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...