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
Søgning i bogen
Den bedste måde at søge i bogen er ved at downloade PDF'en og søge i den.
Derved får du fremhævet ordene visuelt direkte på billedet af siden.
Digitaliseret bog
Bogens tekst er maskinlæst, så der kan være en del fejl og mangler.
K; ; -
388 THE THEORY OF SCREWS. [353,
353. Freedom of the eighth and higher orders.
If the freedom be of the eighth order, then it is easily shown that the
first screw of any other chain may be taken arbitrarily, and that even the
second screw may be chosen arbitrarily from a three-system. Passing on to
the twelfth order of freedom, the two first screws of the chain, as well as the
amplitudes of their twists, may be chosen quite arbitrarily, and the rest of
the chain is fixed. In the thirteenth order of freedom we can take the
two first twists arbitrarily, while the third may be chosen anywhere on a
cylindroid. It will not now be difficult to trace the progress of the chain
to that unrestrained freedom it will enjoy when the mass-chain has 6/x
degrees of freedom, when it is able to accept any displacement whatever. In
the last stage, prior to that of absolute freedom, the system will have its
position defined by 6/x — 1 co-ordinates. A screw-chain can then be chosen
which is perfectly arbitrary in every respect, save that one of its screws must
be reciprocal to a given screw.
354. Reciprocal Screw-Chains.
We have hitherto been engaged with the discussion of the geometrical
or kinematical relations of a mass-chain of y elements : we now proceed to
the dynamical considerations which arise when the action of forces is
considered.
Each element of the mass-chain may be acted upon by one or more
external forces, in addition to the internal forces which arise from the reaction
of constraints. This group of forces must constitute a wrench appropriated
to the particular element. For each element we thus have a certain wrench,
and the entire action of the forces on the mass-chain is to be represented by
a series of y wrenches. Recalling our definition of a screw-chain, it will be
easy to assign a meaning to the expression, wrench on a screw-chain. By this
we denote a series of wrenches on the screws of the chain, and the ratio of
two consecutive intensities is given by the intermediate screw, as before.
We thus have the general statement:—
The action of any system of forces on a mass-chain may be represented
by a wrench on a screw-chain.
Two or more wrenches on screw-chains will compound into one wreuch
on a screw-chain, and the laws of the composition are exactly the same as
for the composition of twists, already discussed.
Take, for example, any four wrenches on four screw-chains. Each set of
four homologous screws will determine a four-system; the resulting wrench-
chain will consist of a series of wrenches on these four-systems, each being
the "parallel projection” of the other.