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Math Help - Mass-spring system with damping

  1. #1
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    Mass-spring system with damping

    The differential equation system is

    m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx(t) = F_o cos(wt)

    To solve this differential equation requires a particular solution xp(t) and two fundamental solutions of the homogeneous equation corresponding X1(t) and X2(t)

    x(t) = xp(t) + C1x1(t) + C2x2(t)

    I find X1(t) and X2(t) for the following cases


    Overdamped:

    x(t) = C1e^{r1t} + C2e^{r2t}

    r1 = \frac{ - \gamma + \sqrt{( \gamma )^2 -4mk}}{2m}
    r2 = \frac{ - \gamma - \sqrt{( \gamma )^2 -4mk}}{2m}



    Underdamped

    x(t) = e^{-(\frac{c}{2m})t} (C1cos(rt) + C2sin(rt))

    r -> imaginary part
    r = \frac{ \sqrt{( \gamma )^2 -4mk}}{2m}



    How do I find the particular solution xp(t) ?
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    The differential equation system is

    m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx(t) = F_o cos(wt)

    To solve this differential equation requires a particular solution xp(t) and two fundamental solutions of the homogeneous equation corresponding X1(t) and X2(t)

    x(t) = xp(t) + C1x1(t) + C2x2(t)

    I find X1(t) and X2(t) for the following cases


    Overdamped:

    x(t) = C1e^{r1t} + C2e^{r2t}

    r1 = \frac{ - \gamma + \sqrt{( \gamma )^2 -4mk}}{2m}
    r2 = \frac{ - \gamma - \sqrt{( \gamma )^2 -4mk}}{2m}



    Underdamped

    x(t) = e^{-(\frac{c}{2m})t} (C1cos(rt) + C2sin(rt))

    r -> imaginary part
    r = \frac{ \sqrt{( \gamma )^2 -4mk}}{2m}



    How do I find the particular solution xp(t) ?
    Use a trial solution for the particular solution of the form:

    x_p(t)=A\cos(\omega t) +B \sin(\omega t)

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Use a trial solution for the particular solution of the form:

    x_p(t)=A\cos(\omega t) +B \sin(\omega t)

    CB
    As long as \omega\ne \frac{ \sqrt{( \gamma )^2 -4mk}}{2m}<br />
.

    If \omega is equal to that you will need to try
    x_p(t)= A t cos(\omega t)+ B t sin(\omega t)
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  4. #4
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    The damped

    m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx(t) = 0


    The characteristic equation

    mr^2 + \gamma r + k = 0

    r1 = \frac{ - \gamma + \sqrt{( \gamma )^2 -4mk}}{2m}
    r2 = \frac{ - \gamma - \sqrt{( \gamma )^2 -4mk}}{2m}


    In overdamped
    ( \gamma )^2 -4mk > 0

    What are the steps now to find the solution:
    X(t) = C1e^{r1t} + C2^{r2t}
    ???


    And how do I get the equations in the case of mechanical vibrations: Beats and resonance ???
    Last edited by Apprentice123; December 19th 2009 at 07:25 AM.
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  5. #5
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    Quote Originally Posted by Apprentice123 View Post
    The damped

    m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx(t) = 0


    The characteristic equation

    mr^2 + \gamma r + k = 0

    r1 = \frac{ - \gamma + \sqrt{( \gamma )^2 -4mk}}{2m}
    r2 = \frac{ - \gamma - \sqrt{( \gamma )^2 -4mk}}{2m}


    In overdamped
    ( \gamma )^2 -4mk > 0

    What are the steps now to find the solution:
    X(t) = C1e^{r1t} + C2^{r2t}
    ???


    And how do I get the equations in the case of mechanical vibrations: Beats and resonance ???
    You have found the homogenous solution. Now you have to add to it the particular solution. You have already been told how to find the particular solution. All this stuff will be found in any decent textbook that covers second order differential equations with constant coefficients (go to your school library) and I'm certain that Google will turn up pages and pages of links.

    Your problem is attempting abstract application questions like these before developing a sufficient grasp of the basic theory in a sufficient number and variety of concrete cases. You are advised to go back and review that theory.
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