_______
50 the theory of screws. [54-
If twist velocities about the three screws A, B, C animate a rigid body,
then these twist velocities will neutralize if they are respectively proportional
to the opposite sides of the triangle.
55. Decomposition of Twists and Wrenches.
The theorems we have just enunciated lead to simple rules for effecting
the composition, or the decomposition of twists or of wrenches. Let a twist
on a screw X be given, and let it be required to find the components of this
twist on any two given screws A, B, all three, of course, lying on the same
cylindroid. Let a> be the amplitude of the twist on X. Then, by the last
article, the following triad of twists on the screws X, A, B, respectively, will
neutralize:—
BX
“ AB'
w,
AX
m AB ’
whence the components on A and B of the twist about X are, so far as
magnitudes are concerned,
BX , AX
“AB and “AB'
A similar proposition holds of course for wrenches.
56. Composition of Twists and Wrenches.
Let two twists, of amplitudes a and ß, about the screws A and B, respec-
tively, be applied to a rigid body. It is required to find the single resultant
screw X, and the amplitude a> of the resulting twist. Divide AB (Fig. .9)
in the point I, so that the segments AI and BI shall be in the inverse ratio
of a to ß. . Bisect the arc AB at H, and draw HI, which will cut the circle
in the required point X.
The value of a> is obtained from the equations
_®_ _ a _ ß
AB BX~AX-