Screws (Part 2): Application to Kinematics

1. Definition: Kinematic Screw

Since the velocity field \(P\in {\cal B} \mapsto {\bf v}_{P/{\cal E}}\) of a rigid body \({\cal B}\) satisfied \({\bf v}_{Q/{\cal E}} = {\bf v}_{P/{\cal E}} + \boldsymbol{\omega}_{{\cal B}/{\cal E}}\times {\bf r}_{PQ}\), it defines a screw, called kinematic screw of body \({\cal B}\) relative to referential \({{\cal E}}\). Recall that \(\boldsymbol{\omega}_{{\cal B}/{\cal E}}\) is the angular velocity of \({\cal B}\) relative to \({{\cal E}}\), defined as the unique vector satisfying \[ \Big(\frac{d{\bf U}}{dt}\Big)_{\cal E}= \boldsymbol{\omega}_{{\cal B}/{\cal E}}\times {\bf U} \] for all vectors \(\bf U\) fixed relative to \({\cal B}\).

Notation: \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \\ {\bf v}_{A/{\cal E}} \end{Bmatrix} \] Recall that:

• the scalar \(\boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot{\bf v}_{A/{\cal E}}\) is an invariant, that is, it is independent of the choice of point \(A\in{\cal B}\).

• the velocity field is equiprojective: \({\bf v}_{A/{\cal E}}\cdot{\bf r}_{AB} = {\bf v}_{B/{\cal E}}\cdot{\bf r}_{AB}\) for any two points \(A\) and \(B\) of \({\cal B}\).

2. Examples

A body \({\cal B}\) in translational motion relative to \({\cal E}\) is characterized by the following kinematic screw (a couple in language of screws) \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} {\bf 0} \\ {\bf v}_A \end{Bmatrix} \] A body \({\cal B}\) in rotational motion relative \({\cal E}\) is characterized by the following kinematic screw (a slider in language of screws) \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} \boldsymbol{\omega} \\ {\bf 0} \end{Bmatrix}_A \] The axis \(\Delta (A, \boldsymbol{\omega})\) represents the axis of rotation of the body. If this representation is only valid at a specific time, the body will be said to be in instantaneous rotation.

3. Instantaneous Screw Axis:

If angular velocity \(\boldsymbol{\omega}_{{\cal B}/{\cal E}}\neq {\bf 0}\) at a given instant, then the kinematic screw can be characterized by the set \(\Delta\) of points of \({\cal B}\) which satisfy \({\bf v}_{Q/{\cal E}} = p \boldsymbol{\omega}_{{\cal B}/{\cal E}}\): we call this set the instantaneous screw axis of the body. The pitch \(p\) is given by \(p = \boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot{\bf v}_{A/{\cal E}}/\boldsymbol{\omega}_{{\cal B}/{\cal E}}^2\).

Then kinematic screw \(\{ {\cal V}_{{\cal B}/{\cal E}} \}\) can be uniquely represented as the sum \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \underbrace { \begin{Bmatrix} {\bf 0} \\ p \, \boldsymbol{\omega}_{{\cal B}/{\cal E}} \end{Bmatrix} }_{\text{translation along }\Delta} +\, \underbrace { \begin{Bmatrix} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \\ {\bf 0} \end{Bmatrix}_{H\in\Delta} }_{\text{rotation about }\Delta} \qquad\qquad (1) \] This sum shows that body \({\cal B}\) is in instantaneous helical motion about \(\Delta\). Axis \(\Delta\) is neither fixed in \({\cal B}\) nor in \({\cal E}\). Hence, this characterization of the motion of \({\cal B}\) is only true for the velocity field (rather that the acceleration field). At any given time, the distribution of velocities of points located in a plane \(\Pi\) perpendicular to \(\Delta\) can be sketched in accordance to (1):

This distribution is invariant under a translation along \(\Delta\) or under a rotation about \(\Delta\).