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Multibody Dynamics

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Equation of Motion

[a][\ddot{x}] + [b][\dot{x}] + [c][\dot{x}] = [f]
Let \dot{x} = \dot{x}_0 + \ddot{x}t
[a][\ddot{x}] + ( [b] + [c] ) ( \dot{x}_0 + \ddot{x}t ) = [f]
\left( [a] + t([b] + [c]) \right) \ddot{x} = [f] - ([b] + [c]) \dot{x}_0

Derivative of Equation of Motion

Let [E] = [a] + t([b] + [c])
Then Derivative of Equation of Motion: [E] \frac{d\ddot{x}}{dt}

Ex.
Let \theta = \frac{d\ddot{x}}{dt} = \frac{\ddot{x} - \ddot{x}_0}{t}

\begin{bmatrix} E_{11} && E_{12} \\ E_{21} && E_{22} \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \begin{bmatrix} E_{11}\theta_1 + E_{12}\theta_2 \\ E_{21}\theta_1 + E_{22}\theta_2 \end{bmatrix}

E_{11}\theta_1 = \begin{bmatrix} e_{00} && e_{01} && e_{02} \\ e_{10} && e_{11} && e_{12} \\ e_{20} && e_{21} && e_{22} \end{bmatrix} \begin{bmatrix} \theta_{1x} \\ \theta_{1y} \\ \theta_{1z} \end{bmatrix} = \begin{bmatrix} e_{00}\theta_{1x} + e_{01}\theta_{1y} + e_{02}\theta_{1z} \\ \vdots \\ \vdots \end{bmatrix}

Force

Forces exerted on a system by bodies external to the system and forces exerted between bodies of the system are called "active" (or "applied") forces.
Forces exerted on the system due to accelerations of its particles are called "inertia" (or "passive") forces.

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