Section 4.2 Basic Operations on Screws
We can define basic operations of the set of screws.
Definition 4.2.1. Equality of two Screws.
Two screws \(\{ \cV \}\) and \(\{ \cW \}\) are equal if their resultants are equal and if there exists a point \(P\) about which their moments are the same vector:
\begin{equation}
\bV = \bW , \qquad \bvv_P = \bww_P\tag{4.2.1}
\end{equation}
It is easy to show that \(\bvv_Q = \bww_Q \) at any other point \(Q\) of \(\cE\text{.}\)
Definition 4.2.2. Sum of two Screws.
The sum \(\{ \cV \} + \{ \cW \}\) of two screws is defined as the screw of resultant \(\bV + \bW\) and whose moment at a particular point \(P\) is \(\bvv_P + \bww_P\text{.}\)
\(\{ \cV \} + \{ \cW \}\) is indeed a screw, since it is readily shown that equation
(4.1.1) is satisfied.
Definition 4.2.3. Multiplication by a scalar.
Given a real scalar \(\lambda\) and a screw \(\{ \cV \} \) of resultant \(\bV\) and moment \(\bvv_P\) about point \(P\text{,}\) screw \(\lambda \{\cV \}\) is defined as the screw of resultant \(\lambda \bV\) and moment \(\lambda \bvv_P\) about point \(P\text{.}\)
These operations endow the set of screws with the structure of a vector space (of dimension 6). In this space, the zero element (corresponding to the vector field \(P\mapsto \bze\)) is denoted \(\{ 0 \}\text{.}\)
Example 4.2.4.
Consider the following two screws defined on the Euclidean space \(\cE (O, \be_1, \be_2 , \be_3)\)
\begin{equation*}
\{ \cV \} = \left\{
\begin{array}{ll}
\be_1 + \be_3 \\
\be_1
\end{array}
\right\}_O , \qquad
\{ \cW \} =
\left\{
\begin{array}{ll}
4\be_1 -\be_3 \\
\be_1 +2 \be_3
\end{array}
\right\}_A
\end{equation*}
where point \(A\) is defined by \(\br_{OA} = \be_1 \text{.}\) Find screw \(\{ \cV \} + 2\{ \cW \} \) resolved at \(O\text{.}\)
Solution.
The resultant is found to be \(\bV + 2\bW = 9 \be_1 -\be_3\text{.}\) To find a moment, we may choose point \(O\) or \(A\text{:}\) if we choose \(O\text{,}\) we need to find \(\bww_O = \bww_A + \bW \times \br_{AO} =\be_1 +2 \be_3
+ (4\be_1 -\be_3)\times (- \be_1) = \be_1 + \be_2 + 2 \be_3\text{.}\) Then, the moment of \(\{ \cV \} + 2\{ \cW \} \) about \(O\) is \(\bvv_O +2 \bww_O = 3 \be_1 + 2\be_2 + 4 \be_3\text{:}\)
\begin{equation*}
\{ \cV \} + 2\{ \cW \} = \left\{
\begin{array}{ll}
9 \be_1 -\be_3 \\
3 \be_1 + 2\be_2 + 4 \be_3
\end{array}
\right\}_O
\end{equation*}
\(\danger\)The error to avoid is to determine the moment of \(\{ \cV \} + 2\{ \cW \}\) about \(O\) as the sum \(\bvv_O +2 \bww_A\text{.}\)
