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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114]
FREEDOM OJF THE FIRST ORDER.
105
Ihen X must be that screw on the cylindroid which is reciprocal to a, for a
wrench on X, and the given wrench on t), must compound into a wrench on p,
whence the three screws must be co-cylindroidal*; also X must be reciprocal
to a, so that its position on the cylindroid is known (§ 26). Finally, as the
impulsive intensity i/" is given, and as the three screws p, X, /x are all
known, the impulsive intensity X'" becomes determined (§ 14).
114. Small Oscillations.
We shall now suppose that a rigid body which has freedom of the first
order occupies a position of stable equilibrium under the influence of a
system of forces. If the body be displaced by a small twist about the screw
« which prescribes the freedom, and if it further receive a small initial twist
velocity about the same screw, the body will continue to perform small
twist oscillations about the screw a. We propose to determine the time
of an oscillation.
is
The kinetic energy of the body, when animated by a twist velocity
MS')’«-
da'
dt
The potential energy due to the position attained by
giving the body a twist of amplitude a away from its position of equili-
brium, is Fv^a" (§ 102). But the sum of the potential and kinetic energies
must be constant, whence
fdax1 „ .
\dt) +
MuS
const.
Differentiating we have
d2a' Fv^ , _
dF Mua1 a
Integrating this equation we have
“ A sin 1 B C,,S
where yl and B are arbitrary constants. The time of one oscillation is
therefore
ua /M
217 va\/F‘
Regarding the rigid body and the forces as given, and comparing
inter se the periods about different screws a, on which the body might have
been constrained to twist, we see from the result just arrived at that the
time for each screw a is proportional to .
We shall often for convenience speak of three screws on the same cylindroid as co-cylindroidal.