Coordinate frames difficulty

1. Dec 5, 2008

chicomore

1. The problem statement, all variables and given/known data

This is a basic question to a kind of complex problem, any help will be deeply appreciatted. I have all the motion equations for the system described below, but i have a problem with the reference frames...

So the problem is as follows:
A small wind generator is protected against high speed winds by autofurling mechanism that depends on the equilibrium of torques between the tail and the moment of inertia generated by wind when it hits the rotor:

the diagrams

THE COORDINATE FRAMES

1. inertial frame $F_A=(\vec a_1, \vec a_2,\vec a_3)$ point in the average direction of the wind.

2 $F_b$ frame of reference attached to nacelle before tilting

$\vec b_1 = \vec a_1 \cos \theta +\vec a_2 \sin \theta$

$\vec b_2 = - \vec a_1 \sin \theta +\vec a_2 \cos \theta$

$\vec b_3 = \vec a_3$ vertical upward

3 $F_C=(\vec c_1,\vec c_2,\vec c_3)$ after tilting

4 $F_D = (\vec d_1, d2,d3)$ aligned with the tail hinge

2. Relevant equations

What are the equations for the different the different coordinate systems?? i have the equations for the coordinate system Fb, but when i derived them following the diagrams i got different relations, are the equations given to me wrong or what i'm doing is wrong?

3. The attempt at a solution

So for the coordinate system $F_B$, according to the diagrams i get that:

$\vec b_1 =\vec a_1 \sin \theta - \vec a_2 \cos \theta$
$\vec b_2 =\vec a_1 \cos \theta + \vec a_2 \sin \theta$
$\vec b_3 = \vec a_3$

since i don't get the same result for reference frame b, i'm prety doubtful of what i get in c and d.