Let us try to understand what actually we
are trying to do. So, we have been talking
about the linearisation of aircraft equations
of motion. And after that, I am also defining
some derivativesSo, let us say where in this
aircraft design cycle, this analysis fits.
Why we are doing this? I want to look at that
briefly, and then we can proceed from where
we left yesterday.
So, you are asked to design an aircraft, given
specifications, performance specifications
or other requirements, for example, you have
to sit 20 passengers right. So, accordingly
you will start planning your geometry of the
aircraft and the weights you can calculate
and then you can look at some of the performance
parameters So, after you have done all that,
you have come down to a basic configuration,
is it not?
So, you have done a paper design of your aircraft;
ofcourse, this will also require lot of repetitions.
So, let us say, we have sort of arrived at
a configuration, geometry I know, and other
information like weight and inertia also we
know. So, we have one prototype right.
Now, this is on paper. We have got all the
numbers on paper and what we want to start
looking from here using whatever we have been
doing so far is, how the aircraft is going
to behave in motion, is it not? The performance
of aircraft or the behaviour of aircraft in
air is going to actually depend upon the configuration
and the aerodynamics around this configuration
right. So, now after you have done this, either
you build one scaled model and put it in the
wind tunnel and compute or measure all the
derivatives. That is a tedious task, is it
not? Finding out all the derivatives at all
flying conditions.
So, from here you can build a scaled model,
and then go for wind tunnel measurements of
the forces and the derivatives 
But when will you do this? Only after you
have frozen your model, because this requires
a lot of effort, is it not? You are not going
to, you know, just like that put your scaled
down model in the wind tunnel, start measuring
something, and then you find that the aircraft
is not giving you the desired performance
in response, then again change this. That
cannot happen right. Only after you have frozen
this, you are going to come down to this and
then to this, and then, use this data to predict
the aircraft motion using aircraft equations
of motion, that is one thing.
That is only when you have frozen your design.
Let us say without doing this, without going
through this route, now because after I have
done this, I cannot go back and change something
on my aircraft, not to a large extent. Only
small variations, or probably, you have to
design control systems to meet the specific
requirements. So, this is one route and this
will not involve cyclic repetition. We cannot
afford that. So, instead what we would want
to do is, we take this prototype, so design
is there, paper design is there, and go for
estimation of the derivatives of the forces,
force moment and their derivatives.
Afterwards, once we have done this, we can
now use the linearised aircraft equations
of motion and look at particular dynamic behaviour
with respect to that, we can look at 
stability and basically stability, and after
meeting the stability criteria, look at some
of the performance objectives. For example,
whether your aircraft is giving you a good
flying or handling quality or not.
So, after we have done this, now everything
is on paper. Paper or you can use computer,
you know, to predict the motion and this will
be using you know both.
Of course, they are not connected like this
but they are using the same set of equations,
is it not? Now, if I have gone through this
route, then I have a choice, because I have
not done this expensive wind tunnel testing,
is it not? Now, the question is how do you
estimate these derivatives, and this is exactly
what I am discussing right now in this, in
the last few lectures.
We want to estimate the derivatives, predict
the motion, predict the stability or dynamic
stability behaviour of the aircraft and see
if I am meeting some of the flying and handling
qualities or not. If I am not, then I have
a choice. I have actually done nothing here,
nothing that requires, you know, nothing that
requires lot of money, is it not? So, if I
am not satisfied, so if you are not satisfied,
then you can (( )) and right here not satisfied,
go back. If you are happy, if you are satisfied,
then you can go through this cycle now because
finally, you would want to look at aircraft
actual motion in air and that we can do using,
by simulating aircraft equations of motion.
So, you understand this part?
So, if you are satisfied, you can go from
here to this point or may be not that but
you can directly go for wind tunnel. So, if
you are satisfied, you can go to build a scaled
down model collect the data through wind tunnel
measurements and repeat this. You will still
have to do this, because wind tunnel measurements
also are going to introduce some errors. That
is one thing, and wind tunnel measurements
will be for the scale down model right. What
here we are doing is, we are looking at you
know different components and trying to find
the derivatives by adding the effect of you
know contributions coming from different components.
But if you are putting the scale down model
in the wind tunnel, then you are measuring
actual data right. Is this alright? Everybody
gets what I am saying?
Now, the question is how do you estimate these
derivatives? How do you go from this point
to this point and that is what we are addressing
right now. That is the question that we are
trying to address, and because we cannot,
you know in a normal class, you cannot do
it for all kind of flying conditions. I am
sticking to one flying condition that we have
taken.
Equilibrium flying condition or equilibrium
flying condition is, and this is one for example,
is it not? I am doing this part only for one
particular equilibrium flying condition; you
may have to repeat it for several flying conditions
right.
So, yesterday we were looking at the derivatives
of forces with respect to the state variables
of aircraft right. So, we already found an
expression for X u It came from 
it came from this balance of forces. Let us
say, the thrust is acting along the X axis
of the body and you have X equal to T plus
L sin alpha minus right. So, finally we have
found, if I include this thrust, we can also
keep this thrust separately. Right now, I
am including this thrust in this X force We
have already separated the gravity, gravitational
forces. I am not including that in this force
because that is not changing aerodynamically
right. Thrust also we will have to see, whether
it is changing with velocity or not, if it
is, or angle of attack, right. If it is not,
then we can drop this.
So, we derived expression for X u and that
was for this equilibrium condition. So, you
have to keep that in mind. Where C D u is
, 
So I want to find out, this is a constant,
so I am non-dimensionalising this derivative
with respect to u over u naught right. C D
has no dimension and this will also have no
dimension. So, we can find this as a constant.
C T u is actually 0 for an aircraft powered
by jet engines right. We do not have thrust
changing much with forward speed, so C T u
is actually 0 for and it is also 0 for and
it is also 0 in gliding flight right. You
have switched off your engine, or glider will
not, you know, many gliders. Now-a-days we
have motor gliders, which have engine also
there, but there are gliders which are not
powered or your aircraft may be gliding. You
know, you have switched off your engine, when
you are trying to land your aircraft.
So, C T u is going to be 0 in these two cases
and for variable pitch propeller, for aircraft
with variable pitch propeller engine, C T
u is nothing but minus CD naught. This is
again an estimate, it is only an estimate.
If you want to get an accurate measurement,
then you have to, you know, do a wind tunnel
testing.
So, in this case, X u becomes . 
What about this term? This term is change
in CD with respect to a change in non-dimensional
forward speed 
When is this going to be important? Only for
the flight in the compressible range. Actually,
it is going to be effective only when you
are flying beyond Mach number 0.5. So, you
have to remember that you have to actually
include this in your equations when you, if
you want to accurately predict the motion
right.
So, you can never actually predict motion
using this, you know, 100 percent accurate
but you want to get a close estimate, a close
prediction.
Yeah. Even for lower velocities, how do we
say only for Mach greater than 0.5 and different
Mach but how is that happening? Because that
that learner CDu. Yeah. Once another keeps
increasing, the CDu will keep decreasing at
least zero point.
So, only in this part it becomes, only beyond
Mach number 0.5, it becomes, you know, I am
not, remember I am saying that if it is not
important, you should not include it. If the
aerodynamics is telling you that this number
is significant at your flying condition, then
you have to include it to predict your aircraft
motion accurately.
Now, we can repeat the exercise that we did
yesterday using this balance of forces. We
can find out what is X w.
So, remember this 0 is the equilibrium flying
condition. This CL naught is corresponding
to that particular equilibrium flying condition,
which is also alpha naught equal to 0 and
Q naught is half rho u naught squared .
Now, let us try to write down the equation
in this Z direction. Then, we can find out
the Z derivatives. So, without repeating this
exercise, now I am going to write down what
these derivatives are.
Actually, there is one more thing we could
have done. You know, I already wrote force
equations in two different axis systems if
you remember right. Now, you will think why
I am doing this. If I had just written my
force equations in the wind fixed axis system,
I would not have really bothered about the
derivative of Z with respect to u I would
have only talked about CL and CD, is it not?
Because there we do not see this notation.
Let us look at what C L u is? Again, this
is going to be only significant when you are
in this range of speed Mach number greater
than 0.5. So, there is a formula given by
Prandtl and Glauert to account for the compressibility
correction .
From here, you can find out what C L u is.
C L u is Now, we are talking about derivatives
which are non-dimensional. So, as I said,
now each of these forces are going to depend
upon each of the variables; u, v, w, p, q,
r. Now, it depends which flight regime you
are designing your aircraft for right. So,
if you are designing your aircraft for supersonic
speeds, then you have to see, in supersonic
speeds, which are the derivatives which become
important.
Significant derivatives, you have to look
at the magnitudes, right and depending upon
their relative magnitudes, you may want to
keep them or you may want to drop them. But
if I am doing a numerical simulation, I may
just want to keep all of them right. Even
if the effect is smallest. But if I am doing
something analytically, I want a simpler model
and there I will only keep the significant
terms. Is it not?
See whenever we are talking about flight regimes,
we are talking about two parameters. One is
angle of attack and the other one is the speed
So, alpha and Mach number are the two parameters
defining the flight regime You can be flying
at high alpha but in the subsonic speed regime
At a higher Mach number, you may not want
to try high alpha maneuver. You know, supersonic
speeds you cannot actually change alpha the
way you can change in the subsonic speed,
is it not? Because large momentum in the forward
direction, if it is flying at supersonic speeds.
So, there these derivatives with respect to
Mach numbers will become important and you
can keep alpha as low alpha flights.
So, let us say somebody comes to you and asks
you to design a four seater aircraft So, you
can figure out, you know, there will be many
four seater aircraft, the geometry for which
will be available, but aerodynamic data will
not be available. That is the trade secret,
because that is the one which requires the
maximum amount of work right and also classified.
So, that is one thing which is classified
about the aircraft. Nobody gives you the aerodynamic
database just like that So, I am going to
look at some more derivatives because it may
not be possible to look at all of them.
Let us look at q derivatives. So, the aircraft
is having a steady pitch rate and now, we
want to find out how that is going to affect
the forces and moments So, what I am interested
in finding out is this How C m is changing
with respect to this steady pitch rate? How
C L is changing with respect to this pitch
rate or how C z q, if you want to talk about
this normal force coefficient. So, your aircraft
is steadily pitching up. We will only include
the significant parts. We are only trying
to get an estimate In general there will be
contributions coming from other components
also but we are looking at major contributions.
When you are having a steady pitch up, what
is happening is, this tail is going down right.
It is going to push air down and then you
see a relative wind upwards. So, if this distance
is l t, then this tail is going to see upward
wind, which is
So, aircraft is flying at this forward speed
and then you see a steady pitch, and this
can be around your equilibrium condition We
are talking about both, flying equilibrium
condition and also the perturbed flight condition
right. So, this q may just come from because
of the perturbed condition, is it not?
So, what is going to change here because of
this?
So, what is going to change here because of
this w, is the angle of attack. Angle of attack
at the tail is going to change, it is going
to increase, is it not? If I have positive
pitch rate, then this alpha, increase in angle
of attack at the tail is going to be positive
right. So, there is a change in overall lift
of the airplane and that is because of the
change in lift at the tail due to this increase
in angle of attack.
So, this eta t is the ratio of the dynamic
pressures St is the tail platform area, Sw
is the wing platform area, a t is the lift
curve slope of the tail, into this increase
in angle of attack due to pitch rate. So,
the change in lift because of this pitch rate
is eta t .
Now, this is not a non-dimensional quantity
Q is, C L is a constant and q has some unit.
So, I want to non-dimensionalise it. I want
to find out an expression for C L q actually
and this has to be a constant.
So, now I am introducing some parameter which
is q c bar over 2u naught, which will have
the same, if you look at the unit of 2u naught
or u naught over c bar, it will have the same
unit as this q, q is in radian per second
right. So, that is how you non-dimensionalise
the rate derivatives.
Remember, if you are not talking about these
derivatives which are damping derivatives,
then you are not talking about dynamic stability
right. We already did a static stability analysis
where we talked about the stiffness term What
we were missing there are these derivatives.
So, in this particular condition, when I am
doing everything around alpha naught equal
to 0, I am trying to find out derivatives
only for this trim condition. Then, C L q
is also C Z q. There should be a minus sign
here because lift is acting upward and Z force
is acting down.
What is happening to the pitching moment,
because of this pitch rate? .
So, I am including this angle which, you know,
this expression we have written already, expression
for C m. Now, I am writing down the expression
for change in this C m because of change in
angle of attack at the tail due to the pitch
rate
So, this was the steady pitch rate. It is
going to change the angle of attack at the
tail, but there is another derivative, which
looks similar to this, is due to the unsteadiness
in the flow.
So, it is because of the (time) rate of change
of angle of attack.
So, you are in a perturbed condition. Imagine
where all you will need these derivatives?
You are flying in equilibrium condition and
then a gust will hit your aircraft. Let us
say you are flying a cruise condition. Gust
will hit your aircraft. So, you are going
to see a perturbed motion Steady condition
is when you are sitting in your aircraft,
you actually feel nothing, is it not? You
will feel as if you are sitting in this room,
unless you see something outside moving relatively.
Otherwise, you know you are actually sitting
in a room and whenever there is a gust hitting
the aircraft, the aircraft does some motion
and that is the motion that we are talking
about. So, derivatives due to rate of change
And this is primarily because of the unsteadiness
in the flow.
So, there is a change in angle of attack,
rate of change of angle of attack is involved
and 
the flow becomes unsteady for various reasons.
When your aircraft is pitching up, you know
it is almost like pitching up when I am changing,
when I am having a rate like, you know, rate
of change of angle of attack, then actually
the aircraft is pitching up right. That is
not a steady pitching. You know it is a change
in angle of attack. So, what happens is, the
flow field gets disturbed and there is a time
lag involved in the movement of the flow.
So, the flow, that tail is going to see is
going to be after time lag.
What this time lag does is, it introduces
a rate in the change of downwash angle with
respect to time. Earlier, we have assumed
that, in steady conditions, downwash angle
is only going to depend upon the steady alpha
right, but because of this unsteadiness in
the flow and due to this alpha dot, this downwash
angle becomes a function of alpha which is
delayed. So, tail is not going to see the
flow at the same angle as the wing is going
to see, and you know, it is going to see a
delayed change in angle of attack
So, alpha at the tail is going to be a function
of time now.
So, I will just write down now the expression
for the derivatives. .
This you can find out from this expression
right. So, it is going to be
So, with this backdrop now, I am going to
move forward. I am not going to, you know,
keep doing this. So, now whatever we have
done, the derivatives, there are some more
derivatives, the physical explanation of arriving
at any formula for those derivatives are given
in the book. I would not go into describing
all of them. I will assume that we are knowing
how to find the expressions for them and then
move on from here.
