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Das Doppelpendel:

 

Zwei masselose Stangen der Längen l1 und l2 verknüpft, an welchen 2 Massen, m1 und m2 hängen, bilden ein Doppelpendel.

Hier wird die Aufstellung der Bewegungsgleichungen unter Zuhilfenahme der 2. Lagrangegleichungen behandelt.

Generalisierte Koordinaten:

  1. Masse m1:
    \[x_{1}=\ell_{1}\sin\varphi\]

    \[y_{1}=-\ell_{1}\cos\varphi\]

    \[z_{1}=0\]
  2. Masse m2:
    \[x_{2}=\ell_{1}\sin\varphi+\ell_{2}\sin\psi\]

    \[y_{2}=-\ell_{1}\cos\varphi-\ell_{2}\cos\psi\]

    \[z_{2}=0\]

Geschwindigkeiten:

  1. Masse m1:
    \[\dot{x}_{1}=\dot{\varphi\,}\ell_{1}\cos\varphi\]

    \[\dot{y}_{1}=\dot{\varphi}\,\ell_{1}\sin\varphi\]
  2. Masse m2:
    \[\dot{x}_{2}=\dot{\varphi\,}\ell_{1}\cos\varphi+\dot{\psi}\,\ell_{2}\cos\psi\]

    \[\dot{y}_{2}=\dot{\varphi}\,\ell_{1}\sin\varphi+\dot{\psi\,}\ell_{2}\sin\psi\]
Lagrangefunktion aufstellen:

\[\mathfrak{L}=T-V=\frac{m_{1}}{2}(\dot{x}_{1}^{2}+\dot{y}_{1}^{2})+\frac{m_{2}}{2}(\dot{x}_{2}^{2}+\dot{y}_{2}^{2})-(m_{1}\, g\, y_{1}+m_{2}\, g\, y_{2})=\]


\[\frac{m_{1}}{2}(\dot{\varphi}^{2}\ell_{1}^{2}\cos^{2}\varphi+\dot{\varphi}^{\text{2}}\,\ell_{1}^{2}\sin^{\text{2}}\varphi)+\frac{m_{2}}{2}((\dot{\varphi\,}\ell_{1}\cos\varphi+\dot{\psi}\,\ell_{2}\cos\psi)^{2}+(\dot{\varphi}\,\ell_{1}\sin\varphi+\dot{\psi\,}\ell_{2}\sin\psi)^{2})\]


\[+m_{1}g\,\ell_{1}\cos\varphi+m_{2}g\,\ell_{1}\cos\varphi+m_{2}g\,\ell_{2}\cos\psi=\]


\[\frac{m_{1}}{2}(\dot{\varphi}^{2}\ell_{1}^{2})+\frac{m_{2}}{2}(\dot{\varphi}^{2}\ell_{1}^{2}\cos^{2}\varphi+\dot{\psi}^{2}\ell_{2}^{2}\cos^{2}\psi+2\dot{\varphi}\dot{\psi}\ell_{1}\ell_{2}\cos\varphi\cos\psi+\dot{\varphi}^{2}\ell_{1}^{2}\sin^{2}\varphi+\dot{\psi}^{2}\ell_{2}^{2}\sin^{2}\psi\]


\[\dot{\varphi}^{2}\ell_{1}^{2}\sin^{2}\varphi+\dot{\psi}^{2}\ell_{2}^{2}\sin^{2}\psi+2\dot{\varphi}\dot{\psi}\ell_{1}\ell_{2}\sin\varphi\sin\psi)+(m_{1}+m_{2})g\,\ell_{1}\cos\varphi+m_{2}g\,\ell_{2}\cos\psi=\]


\[\opaque\frac{m_{1}}{2}(\dot{\varphi}^{2}\ell_{1}^{2})+\frac{m_{2}}{2}(\dot{\varphi}^{2}\ell_{1}^{2}+\dot{\psi}^{2}\ell_{2}^{2})+m_{2}\ell_{1}\ell_{2}\dot{\varphi}\dot{\psi}\cos(\varphi-\psi)+(m_{1}+m_{2})g\,\ell_{1}\cos\varphi+m_{2}g\,\ell_{2}\cos\psi\]

Bewegungsgleichungen:

Lagrangegleichungen:

\[\frac{d}{dt}\frac{\partial\mathfrak{L}}{\partial\dot{q}_{i}}-\frac{\partial\mathfrak{L}}{\partial q_{i}}=0\]

  1. Gleichung für
    \[\varphi\]
    :

    \[\Rightarrow\frac{d}{dt}\frac{\partial\mathfrak{L}}{\partial\dot{\varphi}}-\frac{\partial\mathfrak{L}}{\partial\varphi}=\frac{d}{dt}\left\{ m_{1}(\dot{\varphi}\ell_{1}^{2})+m_{2}\ell_{1}\ell_{2}\dot{\psi}\cos(\varphi-\psi)\right\}\]

    \[-\left\{ -m_{2}\ell_{1}\ell_{2}\dot{\varphi}\dot{\psi}\sin(\varphi-\psi)-(m_{1}+m_{2})g\,\ell_{1}\sin\varphi\right\} =0\Leftrightarrow\]

    \[\opaque(m_{1}+m_{2})\ddot{\varphi}\ell_{1}^{2}+m_{2}\ell_{1}\ell_{2}\left[\ddot{\psi}\cos(\varphi-\psi)-\dot{\psi}\sin(\varphi-\psi)(\dot{\varphi}-\dot{\psi})\right]\]

    \[\opaque=-m_{2}\ell_{1}\ell_{2}\dot{\varphi}\dot{\psi}\sin(\varphi-\psi)-(m_{1}+m_{2})g\,\ell_{1}\sin\varphi\]
  2. Gleichung für
    \[\psi\]
    :

    \[\Rightarrow\frac{d}{dt}\frac{\partial\mathfrak{L}}{\partial\dot{\psi}}-\frac{\partial\mathfrak{L}}{\partial\psi}=\frac{d}{dt}\left\{ m_{2}\ell_{2}^{2}\dot{\psi}+m_{2}\ell_{1}\ell_{2}\dot{\varphi}\cos(\varphi-\psi)\right\}\]

    \[-\left\{ m_{2}\ell_{1}\ell_{2}\dot{\varphi}\dot{\psi}\sin(\varphi-\psi)-m_{2}g\,\ell_{2}\sin\psi\right\}=0\Leftrightarrow\]

    \[\opaque m_{2}\ell_{2}^{2}\ddot{\psi}+m_{2}\ell_{1}\ell_{2}\ddot{\varphi}\cos(\varphi-\psi)-m_{2}\ell_{1}\ell_{2}\dot{\varphi}\sin(\varphi-\psi)(\dot{\varphi}-\dot{\psi})\]

    \[\opaque -m_{2}\ell_{1}\ell_{2}\dot{\varphi}\dot{\psi}\sin(\varphi-\psi)+m_{2}g\,\ell_{2}\sin\psi\]

Video:

{youtube}z3W5aw-VKKA{/youtube}

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