Screws (Part 1): A Powerful Tool for Rigid Body Mechanics

1. Definition: What is a screw?

A screw is a mathematical representation of a vector field \(P \in {\cal E} \to \boldsymbol{v}_P\) which satisfies the relationship

\[ \boldsymbol{v}_Q = \boldsymbol{v}_P + \boldsymbol{V} \times {\bf r}_{PQ}\]

where vector \(\boldsymbol{V}\) is independent of position. Such fields play an important role in rigid body mechanics. We can take advantage of the fact that they are entirely defined by the knowledge of \(\boldsymbol{V}\) and \(\boldsymbol{v}_A\) (where \(A\) is a particular point) to use the following notation: \[ \{ {\cal V} \} = \begin{Bmatrix} \boldsymbol{V} \\ \boldsymbol{v}_A \end{Bmatrix} \qquad\qquad\quad (1) \] Vector \(\boldsymbol{V}\) is referred to as the resultant of screw \(\{ {\cal V} \}\). Vector \(\boldsymbol{v}_A\) is referred to as the moment of screw \(\{ {\cal V} \}\) about point \(A\). We can then define operations on the sets of screws (which is a vector space of dimension 6). Note that all vector fields satisfying \(\boldsymbol{v}_A\cdot{\bf r}_{AB}=\boldsymbol{v}_B\cdot{\bf r}_{AB}\) (equiprojectivity) necessarily define screws (the converse is of course true).

It is possible to define operations on the set of screws (addition and multiplication by a scalar) which can be shown to be a vector space (of dimension 6).

Note that the scalar quantity \(\boldsymbol{V}\cdot \boldsymbol{v}_A\) is invariant, that is, it is independent of the choice of point \(A\).

2. Example:

Given a right-handed basis \((\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}})\) of unit vectors of \(\cal E\), the screw \(\{ {\cal V} \}\) defined by \[ \{ {\cal V} \} = \begin{Bmatrix} \boldsymbol{\hat{\imath}}-\boldsymbol{\hat{k}} \\ 2\boldsymbol{\hat{\jmath}} \end{Bmatrix}_O \] is the vector field \(P \in {\cal E} \to \boldsymbol{v}_P = 2\boldsymbol{\hat{\jmath}}+ (\boldsymbol{\hat{\imath}}-\boldsymbol{\hat{k}})\times {\bf r}_{OP}\). Note that we must indicate the point (here point \(O\)) about which the screw has been resolved.

3. Two Special Classes of Screws:

Two types of screws (1) play a special role in mechanics:

1. Couples: A couple \(\{ {\cal C} \}\) is a screw with vanishing resultant: \[ \{ {\cal C} \} = \begin{Bmatrix} \boldsymbol{0} \\ \boldsymbol{v}_O \end{Bmatrix} \] This represents the uniform field: \(\boldsymbol{v}_P= \boldsymbol{v}_O\) for all points.

2. Sliders: A slider \(\{ {\cal S} \}\) is a screw of resultant \(\boldsymbol{S}\neq 0\) and with a vanishing moment about a particular point \(A\): \[ \{ {\cal S} \} = \begin{Bmatrix} \boldsymbol{S} \\ \boldsymbol{0} \end{Bmatrix}_A \] All points of line \((A,\boldsymbol{S})\) have vanishing moment. This line is referred to as the axis of slider \(\{ {\cal S} \}\).

4. Sum of Sliders:

It is possible to show that the sum of two sliders is a slider if and only if their axes intersect or are parallel.

5. General Representation of Screws

It can be shown that any screw \(\{ {\cal V} \}\) of non-zero resultant can be decomposed in a unique way as the sum of a slider \(\{ {\cal S} \}\) and a couple \(\{ {\cal C} \}\). The geometric object which defines \(\{ {\cal S} \}\) and \(\{ {\cal C} \}\) is the set of points \(Q\) whose moments are colinear to the resultant \(\boldsymbol{V}\) of screw \(\{ {\cal V} \}\): this set is in fact a straight line \(\Delta\) called screw axis whose direction is defined by vector \(\boldsymbol{V}\). All points \(Q\) of \(\Delta\) satisfy \(\boldsymbol{v}_Q = p \, \boldsymbol{V}\) where \(p\) (called pitch of the screw) is the scalar given by \[ p = \frac{\boldsymbol{V}\cdot \boldsymbol{v}_O}{\boldsymbol{V}\cdot\boldsymbol{V}} \] where point \(O\) is arbitrary (the scalar \(\boldsymbol{V}\cdot \boldsymbol{v}_O\) is an invariant). Then screw \(\{ {\cal V} \}\) can uniquely expressed as the sum \[ \{ {\cal V} \} = \underbrace{ \begin{Bmatrix} \boldsymbol{0} \\ p \, \boldsymbol{V} \end{Bmatrix}} _{\{ {\cal C} \}} \,+\, \underbrace{ \begin{Bmatrix} \boldsymbol{V} \\ \boldsymbol{0} \end{Bmatrix}_{H\in\Delta} }_{\{ {\cal S} \}} \qquad\qquad(2) \]

Note that, when \(p=0\), the screw is necessarily a slider and the screw axis is then the set of points \(Q\) satisfying \(\boldsymbol{v}_Q ={\bf 0}\).

Equation (2) yields a simple geometric representation of the values of the field \(P \in {\cal E} \to \boldsymbol{v}_P\) in relation to its screw axis \(\Delta\) as shown in this sketch:

There are many other mathematical properties satisfied by screws. In the next post, we will see the role played by screws in kinematics.