
#1
Apr1612, 08:49 PM

P: 751

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 



#2
Apr1712, 11:40 AM

HW Helper
P: 6,189

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: $$\frac{ds}{dt}=\frac{r d\phi}{dt}=r \frac{d\phi}{dt} = r \omega$$ 



#3
Apr1712, 09:16 PM

P: 751

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.




#4
Apr1712, 11:02 PM

Mentor
P: 14,462

Euler angles and angular velocity
Exactly. The time derivatives of a set of Euler angles (better said: TaitBryan angles, Bryan angles, or Cardan angles; Euler angles are a zxz rotation) are not angular velocity.




#5
Apr1812, 01:22 AM

P: 67





#6
Apr1812, 01:38 AM

P: 67

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



#7
Apr1812, 02:27 PM

P: 67

Oh jeeze. I came to check this just now and I realized I forgot one line. :)
Here it is: [tex] \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) [/tex] Change this line in my above post. This can then deduce the last equation after realizing that the unit vectors have a fixed length. 


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