Euler angles and angular velocity

[Curl]

How do you prove that angular velocity is just the time derivative of each Euler angle times the basis vector of its respective frame?
I remember it used to be perfectly clear to me a while back, but now I don't remember how the result was derived, and I couldn't find it in any of my books I looked so far.
Does anyone remember how the result was derived?

Thanks

 

[I like Serena]

Hi Curl!

Each euler angle is measured in a plane through the origin.
In other words, one specific Euler angle behaves exactly as if it was a 2 dimensional polar coordinate.
If an angle changes by an amount $d\phi$ in an time interval $dt$, the position changes by $r d\phi$.
In other words:
$$\frac{ds}{dt}=\frac{r d\phi}{dt}=r \frac{d\phi}{dt} = r \omega$$

 

[Curl]

the problem I have is that in order to add the 3 angular velocity vectors (one in each frame) implies that the rotation can be represented as the sum of the three individual rotations. But rotation order matters, which makes things confusing.

 

[D H]

Exactly. The time derivatives of a set of Euler angles (better said: Tait-Bryan angles, Bryan angles, or Cardan angles; Euler angles are a z-x-z rotation) are not angular velocity.

 

[FeX32]

How do you prove that angular velocity is just the time derivative of each Euler angle times the basis vector of its respective frame?

I think you mean the 'cross product', not times.

 

[FeX32]

Well that sentence is flawed actually.
Say you have a vector $A=A_x\hat{i}+A_y\hat{j}+A_z\hat{z}$ in a Cartesian frame. Knowledge of its velocity in this frame is obtained through its time derivative. This yields
$$\frac{d\hat{A}}{dt}=\frac{dA_x}{dt}\hat{i}+\frac{dA_y}{dt}\hat{j}+\frac{dA_z}{dt}\hat{z}$$
which can then be deduced to
$$\frac{d\hat{A}}{dt}=\dot{A} \vec{A}+\vec{\omega} \times \vec{A}$$
In the above, $\vec{\omega}$ is what you call angular velocity.

Cheers,

 

[FeX32]

Oh jeeze. I came to check this just now and I realized I forgot one line. :)

Here it is:
$$\frac{d\hat{A}}{dt}=\frac{dA_x}{dt}\hat{i}+\frac{d A_y}{dt}\hat{j}+\frac{dA_z}{dt}\hat{z}+\left( A_x \frac{d\hat{i}}{dt}+A_y\frac{d\hat{j}}{dt}+A_z \frac{d \hat{z}}{dt}\right)$$

Change this line in my above post. This can then deduce the last equation after realizing that the unit vectors have a fixed length.