3d rigid body kinematics: multiple frames acceleration
In this blog post, I explore the nuances of angular acceleration when dealing with multiple rotating frames of reference. A common misconception is that angular accelerations simply add up like angular velocities. However, this isn’t the case. I aim to clarify why, introducing the concept of the gyroscopic term.
Let’s consider two frames, Frame 1 and Frame 2. The angular velocity of Frame 2 relative to Frame 1 is denoted as \boldsymbol{\omega}_{2/1}. The angular acceleration, \boldsymbol{\alpha}_{2/1}, is the time derivative of \boldsymbol{\omega}_{2/1}.
When observing this from different frames, the transport theorem states:
\frac{d\boldsymbol{\omega}_{2/1}}{dt}\bigg|_1 = \frac{d\boldsymbol{\omega}_{2/1}}{dt}\bigg|_2 + \boldsymbol{\omega}_{2/1} \times \boldsymbol{\omega}_{2/1}
Because \boldsymbol{V} \times \boldsymbol{V} = \boldsymbol{0}, this simplifies to:
\boldsymbol{\alpha}_{2/1} = \frac{d\boldsymbol{\omega}_{2/1}}{dt}\bigg|_1 = \frac{d\boldsymbol{\omega}_{2/1}}{dt}\bigg|_2
This tells me that the angular acceleration is the same whether observed from Frame 1 or Frame 2.
Now, let’s consider three frames: Frame 1, Frame 2, and Frame 3. We have the addition theorem for angular velocities; the relationship for angular accelerations is:
\boldsymbol{\alpha}_{3/1} = \boldsymbol{\alpha}_{3/2} + \boldsymbol{\alpha}_{2/1} + \boldsymbol{\omega}_{2/1} \times \boldsymbol{\omega}_{3/2}
The term \boldsymbol{\omega}_{2/1} \times \boldsymbol{\omega}_{3/2} is the gyroscopic term. This term is a consequence of Frame 2 rotating relative to Frame 1, influencing the observed change in \boldsymbol{\omega}_{3/2}. It’s this term that distinguishes the addition of angular accelerations from the simpler addition of angular velocities.
For more insights into this topic, you can find the details here.