>> Yo, what up y'all?
Welcome back.
In this video, we're going to
continue our conversation about how
to translate the coupled pendula problem
into a standard eigenvalue problem.
Specifically, we're going to talk
about how to mathematize this problem.
When we mathematize the problem, what we mean is
turn it into mathematical symbols and equations
that capture the relevant features
of the problem we're trying to solve.
For students in my class, I'm going to
give you access to a separate playlist
that gets you ready to do lab
work on a McCusker Apparatus.
I'm going to bring one into class.
You'll have time to play with it, to set it
into motion, to capture video using your iPhone
or smart phone and then import
that video into Tracker.
In this video, though, we're going to
focus on observing some of the features
of the McCusker Apparatus that are most
relevant to mathematizing this problem.
Let's start our exploration by looking at a
schematic describing the McCusker Apparatus.
You see that schematic on the
left-hand side of the screen.
On the right-hand side is an
ideal diagram of the components
that are most important for
the mathematical model.
When we actually build one of these
apparatus, some features come into play.
First, the rods that define the ideal
pendulum are created using thin wire almost
like the wire used on a bike
for the brake cables.
That wire actually comes down in through a small
particle board that's cut very thin and then
up through the other side
hanging off a stable structure.
On top of the particle board, which Mike
calls a sled, we place masses or weights.
A creative use of those masses or weights
actually combines small weights used
in a free-weight gym with small
containers filled with sand
so that the two masses can be designed to
have identical mass measurements if desired.
In the ideal diagram, notice that we're thinking
about each of these masses at a point mass,
where in the actual apparatus, the masses are
distributed there are small center-of-mass
calculations that we'll do when we actually
do our modeling problem to allow us to figure
out how long from the point at which
the cables are attached to the center
of a mass horizontally is that value l?
When setting up an eigenvalue experiment,
we attach a camera from the top-down view
so that it actually films from
above the movement of these masses.
Moreover, we put a black dot in
the center of the mass on above
so that the camera picks up that black dot.
Finally, we move one of the masses
along this direction labeled here
as the direction of travelled.
Over here, we see that direction
actually has a metric ruler assigned.
Once we've moved that mass, we
let both masses swing freely,
and we capture the dynamics from above.
We capture the movement of the center
of masses using our camera from above.
From the standpoint of the
mathematical model, what's important is
that these cables act as a stiff rod.
They do not bend and buckle.
That the masses move in small
angle approximation.
In other words, these are not
wild swings, but very small swings
around the center of the resting mass.
Third, that the spring acts
as an extension spring.
So there's never a time where
the masses get so close together
that the spring coils touch each other
and turns into, like, a solid cylinder
and instead it always a little bit extended.
And then last that we're not losing a
ton of energy to the support structure.
Another way to say that is that the
structure itself is quite stable.
There's not a lot of vibrations
in the structure.
The only thing that's shaking
back and forth are the masses.
One of the beautiful thing about Mike
McCusker's design is that he captured all
of those specifications at a very low cost.
I think he made his for something
like 40 or 50 bucks.
And we could actually capture the fundamental
quantities needed for the mathematical modeling
as the length of each rod, the spring constant
in the spring and then each individual mass.
And that's all we'll need in order to
set up our entire mathematical model.
To bring this point home, let's
take a look at a side angle
of the McCusker apparatus that I built.
Then we'll look at a video
of a mixed-mode oscillation
of an eigenvalue experiment shot from above.
We'll start with a side view of
the real apparatus in action.
>> Here we have it.
This thing's ready to go.
This is an eigenvalue problem maker.
Check it out.
I can go ahead and put it in the
mixed mode, the normal mode 1,
normal mode 2, and we're ready to go.
>> Here is a bird's-eye view
of the motion of the masses.
This camera is attached to the top
of that structure shooting downwards.
We see that the center of each
mass is marked with a black dot.
There's a ruler attached at the bottom of the
screen, and that ruler actually has some marks
to say "Those two blue marks
are 10 centimeters away."
Then we see that the spring
always acts in extension.
Notice, it never gets close enough to the
other mass so that it turns into a cylinder.
This is really interesting if we
see the motion in the mixed mode--
This has two different things
going on, which we'll see later--
the energy is kind of transferring
from one mass to the other.
You see that?
So here the right mass is moving.
Mass 2 is moving.
And then all of a sudden, the spring
transfers that energy over to mass 1.
Mass 2 stopped moving.
The question that we're going to ask ourselves
is if we were to think about that center of mass
as being positioned along this ruler, the claim
is that we could actually map out that position
over time, and it would create
a mathematical function.
And our question in the coupled
pendula problem is
"Can we get an explicit description
of that function?"
Let's take a look at an ideal representation
of this problem that I created in Mathematica.
The point of the image that
you see is to slow things
down a little bit and let
us play with parameters.
I hope that in doing so we can get a very
accurate idea of the problem of trying
to describe the displacement
of the masses using functions.
In this case, what we've done is we've rotated
the masses so we get a side view of this.
The red mass will be the left mass in the video.
The blue mass will be the right mass.
And then we see that we have a
metric ruler on the left-hand side.
What we can now do is kind of play
around with the initial condition.
So I can change the location of the left mass.
Imagine that I pushed the left mass
towards the right mass to start with,
and then the question is
"Well, what happens over time?"
And indeed, what we see is
there's this function behavior.
Check that out.
I can actually play this over time.
Changing the initial position
of my left mass changes the way
that that actual function plays out.
Now, if I don't perturb either of
them, that's called stable equilibrium.
There's no displacement.
It just stays where they are.
But if I displace one of the masses to begin
with-- In this case, I've moved the mass one,
the red mass towards the
right mass to begin with.
And then we see that over time
these masses move back and forth,
and there is this energy transfer.
See, when this energy is
low, when the mass is moving
around that equilibrium position,
this one is high and back forth.
The question is "Well, what
happens when I move either one?"
And check this out.
There's this really interesting thing.
If I were to move both masses-- So
let's go ahead and restart this.
We see under the original condition,
if I don't displace either mass,
I have no change in displacement, and
my position functions are really boring.
They're constant.
If I change the position of one mass--
Let's say that I move the right
mass away from the left mass--
there's this really, really complex behavior
where the energy is being transferred
from the right to the left and back and forth.
And you kind of see that over here.
In fact, I can up the spring constant.
Yeah, there it is.
You see that?
When I have large oscillations
in the right mass,
I have small oscillations in the left mass.
And then vice versa, so there's
this energy transfer.
One of the claims I'm going
to make is "What if I were
to actually displace both masses the
same amount in the left and the right?"
Look at that beautiful structure.
Doesn't that look like a cosine curve?
And in fact, this is a demonstration
of what we're going to learn later
as called the first natural
frequency of the system.
The period of this frequency from peak
to peak, the period of that cosine curve,
is very much related to the first
eigenvalue and eigenvector of this system.
On the other hand, if instead of
moving both masses the same amount--
So I'm pushing both masses
here, and then they just swing
as a normal pendulum would swing
that's not connected by a string.
If I were to push the masses away
from each other the same amount--
In other words, I displace the right mass
by, in this case, positive 0.1 meters,
and I displace the left mass
by negative 0.1 meters.
Notice, these are antisymmetric.
They move towards each other
and away from each other,
towards each other and away from each other.
This is called the second natural
frequency or the second mode of this system.
It is also directly connected
to the second eigenvalue
and eigenvector pair of the
associated matrix model.
This antisymmetric motion stands in
contrast to the symmetric motion.
Notice that when we displace the right and
left mass together, they move as a pair.
So, literally, they're just moving
as a kind of coupled set of pendulum.
And the claim that we're going to make is
that when we move one mass a little bit
and move the second mass in a different way,
the function that gets out of this is going
to be related to something called a
linear combination of eigenstates.
Now, I'm stealing your thunder a little bit.
The point of this though is, in
this diagram, what we're seeing--
The goal of the coupled pendulum problem is
actually to come up with a function definition.
Like, describe this thing in terms of sines and
cosines, probably, because it looks periodic.
Can we come up with an explicit definition
of what this function is only knowing the
spring constant, the masses on each sled
and then also the length of the pendulum?
Of course, our solution has to take
into consideration the initial position
because that very much affects the dynamics.
But that is the coupled pendulum problem.
Trying to figure out the motion of those
masses under different parameters and come
up with an explicit function
to describe that motion.
