Developing a Proper Suspension Model

[Shankenstein]

Originally Posted by zohare

Updates anyone?

Sorry for leaving the thread unfinished. Been busy at work and weekends have been crazy. AutoX 2 weekends ago, my old lady got her PhD last Friday (woot! for smart womenfolk). I also started a thread of race videos, which are a wonderful way to get faster without spending any money.

Suspension points will come as soon as I get a free Saturday. I'm looking forward to some pretty camber curves and roll centers, so that we can start picking them apart.

Eventually, it would be nice to connect an optimized model result with coilover conditions (spring rate, damping rates, ride height, free camber angle, etc).

@robispec , thanks for all of the info and feedback. You're doing a great service to the 86 community with all of the test-n-tune time... and your willingness to communicate with us.

In the mean time, here's a puppy!

[robispec]

Well I never mind being beat by customers....until my fat old slow ways catch up with me I share enough to attract customers...lol...
My car is a moving target though...lol I have MAJOR ADD when it comes to freezing a project!

Robi

[robispec]

new products

STREET no sheetmetal triming
Adds 2" rear travel



RACE requires rear sheetmetal trimming
Adds 3" rear travel


Last edited by robispec; Today at 10:50 PM.

[EarlQHan]

Originally Posted by Shankenstein

While we look forward to RobiSpec's analysis, I'll provide some brain food.

There are multiple systems interacting in a car suspension, with different natural frequencies and damping characteristics. The lowest decade (1-10Hz) is responsible for almost all system dynamics, but all of them should be mentioned. The system looka like dis:


1) Upper system (fix x_2 and w):
Sprung weight + suspension spring. This motion is damped by the suspension damper. These are very low frequencies (0.5-5 Hz).

f_nat1 = constant * sqrt(spring rate / sprung mass)
b_1 = damper constant / sqrt(spring rate / sprung mass)

2) Middle system (fix x_1 and w):
Unsprung weight + suspension + tires. This motion is damped by both the suspension damper and tire damping. These are generally much higher frequencies.

f_nat2 = constant * sqrt([spring rate + tire rate] / unsprung mass)
b_2 = damper constant / sqrt([spring rate + tire rate] / unsprung mass)

3) Lower system (fix x_1 and x_2)
If we assume that all of the car's weight is sent to the ground, the parts involved are the sprung weight + unsprung weight + tires. This motion is damped by the tire, which is generally a very low damping factor (tires bounce).

f_nat3 = constant * sqrt(tire rate / [sprung mass + unsprung mass])
b_3 = tire damping constant / sqrt(tire rate / [sprung mass + unsprung mass])

Due to the low damping capabilities of tires, it's best to let the dampers handle vibration control. For tire dynamics to minimally affect suspension dynamics, a decade of frequency separation should be sufficient.

F_tire > 10 * F_susp
sqrt(tire rate / unsprung mass) > 10 * sqrt(spring rate / sprung mass)
If both values are more than one,
tire rate / unsprung mass > 100 * spring rate / sprung mass
tire rate / spring rate > 100 * unsprung mass / sprung mass

for our example:
6500 / 131 > 100 * 83 / 618
49.6 > 13.43 --> sufficiently separated

I guess I should amend the above statement. Thanks for pointing it out!

Continuing this thought:
If we calculate the max spring rate that can be used without being affected by tire dynamics (at stock pressures):
max front wheel rate = 484 lbs/in
max front spring rate = 526 lbs/in
max rear wheel rate = 422 lbs/in
max rear spring rate is = 548 lbs/in

At autox pressures, max spring rates would be 809 (front) and 843 (rear). In metric, that's 14.2k and 14.8k. Interesting, not that anyone would want to run them that stiff anyways.

Since tire damping is so low and so so very hard to quantify, you can use this simplified model for the transfer function:



NOTE: Should say MASS not WEIGHT

[plucas]

Originally Posted by EarlQHan

Since tire damping is so low and so so very hard to quantify, you can use this simplified model for the transfer function:



NOTE: Should say MASS not WEIGHT

This is basically what I did for my capstone. However it was damping a chassis (no suspension, only chassis and tires for the spring). I used Matlab for the transfer functions, and I used the root locus method to design the desired damping ratio and frequency of the system.

[EarlQHan]

Originally Posted by plucas

This is basically what I did for my capstone. However it was damping a chassis (no suspension, only chassis and tires for the spring). I used Matlab for the transfer functions, and I used the root locus method to design the desired damping ratio and frequency of the system.

NERD! So that's what you and Jeff did, eh?

[Shankenstein]

Originally Posted by EarlQHan

Since tire damping is so low and so so very hard to quantify, you can use this simplified model for the transfer function:



NOTE: Should say MASS not WEIGHT

Everything checks out, to my eyes. That's the traditional generalization about suspensions... all masses are point masses, all forces are 1D, all spring rates are constant.

If you want to trick it up, you can do:
- progressive spring rate = f(x-x_0)
- linear damping = k * (x-x_0)'
- digressive damping = f((x-x_0)')
- damper hysteresis = nasty equations)
- pressure-sensitive tire rate = k_sidewall + k_air * pressure (absolute)
- temperature-sensitive tire rate = k_sidewall + k_air * n*R/V * temperature (absolute)
- tire temperature estimation = T_atm + k * log (T_tread - T_air_in_tire)
- tire tread temperature = T_atm + k * (heat generated - heat dissipated)
... lots of ways to complicate the problem.

[EarlQHan]

Originally Posted by Shankenstein

Everything checks out, to my eyes. That's the traditional generalization about suspensions... all masses are point masses, all forces are 1D, all spring rates are constant.

If you want to trick it up, you can do:
- progressive spring rate = f(x-x_0)
- linear damping = k * (x-x_0)'
- digressive damping = f((x-x_0)')
- damper hysteresis = nasty equations)
- pressure-sensitive tire rate = k_sidewall + k_air * pressure (absolute)
- temperature-sensitive tire rate = k_sidewall + k_air * n*R/V * temperature (absolute)
- tire temperature estimation = T_atm + k * log (T_tread - T_air_in_tire)
- tire tread temperature = T_atm + k * (heat generated - heat dissipated)
... lots of ways to complicate the problem.

Too many deltas! I was thinking state-space might be a better way to model the suspension so you can do transient analysis. This, including modeling the driver-vehicle feedback loop as the base block diagram is what I want to do my masters (or possibly doctorate) research and thesis on. But that's just too much math and I'm going to say EF IT for now lol.

[Shankenstein]

Originally Posted by EarlQHan

Too many deltas! I was thinking state-space might be a better way to model the suspension so you can do transient analysis. This, including modeling the driver-vehicle feedback loop as the base block diagram is what I want to do my masters (or possibly doctorate) research and thesis on. But that's just too much math and I'm going to say EF IT for now lol.

This IS state-space. People like to use that term to indicate magic.

Variables take specific states, which interact based on a set of differential equations (some complex some simple).

What are your states? You can consider either the ground or the car body to be fixed... it's just perspective (although it matters that you pick one and stick with it). We care about tire deflection (x) and shock displacement (y). Thus the states are:
Q =
[ x ]
[ y ]
[ x' ]
[ y' ]
[ x'' ]
[ y'' ]

Your state space matrix will have diagonal elements to define the derivatives:
Q' =
[ 0 0 1 0 0 0 ] * Q
[ 0 0 0 1 0 0 ]
[ 0 0 0 0 1 0 ]
[ 0 0 0 0 0 1 ]
[ 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 ]

You can fill in the rest of the elements using the force balance equations in any level of complication you desire... but stiffness and damping components will appear, scaled by mass.

Solving the state space version is the same as solving the diff-eq version is the same as solving the numerical version. I'm sure you know all this, but it's good to lay it out there for the forum to glance over.

Good luck with the Master's btw. Where are you planning to go to school (or are already there)?

[7thgear]

Originally Posted by robispec

new products

STREET no sheetmetal triming
Adds 2" rear travel



RACE requires rear sheetmetal trimming
Adds 3" rear travel


Last edited by robispec; Today at 10:50 PM.

this is awesome

however a question/concern

with these, the wheel can now compress 2-3" more than it could in stock, is there enough clearance for the tire? Furthermore, the control arms too can now move significantly more upwards into the chassis.

is none of this a concern? how does this affect geometry under extreme cornering?

or am i overthinking this?

[SubieNate]

@Shankenstein @EarlQHan @plucas

I have a copy of Matlab and I'm a relatively decent (if rusty) coder. If we want to collaboratively use it as a tool to model our suspension mathematically I'd be more than happy to help.

Cheers
Nathan

[EarlQHan]

Originally Posted by Shankenstein

This IS state-space. People like to use that term to indicate magic.

Variables take specific states, which interact based on a set of differential equations (some complex some simple).

What are your states? You can consider either the ground or the car body to be fixed... it's just perspective (although it matters that you pick one and stick with it). We care about tire deflection (x) and shock displacement (y). Thus the states are:
Q =
[ x ]
[ y ]
[ x' ]
[ y' ]
[ x'' ]
[ y'' ]

Your state space matrix will have diagonal elements to define the derivatives:
Q' =
[ 0 0 1 0 0 0 ] * Q
[ 0 0 0 1 0 0 ]
[ 0 0 0 0 1 0 ]
[ 0 0 0 0 0 1 ]
[ 0 0 0 0 0 0 ]
[ 0 0 0 0 0 0 ]

You can fill in the rest of the elements using the force balance equations in any level of complication you desire... but stiffness and damping components will appear, scaled by mass.

Solving the state space version is the same as solving the diff-eq version is the same as solving the numerical version. I'm sure you know all this, but it's good to lay it out there for the forum to glance over.

Good luck with the Master's btw. Where are you planning to go to school (or are already there)?

I am literally the worst at controls. It doesn't help my controls prof got canned the semester after I had him since he didn't pass his reviews. I passed that class by the skin of my teeth. As for matrix math, I blame dyslexia... which I don't have...

I've been accepted to Cranfield University's Advanced Motorsports Engineering program in England. I was also accepted to Oxford-Brookes Masters in Motorsports Engineering, also in England, but I turned down their offer in favor of Cranfield. I may defer though...

Originally Posted by SubieNate

@Shankenstein @EarlQHan @plucas

I have a copy of Matlab and I'm a relatively decent (if rusty) coder. If we want to collaboratively use it as a tool to model our suspension mathematically I'd be more than happy to help.

Cheers
Nathan

I'd be game. Nathan, I'm also going to shoot you a PM about composites...

[plucas]

Originally Posted by SubieNate

@Shankenstein @EarlQHan @plucas

I have a copy of Matlab and I'm a relatively decent (if rusty) coder. If we want to collaboratively use it as a tool to model our suspension mathematically I'd be more than happy to help.

Cheers
Nathan

I would be up for this

[SubieNate]

I did some of my elective work in controls, it would take a bit of work for me to recall everything but I'm sure I could get back into it.

@EarlQHan-PM replied, we should setup a skype session or something on the suspension modelling front.

Cheers
Nathan