Welcome back after this practice section where we have found out that the eigenvalues of the Matrix
eight are given by these expressions here.
So you see here on the numerical values, but there's point for one, for two, this is clearly a square
root of two.
And you could also figure this out by just calculating analytically the roots of this polynomial here,
which I have done, and the analytical solutions are indeed two plus get rid of two, two and two minus
square root of two.
And we've learned before that the eigen frequencies omega aren't given by the square root of K over
m times lambda and in our case, Keenum are one.
So we just have to calculate here to square roots of these values.
OK.
So these are the eigen frequencies, and we need to understand now what these IGen frequencies mean.
And one thing that we can do before is we can analyze the EIGEN system.
So maybe you have heard from this by mathematics.
That's a matrix does not only have eigenvalues, but it also has eigenvectors which are closely related
to the transformation that we have talked about before.
And to calculate the system, we have to write and p dot pulling out.
Off our matrix and then dot h.
So not Ike, but just Ike.
And this gives us the whole eigen system where these ones here are the eigenvalues.
So the same as before.
And these ones here are the corresponding eigenvectors.
So this is here's what you get analytically.
These are the three vectors and you can see they correspond to these components that you can see here.
One over square root of two is approximately zero point seven.
Oh, seven.
OK, so these basically correspond one to one to these frequencies.
So the first vector corresponds to the largest frequency, which would be this one here.
And the values here day describe how the individual oscillators move with respect to each other.
So a preferred mode of oscillation would be each one of those three.
But depending on the frequency, the oscillators move differently.
For example, for the middle frequency here, which is square root of two, which would correspond to
this, I can I convey a value here which you still have to calculate the square root.
This would correspond to the second factor here, which is this one.
And it indicates that only the first and the third oscillator, they move and they move along the same
direction and the middle oscillator doesn't move at all.
This is probably because that's a mode where these two oscillators here, they both move and they move
in such a way that they leave this point in variance.
And this is a preferred mode of isolation of the system.
So you see the other two modes with the other frequencies are a bit more difficult because here are
all three oscillators move and they move also along opposite directions partially.
And you see that the middle oscillator moves a bit stronger, has a higher amplitude because a 0.7 is,
of course, larger than 0.5.
So in the following two lectures, we will use a four year transform and we will use a fit methods to
figure out how we can find these IGen frequencies that we have determined that we can identify them
with our numerical solution.
And after these two lectures, we will see that this pretty chaotic behavior here is, of course, not
really chaotic from a mathematical sense.
But this difficult looking behavior here, which is not at all harmonic, can be represented by the
superposition of three harmonic sine function, for example.
So these are just two superposition of three sine functions, and you may guess it already, since these
are the EIGEN frequencies.
These will be the frequencies for the three periodic functions, the three sine functions.
We will first analyze this by calculating the year transform of the spectrum here, and then later on,
we will establish a fit that will fit the three functions to the numerically generated data.
And then we really know what the numerical solution means.