Planar rigid body kinematics: velocity
This post explains how to analyze a point’s velocity observed from two frames, one moving and rotating relative to the other. I derive the velocity transformation equation.
Consider point P observed from a fixed frame (1) and a moving frame (2). The position of P relative to frame 1 is:
\mathbf{r}_{O_1P} = \mathbf{r}_{O_1O_2} + \mathbf{r}_{O_2P}
Differentiating with respect to time (from frame 1):
\frac{d\mathbf{r}_{O_1P}}{dt} \bigg|_1 = \frac{d\mathbf{r}_{O_1O_2}}{dt} \bigg|_1 + \frac{d\mathbf{r}_{O_2P}}{dt} \bigg|_1
Accounting for frame 2’s rotation (using the general derivative formula):
\frac{d\mathbf{r}_{O_2P}}{dt} \bigg|_1 = \frac{d\mathbf{r}_{O_2P}}{dt} \bigg|_2 + \boldsymbol{\omega}_{2/1} \times \mathbf{r}_{O_2P}
The velocity transformation becomes:
\mathbf{v}_P = \mathbf{v}_{O_2} + \mathbf{v}_{\text{rel}} + \boldsymbol{\omega} \times \mathbf{r}
where:
- \mathbf{v}_P: absolute velocity of P,
- \mathbf{v}_{O_2}: absolute velocity of frame 2’s origin,
- \mathbf{v}_{\text{rel}}: relative velocity of P,
- \boldsymbol{\omega}: angular velocity of frame 2,
- \mathbf{r}: position of P relative to frame 2’s origin.
The \boldsymbol{\omega} \times \mathbf{r} term captures the effect of rotation. This is a general 3D equation.
For more insights into this topic, you can find the details here.