Extensions to System of Rigid Bodies

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Section 9.8 Extensions to System of Rigid Bodies

Unlike kinematic screws, kinetic and dynamic screws can be defined for systems of rigid bodies. This is not surprising since these quantities were first defined for arbitrary material systems of constant mass.

Theorem 9.8.1. Kinetic/Dynamic Screws and Kinetic Energy of a System of Rigid Bodies.

The kinetic screw, the dynamic screw and the kinetic energy of a system $\Sigma$ of $N$ rigid bodies $\{ \cB_1 , \cB_2 , \ldots \cB_N \}$ relative to a referential $\cE$ are given by

\begin{equation} \{ \cH_{\Sigma / \cE} \} = \{ \cH_{\cB_1 / \cE} \} +\{ \cH_{\cB_2 / \cE} \}+ \cdots + \{ \cH_{\cB_N / \cE} \}\tag{9.8.1} \end{equation}

\begin{equation} \{ \cD_{\Sigma / \cE} \} = \{ \cD_{\cB_1 / \cE} \} +\{ \cD_{\cB_2 / \cE} \}+ \cdots + \{ \cD_{\cB_N / \cE} \}\tag{9.8.2} \end{equation}

\begin{equation} \kin_{\Sigma / \cE} = \kin_{\cB_1 / \cE} + \kin_{\cB_2 / \cE} + \cdots + \kin_{\cB_N / \cE} \tag{9.8.3} \end{equation}

Proof.

This is simply a consequence of writing the integral $\int_\Sigma\br_{AP} \times {\bf u}_P \, dm $ as the sum $\int_{\cB_1} \br_{AP} \times {\bf u}_P \,dm + \int_{\cB_2} \br_{AP} \times {\bf u}_P \, dm + \cdots + \int_{\cB_N} \br_{AP} \times {\bf u}_P \, dm\text{.}$ The same applies to the kinetic energy of $\Si\text{.}$

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