Stationary Precession of a Spinning Top

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The stationary precession motion of a frictionless symmetric top under the influence of gravity can be described by a nonlinear ODE:

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,

where and are parameters that depend on the initial conditions, the variables , , and are, respectively, the nutation, precession, and rotation angles (Euler angles), is the height of the center of mass, and the parameters and are the transverse and longitudinal moments of inertia of the top, respectively.

The precession angle is then calculated by

.

With and it is possible to determine the position of the top.

For a stationary precession motion, given the other parameters, the initial precession velocity must be such that

,

which has real solutions only if

.

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Contributed by: William C. Guttner (March 2009)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshots 1 and 2: adding an initial nutation velocity, the motion becomes a combination of a stationary precession and a nutation motion

Snapshot 3: increasing the initial spin, the "gyroscopic stability" increases, which reduces the precession velocity:

Reference: C. P. Pesce, Dinâmica dos Corpos Rígidos, Escola Politécnica da Universidade de São Paulo, 2004.



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