Gyroscope

1
Gyroscope
Principle axes
q
z is the axis along Izz
f is the precession of the z axis
q is the angle between z and the direction of
gravity
2
Euler Angles
z

• Rotate around z such that x is perpendicular to
  z
• Rotate around x by q
• Rotate around z by y

z
y
q
y
z is the axis along Izz
y
f
x
x
f is the precession of the z axis
x
q is the angle between z and z (the direction of
the torque)
Axes z, z, x Non orthogonal and a function
of time!
3
w in Coordinate System
z
Body centered axis
z
y
q
y
y
f
x
x
x
4
Kinetic Energy
(Note This is an orthogonal coordinate system.)
Assume
5
Lagrangian
6
Energy Solution
7
Effective Potential
Veffective
8
Precession
Precession angular frequency
precision can occur.
If
9
Effective Potential (again)
Veffective
E
Stable angle
Excursions around precession angle -- nutation
Linking between pf and py give complicated
precession patterns
10
Vertical Gyroscope
where
Stable vertically as long as