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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97]
THE PRINCIPAL SCREWS OE INERTIA.
85
97. Co-ordinates of Impulsive and Instantaneous Screws.
Taking as screws of reference the n principal screws of inertia (§ 84), we re-
quire to ascertain the relation between the co-ordinates of a reduced impulsive
wrench and the co-ordinates of the corresponding instantaneous screw. If the
co-ordinates of the reduced impulsive wrench are y"', ...i?»"', and those of the
twist velocity are d1; d2)... ån, then, remembering the property of a principal
screw of inertia (§ 78), and denoting by ult ... un, the values of the magnitude
u (§ 89) for the principal screws of inertia, we have, from § 90,
• ^l2 1 H>
aiP1=MV1 ’
whence observing that å1 = åa1; ... = da„, we deduce the following theorem,
which is the generalization of § 80.
If a quiescent rigid body, which has freedom of the nth order, commence
to twist about a screw a, of which the co-ordinates, with respect to the
principal screws of inertia, are an and if plt ... pn be the pitches, and
wi, ... un the constants defined, in § 89, of the principal screws of inertia,
then the co-ordinates of the reduced impulsive wrench are proportional to
Let T denote the kinetic energy of the body of mass M when animated
by a twisting motion about the screw a, with a twist velocity a. Let the
twist velocities of the components on any n conjugate screws of inertia be
denoted by d1( d2,... an. [These screws will not be co-reciprocal unless in the
special case where they are the principal screws of inertia.] It follows (§ 88)
that the kinetic energy will be the sum of the n several kinetic energies due
to each component twisting motion. Hence we have (§ 89)
T = Mu^äi + ... + Mu„2ån2,
and also wa2 = u2a2 + ... + un2a»2-
Let a1; ... an and ßlt ... ßn be the co-ordinates of any two screws belong-
ing to a screw system of the nth order, referred to any n conjugate screws of
inertia, whether co-reciprocal or not, belonging to the same screw system, then
the condition that a and ß should be conjugate screws of inertia is
Wi2«iA + • • • + ‘Un«nßn = 0.
To prove this, take the case of n = 4, and let A, B, C, D be the four screws of
reference, and let A,, ... Ae be the co-ordinates of A with respect to the six
principal screws of inertia of the body when free (§ 79). The unit wrench on
a is to be resolved into four wrenches of intensities a,, ... a, ou A, B, C, D:
each of these components is again to be resolved into six wrenches on the