Relative motion and frames

Reference frames and coordinate systems

A reference frame (or simply a frame moving forward) is a view that allows us to make observations and measurements regarding the kinematics and kinetics of a system. Solving a dynamical system always starts with defining the necessary frames.

Frames should not be confused with coordinate systems, which is the set of scalars used to describe the location of a point relative to another point in a frame. Coordinate systems are complementary to reference frames.

The figure shows two frames, $\mathcal{I}, \mathcal{F}$, and their coordinate systems. Frame $\mathcal{I}$ has origin $O$ and uses the Cartesian coordinate system $(x, y, z)$, and frame $\mathcal{F}$ has origin $O\,'$ and uses the coordinates $(u, v, w)$ for its coordinate system.

As showcased in the figure above, a well-defined frame has the following:

  • A labeled origin.
  • Three orthogonal directions/axes.
  • A coordinate system.

To distinguish between the frames used when it comes to coordinates, we will specify the frame with a subscript. For example, $(x, y, z)_{\mathcal{I}}$ represents the Cartesian coordinates with respect to frame $\mathcal{I}$, and $(u, v, w)_\mathcal{F}$ represents the coordinates $(u, v, w)$ with respect to frame $\mathcal{F}$.

All in all, we will express our frames in the following way:

Frame notation.

\[ \mathcal{I} = \left[O, \hat{e}_1, \hat{e}_2, \hat{e}_3\right] \]

$\mathcal{I}$ is the frame name, $O$ is the origin of the frame, and $(\hat{e}_1, \hat{e}_2, \hat{e}_3)$ are the three orthogonal directions of the frame (the basis of the frame).

As stated previously, solving dynamical systems begins with defining the necessary frames.

Example Problem: Precession of a circular body.

Consider the following system, which shows the precession of a circular rigid body. Identify and label the appropriate frames needed to model the equations of motion of point $P$ on the body.

To model the dynamics of point $P$ on the body, we need 3 frames: the fixed frame, the polar frame to model the rotation of the body about point $O$ through the axis $\hat{k}$, and the rotation of point $P$ about the center of mass $C$ through the axis $\hat{k}$ (or more generally, the axis $\hat{e}_r \times \hat{e}_\theta$). We will denote the fixed, polar by $\mathcal{I}, \mathcal{B}$ respectively, and the final frame $\mathcal{F}$. Using frame notation, we have: \[\begin{aligned} \mathcal{I} &= [O, \hat{\imath}, \hat{\jmath}, \hat{k}] \\ \mathcal{B} &= [O, \hat{e}_r, \hat{e}_\theta, \hat{k}] \\ \mathcal{F} &= [C, \hat{e}_R, \hat{e}_\phi, \hat{k}] \end{aligned}\]

Later on, we will derive the equations of motion of point $P$.

We will be redefining a lot of terms in this page, and we will be emphasizing the importance of notations (particularly when it comes to labeling frames). When many different frames are used within a problem, labeling those frames will become very important.

Velocity and acceleration notation

We will be defining the notation used throughout this page explicitly here, specifically the velocity and acceleration with respect to a particular frame.

The velocity and acceleration of a point $P$ with respect to a specific frame is the following:

Velocity and acceleration of point $P$ with respect to frame $\mathcal{I}$.

\[\begin{aligned} \vec{v}_\mathcal{I}(P) &= \left.\frac{d}{dt}\right|_\mathcal{I}\vec{r}_{OP} \\ \vec{a}_\mathcal{I}(P) &= \left.\frac{d^2}{dt^2}\right|_\mathcal{I}\vec{r}_{OP} \end{aligned}\]

Warning: Only use dot notation when working with coordinates.

When working with vectors in different frames, it is best to express the velocity in fraction-derivative form, as it may look messy and confusing otherwise, particularly when it comes to expressing the acceleration as a time derivative of velocity, which is also relative to a frame. Using dots should be reserved to time derivatives of coordinates. For example: \[ \frac{d}{dt}(x, y, z)_{\mathcal{I}} = (\dot{x}, \dot{y}, \dot{z}) \]

The frame identification could be dropped as they are scalars.

  • Many textbooks use the notation $\vec{r}_{Q / P}$ to denote the position of point $P$ relative to point $Q$. This is equivalent to $\vec{r}_{PQ}$ in these pages (i.e the vector from $P$ to $Q$).

  • Many textbooks also use the notation $^{\mathcal{I}}\vec{v}_P$ to denote the velocity of point $P$ with respect to frame $\mathcal{I}$. Another common notation is $\vec{v}_{P / \mathcal{I}}$.

Relative motion

We will examine different types of motion of a frame relative to another. We will start with the motion of one frame at constant velocity relative to another frame, as shown below.

The figure showcases two frames, $\mathcal{I}, \mathcal{F}$, and their coordinate systems. Frame $\mathcal{I}$ has origin $O$ and uses the Cartesian coordinate system $(x, y, z)$, and frame $\mathcal{F}$ has origin $O\,'$ and uses the coordinates $(u, v, w)$ for its coordinate system.