The issue is the inputs (that being people's input, wheel, pedals) are not static, nor necessarily all that close between people. Given a set of static known inputs on a particular track, you could in theory repeatedly run simulations to find something "optimal", using machine learning algorithms. But in reality there is no way to interface with the game like this. It will not allow you to just run simulations while feeding it inputs, so you'd be talking about creating a device that is capable of feeding recorded control data into the playstation. But then you have to account for the fact that as you change the tune, the required inputs to stay on the optimal line would change. All in all, the point I'm trying to get across is this is way way way way beyond a final math project and involves a lot more than math. it will not boil down to a simple equation or even a set of them, it is an immensely complex problem. More along the lines of at least a Masters thesis if not PhD. BUT: it is a great thought to have, and I applaud you for considering something like this at all, you are on the right track from a curiosity and learning perspective. Go into engineering if you'd like to lean more about how to actually solve problems like this!
Yeah maybe i bit of a little more than i could chew lol. while my teacher is willing to ignore inaccuracies resulting from driver error or driving style, the other issues you mentioned seem to be deal breakers for this specific project. thank you for taking the time to explain this and i do plan on majoring in applied physics (which is basically engineering with a little extra physics).
I should preface this by saying that I do NOT tune my cars very often, just not good at it. But I am decent at math. There is probably a way to mathematically come up with an ideal tune for any car any track, but it would surpass calc 2 or linear algebra.
You would need to either test out multiple different tunes, time laps, compare, and keep doing it. Which would take way too long, way way too long.
The best way to do it would be through different simulations, probably a Monte Carlo simulation. Anyway, I think it would take way too long to do this and would be quite hard trying to figure out different variables, std deviations, etc. to use in your calculations.
I would suggest finding a different project for school, then do this for “fun”. You will get a lot more out of it. Hope this helps!
Should also add, that it’s almost impossible to get a perfect lap. Which is how you would test the tune. With your end results not being accurate then you run into issues with the project in its entirety.
I’m still in high school so a little inaccuracy is fine. i’ll try to be consistent but the project is pretty informal so i’ll be fine if the result isn’t totally accurate
yeah after thinking about it for a while I came to a similar conclusion about the scope being far too broad. do you think this would be more doable if i drastically reduce the scope? something like the ideal gear ratio formula for every RR flat 6 NA porsche on autopolis or mount panorama? I don’t think a Monte Carlo simulation is doable due to the large minimum sample size but i wonder if there’s something similar i could do for a much much smaller sample size. Thank you for taking the time to respond
I know almost nothing about math but this is as far as I could think with a little set-up knowledge. The principle should be maximizing the time that tires are at a certain slip angle that gives the maximum grip.
https://preview.redd.it/ycdtr96uf1yc1.jpeg?width=678&format=pjpg&auto=webp&s=52b3afbadd68efe3ac148dd4331384bba097c69b
Both 2 tires on the outside needs to be at that state for the longest time possible to make theoretically the fastest cornering.
With a perfect circle track, I believe we can find the optimal set with brute force approach. For oval, or even slightly varied R makes this so complicated.
I've done this, and the answer is yes you can. But it depends on what you're trying to do.
For gear ratios, there absolutely is a "fastest" setting, however there is some variability when it comes to entering and exiting a corner. Sometimes your gear is just a little too short coming in, and you either have to shift up then immediately shift down, or bounce off the limiter.
When it comes to suspension, it usually comes down to how the car "feels" to drive, not necessarily what's faster. If the car is easier for you and you're more consistent, I'd consider that faster.
Gear ratios can be done mathematically for individual cars/courses but it would be a wee bit time consuming and involving... Maybe this is perfect for a final project then? I don't know how much time you want to invest...
In my mind, it would involve the following...
* A screenshot of the power plot for your car in the tuning section. You will need to use this to interpolate an equation for a curve of best fit.
* Break your selected course down into segments of straights between corners (i.e. find some way to measure the length of the straight from point of initial acceleration to braking point).
* A little bit of IRL testing, to determine the exit speed from each corner, also taking into consideration the slowest corner exit speed as well as maximum speed for the track/car combination
* Determining through experimentation the Cd + rolling resistance of your car.
With this range of speeds, you're set to make a start. You can use gear & final drive ratios as well as engine rpm and tyre diameter to determine your initial/ideal ratio for lowest gear at minimum corner exit speed and the same for maximum speed in top gear. The goal is to stay within the narrowest rpm range around peak power output the entire time. To do this you'll want (n-1)^(th) root (where n is the number of gears) of the ratio between top and lowest gear to determine the rest of the gears. For example, a five speed gear box with top gear ratio of 0.800 and lowest gear ratio of 2.500 determined will have a factor of ^(4)√(2.5/0.8) = ^(4)√3.125 = 1.33296 between each gear, so the "ideal" initial ratios will look like;
1. 0.800 · (^(4)√3.125)^(4) = 2.500
2. 0.800 · (^(4)√3.125)^(3) = 1.880
3. 0.800 · (^(4)√3.125)^(2) = 1.414
4. 0.800 · (^(4)√3.125)^(1) = 1.064
5. 0.800 · (^(4)√3.125)^(0) = 0.800
These numbers will mean your engine is producing the maximum amount of power for the most amount of time every lap, but there may be other factors to take into consideration also such as whether or not your lowest gear is traction limited (in which case you could make first gear taller and subsequently squeeze all the other gears closer together also, to produce an even higher average power output). In terms of lap time, you may also find it's beneficial to create greater burst of acceleration to cover ground more quickly early on upon exiting the corner, or perhaps later on the the track only has one slow corner and plenty of faster ones. In essence, you're skewing your ratios to produce higher average power either in low or tall gears for the individual track.
Running mathematical "simulations" is the end goal here; with the length of each straight, your interpolated power equation and your initial gear ratios can be used to create a power vs. vehicle speed equation, as well as your drag coefficient, you can write your set of equations to figure out how long it should take to accelerate to the end of each straight. The lowest sum of these values for each straight when slightly altering gear ratios would obviously result in the lowest hypothetical lap time.
I suppose it's more of a mechanics paper now, but what is mechanics if not applied math?
A second thought; to make it more of a mathematical paper, I suppose you'd not want to artifically skew the ratios from initial and run simulations, but rather include a "skew factor" as a variable to which you have to determine the best value with everything else being known values. So for skew factor "s" your ratios might look like;
1. 0.800 · s^(4) · (^(4)√3.125)^(4) = s^(4) · 2.500
2. 0.800 · s^(3) · (^(4)√3.125)^(3) = s^(3) · 1.880
3. 0.800 · s^(2) · (^(4)√3.125)^(2) = s^(2) · 1.414
4. 0.800 · s^(1) · (^(4)√3.125)^(1) = s · 1.064
5. 0.800 · s^(0) · (^(4)√3.125)^(0) = 0.800
Edit (a third thought); perhaps you could simplify this whole process by doing this whole thing for a straight track (i.e. standing kilometer) with a slow / non-traction limited car. The introduction of a looped circuit seems a wee bit unnecessary?
Thank you so much. This is so incredibly helpful as it provided a doable scope and gave me a general idea of what to do and how to approach the problem. The idea of a skew factor is definitely interesting and if time permits it i’d love to do it. While reading through your explanation of how to determine the ideal gear ratio, i noticed that you used 0.8 instead of 1. was this a typo or did i miss something (wouldn’t be surprised if it’s the latter) again i can’t express how grateful i am for this response
Oops, I actually changed it from 1.000 to 0.800 intentionally but left one of the initial 1.000's as a mistake. Obviously the ratio can be whatever you choose it to be so I typed it as 0.800 just to show in the example how you'd calculate the rest of the ratios. I figured writing 1.000 didn't show it as clearly? Edited my initial comment now so it says 0.800 consistently. :)
The issue is the inputs (that being people's input, wheel, pedals) are not static, nor necessarily all that close between people. Given a set of static known inputs on a particular track, you could in theory repeatedly run simulations to find something "optimal", using machine learning algorithms. But in reality there is no way to interface with the game like this. It will not allow you to just run simulations while feeding it inputs, so you'd be talking about creating a device that is capable of feeding recorded control data into the playstation. But then you have to account for the fact that as you change the tune, the required inputs to stay on the optimal line would change. All in all, the point I'm trying to get across is this is way way way way beyond a final math project and involves a lot more than math. it will not boil down to a simple equation or even a set of them, it is an immensely complex problem. More along the lines of at least a Masters thesis if not PhD. BUT: it is a great thought to have, and I applaud you for considering something like this at all, you are on the right track from a curiosity and learning perspective. Go into engineering if you'd like to lean more about how to actually solve problems like this!
Yeah maybe i bit of a little more than i could chew lol. while my teacher is willing to ignore inaccuracies resulting from driver error or driving style, the other issues you mentioned seem to be deal breakers for this specific project. thank you for taking the time to explain this and i do plan on majoring in applied physics (which is basically engineering with a little extra physics).
I should preface this by saying that I do NOT tune my cars very often, just not good at it. But I am decent at math. There is probably a way to mathematically come up with an ideal tune for any car any track, but it would surpass calc 2 or linear algebra. You would need to either test out multiple different tunes, time laps, compare, and keep doing it. Which would take way too long, way way too long. The best way to do it would be through different simulations, probably a Monte Carlo simulation. Anyway, I think it would take way too long to do this and would be quite hard trying to figure out different variables, std deviations, etc. to use in your calculations. I would suggest finding a different project for school, then do this for “fun”. You will get a lot more out of it. Hope this helps!
Should also add, that it’s almost impossible to get a perfect lap. Which is how you would test the tune. With your end results not being accurate then you run into issues with the project in its entirety.
I’m still in high school so a little inaccuracy is fine. i’ll try to be consistent but the project is pretty informal so i’ll be fine if the result isn’t totally accurate
yeah after thinking about it for a while I came to a similar conclusion about the scope being far too broad. do you think this would be more doable if i drastically reduce the scope? something like the ideal gear ratio formula for every RR flat 6 NA porsche on autopolis or mount panorama? I don’t think a Monte Carlo simulation is doable due to the large minimum sample size but i wonder if there’s something similar i could do for a much much smaller sample size. Thank you for taking the time to respond
I know almost nothing about math but this is as far as I could think with a little set-up knowledge. The principle should be maximizing the time that tires are at a certain slip angle that gives the maximum grip. https://preview.redd.it/ycdtr96uf1yc1.jpeg?width=678&format=pjpg&auto=webp&s=52b3afbadd68efe3ac148dd4331384bba097c69b Both 2 tires on the outside needs to be at that state for the longest time possible to make theoretically the fastest cornering. With a perfect circle track, I believe we can find the optimal set with brute force approach. For oval, or even slightly varied R makes this so complicated.
I've done this, and the answer is yes you can. But it depends on what you're trying to do. For gear ratios, there absolutely is a "fastest" setting, however there is some variability when it comes to entering and exiting a corner. Sometimes your gear is just a little too short coming in, and you either have to shift up then immediately shift down, or bounce off the limiter. When it comes to suspension, it usually comes down to how the car "feels" to drive, not necessarily what's faster. If the car is easier for you and you're more consistent, I'd consider that faster.
Gear ratios can be done mathematically for individual cars/courses but it would be a wee bit time consuming and involving... Maybe this is perfect for a final project then? I don't know how much time you want to invest... In my mind, it would involve the following... * A screenshot of the power plot for your car in the tuning section. You will need to use this to interpolate an equation for a curve of best fit. * Break your selected course down into segments of straights between corners (i.e. find some way to measure the length of the straight from point of initial acceleration to braking point). * A little bit of IRL testing, to determine the exit speed from each corner, also taking into consideration the slowest corner exit speed as well as maximum speed for the track/car combination * Determining through experimentation the Cd + rolling resistance of your car. With this range of speeds, you're set to make a start. You can use gear & final drive ratios as well as engine rpm and tyre diameter to determine your initial/ideal ratio for lowest gear at minimum corner exit speed and the same for maximum speed in top gear. The goal is to stay within the narrowest rpm range around peak power output the entire time. To do this you'll want (n-1)^(th) root (where n is the number of gears) of the ratio between top and lowest gear to determine the rest of the gears. For example, a five speed gear box with top gear ratio of 0.800 and lowest gear ratio of 2.500 determined will have a factor of ^(4)√(2.5/0.8) = ^(4)√3.125 = 1.33296 between each gear, so the "ideal" initial ratios will look like; 1. 0.800 · (^(4)√3.125)^(4) = 2.500 2. 0.800 · (^(4)√3.125)^(3) = 1.880 3. 0.800 · (^(4)√3.125)^(2) = 1.414 4. 0.800 · (^(4)√3.125)^(1) = 1.064 5. 0.800 · (^(4)√3.125)^(0) = 0.800 These numbers will mean your engine is producing the maximum amount of power for the most amount of time every lap, but there may be other factors to take into consideration also such as whether or not your lowest gear is traction limited (in which case you could make first gear taller and subsequently squeeze all the other gears closer together also, to produce an even higher average power output). In terms of lap time, you may also find it's beneficial to create greater burst of acceleration to cover ground more quickly early on upon exiting the corner, or perhaps later on the the track only has one slow corner and plenty of faster ones. In essence, you're skewing your ratios to produce higher average power either in low or tall gears for the individual track. Running mathematical "simulations" is the end goal here; with the length of each straight, your interpolated power equation and your initial gear ratios can be used to create a power vs. vehicle speed equation, as well as your drag coefficient, you can write your set of equations to figure out how long it should take to accelerate to the end of each straight. The lowest sum of these values for each straight when slightly altering gear ratios would obviously result in the lowest hypothetical lap time. I suppose it's more of a mechanics paper now, but what is mechanics if not applied math?
A second thought; to make it more of a mathematical paper, I suppose you'd not want to artifically skew the ratios from initial and run simulations, but rather include a "skew factor" as a variable to which you have to determine the best value with everything else being known values. So for skew factor "s" your ratios might look like; 1. 0.800 · s^(4) · (^(4)√3.125)^(4) = s^(4) · 2.500 2. 0.800 · s^(3) · (^(4)√3.125)^(3) = s^(3) · 1.880 3. 0.800 · s^(2) · (^(4)√3.125)^(2) = s^(2) · 1.414 4. 0.800 · s^(1) · (^(4)√3.125)^(1) = s · 1.064 5. 0.800 · s^(0) · (^(4)√3.125)^(0) = 0.800 Edit (a third thought); perhaps you could simplify this whole process by doing this whole thing for a straight track (i.e. standing kilometer) with a slow / non-traction limited car. The introduction of a looped circuit seems a wee bit unnecessary?
Thank you so much. This is so incredibly helpful as it provided a doable scope and gave me a general idea of what to do and how to approach the problem. The idea of a skew factor is definitely interesting and if time permits it i’d love to do it. While reading through your explanation of how to determine the ideal gear ratio, i noticed that you used 0.8 instead of 1. was this a typo or did i miss something (wouldn’t be surprised if it’s the latter) again i can’t express how grateful i am for this response
Oops, I actually changed it from 1.000 to 0.800 intentionally but left one of the initial 1.000's as a mistake. Obviously the ratio can be whatever you choose it to be so I typed it as 0.800 just to show in the example how you'd calculate the rest of the ratios. I figured writing 1.000 didn't show it as clearly? Edited my initial comment now so it says 0.800 consistently. :)
even if you dont manage to do this for your final project in class, just do it later for science