Relative and absolute velocity in rotating frames
When analyzing motion in mechanics and dynamics, distinguishing between relative and absolute velocities is essential, especially when dealing with rotating reference frames. In this post, I outline the key concepts and provide a concise derivation of the velocity relationship.
I begin by expressing the position vector of a point P in terms of two frames: a stationary frame and a rotating frame. This can be written as:
\mathbf{r}_{O_1P} = \mathbf{r}_{O_1O_2} + \mathbf{r}_{O_2P}
Differentiating this position vector with respect to time yields the velocity equation, which accounts for both translational motion and the effect of frame rotation. A crucial step is recognizing that vectors expressed in the rotating frame require additional treatment during differentiation. Specifically, the angular velocity vector \boldsymbol{\omega} contributes to the final velocity expression.
The resulting velocity relationship for a point P is:
\mathbf{v}_P = \mathbf{v}_{O_2} + \mathbf{v}_{\text{rel}} + \boldsymbol{\omega} \times \mathbf{r}
Here, \mathbf{v}_P is the absolute velocity of the point, \mathbf{v}_{O_2} represents the absolute velocity of the origin of the rotating frame, \mathbf{v}_{\text{rel}} is the relative velocity within the rotating frame, and \boldsymbol{\omega} \times \mathbf{r} captures the rotational effect.
This formula is particularly useful for analyzing planar motion, where the angular velocity vector simplifies to a scalar.
For more insights into this topic, you can find the details here.