Natural Frequency Balance and Spring Rates

[Dieselboy]

For those of you following my build thread here, you know that it's not an average track build and thus presents some unique challenges with regards to suspension tuning.

I've made a few assumptions on vehicle constants based on research from fe1rx and Orb.

For the E82 I am making these general assumptions:

Motion Ratios:

• Front Spring - 0.96
• Rear Spring - 0.563
• Front Damper - 0.96
• Rear Damper - 0.813
• Front Sway - 0.96
• Rear Sway - 1.0

Weight

• Unsprung Weight, Front - 115 lbs/side
• Unsprung Weight, Rear - 121 lbs/side
• Weight Distribution, Front - 52.3%
• Total Weight - 3300 lbs

Center of Gravity:

• Sprung Weight Center of Gravity Height - 21.5 in
• Unsprung Center of Gravity Height - 13 in
• Roll Center Height, Front - 0.79 in
• Roll Center Height, Rear - 3.93 in

ARB

• Stock Roll Bar Rate, Front - 227.4 lb/in
• Stock Roll Bar Rate, Rear - 13.9 lb/in

Specific to my build:

• Track Width, Front - 62.0 in
• Track Width, Rear - 62.5 in
• Spring Rate, Front - 457 lb/in
• Spring Rate, Rear - 800 lb/in

Components:

• Öhlins "Road and Track RTR" Suspension Kit (3DM "TrackDay")
• Vorshlag racing camber plates w/65mm perches
• AKG Sperical bearing lower rear Shock Mount
• 1M/M3 front control arms

Using these values, I've calculated a few important results (at least what I understand to be important anyway):

• Natural Spring Freq, Front - 2.36Hz
• Natural Spring Freq, Rear - 1.94Hz
• Front Weight Bias - 66.93% (I calculated 70.5% with a stock E82 setup which seems to jive with Fe1rx's value)
• "Magic Number" - 14.63% (Front Weight Bias - Front Weight Distribution)
• Total Body Roll per 1G - 1.62 degrees

Discussion:
From what I've researched, natural frequencies of 2.0 - 2.5Hz are in the range for what I'm building (coupe racecar with moderate downforce). It has been mentioned in my previous thread that the 800 lb/in rear spring rate is under sprung. Now, should I be looking for a "flat ride" frequency, or bias towards the front for faster transient response at corner entry, less ride height variation on the front (the aerodynamics are usually more pitch sensitive on the front of the car) and to allow for better rear wheel traction on corner exit? A 1000 lb/in rear spring will give me a rear natural frequency of 2.17Hz, so how stiff is too stiff for the rear?

Also, to bring the "magic number" closer to 5% (which seems to be the desired result) I can use a 20mm rear sway bar (rate of 102 lb/in) which would bring the "magic number" to 4.2% (Front Weight bias to 56.5%) and reduce the 1G lateral body roll to 1.33 degrees.


So, what I am trying to determine here is:

• Are my assumptions correct?
• Will a 1000 lb/in rear spring rate and a 20mm rear sway be the optimal theoretical baseline?

[135]

Originally Posted by Dieselboy

Discussion:
From what I've researched, natural frequencies of 2.0 - 2.5Hz are in the range for what I'm building (coupe racecar with moderate downforce). It has been mentioned in my previous thread that the 800 lb/in rear spring rate is under sprung. Now, should I be looking for a "flat ride" frequency, or bias towards the front for faster transient response at corner entry, less ride height variation on the front (the aerodynamics are usually more pitch sensitive on the front of the car) and to allow for better rear wheel traction on corner exit? A 1000 lb/in rear spring will give me a rear natural frequency of 2.17Hz, so how stiff is too stiff for the rear?
…
So, what I am trying to determine here is:

• Are my assumptions correct?
• Will a 1000 lb/in rear spring rate and a 20mm rear sway be the optimal theoretical baseline?

This thread (https://www.1addicts.com/forums/show...591&page=2) discusses "flat ride". I think it's fine for a street car but I don't subscribe to that theory for the track. Each person is going to have their preference for how their car handles in different situations.

I prefer my car setup for pitch (especially on the track), currently similar to yours with 400/800 F/R linear springs, E93 M3 front swaybar and standard E82 rear swaybar and solid RSFBs/diff/engine/trans mounts, rear adjustable bars and soon to be installed SPL Parts front kit. Last time I did my calcs, it was front 1.98 Hz ± 0.03 Hz and rear 1.80 ± 0.04 Hz.

My preference is for the rear to be 10% less (or within a range or 0.2-0.3 Hz less) than the front.

I was looking to change to 571/1142 F/R springs (originally to support slicks and aero), which would result in a natural frequency of front 2.29 Hz ± 0.05 Hz and rear 2.07 Hz ± 0.05 Hz. Alternatively a 1028 lbs rear spring would result in 1.99 Hz ± 0.05 Hz.

I have an LSD and don't plan on changing my rear swaybar from the stock item.

[Phloozy]

I went through this a ton. I ran 450f/1050r for a while with moderate success for the same reasons you are finding. Once I added aero I bumped the rear up slightly as well as the front. I am at 650f/1100r now and the balance is quite good. It sounds like you are on the right track and I would recommend trusting the math and trying it out.

[tsk94]

I've also gone through this a ton myself. Crunching numbers and experimenting on the car. What I've realized is that what on paper might be optimal (flat ride - according to some), doesn't always translate into real life.

Front McPherson suspension has terrible roll geometry. And it only gets worse as you lower them. The rear doesn’t suffer the same issue (or at least not nearly as much as the fronts). So, we need to use more spring rate in the front to properly control the rate of roll relative to the rear. Sway bars help, but only can do so much. This is where I start seeing issues with flat-ride philosophy. To run flat-ride with adequate front spring rate, the rear spring rates would need to be so high – in the realm of 1500-2000+ (divorced) to pair to a ~700-900 front spring. That’s simply not realistic, the rear would have 0 traction. While we want higher front spring rates to control roll and pitch under braking, we also want some compliance in the rear to allow the car to squat under acceleration for traction and easier drivability. Again, I don’t see how this is possible with a flat-ride philosophy that has proper front roll control.

For flat-ride to work properly, you run a lower than conventional (compared to a pitch setup) front spring rate and pair it with a massive front bar (to maintain adequate roll control). Subsequentially, you can’t run the car as low with this setup, which in my eyes is a serious negative. Pretty simple, you can’t run the car as low on a lower spring rate as you can on a stiffer spring rate. I believe you want to run the car as low as you can while maintaining enough bump travel and without impeding wheel/tire clearance. And back to the rear, I still think the rear spring rate would be too high and the car would be nervous at the limit – at least in a road course environment (autox may be a slightly different story).

Do I think a flat-ride setup cannot be made to work well? No. I’m sure there’s all sorts of trickery you can do with bumpstops to increase effective front spring rate and so on. But I really think a pitch biased setup is more optimal for a road course setup. And I believe there are enough examples of well setup and FAST cars that utilize a pitch setup (fast regional club racers, GT3 Cup cars, M3/M4 GT4’s, etc.) to prove this.

[Dieselboy]

Really great information here. I agree in that a pitch biased towards the front is likely optimal on a big track, which is my preference.

Switching to a 1028 lb/in rear spring and a 20mm rear sway gives me:

• Natural Spring Freq, Front - 2.36Hz
• Natural Spring Freq, Rear - 2.20Hz
• Front Weight Bias - 56.14%
• "Magic Number" - 3.84%
• Total Body Roll - 1.32 degrees/G

On paper, this seems like a good baseline.

Seems that the Swift 65mmx228mm spring rates only go to 160n/mm (913 lb/in). What springs are you guys using to get higher rates in the rear?

[tsk94]

Originally Posted by Dieselboy

Really great information here. I agree in that a pitch biased towards the front is likely optimal on a big track, which is my preference.

Switching to a 1028 lb/in rear spring and a 20mm rear sway gives me:

• Natural Spring Freq, Front - 2.36Hz
• Natural Spring Freq, Rear - 2.20Hz
• Front Weight Bias - 56.14%
• "Magic Number" - 3.84%
• Total Body Roll - 1.32 degrees/G

On paper, this seems like a good baseline.

Seems that the Swift 65mmx228mm spring rates only go to 160n/mm (913 lb/in). What springs are you guys using to get higher rates in the rear?

Eibach would be the easiest to source.

That sounds like a good starting point. I bet you'll end up going stiffer in the front in the long run, but that's somewhat driver and circuit dependent as well.

[Driven5]

I'm running 18k (1000 lb) springs on the rear of my street driven E88. The problem with doing that in these cars is not the 'high' rate of the springs themselves, but rather the shitty rear suspension/subframe design that leverages large motion ratios against soft materials. With the RSFB and shock mounts addressed, although I don't see upper mounts mentioned in your build list, I would not fear them for a track car. Is there a reason you are wanting 228mm springs? I would think the 203mm should provide sufficient usable travel.

An easy thing to do would be to run the flat ride numbers. Start with a 1400 pound rear spring. That would get you a reasonable flat ride ratio. Now, keeping some cushion available for dialing things in, how much front sway bar would you need to get your 'magic number' to 10%?

So if you consider the 18k rear springs and 20mm sway as your most aggressive setup, what 'magic number' do you get for the most conservative setup with the 18k springs and stock rear bar? And where in between does the 18k springs, 20mm rear bar, and any of the front bar upgrades (E93 M3, H&R, Eibach, etc) get you?

If you really get the aero package working, I wouldn't be surprised to see you going stiffer all around in the future. Coming from a non-BMW background, I can't help but be surprised every time I see dedicated racers seeming to fear exceeding 2Hz in the rear.

[Mach3M3]

I use 550/1000 lb/in springs front/rear
dynan 28 mm tubular bar front set to stiff
now discontinued superpro 20 mm rear bar also set to stiff

Alignment will also play a role in how the car feels.
Going from around 3.5* to around 4* of front camber (2* rear camber, 1/8 toe in) resulted in me having to stiffen the front sway bar. and in the video below, I think the car was still too loose on track (see video below) and I actually need to max out the rear camber (to 2.5*) to neutralize it more. Plus, I still need seat time to get the shocks adjusted to the right settings.

Hyperco is another choice for spring manufacturer and thats what i use.

video for reference:

[tsk94]

Originally Posted by Driven5

I'm running 18k (1000 lb) springs on the rear of my street driven E88. The problem with doing that in these cars is not the 'high' rate of the springs themselves, but rather the shitty rear suspension/subframe design that leverages large motion ratios against soft materials. With the RSFB and shock mounts addressed, although I don't see upper mounts mentioned in your build list, I would not fear them for a track car. Is there a reason you are wanting 228mm springs? I would think the 203mm should provide sufficient usable travel.

An easy thing to do would be to run the flat ride numbers. Start with a 1400 pound rear spring. That would get you a reasonable flat ride ratio. Now, keeping some cushion available for dialing things in, how much front sway bar would you need to get your 'magic number' to 10%?

So if you consider the 18k rear springs and 20mm sway as your most aggressive setup, what 'magic number' do you get for the most conservative setup with the 18k springs and stock rear bar? And where in between does the 18k springs, 20mm rear bar, and any of the front bar upgrades (E93 M3, H&R, Eibach, etc) get you?

If you really get the aero package working, I wouldn't be surprised to see you going stiffer all around in the future. Coming from a non-BMW background, I can't help but be surprised every time I see dedicated racers seeming to fear exceeding 2Hz in the rear.

One issue going much stiffer than ~1200 in the rear is the rear control arms. At least on the M3's, much more than 1300 can have them fail. At that point, you need to go to rear coilover. And to do that, you should reinforce the rear strut towers.

I don't know if the non-M rear lower control arm (spring arm) doesn't suffer the same issue or not.

[Ginger_Extract]

Flat ride is just a general guide, at some point you have to take some educated guesses and test. I think I mentioned it in your thread, or someone else's, but if I was doing another E82 for track, I would start at 500/900 (assuming stock bars) and adjust from there. I ran a variety of springs on my car, chasing balance around a 700# rear spring, and had such awful tire wear all around, even with -3.5 camber all around, because the car was so undersprung. It's a big, heavy and powerful car, physics will be unkind.

Anecdotally: here's hoping you have a clutch type diff. Helical doesn't work for shit with mega stiff springs and an even bigger rear bar.

[135]

Originally Posted by Dieselboy

Really great information here. I agree in that a pitch biased towards the front is likely optimal on a big track, which is my preference.

Switching to a 1028 lb/in rear spring and a 20mm rear sway gives me:

• Natural Spring Freq, Front - 2.36Hz
• Natural Spring Freq, Rear - 2.20Hz
• Front Weight Bias - 56.14%
• "Magic Number" - 3.84%
• Total Body Roll - 1.32 degrees/G

On paper, this seems like a good baseline.

Seems that the Swift 65mmx228mm spring rates only go to 160n/mm (913 lb/in). What springs are you guys using to get higher rates in the rear?

Originally Posted by tsk94

Eibach would be the easiest to source.

That sounds like a good starting point. I bet you'll end up going stiffer in the front in the long run, but that's somewhat driver and circuit dependent as well.

Swift Springs (which I currently use), Hyperco, Eibach and King Springs don't make 9" (rear) springs in the spring rates we require, and have otherwise limited offerings, e.g.

Swift Sprints 65mm ID:
203mm (8") 1008 (1028), 1120 (1142), 1232 (1256), 1344 (1370) lbs/in
Note: advertised spring rates with actual spring rate in brackets

Hyperco 2.5" (63.5mm) ID:
203mm (8") 1000, 1100, 1200 lbs/in
Note: Doesn't make a 9" length spring
Note: May make custom spring lengths/rates

Eibach 65mm ID:
200mm (7.87") 1142 lbs/in
230mm (9.05") 1313 lbs/in

King Springs 65mm ID:
200mm (7.87") 1000, 1100, 1200, 1300, 1400, 1500 lbs/in
250mm (9.84") 1000 lbs/in
Note: Can make custom spring lengths/rates

I'm not sure if Swift or Eibach will make custom spring lengths/rates.

There aren't any 60mm ID options, except custom-made King Springs.

[Mach3M3]

I run 8" long springs in the back of my car and it seems fine, why do you think you need 9" length?

For reference, my fronts are 6" and I run helpers with them.

[Dieselboy]

Are there benefits/drawbacks to running an 8" rear spring over the 9"? I was looking at the Swift 65mmx203mm which would get me the 1028 lb/in rear. Not sure what problems the 8" spring may cause though.

[Driven5]

As long as you don't need more travel than the spring provides, there's no real disadvantage. The advantage is greater availability and reduced weight. The Swift 18k spring lists a 2.9 usable travel and a 4.6 max stroke. At the wheel, that's a little over 5" usable and a little over 8" to coil bind. You'll never coil bind it, and even with some aero I expect that would be more than enough usable travel... Assuming your bump stop hasn't intervened before that anyway.

[fe1rx]

Originally Posted by Dieselboy

For the E82 I am making these general assumptions:

Motion Ratios:[LIST][*]Front Spring - 0.96[*]Rear Spring - 0.563[*]Front Damper - 0.96[*]Rear Damper - 0.813

Specific to my build:

• Track Width, Front - 62.0 in
• Track Width, Rear - 62.5 in
• Spring Rate, Front - 457 lb/in
• Spring Rate, Rear - 800 lb/in

Something to consider is that motion ratios are affected by track width because if you widen the track width you give the tire more lever arm relative to the spring and damper movement, Thus if you widen the track, you effectively soften the suspension (springs not changed) and your car will ride lower. You can check this by checking your ride heights with stock wheels, and then with your wider wheels. Your wheel rates and ride frequencies are likely lower than you are calculating.

Also, on the subject of "flat ride", this is only useful in lightly damped (i.e. street) vehicles. To quote from Milliken and Milliken's "Race Car Vehicle Dynamics" (RCVD) (page 796, if you have a copy):

"To obtain the flat ride in a lightly damped vehicle it is important to have the static deflection at the front greater than the static deflection at the rear. With heavily damped vehicles, where the main ride resonance is well suppressed, this requirement is less important (race cars, for example)."

Static deflection is related to ride frequency (CPM = 188 / SQRT(static deflection), so Flat Ride requires a lower front ride frequency than rear. This is in direct conflict with the general race car guidance that says race cars should have higher front ride freqencies than rear (e.g. RCVD Table 16.5) so flat ride really is a useless diversion when it comes to dedicated track cars.

[Driven5]

Originally Posted by fe1rx

Something to consider is that motion ratios are affected by track width because if you widen the track width you give the tire more lever arm relative to the spring and damper movement.

That being said, I wouldn't start using the wheel center line as your force application point for motion ratios and wheel rates either. The further your suspension is from swing axle geometry, the less true this becomes. The majority of (non-swing axle) suspensions are intentionally not particularly close to swing axle geometry. So while it may be a little softer than calculated, the true 'force application point' will still generally be closer to the pivot than the wheel center line for the typically used simplified calculations.

[fe1rx]

Originally Posted by Driven5

The further your suspension is from a swing axle geometry, the less true this is. The vast majority of suspensions are not particularly close to swing axle geometry. Thus the true 'force application point' will generally be closer to the pivot than the wheel centerline.

The instant centre concept implies that all suspensions act like swing arms, with a virtual swing arm (of non-constant length with suspension travel) pivoting about a virtual pivot point (the instant centre, which has a non-constant location with suspension travel).

What do you mean by "true force application point"? The ground acts at the centre of the contact patch, which is more-or-less on the wheel centre line, but is affected by camber.

[Driven5]

If you do a free body diagram on a swing axle, with the instant center at the inner pivot, you'll see that 100% of any offset change affects the motion ratio.

If you do a free body diagram on a pure trailing arm or equal length parallel double wishbone, with the instant center at infinity, you'll see that 0% of any offset change affects the motion ratio.

If your instant center is somewhere in between, so is the effect from the offset. So how does one factor that in correctly? I'm glad you asked.

In addition to the spring angle correction factor, most common motion ratio equations and calculators use the length from the inner pivot to the spring center line divided by either the length from the inner pivot to the outer pivot or the length from the inner pivot to the wheel center line. The former is incomplete by virtue of not accounting for the instant center, but is a reasonable approximation if the actual instant center location is not actively known. The latter is simply wrong.

The complete answer is the length from the inner pivot to the spring center line divided by the length from the inner pivot to the outer pivot, then multiplied by the length from the instant center to the outer pivot divided by the length from the instant center to the wheel center line. Because most suspensions use an instant center at least 3x-4x longer than the control arm, that means the the wheel center line (offset) effect is relatively minimal... Especially if you're not also getting to the level of factoring the center of pressure location change (moves inboard from wheel center line) on the tire resulting from both static and dynamic camber changes. The rest of the effects from the offset are simply a moment created that loads the control arms axially with no effect on motion ratio or wheel rate.

I suppose my reference to a 'force application point' was a bit vague. What I was getting at is a virtual outboard point that would provide the true motion ratio relative the the spring center line. For most applications it will be much closer to the outer pivot than the wheel center line.

[fe1rx]

Originally Posted by Driven5

If you do a free body diagram...

The instant centre has no relevance to a free body diagram because no loads are reacted there. It is just a useful construct to visualize the (instant) camber gain curve and (instant) roll centre location.

For a Macpherson strut, the spring is inclined in both the side view (due to caster) and the front view (camber plus strut inclination angle relative to the wheel plane). Both inclinations need to be accounted for. The wheel centre moves forward in bump due to anti-dive. A simple 2-D front view calculation of motion ratio will not be accurate because it ignores of the above.

For our Macpherson struts, the outer ball joint is virtual and its locaation cannot be solved graphically, and the location changes continuously with suspension movement. Accordingly no graphical solution (2-D or 3-D) is practical in our case. (This is doubly true of the rear suspension.)

Suspension analysis software is the only practical way of calculating the motion ratio and instant centre for our suspensions. The input values are difficult to measure, and critical to getting good results. Measuring the motion ratio is probably easier, but is far from simple.

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

Back to free body diagrams - how do you draw one with a point at infinity?

[Driven5]

While the instant center is not used as part of the free body diagram, the geometry required to achieve a given instant center does affect the free body diagram results... And in doing so reinforces the point that seems to have been lost here:

That motion ratios are not particularly sensitive to track width variations on most suspensions, and the simplified calculations can provide reasonable approximations within a couple percent as long as they're performed correctly.

It's awesome that there are those with the tools and inclination to analyze things within a gnats ass, but there are also easier ways to get into the same ballpark... Where all of that academic theory ends up taking a backseat to reality anyway.

[rac]

Originally Posted by Dieselboy

Are there benefits/drawbacks to running an 8" rear spring over the 9"? I was looking at the Swift 65mmx203mm which would get me the 1028 lb/in rear. Not sure what problems the 8" spring may cause though.

I currently run Swift 65mm x 203mm - 22kg (1232lb) in the rear.
Depending on how much height adjustment you have and your desired ride height you might need a spacer with the suspension completely unloaded (wheel in the air).

[135]

Originally Posted by Mach3M3

I run 8" long springs in the back of my car and it seems fine, why do you think you need 9" length?

Originally Posted by Dieselboy

Are there benefits/drawbacks to running an 8" rear spring over the 9"? I was looking at the Swift 65mmx203mm which would get me the 1028 lb/in rear. Not sure what problems the 8" spring may cause though.

Originally Posted by rac

I currently run Swift 65mm x 203mm - 22kg (1232lb) in the rear.
Depending on how much height adjustment you have and your desired ride height you might need a spacer with the suspension completely unloaded (wheel in the air).

The 9" spring will become loose at full droop so an 8" spring will become loose even earlier though the range of droop.
As @rac said, it depends on the ride height but I would expect an 8" spring to still be loose during droop, irrespective of the ride height.