Welcome to AE Resources

Converted document

PROPERTIES OF ANGULAR VELOCITY

Recall that we first began our 3D analysis with:

$\dot{\vec{Q}}^{A} = \dot{\vec{Q}}^{B} +( Q_{x}\dot{\vec{i}}^{A} + Q_{y}\dot{\vec{j}}^{A} + Q_{z}\dot{\vec{k}}^{A})$

$\dot{\vec{Q}}^{A} = \dot{\vec{Q}}^{B} + Q_{x}(\omega_{B/A}\times\vec{i}) + Q_{y}(\omega_{B/A}\times\vec{i}) + Q_{z}(\omega_{B/A}\times\vec{i})$

$\dot{\vec{Q}}^{A} = \dot{\vec{Q}}^{B} + \omega_{B/A}\times\vec{Q}$ (1)

We have seen something similar relating different points in planar dynamics.

$\dot{\vec{Q}}_{A} = \dot{\vec{Q}}_{B} + \omega\vec{k}\times\vec{Q}_{BA}$

Recalling this, we can prove that ω_B/A is unique and that ω_B/A= -ω_A/B

Notice what happens to equation (1) if B remains fixed in time with respect to A, ω_B/A=0. What happens when we have multiple frames of reference, as in many moving parts in a system? Using our basic formula, we can write the relationship of the various systems to each other:

$\dot{\vec{Q}}^{A} = \dot{\vec{Q}}^{B} + \omega_{B/A}\times\vec{Q}$ (1)

$\dot{\vec{Q}}^{A} = \dot{\vec{Q}}^{C} + \omega_{C/A}\times\vec{Q}$ (2)

$\dot{\vec{Q}}^{B} = \dot{\vec{Q}}^{C} + \omega_{C/B}\times\vec{Q}$ (3)

Replacing (3) into (1), we have:

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

Comparing with (2) we see that ω_C/A=ω_C/B + ω_B/A. This can be easily shown to be extensible to multiple axes.

If the k axis in both frames A and B remains the same, then we have simple planar motion. In that case,

$\omega_{B/A} = \dot{\theta}\vec{k} = \omega\vec{k}$ (Simple angular velocity)

However, this result can also hold if A is moving along k, but such that k_A still remains parallel to k_B. This is translational motion in k, but the bodies no longer remain in the same plane.

Note that you can have simple angular velocity in i and j as well;

$\omega_{B/A} = \dot{\theta}\vec{i}$

// to i axis

$\omega_{B/A} = \dot{\theta}\vec{j}$

// to j axis

Finally, $\dot{\omega}_{B/A} = \dot{\omega}_{A/B}$

The derivative of the angular velocity will rpovide the same result no matter which frame of reference it is computed in.

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