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Stability regions for an unsymmetrical rigid body spinning about an axis through a fixed point

Stabilitätsgebiete der Drehbewegung eines unsymmetrischen starren körpers mit festem Aufhängepunkt

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Summary

The classical stability problem of the spinning rigid body with a fixed point under the influence of uniform gravity, is examined by several approaches. First, an inertia parameter is introduced which simplifies and clarifies investigation of the problem for all possible rigid bodies. Instability zones are then found using the Floquet theory. The results of Rumiantsev's Lyapunov analysis are used to provide part of the stable zone. Finally, the remaining regionR, for which a stability (in the large) decision is not guaranteed either by the Floquet theoryor the Lyapunov analysis, is discussed.

Zusammenfassung

Das klassische Stabilitätsproblem, ein starrer Körper rotiert im Schwerefeld um eine durch den festen Aufhängepunkt des Körpers gehende Achse, wird untersucht. Zuerst wird ein die Untersuchung vereinfachender und übersichtlich gestaltender Trägheitsparameter eingeführt. Instabilitätsbereiche werden dann unter Verwendung der Floquetschen Theorie gefunden. Von Rumjanzew mit Hilfe von Ljapunovschen Funktionen gefundene Resultate werden zur Ermittlung von Teilen der Stabilitätsgebiete verwendet. Abschließend wird das verbleibende GebietR, für das Stabilitätsaussagen (im Großen) weder durch die Floquetsche noch durch die Ljapunovsche Theorie gegeben werden können, diskutiert.

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Abbreviations

B :

Rigid body

C :

Mass center ofB

d :

Distance fromC to 0

{ei91-1}:

Matrix defined in Eq. (25)

{ei91-2}:

Angular momentum ofB about 0

i x :

=I xo /I zo

{ei91-3}:

Unit vectors parallel to body axes

<K :

Unit vector, vertically upward

k :

=I yo /I xo

{ei91-4}:

Moment of external forces about 0

0:

Fixed point ofB

R :

Inertial reference frame

S :

Speed parameter, {ei91-5}

t :

Time

V :

Lyapunov function

V i :

Integral of perturbed differential equations

(x 0,y 0,z 0):

Body-fixed axes

(X, Y, Z):

Axes ofR

(α, β, ω):

Perturbations of {ei92-1}

(γ, δ):

Angular rates, {ei92-2} respectively

{ei92-3}:

Direction cosines between<K and (î, j, <k)

{ei92-4}:

Perturbations of {ei92-5}

{ei92-6}:

Angular rotations (in that order) that positionB inR

τ:

Nondimensional time {ei92-7}

{ei92-8}:

Perturbations of {ei92-9}

{ei92-10}:

Initial spin speed ofB aboutZ

{ei92-11}:

Angular velocity vector

{ei92-12}:

Components of {ei92-13}

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Authors and Affiliations

  1. School of Engineering Science and Mechanics, Georgia Institute of Technology, 30332, Atlanta, GA, U.S.A.

    D. J. McGill &  L. S. Long III

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  1. D. J. McGill
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  2. L. S. Long III
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McGill, D.J., Long, L.S. Stability regions for an unsymmetrical rigid body spinning about an axis through a fixed point. Acta Mechanica 22, 91–112 (1975). https://doi.org/10.1007/BF01170620

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  • DOI: https://doi.org/10.1007/BF01170620

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