Left last lecture, essentially, we got the
expression for the thrust in forward flight,
but the rotor disk is in a state of climbing
as well going forward. That is why, we put
the V cos alpha V sine alpha and the total
inflow through the rotor disk, you will have
the component of this nu and this is due to
the flight speed. So, this is the V sine alpha,
as well as, nu.
Now, usually the expression, that is the thrust,
which is written as rho A nu v cos alpha square
plus identical to your similar to momentum
theory. The only difference is, now let us
plot certain variation of the inflow because
in hover it is supporting the same weight
of the helicopter. So, thrust is same even
if it is climbing forward or moving forward.
So, in hover we write. So, please note, that
I am putting a subscript H to denote the difference
between the hover inflow and the forward flight
inflow. But since the rotor is supporting
the same weight, so what you have to do is
you equate both of them and write an expression,
which is given their nu is nu H square or
you can, because when you equate both, you
will get nu is nu H square over this. Ok,
straight away.
And then, you will write this expression.
So, this is just a modified form of the same
inflow expression in forward flight, only
thing is, it is non-dimensionalized with respect
to the hover inflow.
Now, if you just plot it, because this curve
is plotted, how it is done is, I will erase
this part because you can plot like this,
take it. This is nu over nu H and on the X-axis
you have, this is plotted for, because please
understand, it is just an indication, what
is the value of alpha, that is the key. Alpha
is the oncoming flow with respect to the horizontal.
Suppose, let us take alpha is 0, that is easy,
if we take alpha 0, that means, it is, the
disk is flying forward straight. Now, you
can set alpha 0, then this is, this term will
go of V over nu H square plus nu over nu H
whole square.
Now, this curve can be plotted, may be 0.5,
1, 1.5 and 2 and here X-axis, you can have
V over nu H, which is 1, 2, 3. So, I am just
for alpha 0. This curve for various values
of V over nu H because alpha is 0, means this
is, this will be, because you know, hover
the value is 1 nu over nu H, the curve will
start and go like this. I, may be I am, it
is a little sharper, it could be a little;
this is the momentum theory in forward, nothing
else, how inflow varies with forward speed,
that is all.
Now, the another question, which we can have
is, because we were asking last class, when
the rotor will behave like a wing, at what
speed, because we mentioned last class, at
high speed the rotor behaves like a wing,
in which case, V is much larger than inflow
velocity. Therefore, you can write, at, at,
for V very large. I am just saying very large,
means, how large?
You will find out the thrust becomes, only
this term V cos alpha, but now when alpha
is 0, that is, the disk is flying horizontally,
this term will become 1. So, my thrust is
this expression. Now, you equate these two
because thrust is, it is supporting the same
weight of the helicopter. So, you will have
nu nu H, sorry, nu into V is nu H square.
So, you will have…, you take 1 nu H this
will lead to…, very simple.
So, you see, I am making an assumption, that
is, flying at high speed and if you plot this
curve, this is the exact, whatever momentum
theory says, this is, if I plot this curve,
this is again, nu over nu H 1 over v, this
is like a 1 over x type of curve, that 1 over
x type curve will come, it will go, this is
directly high speed assumption. This is high
speed, this curve because you know, that as
V become 0, this going to become asymptotically
infinite, but in general, the deviation will
start V over nu H greater than 1.5. So, this
is, now your question is V over nu H, you
can make this assumption greater than 1.5;
the high speed assumption is fine. Assumption
of high speed, that is, I do not bother about
this complicated, but this is only for very
basic level.
Now, let us look at in terms of non-dimensional
quantities, non-dimensional quantity you will
have, because V over omega R, this is nothing,
but approximately, you take it the V cos alpha,
alpha is 0 we have taken. So, this is basically,
nu over lambda hover. Now, in non-dimensional,
usually forward speed, rather than specifying
forward speed, you put it in non-dimensional
forward speed, which is in terms of advanced
ratio.
Because we defined last time, advance ratio
is V cos alpha over omega R, we took it the
velocity in the disk divided by. Now, for
what non-dimensional value you can make the
assumption of almost the wing, sorry, the
rotor acts like a wing.
Usually, lambda H is a hover inflow, we know
lambda H is C T by 2 and if C T is, I am saying
up 005, then lambda H is 0.05. Now, if you
take 0.05 into 5, 0.075. So, usually they
say, mu greater than 0.1, greater than 0.1,
then you can use the high speed assumption
for inflow. The wing, basically the rotor
behaves like a circular wing, reasonable,
but in between 0 to 0.1 in the forward speed.
Because this is forward speed advance ratio
in the range 0 to 0.1, but mu is greater than
0, yeah, in this range, usually it is called,
rotor is called in transition.
The 
time the rotor wake, I will show, because
one, you start the forward speed, now if you
look at this, what really happens is, initially
the rotor wake, wake in the sense, whatever
is the disturbance, which is coming out from
the rotor blade, the lifting theory or anything,
the wake comes below the rotor straight. But
once you start moving forward, the wake gets,
it is in a skewed cylinder.
When you go in a skewed cylinder, as you increase
the speed, this angle is going to keep on
changing. So, that is why, in this range of
0 to 0.1. Let us write this as a tan from
this diagram, tan chi, this is called the
wake skew angle; tan chi is basically nu over
lambda wake skew angle.
And if we apply this rule, we said, that 
nu over lambda H, approximately 1.5. If you
take it, usually the angle of that becomes,
see, 0; when nu is 0, tan chi is 0, that is
straight. When nu is very large, the wake
is swept and this angle is almost 90 degree.
If you go around advance ratio 0.1, then that
angle is close to around 60 degrees. So, in
the range of 0 to 60, that is the transition,
in the sense, the wake is not completely swept
behind the rotor disk, it is not straight
below also, but it is very close to that and
then, it is skew.
And this really complicates the problem because
the wake is close. When the blade travels,
the wake from the front blade will close to
the rear blade, usually you will have high
vibration, etcetera in that transition zone.
Normally, helicopter you do not fly in this.
Of course, you have to, when you increase
the speed you will be going in that, but you
do not try to operate fully flying.
Later, when you go to the power curve, if
you want to fly at a minimum power, minimum
power point is actually beyond this 0.1, it
is around 0.15, somewhere around that. But
if you want good range, then there is another,
which is still further. So, you find that
this zone, where it is 60, 0 to 60, you have
lot of variation in the vibration, but the
important thing is, the rotor behaves like
a circular wing and I said, aspect ratio is
only very small. And as a result, there is
a lot of variation in the inflow.
Our assumption of uniform inflow over the
entire disk is not really valid because you
have done one homework, even in the hover
case it is not, but in forward flight it is
still never. Now, how do we get the varying
inflow in the rotor disk? But please remember,
this equation or the earlier expression, which
I showed you, they are used to get the, this,
this relationship, please understand.
This is induced flow, nu over omega R. This
I wrote last time, this is essentially the
equation, which you use it to get the, but
please remember, here it is non-dimensionalized
with respect to omega R, tip speed. Here,
when we did, we did the, this curve non-dimensionalized
with respect to hover inflow. Of course, you
can divide by omega R, both of them, then
it will become lambda I over lambda H. Here
you can write it as mu over lambda H. So,
these are different ways of...
This curve, see this curve is essentially,
I wrote the high, the high speed assumption,
that is, v over… high speed. Alpha is, see
alpha is small, you can take it because cosine
alpha is almost, but it is the high speed
assumption and if I plot this curve, please
understand, I have plotted this with alpha
0, exactly this is the curve, this continuous
line.
Whereas, when I plot this, is almost come
close up to 1.5, then only it starts deviating.
This is, this curve, this is, this curve with
alpha 0.
This one, no, this is not high speed break.
Why, I can fly horizontally, that is not high
speed. Please understand, high speed is when
V is much larger than nu, it is not nothing
do with alpha. Please understand, alpha does
not define the high speed, it is the velocity
defines the high speed. Alpha defines the,
how, whether you are climbing and moving forward,
that is all. I can have alpha 0 and the disk
can go like this, is it clear?
That is, based on that assumption, you draw
this curve, but later you realize that things
are slightly different. This is just for understanding;
that is why, your inflow quick calculation.
If you want to know if the velocity is non-dimensional,
more than 0.1 mu, then you can take it my
inflow calculation is pretty quick. But you
have some question?
No, no, this is, no, this is V, V, this is
nu over nu H.
Yeah, when it becomes low, it shoots. This
is nu, this is y, when it is 0 what happens?
Yeah, high speed, V is higher, this is the
x. I am plotting a curve Y equals 1 over x.
1 over x curve will just go like this. What,
you guys have a doubt? See, I am just plotting
Y equals 1 over h 1 over x, how this curve
will be?
Yeah, V is higher only, when v is higher,
v is higher means, this expression is same
as this expression; same means, they give
almost same values, is it clear.
See, this is exact, as far as in a momentum
theory. I do not make an assumption that my
nu is much smaller than v cos alpha or anything
like that, only thing is. Alpha equal to 0
because I have to assume them, alpha, I took
it, that 0 value and I am getting inflow purely
from this equation, that is, when alpha is
0, my inflow will be…
But this I cannot directly solve because it
is in both sides, I have to do iterative,
is it clear. Because nu over nu H is on right
hand side also, otherwise what you will say,
I will square it, make and then as a, what
it is, become a 4th, 4th order and then, try
to get the roots.
That is different thing, you can do it, but
then when you do, you do a meaningful root,
otherwise you just take this, this is plotted;
is it clear or doubt?
Wait, wait, wait V should be…
Higher, yes, V is higher means, V are definitely,
it is greater than 1.
Dotted line, when V tending to 0, it is going…
Yes, high speed assumption is only on this
side I am saying that, but is it still clear
or not clear? I am plotting two curves, one
curve with high speed assumption, another
curve no assumption, straightaway, directly
from here; it is the momentum theory. Now,
I plot this curve, this curve is this.
When I make a high speed assumption, that
is, this curve, now at what point I can make
the high speed assumption, at what velocity?
That was the question, the velocity at which
you can make the high speed assumption.
You are not making lot of error is when V
over nu H is greater than 1.5, that corresponds
to, if you take nu H as, sorry, lambda H is
root of C T by 2, so you get this. So, V over
nu H, you can non-dimensionalize as mu over
lambda H. It is greater than 0.075 precisely,
but you normally take greater than 0.1, 0
to 0.1 forward speed, which is the advance
ratio, forward speed divided by tip speed.
If it is in the range of 0 to 0.1, you say,
that is transition, is it clear.
But during the time, that region, how the
wake, wake is close to the rotor and it is
of course, getting skewed. As you increase
the speed, the wake goes back. So, that is
why the assumption of, because if you want
to get quick answer what is the inflow, you
cannot make that assumption in this zone,
beyond that it is ok, is it clear, alright.
Now, the question is, rotor behaves like a
circular disk, now circular disk, we in this,
we assume, that the inflow is constant everywhere,
but which is not true. There will be substantial
variation in the inflow over the radial as
well as Azimuthal location. Then, you have
to have a theory, but this is, I will briefly
give a, this is only history, what we are
going do is we will follow it.
Well, I am going to show, this is how the
wake, I have just drawn for two blades, how
they keep moving behind because each blade
will give out import x, as well as, you will
have a x also. I showed you, you will have
trailing as well as shed.
But if you make the assumption, that my loading
is constant that means, my circulation gamma
is constant over the span of the blade. That
means, you will have only tip vortex and the
tip vortex is what is shown here. So, you
will find, they will come in each blade, will
give and they may go and interact with the
other blade. So, you will have lot of helix
inter point initially.
Now, I am just giving you a brief history,
how do you get the inflow? So, it is, it is
a very interesting part, what you do is, you
assume actuator disk. Actuator disk win, it
has infinite number of blades and every blade
is giving out. That means, it is almost like
a skewed cylinder, but assume, uniform loading
gamma is constant, that means, it will be
like a lot of wake semi-infinite skewed cylinder.
With this assumption, I will just write the,
because I will not go into the details of
this formulation because these are all a little…
So, this is actuator disk first, that means,
infinite number of blades momentum, similar
because if you want to get momentum theory,
does not give you the very variation here.
Please understand, you do not start using
the differential momentum theory what we derived
in the class. That lambda I, which is the
function of the blade pitch angle, at every
section we had, that sigma A over 16, etcetera,
right, do not use that here.
So, you assume, the wake is, this is the cylinder,
that is all. So, you have, using this kind
of, you find out this wake structure, you
assume uniform loading, which means gamma
is constant.
How it became like this? Purely, because of
the forward speed. So, this was done in 1945,
that is, by Coleman, Feingold and Stempin.
1945 they used this, they got some closed
form expression, I am not, I am leaving it,
just giving in terms of a closed form expression
for the inflow variation along the rotor,
only in the 4 and half direction. Then, in
1949, I think that is 49, yes 1949, he made
a slightly different assumption. He said,
that hey, I am not going to assume uniform
loading, but my loading is going to vary sinusoidally.
So, he had a trailing as well as shed vortex
that means, his circulation is constant along
the blade span, but it varies with Azimuth
location. That means, you will have a trailing
as well as shed vortex, with that he made,
this is called Drees model. He again got the
inflow variation on that.
Then, in 1950s, using the potential flow theory,
the 49, 50, 52, that period, but please note
all of them are actuated this theory. 1950,
52, I will put it; this is Mangler and Squire
using potential flow in a pressure field.
So, this is like a disk, it is the potential
flow assumption and then, get the pressure
difference across it and he solved with, he
got a series expression for inflow series,
long series. Now, these are all till 50. Simultaneously,
slowly, the computational capability because
they all got close form, please understand,
closed form solution. This assumes actuator
disk, but that is not the real situation.
Real situation, you have number of blade 2
or 3 or 4 or 5, something like that.
So, you have vortex theory, similar to fixed
wing, lifting line. But vortex theory, you
need to have, this is also vortex theory only,
but it assumes infinite number of blades,
whereas here finite number of blades; vortex
theory, finite number of blades, which is
a realistic situation. But then, what is the
structure of the wake?
This is the structure I have assumed, that
means, I will take something similar, like
this. Structure is decided, only thing is,
I do not know the strength and this is the
prescribed wake analysis, prescribed. That
means, you prescribe the, basically prescribed
wake, wake structure, but another one is free
wake. That means, you allow completely, that
is, wake can interact with another wake and
finally, it will come to its own equilibrium
for a given condition. These are all computationally
more involved and of course, prescribed wake
is reasonably, you know, better than free
wake; free wake computationally, very, you
know, time consuming. But these are research
fields, which people have been working on.
Now, all of them, please understand, this
is very, very interesting, one is finite number
of blades, this is infinite blades; they got
in the 50s, these are all even. Now, it is
going on, people have developed models, some
people use it, but some people use even this.
We use this and these results have, industry,
they may have some models, if they want, they
will use this part if it is available, otherwise
these are good, absolutely no problem, only
thing is, more and more accurate determination
of vibratory loads. Yes, then still it is
the question of debate, where I go and then
improve my model. So, I am talking about only
inflow modeling, nothing more, inflow modeling
in forward flight steady case.
Now, in this course, what we will do? I will
briefly mention what these are, how all of
them ultimately is represented, wake skew
angle, that is all these. What is the only
parameter is wake skew angle.
Because you write your nu… k x, k x…,
alright. Here, what is this nu naught? So,
this is my rotor disk and this is my, this
is the traction of flight, this is x-axis,
this is y-axis. This angle is psi and this
is the radiant, that means, please understand,
here psi represents the Azimuthal location
in this disk r over R. And basically, r represents
wherever in the distance, nu naught is some
kind of a mean value, mean inflow. But please,
now I am using one more symbol, nu sub-zero,
but this is in forward flight. That means,
inflow is basically the value you obtain from
the momentum theory. So, that is this value.
So, now, you nu 0 is this quantity, though
I write is nu because this assumes uniform
inflow. Therefore, nu 0 or in other words,
if it is a, you can use this expression lambda
I if you know C T directly. So, even though
I put I, it now I am changed to 0 lambda I
into omega R, that becomes nu naught. So,
please understand, there are different symbols
which may be confusing, but the idea is, you
have to understand clearly, that what we are
using; is it clear?
Because you assume there is a uniform inflow,
which you get it from momentum theory, this
is, I obtain these two quantities: k x, k
y represent essentially the variation with
respect to r and psi and that is all. And
what are these? This I will write it, I wrote
one of them, but I will give the other one,
what just for reference.
He obtained is k x, please understand, this
is tan chi by 2 and k y is 0. That means,
this is not there, this term is, this term
is there and this given directly in terms
of chi by 2, which is also given by this this,
you know, this you can get it because tan
chi is nu over lambda. But please understand,
in this the lambda includes V sine alpha plus
nu. Do not think, that, that is only nu because
this is the total lambda, please understand;
this is not only the nu V sine alpha you take.
Now, you see, I define in terms of mu and
lambda mu is in the plane of disk lambda,
is normal to the disk, that is all you follow,
these are the two quantities.
Now, Drees model, he gave k x 4 over 3 minus
1.8, which is written as 1 minus 1.8 nu square
root of 1 plus lambda over nu whole square
minus, you can say lambda over nu, that is
enough. And then, k y, he also gave k y, which
is minus 2 nu. But please understand, k x,
this is value only forward flight, you cannot
apply it for hover because psi is 0, otherwise
you will get infinity only for forward flight.
So, here, he says k x is 0 for hover, that
you have to take it.
But please understand, these models, this
model is very good; Drees model is a good
model. How do you know, whether it is good
or bad, that we will come later. See, when
I define, it is a good model. Of course there
are better models are now, we can use a better
approximation, which have come, which we use
it in our thesis. Of course, we have option
to use this, but please understand, in all
these models if you want to use, you still
have to go and get this nu naught from momentum,
that relation you have to get that. So, you,
I will give you, I just gave a brief, no I
would say the overview, because if you want
little bit more detail, now it is part of
one of the my student’s PhD thesis, no,
it is like a history because even the reviewer
says, said it is pretty neat, good. Just inflow
calculation, how it all started, but this
is 45, this is 49, 50, but helicopters flew
around 39 to 42, but people were struggling
to get expressions.
But today, of course, computational capability
has improved, but if one wants to use, he
has to develop his own computational tool
to get an inflow, but some are good physical
models and they are able to get results, which
are only after getting this you realize, if
I use this model I will be able to get certain
results, which are resembling the flight test,
that is the key. Ultimately, the whole thing
is, whether these are valid or not valid,
purely depends on how would the results match
with the experiments. So, I leave at that
part.
Now, we will go in this course, please understand,
we will just use this expression. So, if I
say Drees model, you know, that this is the
Drees model. So, this is, this is sometimes
people use, it is old, but Drees model is
reasonable, but in all our formulation, please
understand, I do not include this because
it makes simpler, because most of the expressions,
which would derive in the books, whatever
you get, it assume uniform inflow, is it clear,
which is a gross assumption.
But if you want to get some closed form results
in the reasonable length of mathematics, basically
algebra, then make an assumption. It is, my
inflow is uniform over the whole disk even
though it is not correct, is it clear, because
this you have to know, because there is this
is not the end of it, this is a good approximation
for realistic problems, but this not just
sufficient. We need to take more time variation
also into account here, that becomes, those
all started later and today it is at some
stage, where we can model the time variation
also in an approximate version time variation
of inflow.
Please understand this is a constant. When
I say constant, it does not vary with time;
when I say uniform, it is uniform everywhere;
that is all. But uniform, but it can be time
varying, that is one model, it can be uniform
constant, it can be just constant, but need
not be uniform, that is what this model is.
But then, the most general is its time varying
and it is not uniform and that is the much
better approximation. If you want to get some
more refined, I would call it more refined
results, you need to have that models, but
those models are time consuming. You cannot
put it, you know, simple expression, you have
to do computationally, that is why, for this
course, because we need to get the loads because
hover we learnt, forward flight we need to
go and then analyze.
What do we analyze in forward flight? First
is, you say, I want to fly at that speed,
what should be the control because you need
to fly? Whether you were able to achieve equilibrium,
equilibrium of forces and moments? Very simple,
for the body, for the helicopter to fly at
just, take level flight, that is the first.
Second one is, if there is a disturbance,
is it stable? That is the stability part,
that is the flight dynamics, but even the
first part, which is called level flight or
anything, it is called a trim or equilibrium,
but the trim equilibrium part itself is a
aero-elastic problem.
Then, you will slowly realize, in forward
flight things become much more complicated
and then, you start analyzing only some narrow
portion for a specific. It is not, that you
include everything in your and then analyze
it, this is what happens. Now, we got an expression
for inflow in forward flight good.
Let us now go back, actually start, we learnt
momentum theory, how do we apply because you
know, that this is for you to get this nu
naught, you need to know C T and the C T was
given here. Please remember, C T, C T is given
here, lambda I to C T expression is right
there. Now, I will close this here; let us
go back to the rotor blade.
So, you got a, we said, in forward flight
because of time varying, this side is more
velocity, this is advancing and this is retreating;
advancing side more velocity, retreating side
less velocity, oncoming flow, I am neglecting
radial flow.
But later we will see how we very crudely
take the radial flow. Radial flow means, because
this flow if I resolve it, one will come here,
another will go, this is along the span of
the blade, this is like a swept wing flying
forward. That means, there is a flow over
the surface not along the aerofoil card wise,
but in the radial direction, it neglected
or we can take approximately some, that we
will come to that later.
Now, you know, that this blade is going to
experience load and it is going to, we will
consider very simplistically, this velocity
varies, we know that, sine psi. Now, the lift
is going to be, because it is a square of
the velocity, so you have harmonic variation
of lift. So, the blade is going to go up and
down.
Let us consider only the flapping motion,
flap. When you say flap that is the out of
plane motion of the blade only; out of blade
motion. How it will vary because you know,
something is sinusoidal or I would say not
say periodic, therefore you expect, that flap
motion is also periodic, but we will call
the flap motion by the symbol beta.
Let us take a very simple case, the blade
is flapping and the flap, let us say, this
is the hub and blade is I call it beta. But
this is one approximation I am showing because
this is rotating. Later, we will see how to
represent the blade. Now, let us say, the
flap motion because it is excited by a periodic
load, any periodic function you can represent
by a Fourier series. Therefore, I am representing
the flap motion also as a Fourier series 
and so on, so on, so forth, which is, write
it as beta naught plus summation 1 to cosine
n psi plus. So, my flap motion is a fourier
series. So, we got inflow, which is the time
varying function, in the sense, sorry, not
time varying, azimuth varying. Now, you have
flap motion which is represented like this,
which has all the harmonics, but what we will
do is in this course, we will say, we are
not going to be bothered about this 2nd harmonic
onwards.
But please, I would like to caution you, 2nd
harmonics is important when you go to slightly
high speed; high speed means, I am talking
about 0.3 non-dimensional mu, about 0.3. Then,
you will find the 2nd harmonic content is
more than the 1st harmonic content, but for
the present, assume we neglect everything.
So, there is a beta naught beta 1c cosine
psi beta 1s sin psi, only three term approximation.
Please understand this is an approximation
I am making for the motion of the blade, physically
we will look at it, how it really represent.
What it really means, this is beta is for
one blade, but if I want to look at all the
blades and if I index them, then I will be
giving k, beta k, this is where, you know,
this is the psi k. If what psi k becomes psi
plus 2 pi, which is the number of the blades.
So, if this is the 1st blade, this will be,
psi 1 is nothing but psi 2, but this is omega
t.
Now, what happens, each blade is represented
as Fourier series because we have to know
each blade is independent, we are not teetering
rotor, it is an independent. This blade behaves,
that blade behaves, but all of them have the
same expression that means, when it rotates,
all the blades execute identical motion. When
it comes to a particular value of the azimuth,
will have only one value, it is not, that
each one is doing independently when it goes
round and round. Now, you see, this is the
blade beta naught means, it has gone up, that
is, all the blades have gone up by the same
amount and this is called the coning; all
of them have gone up same value.
Now, when you look at c, 1 c represents what?
If beta 1 c is positive, that means, this
is my psi 0 position, this is 180, this is
90, this is 90 degree, this is 180. That means,
when the blade comes back to this point, if
this is the, this will be like this, it will
do like this, at the back it will go up, when
it comes to the front it will come. So, when
it rotates it will do like this, that means,
the disk is tilted in the forward direction.
So, one is, I took the disk up then I tilted
in the forward, but 1s represent the tilt
in the, because here it is up, here it is
down, that is, tilt in the lateral direction.
Now, essentially, what this represents is
these three quantities only. The three quantities,
the orientation of the disk, disk in the sense,
you are outside, you are looking at the tip
path plane because all the blades are going
round and round, the tip path plane of the
rotor disk is like a circular disk. How the
circular disk is, circular disk, it can go
up, tilt, it can tilt, but when you go to
higher harmonics, these are all like warping
of the plane. It is like, you know, corrugated
sheet, how they will have it is, that motion.
Now, you see, that the complex representation
of that plane, I do not look at the warping;
I look at only the beam path. So, the rotor
disk, now please understand, without this
you cannot fly the helicopter, you need flap
because the flap is the one, which is tilted,
your thrust vector, whatever is there, you
tilt the thrust vector this way, then you
fly forward; you tilt that thrust vector in
the other direction, you fly. The tilting
of the thrust vector is obtained by flapping.
Now, how do we achieve the flapping? Because
pilot gives only pitch angle change, you know,
that collective pitch angle, all the blades
experience the same change in pitch, that
means, thrust will go up, more weight, more
lift, but when you do cyclic, swash plate,
cyclic plate because pilot sticks left side
or front, is basically tilting the swash plate,
as a result, is giving a cyclic variation
in the pitch angle. The cyclic variation of
the pitch angle is responsible for tilting
the disk because one side increases more lift,
other side lifts. So, it kind of tilts, but
there is a dynamics involved because it is
not just simple because the blade is go.
Now, pilot collective and two cyclic for one
main rotor. So, he gives, he can control these
three technically. Now, the question is, you
said, like this pilot is giving a pitch angle
in one frame and you defined your lambda mu
in some, we put, we drew a line rotor disk,
this is the rotor disk line, what is that
disk? Is it the tip path plane or is it the
hub plane or is it some control plane, what
is that plane? Now, you see, you can define
several reference axis and this will complicate
the problem. I will briefly tell you, there
are different types of frames of reference
because when I said flap, flap is with respect
to what plane? Is it with respect to…
I will briefly show one diagram and just to
indicate, that very simplistically, do not
bother about all these things. This is the
tip path plane; tip path plane means, you
draw a plane, which actually, the blade is
going only in that plane no matter where it
is, it is going in that plane. The root, do
not bother because the blade can flap up,
that is why, you are not bothered about the
root part. You are looking at the tip of the
blade, how it goes round, all the blades will
follow the same path because this is the same
Fourier expression. These are assumptions
we are making because all blades perform the
same motion.
Now, that is the tip path plane, then you
have shaft, the shaft axis and the hub, that
is the hub plane. Now, the hub can be, this
is the shaft axis, the shaft axis need not
be perpendicular to the tip path plane because
the shaft axis can be like this, the blade,
the tip path plane can be like this. That
means, shaft axis is another one; tip path
plane if you want to take that is another
axis. Similarly, you can define one more axis,
which is called the no feathering plane, no
feathering plane, NFP.
So, I will write here some three, four planes.
So, one is tip path plane, this we call it
TPP. Then, another one is shaft…
This we call it hub plane because the shaft,
hub is perpendicular to the shaft and the
shaft is attached to the helicopter. So, the
helicopter tilt, shaft also tilts. Then, you
also define no feathering plane, this is NFP.
Then, you can have one more plane, which is
control plane. Now, the, what is the control?
You have to know what each one of them is
and you can choose any one of the planes for
your problem, but it actually is, everything
has its own complexity, but finally, you choose
one. That is why, in the text book, some books,
old books if you see, you really do not know
what they are using unless you are very clear
about. Later, after more and more of understanding,
oh this is what you see, this is what tip
path plane you have agreed.
What happens if you are in that plane? There
is no flapping because the blade is only in
that plane, blade will not come out of that
plane, so there is no flapping in tip path
plane, agreed. Because you have a plane, as
a blade tip is going in that plane means what?
There is no flapping with respect to that
plane, but the pitch angle of the blade can
change as it goes around in that plane, is
it clear.
Now, you choose no feathering plane. In this
plane the pitch angle of the blade is constant
in that plane; the pitch angle of the blade
is a constant. This is, sketching is, maybe
we can with computer, three-dimensional we
should be able to make it. I can briefly describe
you with this diagram, a very simplistic diagram.
That is, you take this, our theta is what,
theta naught, this is what? The pitch angle,
let us say this is 0, just for simplicity.
When it is 90 degree, the angle is theta naught
plus theta 1s. When it is here, it is theta
naught, that means, the same aerofoil. If
you look here, it will have a large pitch
angle, which is plus theta 1s; when you come
here, it will have minus theta 1s, less pitch
angle. But you want to get a plane in which
the blade has theta naught only, that means,
what you do?
If you tilt this plane up like this, what
will happen? That means, this will go up,
here it will come down. So, you will find,
that the pitch angle here will reduce, here
will increase till you tilt it, such that
the pitch angle is same both sides. That is
what is shown here, the theta 1s, just for
indication.
Now, this is same as, then the, I mentioned
control plane. What is control plane? Control
plane is basically the plane of the swash
plate, but please understand, swash plate
I mentioned here because swash plate is there
for these helicopters, swash plate plane.
Now, if you do not have any other additional
mechanism, please understand, if you do not
have any other additional mechanism to introduce
pitch in the blade because sometime you can
have some coupling, which you will learn later,
or you can put a trailing edge tab and that
will twist the blade. Your swash blade may
be here, trailing edge tab may be twisting,
but if you do not have any additional mechanism
to, twist, change angle of attack of the blade,
these two are same, you follow.
So, here we have several, four planes, whether
all of them can be same or all of them cannot
be same? In certain specific situations, you
will find two planes, may be same, otherwise
each one is independent. Now, whole point
is, what reference access I must choose for
defining my motion of the blade defining because
I have to get the aerodynamic load. Please
understand, I have to get the aerodynamic
load, I have to define my inflow, you follow.
You have so many planes your hanging around,
so what we choose in this class? Hub plane,
but the complexity in hub plane is, it will
have both flapping as well as feathering.
Feathering is basically pitch angle change;
your blade will have both flapping as well
as feathering, that means, you have to take
both of them into account, it becomes, expressions
become more complex. On the other hand, if
you choose this plane, there is flapping only;
fathering, if you choose this plane, there
is no feathering, but there is only flapping.
So, one of them can be made simpler, but here,
you have to consider everything, but his is
more systematic and you will not make any
mistake. And most of the research we use this
plane because that is very systematically
you can develop, because these planes are
fixed with respect to the helicopter, whereas
these are all not fixed with respect to the
helicopter, you understand. Now, we will use
only hub plane.
Now, the whole point goes, what is my advance
ratio? What is my inflow? Everything is referred
with respect to hub plane. So, this is the
shaft, this is the hub, which is 90 degree.
So, you define what is the in the velocity,
in the plane of the hub, normal to the hub,
that is all. In the plane of the hub is your
forward speed, normal to the hub is you say
inflow, that is all.
But in this plane, the blade may be like this,
it may be twisted, does not matter, but when
you go and calculate aerodynamic load on the
blade, please understand, you have to define
proper coordinate system and this is where
you have to have a very systematic development.
If you follow the systematic development it
is clear to you because if, most of the, actually
all books, material, they do not give the
systematic, they will give final expression.
They will say take this, actually they make
lot of approximations in getting those, unless
you are sure what is being done you will say,
why it is taken like this.
So, what I thought, in my course, in this
notes also, which I am giving, we will follow
pure, like basic mechanics, like dynamics,
we will define a coordinate system, we will
define a position vector. Then, we will get
the velocity and then you we will get the
acceleration, we will define every point.
And then, we will say, at every stage it will
become complex, but we will say throw this,
throw this, make approximation and then you
will get a neat expression, which is to the
what you see in some other publications, books,
they make lot of assumptions.
