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Math Help - Equation for free mechanical vibration

  1. #1
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    Equation for free mechanical vibration

    Question:
    Consider the equation for free mechanical vibration, my'' + by' + ky=0, and assume the motion is overdamped. Suppose y(0) > 0 and y'(0) > 0. Prove that the mass will never mass throught its equilibrium at any positive time.

    Really stuck on this question, all i know is for over damped:
    b^2 > 4mk
    General solution: y(t)=C1e^(r1t) + C2e^(r2t) --> y(0) = C1 + C2 > 0
    y'(t)=C1r1e^(r1t) + C2r2e^(r2t) --> y'(0) = C1r1 + C2r2 > 0

    Any help with this question would be really appreciated thanks.
    Last edited by kaboose786; February 11th 2010 at 09:23 AM.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    See attachment
    Attached Thumbnails Attached Thumbnails Equation for free mechanical vibration-mechanical.jpg  
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  3. #3
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    thanks for replying is it possible if u could explain the steps after A-B .. really confusing
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  4. #4
    MHF Contributor Calculus26's Avatar
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    A + B >0 comes from y(0) > 0

    A - B >0 comes from y ' (0) > 0

    So A and B can't both be negative further A > B so A > 0

    A > -B

    1> -B/A

    I then set y equal to 0

    e^(-bt/2m) > 0

    so if y = 0 then Ae^(rt) +Be^(-rt) = 0

    multiply by e^rt

    A e^(2rt) + B = 0

    e^(2rt) = -B/A

    take logs

    2rt = ln(-B/A) if B s also postive no solutions.

    if B < 0 -B/A < 1 from above and ln(x) < 0 if x < 1

    hope this clears things up
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  5. #5
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    thx for replying ... i finally understood ... btw its ur own solution right ? not from some solution manual ? thx
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