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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104 the THEORY OF screws. [Ill-
reference, we shall have for the intensity of the component of the resultant
wrench on wn—
6"0n + </>"</>„.
Hence the co-ordinates of the resultant wrench are proportional to
... 6"6s + </>"</>6.
1 or equilibrium this screw must be reciprocal to a, whence we have
P& (Ø'^ + </>"&) + ... + (0"0S + = 0,
or> = 0.
This equation merely expresses that the sum of the works done in a
small twist about a against the wrenches on 0 and is zero.
We also perceive that a given wrench may be always replaced by a
wrench of appropriate intensity on any other screw, in so far as the effect
on a body only free to twist about a is concerned.
It may not be out of place to notice the analogy which the equation just
written bears to the simple problem of the determination of the condition
that two forces should be unable to disturb the equilibrium of a particle
only free to move on a straight line. If P, Q be the two forces, and if I, m
be the angles which the forces make with the direction in which the particle
can move, then the condition is—
P cos I + Q cos m — 0.
This suggests an analogy between the virtual co-efficient of two screws,
and the cosine of the angle between two lines.
112. Particular Case.
If a body having freedom of the first order be in equilibrium under the
action of gravity, then the vertical through the centre of inertia must lie
in the plane of reciprocal screws of zero pitch, drawn through the centre
of inertia.
113. Impulsive Forces.
If an impulsive wrench of intensity act on the screw >/, while the
body is only permitted to twist about a, then we have seen in § 90 how the
twist velocity produced can be found. We shall now determine the impulsive
reaction ot the constraints. This reaction must be an impulsive wrench of
intensity X'" on a screw X, which is reciprocal to a. The determination of X
may be effected geometrically in the following manner:—Let p be the screw,
an impulsive wrench on which would, if the body were perfectly free, cause
an instantaneous twisting motion about a (§ 80). Draw the cylindroid (y, p).