Hello, welcome to basics of finite element
analysis, today is the second last day of
this week that is the seventh week, and in
today’s lecture, tomorrows lecture and maybe
in the third lecture also we will discuss
Eigenvalue problems.
Or how what they mean, how they are formulated
and how are they analyzed, so that is.
Going to be our focus Eigenvalue problems
okay.
Now this term Eigen it comes from German,
and it is also there in Dutch language 
and it has several meanings but they are all
related, first meaning is own okay, second
meaning is unique, another interpretation
is characteristic, another interpretation
is peculiar or self.
So these are the ideas which when a German
speaks this term Eigen they come to his mind
own, self, unique, characteristic, peculiar.
In context of mechanical engineering problems
or any problem for the mathematical problems,
we say that system has an Eigenvalue.
It means that there may be some number which
is peculiar or it is a characteristic of that
system, so it will not depend on some external
force.
But it is, it is independent of some external
force because if I am putting some force on
the system then that number may change, but
there may be some number or some property
of that system which is unique about itself
which I which is its identity, another interpretation
of Eigen is identity.
So I will give you an example, first example
so I have a spring, mass and I apply a force
and let us say I apply a harmonic force ei?t
okay.
Now when I am exciting it with this force,
when I am exciting this system with this force
whose frequency is ? then the system will
get excited at the same frequency ? so this
?.
Is not a characteristic of the system okay,
but suppose I remove this force, so case when
F0 is equal to 0, and then if I disturb this
mass by a little amount then the system will
vibrate back and forth at a unique frequency.
Let us call that unique frequency as ?.
That unique frequency will depend on mass,
stiffness and the boundary condition of the
system and that unique frequency will be called
Eigenvalue, and Eigenvalue is one word of
the system.
So it does not depend on an external factor,
it depends on everything which is internal
to the system and what are the things internal
system boundary conditions, and in this case
a stiffness and the mass.
In another system it may be something different
but some things which are unique about an
internal to the system.
So Eigen values are related to something internal
things related to system.
Another example so, so before I go to another
example here the governing equation is m?+kx
is equal F0 ei?t okay, and if F0 is equal
to 0 if F0 equals 0 then my equation becomes
m?+ kx equals 0 or I can express it as ? equals
minus k over m times x or I can write this
as a function of x which is the right side
equals.
Some constant ? and this ? maybe an Eigenvalue,
so if it, so sorry before I use that, times
another function of x which is B(x), so if
I can express so if there is a system which
does not have which is not seeing any external
force and if I can represent that system as.
In this form A(x) equal to ? times B(x), then
? so A will depend only on the internals of
the system there is no external force in the
system, B also depends only on the internals
of the system, so and these two sides will
be equal if there is appropriate value of
?, so this ? is known as Eigenvalue of the
system, so this is Eigen value of the system.
So this is a very general statement for one-dimensional
Eigen value problems, it is very general statement.
(Refer Slide Time: 07:06)
We will see another example, axial motion
of bar okay, so you have the governing equation
is ? A, second derivative of displacement
so u is the displacement not velocity.
So ? A times acceleration of particle minus
EA du over dx and I am differentiating this
whole thing with respect to x, and on the
right side I have the external force, so how
is this, suppose this is a bar 
this small element of the bar maybe there
is some q, so this q could be a function of
x and t, this is my area of cross section,
E is the Young’s modulus, ? is the density
so the axial motion of this bar is governed
by.
This differential equation, if I have to find
the Eigenvalue of the system what should I
do?
I should make q to be 0 and then solve the
equation then the solution will give me the
Eigen values of the system, because when I
am solving it using q it is not everything
is not internal to the system right, so when
I am doing.
Eigenvalue of this then so far EV, first thing
I do is q I remove q, so I said q to be zero,
so ?A ü –(EAu') and then again I differentiate
this is equal to 0, and if I mathematically
process or manipulate this equation I can
again express this equation as A, so as a
function of, so A which is a function of u
is equal to some number times another function
of u.
I can express this as like this okay.
So this is, so I am again able to, so this
is my Eigenvalue formulation of the system
so this is again same format.
Which we saw earlier.
How do we get there we will actually see it,
but I can express it in this form, I will
do another example, so the other thing I wanted
to mention is that Eigen values are not only
not always about vibrational frequencies.
But they reflect something internal and unique
about the system.
We will see one example where they are not
the frequencies of the system.
But right now we will discuss beam vibrations
okay, so for a beam which is a cross section
A and a moment of inertia I and Young’s
modulus e and density is ? the governing differential
equation, the dynamic dynamical equation is
the ?A ?, second derivative of W, so W is
reflection plus EI(w?) equals f, f is the
distributed force 
on the beam, again if I have to find the Eigenvalue
of this system the first step I have to do
is I have to, I have to equate.
f to be 0 so for Eigen value 
calculation f(x) we make it 0 so my equation
is reduced to ?Aw?+ EI w excuse me, ?A times
second derivative of w + EI w? the entire
thing differentiated twice in x, this equals
zero okay, and again by mathematical manipulation
I can express this thing as in this form,
how do I do it, I can consider w as a function
of t and x, I can separate the variables and
I do all the mathematics I can express it
in this form by variable separable method.
I can use the same variable separable method
and I can get it here also okay, so that is
not a problem.
So in all these examples.
This example spring-mass system.
Axial motion of the bar and beam vibrations,
it turns out that ? represents the angular
frequency square of angular frequency okay,
that is what it turns out.
But now we will consider case where?
That is not the case where it is different,
so here we will discuss.
Buckling of a beam bar, what is it?
So I have a bar and I am applying some force
and if I keep on pressing the bar initially
the bar will just get compressed in the axial
direction.
But once the bar the force exceeds a particular
threshold.
The bar will do this, so it will buckle when
the compressive force P exceeds a particular
number, that particular number depends on
the characteristic of the system it does not
depend on P, so this is one case where the
Eigenvalue is not related to the.
Angular frequency of the system but something
different, so for this case the governing
equation is (EI w?)
+ Pw?
0 or in this case this is very straightforward,
EI w?
differentiated twice and the whole thing is
differentiated twice is equal to minus P w?.
Now here we directly see that this is nothing
but of the form A(u), A(w) is equal to ?(w),
it is of that form.
So this is again an Eigenvalue problem ?B(w),
so this is one example where the Eigenvalue
is not related to the natural frequency of
the system it is a different thing, here Eigenvalue
corresponds to the buckling load of the system.
And a last case is for heat conduction 
in a bar, so here if I have 
bar with cross sectional area A then the governing
equation is ?CA du over dt, so here u is equal
to temperature in this case okay.
And this minus KA okay so I have to make a
small correction it is now partial and same
thing here, ?u over ?x and the entire thing
is differentiated with respect to x, and this
equals heat which is generated internally
to the system per unit volume so this is the
differential equation.
Now in this case also, if I have to compute
the Eigenvalue this is the thing which is
the q is the term which is external to the
system right, so if q is equal to 0 then solution
of this differential equation 
will give us Eigenvalue okay, it will give
the Eigenvalue and again in this case also,
if we do this variable separable method we
will be able to express the overall equation
as A(u) equals ? B(u) this form.
So in a sense a quick summary of this thing
is that when we are talking about Eigen values
we are talking about finding some numbers
which are specific to a particular system
and they can depend on the material properties
of the system, they can depend on the geometry
of the system and they will depend on the
boundary conditions of the system.
What they will not depend is on any other
external any external factor, which could
be a force or a heat source or an excitation
or whatever okay.
So that is the overview of Eigen value problems,
next what we will do is we will actually solve
a particular Eigenvalue problem, and that
is for the vibrations in a bar that is what
we will solve, and that is going to be the
focus of ours for tomorrow's lecture.
So today what we have done is we have explained
what an Eigenvalue problem is, and tomorrow
we will actually solve one of these Eigenvalue
problems using the finite element method.
So that concludes our discussion for today
and we will meet tomorrow.
Thanks.
