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

Screws form a special class of vector fields (known as fields of moments). The fields of velocity, angular momentum, dynamic moment, and moments of mechanical actions all define screws. Screws provide a simple formalism which unifies all aspects of 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.