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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188
THE THEORY OF SCREWS.
[186-
If we take three conjugate screws of inertia from the screw system as
screws of reference, then we have seen (§ 97) that, if 0lt 02, 0S, be the co-
ordinates of a screw 0, we have
= uf0^ + u.?0.f + u202,
where ult u2, us are the values of ue with reference to the three conjugate
screws of inertia.
Draw from any point lines parallel to 0, and to the three conj ugate screws
of inertia. If then a parallelepiped be constructed of which the diagonal is
the line parallel to 0, and of which the three lines parallel to the conjugate
screws are conterminous edges, and if r be the length of the diagonal, and
x, y, z the lengths of the edges, then we have
We see, therefore, that the parameter w appropriate to any screw 0 is
inversely proportional to the parallel diameter of the ellipsoid
u?a? + u.?y- + ujz2 = H,
where II is a certain constant.
Hence we have the following theorem:—The kinetic energy of a rigid
body, ruhen twisting with a given twist velocity about any screw of a system
of the third order, is proportional to the inverse square of the parallel
diameter of a certain ellipsoid, which may be called the ellipsoid of inertia;
and a set of three conjugate diameters of the ellipsoid are parallel to a set
of three conjugate screws of inertia which belong to the screw system.
We might also enunciate the property in the following manner:—Any
diameter of the ellipsoid of inertia is proportional to the twist velocity with
which the body should twist about the parallel screw of the screw system, so
that its kinetic energy shall be constant.
187. The Principal Screws of Inertia.
It will simplify matters to consider that the ellipsoid of inertia is con-
centric with the pitch quadric. It will then be possible to find a triad of
common conjugate diameters to the two ellipsoids. We can then determine
three screws of the system parallel to these diameters (§ 180), and these
three screws will be co-reciprocal, and also conjugate screws of inertia.
They will, therefore, (§87), form what we have termed the principal screws
of inertia. When the screw system reduces to a pencil of screws of zero
pitch passing through a point, then the principal screws of inertia reduce
to the well-known principal axes.