How Do Coordinate Frames Influence Equations in Wind Generator Dynamics?

[chicomore]

Homework Statement

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

Homework 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?

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.

Thanks in advance...

 

[Jhero]

Hello,

Thank you for sharing your problem and the diagrams. It seems like you are trying to understand the reference frames and their equations for a small wind generator protected by an autofurling mechanism. From the diagrams, it looks like there are four different frames of reference involved: F_A, F_B, F_C, and F_D.

The first frame, F_A, is the inertial frame and it is defined by three unit vectors \vec a_1, \vec a_2, and \vec a_3. These vectors point in the average direction of the wind, which is the direction of the wind at a particular location and time. This frame is used to describe the motion of the wind itself.

The second frame, F_B, is attached to the nacelle before tilting. It is defined by three unit vectors \vec b_1, \vec b_2, and \vec b_3. From the diagrams, it looks like this frame is rotated by an angle \theta with respect to the inertial frame \vec a_1 and \vec a_2. This means that the equations for this frame will have trigonometric functions of \theta.

The third frame, F_C, is defined by three unit vectors \vec c_1, \vec c_2, and \vec c_3. This frame is after tilting, so it is rotated by an angle \phi with respect to the frame F_B. The equations for this frame will also have trigonometric functions of \phi.

Finally, the fourth frame, F_D, is defined by three unit vectors \vec d_1, \vec d_2, and \vec d_3. This frame is aligned with the tail hinge, so it will have its own set of equations.

Now, to answer your question about the equations for the different coordinate systems, here is my attempt at a solution:

1. F_B frame of reference:
\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

2. F_C frame of reference:
\vec c_1 = \vec b_1 \cos \phi + \vec b_3 \sin \phi