Planar rigid body kinematics: acceleration
In this blog post, I explore the concept of acceleration when observed from different reference frames. I will present a thorough mathematical derivation to show how acceleration changes between frames, considering both translational and rotational motion of the frames. The analysis includes the introduction of essential terms such as the Coriolis acceleration and centripetal acceleration and shows how they arise from the mathematical formalism. The outcome is the general equation for relating accelerations between two moving reference systems. This article provides a clear, step-by-step process to achieve the final formula and discusses each term along the way.
When dealing with motion, it is often necessary to describe it relative to different points of view. Consider a point \mathbf P and two reference frames. Frame 1 is considered the absolute frame, and frame 2 moves relative to frame 1. My aim here is to determine the acceleration of \mathbf P as seen by an observer in frame 1, using the motion of \mathbf P as seen by an observer in frame 2 and the motion of frame 2 itself.
Starting with velocities, the velocity of \mathbf P in frame 1, \mathbf{v}_P \big|_1, is given by:
\mathbf{v}_P \big|_1 = \mathbf{v}_{O_2} \big|_1 + \mathbf{v}_P \big|_2 + \boldsymbol{\omega} \times \mathbf{r}_{O_2P}\big|_2
where \mathbf{v}_{O_2} \big|_1 is the velocity of the origin of frame 2, \mathbf{v}_P \big|_2 is the velocity of \mathbf P relative to frame 2, \boldsymbol{\omega} is the angular velocity of frame 2, and \mathbf{r}_{O_2P} is the position of \mathbf P relative to the origin of frame 2.
To find the acceleration in frame 1, I need to differentiate this velocity equation with respect to time, considering that the derivative must be taken in frame 1. Using the transport theorem, which states that the derivative of a vector \mathbf{V} in frame 1 is related to the derivative in frame 2 by \frac{\mathrm{d}\mathbf{V}}{\mathrm{d}t} \big|_1 = \frac{\mathrm{d}\mathbf{V}}{\mathrm{d}t} \big|_2 + \boldsymbol{\omega} \times \mathbf{V}, I find:
\mathbf{a}_P \big|_1 = \frac{\mathrm{d}}{\mathrm{d}t} \left( \mathbf{v}_{O_2} \big|_1 + \mathbf{v}_P \big|_2 + \boldsymbol{\omega} \times \mathbf{r}_{O_2P}\big|_2 \right) \bigg|_1
After some steps, the full expression for the acceleration becomes:
\mathbf{a}_P \big|_1 = \mathbf{a}_{O_2} \big|_1 + \mathbf{a}_P \big|_2 + \boldsymbol{\alpha} \times \mathbf{r}_{O_2P}\big|_2 + 2\boldsymbol{\omega} \times \mathbf{v}_P \big|_2 + \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf{r}_{O_2P}\big|_2 \right)
where:
- \mathbf{a}_P \big|_1 is the acceleration of \mathbf P in frame 1,
- \mathbf{a}_{O_2} \big|_1 is the acceleration of the origin of frame 2 with respect to frame 1,
- \mathbf{a}_P \big|_2 is the acceleration of \mathbf P with respect to frame 2,
- \boldsymbol{\alpha} = \frac{\mathrm{d}\boldsymbol{\omega}}{\mathrm{d}t} \big|_1 is the angular acceleration of frame 2 with respect to frame 1.
The additional terms are:
- \boldsymbol{\alpha} \times \mathbf{r}_{O_2P}: tangential acceleration,
- 2\boldsymbol{\omega} \times \mathbf{v}_P \big|_2: Coriolis acceleration,
- \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf{r}_{O_2P}\right): centripetal acceleration.
This final expression shows that the absolute acceleration of a point in a moving frame is not simply the acceleration in the moving frame plus the acceleration of the moving frame’s origin. Extra terms appear due to the rotation of the moving frame, namely the Coriolis and centripetal accelerations. These terms become particularly relevant when dealing with rotating systems such as in geophysics or machinery design.
For more insights into this topic, you can find the details here.