OptimumG Magic Number Approach to Suspension Tuning

[fe1rx]

You may have heard of the concept of a Magic Number approach to suspension design and tuning of track cars. The notion relies on the concept that matching the front to rear lateral weight transfer (essentially the roll stiffness distribution front to rear) with the front to rear weight distribution will result in neutral steady state cornering balance. As a starting point a Magic Number 5% higher than the percentage of mass on the front axle is recommended. Given that our cars have approximately 52% of their mass on the front axle, a Magic Number close to 57% would be desirable. As Magic Numbers get larger, steady state handling moves further toward understeer.

For discussion purposes, I have calculated the Magic Number for 4 configurations:

1) OE configuration (OE springs, bars and ride height) – Magic Number = 71%
2) 60 N/mm (342 lb/in) front springs and 70N/mm (400 lb/in) rear – Magic Number = 74%
3) 60 N/mm (342 lb/in) front springs and 120N/mm (685 lb/in) rear – Magic Number = 62%
4) as above with 20 mm rear bar – Magic Number = 57%
5) as above with 140 N/mm (800 lb/in) rear springs – Magic Number = 56%

For simplicity the above has assumed no change in ride height. These examples are meant to show the trend associated with spring and bar changes rather than to introduce too many other changes to confuse the trend. The numbers show clearly why the basic 135i has such terrible limit understeer out of the box, and why the kind of spring rates track junkies have settled on tend to actually work well.

This analysis ignores the role of alignment settings, tire size, compliance, non-linear springs, bump stops and dampers play on the total handling package, but without the basic steady state balance being right, fiddling with these other factors will not provide an optimum handling solution. Of note, putting a big front bar on our cars will make the Magic Number worse, but may actually improve handling if it prevents the front tires rolling into positive camber under cornering loads. The Magic Number really is only one part of the big picture, but it should be the place you start if you are trying to optimize a track car.

For those that care to follow along, I will dig deeper into the weeds:

1) Summary of Magic Number calculations:



2) Sample Magic Number spreadsheet for OE configuration:





3) Explanation of Magic Number Input Variables
- front spring motion ratio: OptimumG uses (wheel movement/spring movement) as the definition for MR. Many others use the inverse as the definition. For our vehicles front MR = 1.04 (or 0.96 using the conventional definition)
- OE front springs have been reported to be 21 N/mm (120 lb/in). I have assumed this value without independently verifying it.
- front bar acts at the strut so has the same MR as the spring
- I have previously measured the 26.5 mm OE front bar stiffness as being 40 N/mm (227 lb/in)

https://www.1addicts.com/forums/show...3&postcount=37

- Front ride height is assumed to be 592 mm to wheel rim (345 mm to wheel centre) front and
- Front and rear tires are assume to have an unloaded radius of 322 mm and a spring rate of 373 N/mm (2130 lb/in) which is typical of what I have measured.

https://www.1addicts.com/forums/show...5&postcount=17

- Rear spring motion ration is 1.76 (0.563 by the conventional definition)
- OE rear springs have been reported to be 61 N/mm (350 lb/in). I have assumed this value without independently verifying it.
- Rear bar acts directly at the wheel hub so its motion ratio is 1.0
- I have previously measured the 12 mm OE rear bar stiffness as being 2 N/mm (13 lb/in) and an H&R 20 mm rear bar stiffness to be 18 N/mm (102 lb/in). (Earlier link shows 15 lb/in but later testing refined this to 13 lb/in.)
- Rear ride height is assumed to be 584 mm to wheel rim (337 mm to wheel centre)
- Corner weights have been simplified to 409 kg (902 lbs) on each front corner and 373 kg (823 lbs) on each rear corner. This gives a total weight of 1564 kg (3448 lbs) with 52.3% on the front axle. This is representative of the car with fuel and driver.
- I have previously measured unsprung masses front and rear and assumed them to be 52 kg (115 lbs) front and 55 kg (121 lbs) rear. These change with wheel, tire and BBK modifications but not in a way that is materially significant to the overall conclusions of this simplistic analysis.
- CG height is assumed to be 517 mm (20.4 in) above the ground. I will outline later in the thread how I arrived at this assumption.
- Roll centre heights have been assumed to be 20 mm front and 100 mm rear. I will outline later in this thread where these assumptions come from.

[fe1rx]

Measuring CG Height

Car and Driver recently added cg height measurement to its toolkit.

https://www.caranddriver.com/news/a1...ravity-height/

Their summary of the math is correct, but they ignore the details that make the process much more complicated, if accurate results are needed.

In fact, a study by the University of Michigan had the NHSTA, GM, Ford and Chrysler labs perform cg height measurements on 4 sample vehicles to examine the accuracy of their various approaches and procedures revealed huge variation in the results between organizations and the reasons for them:




For those interested here is a link to the full report:

https://www.google.com/url?sa=t&rct=...wcU3CyDJg-oEw9

I performed my own measurements to determine the cg height of my 135i, using a variety of rear wheel heights, with the expectation that as the rear wheel height increases, the accuracy would increase too. I used a magnetic digital protractor stuck on the roof to measure angles.



The results did, indeed, change with height and appear to be converging toward a value of approximately 21 inches. You can see that the change of weight on the front axle with increasing angle is very small making the method very sensitive to weighing inaccuracies. Contrary to recommendations, I weighed with approximately ¾ fuel.



Longacre racing provides their own instructions for the process, and they add a very important caveat – “replace each shock absorber with a solid link to eliminate suspension travel”. Needless to say I didn’t do this (and neither does Car and Driver), rendering our measurements further suspect.

http://www.longacreracing.com/techni...spx?item=42586

Obviously this is more than a little inconvenient, but as my results have shown, without it the results are suspect. Even with that precaution, results may be of dubious accuracy and repeatability.

Warren Rowley in “An Introduction to Race Car Engineering – Book One” states:

“It is very difficult to accurately determine the z coordinate of the chassis’ total mass (extreme understatement). Even though the CG height is one of th most important number the race engineer needs when calculating chssis dynamics, it remains one of th most difficult number to accurately obtain …”

He goes on at some length as to why the number is so difficult to obtain accurately, but one reason is that the method is so sensitive to weight variation that setup scales (accurate to 1 lb each) are not accurate enough. The University of Michigan study confims his assertion that we should approach all vertical cg height measurements with some skepticism.

My actual approach to estimating the cg height relies on SAE report 1999-01-1336, which shows by empirical analysis that the cg height of a car with driver is on average 37-38% of the overall height.



Here is a link to the complete report:

https://www.google.com/url?sa=t&rct=...lxiNuKNhezw4qe

Using this approach I have assumed a cg height of 517 mm (20.4 in.), which is slightly lower than my own weighing would suggest.

[fe1rx]

Roll centre heights deserved their own thread:

https://www.1addicts.com/forums/show....php?t=1604010

[rac]

Originally Posted by fe1rx

The notion relies on the concept that matching the front to rear lateral weight transfer (essentially the roll stiffness distribution front to rear) with the front to rear weight distribution will result in neutral steady state cornering balance. As a starting point a Magic Number 5% higher than the percentage of mass on the front axle is recommended. Given that our cars have approximately 52% of their mass on the front axle, a Magic Number close to 57% would be desirable.

nice post, I should stay in touch with this forum more regularly.

I wish I did all this stuff years ago when I first upgraded my suspension, which is now horribly off where it should be based on the recommendations of the time (considering correct basic setup starting points based on weight transfer and natural frequencies).

Just curious, I calculate essentially the same numbers as you on a couple of your scenarios but then differ on others - using only using basic roll stiffness ratio front to back from wheel, arb rates and track length.

What other inputs is the worksheet sensitive to?

For Case 1, I calculate the same ratio 71%, for the other cases I am between 2 - 6% higher.

The difference could mean being slight U/S bias based on my calcs actually makes me slightly O/S biased based on your more detailed calculation.

Also - any consideration for chassis stiffness effects? I wonder, could the actual front to back load transfer ratio of the complete system be confirmed with corner scales and jacking the car up in two different known locations?

R.

[rac]

Originally Posted by rac

I wonder, could the actual front to back load transfer ratio of the complete system be confirmed with corner scales and jacking the car up in two different known locations?

R.

just thinking out loud - if the car is on corner scales, and you jack it up on one side along the same plane as the CoG (length along wheelbase can be calculated), then the moment is being applied in the same plane i.e. same distance from either axle as the CoG position, then the ratio of weight gain on the two outside wheels will be the same as if the load was acting through the CoG, no? Doesn't really matter what the moment arm or total moment is is because we just want to know the relative load transfer is front to back from any moment applied in that plane?

[fe1rx]

Originally Posted by rac

just thinking out loud - if the car is on corner scales, and you jack it up on one side along the same plane as the CoG (length along wheelbase can be calculated), then the moment is being applied in the same plane i.e. same distance from either axle as the CoG position, then the ratio of weight gain on the two outside wheels will be the same as if the load was acting through the CoG, no? Doesn't really matter what the moment arm or total moment is is because we just want to know the relative load transfer is front to back from any moment applied in that plane?

Interesting idea. Ideally you would apply a pure moment, not an offset jacking force.

Here is one way to do it - build an outrigger, centered on the driver's location fore and aft. Weigh the car with ballast in the driver's seat and get corner weights. Weigh it again with the ballast as far outboard on the outrigger as possible and compare the corner weights to find out how much of the moment is reacted at the front axle and how much at the rear axle. Total weight and longitudinal cg is the same in both cases. Only difference is that outrigger case has an applied rolling moment.

Big rolling moments (which will be difficult to apply safely) will be more accurate. Shocks will have to be dialed full soft to minimize stiction. Comparing ballast full left to ballast full right on the outrigger is probably a good idea.

I might try this one day ...

[fe1rx]

Originally Posted by rac

nice post, I should stay in touch with this forum more regularly.

What other inputs is the worksheet sensitive to?

For Case 1, I calculate the same ratio 71%, for the other cases I am between 2 - 6% higher.

The difference could mean being slight U/S bias based on my calcs actually makes me slightly O/S biased based on your more detailed calculation.

Also - any consideration for chassis stiffness effects?s?

R.

Are you using the Optimum G spreadsheet? If you don't have it and would like it send me a PM.

Definitely no consideration of chassis stiffness.

[rac]

Originally Posted by fe1rx

Interesting idea. Ideally you would apply a pure moment, not an offset jacking force.

Here is one way to do it - build an outrigger, centered on the driver's location fore and aft. Weigh the car with ballast in the driver's seat and get corner weights. Weigh it again with the ballast as far outboard on the outrigger as possible and compare the corner weights to find out how much of the moment is reacted at the front axle and how much at the rear axle. Total weight and longitudinal cg is the same in both cases. Only difference is that outrigger case has an applied rolling moment.

Big rolling moments (which will be difficult to apply safely) will be more accurate. Shocks will have to be dialed full soft to minimize stiction. Comparing ballast full left to ballast full right on the outrigger is probably a good idea.

I might try this one day ...

yeah stiction could be an issue. would be interesting in itself to know.

not sure I follow why a jack wouldn't suffice, broken down to a freebody diagram the result should be the same - what am I missing? it does assume the chassis is a rigid body.

I thought about the how big the lift would need to be - in terms of garage safety. I did some quick calc's and it seems feasible to result in measurable load changes with basic equipment, unless your looking for 1% but that wasn't my intent.

[rac]

Originally Posted by fe1rx

Are you using the Optimum G spreadsheet? If you don't have it and would like it send me a PM.

Definitely no consideration of chassis stiffness.

no, I made my own spread sheet out of interest. its basically just the roll stiffness ratio front to back. didn't think any other variables would have a significant impact on weight transfer ratio.

couldn't send you a pm - your inbox is full until you clear some space!

[Tambohamilton]

Been playing around with this a bit - made my own spreadsheet which is now churning out sensible numbers for ARB stiffness, but it needs to account for bushing windup and bump stops, for a start...

How should those be implemented in a spreadsheet like this? I know how to apply bushing windup to the ARB, and I'll figure out bump stops... I guess my question is; how does bushing rate affect wheel rate and therefore ride frequency? And the same with bump stops? From what I see above, they're not included in the OptimumG sheet?

[fe1rx]

Originally Posted by Tambohamilton

Been playing around with this a bit - made my own spreadsheet which is now churning out sensible numbers for ARB stiffness, but it needs to account for bushing windup and bump stops, for a start...

How should those be implemented in a spreadsheet like this? I know how to apply bushing windup to the ARB, and I'll figure out bump stops... I guess my question is; how does bushing rate affect wheel rate and therefore ride frequency? And the same with bump stops? From what I see above, they're not included in the OptimumG sheet?

The OptimimumG approach is looking at steady state handling balance. I think it is fair to say that track cars should not be on their bump stops under steady state maximum cornering conditions, except when they encounter a bump.

Generally though, bushing windup acts like a spring in parallel with the main spring, so the wheel rates from each source simply add together. The same is true of a bump stop, when it is engaged.

Bushing windup of the ARB bushings should be negligible. The ARB should rotate in the bushings, not bind to them.

Another confounding factor is that bushing rates can be expected to be both temperature and rate dependant (I expect stiffer at higher rates and softer at higher temperatures). This is certainly true of bump stops.

[Tambohamilton]

Originally Posted by fe1rx

The OptimimumG approach is looking at steady state handling balance.

Good point.

Originally Posted by fe1rx

I think it is fair to say that track cars should not be on their bump stops under steady state maximum cornering conditions, except when they encounter a bump.

Yes, agreed. As stock, it only takes a few mm (if any) bump travel to engage the "auxiliary springs", though... Something I'll change a little with all of this, but currently that's how it is for me.

Originally Posted by fe1rx

Generally though, bushing windup acts like a spring in parallel with the main spring, so the wheel rates from each source simply add together. The same is true of a bump stop, when it is engaged.

Thanks. That's what I needed to know! I just couldn't settle on how the interaction would go numerically, in my head, and nobody at google seemed to have an answer either

Originally Posted by fe1rx

Bushing windup of the ARB bushings should be negligible. The ARB should rotate in the bushings, not bind to them.

I have an OE E93 M3 rear ARB on my car right now...the bushings for it are the same as the E90/92 M3 items, which fit a 20mm (vs 23mm?) bar. The bushings definitely clamp down hard onto the E93 bar, and I'm fairly sure would add significantly to the rate. Next time I have it apart, I really should drill out the mounts a little, to ease the pressure.

Originally Posted by fe1rx

Another confounding factor is that bushing rates can be expected to be both temperature and rate dependant (I expect stiffer at higher rates and softer at higher temperatures). This is certainly true of bump stops.

Yup...I won't even be thinking about that - beyond the scope of my project, and my driving


Thanks for the input!

[rac]

https://www.chassissim.com/lateral-load-transfer-at-the-front-magic-number-what-really-drives-it/

Interesting add on to the magic number, the stability index as a function of magic number.

[Tambohamilton]

He's so difficult to listen to!!! Causes me great confusion. Good find though - I'll definitely dig into it more some other time.

Actually had a guest lecture from him at uni once...

[rac]

Originally Posted by Tambohamilton

He's so difficult to listen to!!! Causes me great confusion.

Australian English?

ok for me....

[Tambohamilton]

Nah, my mum is Aussie so the accent is no bother. It's when he umms and ahs and skips back a few words...while talking through an equation or something where the exact sequence is critical! Totally derails my train of thought, every time! Not saying I'd be better than him; just saying I find it hard to follow.

[rac]

Originally Posted by Tambohamilton

Nah, my mum is Aussie so the accent is no bother. It's when he umms and ahs and skips back a few words...while talking through an equation or something where the exact sequence is critical! Totally derails my train of thought, every time! Not saying I'd be better than him; just saying I find it hard to follow.

Have a go at these equations then...


https://www.chassissim.com/dans-vehi...car-stability/

Some more insight to the stability index and what pro drivers can tolerate, which I guess provides context to why oem magic numbers are so high and why 5% or so is the magic number.

The first link has a downloadable spreadsheet you can populate to and plots a stability index curve.

[rami88]

Originally Posted by fe1rx

You may have heard of the concept of a Magic Number approach to suspension design and tuning of track cars. The notion relies on the concept that matching the front to rear lateral weight transfer (essentially the roll stiffness distribution front to rear) with the front to rear weight distribution will result in neutral steady state cornering balance. As a starting point a Magic Number 5% higher than the percentage of mass on the front axle is recommended. Given that our cars have approximately 52% of their mass on the front axle, a Magic Number close to 57% would be desirable. As Magic Numbers get larger, steady state handling moves further toward understeer.

For discussion purposes, I have calculated the Magic Number for 4 configurations:

1) OE configuration (OE springs, bars and ride height) – Magic Number = 71%
2) 60 N/mm (342 lb/in) front springs and 70N/mm (400 lb/in) rear – Magic Number = 74%
3) 60 N/mm (342 lb/in) front springs and 120N/mm (685 lb/in) rear – Magic Number = 62%
4) as above with 20 mm rear bar – Magic Number = 57%
5) as above with 140 N/mm (800 lb/in) rear springs – Magic Number = 56%

For simplicity the above has assumed no change in ride height. These examples are meant to show the trend associated with spring and bar changes rather than to introduce too many other changes to confuse the trend. The numbers show clearly why the basic 135i has such terrible limit understeer out of the box, and why the kind of spring rates track junkies have settled on tend to actually work well.

This analysis ignores the role of alignment settings, tire size, compliance, non-linear springs, bump stops and dampers play on the total handling package, but without the basic steady state balance being right, fiddling with these other factors will not provide an optimum handling solution. Of note, putting a big front bar on our cars will make the Magic Number worse, but may actually improve handling if it prevents the front tires rolling into positive camber under cornering loads. The Magic Number really is only one part of the big picture, but it should be the place you start if you are trying to optimize a track car.

For those that care to follow along, I will dig deeper into the weeds:

1) Summary of Magic Number calculations:

Attachment 2024726

2) Sample Magic Number spreadsheet for OE configuration:

Attachment 2024727

Attachment 2024728

3) Explanation of Magic Number Input Variables
- front spring motion ratio: OptimumG uses (wheel movement/spring movement) as the definition for MR. Many others use the inverse as the definition. For our vehicles front MR = 1.04 (or 0.96 using the conventional definition)
- OE front springs have been reported to be 21 N/mm (120 lb/in). I have assumed this value without independently verifying it.
- front bar acts at the strut so has the same MR as the spring
- I have previously measured the 26.5 mm OE front bar stiffness as being 40 N/mm (227 lb/in)

https://www.1addicts.com/forums/show...3&postcount=37

- Front ride height is assumed to be 592 mm to wheel rim (345 mm to wheel centre) front and
- Front and rear tires are assume to have an unloaded radius of 322 mm and a spring rate of 373 N/mm (2130 lb/in) which is typical of what I have measured.

https://www.1addicts.com/forums/show...5&postcount=17

- Rear spring motion ration is 1.76 (0.563 by the conventional definition)
- OE rear springs have been reported to be 61 N/mm (350 lb/in). I have assumed this value without independently verifying it.
- Rear bar acts directly at the wheel hub so its motion ratio is 1.0
- I have previously measured the 12 mm OE rear bar stiffness as being 2 N/mm (13 lb/in) and an H&R 20 mm rear bar stiffness to be 18 N/mm (102 lb/in). (Earlier link shows 15 lb/in but later testing refined this to 13 lb/in.)
- Rear ride height is assumed to be 584 mm to wheel rim (337 mm to wheel centre)
- Corner weights have been simplified to 409 kg (902 lbs) on each front corner and 373 kg (823 lbs) on each rear corner. This gives a total weight of 1564 kg (3448 lbs) with 52.3% on the front axle. This is representative of the car with fuel and driver.
- I have previously measured unsprung masses front and rear and assumed them to be 52 kg (115 lbs) front and 55 kg (121 lbs) rear. These change with wheel, tire and BBK modifications but not in a way that is materially significant to the overall conclusions of this simplistic analysis.
- CG height is assumed to be 517 mm (20.4 in) above the ground. I will outline later in the thread how I arrived at this assumption.
- Roll centre heights have been assumed to be 20 mm front and 100 mm rear. I will outline later in this thread where these assumptions come from.

How can I get the spread sheet for calculation magic number and Springs/ARB/Load Transfer?

[bbnks2]

Originally Posted by rami88

How can I get the spread sheet for calculation magic number and Springs/ARB/Load Transfer?

Fatcat motorsports and Flyin Miata both have spreadsheets. You just need to change the numbers. Here is Rob Binette's: https://robrobinette.com/Suspension_Spreadsheet.htm

I would take their subjective comments with a grain of salt. My personal experience, and I think most people would agree with me, is the opposite of what is being said at the bottom of that spreadsheet about total roll couple. In autocross, I believe there is LESS importance on having higher front roll couple. You can run the car more neutral to get the low speed agility you need without under-steering everywhere. They recommend 70%+ for autocross and that just makes no sense to me. Seems to me on a road-course is where you want the higher front roll couple since you are dealing with tossing the car into corners at higher velocities. You need more front roll couple. In autocross you need more front grip which doesn't come from running a heavily under-steer oriented setup. best setup I have used in autocross so far has been 80N/mm / 160N/mm with M3 sways F/R. Decent amount of front toe-out. I have just switched to the Ohlin's dedicated track 120N/mm / 180N/mm and now I have very little front-end grip. I am basically drifting aound autocross courses again. The 120/180 setup does feel better on a road-course though. The former setup is in the 60-70% range and the latter setup is in the 70%+ range like a stock car that everyone complains about understeer in...

[Driven5]

Originally Posted by bbnks2

Here is Rob Binette's: https://robrobinette.com/Suspension_Spreadsheet.htm

I would take their subjective comments with a grain of salt.

I would take everything on that site and throw it out the window. While the spreadsheet formulas themselves may actually be correct, the stock NC Miata outputs are wildly inaccurate. So the question is, was the the inputs (garbage in = garbage out), the formulas themselves, or a combination of the two. Without manually going through and validating it line-by-line, I wouldn't venture to guess. There is also way too much seemingly plausible but (f)actually erroneous information being propagated there, like the multiple of the measurement methods are wrong in addition to the FRC explanation. It would be much easier to stick to getting the same basic set of calculations from a more reputable site.