# Euler rotations and acceleration

I have a system determined by 3 Euler angles that describe the orientation with respect to a fixed coordinate system XYZ.

The angular velocities are: $$\omega_2$$ (precession), $$\omega_1$$ (nutation) and $$\omega_z$$ (spin). The angular acceleration of $$\omega_z$$ is $$\alpha_z$$. The other angular accelerations are 0.

How can I find the acceleration of a point (e.g. A on the spinning reference frame xyz)? I am looking for the procedure to follow.

## 1 Answer

First of all you need to be about which reference frame you are estimating the acceleration of A. I assume you need the accelation of A wrt to the inertial frame of reference.

The most generic way is to link the acceleration of A to B. Then find the acceleration of B with respect to O.

See the following link for a starting point

The bottom line is that if you need the acceleration of P with respect to O, then you'll probably just need the following two equations:

$$\vec{v}_P =\vec{v}_O + \vec{\omega}\times \vec{r}_P$$ $$\vec{a}_P =\vec{a}_O + \dot{\vec{\omega}}\times \vec{r}_P + \vec{\omega}\times \left( \vec{\omega}\times \vec{r}_P\right)$$

• I need it indeed wrt the initial reference frame. This was the confirmation. I needed. Thanks. Aug 11 '20 at 20:47