The Dynamic Screw of a Material System

Section 9.3 The Dynamic Screw of a Material System

Definition 9.3.1. Dynamic Screw.

The dynamic screw of material system \(\Si\) relative to referential \(\cE\) is the screw denoted \(\{ \cD _{ \Si / \cE } \}\) corresponding to the vector field \(A \mapsto \bD_{A, \Si / \cE} = \int_{\Si} \br_{AP} \times \ba_{P/\cE} \, dm\text{.}\) Its resultant is the vector

\begin{equation} \int_{\Si} \ba_{P/\cE} \, dm = m \ba _{G/\cE} \tag{9.3.1} \end{equation}

Moment \(\bD_{A, \Si / \cE}\) is called dynamic moment of \(\Si\) relative to \(\cE\) about point \(A\text{.}\)

Again, we may denote dynamic moments simply as \(\bD_{A, \Si}\text{,}\) or even \(\bD_A\text{.}\) Dynamic moments about any two points are related by the formula

\begin{equation} \bD_{A} = \bD_{B} + \br_{AB} \times m \ba _{G/\cE}\tag{9.3.2} \end{equation}

The dynamic screw plays a fundamental role in the laws of motion of material systems, as will be seen in Chapter 11. Its determination is easily made by relating the dynamic moment to the corresponding angular momentum. This relationship is defined in the following theorem.

Theorem 9.3.2. Dynamic Screw from Kinetic Screw.

The dynamic screw of system \(\Si\) is the time-derivative of its kinetic screw:

\begin{equation} \{ \cD _{ \Si / \cE } \} = \begin{Bmatrix} \frac{d}{dt}\cH _{ \Si / \cE } \end{Bmatrix}_\cE\tag{9.3.3} \end{equation}

according to definition of the time-derivative of a screw. Hence, dynamic moment \(\bD_{A}\) about arbitrary point \(A\) can be found from angular momentum \(\bH_{A}\) according to

\begin{equation} \bD_{A} = \left( \frac{d\bH_{A}}{dt} \right)_{\cE} + \vel_{A/\cE} \times m \vel_{G/\cE} \tag{9.3.4} \end{equation}

Proof.

We can relate dynamic moment to angular momentum by taking the time derivative (relative to \(\cE\)) of the expression \(\bH_{A} = \int_{\Si} \br_{AP} \times \vel_{P/\cE} \, dm\) to obtain (using property (9.1.1))

\begin{align*} {d \over dt}\bH_{A} \amp = \int_{\Si} {d \over dt}(\br_{AP} \times \vel_{P/\cE}) \, dm \\ \amp = \int_{\Si} ( \vel_{P/\cE} - \vel_{A/\cE} ) \times \vel_{P/\cE} \, dm + \int_{\Si} \br_{AP} \times \ba_{P/\cE} \, dm \end{align*}

The first integral is recognized to be \(m \vel_{G/\cE} \times \vel_{A/\cE}\) since \(\int_{\Si} \vel_{P/\cE} \, dm = m \vel_{G/\cE}\) as obtained by taking the time derivative of \(m \br_{OG}\text{.}\) The second integral is recognized to be the dynamic moment \(\bD_{A}\) about point \(A\text{.}\) Finally, we obtain for any point \(A\) (in motion or not) the following formula (9.3.4).

The r.h.s. of (9.3.4)} is precisely the moment of screw \(\Big\{\frac{d}{dt}\cH _{ \Si / \cE } \Big\}_\cE\) about \(A\text{.}\) It is straightforward to prove the equality of the resultants in (9.3.3).

\(\danger\) Whenever formula (9.3.4) is applied to a rigid body \(\cB\text{,}\) velocity \(\vel_{A/\cE}\) must be found as \((d\br_{OA} / dt )_{\cE}\) and not be confused with \(\vel_{A\in \cB /\cE}\text{:}\) point \(A\) is not necessarily attached to \(\cB\text{.}\)

Remark 9.3.3.

Formula (9.3.4) is important in practice: it shows that the dynamic moments are easily found from the corresponding angular momenta.

Remark 9.3.4.

Note the following special cases of equation (9.3.4):

If point \(A\) is fixed relative to \(\cE\text{,}\) \(\vel_{A/\cE}=\bze\text{.}\) Then
\begin{equation} \bD_{A} = \left( \frac{d\bH_{A}}{dt} \right)_{\cE} \tag{9.3.5} \end{equation}
If \(A\) is the mass center \(G\text{,}\) then \(\vel_{A/\cE} \times m \vel_{G/\cE} = \bze\) leading to
\begin{equation} \bD_{G} = \left( \frac{d\bH_{G}}{dt} \right)_{\cE} \tag{9.3.6} \end{equation}

Remark 9.3.5.

If we combine equations (9.3.6) and (9.3.2) of the dynamic screw, we obtain the relationship

\begin{equation} \bD_{A} =\left( \frac{d\bH_{G}}{dt} \right)_{\cE} + \br_{AG} \times m \ba_{G/\cE}\tag{9.3.7} \end{equation}

This relationship is very useful since it allows the determination of any dynamic moment from the knowledge of kinematics and kinetics about the mass center \(G\text{.}\)

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