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I Electrical circuit differential equation

  1. May 13, 2017 #1
    q''+ 20 q = 100 sin(ωt)

    I have been asked to find all mathematically possible values of ω for which resonance will occur. From the homogeneous solution, q(t) = Acos(√20 t) +Bsin(√20 t), I can see that resonance occurs when ω=√20. My question is, should I also consider -√20? And if so, what is the significance of a negative angular frequency?

    Thanks.
     
  2. jcsd
  3. May 13, 2017 #2

    BvU

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    You sure you have the full solution ?
    What determines A and B ?
    Is ##C\cos(-\sqrt{20} \, t) + D\sin(-\sqrt{20} \,t )\ ## much different from ##A\cos(\sqrt{20} \, t)+ B\sin(\sqrt{20} \,t ) \ ## ?

    Likewise: what's the difference between ##100 \sin(\omega t)## and ##100 \sin(-\omega t)## ?
     
  4. May 13, 2017 #3
    Thank you for your reply BvU. The only difference is that the sine term will have a negative in front, which doesn't really make a difference since A, B, C and D are arbitrary constants. So my particular solution in both cases (ω=+√20 and ω=-√20 ) will be of the form q(t) = t(acos(√20 t) ± bsin(√20 t)), which means the charge will oscillate with increasing amplitude and without bound as t increases. So these are the two (and only two) ω values for which resonance occurs in this system?
     
  5. May 13, 2017 #4

    BvU

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    (If that is what you mean) Doesn't look correct.
    General solution = one particular solution + solution of homogeneous equation ,
    the first without integration constants, the second with
    is what I remember.
     
  6. May 13, 2017 #5
    Sorry, that is not what I meant. I should not have written it like that. As you said, the general solution contains a particular solution and the homogeneous solution. So the general solution will be of the form q(t) = Acos(√20 t) +Bsin(√20 t) + t(acos(√20 t) + bsin(√20 t)).

    So the values of ω that will give resonance are both +√20 and -√20, right? I just need some confirmation as I haven't been given a problem like this before and I have little knowledge of electrical circuits.
     
  7. May 13, 2017 #6

    BvU

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    Does that satisfy the differential equation ? I don't think it does...
     
  8. May 13, 2017 #7
    It does after solving for the constants. Thanks for your time BvU. I think I've got it now :)
     
  9. May 14, 2017 #8

    BvU

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    Also for ##\omega\ne \omega_0## ?

    And: you're welcome.
     
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