A Special Class of Screws

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Section 4.8 A Special Class of Screws

We can generalize the notion of screw defined as the discrete sum of bound vectors (sliders) to the case of a continuous distribution of bound vectors over a region \(\Si\) of a Euclidean space \(\cal E\text{.}\) In practice, \(\Si\) represents a material system (a continuum). Given a vector field \(P\in \Si \mapsto \bvv_P\) defined over \(\Si\text{,}\) consider the following field

\begin{equation} A \in \cE \mapsto \bM_A = \int_\Si \br_{AP} \times\bvv_P \; dV\tag{4.8.1} \end{equation}

where \(dV\) is an infinitesimal volume element at point \(P\text{.}\) It is easy to show that equation (4.8.1) defines a field of moments, and hence a screw whose resultant is given by

\begin{equation} \bR = \int_\Si \bvv_P \; dV\tag{4.8.2} \end{equation}

Indeed, we have

\begin{equation*} \bM_B = \int_\Si \br_{BP} \times\bvv_P dV = \int_\Si (\br_{BA}+\br_{AP})\times\bvv_P dV = \bM_A + \bR \times \br_{AB} \end{equation*}

This definition can also be extended to the case of vector fields defined over a surface or a curve of \(\cE\text{.}\)

Several fundamental screws of this type will be defined in later chapters, \(\Si\) representing one or more rigid bodies.