Differential Equations and Damper Curves

[aeb2335]

TL;DR

How to solve harmonic motion equations with non constant damping

Good evening,

I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist!

I am assuming a quarter car type model and a step input is applied to a simple damped harmonic oscillator which, is often presented in the following form: m*(d2x/dt)+c*(dx/dt)+k*x=0; however, after a lot of searching I have never seen this equation solved for when c is not constant i.e c(F(dx/dt)) .

The reason I ask about c being non constant is in automotive engineering the primary suspension damper often has a knee point (see attachment); I am reasonably comfortable with coming up with equations to describe c as a function of force and velocity but, I don't really know what the next step should be.

Now if this were just a spring and a mass the energy stored is fairly straight forward: Wspring=.5kx^2 and KE=1/2mv^2
I assume for the damper the energy could be expressed as Wdamper=integral(c(F,v)*v^2dt) but, I am not totally sure. I am even less sure if that is the correct direction I want to go.

For the whole of my education dampers have been linear so this would be really cool to get understand.

 

[BvU]

Hi,

I don't really know what the next step should be

Don't think you get much further analytically. My next step would be to go numerical.

damper often has a knee point (see attachment)

Would that be real, or also an approximation ? How were these curves determined ?

 

[aeb2335]

Firstly, Thanks for your response!

As usual there is more context to this problem! I work in multi-body systems and I am trying to come up with a "rule of thumb" or sensible hand(ish) calc in order to tune the energy absorption of the various suspension components.

So unfortunately, I am trying to get away from numerical simulations as, effectively I already have the answer as it were. I find it extremely interesting that there is so much literature around this idealisation when in the real world the correlation is actually extremely poor; at least for automotive dampers.

As for damper behaviour unfortunately, they are significantly more complex than a linear relationship suggested by a constant "C" and the knee point is very real. Knee point(s) or indeed any changes in the damper curve can be caused by caviation or by a value that has been deliberately engineered into the damper.

I have seen programs and work similar to below that suggest this problem has been solved analytically but, I have no idea how!

https://royalsocietypublishing.org/doi/full/10.1098/rsta.2014.0402#d3e639

This publication seems to be close to what I am on about but, I am not sure how to proceed; Any thoughts?

 

[BvU]

I'm afraid you are the expert here . Interesting world opening when I looked at your link and at ref 13 therein (Surace). But no sharp knees found.

Did find some support for my answer, though

(c) Numerical simulation for a single-degree-of-freedom system
In practice, for higher order forms of nonlinear damping, deriving an analytical expression for the response becomes more difficult, but direct time domain simulations can still be used to calculate the result

In my naive picture of dampers I thought the nonlinearity in fluid friction due to turbulence caused the deviation from linearity.​