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.
412
[375,
THE THEORY OF SCREWS.
and as 0/, 02', &c., are indefinitely small, this reduces to
^ = 0,
where T7, is a homogeneous function of 0lt 02, ... 0n in the second degree.
For the study of the permanent screws we have, therefore, n equations
of the second degree in the co-ordinates of the instantaneous screw-chain,
and any screw-chain will be permanent if its co-ordinates render the several
differential coefficients zero. We may write the necessary conditions that
have to be fulfilled, as follows:—
Let us denote the several differential coefficients of T with respect to the
variables by I, II, III, &c. Then the emanant identity is
ØJ + ØJI+03111+..^O,
and we may develop any single expression, such as III, in the following
form :—
in = innfc + iii22^ + + aiu.M +... 211MA
As th© cnianant is to vanish identically, wo must have the coefficients of
0i202) ØiØ-203, &c„ all zero, the result being
the several terms, such as 0^,
three types of equation—
In = 0, I22 + II12 = 0, i23+iiis+iii12
II22 = 0, 1111 + 112 = 0, &c.,
in33 = o, Us + IU23 = 0, &c.,
IV« = 0, &c., &c.
&c.
= 0,
Of the first of these
there are n(n — 1), and of the third, n^n , -n aj^
1.2.3
classes of equations, Iu = 0, there are n, of the second
n (n +l)(n+2)
1.2.3 ‘
376. Another identical equation.
Let T be the kinetic energy of a perfectly free rigid body twisting for
the moment around a screw 0. It is obvious that T will be a function of
the six co-ordinates, 0/,... 0e', which express the position of the body, and
also of 01; ... 0e, the co-ordinates of the twist velocity,
T=f(01',...06/, 01)...06).
\\ e may now make a further application of the principle employed in
§ 362. The kinetic energy will be unaltered if the motion of the body be
arrested, and if, after having received a displacement by a twist of amplitude
e about a screw of any pitch on the same axis as the instantaneous screw,
the body be again set in motion about the original screw with the original
twist velocity. This obvious property is now to be stated analytically.