A third conformation with the screw systems: Finite twist displacements of a directed line and point

Abstract

It is well-known that physical wrenches and screw-velocities, when added to form a resultant, conform with the linear combination rules applicable to screws which are expressed in Plücker co-ordinates. This paper establishes that, in a certain generic situation which occupies a theoretically central position, the axes of finite twist displacementshave similar conformation with linear combinations of screws. The situation referred to is that in which a directed line with an identified point upon it is displaced from one spatial location to another. The displacement is considered to arise solely as the result of a finite twisting motion which consists of a translation along and a rotation about some suitable line as twist-axis. The paper shows that, when invested with an appropriate measure of pitch, the twist-axis screws on such lines form a linearly dependent set, namely, a 2-system of screws. The appropriate measure of pitch for a finite twist is found to be half-translation distance tan(half-rotation angle) , in distinction from the standard measure translation distance/rotation angle which has previously been proposed for use. The new measure is shown here to have a distribution which conforms with the generator pitches of the relevant 2-system and gives other indications that, for theoretical purposes at least, it is the more suitable measure for use. A straightforward means if given for deriving two basis screws of the relevant 2-system (and hence, by linear combination, all other twist-axis screws) for the situation in which two directed lines with points are given as specifying a displacement of the kind prescribed.