Suspension by the Numbers - Car Setup How To

Discussion in 'General – Setup & Tuning, PSA Knowledgebase' started by phate, Mar 16, 2016.

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  1. phate

    phate Motorhead

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    We're talking about turning. Autocross, road racing, canyons and mountains. Drag racers need not read any further.

    What if I told you general car setup involved nothing more than a few things added, subtracted, multiplied, or divided? General car setup is that simple, no black magic, no ridiculous formulas looking at anything but the basics of springs and levers. This post is probably going to take a few installments, because it will be lengthy. At the end, though, if you apply these principles you can use them to assess your current setup and maybe even figure out why your car is behaving the way it is, and most importantly - how to fix that behavior and go faster.

    The first step in this is to look at steady state cornering behavior. That means after the car has taken set and you aren't trying to flick the back end around or anything like that, and then seeing if it's understeering or oversteering. That also means we aren't including shock forces, since they only contribute force during transient motion (they're velocity sensitive). I highly suggest you get the idea of 'neutral' handling out of your head. Just assume it doesn't exist, and if you think your car is 'neutral', then assume that you weren't driving it hard enough.


    I'm going to make a couple claims without backing them up :) The main thing we want to know is how stiff the car is in the front vs the rear, and how that relates to the turning behavior of the car. We end up comparing the % stiffness contributed from the front end to the % front weight of the car. Let's call front % stiffness the roll bias, and let's call the front % weight the weight bias. It comes down to this general idea:



    • If roll bias = weight bias, neutral handling (doesn't exist, so go to the next line)
    • If roll bias > weight bias, understeer (I'm looking at you, MS3's)
    • If roll bias < weight bias, oversteer

    Quick example of an MS3:

    Weight bias: 60%
    Roll bias: 80% <--yes, that's close to actual

    Result: UNDERSTEER. We've all experienced it, we all know it to be true.


    So how do we fix that? Well, we can't really shift the weight bias, so it comes down to roll bias. Everyone wants a bigger rear bar, and this is the real reason. More rear stiffness decreases the roll bias, and when roll bias gets close to weight bias we start to get a less understeering monster, and we can even get into the realm of steady state oversteer if things are stiff enough.

    Finding weight bias is easy. Get on some scales and math it out or maybe the scales even have a front weight readout. Easy peasy.


    Finding Roll Bias is what we're really trying to do, here. Yes, you can slap on different bars and springs and bump stops until you find something that works, but why waste that money when you can get it very, very close just using some math? At the end of this, we'll be able to throw this into a spreadsheet and you can see exactly what is going on and change whatever parameters you want. Door cars lend themselves to the analysis I'm about to do because the front and rear track widths are equal or very close to equal. When that's the case, you don't need to go into actual weight transfer stuff or into roll angles or anything else, that's just extra if you want it...

    So where does roll stiffness (and bias) come from? Well, it comes from the things that resist roll:

    • Main Springs
    • Anti-Roll Bars (ARB, aka Sway Bar)
    • Bump Stops
    • Shocks (which we're ignoring!)
    • Tires - tires are not rigid cylinders, they compress quite a lot. Check this out, they're all sorts of floppy


    Now how do we add all that stuff together to give us the right numbers? They attach at different points on the control arms and some even at angles (think the shocks and struts). To put everything onto a level playing field, we're going to normalize everything to act as if it were working directly in line and completely vertical with the center of the tire.


    1-3_zpspkbancma.PNG


    MOTION RATIO
    Pretty simple idea, right? So let's see how we actually do that. First, let's talk about motion ratio. The motion ratio is just that, a ratio of movement between two points on a single object. That object, for us, is the lower control arm. The points we are comparing are always something versus the vertical wheel movement - that means springs, sway bars, shocks, and bump stops. For example, we would like to express spring movement - how much it compresses - versus the wheel movement.

    Expressing this motion ratio mathematically is just a little bit of trig:
    1-2_zps5nc5vgyw.PNG

    There's one nuance with springs and motion ratios. Springs essentially lose leverage twice by a factor of the motion ratio. This is because force is lost by a factor of the motion ratio, and the distance displaced by the wheel is inversely proportional to that of the spring by a factor of the motion ratio. So the reduced force is divided by the increased distance...If you would like to see an example and proof of this, click here.

    Now that we know what motion ratios (henceforth referred to just as MR) are and where they come from, let's look at what has an MR and how we're going to keep everything straight:

    1-4_zpsqu8breqc.PNG
    ^In this thread text, I'll use something like MR_F,K if I need to type something out.




    A slight detour
    Essentially, we just want to add up everything for the front, and add up everything for the rear, and compare.
    4-1_zpsdff2amt1.PNG
    For the main spring, ARB*, and bump stop, we can simply add them up for a combined spring rate. These act like parallel springs, meaning they just work and add together directly. One thing we haven't considered yet is tire stiffness! If you watched the video above, you have seen just how floppy tires are. They act just like springs! This spring, though, acts in series between the wheel and other springs we've already calculated. (*see Spring Specifics section for details)

    You say it doesn't matter? I disagree, and I'll use my car as an example. I hadn't been using tire spring rates in my calculations and they were telling me that I was still pretty far away from the neutral steer point on the understeer side. But, slight changes to rear bias were getting appreciable oversteer results so it seemed we were closer to the neutral point than the numbers indicated. I actually tested my tire spring rate (more on that testing later) and recalculated the front bias. The numbers were revealing:

    006_zpsyebksahw(1).JPG
    So just by including one more piece of information we found that calculated results had become much closer to what we were experiencing. This is with a fairly stiff tire, too (Hankook RS3V2). The softer a tire is, the more pronounced the effect will be. The softer a tire is, the more it will reduce the overall rate - the overall spring rate of springs in series will be lower than the lowest spring's rate.

    We'll talk about tire rate testing in a later post. For now, just remember that it can influence the stiffness quite a bit. The formula for the overall spring rate for two springs in series is simple:

    Overall rate = (k1 x k2) / (k1 + k2)





    Spring Specifics

    Alright, there's a couple caveats with each type of spring that we need to talk about.

    Sway bars - henceforth referred to as anti-roll bars or ARBs. How sway bars are rated can be extremely misleading. Even the formula for sway bar stiffness can be misleading since it gives you a rate in lb/in, which seems perfect for what we need.

    ARB's are rated by a bench rate
    . A bench rate is the rate of the bar with one end held in place and the other end deflected. The problem with that is how it actually works on a car. When we enter a turn and the chassis rolls a little bit, one side compresses, but the other side extends. This means the ARB is actually twisting twice the amount**. (**It's not really twice the amount since the droop side may not extend as much as the other side went into bump, but we'll assume twice the amount for now.

    The theoretical bench rate of a sway bar is easily calculated for the MS3 bars and the front MS6 bar. They are simple, geometrically, and this formula is fairly accurate:
    Puhn%20Sway%20Bar%20Formula_zpsfqthgjjy.PNG
    ^Note those variables are in inches, and the formula yields results in units of lb/in.

    That formula comes from combining three springs together, as detailed in this link. There's a factor of 10 error in that paper that I'll let you find :)

    This formula assumes rigid bushings, meaning no squish or deflection in them. That is NOT how our sway bars are set up, they have soft rubber or polyurethane bushings. I have actually strapped ARB's into test rigs to verify the reduction in stiffness. It can be immense. Like the above note says, for thick rubber bushings (stock), the rate is reduced upwards of 50%, and even with 'stiff' polyurethane bushings the rate was reduced nearly 40%. Like the tire spring rates, it's another thing to consider that will make our calculations more accurate.

    Bump Stops - bump stops can be tricky to incorporate into this equation. They don't touch at ride height, so they begin to work somewhere in roll. Good bump stops will have a linear-ish spring rate range, but if you compress them enough they will go solid. Solid means infinite spring rate, and that means your bias goes way up or way down very quickly. Bump stops are a great tuning tool, but not so much a great main spring (like how the MS3 and MS6 utilize them in stock form).

    Main Springs - No real caveats here. If you use squishy rubber spring seats they act like springs in series, again...meaning it lowers their rate a touch. They go solid pretty quickly so I probably wouldn't worry about this too much.

    I have rate tested some springs and have found they aren't completely linear, even if advertised as linear. They have a linear range, but no two springs are perfect. You can test them yourself (more on this later), or you can pay some shops to rate test them for you. It's another thing to do to make your calculations a little more accurate.



    Let's put it all together

    Since door cars are close to equal left to right, we can simply take a front corner's stiffness and a rear corner's stiffness for this analysis. We calculate the total corner stiffness from everything contributing:

    4-2_zpsmkbnt0rd.PNG
    ^The extra 'e' designation in that equation for each of the spring rates just means "effective", as in the effective rate acting at the tire.

    Then you simply compare the front stiffness to the total front and rear stiffness to get the % front bias:

    7-2_zpsp5razcyl.PNG

    And then we're back to comparing % front bias to % front weight...

    • If roll bias = weight bias, neutral handling (doesn't exist, so go to the next line)
    • If roll bias > weight bias, understeer (I'm looking at you, MS3's)
    • If roll bias < weight bias, oversteer

    Looking forward a bit, you can see how easy this sort of analysis would be to implement in Excel. If you wanted to get fancy, you could add drop downs for selecting different bars, springs, tires, or anything mentioned so far. I have a calculator built for this (and more), already, and I'll share some of it in here.

    I'll also show some examples of how I have tested various components. I've had sway bars, springs, and tires in test rigs trying to get accurate numbers for them and have learned some things along the way that I'll share.
     
    Last edited: Sep 19, 2018
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  2. ibcrusn

    ibcrusn Silver Member

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    Tons of good info....my head hurts, haven't done trig in a long time. Will be back for more soon.
     
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  3. SIXual Panda

    SIXual Panda Greenie Member

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    good write up @phate !
    definitely keeping this thread in the back of the vault.
     
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  4. Nliiitend1

    Nliiitend1 aka "Nintendo" Greenie Member

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    :cool::thumbsup:
     
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  5. Darthxar

    Darthxar Gold Member

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    Might have to fire up Excel, I'm planning on doing some suspension work soon :)

    Great post.
     
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  6. Naoandlater

    Naoandlater Greenie Member

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    Sweet. I'll take a look into making an excel when I have some time.
     
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  7. Redline

    Redline I done fucked up for the last time. BANNED Greenie Member

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    Maybe this is faaaar too simplified a question, but what spring rates would you recommend for a performance-oriented DD? Feels like the 252 lbs/in are too little. I've heard lots of great things about the Mazdaspeed Coilovers. I realize there's a lot more to setup than just springs/dampers, but I thought I'd ask. Thanks for the significant knowledge dropping :D
     
    Last edited: Mar 18, 2016
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  8. phate

    phate Motorhead

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    There's a second method of analysis for getting spring rates "close", which is the frequency method. I'll go into that in my next big post, I think.
     
    phate, via a mobile device, Mar 18, 2016
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  9. Rokusek

    Rokusek Platinum Member

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    Great work as always Clint.
     
    Rokusek, via a mobile device, Mar 22, 2016
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  10. phate

    phate Motorhead

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    FREQUENCY CALCULATIONS

    Ok, let's get into the frequency method for spring selection. The idea of using frequencies is great for comparing setups between cars and simplifies figuring out what will work for your car. Me saying I use an 850 lb/in spring in the front of my MS6 means nothing to someone running an MS3 because the motion ratios are different and the weight is different. Remember back to high school or college physics and you'll recall that spring frequency is dependent upon its rate and the weight it is suspending by:

    ωn = sqrt(k / m), where ωn is the natural frequency of the system (in rad/s), k is the spring rate, and m is the mass. We're going to use this exact same equation and just convert it to Hertz because people are much more familiar with Hz than rad/s. 1Hz = 1 full oscillation per second, 2Hz = 2 oscillations per second, etc.

    We like to simplify things and work in weights and english units, yeah? All you need to do to convert that equation to Hz is multiply it by 187.9/60, or just use 3.13. Now we have:


    Frequency (Hz) = 3.13 x sqrt(ke / Wc,x)

    Now when I say that my MS6's front end frequency is 2.2 Hz, you have something that is directly comparable to your or any other car out there :)

    Wc,x is the corner weight in pounds, and ke is the corner's effective stiffness in pitch or two wheel bump (meaning no bar rate added since they only contribute in roll). That leaves just the main spring, bump stops, and the tire spring rate. For general ride frequency, we'll hope that you aren't constantly on the bump stops so we'll remove them from the equation. Many, many, many cars have been set up without using tire spring rate...so I'll leave that up to you. I'll link to an example where I figure out spring rate with both methods; the difference can be pretty surprising. [I didn't have tire spring rates when I did my suspension, so I used only the main spring.]

    Example without tire spring rate
    Example with tire spring rate




    TARGET FREQUENCIES

    That leaves us with what we should target. If you head over to Far North Racing's site (which is all an incredible read), you'll find these ranges:

    Street Car - 0.8 Hz
    Occasional Autocross/Track: 1-1.5 Hz
    Full Bore Autocrosser: 2.2-2.5 Hz

    I tend to agree with those ranges, though nice shocks will allow higher frequencies without penalizing ride quality.

    Another method goes hand in hand with the above, and that is flat ride. Flat ride is just a method that allows the back end to settle at the same time as the front end as you're cruising down the road and hit a bump. The Suspension Truth channel on YouTube has a number of great videos, and a couple specifically about flat ride. I highly recommend checking them out. One interesting tidbit that he drops in one of the videos is that cars that achieve flat ride need less pitch damping...I haven't modeled that, but the results of their suspensions speak volumes, they know what they're doing.

    Anywho, imagine driving at 60mph and you hit a bump with the front axle, then the rear. If the front and rear have the same frequency, they'll settle at different times equal to the difference in time between when the front and rear axle hit the bump. So you can speed up the rear axle's frequency so that they settle at the same time within some specific speed range.

    For autocross, I've found a good frequency split for the MS6 is ~12% (rear freq is 12% higher than the front). It achieves flat ride around 60mph and works great for autocross and cruising down the road. 60mph, imagine that, it's around the same speed as many autocrosses...For the MS3, since it has a slightly shorter wheelbase, might need just a slightly smaller difference between the front and rear at the same speed.



    The big caveat: Ride quality is more a function of the shocks than the spring rates and frequencies. If you have shocks that have ridiculous amounts of high speed damping, the car is going to ride like crap (I'm looking at you...every...budget (and even some expensive) coil over kit out there). As a reference, my MS6 is set up at 2.2 Hz front and ~2.47 Hz rear and the ride quality is decent. It's not uncomfortable except for some weird mid-speed bumps. Sharp bumps get sucked up like you wouldn't believe, and low speed transitions are great. It's all about the shocks.
     
    Last edited: Mar 23, 2016
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  11. AYOUSTIN

    AYOUSTIN Greenie Member

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    Awesome read Clint! I'm definitely staying tuned. Hoping to develop my understanding of suspension tuning this summer.

    Two questions, as far as tire spring rate goes, is it a linear relationship to air pressure? Have you seen/done any testing to see if runflats have a significantly higher spring rate?
     
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  12. phate

    phate Motorhead

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    Well, let's just talk about tire testing, yeah? I like to know how things work and what all is going on with my suspension bits so I can take a methodical, mathematical approach to setup. Laboratory tire testing yields some incredible data that would be invaluable for setup...but no manufacturer releases that data. Calspan has a tire research facility where they independently test tires...but most of us don't have access to that data, and they don't really test the tires we're interested in, anyway.

    So, with that said, tire vertical spring rate is just a tiny piece of the puzzle and isn't a 100% accurate way to describe tire behavior. Tires are subjected to pretty significant lateral forces during cornering, which likely throws off these measurements a little bit (or maybe a lot, who knows). But, like most everything else here, it moves us a step in the right direction and gives us a little bit more accurate model to rely on.

    So, how I've tested tires. Springs just store energy depending on how much we displace them from their free length, and we rate that in force/displacement. We just need to be able to accurately measure force and displacement, and we can measure spring rate of just about anything. I literally just took a tire and stuck it in my hydraulic press:

    [​IMG]

    I used one of my corner scales to measure the force, and that awesome vice grip clamp on snake thing with a dial indicator is how I measure displacement. I start at a "0" point with just a tiny force, then move the press .050" or .100" at a time and take a reading from the scale. My scales max out at 1,250 lb, so I just went until the scale was close to maxed out. I tested spring rate at different pressures, starting at 50 or 55 psi (depending on tire) and went in 5psi increments all the way to 25 psi. So far, I've done this for 255/40/17 Hankook RS3V2's and 275/40/17 Hoosier A6's. As I get more tires, I'll test and record their rates to compare.

    One thing to remember when testing a tire like this is that you are deflecting both the top and bottom sidewalls at the same time. When the tire is on the car, we hope that it's only touching the road. That means that we're effectively testing a set of springs in series. Going back to the series spring equation, kt=(k1*k2)/(k1+k2), where k1=k2 (because the tire should be the same spring rate everywhere). We are directly measuring kt, so with a little bit of math, we see that kt=0.5k1, so that means that in order to have the single spring rate we need to double our force readings. Said another way, you are deflecting each of the top or bottom of the tire only half of the measured amount.

    When you plot a tire's data (with the doubled rate) of force vs displacement, you come up with a graph that looks like this for each pressure test:

    [​IMG]

    I've added a linear trendline to this graph because the slope of that line is the spring rate of the tire in lb/in. And if we graph all of the tests together, we get something like this:

    [​IMG]



    And this is what the Hoosier A6's look like graphed together:
    [​IMG]

    Getting back to @AYOUSTIN's question about pressure vs spring rate linearity...looking at data like this leaves us interpreting the tires pressure vs spring rate by eye. If we simply take the slope of each line (its spring rate) and the corresponding pressure and graph them so that we have a pressure vs spring rate graph, we can see the relationship plain as day:

    [​IMG]

    With that r-squared value included, we see that the Hoosiers are described extremely well by a linear line, and the RS3V2's are described decently by a linear line. The difference probably lies in my error measuring the RS3's, as it was the first time I had done this sort of thing and it was a lot of trial and error before getting a consistent setup.

    So, yes, tire spring rate vs pressure is pretty linear. That trendline equation is invaluable, the slope of the trendline is telling us how quickly the tire spring rate increases per pound of pressure (x). The units would be (lb/in)/psi. Seems like we might be able to put a simple equation into our developing spreadsheet to simplify analysis ;)

    For the run flat v non-run flat spring rate, I don't know. I've only tested these two tire types. If I had to guess, I would say that the run flat version would be stiffer, but in the grand scheme of things I doubt they're as stiff as a competition-oriented tire.
     
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  13. Maisonvi

    Maisonvi Silver Member

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    Well shit, I have a lot of reading to do before I hit the track this summer.

    Thanks for all the work clint. Wanna come set up my car for me?

    Sent from my SM-G900V using Tapatalk
     
    Maisonvi, via a mobile device, Mar 26, 2016
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  14. helrich

    helrich Greenie N00B Member

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    This is great!

    On a side note, I would think calculating roll bias couldn't be done without adding some coefficient of friction with respect to weight bias for tire grip. This past weekend was my region's first auto-x and I put much better tires on than what I ran last season, and it completely changed the handling characteristics of the car. I went from having loads of understeer to having a rather unsettled rear end, willing to trade sides with the front at every corner. My RSB is at full stiff and I have 800lb/in springs and I think due to these new tires I will need to soften at least one of those elements to get the car back to neutral.

    I think this behavior can at least partially be attributed to the weight distribution of the car. With so much of the weight over the front wheels, changes in grip have a more pronounced effect than they do to the rear, and so where the shit tires caused the car to push, the better ones brought the front to terms.
     
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  15. ConeKiller

    ConeKiller Motorhead Greenie Member

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    THIS
     
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  16. phate

    phate Motorhead

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    In short, you don't need it. I can go into the math for it and the assumptions made later if you guys are interested (it's a part of the big red section in the first post :) ).
     
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  17. John

    John Full Fledged Member Platinum Member

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    I'm interested in all the details, or simply references thereto would be fine.
     
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  18. phate

    phate Motorhead

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    Alright, it'll be next week-ish at the earliest. Trying to finish flares and get the car ready for autocross this Sunday. Mostly just an asphalt shakedown for later events.
     
  19. Gandalf

    Gandalf Greenie Member

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    You lost me at Hello. I don't want to understand your voodoo, just want you to tell me what to buy. Geez.
     
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  20. Redline

    Redline I done fucked up for the last time. BANNED Greenie Member

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    Mazdaspeed coilovers (the MSM re-badged/re-tuned KWv3s), a good RSB, sticky tires with the right width rims, and a decent set of brake pads with some brake fluid that doesn't boil easily. ;)
     
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