are you trying to determine the apparent position of a flagger relative to the driver's field of vision (as in the first figure)? Please be specific - would love to help, but need to understand precisely what you are asking.
 Apr 1st 2017, 09:51 AM #3 Junior Member   Join Date: Mar 2017 Posts: 9 Have no idea what questions to ask. It was just a guess this is the proper forum or if anything about my interest is proper at all. No clue even what is the topic. Appreciate ChipB that you took the time to reply. Noticed being able to pick a specific point at exit and set up a trajectory during entry that will get there. Pretty sure this is accomplished by offsetting the centers of mass for the front and rear suspensions and for the car. As you can see by the steering wheel in the Aremberg pic the screenshot was made at the exact instant the exit point was set. The next motion is to set the opposite point, (the ghost point), which the sprung weight will bounce from to arrive at the exit point. From then on the steering wheel is held consistently against the camber of the, (not used to "steer" the car but rather to induce front grip), and throttle is used to push the car around the curve, with the intent of establishing the same slip angle on all four wheels. This allows the car to maintain the set trajectory thru the turn with no or little change in driver inputs. This happens so certainly, (the red line), there must be some math and physics that would be useful to a driver in real time, using the driver's POV, compared to an "academic" analysis by an observer, as when a how-to-drive the racing line is presented using a plan view and a diagram similar to the 4th pic above, "Funny Calculus." Otherwise for the driver typical instructions are things like, "In slow, out fast," and "start on the outside, come in toward the apex, then exit back to the outside." Just seems not very technical or specific, while the certainty of the trajectory is absolute, in that once the car is set in its condition only the most extreme inputs will cause it to veer off course. Do understand the math and physics of race engineering and car setup is absolutely and very specific. Trying to find what there is about being the line compared to observing it. Or is this just all silly fantasy? No doubt some of it is. As the car travels on the red line, Y continues to equal zero? X equals infinity, (the point farthest away which is never reached)? Even I know that's not right from an observer's POV and maybe any POV. Seems pretty silly even to me. I guess, looking for answers to questions I don't know enough to ask that apply to a driver's POV and would be useful in car at speed. Last edited by simracer; Apr 1st 2017 at 05:03 PM.
 Apr 1st 2017, 06:02 PM #4 Physics Team     Join Date: Jun 2010 Location: Naperville, IL USA Posts: 2,225 Sorry, but I am at a loss as to what you are trying to accomplish. I understand the idea of setting up the entry to the corner so as to hit the desired exit point, but trying to derive a formula for it would be incredibly complicated. It depends on factors such as tire grip versus slip angle, how the driver modulates the throttle through the turn, what track conditions are like, etc. I don't know what you mean by "opposite point" or "ghost point:" - these are not terms I've heard before. And I don't understand what you mean by the driver's point of view versus the "academic analysis" - please clarify. I would suggest that you look into a few books that delve into the physics of track driving. Check out: "Speed Secrets, Professional Race Driving Techniques" by Ross Bentley. Basic info on late apexing, smooth braking, "slow in, fast out," etc. "Going Faster, Mastering the Art of Race Driving" by the Skip Barber Racing School. A bit more technical than "Speed Secrets." And finally: "How to Make Your Car Handle" by Fred Puhn. This last one goes into some pretty good details of suspension geometry, tire friction, and weight transfer that may get at what you're looking for. Good luck!
 Apr 1st 2017, 07:09 PM #5 Junior Member   Join Date: Mar 2017 Posts: 9 Thanks ChipB, have had and used Puhn for several years. He just recently passed, sad to say. Have read Bentley's when it was loaned by a friend. "Academic," as I meant it, means an objective non-intuitive analysis from the viewpoint of an observer. It is the POV of looking at a map or Cartesian quadrants with 0,0 or 0.0.0 in center. When you look at the racing line projected on either, you see a line. Neither are the POV of a driver, which seems more appropriate looking "down" the Y-axis as in the Aremberg pic. Is there any other imaginary structure that can be applied in real time to real space, or virtual space, and can be used to determine a racing line? The ghost point is similar to the ghost pocket used for many years to determine a bank shot on a pool table. It is called the ghost, I guess, because it doesn't exist yet can be used as a target. On the racing line, it is the point the suspension will bounce the centers of mass away from and with no steering input reflect them back to the exit point. Over the years I have received a lot of grief about these ideas, mostly in good humor but to my knowledge nothing from qualified experts. Attached Thumbnails   Last edited by simracer; Apr 2nd 2017 at 09:06 AM.
 Apr 2nd 2017, 09:04 AM #6 Junior Member   Join Date: Mar 2017 Posts: 9 Maybe this? Hm...now see driver POV is more appropriately Z-axis? But it is more fun when it is Y = You. Attached Thumbnails   Last edited by simracer; Apr 2nd 2017 at 09:20 AM.
Apr 3rd 2017, 07:26 AM   #7
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 Originally Posted by simracer Is there any other imaginary structure that can be applied in real time to real space, or virtual space, and can be used to determine a racing line?
It depends on the specific geometry of the track. First, I believe the fastest racing line is the one that gives the largest radius of curvature around a corner. This means entering the turn at the outside edge, then setting a constant radius turn so that the car skims the inside of the track at the apex then continues maintaining that radius until track-out at the outside wedge as you exit the turn. The exact point of turn-in depend on the radius of the corner and the width of the track - see the attached sketch. If R_i = radius of curvature of the turn (measured at the inside edge of the track), and W = width of the track, then the largest possible turn radius for a 90 degree corner is $\displaystyle 2W(1+\frac {\sqrt 2}2 ) + R_i$. I'm happy to walk you through the geometry which leads to this.

In terms of the POV of the driver, he should commence his turn when the apex is 22.5 degrees to the side of his straight-ahead view. Alternatively, if you know where the exit point is you can turn in when that point is at 45 degrees from straight ahead.

This assumes (a) a perfect 90-degree corner, (b) no elevation change, (c) constant width track and radius of the corner, and (d) that the objective is to get through the corner in the shortest possible time. But in reality things may be quite different:

1. The above leaves no margin for error. In reality drivers don't follow this line because if you're off just a bit you run the risk of crashing into the outside wall.

2. Quite often the objective is not to maintain a constant speed through the turn, but to find a line that yields the highest possible speed at the exit of the turn (slow-in, fast-out). Constant speed is a good strategy for low-horse power cars that need to maintain momentum (think Mazda Miata for example). But for high HP cars it may be more advantageous to do the late turn-in/late apex method, which results in slower speeds at the start of the turn but faster exit.

3. If the corner in question leads to a long straight the late apex approach is probably better. But if the turn leads to a second turn without a long straight it may be more important to set the car up for that next turn rather than try to speed through this first turn and end up on the wrong side of the track for entry to the next.

 Originally Posted by simracer The ghost point is similar to the ghost pocket used for many years to determine a bank shot on a pool table. It is called the ghost, I guess, because it doesn't exist yet can be used as a target.
Sorry, but I still don't see what this has to do with driving around a corner. As noted above you can set a visual target if the apex and/or ideal exit point is clearly marked.

 Originally Posted by simracer On the racing line, it is the point the suspension will bounce the centers of mass away from and with no steering input reflect them back to the exit point.
I don't understand what you're trying to say here. Please clarify.

I see in one of your diagrams a reference to a Brachistochrone curve. I don't see how that would apply at all - a Brachistochrone curve has to do with the fastest route for an object to slide downhill on a frictionless surface. The mathematics behind it depends on constant gravity g, acting in a constant vertical direction, and where the kinetic energy gained as the object slides is dependent on vertical distance traveled. That's totally different than what happens with a car in a curve, where g forces act away from the center of curvature of the turn (i.e. the direction of g-forces changes throughout the turn), and KE depends on engine torque (which varies with speed) minus friction forces of tire-to-pavement (which depends on rolling resistance and slip angle of the tires), internal resistance of the drive train, and wind resistance. It's a much more complicated situation.
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Last edited by ChipB; Apr 3rd 2017 at 08:01 AM.

 Apr 3rd 2017, 09:23 AM #8 Junior Member   Join Date: Mar 2017 Posts: 9 Appreciate your comment about the difference in driving lines for different cars. Many years ago replaced the motor in an 82 Cutlass Calais with a 72 Olds 455 Rocket V8. It bolted right in, even the cross member holes matched up perfectly. 200 more horsepower in less than six hours plus tidying up time. The motor was so heavy it squished the car down and the tires pressed on the wheel wells, so the front springs had to be replaced. At speed thru a wide sweeping curve on a remote country road at entry the motor could be thrown to the exit point by juggling the suspension a bit with slight inputs to steering, brake, and throttle. The car would exactly follow this toss no matter what. The first recognition the car was following the motor thru the turn was a discovery not an experiment. At somewhere around 70mph the motor was not going to an exit point on the road. It was going over a ditch thru a fence into to a field on the outside. Fortunately the technique was not discovered by accident. It was a big aha moment. The evidence was indisputable to me. Toss the motor and follow it where it will surely go, by not steering except to hold the car on that line, (just hold the wheel steady), not braking for sure, and using throttle to accelerate or at least maintain momentum. It was pure luck the stiffer front springs alone enabled the car to power slide like you wouldn't believe. Now using a simulator find it is also possible to toss the car's center of mass to a point symmetrically opposite the exit point and the suspension will bounce away from the point of toss exactly. That point is what I call the ghost point. Note this is not a geographically symmetrical point or only would be I guess given a 90deg turn as in ChipB's post. The toss creates the trajectory. Steer or brake at your peril. If in doubt, accelerate. So maybe, really interested in the physics/math of the toss? ChipB, thanks very much for your reply. Last edited by simracer; Apr 3rd 2017 at 09:45 AM.
 Apr 5th 2017, 05:29 PM #9 Junior Member   Join Date: Mar 2017 Posts: 9 ChipB, Have you seen this? https://dspace.mit.edu/bitstream/han...pdf?sequence=2 The title is "Racing Line Optimization." Do you make use of the 22.5deg or 45deg lines when on track?
Apr 5th 2017, 07:12 PM   #10
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 Originally Posted by simracer ChipB, Have you seen this? https://dspace.mit.edu/bitstream/han...pdf?sequence=2 The title is "Racing Line Optimization."
Interesting paper. But it doesn't really answer the question it poses - just presents several analytical techniques for finding the fastest line around a couple of different track configurations.

 Originally Posted by simracer Do you make use of the 22.5deg or 45deg lines when on track?
No - there aren't many turns that are perfect 90 degrees and that don't involve elevation change. Plus (a) it's often difficult to see the inside midpoint of the turn from the turn-in point, or the exit point and (b) most times it's best to use a late apex line in order to increase corner exit speed. In other words rather than maintain constant speed throughout the turn you start at a lower speed and tighter turning radius and increase speed as you hit the apex and unwind the steering wheel on the second half of the turn. As previously noted determining this optimal line mathematically would require all sorts of detailed assumptions on acceleration versus velocity and turning angle, and an understanding of the sideways g-forces that the car can sustain without losing grip either front or rear under various conditions of weight transfer front-to-rear and side-to-side. If you watch races on TV you know they're always talking about how the car is understeering or oversteering (they use terms like "tight' or "loose" or say the car is "pushing" of "free"), and these behaviors can vary at different portions of the same turn. It might be interesting to try a few mathematical simulations, but I am skeptical that the results would have much practical use.

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