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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192
THE THEORY OF SCREWS.
[193-
193. Quadric of the Potential.
A body which has freedom of the third order is in equilibrium under
the influence of a conservative system of forces. The body receives a twist
of small amplitude 3 about a screw 3 of the screw system. It is required
to determine a geometrical representation for the quantity of work which
has been done in effecting the displacement. We have seen that to each
screw 3 corresponds a certain linear parameter vg (§ 102), and that the work
done is represented by
Fvg3'-.
We have also seen that the quantity vg may.be represented by
v^ + v^+v.^-
where 3lt 32, 0S are the co-ordinates of the screw 3 referred to three conjugate
screws of the potential, and v2, v3, denote the values of ve for each of the
three screws of reference (§ 102).
Di awing through the centre of the pitch quadric three axes parallel to
the three screws of reference, we can then construct the quadric of which
the equation is
if a? + vff + v32z2 = H,
which proves the following theorem :—
The work done in giving the body a twist of given amplitude from a
position of equilibrium about any screw of a system of the third order, is
proportional to the inverse square of the parallel diameter of a certain
quadric which we may call the quadric of the potential, and three conjugate
diameters of this quadric are parallel to three conjugate screws of the
potential in the screw system.
194. The Principal Screws of the Potential.
The three common conjugate diameters of the pitch hyperboloid, and
the quadric of the potential, are parallel to three screws of the system
which we call the principal screws of the potential. If the body be
displaced by a twist about a principal screw of the potential from a
position of stable equilibrium, then the reduced wrench which is evoked
is upon the same screw.
The three principal screws of the potential must not be confounded with
the three screws of the system which are parallel to the principal axes of
the ellipsoid of the potential. The latter are the screws on which a twist
of given amplitude requires a maximum or minimum consumption of
energy, and they are rectangular, which, of course, is not in general the
case with the principal screws of the potential.