Fundamental Decomposition

Section 5.2 Fundamental Decomposition

The velocity of any point of \(\cB\) can be written as the sum of two orthogonal vectors (see Figure 5.2.2):

\begin{equation} \vel_{P \in \cB / \cA} = p \, \bom_{\cB/\cA} + \bom_{\cB /\cA} \times \br_{HP}\tag{5.2.1} \end{equation}

where point \(H\) denotes the projection of \(P\) onto instantaneous screw axis \(\Delta\text{.}\) Note that, in the decomposition of equation (5.2.1),

the first vector is independent of \(P\text{:}\) it is the (instantaneous) velocity of points located on axis \(\Delta\text{,}\)
the second vector is the (instantaneous) velocity of a point in rotation about \(\Delta\text{:}\) any point of \(\Delta\) can be substituted for point \(H\) in equation (5.2.1).

Theorem 5.2.1. Fundamental Decomposition of the Kinematic Screw.

At any instant for which instantaneous screw axis \(\Delta\) is not the empty set, the motion of \(\cB\) relative to \(\cA\) can be decomposed as the sum of a translation along and a rotation about \(\Delta\text{,}\) corresponding to the unique decomposition of the kinematic screw \(\{ \cV_{\cB / \cA} \}\) according to

\begin{equation} \{ \cV_{\cB / \cA} \} = \underbrace{ \left\{ \begin{array}{ll} \bze \\ p \bom_{\cB/\cA} \end{array} \right\} }_{\text{translation along }\Delta} + \underbrace{ \left\{ \begin{array}{ll} \bom_{\cB/\cA} \\ \bze \end{array} \right\}_{H\in \Delta} }_{\text{rotation about }\Delta}\tag{5.2.2} \end{equation}

Figure 5.2.2. The velocity field in relation to the instantaneous screw axis.

Remark 5.2.3.

At any given time, the minimum speed \(|\vel_{P\in\cB /\cA}|\) of points of \(\cB\) is reached on the instantaneous screw axis \(\Delta\text{.}\)

Remark 5.2.4.

For points far away from \(\Delta\text{,}\) the contribution of the term \(p\,\bom_{\cB/\cA}\) becomes negligible compared to that of \(\bom_{\cB /\cA} \times \br_{HP}\text{:}\) far away from axis \(\Delta\text{,}\) points of \(\cB\) are approximately in rotation about \(\Delta\text{,}\) that is, \(\vel_{P\in\cB /\cA}\approx \bom_{\cB /\cA} \times \br_{HP}\text{.}\)

Remark 5.2.5.

Figure 5.2.2 shows the distribution of velocity vectors of points along a line \(PQ\) of plane \(\Pi\) perpendicular to \(\Delta\text{.}\) This distribution is invariant by translation along \(\Delta\text{.}\) Furthermore, the velocity distribution along any line \(P'Q'\) of plane \(\Pi\) obtained by rotation about \(\Delta\) of line \(PQ\) is found by the same rotation of the distribution along \(PQ\text{.}\)

In conclusion, whenever \(\bom_{\cB/\cA}\neq \bze\text{,}\) rigid body \(\cB\) appears to be in instantaneous helical motion about axis \(\Delta\) relative to \(\cA\text{.}\) This characterization of the motion of \(\cB\) only applies for its velocity field. The acceleration field \(P\in\cB \mapsto \ba_{P \in\cB /\cA}\) of \(\cB\) is not that of a helical motion about \(\Delta\text{.}\)

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