Friction constant minimizing the duration of vertical motion

[TheRealPhone]

Homework Statement

The mass of a car that acts on one wheel is 100 kg. The elasticity (spring) constant in the suspension system of that wheel is k = 10^4N/m. Design the strut (find the friction/resistance constant c) such that any vertical motion of the wheel (set up for example by going over a bump or pothole on the road) will die out in the shortest amount of time.

Homework Equations

mx′′+cx′+kx=F(t)

The Attempt at a Solution

I have determined that the equation that we will most likely be using is mx''+cx'+kx=F(t) where F is some force. I at first thought that I should look at it at the critically damped point or where c^2-4mk=0 but I thought it was to simple and didn't make sense for this situation. Overall very confused and could use some assistance.

 

[BvU]

Hi Phone, welcome to PF !

Critical damping ( $\zeta =1$ ) is often a bit too much and too stiff, but what damping ratio (*) is optimum depends on the criteria (an error band, or the integral of deviation squared or something). Since your exercise asks for a single constant, alternatives like nonlinear damping are ruled out. So depending on your context, the simple solution might be the right one. The "critically damped response is the response that reaches the steady-state value the fastest without being underdamped" (from wiki). Key is this last restriction: you can accept some underdamping if it makes the system recover faster from a bump.

(*) Note that the picture here is a step response; you probably want to optimize the pulse response.

 

[TheRealPhone]

Thanks for the response,

I tried setting up a system of equations that may indeed minimize it using underdamping aswell. The problem is that while c^2-4mk <= 0 the functions is obviously decreasing but when you look at when it is greater then 0 it ends up increasing which would be the damping from the bottom side. I'm having trouble now finding where I could minimize this thought so that the time could be minimized.

 

[BvU]

Underdamping is not the same as negative damping ! For all $\zeta > 0$ there is damping !