Question

Q5: A space station shaped like a long right cylinder is originally spinning symmetry (defined asXy). ItsTocated far out in space. At t-0, two ( thrusters which are located along x1 axis and pointing in opposite directions are turned on giving a constant torque N about the x2 axis of he the station. ar velocity 00_ precisely aligned along its axis of a) Find and solve the equations of motion describing the an velocity in the body coordinates. b) If the thrusters are shut off after a time t, describe the subsequent motion of the angular velocity in both body and the inertial frames. (8 points) 5 Good Luck ルV

Answer 1

Euler's equations

I1w1ww(I3 - In) = N1

and similarly for 2 and 3. By symmetry and given conditions,

Lo, 112=1, N1 = Ng= U

The '3's equation gives that

w3 constant, w

The other equations are

Iw1wwo(Io - I) 0

and

Iw2ww(I - Io) = N

Just solve these two simultaneously.

1 w P(0- 1)N I2

where

(I - Io)wo

The equation is a SHM. Solution will be of the form

[1 cos t wi(t)wolIIo)

assuming 0 initial angular velocity along the axis 1, and

sin t 2(t)wolIIo)