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Damped Oscillator and Oscillatory Driving Force

  • #1

Homework Statement:

The equation is FCoswt = mx'' + myx' +mw_0^2x
- Find the steady state solution for the displacement x and the velocity x'
- Sketch the amplitude and phase of x and x' as a function of w
- Determine the resonant frequency for both the displacement and the velocity
- Defining deltaw as the full width at half maximum of the resonance peak, calculate deltaw/w_0 to leading order in y/w_0
- For a lightly damped driven oscillator near resonance, calculate the energy stored and the power supplied to the system. Confirm that Q = w_0/y.

Homework Equations:

Steady state solution is the particular solution
I found the steady state solution as
F_0(mw_0^2 - w^2m)Coswt/(mwy)^2 + (mw_0^2 -w^2m)^2
+ F_0mwySinwt/(mwy)^2 + (mw_0^2 -w^2m)^2
But I'm not sure how to sketch the amplitude and phase? Do I need any extra equations?
 

Answers and Replies

  • #2
BvU
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Well, what is the expression for amplitude ?
And what is the expression for the phase ?

Oh, and
please use ##\LaTeX## so one can read your solution

$$
F_0(m\omega_0^2 - \omega^2m)\cos\omega t/(m\omega \gamma)^2 + (m\omega_0^2 -\omega^2m)^2
+ F_0m\omega \gamma\sin\omega t/(m\omega \gamma)^2 + (m\omega_0^2 -\omega^2m)^2 $$

which, if I reproduced it right, as it looks now, does not look familiar at all... did you check it satisfies the equation $$ m\ddot x + m\gamma \dot x +m\omega_0^2\;x = F\cos\omega t \ \ ?$$
 
  • #3
vela
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There seem to be missing parentheses. I think the OP meant
$$\frac{F_0 m(\omega_0^2-\omega^2)}{(m\omega\gamma)^2 + m^2(\omega_0^2-\omega^2)^2}\cos\omega t + \frac{F_0 m\omega\gamma}{(m\omega\gamma)^2 + m^2(\omega_0^2-\omega^2)^2}\sin\omega t$$
 
  • #4
BvU
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I know, but that only became painfully clear once I typeset the litteral text in post #1.

After @physconomics posts the relevant equations for amplitude and phase we can proceed with this thread
 
  • #5
$$
I know, but that only became painfully clear once I typeset the litteral text in post #1.

After @physconomics posts the relevant equations for amplitude and phase we can proceed with this thread
Yes, Vela was right. Sorry I'm new here and had no idea I could use LaTeX.
To get the amplitude and phase would I have to move it into the form ##x = Cos(\omega t - \phi)##?
 

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