RX-7 Tuning Equation Demonstration
OTI equations provided by budious
If you have been following, or have attempted to follow, the second post in this thread then you already know about the Open Tuning Initiative. I know at the moment it is very disorganized, ambiguous, and in some cases inaccurate. However, I have prepared a short demonstration of the techniques that I do know work fairly well at this point. I hope after you try the following guided demonstration you can appreciate and better comprehend the equations I have published to your own benefit.
Test Car: Mazda éfini RX-7 Type R (FD) '91
- Premium car, purchase at new car dealership.
- Equipped at stock weight with upgrades to approximately 300HP.
Additional Purchases:
- GT Auto: Oil Change
- Tune Shop: Sport Soft
- Tune Shop: Fully Customized Suspension
- Tune Shop: Low RPM Turbo
- Tune Shop: Sport Exhaust
Deep Forest Raceway Testing:
- Equip the car with all purchased parts except return to factory suspension; drive a few laps around Deep Forest while noting the handling characteristics of the car.
- Equip the car with all purchased parts including the fully customized suspension; zero out the rear positive toe, then drive a few laps around Deep Forest while noting the handling characteristics of the car.
Examining the defaults on the Fully Customized Suspension for the RX-7:
Damping: For the default values on the RX-7, both extension and compression are set to values of five. These values have multiple roles in the overall physics engine, but for the particular sub-component determining overall suspension balancing, we need to compute the LOG of each.
Dampers:
- LOG(1) = 0
- LOG(2) = 0.301
- LOG(3) = 0.477
- LOG(4) = 0.602
- LOG(5) = 0.699
- LOG(6) = 0.778
- LOG(7) = 0.845
- LOG(8) = 0.903
- LOG(9) = 0.954
- LOG(10) = 1
RX-7's Damping Efficiency Factor, or
DEF = LOG(5) + LOG(5) = 1.398
Stabilizers: For the default values on the RX-7, the anti-roll bars are set to values of four. These values have multiple roles in the overall physics engine, but for the particular sub-component determining overall suspension balancing, we need to compute the Natural Log, or LN, of the stabilizer bar.
Anti-Roll Bars:
- LN(1) = 0
- LN(2) = 0.693
- LN(3) = 1.099
- LN(4) = 1.386
- LN(5) = 1.609
- LN(6) = 1.792
- LN(7) = 1.946
RX-7's Stabilizer Efficiency Factor, or
SEF = LN(4) = 1.386
Spring Rates (and Ride Height): As for the defaults on all cars in the game, ride height is zeroed. The spring rates on the RX-7 default setup are 8.4 front and 6.3 rear. As all variables in the suspension setup, the following calculation is not the single condition nor only contributing factor for what the maximum spring rate should be; it is only a determination of what the minimum spring rate should be.
RX-7's Spring Rate Stiffness Factor, or SF(SR) is calculated as follows:
(Front Spring Rate + Rear Spring Rate) / (Car's Weight KG / (Base Ride Height + Ride Height Change))
(8.4 + 6.4) / (1260/(100+0)) = 1.167
This poses a problem because the spring rate stiffness factor, or SF(SR), is below the ~1.4 threshold we are looking for. So how do we achieve a SF(SR) of 1.4? For the simple purposes of testing our theory we can start with a ride height adjustment.
(8.4 + 6.4) / (1260/(100+(+20))) = 1.400
Perfect, our formula reveals that adding +20 ride height to the car will achieve the target SF(SR) we were looking for. Let's test it on the track, watch the tire temperature indicator to realized how much grip has been optimized on your first lap. Things will probably get a little sloppy on the second lap but settle down if you keep driving. This setup achieves the intended effect we were looking for by increasing SF(SR) to 1.4 but using this method has increased lateral weight transfer on the car with the increase in ride height.
So how do we achieve a SF(SR) of 1.4 but closer to the normal ride height, or any desired ride height, for our car? There's an equation for that... and hopefully soon enough, an equation app for that... in the meantime, break out the calculator and some scrap paper.
First, we need to calculate the relative distribution of supported weight on the car's springs; this should not to be confused with the actual weight distribution of the car. This is a fairly simple process.
% Weight Distribution Front Springs = Front Spring Rate / (Front Spring Rate + Rear Spring Rate)
% Weight Distribution Rear Springs = Rear Spring Rate / (Front Spring Rate + Rear Spring Rate)
8.4 / (8.4 + 6.3) = .571 (57.1%)
6.3 / (8.4 + 6.3) = .429 (42.9%)
You really only need to do one to find both, simply subtract the one you do first from 1.00 to get the other. Next, you need to multiply those figures by the weight of the car to find the supported weight in kilograms on each axle.
KG Weight Distribution Front Springs = (Front Spring Rate / (Front Spring Rate + Rear Spring Rate)) x Car's Weight
KG Weight Distribution Rear Springs = (Rear Spring Rate / (Front Spring Rate + Rear Spring Rate)) x Car's Weight
8.4 / (8.4 + 6.3) = .571 x 1260 = 720 KG = 1.00 SF(SR)
6.3 / (8.4 + 6.3) = .429 x 1260 = 540 KG = 1.00 SF(SR)
The resulting figures represent the weight supported at a spring rate stiffness factor, or SF(SR), at 1. This is because the weight you multiplied against the percentage of distribution was the car's actual weight. You could have compiled the previous step with the following step, but for comprehension it was broken down into an extra step. To bump SF(SR) up to the desired 1.4 all we need to do is multiply the previous results by the new desired factor.
8.4 / (8.4 + 6.3) = .571 x 1260 = 720 KG x 1.4 SF(SR) = 1008KG
6.3 / (8.4 + 6.3) = .429 x 1260 = 540 KG x 1.4 SF(SR) = 756KG
Now we are ready for the final step. We need to divide these figures by the desired ride height to determine the final spring rate at that ride height. This can be a bit frustrating to do with a standard calculator. However, there are shortcuts for a graphing calculator and I will update this post and the OTI post with these at a later date. If your standard calculator supports returning to the previous line and simply overwriting the ride height value then this following task is much easier. Note: Base Ride Height is not always 100; a few cars may have a non 1:1 motion ratio on one linkage; a few others may have non 1:1 motion ratios for both linkages (ie. Formula cars are 2:1 wheel rate to spring compression). However, for the vast majority of cars in the game, a motion ratio of 1 for front and rear, and a Base Ride Height of 100 can be safely assumed.
Stiffness Factored Weight / (Base Ride Height + (Change in Ride Height))
720 KG x 1.4 SF(SR) = 1008KG / (100 + (+5)) = 9.6 kgf/mm
540 KG x 1.4 SF(SR) = 756KG / (100 + (+5)) = 7.2 kgf/mm
These are your new spring rates at the indicated ride height. I prefer to attempt to get these figures as close to but under a whole value to a tenth of decimal accuracy as this is the limitation of the spring rate tuning allowed in the game. I also attempt to get them with no to as little rake as possible involved because rake involves shifting weight transfer and I have not yet produced a variant on the formula to account for these other variables, among many others. Basically, what I attempt to do is divide by ride heights until I find one that will make both the front and rear at the same ride height level no less than five hundredth under or one hundredth over the next closest settable value. (ie. 12.95 - 13.01 would be set to 13.0 kgf/mm)
The final suspension configuration for SF(SR)=1.4 @ +5mm Ride Height is as follows:
Ride Height: +5 / +5
Spring Rate: 9.6 / 7.2
Extension: 5 / 5
Compression: 5 / 5
Anti-Roll Bars: 4 / 4
Why did this work? Let's look at the much larger, but still limited, view I have assembled so far (as theorized) for the internal functioning of the suspension balancing and grip determination mechanisms of the physics engine.
Stiffness Factor of Spring Rates ^ Natural Log of Anti-Roll Bar
----------------------------------------------------------------------
Sum of the Damper LOGs ^ Natural Log of the Anti-Roll Bar
or
(SF(SR)^LN(ARB)) / (LOG(Extension) + LOG(Compression) ^ LN(ARB))
1.4^1.386
---------------------
(.698+.698)^1.386
1.594
------- = 1.002 (100.2% optimized)
1.591
Now keep in mind that all this does is establish the minimum stiffness factor of spring rates, SF(SR), to be used on the car you are tuning. You can increase the final ratio anywhere above 1.000 (~100%) and the car remains drivable but there is probably another such equation to determine upper range efficiency and it is probably track specific or natural frequency of the course specific to determine the new optimization. Though, I think with the above rationalization, that the car will achieve its maximum grip efficiency while increasing the stiffness factor of spring rates beyond that threshold will continue to improve lap times in a trade off for less grip. At this point much remains speculative, I only have what works on one hand, what doesn't on another, and a whole lot at my feet I haven't gotten around to just yet.
Final Note: If you are intending to reproduce this exercise on other cars keep in mind the following things.
- See the OTI post on the Supra example for how to deal with motion ratios if the car you are tuning uses one.
- Chassis rigidity factors into this equation somewhere; attempt with new dealership purchases without chassis reinforcement for most normalized results.
- Stiffness Factor of Spring Rates >= Damping Efficiency Factor >= Anti-Roll Bar Efficiency Factor