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 $$\Delta$$.