358 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
is the configuration space for the vehicle using reduction theory for mechanical
systems with symmetry.
Notation and assumptions.
- neutrally buoyant and ellipsoidal vehicle
- centers of gravity and buoyancy need not be coincident
• inertia matrix of the body-fluid system: J = diag(fi, h, !3)
• mass matrix M = diag(m 1 , m 2 , m 3 ) (J and M include the "added" inertias
and masses due to the fluid)- the mass of the body alone is m, and the acceleration of gravity is g
• l = distance between center of buoyancy and center of gravity.
Relative equilibrium 1. In the first relative equilibrium we look at, the energy
momentum technique reproduces a nonlinear extension of a classical result that
may be found in Lamb [1932]. We assume the following:- the vehicle is symmetric about the third (principal) axis
- it has coincident centers of buoyancy and gravity
- it translates with momentum (impulse) P~ along the third axis and rotates
with angular momentum (impulse) rrg about the same axis (see Figure 2.2).
The techniques outlined above applied to this case show that: This motion is stable
modulo rotations about the third axis and translations in any direction provided that
(Prr~) 3 2 > 4h (-m3 1 -J_)m2.
Roughly speaking, this means that "blunt" motion is stable, while "streamline"
and slowly spinning motion is unstable.~
:--~~~~+--~~ pg--~
Figure 2.2. Translating and spinning ellipsoid.R elative equilibrium 2. The second sample relative equilibrium has the following
features:
- The vehicle has noncoincident centers of buoyancy and gravity oriented with
the third principal axis parallel to the direction of gravity
• It is translating (not spinning) with momentum P~ along the second prin-
cipal axis