Maribor, December 10, 2010 2 Stability of relative equilibria in spinning tops 9th Christmas...

Maribor, December 10, 2010 2

Stability of relative equilibria in spinning tops

9th Christmas Symposium dedicated to Prof. Siegfried Großmann

CAMTP, University of MariborDecember 10, 2010

Peter H. Richter University of Bremen


Outline

• Rigid body dynamics - parameter space- reduced phase space

• Relative equilibria: Staude solutions- bifurcation diagrams

• Stability: Grammel analysis• Results and Todo

Thanks to my student Andreas Krut


Rigid body dynamics

two moments of inertia

two angles for the center of gravity s1, s2, s3

4 essential parameters after scaling of lengths, time, energy:

One point fixed in space, the rest free to move

planar

planar

plan

ar

linear

linearlinea

r

3 principal axes with respect to fixed pointcenter of gravity anywhere relative to that point


Euler-Poisson equations

sAA

321 ,,

321 ,,

Al 332211 ,,

coordinates

angular velocity

angular momentum

1 ll

Casimir constants

sAh

21energy constant

→ four-dimensional reduced phase space with parameter l


Relative equilibria: Staude solutions

0

0 sAA

angular velocity vector constant, aligned with gravity

high energy: rotations about principal axes

low energy: rotations with hanging or upright position of center of gravity

possible only for certain combinations of (h,l ): bifurcation diagram


Typical bifurcation diagram

A = (1.0,1.5, 2.0) s = (0.8, 0.4, 0.3)h32h22

h12

l

h

hstability?


Stability analysis: variational equations (Grammel 1920)

0

0

0

12

13

23

0

0

0

12

13

23

ll

ll

ll

0

0

0

12

13

23

ss

ss

ss

S

Al relative equilibrium:

lll variation:

lJ

lAS

A

l

1

1

variational equations:

J: a 6x6 matrix with rank 4 and characteristic polynomial g06 + g14 + g22


Stability analysis: eigenvalues

2 eigenvalues = 0

4 eigenvalues obtained from g04 + g12 + g2

The two 2 are either real or complex conjugate.

If the 2 form a complex pair, two have positive real part → instability

If one 2 is positive, then one of its roots is positive → instability

Linear stability requires both solutions 2 to be negative: then all are imaginary

We distinguish singly and doubly unstable branches of the bifurcation diagram depending on whether one or two 2 are non-negative


Typical scenario

• hanging top starts with two pendulum motions and develops into rotation about axis with highest moment of inertia (yellow)

• upright top starts with two unstable modes, then develops oscillatory behaviour and finally becomes doubly stable (blue)

• 2 carrousel motions appear in saddle node bifurcations, each with one stable and one singly unstable branch. The stable branches join with the rotations about axes of largest (red) and smallest (green) moments of inertia. The unstable branches join each other and the unstable Euler rotation


Orientation of axes, and angular velocities

1

stable hanging rotation about 1-axis (yellow) connects to upright carrousel motion (red)

3

stable upright rotation about 3-axis (blue) connects to hanging carrousel motion (green)

2

unstable carrousel motion about 2-axis (red and green) connects to stable branches


Same center of gravity, but permutation of moments of inertia


M


ResultsThe stability analysis is surprisingly simple, but the complexity of its results exceeds that of the bifurcation diagrams.

Todo

A number of typical scenarios have been identified, and Andreas Krut has written a powerful Matlab program.

The complete parameter dependence in the 4D set of moments of inertia and center of gravity locations seems within reach and should be established.

In addition to the eigenvalues, the eigensolutions of the variational equations should also be determined.

But keep in mind:

Relative equilibria are only the simplest aspect of rigid body dynamics


Siegfried Großmann 1990

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