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Math Help - Second order ODE with general forcing function of two variables

  1. #1
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    Second order ODE with general forcing function of two variables

    Hi,

    I'm trying to solve for the modal transient dynamic response of a beam subjected to arbitrary loading in t (time) and x (measurement along the length of the beam). I've already solved for the eignefunctions that describe the beams displacement. I'm now trying to solve for the temporal solution. I get to

     \ddot{T_{n}}(t) + T_{n}(t) = R(x,t)

    where  R(x,t) is an arbitrary function determined from some integration and operation on the forcing fuction.

    Question: Does anyone know how to obtain a solution to this equation for  T_{n}(t) ? I'm not sure what to use as an initial solution guess.

    Any help is much appreciated.

    Thanks.
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  2. #2
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    That's just the standard y''+y=f(t) which in your case is T''+T=f(t,x) and that has the general solution:

    T(t)=c_1\cos(t)+c_2\sin(t)+\int_{a}^t f(\beta,x)\sin(t-\beta)d\beta

    assuming f(t,x) is integrable in the interval a<t<b. That solution is via variation of parameter
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  3. #3
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    Hi Shawsend,

    I went through my old ODE book and found variation of parameters. Thanks for the guidance. I'll give this a shot.
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