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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128]
FREEDOM OF THE SECOND ORDER.
115
the screw a with the unit of twist velocity, the kinetic energy is found by
multiplying the mass of the body into the square of the line ua.
We are now going to consider the distribution of this magnitude ua on
the screws of a cylindroid. If we denote by ult u2 the values of ua for any
pair of conjugate screws of inertia on the cylindroid (§ 81), and if by a„ a2
we denote the intensities of the components on the two conjugate screws of
a wrench of unit intensity on a, we have (§ 97)—
wa2 = W;2«!2 + w22a.,2.
From the centre of the cylindroid draw two straight lines parallel to the
pair of conjugate screws of inertia, and with these lines as axes of x and y
construct the ellipse of which the equation is
UfX1 + Z/227/3 = H,
where II is any constant. If r be the radius vector in this ellipse, we have
(§ 35)
« . y
- = «1 and ~ = a2;
whence by substitution we deduce
which proves the following theorem : —
The linear parameter ua on any screw of the cylindroid is inversely
proportional to the parallel diameter of a certain ellipse, and a pair of
conjugate screws of inertia on the cylindroid are parallel to a pair of
conjugate diameters of the same ellipse. This ellipse may be called the
ellipse of inertia.
The major and minor axes of the ellipse of inertia are parallel to screws
upon the cylindroid, which for a given twist velocity correspond respectively to
a maximum and minimum kinetic energy.
An impulsive wrench on a screw y acts upon a quiescent rigid body
which has freedom of the second order. It is required to determine the
screw 0 on the cylindroid expressing the freedom about which the body
will commence to twist.
The ellipse of inertia enables us to solve this problem with great facility.
Determine that one screw </> on the cylindroid which is reciprocal to y (§ 26).
Draw a diameter D of the ellipse of inertia parallel to </>. Then the required
screw 0 is simply that screw on the cylindroid which is parallel to the
diameter conjugate to D in the ellipse of inertia.
The converse problem, viz., to determine the screw y, an impulsive wrench
8—2