Euler's equations for constant torque?

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AI Thread Summary
The discussion revolves around solving Euler's equations for an axially symmetric space station experiencing a constant torque about its symmetry axis. The initial angular velocity is given as ω = (ω1, 0, ω3) at t = 0. Participants are trying to determine which components of Euler's equations can be set to zero, particularly focusing on the torque component τ related to the symmetry axis. There is a consensus that the third torque component, Γ3, equals τ, suggesting ω3 remains constant during the motion. The challenge lies in accurately applying the equations to describe the resulting motion of the space station.
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Homework Statement



An axially symmetric space station (principal axis e3, and λ1 = λ2) is floating in free space. It has rockets mounted symmetrically on either side that are firing and exert a constant torque τ about the symmetry axis. Solve Euler's equations exactly for ω (relative to the body axis) and describe the motion. At t = 0 take ω = (ω1,0,ω3).

As well I am told to introduce the constants
(λ3-λ1)ω3/λ1

α = (λ3-λ1)τ/(λ1λ3)


Homework Equations



SDkSib1.jpg

n=ω1+iω2

The Attempt at a Solution


I tried several different things trying to solve the problem but unfortunately I could never quite figure out which parts of Euler's equation are zero. I think that the third τ in Eulers equation is zero which would make ω3 constant but then I'm not really sure where to go.
 
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x3 is the symmetry axis. The problems states that the torque τ is about this axis. That means that \Gamma3 = τ
 

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