Dynamics 28 - Relative Motion Analysis Using Rotating Axes

Video

#Physics #Engineering #Engineering_Mechanics #Dynamics

↩️ Go Back

Table of Contents:

A) Part 1: Relative Motion with Translating frame

When we study the relative motion of 2 particles (let's say "A" and "B"), we can begin our analysis by only using "translating" frames of reference. In other words, the motion of the moving coordinate system, let's say {xyz} is only "translating" with no "rotation".

In this example, ${\overrightarrow{r}}_A$ and ${\overrightarrow{r}}_B$ are absolute positions with respect to the fixed reference frame.

And ${\overrightarrow{r}}_{B/A}$ is the relative position vector of the two particles ($B$ relative to $A$), and it is simply the difference of the two absolute position vectors.

Also, if we take the time derivative of the relative position equation, we get the equation of relative velocity.

However, let's not forget something...

If we express the vector ${\overrightarrow{r}}_{B/A}$ with respect of {xyz}, we have to use its unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$.

And if we take the time derivative,

So, there is no need to apply the Product Rule when we take the time derivative, since only $x$, $y$, and $z$ positions are functions of time,

B) Part 2: Relative Motion with Rotating frame

So now let's revisit this scenario,

This time the moving coordinate system represented by the small case letters $x$ and $y$, is rotating with angular velocity $\overrightarrow{\mathrm{\Omega }}$ (and angular acceleration $\dot{\overrightarrow{\mathrm{\Omega }}}$) with respect to the fixed frame.

In this example, ${\overrightarrow{r}}_A$ and ${\overrightarrow{r}}_B$ are absolute positions with respect to the fixed reference frame (just like before).

And ${\overrightarrow{r}}_{B/A}$ is (still) the relative position vector of the two particles ($B$ relative to $A$), and it is simply the difference of the two absolute position vectors.

Now, to get the relative velocity equation (again), let's take the time derivatives of the relative position equation,

Remember, the relative position vector ${\overrightarrow{r}}_{B/A}$ can be represented in either the fixed frame or the moving frames. Let's write it with respect to the moving frame,

And now something very important has happened, because the {xy} frame "rotates", $\hat{i}$ and $\hat{j}$ are now variables!

Therefore when we take the time derivative of the relative position vector ${\overrightarrow{r}}_{B/A}$ we must apply the product rule,

And something looks familiar,

But what about the other terms?

Let's consider again the diagram that showed pure rotation,

First we have the rotating frame, with angular velocity $\overrightarrow{\mathrm{\Omega }}$,

After some time $dt$, we have a new orentation,

The magnitud of the vector $d\hat{i}$ can be approximated as the lenght of an arc with central angle $d\theta$,

Therefore, the time derivative of $\hat{i}$ is,

But we can also write it as a cross product,

And we can derive the time derivative of $\hat{j}$  in a similar way,

Now we can come back to our relative velocity equation,

And plug our new expressions for the derivatives of $\hat{i}$ and $\hat{j}$ ,

Therefore we get the final equation,