** **

** **

We will study **coupled oscillations** of a linear chain
of identical non-interacting bodies connected to each other and to fixed
endpoints by identical springs.

First, recall Newton’s second Law of Motion:

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** Newton’s second Law of Motion** Everyone
unconsciously knows this Law. Everyone knows that heavier objects require more
force to move the same distance than do lighter objects. The Second Law,
however, gives us an exact relationship between force, mass, and acceleration:

In the presence of external forces, an object experiences an acceleration directly proportional to the net external force and inversely proportional to the mass of the object.

This
Law is widely known with the following equation:

Where *F* is the net force, *m*
is the mass of the object which the force F acts upon and *a* is the
acceleration of the object. Since the acceleration is the second derivative of
the distance with respect to time, the above law can be stated as

_{}

where _{} stands for the
second derivative of *x* with respect to time *t.*

The velocity, force, and acceleration have both a **magnitude**
and a **direction** associated with them. Scientists and mathematicians call
this **vector quantity** (magnitude plus direction). The equation shown
above is actually a vector equation and can be applied in each of the component
directions.

A second law that we will need is Hooke’s law

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** Hooke’s law** discovered by the English scientist Robert Hooke
in 1660--states that the force

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Let us consider first the simple case of one mass attached to a spring
one end of which is attached to a vertical wall:

If *x(t)* is the position of the mass *m* from the equilibrium
position at time *t *and *k* is the spring constant, then Newton’s
second law of motion together with Hooke’s law give:

_{}

or equivalently,

_{}

This is one of the most famous equations of all physics. It is known as
the **harmonic equation. **Let

_{}

then (*w _{0}* is called the

_{}

where *A _{0}* is a positive real number representing the
maximum value of

_{}

where _{}and _{}are the values of the position of the mass and its velocity
at time *t=0* respectively. The **period** of the oscillation
described by formula (1) is

_{}

and the quantity

_{}

is called the **natural frequency **of the
oscillation.

In what follows we treat some more complicated cases of oscillations._{}

** 1) Case of two masses** Consider two identical bodies joined
up with identical springs on a frictionless track as follows:

Here A and B represent the
equilibrium positions of the two masses. Let *x _{1}(t) *and

The force acting on the first
body has two parts by Hooke’s law: the first part is *–kx _{1}* due
to the leftmost spring and the second part is

_{}

Similarly,
the net force acting on the second mass is

_{}

Applying
the second Newton’s of motion gives the following system of differential
equations:

_{}

This
can be written in matrix form as follows:

_{}

Let
us now find the eigenvalues of the matrix

_{}

that
appears in the above equation.

Recall
that the eigenvalues of *A* are those values of λ satisfying the
equation:

_{}

where *I* is the identity
matrix of the same size as *A*:

_{}

Let us now find corresponding
eigenvectors. If _{} is an eigenvector of *A
*corresponding to the eigenvalue 1, then one would have_{}, or *(A- I)X=0*:

_{}

which means that *a=b*. So

_{}

is a basis for the eigenspace corresponding to 1. Similarly, one can show that
the vector

_{}

is a basis for the eigenspace corresponding to 3.

Now let

_{}

then _{} where

_{}

is the “diagonal form” of *A*. Note that _{} so equation (*) above
becomes

_{}

Now, let us consider the following change of variables:

_{}

so,

_{}

Taking the second derivatives gives:

_{}

so equation (**) above becomes

_{}

after simplification. This gives the following system of harmonic equations:

_{}

_{}

that we know how to solve by the above simple case of a single mass attached to a spring.

**So, what is the physical
interpretation of all this? **

Well, it is not difficult to see that there are two special kinds of motions that one can easily describe:

*1. *Let
us look again at the eigenvector

* *

_{}

corresponding
to the eigenvalue 1. The fact that the components are equal tells us that *x _{1}*

* *

_{}

* *

2. In the case of the second eigenvector

_{}

*x _{1}*

_{}

Note
that

_{}

is an eigenvector corresponding to the
eigenvalue 3, this is why the _{} appears in the
frequency above.

These two particular cases are called the
**normal modes **of the system. As you ca guess, they have the property that
if the system starts out in one of these modes, it will remain in this mode.

Of course, the above problem involving
two masses can be dealt without talking about eigenvalues and eigenvectors. The
benefit of using that algebraic technique is more apparent in the complicated
cases of more than two masses.

** 2) Case of three masses **Let us consider the case of three masses:

Repeating the same argument as in the
previous case gives the following system

_{}

with

_{}

as usual. One can show that (please work out the details) that the
eigenvalues of the matrix:

_{}

are

_{}

and that

_{}

are corresponding eigenvectors.

One motion easy to describe is the one corresponding
to the eigenvalue 2: The mass in the middle is not moving and the other two are
moving in opposite directions. Each one of these masses has two springs
attached to it; this explains the eigenvalue 2. The other two motions are
little harder to describe.

Let us look now at a vibration example described by
the following diagram:

Using

_{}

the system can be represented by the matrix
equation:

_{}

where, as usual, the symbol _{}represents the second derivative of *x *with respect to
time.

Let us take this time the matrix *A* to be:

_{}

then its eigenvalues are _{}and _{}, and the corresponding eigenvectors are:

_{}

So, the **normal
frequencies **of the vibration are _{}and the **normal
modes **of vibration are as follows: