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Damping - Differential Equations

I've been working on this problem for a while, and I'm not sure how to work out the damping of an SHM system (spring with mass attached on end).

The differential equation is (d2x/dt2) + 54.199x = 0 and the particular integral is x = -41.84cos(7.362t)

How do I go about working out the (dx/dt) term in the DE to incorporate underdamping?

Please help, I'm reaching the end of my tether :frown:
If it's underdamped, oscillations will still take place - what does this tell you about the form of the complementary function? And what does that tell you about the discriminant of the auxillary equation (and hence the coefficient of the dx/dt term)?
Reply 2
Avatar for i.am.lost
i.am.lost
OP
18
matt2k8
If it's underdamped, oscillations will still take place - what does this tell you about the form of the complementary function? And what does that tell you about the discriminant of the auxillary equation (and hence the coefficient of the dx/dt term)?

Well if it's underdamped then the discriminant is negative, so the dx/dt coefficient is <2&#969;. But how do I find it's exact value?
Reply 3
Avatar for Scipio90
Scipio90
You can't - underdamping is just a general condition, not a specific value. Are you sure you don't mean critically damped?
i.am.lost
Well if it's underdamped then the discriminant is negative, so the dx/dt coefficient is <2&#969;. But how do I find it's exact value?

There will always be a range of values.

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