Flight Dynamics II is the name of the course.
In this course, we are studying motion of flying objects in air. The motion that we
will study is about the equilibrium condition. Let us say we assume that, the object that
we are looking at, is flying steady equilibrium motion. So there is an equilibrium state,
let us say this object is flying. And what we want to look at is what happens to this
motion, when aircraft or any flying object, is disturbed from this equilibrium state.
So, what we actually interested in looking at is, the perturbed motion of the flying
object from its steady equilibrium flying condition. This perturbed motion is introduced
because of the unsteadiness in wind. Several wind conditions, for example,
gust or may be a different wind condition, which is suddenly encountered.
There are two things; one is to look at the small motion around the equilibrium state,
so this motion is mainly caused by small disturbances, and the other one is related to large motion.
So, this is small motion around the equilibrium state and large motion, which requires the
control effort. Some of the examples of flying objects are: birds,
insects. These are living examples. And there are examples which are man-made machines.
In this category, you will have several types of objects: gliders, spacecraft, missiles,
and aircraft and so on. In this course, we look mainly at the class
of objects, which are using its interaction with air to produce forces and moments to
sustain its motion through air. So, we basically are looking at
atmospheric flight, and majority of this course will be about aircraft (airplanes). So, we
are basically looking at aircraft atmospheric flight dynamics.
An aircraft, we know that, it uses its aerodynamic shape to create forces and moments which sustain
the motion of aircraft in air and what we want to look at is, how the sustained motion,
which is, for example, you can talk about a level flying condition where aircraft engine
produces thrust, which gives it a forward motion to overcome drag, and because of the
aerodynamic shape of the aircraft, you have lift produced which balances the weight. This
is the one such flying condition which is cruise flying condition, where lift is balancing
the weight of the aircraft and thrust is providing the forward speed, overcoming the drag.
Of course, there are many parameters which are going to affect this balance of forces
and also of moments. There is no unbalanced moment acting here which will try to give
aircraft also angular motions. So, here we are assuming that, the moments are all balanced
and forces are also balanced, so that aircraft is in steady condition.
This motion that we are talking about, small perturbed motion or in general, even the steady
flight condition, is going to depend upon various factors. For example, whether the
aircraft is flying at sea level conditions or at some altitude which is quite large,
will depend upon, what surrounding atmosphere is. Atmosphere which changes with altitude affects
the motion of aircraft. Motion is small motion around equilibrium state or the large motion
from an equilibrium state. A typical parameter which is, the density, the variation of density
with altitude is quite significant .... And if you look at the order of magnitude
of density at different altitude levels: at sea level, I am going to very soon tell you
where this density is appearing in these forces. So, let us look at how this density is changing
with altitude. We know that the value of density of air at
sea level condition is 1.225 kg per meter cube, and let us look at how it changes over
different altitudes. At 20 kilometers, it is roughly 0.088 kg per meter cube, and at
47 kilometers, it is 0.00147, 61 kilometers it is 0.00025 and, at 80 kilometers to 90
kilometers altitude, it is of the order of 10 raised to minus 5 (unit kg per meter cube).
With such large variation in density with respect to the altitude, what changes is the
aerodynamic forces and also the moments. So, this is one parameter which is going to affect
the motion of the aircraft, through aerodynamic forces and moments that are developed on the
aircraft when it is going through air. The other things that can affect the motion of
aircraft are its own parameters. For example, each of the components of aircraft . For example, the fuselage, which we designed
based on our mission requirements. A four seater aircraft will have some fuselage geometry,
or large transport will have a longer geometry. Depending upon how the shape of the fuselage
looks like, it is going to contribute to the aerodynamics of aircraft. Engine plants, which
is an integral part of aircraft, how it interacts with air, that is also going to play a major
role in deciding the motion of the aircraft. And there are several other surfaces, which
are control surfaces, mounted on (lifting surfaces of aircraft), airfoil like shapes.
You have wing, tails; which are horizontal tails and vertical tails. So this is horizontal
tail and this is vertical tail. And control surfaces, these are aerodynamic control surfaces
which manipulate air passing over an aircraft to change lift and moments. Control surfaces
are: for example, ailerons mounted on the outboard side of the wing. Its a hinged surface,
which can be moved about the hinge line, rotated about the hinge line, so that, it affects
the flow field and thus results in change in aerodynamic forces and moments.
Aileron is the control surface mounted on the wing, elevator is, all of these are small
flaps mounted on the major surfaces. Elevator is the small flap attached to the trailing
edge of the horizontal tail and, rudder is mounted on the vertical tail. So, these are
major components of aircraft and they contribute actually to the motion of aircraft in steady
condition or perturbed conditions. What we are going to do is, we are going to
first try to understand aircraft parameters. And in order to do so, let us first define
an axis system (fixed) to the aircraft with origin at the center of gravity which is also
the center of mass of the aircraft. XB-YB-ZB is an orthogonal axis system
fixed to aircraft with origin at its center of mass, which is also
the center of gravity. We will assume the aircraft to be a rigid
aircraft, it will have 6 degrees of freedom in motion, 3 degrees of freedom for translational
motion and 3 for rotational motion. These translational motions can be along the 3 axes
of the aircraft . Aircraft has velocities along its axes which are u, v and w. It can
also have rotations about the axes. Rotation about the X-axis is called the roll motion.
A positive roll motion would be like this (Y axis moving towards Z axis). And the rate
associated with this motion is called the roll rate . Similarly, we have two other rotational motions.
A pitch motion is the motion which is about the Y-axis of this aircraft.
A positive pitching motion is when ZB is moving towards XB and the variable associated with
that is called pitch rate. So, aircraft is rotating about the YB axis in this XZ plane
and that is what is pitching motion. Yawing motion is the motion, rotational motion,
about the ZB axis. So, this X axis is moving towards Y in XY plane about this Z axis. A
positive yawing motion is when XB is moving towards the YB axis and that rotation is about
Z axis. So, this is yaw and the variable associated with this motion is r, which is called yaw
rate . Aircraft has three (linear) velocities u,
v and w along its axes, which describe the translation motion of aircraft and 3 rotational
motion about the 3 axes XB, YB and ZB, given by these variables p, q and r which are roll
rate, pitch rate and yaw rate. Now, this motion is being caused because of the forces and
moments acting on the aircraft. So let us look at expressions for those forces and moments. X is the axial force, it is sum of all the
forces acting on the aircraft along its X axis and that is called axial force, which
is equal to Q into S into Cx. Where Q is the dynamic pressure, expression for which is
half rho V squared, where V is the absolute velocity, also the relative wind speed, S
is the wing planform area and Cx is a non-dimensional coefficient.
In this case, Cx is axial force coefficient. The force along the Y axis is called side-force
which is equal to Q into S into Cy, where Cy is side-force coefficient. Z is the force in
the vertical direction along ZB axis . Cz is vertical force coefficient or we should
say normal force coefficient . So, these are the three forces acting along X, Y and Z axis
of the aircraft and there are three moments. These moments are: rolling moment, the expression
for which is, L is the rolling moment, the sum of all the (aerodynamic) moments acting
about the X axis of the aircraft is rolling moment. So, everything else you know, Cl is
the rolling moment coefficient and b is the span
of the wing. Expression for the pitching moment.
Pitching motion is motion about the Y axis you know. Sum of all the (aerodynamic) moments
which are resulting in the pitching motion of the aircraft is given by this M, which
is equal to Q into S into Cm into c bar. Cm here is pitching moment coefficient and c bar is the
mean aerodynamic chord of the wing . Expression for yawing moment is given by N
equal to Q into S into Cn into b, where Cn is yawing moment coefficient . So, axial forces
are actually the resultant forces, which are going to depend upon forces due to, gravitation,
aerodynamic forces acting on the aircraft. This force for example, the axial force is
the sum of its aerodynamic components, thrust component
and the gravitational component. The aerodynamic components (of) the forces
and moments are going to depend upon how the wind is oriented with respect to the aircraft.
Orientation of wind, relative wind coming onto the aircraft, is described by two angles:
which are alpha; which is the angle of attack and beta, which is the angle of sideslip.
Let us look at what these angles are in terms of the velocity components of aircraft along
its axes. The relative wind coming onto the aircraft,
or the aircraft is going into the wind with this velocity V, with its component u, v and
w. The aircraft velocity vector makes an angle beta, with its projection on the XZ plane
(of the aircraft) . XZ plane is also known as
the longitudinal plane of the aircraft and the aircraft is assumed to be symmetric about
this plane. Or we can say that the axis system is so defined that, this X Z plane is the
plane of symmetry of the aircraft. So, beta is the angle which the plane, this shaded
plane, makes with the XZ plane . In terms of beta and the angle of attack,
which is defined as this (Refer to the sketch on board Time 33:28) . In this XZ plane, the
component of velocity vector is in this direction and the angle, that projection of V onto X
Z plane makes with XB axis is the angle of attack . Alpha is the angle of attack, which
is defined as the angle between the projection of velocity V onto the XZ plane and the X
axis of the aircraft. Sideslip angle is defined as
the angle between , shaded plane in this picture and the X Z plane . So, in terms of the velocities,
these angles are given as beta, sin beta equal to v over capital V, and tan alpha is w over
u. Clearly this part of the axial force, the
aerodynamic part, this is coming from the lift and drag components, this clearly depends
upon what these angles are. How wind is or aircraft is oriented with respect to the wind,
this component is going to depend upon that, and it also depends upon the Mach number.
If you also want to include the altitude in this function, then, the force is, aerodynamic
force is actually depending upon the Reynolds number, which includes the velocity and the
density, both. There are many popular books on this subject
available now. But in this course, the books that I am going to refer to are the following
. - Flight Stability and Automatic Control by
Robert C Nelson published by McGraw-Hill , - Dynamics of Atmospheric Flight by Bernard
Etkin, published by Dover publications , - Mechanics of Flight by Warren F. Phillips
published by John Wiley publications , - Airplane Performance Stability and Control
by C D Perkins and R E Hage, the first edition of this book appeared in 1949
and this is still one of the classics , and
- Performance Stability Dynamics and Control of Airplanes by Bandu N Pamadi, published
by AIAA education series of books. So, throughout this course, we will be taking
contents from one of these books for different topics and many of the examples (problems)
actually will come from this book by Robert C Nelson. I think we will stop.