Suppose you’re designing something to damp mechanical oscillations, and you’d li

Question

Suppose you’re designing something to damp mechanical oscillations, and you’d like them to dissipate as quickly as possible. (Perhaps you’re building a car’s shock absorbers, or an optics bench for LIGO.) The mass m and spring constant k are given, and you want to choose an appropriate value of the drag co¨efficient b. Whether your system is over-, under- or critically damped, the amplitude dies away exponentially with time, at some rate . (In the overdamped case you have the sum of two exponentials, but one of them is slower than the other, so examine that one.) Draw a sketch of as a function of b. What value of b guarantees that even the slowest-dying part of the solution dies as quickly as possible? If you can only build your damper with approximate precision, is it better to have b a little too small or a little too large?

Explanation / Answer

Given, damping constant = b
and rate of dfecay of amplitude = alpha

now for underdamped osscilators, amplitude as a funciton of time is given by
A(t) = e^-gamma*t
where gamma = b/2m [ m is mass of osscilator ]

for overdamped case, A(t) = e^([-b + sqroot(b^2 - 4*k/m)]*t/2)Ao

so underdamped case
dA(t)/dt = -b*A(t)/2m
overdamped case
dA(t)/dt = (-b + sqroot(b^2 - 4k/m))*A(t)/2

for dA(t)/dt to be large, b should be really large in case of underdamped osscilations, and in second case as well
but in overdamped case it takes a lot of time to come back to equilibrium
and in underdamped case
2*sqroot(mk) > b
b^2 < 4mk

hence keeping b equal to sqroot(4mk) will make the system critically damped and it will return to its equilibrium position as quickly as it can. To make up for approximations, one can keep b a b it lower than this value but not more, because when b overshoots this critical value, the system becomes overdamed and hence takes a lot of time to come back to equilibrium position

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