Section 4.1 Definition
We start with a definition.
Definition 4.1.1. Screw.
A screw, denoted \(\{ \cV \}\text{,}\) is a vector field \(P \in \cE \mapsto \bvv_P\) satisfying the relationship
\begin{equation}
\bvv_Q = \bvv_P + \bV \times \br_{PQ}\tag{4.1.1}
\end{equation}
between any two points \(P\) and \(Q\) of \(\cE\text{.}\) Vector \(\bV\) is called the resultant of screw \(\{\cV \}\text{.}\) The vector \(\bvv_P\) is called moment of screw \(\{ \cV \}\) about point \(P\text{.}\)
Hence screws can be construed as short-hand notations of vector fields which satisfy equation
(4.1.1) (field of moments). A screw is thus entirely specified by its resultant
\(\bV\) and by a moment
\(\bvv_A\) at a particular point
\(A\text{.}\)
The two quantities \((\bV, \bvv_A)\) are called the elements of reduction of the screw. To take this feature into account, a screw is written in the following array form
\begin{equation}
\{ \cV \} =
\left\{
\begin{array}{ll}
\bV \\
\bvv_A
\end{array}
\right\}
=
\left\{
\begin{array}{l}
V_1 \be_1 + V_2 \be_2 + V_3 \be_3 \\
v_{A1} \be_1 + v_{A2}\be_2 + v_{A3} \be_3
\end{array}
\right\}_A\tag{4.1.2}
\end{equation}
The upper element in this array is the resultant vector
\(\bV\) of the screw. The lower element denotes the moment about the particular chosen point
\(A\text{.}\) It is often necessary to specify (for instance, in the lower left side of the array) the particular point about which the moment has been determined. This is shown in the second equality of
(4.1.2) where screw
\(\{\cV\}\) is written in terms of the components of
\(\bV\) and
\(\bvv_A\) on a particular orthonormal basis
\((\be_1, \be_2 , \be_3)\) of
\(\cE\text{.}\) We then say that screw
\(\{\cV\}\) is resolved at point
\(A\text{.}\)
Example 4.1.2.
Consider the following vector field \(P \mapsto \bvv_P\) defined in space \(\cE (O, \be_1,
\be_2 , \be_3)\) by
\begin{equation*}
v_1 = 1+ x_2 + \lambda x_3, \quad v_2 = 2\lambda - \lambda x_3 -x_1,
\quad
v_3 = -1 - \lambda x_1 + \lambda^2 x_2
\end{equation*}
where \((x_1,x_2,x_3)\) are the Cartesian coordinates of point \(P\) and where \(\lambda\) is a parameter. Show that there exist values of \(\lambda\) for which this vector field defines a screw. In each case, find the corresponding resultant \(\bV\text{.}\)
Solution.
Note that the mapping
\(P(x_1,x_2,x_3) \mapsto \bvv_P\) must necessarily be linear in
\((x_1,x_2,x_3)\) to define a screw. Indeed field
\(\bvv\) must satisfy
\(\bvv_P = \bvv_O + \bV\times \br_{OP}\text{.}\) Of course, being linear is not sufficient. We must show the fundamental property
(4.1.1). Given two arbitrary points
\(P (x_1,x_2,x_3)\) and
\(P' (x'_1 ,x'_2,x'_3)\text{,}\) the identity
\(\bvv_{P'} - \bvv_P = \bV \times \br_{PP'}\) can be written as
\begin{equation*}
\begin{array}{lll}
\lambda (x_3'-x_3) +(x_2'-x_2) = V_2 (x_3'-x_3) - V_3 (x_2'-x_2) \\
-\lambda (x_3'-x_3) - (x_1'-x_1) = V_3 (x_1'-x_1) - V_1 (x_3'-x_3) \\
-\lambda (x_1'-x_1) + \lambda^2 (x_2'-x_2) = V_1 (x_2'-x_2) -V_2 (x_1'-x_1)
\end{array}
\end{equation*}
where \((V_1, V_2, V_3)\) are the components of vector \(\bV\text{,}\) yet to be found. We can easily see by inspection that these equations are satisfied for arbitrary \(P\) and \(P'\) if only if \(\lambda = 0\) or \(\lambda = 1\text{.}\) For these values of \(\lambda\text{,}\) the vector field \(\bvv_P\) defines a screw \(\{\cV\}\) whose resultant is
\begin{equation*}
\bV = -\be_3 \quad (\lambda = 0) , \qquad
\bV = \be_1+ \be_2 -\be_3 \quad (\lambda = 1)
\end{equation*}
Screw \(\{\cV\}\) can then be written as
\begin{equation*}
\{ \cV \} =
\left\{
\begin{array}{ll}
-\be_3 \\
\be_1 - \be_3
\end{array}
\right\}_O \quad (\lambda = 0) , \qquad
\{ \cV \} =
\left\{
\begin{array}{ll}
\be_1+ \be_2 -\be_3 \\
\be_1 +2\be_2 - \be_3
\end{array}
\right\}_O \quad (\lambda = 1)
\end{equation*}
where we have chosen point \(O\) to resolve \(\{\cV\}\text{,}\) although any other point can be considered.
