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Math Help - second order problem

  1. #1
    ain
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    second order problem

    I have a dynamic model like below:

    xddot = u1;
    thetaddot=u2;
    yddot=u1 tan (theta) + (xdot tan (theta) - ydot)

    I want to convert it into the second order chained form as:
    y1ddot = v1;
    y2ddot = v2;
    y3ddot = y2v1 + (additional terms)

    hope someone can give an idea how to start to get the derivation in second order chained form pattern.
    Thank you very much.
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  2. #2
    A Plied Mathematician
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    So, as I understand it, your current problem is this:

    \ddot{x}=u_{1}

    \ddot{\theta}=u_{2}

    \ddot{y}=u_{1}\tan(\theta)+\dot{x}\tan(\theta)-\dot{y},

    and you want to convert to

    \ddot{y}_{1}=v_{1}

    \ddot{y}_{2}=v_{2}

    \ddot{y}_{3}=y_{2}v_{1}+\dots

    Is that correct? If so, can I ask if you have the freedom to assign the y_{k} and v_{k}? If so, I would go with the following assignments:

    y_{1}=x

    v_{1}=u_{1}

    y_{2}=\tan(\theta).

    Differentiating y_{2} yields the following:

    \dot{y}_{2}=\sec^{2}(\theta)\,\dot{\theta}, and

    \ddot{y}_{2}=\sec^{2}(\theta)(\ddot{\theta}+2 \dot{\theta}^{2}\tan(\theta))=\sec^{2}(\theta)(u_{  2}+2\dot{\theta}^{2}\tan(\theta)).

    So, let

    v_{2}=\sec^{2}(\theta)(u_{2}+2\dot{\theta}^{2}\tan  (\theta)), and

    y_{3}=y.

    We would thus have the following system:

    \ddot{y}_{1}=v_{1}

    \ddot{y}_{2}=v_{2}

    \ddot{y}_{3}=v_{1}y_{2}+\dot{y}_{1}y_{2}-\dot{y}_{3}.

    Is that what you were seeking?
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  3. #3
    ain
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    Dear Sir Ackbeet,

    You are correct. That's what i'm seeking. I still need help to convert another system to that form, but I will post that in another thread.

    Thank you very much Sir.

    Quote Originally Posted by Ackbeet View Post
    So, as I understand it, your current problem is this:

    \ddot{x}=u_{1}

    \ddot{\theta}=u_{2}

    \ddot{y}=u_{1}\tan(\theta)+\dot{x}\tan(\theta)-\dot{y},

    and you want to convert to

    \ddot{y}_{1}=v_{1}

    \ddot{y}_{2}=v_{2}

    \ddot{y}_{3}=y_{2}v_{1}+\dots

    Is that correct? If so, can I ask if you have the freedom to assign the y_{k} and v_{k}? If so, I would go with the following assignments:

    y_{1}=x

    v_{1}=u_{1}

    y_{2}=\tan(\theta).

    Differentiating y_{2} yields the following:

    \dot{y}_{2}=\sec^{2}(\theta)\,\dot{\theta}, and

    \ddot{y}_{2}=\sec^{2}(\theta)(\ddot{\theta}+2 \dot{\theta}^{2}\tan(\theta))=\sec^{2}(\theta)(u_{  2}+2\dot{\theta}^{2}\tan(\theta)).

    So, let

    v_{2}=\sec^{2}(\theta)(u_{2}+2\dot{\theta}^{2}\tan  (\theta)), and

    y_{3}=y.

    We would thus have the following system:

    \ddot{y}_{1}=v_{1}

    \ddot{y}_{2}=v_{2}

    \ddot{y}_{3}=v_{1}y_{2}+\dot{y}_{1}y_{2}-\dot{y}_{3}.

    Is that what you were seeking?
    Last edited by Ackbeet; May 19th 2011 at 07:48 PM.
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  4. #4
    A Plied Mathematician
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    You're welcome!
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