An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

104

VALUE OF GYROSCOPIC RESISTANCE

Multiplying both sides by 20 and integrating we get the
first equation of Art. 99,
while. = GCc(cos 0- cos 0)
A sin' 0
Thus the motion of the axle in the azimuthal plane being
known, and that of the azimuthal plane, the complete motion
of the top is known in terms of 0.
If 00 is zero, and 0 small, the above equation reduces to
Ad = M W sin (I - 2W(1 cos 0)2
~..~-"-.., A0=zMgasin0- Asin
~.~~~~~~C..2
A0=MglaO —,A 0)
_ z N: A neglecting 02 and higher powers,
X, \. o- 0OC2W2 -4MgaA > O
R2 Rz4A2
while
FIG. 56. 2. =W1 say. (Art. 102.)
The above equations show that the axle oscillates about the
vertical with a simple harmonic motion of period r where
092
C2c2 - 4MgaA
W V2-= — 4A2
and the condition for a real oscillation is the condition for
stability obtained in Art. 102.
Since H the head of the top (Fig. 56) describes a simple
harmonic motion in ZR2 the azimuthal plane, while ZR2 rotates
uniformly with angular velocity wo, the velocity of H at any
moment is known, and its path in space completely determined.
Considering the component velocities of H in the directions HZ,
HP, we see that, when viewed from a point in the axis of Z, H
will appear to describe a series of loops as in Fig. 63 in periodic
27r
time-.
0)2
It will be noticed that the point N oscillates in the fixed plane
ZA in periodic time 27. (See Art. 141.) j " d!
09 + -- 2/
107. Motion of the top when G has an initial velocity.
The principles established in Art. 104 show that, whatever be
the initial velocity of G, the top still oscillates between two
limiting positions. These limits can be found analytically as in
Art. 99, by eliminating f between the two equations for
conservation of angular momentum and conservation of energy.