So, what we have is I think three flaps, two
lag, one torsion, one axial. About r 3 appear
3 2 1 1 3 2 1 1 1 or 6 or 7, 8 modes, or 9
modes. We take r 4, flap 2, 3 lag torsion
then we do system.
See we all learnt very simple equation; first
order equation if you take x dot equals a
x a is a constant this you write immediately
the solution as x is equal to some constant
e to the power what right x of t. This is
the solution; the constant is the initial
condition may be at equal to 0 x is equal
x naught. So, you may write this as basically
x naught e to the power a t.
That means this particular term as time evolves
from 0, is changing the initial condition
in the sense not that change initial condition
is changed; initial condition is getting multiplied
to sometime varying function gives you the
response that any specific time. Similarly,
let us now go to this equation. Now, how will
I write my solution a f t can be it is still
a linear system it can be any time varying
function now if you write assume this solution
x of t as e to the power integral t 0 to t
a tau d tau and then x 0 x 0 is a initial
condition.
This is the solution to this equation because,
if you differentiate this what will happen?
x dot e to the power. So, that will stay as
it is. So, e to the power minus t 0 t a tau
d tau then you differentiate this, but that
will be nothing, but a t and x naught and
since this term this is x t. So, you will
find a of t x. So, you stay this is my solution
all right now you just say similar I said
this is x naught if the initial condition
and there is a time varying function which
is getting multiplied to initial condition
to give you response at any particular time.
This is called state transition; if it is
a matrix, it is a matrix problem. Here, it
is just only function and this state transition
phi is e to the power integral some t 0 to
t a of tau d tau. Now, one key difference
between this and that here, this is also you
can write if a is a constant. If a is a constant,
what will happen? This will come out; you
will have t minus t naught. But, t minus t
naught is one term; that means, the difference
between the initial and the response at any
time.
So, if it is a constant thing you will find
the response is only the difference between
the times. Whereas, if a is a general time
function you will find t naught as well as
t, but they it will not be the difference;
it need not be the difference. It is a function
of both t and t naught; it is not the difference
between the initial and final time that is
determining the response. This is the character
and this is called the state transition in
the matrix form because we are, we do not
solve this problem matrix you called a state
transition matrix.
First order one variable you can solve. Now,
you extend it to matrix equation; that is
the starting point. Why we have to go for
matrix formulation is, I am writing now you
take it x dot is a x. This you all know if
a is a this is a, please note this can be
n by 1. This you must have studied; you do
not actually, this is called the state space
representation or a first order representation.
This is basically called state space because,
if you are given a ODE linear all we are talking
linear; if you are given a ordinary differential
equation of any order please understand, it
can be second order it can be third order
can be nth order you can convert that into
this format always I will just do that little
bit.
So that you know that, suppose you take the
simple equation you are all familiar with
the second differential equation m x double
dot a x is a let us take it 0 by f of t means
otherwise that will be a forcing function.
Now, this is a second order equation I can
have third order also. How will I write in
this form? This is what the state-space representation
is.
What you do is, there is a very clear simple
thing; you start, you call this as x 1 you
call this as x 2. Now, your equation x 1 x
2 dot time derivative that is x 1 dot x 2
dot x 1 dot is what X 1 dot is basically x
1 is x, x dot is x 2. So, x 1 dot is x 2 now
x 2 dot is nothing, but x double dot because
x 2 dot, but x 2 is x dot therefore, that
is nothing, but x double dot. So, you can
this stage you take this equation minus e
x dot minus k x right; you know this is x
1 this is x 2 this is x 2 dot. So, you will
have minus this is x 2 this is k by m minus
c over m that is all.
This is the these takes place form because
what we have done is why state is in this
case we are talking about possession is a
state velocity is another state. If I know
possession and velocity then, I know the other
derivative because if I know x and x dot I
know x double dot is not that I have to solve
separately for that. Do you understand? But,
I must know x and x dot, is it clear? Now,
you see this can be converted into matrix
form also what you do is instead of single
values you may have m x double dot. These
are all matrix; now how will you get the form
is you will put like this x 1 x 2 because,
this is again the vector. So, you put that
you will put 0 1 and here you put minus m
inverse k minus m inverse c and you will put
x 1 x 2; that is all do the matrix is it clear.
So, what we have converted? What is the, one
second order differential equation is converted
into two first order differential equations;
but they are coupled is not that they are
independent not like that form. So, one second
order you convert to two first order coupled
if there are third order equation, you can
convert into three first order equation.
Why all this is done is because, then you
have only one solution technique that is it
you solve only this problem. If you know the
solution technique for this problem, every
problem can be handled. Do you understand
the ODE? That is why you learnt; now you will
be able to relate why you learnt matrix algebra.
See these are very… see initially when it
is start - each is start independently then
you say I am able to relate various things
because if I know, because we are going to
have in the aerospace field at least our stability
problems, vibration problems, load anything,
rotor blade, all problems will come under
that. We will try to put it in this; in this
type of format and then immediately convert
to this format to get the solution.
Now, you know how to get the state-space form.
That is why if you are given a differential
equation you should be able to convert to
state-space form. So, this is a standard that
is all you take this take the other term out
a m inverse this goes to m inverse k m inverse
c straight. Now, we will go and then solve
this problem because the solution if a is
a constant matrix .
That is x dot is if A is constant then you
will say solution can be x naught this is
a vector or in other words, you may write
it in this fashion: e to the power a t x 0
e to the power a. Please understand, this
is a matrix; you can put it if you want, you
can put it like this is not that every element
is exponentially increased, please understand.
This you can expand like I plus 
that Euler expansion 1 by 2 I think, 2 factorial.
What a square right? like that it goes this
is the expansion. Now, if I differentiate
this into you will put x 0 I differentiate
this what will happen? This will be 0 this
will be A.
So, if I put x dot this is 0 this will become
A 
plus A what square? t plus A cube t square
over 2; like that it will go right into anyway
x naught will be there. You take out the a
right; when you take out A then what is left
is a this is, this A term will be there because
if you take a outside this side this is nothing,
but 1 plus A t plus the whole thing which
is nothing, but the... So, you will have a
e to the power a t x naught this is nothing,
but A x. So, x dot is A x.
Now, that is why you are doing all these things.
Slowly you will realize you can first thing
is you learn about diagonalisation of a matrix
- real matrix. So, you have the similarity
transformation you get the eigen values eigenvectors
you pre multiply and then do because when
you do that if this matrix you can convert
it like the you know that q inverse what I
think I will put it A is Q inverse lambda
q something like that similarity transformation
right yes or no. Somewhere you have learnt,
yes this is what. Now, if you substitute this
here what will happen is this will become,
you can take out q q the inside one will be
nothing but e to the power lambda t that is
the eigen value.
So, you really know whether the system is
stable or unstable or anything like that because
a square is what a into a into a what will
q q inverse will go up will get lambda square
So, we will have a q inverse.
So, I will write it may be that will become
see you substitute this here then what will
happen x becomes I plus a is Q inverse lambda
Q t plus 1 by 2 factorial Q inverse lambda
square Q because in between Q inverse Q they
will become identity because A square is what
A A you substitute this and like that this
will go on and you will have x naught what
you do is you can write this as Q inverse
Q So, this can come as Q inverse 1 plus or
I you can take it lambda plus lambda square
over into Q x naught this is what.
So, Q inverse a to the power lambda t Q x
naught you follow now you see it is a very
interesting thing the eigenvector please remember
the eigenvectors of the matrix A is the same
as the eigenvector of matrix this e to the
power A t is a matrix only thing is you have
to you do not know how to evaluate that is
all because it is a series right the eigenvector
of this matrix e to the power A t is same
as the eigenvector of A, but eigen values
are not same because this is eigtenvalue is
lambda for this that eigen value is this e
to the power lambda t.
Now, you basically, what you do for this kind
of problems if you are given you want to analyze
stability because all flight mechanics problems
everything you analyze stability you want
to know whether system is stable or not you
look at straight away go solve the eigen values
of this problem once you know the eigen value,
it will be a complex because we are not dealing
with real symmetric positive definite nothing
of that sort because you see this is not the
symmetric matrix first of all of course, this
is the real matrix this is not symmetric.
So, eigen values can be real complex anything.
So, there is no specific condition do you
understand? So, you get the routes and those
routes essentially tell you how the system
is stable or not and if it is stable how much
stable or if it is unstable how far it is
unstable. In the route locus now you know
that why route locus is used and then you
want to change the system because, now you
say I go and change my system a. So, that
makes it stable sometimes you cannot make
it stable then you start feedback control.
This is all the whole thing starts; the entire
subject of then you can have an external loading
then the control system if you go they will
put x dot equal A x plus B U. Now, for helicopters
U is theta naught theta 1 c theta 1 s theta
tail rotor.
So, you use the control angle pilot gives
then, this now you say you want to stabilize
automatic stabilization; that means, you have
to go give a feedback based on how a system
is measured. That we leave it; you will not
bother about the feedback control and other
things, just basic system. This is the system;
is the system inherently stable or unstable
is determinant purely by the matrix A. Here,
we look for the eigen value. Eigen values
will have the complex routes real part imaginary
part.
So, you say your lambda some kth route sigma
k plus minus i omega k. If sigma k is positive
then you say e to the power positive is, it
will grow with time. So, it is unstable; sigma
is negative, it is stable, but this omega
will make it oscillatory. Whether it is oscillatory
mode or it is a non-oscillatory mode, some
modes can be oscillatory some modes need not
be oscillatory you can only real routes; that
means, this will not be there.
So, this is what you analyze. Now, the question
is if a is not constant plus for our if you
take the flap equation itself it is a time
varying just the flap equation because you
can write flap equation in that format. That
state-space form; the moment you put in state-space
form you will have sin psi cos psi which is
basically that is omega t times varying. So,
your matrix system, this is time varying if
it is time varying how will I solve is a question.
Now, time varying arbitrarily is one another
one periodically time varying in the sense
a repeats itself in the flap motion, it repeats.
Now, that is slightly I do not think we will
have time to do that part, but that is very
interesting. How do you do that is the Floquet
theory. Now, we will only discuss I will just
briefly say if a is the function of time how
do we get the solution any arbitrarily function
of that so, I will erase this whole thing.
We will say this is a A of t now we can only
write the solution this is a matrix may be
n by 1 n cross n and n cross 1. I am writing
the solution because, we said the initial
state in getting change that is all. So, I
put this is the initial t naught is the initial
condition phi t is the state transition matrix.
This is a n by n non-singular matrix, but
what is it satisfying? It satisfies phi dot
t comma t 0 is a of t phi t comma t 0; this
has to satisfy this condition satisfying the
condition you can put it this is the condition
please note phi dot this looks like this equation
itself, but here these are states this is
a state transition matrix.
So, you may tell this as a state transition
matrix; now, you can verify why it has to
be that is we know that this condition right.
The first thing is if you set t equals t naught;
that means, this is x t naught x t naught
equal to phi t naught t naught that means,
phi t naught t naught must be I this is I
first condition.
So, you know here phi t naught t naught is
I next you take a derivative of this equation
because we will give a proof that it has to
be like this if you give a proof x dot t is
what d over d t of phi t comma t naught x
t naught which is phi dot t comma t naught
x naught sorry x t naught which is also this
is what this is nothing, but our original
equation A A of t x this is what this equation
is.
Now, what is x x is a t this. So, phi phi
comma t naught x t naught. So, you see this
is equal to this because either that you substitute;
now, in this you can substitute this is what
A t phi t t 0. So, phi t comma t 0 x t 0.
So, you see x dot t is this x dot t is this
which means both are same therefore, the characteristic
of the state transition matrix is it must
satisfy this condition and phi t 0 t 0 is
I now you set the condition, but how do you
get it? There is no easy way to get it; you
cannot write it as a matrix exponential. Why
we cannot write it as a like we wrote e power
A t in the single case we put integral t 1
to some t 0 to t a f tau b tau can we do the
same thing here you can do provided and show
condition is satisfied otherwise you cannot
do is this clear I will erase this part. So,
you are given the proof that phi dot, what
is the condition for that transition matrix
it must satisfy this with this, that is all.
Now, can we write phi as a matrix exponential?
that is I am going to put as you said this
is the equation phi t t naught I am going
to write 
can we write 
e to the power because what was our equation?
Our equation is phi t t naught dot equal A
A t phi t t naught can we write in this fashion?
You can write provided that a f t and integral
commute please note provided 80 and this commute
- commute means this time this is equal to
that time;this a b equal to b a only if they
commute otherwise you cannot do it.
But it is possible only when A is constant
or if A is diagonal otherwise you cannot do
it for a general case you not say A B equal
to B A please note that that in the matrix
2 into 3 number yes 2 3 is 3 2 that is 6 this
is 6, but you cannot put it for matrices.
So, these are all very important that is why
as on operation is it commute. So, provided
it commutes you can write the solutions suppose
if they do not commute sorry this is not the
solution then how do you say they have to
commute? That is purely from the this proof
is here I will give you proof that is you
take you do them expansion this e to the power
this is a matrix only.
So, you can expand it expand it I plus integral
t 0 to phi a f t sorry a f tau d tau plus
2 factorial you will have 2 t 0 t a tau d
tau a tau b tau plus, so on, so on, so on
Now, you see this is the expansion now if
I differentiate phi dot phi dot this is this
is A t, but this will be 1 by 2 factorial
A of t integral t 0 t A tau d tau plus 1 by
2 factorial integral t 0 t A tau B tau and
then A t because this first you take this
thing now if these two commuting then this
is you can take out a outside if they do not
you can write this solution you follow.
Now, this is the main problem, if they do
not commute, how will you write the solution?
If they commute you can put it in this format.
So, do not commute; I will write the solution.
It is no way you can do by hand; it is a you
have to numerically calculate or some simple
problem you can integrate them. So, what I
will do? I will show that part if they do
not commute the general solution this is again
a series.
So, general solution is this is a series expansion:
I plus integral t 0 to t a of tau b tau plus
I will put tau 1 a tau 1 I open a bracket
here I put one more t 0 to tau 1 a tau 2 d
tau 2 close put d tau 1 please note this is
a integration inside you do. And, if you want
to go third term it will be still t 0 to t
A tau 1, you open a bracket t 0 to tau 1 A
tau 2 open one more bracket put integral t
0 tau 2 A tau 3 d tau 3 close d tau 2 then
close d tau 1 plus. So, on. So, on. So, on
this you can show because what you do when
I differentiate this phi dot this is 0 this
is a t here what will happen because the entire
thing is over. So, tau 1 this will become
a t and this will stay as it is because this
is just dummy index like that what will happen
is 1 by 1 it will stay as it is and this is
the c d solution we have to do numerically
only you cannot there is no way; you can compute
this easily this is as far as a general a
of t is concerned.
Now, I will give you one small example. I
will solve after that we will, I will just
briefly mention, but we will not be able to
do it because that derivation takes lot of
time that proof, but that is an interesting
thing.
Suppose you take a simple problem 
dot dot 0 1 0 t; now you try to get the equation
by yourself with t t 0 as 0 compute transition
matrix transition matrix bearingly what is
the solution to this problem. So, you solve
phi t comma t 0. So, what it will have very
first is I I is as it is then the second term
will come A tau d tau you have to put that;
that means, 0 1 0 t put the integral that
will become 0 tau.
So, this will become first terms that is integral
0 to t A tau d tau is nothing, but 0 to t
0 1 0 tau d tau which is 
that is a first part first term this is the
first term now you have to go to the second
term. So, this is I maybe I will write it
this is 1 0 0 1 and this term is 0 t 0 t square
over 2 then plus we have to calculate this
term that will be because we can use tau maybe
I will do it; that term will be 0 to first
evaluate tau 1.
Now, this will be, you 
are going to have this is tau 1 tau 1 square
over 2. This you multiply to now you have
to do this 0 to t A tau 1 A tau 1 will be
0 1 0 tau 1 multiply to 0 tau 1 0 tau 1 square
over 2 d tau 1 because this you put then you
multiply when you that then you take that
matrix multiplication this will become what
integral 0 to t I think 0 0 I think tau 1
square over 2 and then tau 1 cube over 2 d
tau 1.
Now, this again 0 0 this is tau 1 cube by
6 and the limit this is tau 1 4 by 8. So,
that will be t. So, the third term will become
like this 0 t cube over 6 0 t 4 over 8 plus
so and so forth because, I am not this is
what the general solution is.
Now, you add all of them; that is your solution.
So, 1 this term will be 1 here you will have
t t cube by 6, etcetera. Here you will have
1 plus t square over 2 plus t 4 over 8 etcetera.
So, like that you will have a long series
this you have to do only numerically. Now,
the next question comes you quote a very general
A t. I am not interested in a t I am interested
in A t which is periodic, but there are very
special equation. This you may be I do not
know whether you are exposed to or not there
is something called a Mathew equation Mathew.
It is by, I will just write that equation
this is just for your this is Mathew equation
this is one or you can write it in another
form also there are various types of ways
it can be written another way is the write
d square f by d t square sorry d x square
lambda minus h square cos 2 x f 0 any form
maybe all are pretty much same this is called
Mathew equation here it is periodic.
But of course, this is the second order you
have to convert to first order you can do
what initially people are interested in seeing
for what combination of lambda and h square
combination the solution is stable. So, if
you look at some advanced books they call
it some c function s function something like
that though will be there; they will give
the condition of stable condition; that means,
at this for this combination of values I will
get a stable… stable means a periodic solution
and there is a diagram; I cannot draw the
diagram.
The region of stability for what combinations
it will be stable; for what combination the
system is unstable, but it will not tell you
what is the value of damping you can only
say stable, it is unstable. now you may ask
how stable it is am I near the imaginary axis
how close I am. see if you are very far on
either say you are highly stable, but if you
are very near your stability is just a marginal
stable.
That result will not give you that, but it
will tell you the regions; that is the separate
formulation derivation you assume a series
solution and then put the condition it is
stable then what should be the values there
are various the period of the solution can
be t or t by two you know various types of
sin function, cos function, all those things
are there; I leave out that part.
The theory says if x dot is A of t x t and
A of t plus t is a of t periodic because,
we are interested in periodic solution. If
I will not give the proof, the proof I have;
I will simple mention this statement if you
have this condition, what is the form of,
please understand, by transition matrix not
a form, this is my equation which is periodic
please understand.
This is a time varying equation. Only thing
is time varying is rather than very general
I wrote that long; what is that? That series
solution if A of t is periodic, the Floquet
theory; this is what Floquet we call it Floquet
Liapunov theorem. Floquet theorem - anything
you can tell Liapunov you may find it in different
form, but I have it from some notes I am keeping
this way.
What it says is the form of this; there is
a proof I have it will be in this form 
and p of t is p of t plus 
the form of the transition matrix is this
the proof is here. I will not go into the
proof because, it takes about 2 3 pages of
notes, but it is a very interesting proof.
Now, you know this is the transition matrix.
You try to find out basically r is a constant
matrix r is constant matrix. The whole purpose
is finding out the transition matrix. First,
how r is related to the, I will tell you,
what r is related? That is at the end of the
proof it will give you; this R is a constant
matrix which is given by e to the power R
T is phi t comma 0 e to the power capital
R into T 1 period.
So, what is done is, you start at 0 the end
of one period. What is the transition matrix
you evaluate because you are given the equation
all transition matrix have to be, have to
satisfy what this equation right sorry a t
phi t comma t 0. They have to satisfy this
equation; always you try to calculate this
matrix. By solving this differential equation
at the end of one period, you calculate. Once
you calculate you know the relation. You find
out the eigen values of this. This eigen value
is related to this eigen value. Immediately
you will know the eigen values of R; what
is the real and complex. If it is stable,
you will say my system is stable; if it is
not my system is unstable.
So, this is the procedure that is adopted
to find out whether stability of the rotor
blade in forward flight; not for simple flap
problem. Flap lag torsion stability they use
the Floquet theory get the transition matrix
and then roots of that. Theoretically, these
are all done; theoretical calculations industry
whether they use it normally they do not bother.
I will put it away because you try to design
the system size that you do not get into any
of these problems. This has an interesting
result; you want to know what is the damping
that is available; you want to know what is
the frequency, what is that then? Academic
side yes it is very important, but one of
the equation I wrote you, this equation right?
The problem, this is not just arbitrary. There
are several examples for this. One of the
example because you are all structures people,
one of the problem is this I think it is a
some cant beam with the… it is a buckling
problem.
You all know Euler buckling static problem
but, only thing is this tip load is time varying.
If you write the equation for this in fundamental
mode because this is a continuous system you
have to do single mode. You do finally, it
will come to this form; it will come to this
form in one mode. Each mode it will be the
same equation. First mode, second mode, all
modes will then they start looking at whether
stability of this for what values of lambda
x they are related to the this properties
and then p 1 how they vary.
Whether the system will be stable or unstable;
unstable means, it will just vibrate and then
break. So, if you will go to Google search
Mathew equation. They will give various places,
but for structures from the point of view
of vibration and this is the problem. Suppose
I give you this problem, go and derive the
equation. This is an axially loaded beam if
you want to derive this is of course much
easier because I am putting same axial load
throughout. I can vary the axial load; this
you study if you take theory of vibration
next time.
The procedure is get the transition matrix,
relate the eigen value. I am not going to
the detail because little advanced to this
and stability of the system is purely based
on the eigen value of this transition matrix
because composite materials for rotor blade.
You always mention that it has a good of course,
fitting characteristic is good; damage tolerance
is good and you can tailor the properties.
Tailor means you can create coupling bending
torsion coupling anything by keeping the fiber.
But you do not do any such things in the manufacturing
of a actual rotor blade because, you still
do not know what all this coupling effects
will really introduce to the blade dynamics.
So, the composite material is used in such
a way the layup is that the final design will
look like an isotropic plane. So that I am
safe do not try to put too many things and
then finally, you will really have no clue
what is happening because whether it is due
to aero-elastic problems or whether it is
due to this; so, most of the times industry
uses the material for something.
But, tailoring even though it is said make
sure that you do not get into any of these
problem unless you are very thorough about
what happens because, very one interesting
point I write somewhere they replace that
actual blade metal blade. See, this is one
of the projects you say I want to replace
the metal blade by composite blade. You know
the dimension, you know the mass distribution,
you just make a composite blade this is a
major project; that means, all my helicopters
I will throw away, old metal blades; now I
will use composite blades.
So, one of the project they change; what is
that everything should be? You know dimension
should be same; you know the mass distribution
should be same; you want the stiffness also
to be distribution to be the same and then
only you will say dynamically they are same
and they made a blade because dynamically
means how many modes you will make it? First
mode, flap mode, lag mode, torsion, few of
them you say metal blade. Whatever it has
I will have the same thing in the composite
blade design. It is not that easy design;
you made the blade put it on the helicopter;
you fly you find lot of vibrations.
This was in one of those articles; why question
mark? That is all that is the end because
nobody knows what is going on and I say that
everything is same but, I put this. I have
more vibrations you follow what I am saying?
So, there are many things which we do not
know and therefore, but you take the risk,
but very calculated risk. That is why if you
listen to the talk what that visitor Vishwanathan
- Doctor Vishwanathan mentioned it is very
important; but you try to learn more but,
always cautious in your design conservative
thing because you do not know several things.
