# 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.