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Differential equations of forced oscillation and resonance

  1. Jul 25, 2013 #1
    How do I derive A? As you can see in the attachment, I tried to substitute x and expand the equation but I got stuck. How do I get rid of the δ and cos and sin to get the result in the end? Please help!
     

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  3. Jul 25, 2013 #2

    SteamKing

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    I would leave (omega*t-delta) alone while calculating the various derivatives. The sines and cosines with this argument do not need expanding until you have done you calculations.
     
  4. Jul 25, 2013 #3
    OK, what do I do next? I don't expand it, and here's what I got.
     

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  5. Jul 25, 2013 #4

    SteamKing

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    Where did the plain 'ω' come from in the last line of your calculations?

    Specifically, the term (ωe^2 - ω^2)?
     
  6. Jul 25, 2013 #5

    SteamKing

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    Disregard last post. I see now.
     
  7. Jul 25, 2013 #6

    SteamKing

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    Now you expand the cosine and sine terms and equate same to the RHS of the equation.
    In your first attempt, you differentiated δ w.r.t. time. This was incorrect. The phase angle δ is constant w.r.t. time, which was one reason your original derivation got so unwieldy.
     
  8. Jul 25, 2013 #7
    So, you're saying that I don't have to bother the one with the delta? That would mean doing partial differentiation, right?
     
  9. Jul 25, 2013 #8

    SteamKing

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    There is no partial differential involved. If you take the derivative of cos(ωt-δ), you will get -ω*sin(ωt-δ). The δ represents a constant phase angle; it is not a function of t.

    Now that you have your LHS in terms of sin and cos, now is the time to expand, for instance, cos(ωt-δ) using the angle difference formulas. You then solve for the coefficients of the sine and cosine terms on the LHS which correspond to whatever sine and cosine terms you have on the RHS.
     
  10. Jul 25, 2013 #9
    So you mean, equate LHS and RHS and then substitute them back into the equation? But the right hand side only has F/cos(ω_e*t).
     
  11. Jul 25, 2013 #10

    SteamKing

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  12. Jul 25, 2013 #11
    Ok, now what should I do? The right hand side only has cos(w_e*t). And even so, if I did equate the sin and cosine, what should I do with it? I would still have cos(w_e*t) on the right side, don't I?
     

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  13. Jul 25, 2013 #12

    SteamKing

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    If you read the attached notes from post#10, you would see how to handle this.
     
  14. Jul 26, 2013 #13
    Thanks for the post. That really hepls a ton. But my answer is a little different. How come my denominator and numerator are inverted?
     

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    Last edited: Jul 26, 2013
  15. Jul 26, 2013 #14

    SteamKing

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    It's hard to tell from your posted calculations. After expanding the cosine and sine terms which contain the phase angle δ, on the LHS there will be sin δ and cos δ terms mixed in with the cos(ωet) and sin (ωet) terms. You use the phase angle triangle to determine sin δ and cos δ in terms of the other known quantities before solving for A.
     
  16. Jul 26, 2013 #15
    But in my calculation, I already determined the sin δ and cos δ, see I drew the triangle?
     
  17. Jul 26, 2013 #16
    Hey, I got it!!!!!Look! Thank you soooo much!
     

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    Last edited: Jul 26, 2013
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