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## Homework Statement

vinit = sqrt(2g*h); h = drop distance

vfinal = 0;

xinit = 0;

xfinal = 100mm;

a = g;

Issue: non-linear damping.

M*x'' - b*(x')^2 - k*x = 0;

b = 128*mu*(length fluid travels)*(D^4(piston)/[(D(hydraulic)^4)(orifice opening)]

every book I've been reading on vibrations damping says there's no solution for v^2 damping. Currently reading "Influence of Damping in Vibration Isolation" and they give an equivalent linear damping coefficient as:

C(eq) = (D0)/(∏ω(z0)^2); D0 being energy dissipated per cycle, z0 being relative displacement.

Then they go into equivalent damping force being: γ*F; γ= (2/sqrt(∏))*gamma((n+2)/2)/gamma((n+3)/2)...

very long story short. Is there anyway to do a stepwise energy dissipation of a mass/spring/damper problem? Can I use something like: initial energy in - energy to compress spring - energy dissipated by damper = 0.

the issue I think is with this is that I don't know how to figure out energy dissipated by a damper whose dependent on v^2...

can I define a function that says: this system was deflected by 0.1mm at this time and Z amount of energy was taken away from the initial impact. W energy was taken up by the spring, and X was taken by the damper. This is how much much energy was left over at the boundary of this iteration...

Im in analysis paralysis at the moment and I think I'm overthinking this...

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