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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142
THE THEORY OF SCREWS.
[153-
153. Law of Distribution of va.
As we follow the screw a ai’ound the circle, it becomes of interest to study
the corresponding variations of the linear magnitude va. We have already
found a very concise representation of pa and wa by the axis of pitch and the
axis of inertia, respectively. We shall now obtain a similar representation
of va by the aid of the axis of potential.
It is shown (§ 102) that must be a quadratic function of the co-
ordinates; we may therefore apply to this function the same reasoning as
we applied to uA (§ 134). We learn that is at each point proportional to
the perpendicular on a ray, which is the axis of potential.
Thus, if A (Fig. 32) be the screw, the value of va2 is proportional to AP,
the perpendicular on PT; if 0" be the pole of the axis of potential, then,
as in § 59, we can also represent the value of i>„- by the product AO". A A'.
154. Conjugate Screws of Potential,
In general the energy expended by a small twist from a position of
equilibrium can be represented by a quadratic function of the co-ordinates
of the screw. If, moreover, the two screws of reference form what are
called conjugate screws of potential (§ 100), then the energy is simply the
sum of two square terms. The necessary and sufficient condition that the
two screws shall be so related is, that their chord shall pass through 0".
Another property of two conjugate screws of potential is also analogous
to that of two conjugate screws of inertia. If A and A' be two conjugate
screws of potential, then the wrench evoked by a twist round A is reciprocal
to A', and the wrench evoked by a twist around A' is reciprocal to A.