This is the solution of

the characteristic equation that we derived essentially

expressing the relation between natural frequency and

mass one and mass two and the spring constant K1, K2, K3.

This looks rather complicated,

but the first thing we can find out is that,

first natural frequency is

bigger than second natural frequency,

which is obviously following

our general physical concept.

The other thing we can find

out is that equation essentially

shows as mass one and mass two changes.

For example, if mass one getting larger than mass two,

it certainly approach to certain value.

In other words, physically it means that

mass one is very,

very large than mass two,

meaning that the first natural frequency will approach to

the essentially based on

mass and the spring constant of K1,

because mass one is very big.

If mass two is very big compared with the mass two,

essentially vibration characteristics will

be dominated by mass two that expresses in this equation.

This summarizes what we learned before.

First, the case when mass

one and mass two is the same as mass M,

and if all the spring constants are same,

then the first natural frequency will be

K over M because in this case,

the two mass as we demonstrated before,

move with in phase,

in this case their mass is 2M,

and the spring constant over here is K and K,

therefore natural frequencies K over M,

which is fundamental or first natural frequency.

The second natural frequency will approach to

3K over M because in this case,

two masses are moving out of phase,

therefore the natural frequency room approach to 3K over

M. If this mass one and mass two changes,

and the spring constant K1, K2 changes,

then of course the first natural frequency

and the second natural frequency will be changing.

So to see the trend,

our changing of natural frequency compare

with the spring constant,

K1, K2, K3,

this graph essentially demonstrate what was going on,

or what will be changing.

So over here, that is the case when

K3 and K1 is very similar,

then, the stiffness over

here and over here is very similar,

for example, to K2.

Then, that is the case what we explained over here.

So the natural frequency,

omega two and omega one will

be as we demonstrated over here or approach

to this one minus this one.

But as the difference between

K3 and K1 is getting large and large,

then of course the difference

between omega two and omega one

will be getting large and

large as we demonstrated in this graph.

Similar patterns can be seen with respect

to the difference between mass one and mass two.