Gyroscopic Precession

Consider an instant in time at which the Eulerian angle is zero.
This implies that the -axis is horizontal [see Equation (534)], as shown in the diagram.
The gravitational force, which acts at the
center of mass, thus exerts a torque
in the
-direction. Hence, the components of the torque in the body
frame are

(556) | |||

(557) | |||

(558) |

The components of the angular velocity vector in the body frame are given by Equations (537)-(539). Thus, Euler's equations (504)-(506) take the form:

where

(562) |

where

According to Equations (561) and (563), the two quantities and are constants of the motion. These two quantities are the

If there are no frictional forces acting on the top then the total
energy, , is also a constant of the motion. Now,

(565) |

Eliminating between Equations (564) and (566), we obtain the following differential equation for :

(567) |

(568) |

(569) |

where is a

Fortunately, we do not have to perform the above integration (which is very ugly)
in order to discuss the general properties of the solution to
Equation (570). It is clear, from Equation (571), that
needs to be *positive* in order to obtain a physical solution. Hence, the
limits of the motion in are determined by the three roots of the
equation . Since must lie between and ,
it follows that must lie between 0 and 1. It can easily be demonstrated that
as
. It can also be shown
that the
largest root lies in the region , and the two smaller
roots and (if they exist) lie in the region
.
It follows that, in the region
, is only positive between and .
Figure 31 shows
a case where and lie in the range
0 to 1. The corresponding values of -- and , say--are then the limits of the vertical motion.
The axis of the top oscillates backward and forward between these two
values of as the top precesses about the vertical axis. This
oscillation is called *nutation*. Incidentally, if becomes
negative then the nutation will cause the top to strike the ground (assuming
that it is spinning on a level surface).

If there is a double root of (*i.e.*, if ) then
there is no nutation, and the top precesses steadily. However, the
criterion for steady precession is most easily obtained directly from
Equation (559). In the absence of nutation,
.
Hence, we obtain

(572) |

The above equation is the criterion for steady precession. Since the right-hand side of Equation (573) possesses a minimum value, which is given by , it follows that

(574) |

(575) | |||

(576) |

The slower of these two precession rates is the one which is generally observed.