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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36)
SCREW CO-ORDINATES.
35
When the co-ordinates of a screw are given, the screw itself may be thus
determined. Let e be any small quantity. Take a body in the position J,
and impart to it successively twists about each of the screws of reference of
amplitudes ea,, ea2, ... ea6. Let the position thus attained be B; then the
twist which would bring the body directly from J to B is about the required
screw a.
35. Identical Relation.
The six co-ordinates of a screw are not independent quantities, but
fulfil one relation, the nature of which is suggested by the relation between
three direction cosines.
When two twists are compounded by the cylindroid (§ 14), it will be
observed that the amplitude of the resultant twist, as well as the direction
of its screw, depend solely on the amplitudes of the given twists, and the
directions of the given screws, and not at all upon either their pitches or their
absolute situations. So also when any number of twists are compounded, the
amplitude and direction of the resultant depend only on the amplitudes and
directions of the components. We may, therefore, state the following general
principle. If n twists neutralize (or 7i. wrenches equilibrate) then a closed
polygon of n sides can be drawn, each of the sides of which is proportional
to the amplitude of one of the twists (or intensity of one of the wrenches),
and parallel to the corresponding screw.
Let an, bn, cn, be the direction cosines of a line parallel to any screw of
reference co,,, and drawn through a point, through which pass three rect-
angular axes.
Then since a unit wrench on a has components of intensities alt... a6,
we must have
(Mi + ... + a6a„)2 + (&,a, + ... + b.a^2 + (w + ... + c6«8)2 = 1,
whence S«i2 + SSa^cos (12) = 1,
if we denote by cos (12) the eosine of the angle between two straight lines
parallel to &>i and a>2.
36. Calculation of Co-ordinates.
We may conceive the formation of a table of triple entry from which the
virtual coefficient of any pair of screws may be ascertained. The three argu-
ments will be the angle between the two screws, the perpendicular distance,
and the sum of the pitches. These arguments having been ascertained by
ordinary measurement of lines and angles, the virtual coefficient can be
extracted from the tables.
Let a be a screw, of which the co-ordinates are to be determined. The