Planar rigid body kinematics: multiple frames velocity
In this blog post, I explore the extension of rotational kinematics from planar to three-dimensional motion. I focus on the concept of angular velocity as a vector quantity and derive several key properties.
In two dimensions, angular velocity is a scalar quantity representing the rate of change of an angle. However, in three dimensions, rotation becomes more complex, requiring a vector representation. This vector, denoted by \boldsymbol{\omega}, not only gives the rate of rotation but also defines the axis of rotation.
The fundamental equation relating the time derivative of a vector \mathbf{V} as observed in two different frames (1 and 2) is:
\frac{\mathrm d\mathbf{V}}{\mathrm dt} \bigg|_1 = \frac{\mathrm d\mathbf{V}}{\mathrm dt} \bigg|_2 + \boldsymbol{\omega} \times \mathbf{V}
where \boldsymbol{\omega} is the angular velocity of frame 2 relative to frame 1.
A property of angular velocity is its uniqueness. If we assume two angular velocity vectors, \boldsymbol\omega_1 and \boldsymbol\omega_2, describe the same rotation, we have:
Another property relates the angular velocity of frame 2 relative to frame 1 (\boldsymbol\omega_{2/1}) to the angular velocity of frame 1 relative to frame 2 (\boldsymbol\omega_{1/2}):
\boldsymbol\omega_{1/2} = -\boldsymbol\omega_{2/1}
The addition theorem for angular velocities is essential for analyzing systems with multiple rotating frames. It states:
\boldsymbol\omega_{3/1} = \boldsymbol\omega_{3/2} + \boldsymbol\omega_{2/1}
This theorem is useful in situations where I need to determine the overall rotation of a system composed of multiple rotating parts.
For more insights into this topic, you can find the details here.