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Math Help - Could you please help me solve a system of 3 ODEs?

  1. #1
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    Could you please help me solve a system of 3 ODEs?

    Hello. I'm working on a problem involving 3 masses connected by springs. One of the masses has a driving force Focos(wt). These masses move along a line.

    Using the Euler-Lagrange equations (I know some simple classical mechanics) I have derived the equations of motion, shown below:

    [IMG]http://img849.imageshack.us/img849/9243/mathshelp.png[\IMG]

    k is the spring constant, m is the mass of each of the three objects. The displacements from equilibrium for each object are given by x, y and z. At the end I just convert from equilibrium displacement to actual position by adding on the equilibrium positions

    I have spent days trying to figure out how to solve them analytically but I have had no luck. I have been going through a method in Goldstein's Classical Mechanics but cannot get things to work I have put the equations into matrix form, hoping that will help. I would love any help

    I have a numerical solution of this, and I am confident it is accurate, so I have a comparison ready
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    Last edited by ILoveMathematics27; August 30th 2011 at 02:44 AM.
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  2. #2
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    Re: Could you please help me solve a system of 3 ODEs?

    Okay, I have a solution to the homogenous system of equations (so without the driving force) but I am struggling to find a particular solution that will allow me to solve the non-homogenous system of equations.

    I have tried many solutions, as shown below.

    Note that for simplicity I have taken k,m,Fo and omega all to be 1.

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  3. #3
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    Re: Could you please help me solve a system of 3 ODEs?

    Here's a suggestion. Your equations are

    \ddot{x} =\frac{k}{m}( y - x) + \frac{F_0}{m} \cos \omega t

    \ddot{y} = \frac{k}{m}(x+z-2y)

    \ddot{z} = \frac{k}{m}(y-z)

    If you add the three equations together you get

    (x+y+z)'' = \frac{F_0}{m} \cos \omega t.

    This you can integrate twice. You get x + y + z = stuff
    Then eliminate x + z in your second equation to get a single ODE for y. Solve. Then return to the first equation and solve for x.

    On a side note, particular choices for \omega will make a difference so be careful.
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    Re: Could you please help me solve a system of 3 ODEs?

    Thank you for your reply.

    Before reading it though, I had already solved the system of equations. I didn't use your method, I used matrices and such, finding the eigenvalues and eigenvectors and went from there. The results match my numerical solution.

    I get what you are saying about particular choices for omega. My analytic solution won't work for omega = 1, 3^0.5 because those are the natural frequencies. In the numerical solution you do see resonance, as you would hope!

    For all other frequencies i used a particular solution of bfcos(omega*t), where b is a vector which i calculated and f = (Fo/m, 0, 0) is the force vector. That worked fine. For resonant frequencies i tried tbfcos(omega*t) but the maths got out of hand pretty quickly.

    I'll look at your suggestions in the morning, as I am dead tired now.
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