Driven Damped Oscillator problem

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  • #1
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Homework Statement


Given damping constant b, mass m spring constant k,
in a damped driven oscilation system the average power introduced into the system equals the average power drained out of the system by the damping force, for what values of ω does the instantanious damping power = instantaneous drive power



Homework Equations


Total power of system =(-kx - b v(t)+ F0Cos(ωt))v(t)
x(t)= A cos(wt + (phi))


The Attempt at a Solution


Edit: just tried again, I subbed in ma for F and the Still Stuck with amplitudes on one side, and stuck with phi's
 
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Answers and Replies

  • #2
vela
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It would help if you showed your actual calculations.
 
  • #3
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x(t) = Acos([itex]\omega[/itex]t+[itex]\phi[/itex])=Aei([itex]\omega[/itex]t +[itex]\phi[/itex]

Force of Driver = -mA[itex]\omega[/itex]2 ei[itex]\omega[/itex]t

Total force = -kx-bv+ma
total force * x'(t)=x'(t)*(-kx-bv+ma)
divide by m
[itex]\frac{b}{m}[/itex]=[itex]\gamma[/itex]

[itex]\frac{k}{m}[/itex]=[itex]\omega[/itex]02

[itex]\omega[/itex]02Aei([itex]\omega[/itex]t +[itex]\phi[/itex])-A[itex]\gamma[/itex]i[itex]\omega[/itex]ei([itex]\omega[/itex]t +[itex]\phi[/itex])-A[itex]\omega[/itex]2ei([itex]\omega[/itex]
I think the question is asking for what nonzero ω values is the total instantanious power 0
which means

[itex]\omega[/itex]02Aei([itex]\omega[/itex]t +[itex]\phi[/itex])-A[itex]\gamma[/itex]i[itex]\omega[/itex]ei([itex]\omega[/itex]t +[itex]\phi[/itex])-A[itex]\omega[/itex]2ei([itex]\omega[/itex]t =0

Cancel Like Terms

[itex]\omega[/itex]02ei([itex]\omega[/itex]t +[itex]\phi[/itex])-[itex]\gamma[/itex]i[itex]\omega[/itex]ei([itex]\omega[/itex]t +[itex]\phi[/itex])-[itex]\omega[/itex]2ei([itex]\omega[/itex]t) =0

if there was a phi in the force term I could cancel all the e's but according to my book there is no phi term, and phi is not a given so im stuck
 
  • #4
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Remember that [itex]e^{i(\omega t + \phi)} = e^{i\omega t + i\phi)} = e^{i\omega t} e^{i\phi}[/itex].

Then you can cancel the [itex]e^{i\omega t}[/itex]. I'm not sure if this helps too much, but you could also use the vector representation of complex numbers to find [itex]\phi[/itex]?
 
  • #5
vela
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I'm not sure what exactly you mean by "total force." I'd take that to mean "net force" which should be equal to ma. And how is it supposed to be related to the power dissipated by the damping force and the power supplied by the driving force?
 

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