# Determining Analytic Orientation from Angular Velocity

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1. Oct 22, 2015

### dgreenheck

I am doing an analysis concerning the torque-free motion of an axisymmetric body (J1 = J2 != J3).

The angular velocity is given

$\omega(t) = [\omega_t\ sin(\Omega t), \omega_t\ cos(\Omega t), \omega_{z0}]$

where $\omega_t$, $\Omega$ and $\omega_{z0}$ are constants. I would like to determine the orientation of the body at any time $t$ given an initial orientation at $t = 0$. My end goal is to have an analytic representation of the orientation that I can use as "truth" to compare the errors of various numerical methods of estimating the orientation.

I know how to find numerical solutions to this problem using quaternions/direction cosine matrices/rotation vectors, etc., but am not sure how to approach this from an analytic point of view.

2. Oct 23, 2015

### Chandra Prayaga

Unless I have not understood your question, it looks to me that you already have the answer. Since the constants are given, you already know the direction of the angular velocity as a function of time.

3. Oct 24, 2015

### BvU

Yes, but what is the orientation of the spinning body

4. Oct 24, 2015

### Chandra Prayaga

When you use the word orientation, you have to specify what you mean. Usually, orientation means that there is some vector property of the body, which is pointing in some direction. In the case of a spinning body, the angular velocity (or angular momentum) is that vector. The direction of the angular velocity IS the orientation of the body. What you are probably looking for is to specify the geometric shape of the body, such as a cube, and then you can ask a question like, how the edges of the cube are rotating as the object rotates about some axis. The axis in this case, is again, the direction of the angular velocity