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    This is about a system with two pendulums of mass m hanging from strings of length l connected by a spring with spring constant k such that the spring is neutral when the pendulums are vertical. The displacement from equilibrium is x1 and x2.

    The first part of the question asks me to derive the coupled differential equations governing this system in matrix form in the form d2x/dt2=Ax which I have done:



    But the second part asks:

    Substitute a solution in the form x = \psi \cos (\omega t + \phi) to arrive at an eigenvalue-eigenvector problem for the special "spatial configuration" \psi and eigenfrequencies \omega.

    I genuinely do not understand how I'm supposed to go about substituting this solution into here. Could anybody give me some advice? Thanks.
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    (Original post by Plagioclase)
    This is about a system with two pendulums of mass m hanging from strings of length l connected by a spring with spring constant k such that the spring is neutral when the pendulums are vertical. The displacement from equilibrium is x1 and x2.

    The first part of the question asks me to derive the coupled differential equations governing this system in matrix form in the form d2x/dt2=Ax which I have done:



    But the second part asks:

    Substitute a solution in the form x = \psi \cost(\omega t + \phi) to arrive at an eigenvalue-eigenvector problem for the special "spatial configuration" \psi and eigenfrequencies \omega.

    I genuinely do not understand how I'm supposed to go about substituting this solution into here. Could anybody give me some advice? Thanks.
    Well that's a bit vague, isn't it! Is it simply asking you to guess a form of the function \psi to use as it says "a solution in the form...". If so, then just choose x_1 = \cos(\omega_1 t + \psi_1) and x_2 = \cos(\omega_2 t + \psi_2).
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    (Original post by Plagioclase)
    But the second part asks:

    Substitute a solution in the form x = \psi \cost(\omega t + \phi) to arrive at an eigenvalue-eigenvector problem for the special "spatial configuration" \psi and eigenfrequencies \omega.

    I genuinely do not understand how I'm supposed to go about substituting this solution into here. Could anybody give me some advice? Thanks.
    This is incomprehensible as written. You have a normal mode problem, and the usual approach is to use a trial solution of the form

    \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} Ae^{i\omega t} \\ Be^{i \omega t} \end{bmatrix}

    to create an eigenvalue problem - you will need a zero determinant for non-trivial solutions for A and B. If they're not talking about that, then I have no idea what they mean.

    In this case, you should end up with two normal modes, one with both oscillators moving in-phase, and the other with them moving 180 out of phase. The general solution for the motion is an arbitrary linear combination of those modes.
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    Is this further maths?
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    (Original post by Gregorius)
    Well that's a bit vague, isn't it! Is it simply asking you to guess a form of the function \psi to use as it says "a solution in the form...". If so, then just choose x_1 = \cos(\omega_1 t + \psi_1) and x_2 = \cos(\omega_2 t + \psi_2).
    Okay so I've done this...

    \vec{x}'' = A\vec{x}
    (\vec{\psi} \cos (\omega t + \phi))'' = A\cos (\omega t + \phi)
     -\omega ^2 \vec{\psi} \cos (\omega t + \phi) = A \vec{\psi} \cos (\omega t + \phi)
     - \omega ^2 \vec{\psi} = A \vec{\psi}
     (A+\omega ^2 I)\vec{\psi} = 0

    Is that right?
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    (Original post by Plagioclase)
    Okay so I've done this...

    \vec{x}'' = A\vec{x}
    (\vec{\psi} \cos (\omega t + \phi))'' = A\cos (\omega t + \phi)
     -\omega ^2 \vec{\psi} \cos (\omega t + \phi) = A \vec{\psi} \cos (\omega t + \phi)
     - \omega ^2 \vec{\psi} = A \vec{\psi}
     (A+\omega ^2)\vec{\psi} = 0

    Is that right?
    See here for some more info:

    http://www.thestudentroom.co.uk/show...31&postcount=3
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    (Original post by ServantOfMorgoth)
    Is this further maths?
    It's university level physics/applied maths.
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    (Original post by atsruser)
    It's university level physics/applied maths.
    Yeah cause I did this last week and I'm doing mechanical engineering lol

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    Well I did it, turned out it wasn't as hard as I thought, I just wasn't thinking sensibly about it. Thanks for everyone's help, I'll just post my solutions here in case anyone else has the same confusion...

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    (Original post by Plagioclase)
    Well I did it, turned out it wasn't as hard as I thought, I just wasn't thinking sensibly about it. Thanks for everyone's help, I'll just post my solutions here in case anyone else has the same confusion...

    Cool - note that the eigenvectors indicate the two modes that I mentioned earlier - in-phase or out-of-phase motion - by way of the signs.

    (BTW, you edited your earlier post a bit - it didn't make sense to me before, but I see what you are doing now. And did you ever get an answer to your entropy problem?)
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    (Original post by atsruser)
    Cool - note that the eigenvectors indicate the two modes that I mentioned earlier - in-phase or out-of-phase motion - by way of the signs.

    (BTW, you edited your earlier post a bit - it didn't make sense to me before, but I see what you are doing now. And did you ever get an answer to your entropy problem?)
    Thanks. And no, I haven't yet. However, I have arranged a thermodynamics tutorial when I get back to uni so hopefully I'll get it cleared up then, thanks for asking!
 
 
 
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