Rear Spring Location Analysis

[SoundGuyDave]

I'm starting this thread to (hopefully) avoid further diluting Terry Fair's excellent build thread, and to break this fairly narrow (but fascinating) discussion into it's own, separate, thread.

So far, as a recap, we have a few things we are looking at, and that I would like to bring up for discussion:

1) The difference between inboard vs. outboard spring location, and how they impact rear axle wheel rates in both lateral and longitudinal loading.
2) How to calculate the approximate wheel rates for both cases.
3) Attempt to quantify "good" rates and splits through case study and example.
4) How rear roll center location impacts wheel rates, as well as general handling, particularly in terms of how adjustments to RC would affect spring choices.

1) Inboard vs. Outboard spring perch location.
First, we must work with the given values. Based on measurements from multiple sources, the axle-side spring perches are roughly 38.5" apart, the track-width is 62.5" to the tire center (8.5" rim, +45mm offset), the lower shock mounts are 52.5" apart, and the rear shock is inclined at roughly 80* from horizontal.

2) Calculating wheel rates. First, we must start with the motion ratios. The formula is D1/D2, with D1 defined as LRC (Lateral Roll Center) to spring perch center, and D2 as LRC to contact patch centerline. For simplicity's sake, I'm defining LRC as the center of the differential housing. If this is incorrect, I'm all ears! So, for the inboard spring perch, we have 19.25/31.25=0.616 for our motion ratio. For the outboard spring perch, we have 26.25/31.25=0.84, which is certainly NOT an insubstantial difference!
To get approximate wheel rates (not taking swaybars, bushing resistance, etc into account), I'm using the formula WR=(MR2*RATE)*SIN(spring angle). For the inboard case the spring is vertical, thus we can eliminate the SIN term entirely. Based on these assumptions, and using a nominal spring rate value of 300lb/in, we can then calculate the lateral and longitudinal wheel rates for both the inboard and outboard spring locations. I am also taking as a given a front motion ratio of approximately 0.95 for the MacPhereson strut suspension, with a corresponding wheel rate factor of approximately 0.90
Inboard: 114lb/in lateral, 600lb/in longitudinal

Outboard: 208lb/in lateral, 295lb/in longitudinal (inclined springs!)

Thus the inboard has a wheel rate factor of approximately 0.38 lateral, 1.00 longitudinal.
The outboard has a wheel rate factor of approximately 0.69 lateral, and 0.99 longitudinal.


We can start working case studies and extrapolations from here, but this will hopefully keep Terry’s thread clean.

 

[Philostang]

Wow - those are exactly the same figures I got. So we're either both on target or both mutually f'd. =)

One thing, for the Inboard calculations, how did you come up with 600lb/in longitudinal? Shouldn't that have been 300lb/in?

Best,
-j

 

[SoundGuyDave]

Absolutely right, I was doing net force calculations on the side, and transposed some data. In longitudinal motion, since you're compressing two springs simultaneously, it takes 600lbs of load transfer to compress the suspension 1", but the WHEEL RATE remains 300lb/in, since you're still dealing with two wheels... Good catch, and a perfect example of why I need my math checked... I have a liberal arts degree... Would you like fries with that? ;-)

If these numbers prove out, it really explains why I feel like the rear end is just boating over, despite stiffer-than-average rear rates, and a WHOLE lot of bar. Something tells me that I'll be doing a spring relocation in the future. Gotta get some work to afford the new dampers, though!

I also find it VERY interesting that with the inclined springs, you lower the net rear rate (ever so slightly), which gets you a little more longitudinal traction, while still keeping the lateral rates higher to help counter chassis roll. It almost looks to me like this is a no-brainer. That 0.38 lateral factor really sucks!

 

[Whiskey11]

Correct me if I'm wrong but wouldn't moving the springs to the outboard locations over the shocks make the motion ratio greater than one since the spring is mounted behind the center line of the axle? Maximum Motorsports talks about this in the directions for their rear coilover kits for the SN95/Fox cars since they used the springs on the LCA's.

I'm probably speaking about purely in 2 wheel bump and not in roll (although I would think the motion ratio in side view would play into both equally).

It would seem to me that if we could perfectly mount the springs outboard as far as we could on the axle, the math you have is true, but once you relocate it to the shock, I think you start to introduce some errors.

Here is what Maximum Motorsports has to say about their motion ratios in their rear coilover kits for the SN95/Fox cars:

Q. How do I compare a Mustang rear coil-over spring rate to a conventional spring in the stock location on my solid axle equipped Mustang?

A. As with Mustang front spring rates, rear spring rates must be converted into wheel rates. The wheel rate for the rear of a Mustang with a solid axle is approximately 50% of the stock location spring's rate. For example, a 200 lb/in stock location spring has a wheel rate of 100 lb/in. For a rear coil-over suspension, the wheel rate is approximately 110% of the coil-over spring rate (because the spring is actually behind the centerline of the Mustang's solid axle). Also, because the shocks are mounted more outboard than the stock spring location, the rolling rate of the suspension is mildly increased. This helps to reduce understeer.

 

[Norm Peterson]

I get that the 114 and 208 (I get 205 here with a total wheel rate factor of 0.6848 - see below) would be the effective wheel rates in roll due to spring rate (lateral), but I'm not sure that I'm understanding "longitudinal" as well. Running through the numbers, 600 lb/in would be the total rear spring rate either in "ride" or resisting pitch for both 300 lb/in springs the OE spring location and orientation or 300 per side. I get 300 x 0.9848^2 = 291 lb/in for the outboard configuration.

We are assuming that there is no OE spring or shock side view inclinations, or at least that they can be neglected (since the axle migrates a tiny bit longitudinally due to LCA and UCA arcs, the side view spring inclinations generally won't be precisely vertical even if at one point within suspension travel they are). Custom coilover installations might deviate enough to where it might be worth considering.

I think it might be less confusing to refer to the 300 and 291 values simply as being "ride rates". If nothing else, it would be consistent with "ride frequencies" if we get into that later.

I'm sure that in the wheel rate formula we should be squaring the sin(spring angle) term as well. The spring load is increased due to the angle and the spring deflection is reduced in approximately (not exactly) the same ratio. Even at 10° off the vertical, the difference is more important conceptually than numerically.


Norm

 

[Norm Peterson]

Correct me if I'm wrong but wouldn't moving the springs to the outboard locations over the shocks make the motion ratio greater than one since the spring is mounted behind the center line of the axle? Maximum Motorsports talks about this in the directions for their rear coilover kits for the SN95/Fox cars since they used the springs on the LCA's.

YES. This is another component of motion ratio that's easy to overlook, and it has to do with the side view geometry - specifically where the spring is located along the virtual SVSA relative to the axle This individual component of the total motion ratio can exceed 1.00, but you still have to consider the others (the lateral distance ratio and spring angle components) as well. So the overall motion ratio could end up greater than, less than, or equal to 1.00,

It will be sensitive to changes in the LCA and UCA inclinations and would be subject to being "squared" as well.

This did get past me only a few minutes ago as well (shouldn't have).


Norm

 

[Norm Peterson]

I measured again and I have no idea where 34" might have come from.

What I'm getting now is 38" to perhaps slightly less, The inboard edges of the inboard side of the OE LCA brackets are very close to being 41" apart (no more than 41-1/8"). The spring centers are somewhere between 1-1/2" and 1-11/16" inboard of that, per side, putting the spring centers somewhere between about 37-5/8" and 38-1/8" apart.


Norm

 

[SoundGuyDave]

Correct me if I'm wrong but wouldn't moving the springs to the outboard locations over the shocks make the motion ratio greater than one since the spring is mounted behind the center line of the axle? Maximum Motorsports talks about this in the directions for their rear coilover kits for the SN95/Fox cars since they used the springs on the LCA's.

I'll buy that! I don't think it'll be hugely greater than 1.00, because the difference in distance between the shock mount and the LCA mount isn't that large, but I do think that it should be considered. Anybody able to take distance measurements from the LCA chassis mount center to the LCA axle mount center as well as to the shock mount center? With those numbers, we can factor that into the longitudinal as well as lateral calculations.

I'm sure that in the wheel rate formula we should be squaring the sin(spring angle) term as well. The spring load is increased due to the angle and the spring deflection is reduced in approximately (not exactly) the same ratio. Even at 10° off the vertical, the difference is more important conceptually than numerically.

Upon further review, the ruling on the field is overturned... You are, of course, correct. I checked the WR formula on three different websites, and since they all agreed... After you raised the question, I checked several others, and they all agree that you need to square the sine term. Yes, the difference is small, but in the interest of accuracy, we might as well do things the right way. Failing that, we could simply use the 110% listed by MM, which will probably not be far off, even if that is for the SN95 chassis.

I think it might be less confusing to refer to the 300 and 291 values simply as being "ride rates". If nothing else, it would be consistent with "ride frequencies" if we get into that later.

Agreed. My use of the "longitudinal" expression was to clearly identify the difference in pitch and roll axes, rather than having a single "wheel rate" term, which for the purposes of this discussion is useless. I'm basing this on the fact that we put the rear suspension through two differing motions: Longitudinal covers pure acceleration and braking, and Lateral covers pure cornering force. Obviously, there are great periods when both are happening, but this is an attempt to separate the two conditions for examination of rate change ramifications on one condition or the other. The old law of unintended consequences. For example, if you optimize the wheel rates in longitudinal conditions, you may well go with a fairly soft rate to maximize forward bite, but that would be at the expense of lateral roll control, as the rates would now be FAR too soft. Or, if you go the other way, and optimize for lateral loading, the rates would be too stiff in longitudinal loading. In any event, I think it would not muddy the waters to term this rate as a "ride rate."

 

[Norm Peterson]

I'm getting the SVIC at OE ride height being about 84" ahead of the axle center based on LCA and UCA pivot coordinates. If the shock or coilover is located 4" aft of the axle center (strictly guessing here, even before running the MR numbers), the motion ratio for the SVSA effect alone would be 1.048 - which "squared" = 1.098 or right in the same ballpark as the MM number.

So including the spring angle you'd have (0.9848 x 1.0476)^2 = 1.064 for a MR in ride and 319 lb/in wheel rate for 300# springs.


One inch lowered or 1" worth of squat from OE ride height and the SVSA effect on MR becomes something like (56.4 + 4) / 56.4 = 1.0709 (1.0546 with spring angle, squared → 1.112, wheel rate to 334 lb/in).

Effectively, you're getting a mild rising rate.


Norm

 

[Philostang]

Took a measurement this morning and got 4" from the center of the lower shock mount (center of the shock bolt hole) to the axle center (2.5" from bolt hole centerline to the axle tube + 1/2 of the 3" axle tube).

I took the measurement at ride height, but it was a straight horizontal measurement. I didn't think to do it along the line of the lower control arm inclination. So the figure may or may not be exactly what we're looking for. But it should be close.

Best,
-j

 

[JesseW.]

this intrests me

i thought you still have to take into account the spring on the other side of a solid rear axle when calculating wheel rates, it's still contributing to a solid member and that makes them dependent on each other. there is no hard center pivot point to calculate the motion ratios from, and i'm not sure you can use the roll center as the measurement. its the whole, if i hit a bump with one tire, the entire axle moves in that direction bit that the irs guys like to rag on the solid axles about.

correct me if i'm wrong, i need learnin.

 

[Norm Peterson]

A stick axle has one "wheel rate" in ride, another "wheel rate" in "roll", and since a one-wheel bump can be resolved into some amount of ride plus some other amount of roll, you sort of end up with a third "wheel rate".


Norm

 

[JesseW.]

yeah, but even in steady state cornering there are still 2 springs holding the body off the axle, some of the force is transmitted into both of them since they can't move independently.

or am i wrong and the inside spring unloads equally to the ouside loading.

 

[Norm Peterson]

Cornering produces load transfer, which is load coming off the inside spring and being transferred to the outer one. The load transferred has to show up as equal and opposite because you aren't changing the total vertical load back there (IOW, rear weight). So the only place that 200 lbs transferred to the outboard spring can come from is off the inboard spring.


Norm

 

[dontlifttoshift]

Can I dumb this down for a minute to make sure I am following?

By moving the spring from the stock location on the axle to a coilover setup on the shocks we get a slightly rising (wheel) rate? That should correlate to a slightly higher ride frequency as well then correct or am I seeing that backwards?

Then what is the math for roll rate between the two spring locations, leaving the arb out for the time being and assuming stock location on the phb?

 

[Norm Peterson]

Ride frequency would increase slightly as the suspension is compressed. There wouldn't precisely be a fixed frequency even if the springs themselves were single rate. Even though as a practical matter the ride frequency wouldn't normally vary a whole lot.

The math for "roll rate" would have to include consideration of roll center heights at both ends of the car and the front and rear wheel rates as being the most basic approach. If all you're after is the number of degrees roll per lateral g, you'd deduct the unsprung weights, recompute the CG of the sprung mass and work with that instead of the total weight CG location. Typically the sprung mass CG is above and forward of the CG for the total weight.


Norm

 

[Whiskey11]

Question on roll centers, might be a little related.

So with a live axle AND Watts link we have two ways of defining the roll center, either fixed relative to the CG (Chassis mounted, Fays2/Steeda) or fixed relative to the ground (diff mounted, Whiteline, Griggs, Cortex, etc etc).

How does RC height above ground level play into grip? I know that the distance between the CG and the RC is the "moment arm" through which body roll can be guestimated but what about height relative to the ground?

Specifically, in a chassis mounted Watts link, the RC height moving upward under braking and if you add in a little turn would this set up have less grip under trail braking than a differential mounted watts due to the RC raising higher above the ground (but still fixed relative to the CG)?

That is one aspect of RC's that I couldn't figure out. Since not a lot of live axle folks spend a lot of time on the road course relative to those pesky IRS guys, it's hard to Google the information and find something relative to FIXED roll center locations either relative to the CG or the ground level.

 

[Norm Peterson]

It's easier to look at this in terms of "lateral load transfer" and what its sources are. For writing simplicity, the following are taken a 1.0g lateral acceleration.

First, you have lateral load transfer that happens without any roll occurring, and this is equal to the lateral force at (in this case to the rear tires) times the geometric roll center height and divided by the rear track. Lateral force causing roll is due to specifically the sprung mass and its CG, which are somewhat different than for the car as a whole. We'll deal with the unsprung part (wheels, tires, axles, brakes, etc.) separately

Then you have load transfer that is accompanied by roll. This is equal to the roll moment divided by the rear roll stiffness (this being due to the springs and rear sta-bar for a static analysis). This is also based on the sprung mass.

There is a third component of load transfer that you can take as the weight of the rear unsprung mass times the tire rolling radius and divided by the rear track.


To an extent, the first two components of load transfer act in opposition to each other. More of the first via a higher roll center does mean there's less of the second (and a lower rear roll center means more load transfer through suspension roll stiffness). But it's not a total wash because dynamically there is more going on. Load transfer through the geo roll centers happens almost instantaneously, while load transfer via roll stiffness is not fully developed until the car has settled into the steady-state roll angle. Although it is possible to put numerical values on this, it's maybe more of a "feel" thing as the car is transitioning.

Just to note that as the car rolls, you get the axle roll steer contribution to handling feel (though you don't get any lateral load transfer from this).



Norm

 

[Whiskey11]

It's easier to look at this in terms of "lateral load transfer" and what its sources are. For writing simplicity, the following are taken a 1.0g lateral acceleration.

First, you have lateral load transfer that happens without any roll occurring, and this is equal to the lateral force at (in this case to the rear tires) times the geometric roll center height and divided by the rear track. Lateral force causing roll is due to specifically the sprung mass and its CG, which are somewhat different than for the car as a whole. We'll deal with the unsprung part (wheels, tires, axles, brakes, etc.) separately

Then you have load transfer that is accompanied by roll. This is equal to the roll moment divided by the rear roll stiffness (this being due to the springs and rear sta-bar for a static analysis). This is also based on the sprung mass.

There is a third component of load transfer that you can take as the weight of the rear unsprung mass times the tire rolling radius and divided by the rear track.


To an extent, the first two components of load transfer act in opposition to each other. More of the first via a higher roll center does mean there's less of the second (and a lower rear roll center means more load transfer through suspension roll stiffness). But it's not a total wash because dynamically there is more going on. Load transfer through the geo roll centers happens almost instantaneously, while load transfer via roll stiffness is not fully developed until the car has settled into the steady-state roll angle. Although it is possible to put numerical values on this, it's maybe more of a "feel" thing as the car is transitioning.

Just to note that as the car rolls, you get the axle roll steer contribution to handling feel (though you don't get any lateral load transfer from this).



Norm

So then, in conceptual terms (no numbers ) is a chassis mounted watts link going to be less stable (towards oversteer) under braking while turning and more stable under (towards understeer) acceleration while turning due to the change in RC height above the ground even though the RC remains fixed relative to the CG? Or is that over simplifying what is going on a little too much?

I guess what I'm asking is, would a Fays2 be better in situations where more rotation is necessary under braking but more grip for acceleration compared to a differential mounted unit?

My assumption is that with a fixed RC relative to the CG (chassis mounted watts) that your roll contributed to that "arm" between the RC and CG is constant. I get that, but given both, how does the RC height (again, still fixed relative to the CG at all points in roll) above the ground translate into grip under braking and acceleration?

 

[Sky Render]

This article helped me understand roll centers and roll couples:

http://www.motoiq.com/magazine_arti...in-the-geometry-part-one-the-roll-center.aspx