On the top of the fourth page from here, the author trivially derives the components of angular velocity, expressed via Euler angles of the system. I fail to understand how the components of angular velocity were derived. May you please enlighten me!
Answer
I assume you know about rotation matrices, and so for a sequence rotations about Z-X-Z with angles ϕ, θ and ψ repsectively you have
→ω=˙ϕˆz+T1(˙θˆx+T2(˙ψˆz))
The logic here is apply a local spin of ˙ϕ, ˙θ and ˙ψ on the local axes in the sequence.
- Apply spin ˙ϕ about local Z and then rotate by T1
- Apply spin ˙θ about local X (rotated by T1) and then rotate by T2
- Apply spin ˙ψ about local Z (rotated by T1T2).
Update
There is a way to formally derive the above using the identity ˙T=→ω×T but it is rather involved for 3 degrees of freedom.
For two degrees of freedom it goes like this. With a rotation matrix T=T1T2 (defined as above) the time derivative is
dTdt=˙T1T2+T1˙T2=((˙ψˆz)×T1)T2+T1((˙θˆx)×T2)=(˙ψˆz)×(T1T2)+(T1(˙θˆx))×(T1T2)=(˙ψˆz+T1(˙θˆx))×(T1T2)=(˙ψˆz+T1(˙θˆx))×T=→ω×T→ω=˙ψˆz+T1(˙θˆx)
using the distributed property T(→a×→b)=(T→a)×(T→b).
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