- #1
Charlls
- 6
- 0
Hi there,
the Focault compass its supposed to be a fast spinning disk that keeps pointing to the Earth north
the spinning axis is mounted on top of a rotating base, angle which is supposed to oscillate when slighty displaced from the north direction
is this oscilatory behaviour occurring 'only' on the Earth surface, or its true also in inertial far-away-from-earth frames?
i wrote the lagrangian for this thing using omega . I omega
lets assume first the problem in far-awar-from-earth-inertial frames, then i got a fixed basis Ex, Ey and Ez
the gyroscope has its principal axis like Ez, En and Ew, where En = cos(a)Ex + sin(a)Ey
and Ew = -sin(a)Ex + cos(a)Ey
the spinning axis of the gyroscope is En, which is also the symmetry axis
the gyroscope can also rotate in the Ez axis, with an angle a
so i wrote the I tensor in dyadic rep. like I = Iz EzEz + Ir EnEn
but EnEn = cos(a)^2 ExEx + sin(a)cos(a) [ ExEy + EyEx ] + sin(a)^2 EyEy
the rotation of the gyroscope can be represented as:
omega = a' Ez + omega En = a' Ez + omega cos(a) Ex + omega sin(a) Ey
so when you plug this rotation into the inertia tensor to get the kinetic energy, you get at the end:
Lagrangian = a'^2 Iz + omega^2 Ir
(remember that the dyads act with vectors like EiEj * Ek = (Ej . Ek) Ei, where . is the dot product between vectors)
so as you see, my lagrangian does NOT depend on a, so i can't get an oscillatory motion in this system
I am doing something blatantly wrong here?
any insights are welcome
Cheers
the Focault compass its supposed to be a fast spinning disk that keeps pointing to the Earth north
the spinning axis is mounted on top of a rotating base, angle which is supposed to oscillate when slighty displaced from the north direction
is this oscilatory behaviour occurring 'only' on the Earth surface, or its true also in inertial far-away-from-earth frames?
i wrote the lagrangian for this thing using omega . I omega
lets assume first the problem in far-awar-from-earth-inertial frames, then i got a fixed basis Ex, Ey and Ez
the gyroscope has its principal axis like Ez, En and Ew, where En = cos(a)Ex + sin(a)Ey
and Ew = -sin(a)Ex + cos(a)Ey
the spinning axis of the gyroscope is En, which is also the symmetry axis
the gyroscope can also rotate in the Ez axis, with an angle a
so i wrote the I tensor in dyadic rep. like I = Iz EzEz + Ir EnEn
but EnEn = cos(a)^2 ExEx + sin(a)cos(a) [ ExEy + EyEx ] + sin(a)^2 EyEy
the rotation of the gyroscope can be represented as:
omega = a' Ez + omega En = a' Ez + omega cos(a) Ex + omega sin(a) Ey
so when you plug this rotation into the inertia tensor to get the kinetic energy, you get at the end:
Lagrangian = a'^2 Iz + omega^2 Ir
(remember that the dyads act with vectors like EiEj * Ek = (Ej . Ek) Ei, where . is the dot product between vectors)
so as you see, my lagrangian does NOT depend on a, so i can't get an oscillatory motion in this system
I am doing something blatantly wrong here?
any insights are welcome
Cheers