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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219]
FREEDOM OE THE FOURTH ORDER.
229
already seen that two of the roots of this must be infinite, whence this
equation reduces to a quadratic, and its roots are as we have seen equal
but opposite in sign to the pitches of the principal screws of the reciprocal
cylindroid.
After a few reductions and replacing pe by - p we obtain the following
equation
p2sin3 (34) + ptps sin2 (24) + ...)
+ P {PiPiPi + P1P2P4 + Prf>3p4 + PlPiPi)
+ P1P2P3P4 = 0.
We thus deduce from any four co-reciprocal screws the quadratic equation
which gives the pitches of the two principal screws of the cylindroid to
which the given four-system is reciprocal.
218. Equations to the screw in a four-system.
The screws of the four-system are defined by the equations
( pe + a) cos X + z cos p — y cos v = 0,
- z cos X + ( pa + b) cos p + x cos v = 0,
where pe is the pitch where cos X, cos p, cos v are the direction cosines and
where x, y, z is a point on the screw. By these equations the properties of
the various screws of the system can be easily investigated.
If pa be eliminated we obtain
x cos X cos v + y cos p cos v - z (cos2 X + cos2 p) — (a — 6) cos X cos p,
whence we obtain for the equation to the cone of screws which belongs to the
four-system, and has its vertex at x0, y0, z0
Xq (x - x0) (z - z0) + y0( y - y0) (z - z„)
- z0 {(« - xoy + (y - y»)2! - (a - 6) (» - Xo) (y - ?/„) = 0.
This is of course the cone which has been referred to in § 123.
219. Impulsive Screws and Instantaneous Screws.
A body which is free to twist about all the screws of a screw system of
the fourth order receives an impulsive wrench on the screw y, the impulsive
intensity being y". It is required to calculate the co-ordinates of the screw
6 about which the body will commence to twist, and also the initial re-
actions of the constraints.
Let X and p be any two screws on the reciprocal cylindroid, then the
impulsive reaction of the constraints may be considered to consist of
impulsive wrenches on X, p of respective intensities X'", p". If we adopt