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Question on Ordinary Differential Eqn

  1. Mar 20, 2014 #1
    The displacement y(t) of a driven mass-spring system is described by the differential equation 3y" + 14y =7cos(2t) with initial value conditions y(0) = 0, y' (0) = 0

    a.) is this system damped or un-damped?
    b.) Is this system resonant?
    c.) Write the solution to the IVP in terms of a product of 2 sine functions
    d.) What is the frequency of the beats

    Not too sure how to start on this question. Any help to jump start the thinking process would be really appreciated :D.
     
  2. jcsd
  3. Mar 20, 2014 #2

    HallsofIvy

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    An obvious way to start would be to find the solution to the differential equation! Have you done that?

    Do you know what "damped", "un-damped", and "resonant" mean?
     
  4. Mar 20, 2014 #3
    Yup i managed to get the general solution!!

    y(t) = 7/2 Cos(2t) + C1 Cos (*sqre root 14/3* t) + C2 Sin (*sqre root 14/3*)

    Still cant find what is resonant though. does it have something to do with the sine waves?
     
    Last edited: Mar 20, 2014
  5. Mar 20, 2014 #4

    LCKurtz

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    That's correct for the general solution of the DE. But now you need to satisfy the initial conditions to get the specific solution to your problem. Then you will be ready for what's next.
     
  6. Mar 20, 2014 #5
    Is resonant another word for critically damped?

    Either way, if you use undetermined coefficients initially to determine the solution to your homogeneous equation, the way to tell its state of damping it to look at the discriminant for your quadratic equation.

    If it's negative, it's under-damped.

    If it's positive, it's over-damped.

    If it's zero, it's critically damped.
     
  7. Mar 20, 2014 #6

    LCKurtz

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    It has to do with the relationship between the forcing frequency and the natural frequency. Here's a link I think you will find very helpful, although I would expect similar information to be in your text:

    http://www.marietta.edu/~mmm002/Math302/Lectures/Ch4Sec3.pdf [Broken]
     
    Last edited by a moderator: May 6, 2017
  8. Mar 20, 2014 #7

    LCKurtz

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    No. See the link in post #6.
     
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