A Treatise on the Theory of Screws
Forfatter: Sir Robert Stawell Ball
År: 1900
Forlag: The University Press
Sted: Cambride
Sider: 544
UDK: 531.1
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48 THE THEORY OF SCREWS. [52-
screw can be ascertained from the position of its corresponding point on the
circle.
Let us, for instance, seek the shortest distance between the two screws
A and B. Since all screws intersect the nodal axis of the cylindroid at
right angles, the required shortest distance is simply the difference between
the values of m sin 20 for the two screws: this is, of course, the difference
of their abscissae, i.e. the length BQ. Hence we have the following theorem :—
The shortest distance between two screws, A and B, is equal to the pro-
jection of the chord AB on the axis of pitch.
We thus see that eveiy screw j4 on the cylindroid must be intersected
by another screw A', and the chord AA' is, of course, perpendicular to
the axis of pitch. The ray through parallel to the axis of pitch, will give
two screws, L and M. These are the bounding screws of the cylindroid, and
in each a pair of intersecting screws have become coincident. The two
principal screws, U and V, lying on a diameter perpendicular to the axis of
pitch, must also intersect.
If all the pitches be reduced by p0, then the pitch axis passes through
the centre of the circle, and the case assumes a simple type. The extremities
of a chord perpendicular to the axis of pitch define screws of equal and
opposite pitches, and every pair of such screws must intersect. The screws
of zero pitch will then be the bounding screws, while the two principal
screws will have pitches 4- m and — m, respectively.
53. The Angle between two Screws.
This important function also admits of simple representation by the
corresponding circle. Let A, B (fig. 7) denote the two screws; then, if 0
and 0‘ be the angles corresponding to A and B,
AST =20; BST=20',
whence ASB = 2(0- 0f
If H be any point on the circle, then
AHB = 0-0',
and we deduce the following theorem:—
The angle between two screws is equal to the angle subtended in the circle
by their chord.
1 he extremities of a diameter denote a pair of screws at right angles:
thus, A , in fig. 7, is the one screw on the cylindroid which is at light angles
to A. The principal screws, U and V, are also seen to be at right angles.