[ODE] Re: Simulating Wheels

Dave Lloyd dave at chaos.org.uk
Thu Nov 6 13:14:55 MST 2003

Amund Børsand wrote:

>Steve Baker <sjbaker1 at airmail.net> skreiv:
>>Amund Børsand wrote:
>>>When I said real-world parameters, I was primarily thinking about mass,
>>>height of center of gravity, and dimensions.
>>But without real world numbers for springs, dampers and friction, it won't
>>behave realistically.
>I agree - to a certain extend. Even if you could input Nm/s2 and stuff -
>which numbers would you put in? I'm sure you'd have to experiment anyway.
>Springs and dampers is no hokus - if it's too stiff, make it softer, if
>it's too soft - make it stiffer. Make the parameters adjustable as you
>drive and adjust them until you're happy. No problem.
One thing you can do is to keep your sping/damper system critically 
damped as is (I believe) usually the case with car suspension. A 
critically damped system has the damping coefficient related to the 
spring coefficient by
    C = 2sqrt (KM)
where M is the mass of the vehicle - though remember there are four such 
systems in parallel and this leads into the vexed question of what is 
the active mass on one bit of the suspension.
See for example 
http://www.engin.umich.edu/class/me240/Matlab/PMSD/pmsd.html as 
illustration of this.

ODE doesn't really care about C and K directly - it uses ERP and CFM 
derived from these coefficients

    ERP = hK / (hK + C)
    CFM = 1 / (hK + C)

where h is the stepsize (dt - incidentally this means you need a 
constant stepsize for well behaved suspension!)

Substituting in critically damped C gives

    ERP = hK / (hK + 2sqrt(KM))
    CFM = 1 / (hK + 2sqrt(KM))

This depends on only one free parameter - the spring stiffness - the 
mass of the car is already determined.

Setting the stiffness high will give you a sports car feel riding every 
bump while setting the stiffness low will make it soft and soggy and 
handle like a big American saloon! But at least there's now only one 
parameter to experiment with.

The other thing you can use to help guide the numbers is to look at the 
natural frequency of the system
    w = sqrt (K / M)
This tells you how long the suspension will take to recover from a bump 
- a higher stiffness will react faster.  I believe the time constant 
should be between 0.5-3 secs but that's off the top of my head.

Thinking about it, since we can choose our numbers freely we can define 
the suspension entirely in terms of
(*) Z - How critically damped it is (Z < 1 underdamped, Z > 1 overdamped)
(*) T - The reaction time (1 cycle of an oscillation when underdamped)
(*) M - The car mass (or some active component thereof)

W = 2PI/T (frequency)
K = M.W^2
C = 2Z.M.W

ERP = hM.W^2 / (hM.W^2 + 2Z.M.W) = hW / (hW  + 2Z)
CFM = ERP / M.W^2

PS: I hope I got that math right - I'm a bit rusty! Will try this out 

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