It is not necessary, you know, I will just
tell the, see root locus is, you get the,
suppose you have this equation, you get the
homogeneous equation, roots, the roots are
minus zeta omega and plus minus, what is that,
I… this is the root. So, basically, root
locus is, you take the real part, imaginary
part, and real part put a point, imaginary,
that will come. Now, you have some parameter
in this system, which is basically the mu.
For every mu, you will have a value and every
gamma. That means, mu and gamma is one combination,
I gave 4 sets for a given set of mu and gamma,
sorry, not mu, that is, what is that omega
R F.
For omega R F and gamma with varying mu, you
will get this and this. So, you will plot
those points, mu will be varying, but the
other 2 quantities are fixed. Then, you go
to the next set, next set, that is all. You
plot this in this curve, real and imaginary,
only thing is you have to estimate these 2
from your response curves that is all. Basically,
root locus is, how the roots vary with any
parameter because this is standard in control
system or even in a flight dynamics because
flight dynamics, they will always want to
see if I vary some parameter. How my roots
are and you know, that any root, real part,
if it is on the right half, this is, they
call it S plane, real and imaginary, the plus
plane. So, if the root is on the right half,
system is unstable, that is all. So, this
is a standard root locus.
Now, we will start with the blade equations
because you learnt about the, basically the
natural frequency, how it gets influenced
by the hinge offset and the root string, because
hinge offset and root string, they increase
the flap frequency. When the flap frequency
goes up, rotating flap frequency, then the
hub movements go up and you get a more control
movement at the fuselage. So, that is how
the flap frequency dominates your design;
really, flap frequency is very, very important
and you can have different types of rotor
heads, like articulated or hingeless, etcetera.
Now, this is without any, I introduce the
word coupling, only flap motion and pitch
angle is set, but there are certain cases
where they introduce coupling.
Coupling means, I will call it, what type
of coupling you introduce, pitch-flap, then
pitch-lag and flap-lag, that is, structurally
coupling by some structural design, I will
put a structural design, though we do not
call it like this, you can introduce these
couplings. When I say pitch-flap or flap-pitch,
when the blade flaps up, the pitch angle will
change. This coupling you call the, they call
it delta 3, delta 3, alpha 3, something is
there, they just call it delta 3.
And similarly, when you, when the blade goes
back and forth, lead-lag, again the pitch
angle can change. So, this, they call it delta
1 and of course, flap-lag coupling is, there
is no symbol attached to that, it is just
a structural coupling and how that comes,
I will explain. So, these are 3 and these
couplings are very powerful couplings, in
the sense, they alter the dynamics of the
blade frequency-wise as well as stability
wise. So, we will, I will just briefly, this
is not my, I will not be deriving a lot in
these things, just to give you, that these
are very critical.
Our flap equation is this, beta double dot
beta dot beta some beta with respect to lot
number with the forward speed and then, there
is a theta control, control input, which is
independent of flap motion, a pilot gives
that input. Suppose, that input itself is
a function of flap motion, that is, theta
control is proportional to some factor to
flap angle.
Then what will happen? You may have pilot
input plus another term, which is proportional
to the flap deformation. Now, the flap deformation
term, from here will be taken to the left
hand side because that will come as a beta.
So, you find, I can actually alter the stiffness
term because this is related to flap beta,
which is a stiffness term. I can alter the
stiffness term by introducing, that pitch-flap
coupling, but since we have not introduced
the lag dynamics, you can also alter the pitch
of the blade by introducing lag-pitch coupling.
But how do you introduce the coupling is the
question.
Mechanically or I would say structurally,
which I showed there, I will show a diagram
now, that will tell you how that coupling
is… This is the pitch-flap coupling, what
is done is, you have the hinge that is the
flap hinge here. If the flap hinge is like
this, look at only this diagram, this I will
come to the later, flap hinge is like this
when the, where flap motion means blade will
come up and down because this is the hub and
it is the direction of rotation omega.
But if the flap hinge is slightly inclined,
now it can only flap about this hinge, the
blade can only rotate about this hinge because
it cannot rotate about.
Now, what you would say is, I resolve this
rotation q into 2 components, so q cos delta
3. So, I, so I call it beta as q cos delta
3, delta 3 is that angle. And then, the other
component, which is pushing down, that is,
change in pitch angle because the blade pitch
angle is changing, that delta theta is q because
I put a minus sign, because it is nose down
because when it is flapping up, there is a
nose down. So, your delta theta becomes minus,
sorry, equal minus q sine delta 3. Now, you
replace q by beta. So, you will have minus
beta sine delta 3 by cosine delta 3, which
will be...
So, please remember, whenever I call flap,
flap deflection is only about the hub plane
what it is coming out, that angle, not about
what axis it is rotated. That is why, I put
flap beta not in terms of q, q can be something
else, I am replacing this q in terms of beta.
Now, you see, whenever there is a blade is
flapping, you have a change in the pitch angle.
Now, depending on what direction this is,
because it can be like this, it can be like
this also. If it is like this, then when it
flaps up, pitch angle will increase.
So, that is why this is written as minus k
p beta beta; this quantity is your pitch-flap
coupling, this quantity is coupling, but that
minus sign is there, that is how you have
arranged your, the hinge. And you may ask
why should I put it, that we will come to
that later, this is one type of arrangement.
Another arrangement, which is very obvious,
is this. Suppose, I have a pitch horn, pitch
horn means pilot input comes to this point,
it will come out of plane and that means,
my pitch angle changing.
But once pilot has kept a particular pitch,
the spot light is held and assumed, the control
linkage is stiff. Then, when it is flapping,
flap hinge is here, but this point cannot
come out because it is held, which means it
has to rotate about this axis because 2 points
are held and you introduce a delta 3 completely.
So, this happens automatically when your pitch
horn kept it someplace.
So, in the design, it is a little tricky.
Suppose, if you take this horn, instead of
putting it here, you bring it up to this point
and then connect it here, like a 90 degree,
then it flaps. No problem, that is why, if
you see, in most of the design the pitch link,
they will take it like this, bring it back
to the axis and from there link is to go,
then these 2 points are in the same axis,
in the sense, 1 horizontal. So, it will do.
Now, why do you do it is a question? Mathematically,
what we said is this is the change in the
pitch angle. So, I have written here, in the
hover case, from the flap equation just change
my theta control to theta control minus, whatever
is the change, delta theta because delta theta,
because is the change in the angle. So, you
just put it and this delta theta goes there,
if it is flap up pitch down, this is flap
up pitch is down, now what will happen? This
term is flap up, when I put minus, what is,
then this will become plus. So, I should put
a plus sign here because flap up beta is positive.
That means pitch angle is reduced. Now, you
bring this term to the left hand side, add
it with the flap term and then substitute
for delta theta as minus k p beta beta.
So, when the minus, when it comes to the left
hand side it will become plus because here
this plus if I put this directly this will
become minus k p beta beta. So, theta control
become minus, I will take this minus to the
left hand side, that will become plus along
with the aerodynamic terms, that gamma because
it will be theta as a coefficient, that coefficient
term is basically aerodynamic term, that is,
I will show that equation, say all these terms,
all these terms, gamma over 2, entire term
will come.
In hover, mu will go off, only this term will
remain and that will go and get added to the
already existing stiffness term. And I have
written for the case of hover with mu 0, that
coupling will be this term. You see, it is
o.k., I should put a plus sign here, this
is some because repeated use, this is the
additional stiffness term, which comes because
of pitch-flap coupling. If this is positive,
please understand, positive pitch, pitch-flap
coupling means k p beta is positive; that
means, flap up-pitch down.
This is the convention and then this is negative
means flap-up pitch-up, pitch-up. So, you
always do not take this negative into consideration,
only the k p beta term as it is, if it flaps
up, pitch angle will come down automatically,
but it affects; please, the interesting part
is, it affects the stiffness term.
In a simple, I do not take the dynamics of
the torsion here, I am just taking in a study
change in pitch, if it gone up, change, pitch
angle is changed.
Now, you see, this is an aerodynamic term.
So, you have stiffness even in hover forward
flight. Of course, mu sine psi etcetera is
there, this gives additional stiffness to
the blade in flap, but why do you want to
have an additional stiffness to the blade?
Make the flap frequency go very high, but
earlier we said, if you make the flap frequency
go very high your control moment will become
very large.
So, you do not introduce this in main rotors,
you put this in tail rotor because tail rotor
loads are not much. Why you want to put in
tail rotor? They do not want the blade to
flap too much; if you make it stiff, the blade
will not flap-up. But imagine if you have
to put a stiffness that is going to become
structurally very big weight, etcetera. This
is nothing to do with structure, just pure
structural design, that coupling.
So, you introduce a coupling, I have increased
my stiffness of the blade tremendously and
this, if you see in tail rotor blades in cheetah,
chetak it is there. Cheetah, chetak, I do
not know, whether you have seen it, you will
go and see cheetah, chetak, that those helicopter,
they have this, it will… That is because
you do not want the blade to go too much,
it is stiff and it will stay, but main rotor
you do not introduce this intentionally in
articulated blades. But if you introduce,
yes you can. But now you see, by changing
the sign of this, I can increase my flap frequency
or I can decrease my flap frequency, it can
be less than 1 also. That is why, I would
say, flap frequency is normally greater than
1 or it can be less than 1, depending on what
coupling you have put in. So, usually, people
are a flap 1 point something, yes, it is it
is true, but there can be situations where
you can introduce this coupling and influence
the flap. So, that is why, I said, it increases
flap stiffness, hence flap natural frequency.
But you have to rotate in air because gamma
is there. Suppose, if there is no air in vacuuming,
gamma is 0 because rho a c because it is a
ratio of aerodynamics to inertia.
So, this is one coupling, but in hingeless
blades and bearingless blade, this coupling
always exist, you cannot avoid because of
the elastic deformation of the blade. They
come, but you cannot quantify it, is, it is
a bit tricky, that is why they do not introduce
this, but you solve a full elastic problem,
elastic blade problem because depending on
the deformation you can have.
It is like, suppose if this is the beam, the
beam has, because it can have both flap and
lag deformation, so what will happen is the
blade will bend like this. Now, any lift will
give lift force, is acting on the blade. The
pitch link is here, pitch bearing; if the
lift increases, this will give a pitching
moment with respect to how far I am back.
Suppose, if I increase the lag load, then
how much I have flapped up? That much, it
will give the pitch. So, this is both flaps
and lag will introduce this coupling. So,
when they actually met the elastic blade,
this was one of the very important things,
they have to consider because that deformation
you introduce intentionally, you do actually
unintentionally I would put it and that depends
on what is the elastic deformation of the
blade, elastic deformation depends on what
is the load.
So, it is very, very tricky to say, what exactly
is the deformation? You will get coupling,
but one has to know that that this effect
occurs in those places because you may think,
that this is what the blade angle etcetera,
etcetera, but actually what you may get, this
may be totally different in operation. That
is why, they do lot of, you know world tower
test, this test, several test and then check,
match the data at least to a reasonable extent.
Then, they say, you fly; that is why, these
are all the pitch-flap and pitch-lag coupling.
They did research essentially by the, experimentally
they made models and then studied the effect
of this models, these parameters. What it
does to the damping of the blade? This effects
frequency, but this pitch-lag coupling, people
do not use. Now, you see, pitch-lag also you
can give delta 1, lag is what, blade is here,
you have a hinge like this. Now, if this hinge
is tilted like this, then when it, when the
blade goes, it has to rotate, you will get
a change in pitch, but it is not introduced
in blade intentionally. But they studied in
experiments it is very powerful, in the sense,
in influencing the aero-elastic stability,
even though these stills are, but how much
angle.
Now, I will come to delta 3 in the, they use
45 degrees, that mean tan delta 3 is almost
is 1, so tan 45 degree is 1. So, whatever,
if it moves through 1 degree flap, 1 degree
reduction in pitch, so that is 45 degrees
they put. It is not a small 5 degrees, 10
degrees, you say substantially large, but
the lag pitch coupling is very powerful. So,
all these couplings are, they create, please
understand additional problems, additional
effect into the dynamics of the blade.
Now, I will, because pitch-lag is similar,
because there is nothing I cannot show, because
we do not put that in this equation, because
lag dynamics is not part of this, because
this is only flap dynamics, unless you introduce
lag motion you cannot add that coupling. But
just for information I am saying, there is
another coupling, which is called the lag
coupling. Then, you can have flap-lag coupling.
How do you get that flap-lag coupling? See,
one is from inertia that is the Coriolis.
As the blade goes up like this, you will get
Coriolis acceleration, that is, from inertia,
that is very powerful; in addition to that,
structurally can you have flap-lag coupling.
Show you, that one diagram, which you have
seen earlier, this spring assembly, this is
an approximation because I am only giving
an idealization, because you have a soft spot
on the blade about which you want the blade
to flex in a hingeless blade.
Now, whether your pitch angle, whatever the
pilot gives, does he gives inboard of this
or outboard of this. If we change just a pitch
angle outboard of that, that means, the spring
assembly is not rotating because this is lag
spring, this is flap spring. But suppose,
you go and change the spring assembly, if
your pitch angle of the blade is changed inboard,
inboard means inside this, then what will
happen? When you change the pitch angle, the
spring also is going to rotate.
So, when you have a pitch, your spring assembly
can come like this and this is my hub and
this is my theta control or theta pitch angle,
whatever it may be, spring is changed and
my blade may be coming out like this. That
means I am having my pitch bearing inboard
of this.
So, what is flap? Flap is rotation about y-axis,
lead-lag rotation about z-axis. Now, when
I rotate, the spring assembly is now inclined.
Because of this, if I make 1 rotation, you
make, now flap and lag are small angles you
take, whereas theta control can be 10, 15
degrees you can go. So, this you take it as
slightly large, no problem, but flap you take
small angle and assume, that it is a vector,
small angle; you can treat it as a vector,
not large angles. You give, if you say flap,
that is, flap down, I call it beta or if I
want flap up, I can put beta that side.
But sine convention will get a little, in
the sense, this is counterclockwise rotation.
The blade is here, if I do flap, means this
blade is coming down even though this is a
plane of rotation. But if I want to use helicopter
convention, flap up, speed positive, then
this will be other way round and lead-lag
counter clockwise rotation, this you may give
zeta, this is the lead-lag. Now, if I have
these 2, you have to find out, what is the
flap moment, what is the lead-lag moment,
that is, you can say applied external moment.
Because the blade is going down, means you
put some moment about this. So, you may call
it M y and M z, you can have this as one convention,
but you will find, because when I did this
in the eighties, eighties means beginning
eighties, we were discussing lot of this,
how do you get that spring. Then, what different
people had different, but this is one simple
assembly, you have more complicated assembly
also.
That means, you introduce hub stiffness also,
not only blade stiffness put hub stiffness.
That means, you will have 2 sets of assembly
of springs, one hub, which it does not rotate;
another blade, which is rotating like this.
Now, you get an equivalent spring for that
entire assembly, that is there in my, I will
not go into the all the formulation, but this
one simple explain. But here please note,
this is the flap down, I am taking positive
counterclockwise rotation here, counterclockwise
rotation. You know, there is lot of confusion
because flap is up, you know what do I do,
how I take it finally. If you take this way,
you will get one set of equation, which is
consistent, but then you want to introduce
beta, I want beta up is positive, means you
have to change the symbol sine, sine of beta.
When you do that, then you will get one set
up equations. So, different people have used
different, only thing is depending on how
they define this beta. If you follow this
as a conventional, counterclockwise rotation
is positive, you will get one very systematically,
absolutely no problem, that I will write it,
I will write the stiffness. But you will find
in different reports, it may be given in different
format. So, you should not think, that this
right or that is right, if you follow what
assumption he has used, then you will find,
that everything will be fine. So, I am just
using this convention, applied moment M y,
M z, because of that the blade is down and
it is rotating. So, both flap, lag, this is
purely structural coupling because my pitch
bearing is inboard of the spring assembly.
Now, if I have, I am putting in a matrix form,
k, I call this as, this is the flap spring,
k beta; this is the lead lag spring, which
is k zeta. So, k beta cosine square theta
plus… If you are interested, you try, you
derive yourself, this is not very difficult
because I want to discuss this, cosine theta
k beta sine square theta multiplied by beta
zeta, this equation corresponds to this, whatever
this relation. This relation corresponds to
this, it depends on how they, what they, this
is the applied moment; applied moment M y,
M z, flap moment, lag moment, but downward;
please understand. So, the convention everything
you have to be very careful.
Now, in NASA report, what I used was, I used
minus beta and then I get the, not an applied
moment, I used the restoring moment, so I
put one more minus sign. So, you will find,
that there will be some sign difference between
this and there, and some other research paper
you will find there be other sign, but you
have to be careful, that what sign he has
used for everything. Finally, all will come
out to be correct equation, only thing is
do not take it as though this is…
This is following that convention, if you
modify the convention and apply restoring
moment, this is the applied moment; restoring
moment means what, the spring gives on the
blade, this is what externally you apply on
the spring, you can take what is the spring
load, that comes on to the blade. Then, you
will put a minus sign and then, if you say,
that I want to define flap up is positive,
means I put a minus beta. Then what will happen
is some of the terms here change sign; that
is just for information I am giving. If you
want I can write the other expression also,
restoring moment with the minus beta I will
put, this is applied moment with that.
Now, I am saying restoring moment that means,
the spring moment with flap up positive, with
flap up positive; that means this is flap
down is positive. If you want I can write
flap down positive here. Because of my convention,
here if I write this is M y, M z, this will
become k beta cosine square theta plus, but
you will, you have a minus sign here, k beta
sine theta cos theta and this will be same,
this is k beta sine theta cosine theta and
this will be minus sign, minus of k zeta cosine
square theta plus k beta beta zeta.
In this case, beta positive flap up; in this
case, beta positive flap down. This is external
moment; this is the restoring moment. Basically,
what you do is if you want to apply moment
to spring moment, you put a minus sign, everything
minus, minus, minus, minus and then you change
wherever beta is there, that sign will change.
So, this minus, minus, minus, this will become
plus plus, but these two will stay as minus,
that is all.
This is just a sign convention, but what I
want to discuss here is, this is very interesting
discussion you have. If I introduce pitch
bearing inboard of flap and lag hinges, I
introduced flap-lag coupling and all the hingeless
blades because hingeless blade you have here,
pitch bearing is inboard. You change the pitch
here, coupling is always there, you cannot
avoid.
Now, how much coupling comes, this is again,
see lot of debate, see one is what they did
was, whether, whatever pitch angle I give,
whether the same pitch angle goes to the spring
also I can have, if it is, if I rotate by
theta, the spring assembly rotates by R theta
where R is less than 1. That means, I give
a pitch angle of theta, but the spring assembly
does not rotate to same angle, but it rotates
slightly less than that, so they, that is
the coupling. What they did was, they put
R parameter, R. So, theta is R times theta
control, this R can be from 0 to 1; 0 means
pitch bearing is outboard.
So, wherever zeta is there, you put theta.
If R is 1, this is directly theta control;
pitch bearing is inboard. But if it is in
between some value because it does not it
there could be a slight deformation because
this is a structure, I give a rotation it
need not go, faithfully full rotation need
not go and there can be a slight increase.
So, this interesting thing, what happened
is if you look at that coupling term, suppose
I have only flap motion, only flap motion,
I do not have lead lag automatically, I get
a lead lag moment, I get a lead lag moment
because of this.
Now, if I do not want coupling, one is you
take the R is 0, that means, pitch bearing
outboard of the spring assembly. That means,
the spring will not, there is no coupling
between flap and lag structurally. But if
you introduce 
K beta equals K zeta, that means, the flap
stiffness is same as lag stiffness, that is,
structural stiffness, the spring assembly.
Then, I do not have any coupling, no matter
what, I can have a rotation, whether by pitch
bearing is inboard, pitch bearing is outboard,
does not matter.
Now, this, whether you can design a blade?
This is called matched stiffness blade configuration
that flap stiffness, but please understand,
I am deriving all these things with this type
of idealized model elastic blade. You have
to see at the junction where the stiffness
is there, where it is the flexure you have
to see, whether you can get matched stiffness,
usually it is difficult.
But please note, experiments were carried
out to study the effect of these and I will
bring that, that you will find, data will,
results will just change the damp thing data
because I am, I am not gone into aero-elastic
stability part, but these parameters are tremendous
influence on the damp.
Structural coupling, flap lag, but now you
realize elastic blade, hingeless blade, if
I take, I cannot avoid this coupling because
by construction I am getting that and pitch
bearing is inboard. Now, you say, I do not
want stiffness, then can I make the blade
at the point where it is both, the flap stiffness,
lag stiffness must be same. That means it
should be a square section with the same distribution
of material at the point, where it is flexing.
If I can design this, coupling is removed.
Yeah, isotropic, no, dimension, see dimension
also equally matter. See, suppose you take
isotropic, you take straight edge roller,
you can bend one way easily, other way you
can bend it, that is the…
Yeah, yeah, yeah, but the stiffness part,
I am representing the flexural stiffness by
a spring. See, the blade is not going to bend;
you are putting at some place where you deliberately
make it a little soft. Yesterday, he was mentioning
that, so that you create, but the stress levels,
the strain levels are below the limit, but
still it is flexing at that location. You
make these 2 stiffness same, then you can
eliminate the flap lag coupling, but the flap
lag coupling is not, most normally lag stiffness
is high, lead lag stiffness is much higher
than flap stiffness in the design of the blade,
but you find, you try to reduce, but this
coupling will affect.
Now, you know automatically, I cannot analyze
only flap motion because the moment I have
flap, I get a lag moment, lag moment will
make a blade go lead lag and if I have any
pitch associated with that, then that will
create a problem. And then, lead lag motion
will come and give me a flap deformation because
these 2 are coupled lead lag. If I have, I
will get a coupling to lag.
So, flap and lag motion cannot be decoupled
all along. We were, the full course till now
only flap motion we were concentrating, only
flap, we never bothered about lead lag motion.
Actually as a subject or the dynamics of the
helicopter, if you look at it, flap is very
important. But lead lag creates more problems,
but you cannot avoid. So, you have to consider
lead lag; lead lag will come into the picture,
then of course, elastic torsion. So, that
is why, flap lag torsion, that treatment is
the full dynamics.
And all these things start around seventies;
people try to understand what you really going
on in the flap lag those days. It is a rigid
blade with this type of spring assembly only.
All research paper, if you look at it, even
eighties, yes, but now elastic blade models
have come.
Now, I will just, that is a matched stiffness
configuration, which can eliminate the structural
flap lag coupling.
So, I just show here a diagram, whatever I
have written there with the, because here
I put a minus sign; you see, that is the restoring
moment with the flap up positive, that is
slightly modified.
When the blade pitch is theta, spring assembly
rotates through R theta, so I put R. So, experimental
set up was made coincident flap lag springs,
so that the, and blade is made rigid and they
rotated and then studied the, and then the
data was given for correlation. So, those
data I do not, whether I have for some problems,
but not for all the cases, yeah.
Other case, which I want to tell you, how
the flap lag coupling will come is because
of this, this picture, I would know whether
it… See, I will show this diagram, bit messier
taken from some reference, from, to see this
is the original blade, this is the deflection;
this is the deflection, up, down, that dash
line.
Now, lag forces will try to find a pitching
moment lag, lead lag forces. Similarly, when
the blade goes down, that lead lag force will
give a pitching moment, that depends on how
much it has flapped. So, how much you flap
and that becomes a factor with the lead lag
force, that is, the drag force in changing
the pitch angle. Similarly, how much the blade
has gone back and how much is the lift that
will give a pitching? This is an elastic blade,
that is why then it will give.
So, if you do not take these things properly,
your control rod loads, everything will be
wrong. So, I want to tell you, this is structural
part, aerodynamics part is another.
So, usually predicting control rod load is
always tougher, you really do not know because
the elastic blade. Now, you will take more
modes into according, whether this is really
happening, get all the moments properly and
then transfer them. And so, this is just a
picture showing how one load will affect based
on the deformation in another mode to something
else, pitch angle. So, you see, flapped deformation
lag load affecting pitch angle; similarly,
lag deformation flap load influencing pitch
angle.
Now, you see, if you want to analyze a problem,
you cannot neglect anything, you have to have
flap lag torsion problem, which is very complicated.
So, we are not going to do that in this course,
but this is just an introduction to you, to
see the complexity in the factor of safety
for what, for factor of safety. See, its all,
see sometime I can only say what a little
I know because factor safety is always it
should have, infinite life that is the design
usually. But how much load you are getting
in the pitch link is very difficult to predict,
sometimes the error can be a matter of 200,
300 percent, a particularly pitch link load.
So, you design by experience. So, you will
learn now what is that load after you monitor
pitch link load and then, you may replace
it after so many hours. But once the flight,
that is why, initial design you do, they have
done theoretical calculations and theoretical
calculations, if you are a beginner, nobody
is assisting you, nobody has made anything,
but you have to take a risk, you put a pitch,
fly it, you measure it, you see it is now
failing, then you make it bigger.
Then, you will find this is good, after that
only you go by both, this much is a load,
that is coming, where is it coming from, how
it is coming from, but I do not know. Normally,
pitch link may be, they will have safe life,
I do not think they will have infinite life,
I do not know, may be he will be able to tell
you safe life means, you operated for so many
hours, after that even if it looks good, remove
it, put a new one, is that safe life. you
do not take risk in those things because what
you do not know, you have a, you know safe
approach towards those problems.
Now, I will introduce to you the lead lag
dynamics. Now, please I, whether should I
derive equations that you have to tell me.
If you want to derive, that will go into full
derivation, again position vector velocity
acceleration, then inertia moment and write
it, but I would not write any aerodynamic
moment for this because that is a separate
expression, which you have to because that
is a drag load.
I will just briefly tell you that lead lag
dynamics. So, this is isolated, isolated in
the sense, only lag motion, I do not have
any flap motion, please understand, like what
all we derived earlier, it is only flap motion.
Now, I am saying, either only lead lag motion.
So, you put a spring with the hinge offset,
alright. Now, like you have flap frequency,
natural frequency, rotating flap natural frequency,
you will have rotating lag natural frequency
because this is a dynamic system, but what
is that value.
If you derive the equation of motion for this,
please understand, this is I am giving a deformation
zeta, the blade is hinged about this point
and there is a route spring, which is k sub
zeta, which is, so the motion is in the plane
of the diagram, if just.
So, when it goes, rotates, a blade can move
back and forth, this is the equilibrium position.
Now, if you write the dynamics for this, the
equation will be like this, that is what you
can derive it. If you want the derivation,
I will start next class, not today, if you
want the derivation.
This is my inertia, of course there is another
term, which is inertia M X CG, you know the
definition integral, m x d x, m is mass per
unit length.
So, here I b, we know that it is integral
0 to R minus e m x square d x and m x, m is
mass per unit length.
So, if you have uniform mass, uniform, uniform
mass, you will have I b is m R minus 3 I am
just writing, it is what I wrote last class
and M X CG becomes m…, this for uniform
mass. So, if I substitute for uniform mass
case, I divide by I b converted into non-dimensional
time, this is my lead lag rotating natural
frequency.
So, I will put it omega bar RL is k zeta over
I b omega square plus. If you want to put
it as 3 by 2 e bar by l bar that is one form,
that is uniform or else M X CG e over I b,
both are fine and this is power half.
This is my rotating lag, if you, just for
comparison if you take rotating, what we got,
flap you add 1 plus. This is flap, you see
this in flap frequency, you have 1 plus and
of course, I do not have all the pitch, pitch
flap coupling. If I put pitch flap coupling,
that term will come. The one is missing here,
that is not there, that is all.
Now, because of this, you find, that suppose
k zeta is 0, but there is no spring, it is
the articulated blade. Lead lag frequency
rotating depends on what is the hinge offset
at an M X CG RL, this whole term can be replaced
as if it is a uniform mass, it will be 3 by
2 e over R minus e or you can write it as
3 by 2 e bar l bar e bar is e over r and 1
minus e bar is l bar and that is what is written
there.
If this is 0, this is the way and the hinge
offset, usually you will have about 4 percent
5 percent, but sometimes it may go to 10 percent
hinge offset.
So, the lag frequency, non-dimensional, it
is not very large, it is usually of the order
of, I have given here, it is about for articulated
blade, it is in the range 0.25 to 0.3, 0.25
to 0.3. Whereas, flap is one point, something
and hingeless blade because articulated blade,
there is no spring, it is just the hinge.
So, you get a hinge offset with this, that
means, this term is 0, you can find out what
is the corresponding hinge offset. So, immediately
you can calculate, what is the rotating lag
natural frequency, but if it is the hingeless
blade, usually it goes to around 0.7.
Now, there is a one more terminology, which
is there, there is a hingeless blade, I said
0.7, but can I have greater than 1? Yes, you
can have greater than 1, provided you do this
stiffness, you increase.
And suppose, another part, if this omega,
if it is small that is, the rotor rpm, when
you were starting, the blade rpm is low; when
you go to the operational 100 percent rpm,
it will have a much higher value. That means,
my lead lag frequency, I am changing it as
my omega changes the non-dimensional number
itself is changing if I have stiffness, is
it clear? Because as I changed my omega, lag
frequency is changing, so this is a terminology.
If omega bar rotating lag is less than 1,
this is called soft-in-plane blade, this is
the terminology. If it is greater than 1,
it is called stiff-in-plane blade; soft-in-plane,
stiff-in-plane and at the operating rpm. Suppose,
you operate that, I say, like our blade we
operated 1500 rpm small one, a big rotor 300,
400, something like that.
If your lead lag frequency is greater than
1, then it will be called a stiff-in-plane;
if it is less than 1 it is soft-in-plane.
Now, how do you fix, which blade I will, which
soft-in-plane blade or stiff-in-plane blade,
but the same blade if I increase by rpm, what
is originally stiff-in-plane will become a
soft-in-plane.
So, it is really dependent on the operating
rpm, but at the operating rpm which I should
choose.
