>> What's good, y'all?
Welcome back.
We are now ready to formally state the
coupled pendulum problem in a succinct manner.
So suppose that we have a coupled pair of
pendulum, each one composed of a mass attached
to a long rod with length l and
that the two masses on the ends
of these two pendula are connected together
by a spring with spring constant k. Of course,
we can measure each mass using some sort
of a scale, and we say that the left mass
or first mass has mass m1,
and the mass has mass m2.
We attach a ruler below the masses
that we can use to measure positions.
And then the coupled pendulum problem is
if we set one of these masses into motion--
So if I take the right mass and move it and
then let the entire system swing back and forth,
our challenge, if we choose to accept
it is to predict the exact location
of each center of mass along the ruler.
Here's another way to say that.
If I only know masses m1, m2, the length
of the pendulum and the spring constant,
when those things go in motion, I'm going
to get some sort of position function.
So here what I've done is I've taken the
right mass, displaced it a little bit
and then let those masses
swing back and forth over time.
And notice that if we plot the position
versus time or in this case the displacement
versus time, there's a kind of
complex back and forth motion,
and the pendulum problem is
describe those functions explicitly.
Write them, like, I don't know,
u1 equals something times cosine plus something
times sine etc. Not only do we want to be able
to write those functions, but the functions that
we actually converge on, the things that we end
up deciding describe our system, we should
be able to compare to the real dynamics
of the system to measured data on the
McCusker apparatus and see whether
or not our mathematical prediction have value.
We should be able to verify our model.
As I claimed previously, this coupled pendulum
problem is actually an eigenvalue problem
in disguise where the matrix A is 2 by 2
because we have two components in our system.
We have two separate masses that we want to
track the position of as we observe this system.
This is a keystone application.
This problem highlights both themes
that we saw in an introductory video,
and it sets a foundation
for more advanced stages.
As we study this problem and get more familiar
with the dynamics, I want to put a few ideas
in your head so that you can return
to these over and over again.
When we're trying to figure out the
behavior of our system, we're going to focus
on two very special cases that
show up as those masses move.
The first special case is going to be known
as Mode 1 of our system, and that corresponds
to something called the first natural frequency.
In this situation, what will
happen is we will perturb.
We will displace both masses the same
amount, and then as they move together,
they're going to kind of be
coupled so that the function
that describes the left mass is identical to
the function that describes the right mass.
In that case, the first eigenvalue
of the system is going
to be related to the period of that motion.
The second special case we're going to call
Mode 2, and that's the second natural frequency.
We say natural because think about
how a pendulum swings back and forth.
Notice that that pendulum swing kind of
looks like it creates a cosine curve.
So naturally uncoupled, a
pendulum would create a cosine.
So when we're thinking about natural
frequencies, it's the frequency or the function
that the pendulum naturally wants to take.
The fact that the pendulum are coupled
by a spring changes the dynamics
except in these special cases.
So for Mode 2 associated with the
second eigenvector and eigenvalue pair,
lambda 2 times x2, we're going to claim
that the behavior of the displacement,
the behavior of this function matches a pure
cosine curve or a pure periodic function
and the eigenvalue lambda
2 is going to be related
to the frequency of these periodic functions.
One of the most awe-inspiring
features of our process
of translating the coupled pendulum
problem into an eigenvalue problem is
when we have the mixed mode
oscillation, when we set both masses
with separate displacement,
so they're not the same.
And they're kind of doing this
crazy, complex functional behavior.
The claim is that if I have
information about Mode 1 and Mode 2,
the solution to this more
complex behavior is going to come
as a linear combination of the first two modes.
And I literally mean linear
combination as in linear algebra,
which means that we translate a very,
very hard problem into the sum
of two much easier problems.
When we first started this class, I
promised that we were going to see examples
of applied math modeling in action.
And we have this diagram where
we went from a real-world problem
to an ideal mathematical model to an ideal
solution to a meaningful real-world solution.
This coupled pendulum problem relying
on standard eigenvalue problems
is an example of this.
So, for example, when we start with a coupled
pendulum problem, that is a continuous problem.
It exists in time and space, and the behavior
of that problem depends on continuous functions
and what we call differential equations.
One of the things that we're going to see
is we're going to mathematize that problem.
We're going to turn it into an algebraic
problem, so that means turn a problem
from calculus into a problem in algebra.
Turn a problem that involves
continuous function into a problem
that relies only on scalar-valued variables.
Once we do that, we're going to have our
transformed standard eigenvalue problem.
And then we're going to apply eigenvalue
theory from this introductory class to be able
to come up with eigenvalue solutions.
So the two solutions we're going to have
are the Mode 1 and Mode 2 solutions.
That, literally, comes from looking
at a 2 by 2 matrix, not so hard.
Then we're going to do inverse
mathematization or reverse mathematization.
We're going to think about
the structures that we use
to transform our continuous
problem into our discrete problem.
We're going to undo those
structures and interpret the meaning
of these solutions in the continuous paradigm.
When we do that, of course, we
got to make sure that the model
that we've generated actually works,
and I'm really serious about this point.
You should not have to look at me and
trust me when I say this stuff works.
In fact, I want you to actively not believe me.
The real test of this is
not my word, my authority.
The real test is you take your own model.
Look at the original measurements
that we take on the McCusker apparatus
and compare the model behavior
to your own measurements.
And the question is "How closely do they align?"
Where they don't align, why
is the model inaccurate?
With that in mind, by way of our introduction
to the standard eigenvalue problem,
we're going to take our real-world
problem, study it and understand it.
That takes time.
Takes a lot of time to be able to understand
and state a meaningful problem
in the world around us.
Once we do that, we're going to
learn how to transform that problem
into a mathematical statement that looks
like a matrix times a vector
equals lambda times that vector.
That matrix will encapsulate
the dynamics of the system.
We're then going to use theory
from linear algebra
to get an ideal solution called the
eigenvalue eigenvector pairs of that matrix,
and then we're going to map
back and verify our model.
In the next video, we're going to
start our mathematization process
to see how we form this matrix
and set up our eigenvalue problem.
I'll see you there.
