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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120] FREEDOM OF THE SECOND ORDER. 109
Let X, fi, v be three screws upon a cylindroid, and let A, B, G denote the
angles between /z v, between v X, and between X p, respectively. If wrenches
of intensities X", p", v", on X, p, v, respectively, are in equilibrium, we must
have (§ 14):—
> // // //
A. /Z V
sin A sin B sin G'
But we have also as a necessary condition that if each wrench be resolved
into six component wrenches on six screws of reference, the sum of the
intensities of the three components on each screw of reference is zero;
whence
Xj sin A + /i, sin B + sin G = 0,
X6 sin A + pa sin B + v6 sin (7=0.
From these equations we deduce the following corollaries:—
The screw of which the co-ordinates are proportional to u'Xl + bplt ...
aX6 + bpe, lies on the cylindroid (X, p), and makes angles with the screws
X, p, of which the sines are inversely proportional to a and b.
The two screws, of which the co-ordinates are proportional to
aX, + bpi,... aXc + bp,it
and the two screws X, p are respectively parallel to the four rays of a plane
harmonic pencil.
120. Screws on One Line.
There is one case in which a body has freedom of the second order that
demands special attention. Suppose the two given screws 0, (f>, about which
the body can be twisted, happen to lie on the same straight line, then the
cylindroid becomes illusory. If the amplitudes of the two twists be O', $>',
then the body will have received a rotation O' + <f>', accompanied by a trans-
lation O'po + This movement is really identical with a twist on a
screw of which the pitch is:
O'pe + <f>'P<b
0‘ + <p'
Since O', <j>' may have any ratio, we see that, under these circumstances, the
screw system which defines the freedom consists of all the screws with
pitches ranging from — oo to + co, which lie along the given line. It
follows (§ 47), that the co-ordinates of all the screws about which the
body can be twisted are to be found by giving a; all the values from
— oo to + oo in the expressions:
x dR „ x dR
U1 + ApidOi’" ^Ip.dO,’
in which R = (0l + 02)2 + (03 + 04)2 + (05 + 0,;)a.