Stationary Precession of a Spinning Top

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

I (Overscript[θ, ¨])+(((α-β θ)(β-α θ))/(I (sin^2)θ))m g (Subscript[z, G])sin θ,

where α and β are parameters that depend on the initial conditions, the variables θ, ϕ, and ψ are, respectively, the nutation, precession, and rotation angles (Euler angles), (Subscript[z, G]) is the height of the center of mass, and the parameters I and J are the transverse and longitudinal moments of inertia of the top, respectively.

The precession angle is then calculated by

(OverDot[ϕ])=((α-β cos θ)/(I (sin^2)θ)).

With θ and ϕ 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

(OverDot[ϕ])=(1/2)(-((J (OverDot[ψ]))/((J-I)cos θ))±(Sqrt[(((((J (OverDot[ψ]))/((J-I)cos θ))))^2)+((4 m g (Subscript[z, G]))/((J-I)cos θ))])),

which has real solutions only if

(ψ^2)≥-4m g (Subscript[z, G])cos θ (((J-I))/(J^2)).

THINGS TO TRY

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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: J (OverDot[ϕ]) (OverDot[ψ])≈m g (Subscript[z, G])

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

http://www.poli.usp.br/d/pme2200/DCR_Pesce.pdf  (Portuguese)

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Contributed by: William C. Guttner

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