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ANGULAR ACCELERATION VECTOR

As in our planar applications, the angular acceleration can be written as:

$\alpha_{B/A} = \dot{\omega}^{A}_{B/A} = \dot{\omega}^{B}_{B/A}$

Do the theorems we used for angular velocity hold for the angular acceleration? Consider the addition theorem:

$\alpha_{C/A} = \dot{\omega}^{A}_{C/A} = \dot{\omega}^{A}_{C/B} + \dot{\omega}^{A}_{B/A}$

$=\dot{\omega}^{A}_{C/B} + \alpha_{B/A}$

What do we do with the first term? It is a derivative taken with respect to the frame outside of C or B.

Recall our derivative formula:

$\dot{Q}^{A} = \dot{Q}^{B} + \omega_{B/A}\times Q$

If ω_C/B is Q, then

$\dot{\omega}^{A}_{C/B} = \dot{\omega}^{B}_{C/B} + \omega_{B/A} \times \omega_{C/B}$

$\dot{\omega}^{A}_{C/B} = \alpha^{B}_{C/B} + \omega_{B/A} \times \omega_{C/B}$

so that

$\alpha_{C/A} = \alpha_{C/B} + \alpha_{B/A} + \omega_{B/A}\times\omega_{C/B}$

where ω_B/A×ω_C/B is the gyroscopic term. Thus, the additive theorem does not work for angular accleration in three dimensions. In planar motion, the gyroscopic term disappears.