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Next to consider the free damped motion.

Free damped motion means there is some damping but no external force,

so that our differential equation,

becomes x double prime plus 2 lambda x prime plus omega square x is equal to 0.

And its characteristic equation is,

as you can read it from this,

r squared plus 2 lambda times r plus omega squared and that is equal to 0.

The root to sub this quadratic equation is given by,

R is equal to negative lambda plus or

minus square root of lambda squared minus omega squared.

So, depending on the sign of this discriminant lambda squared minus omega squared.

When this is a positive and when this quantity is 0,

and when this is a negative,

we have a three distinct cases as the following.

The general solution, depending on the sign of this,

the discriminant, lambda squared minus omega squared,

we have are three distinct cases like, first,

when the lambda squared minus omega squared is a positive,

which we call as overdamped.

Then, we have a two distinct real roots for the characters to

equation given by this quantity so that the general solution,

the motion of the masses x of t is equals to e to the negative lambda t times c

one exponential square root of lambda squared minus omega squared t and plus c two times

e to the negative square root of lambda squared minus omega squared t. The second case,

when the lambda squared minus omega squared is equal to 0,

then if this is equal to 0 then we have only one double real root,

r is equal to negative lambda.

In that case we quote it as a critically damped,

and the general solution we know is given by the x of t is equal to e to

the minus lambda t times c one plus c two t. Finally,

if this quantity is a negative,

if this lambda squared minus omega squared is a negative,

which we call as the underdamped case,

the solution will be x of t is equal to e to the minus of

lambda t times c one cosine square root of omega squared

minus lambda squared t plus c

two times sine square root of omega squared minus lambda squared

t. So we have those three distinct cases

depending on this discriminant lambda squared minus omega squared.

Graphically, you can see that the first,

the overdamped case, which is the Case 1,

Case 1 down there,

in this case the typical graph is given by,

by this first graph,

which is the overdamped cases.

Second, in the second case,

in which the lambda squared minus omega squared is equal to zero.

So, that solution is given by this one.

Its typical graph is given by this second graph,

and in the third case when lambda squared minus omega squared is negative,

so that underdamped cases in which the general solution

involves not only the exponential term but the cosine and the sine term two,

then this case the graph looks like the last one.

This is the graph for the underdamped cases.

What we have to observe from this three cases,

all these three cases are the following.

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Those motion x of t is non-oscillatory or three in the Cases 1 and 2.

But there is some oscillation in the Case 3 as you can see clearly from the picture.

Second, in all three cases,

in all three cases as you can see,

as you can see from the solution,

from the solution this in the first the case,

in front of we have exponential minus lambda t. In the second case,

we also have exponential minus lambda t. In the third case,

we also have an exponential minus lambda t,

which goes to 0 exponentially when tan goes to infinity.

So, in all three cases we expect,

the limit of x of t,

when t turns to infinity is equal to 0.

And the Case 2, the second graph,

this one is called the critically damped.

It's called a critically dumped because any small decrease in the damping force,

results in the oscillatory motion.

If you decrease the damping force a little bit,

then the sign of the discriminant becomes the

negative so that your general solution will involve the sine or cosine.

In the last Case 3, of which the solution is given by this one.

For this second term in the parenthesis,

you can combine this two into one using the trigonometry identity again in the following.

The last case, the solution can be written as,

A to e times exponential negative lambda t, times

the cosine square root of omega squared minus lambda squared t minus phi,

where as the user the capital A is equal to square root of c one squared plus

c two squared and A times the cosine phi is the c one and the A times the sine phi,

that is equals to c two.

Look at this equation (7).

Because of this exponential term,

the function x of t is not periodic at all.

But we have, by looking at this a second term, cosine term,

this is a periodic function which has a period two

pi over square root of omega squared minus lambda squared.

So this is not genuine,

the periodic function but,

because of this second term,

we call this quantity two pi over

square root of omega squared minus lambda squared, the quasi period,

and it's a reciprocal,

that is square root of omega squared minus lambda squared

over two pi the quasi frequency.

In Fact, the quasi period is

the time interval between any two successive maxima of x of t or minima.

As you can see from the picture,

from this point to another minimum point,

so from this to that,

the time needed, that is a quasi period.

And finally, let's consider the forced damped motion.

So we now consider the non homogeneous differential equation.

To say this is the equal to,

you can remind x double prime t plus two lambda x the prime of t and

plus omega square x of t and that is equal to g. And here we are we are shown that,

there is some external force, f,

which is not necessarily zero.

So that g is not equal to 0.

So that's a non-homogeneous second order constant co-efficients differential equation.

And we know that,

the general solution of this non-homogeneous differential equation,

x of t can be decomposed as the sum of x of sub c of t plus x of sub p of t,

Where x of sub c of t is a general solution of

the corresponding homogeneous differential equation when g is equal to 0.

So, in other words, this is a general solution of the free undamped motion.

On the other hand, the x of p of t is

any particular solution of ordinary differential equation.

What I mean is,

x of c of t is a general solution of

corresponding homogeneous equation that is equal to 0.

From this you are going to get x of c of t and x of p of t is

the particular or solution of this given non-homogeneous differential equation.

In particular, if this external force is the periodic force,

given by a cos gamma t plus b sin gamma t,

which is a periodic,

then there is a particular solution of the form.

x of p of t equals to a cos alpha gamma t plus B times the sine gamma of t.

We know the form of this particular solution by

the method of the undetermined coefficients.

And also we know that the corresponding homogeneous solutions say,

x of c of t, the complementary solution,

it goes to 0, when t turns to infinity.

So, that means what?

For the solution given by x is equal to x of c plus x of p,

let t turns to infinity.

And we know that this turns to zero so that for large enough the time,

x behave like the x of t of p,

for t big enough.

For t large enough, the our solution x of t,

it just to behaves like x of p of t. In this sense,

we call the complementary solution x of c of t, a transient solution,

solution value for the finite time and the particular solution x p of t,

a steady state solution of differential equation.