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LECTURE 1

Introduction. External forces acting on an aircraft.

Flight dynamics is the science of the aircraft motion in airspace by the action of

external forces applied to it.

Flight dynamics is, on the whole, a combination of three classical branches of

science, such as solid mechanics, fluid and gas mechanics and mathematics.

Being a theoretical subject, flight dynamics finds an important practical

application for aircraft design at the same time. The tasks of the aircraft design can be

classified into three main groups, which come to ensuring: general characteristics,

structural strength and flight performance. By studying flight dynamics we determine

the aircraft flight performance and loads acting on its structure during the flight in

disturbed atmosphere or during manoeuvring. The importance of this subject is defined

even by the fact that studying the flight dynamics issues is an integral part of all

programmers of preliminary design work preceding all flight tests.

Handling the flight dynamics problems is possible from two positions.

1. It is possible to set up exact general equations the solution of which can be

obtained only by computers. However using the computers can never substitute

for mathematical and physical researches.

Analytical solutions are of special importance for engineers when the application

field of these solutions is known, that is the application field of simplified

assumptions used for solving a problem is known.

However it is necessary to keep it mind that the analytical model does not always

reflect a real phenomenon exactly and there are processes when using numerical

methods is difficult or impossible, e.g., in connection with necessity of finding a

solution near a peculiar point.

2. In the case when neither analytical nor numerical methods of the engineering

analysis give the grounds sufficient for an exact quantitative characteristic of

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motion process, an experiment should be used. The experiment is made both in

the process of designing and at the end of it.

Basic laws of dynamics. Dynamical problems. Fundamentals and definitions

Basic laws of dynamics were suggested by Newton in his work Mathematical

principles of natural philosophy.

1. The law of inertia (the first Newtons law):

Every body remains state of rest or uniform motion in a straight unless some

force to change that state.

The reference system in respect to which the first Newtons law is acting is

called the inertial one.

2. Basic law of dynamics (the second Newtons law):

jmFrr

==== . Any change of a body motion is proportional to the applied force and

acts in the same direction.

3. The third Newtons law:

Actions of two bodies on each other are always equal in quantity and opposite in

direction: 21 FFrr

==== .

Having stated this law Newton first formulated the principles of jet propulsion.

Knowing these laws on the whole, it is possible to turn directly to the

problems of flight dynamics which can be stated in such a way:

1. knowing the laws of the aircraft motion (path), determine the forces acting on it,

which ensure the action of this law of motion.

2. knowing the forces acting on the aircraft determine the law of the aircraft motion.

If the forces and moments or the law of the aircraft motion are completely

specified, such problems are called definite or related ones. As shown further, the

behaviour of the aircraft in space is described by the system in which there are fewer

equations than unknown quantities. Therefore there appears the necessity of specifying

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one of the parameters of motion in the function of any other parameter (including the

time) for closing the system.

Since there are fewer unknown quantities in the equations than the equations

themselves, one or several parameters can arbitrarily vary. Thus, the problems on

determining extreme values of this or that parameter of motion reached during the

flight are of special importance in flight dynamics.

In this case the path type, the law of varying the forces and moments acting

on the aircraft in the course of time are unknown beforehand. Only the so-called

boundary and edge conditions, initial values of the motion parameters, as well as

extreme conditions (min time, min fuel consumption, etc.) are prescribed.

The motion of the aircraft can be straight-line, curvilinear, steady and unsteady

motion.

To assess the possibility of handling the problem stated for the aircraft it is

necessary to know its performance and stability, controllability characteristics. The

performance of the aircraft under consideration is defined by possible speed range

(absolute ceiling, zoom altitude and service ceiling), range ( techL , maxL , realL ), time in

flight, manoeuvrability, etc.

Minimum flight speed ( minV ) is the lowest flight speed, at which the aircraft is

still in a horizontal uniform flight.

Maximum speed ( maxV ) is the speed of steady horizontal flight, which can be

provided at the completely open engine throttle (maximum engine power setting).

The static ceiling ( staticH ) is a maximum altitude, at which the aircraft of the

predetermined mass can fulfil the level flight at a constant speed, when the throttle is

completely open.

Technical range ( techL ) is the range that can be reached at the complete fuel

utilization.

Maximum range ( maxL ) is the distance that can be covered by the aircraft at the

complete fuel utilization and under optimum flight condition.

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Service range ( realL ) is the distance covered by the aircraft keeping abroad some

fuel reserve that amounts to 5-10% of the initial fuel capacity.

The aircraft endurance is the time in flight under conditions when the min. fuel

consumption (kg/hr) is realised.

The aircraft manoeuvrability is the ability of the aircraft to change its position in

n-dimensional parameters space in a definite time interval (velocity, altitude and flight

direction change).

The aircraft manoeuvrability is characterised by the load factor vector, because it

takes into account the forces magnitude and direction, the change of which makes it

possible to control the flight.

G-Load is the ratio of thrust vector sum and the total aerodynamic force to the

gravity value:

G

R

mg

RPn A

rrrr

====++++

====

By projecting the load factor vector on the coordinate axis we obtain load

factor components along the axes. Thus, the load factor vector projections on the axis

of the rate coordinate system are as follows:

G

Rn a

a

xx ==== - tangential,

G

Rn a

a

yy ==== - normal (in wind coordinate system),

G

Rn a

a

zz ==== - lateral.

In body axes projections the load factor vector may be presented by xn , yn , zn

components called longitudinal, normal and lateral load factor, respectively.

Load factor sign in the body axes are introduced in such a way:

0nx >>>> , if the pilot is pressed to the seat back, acceleration;

0nx

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0ny >>>> , if the pilot is pressed down to the seat;

0ny > , if the pilot is pressed to the left hand side and vice versa.

The aircraft controllability is its capability to respond to the force applied to

control levers by the pilot with a corresponding movement in space.

The aircraft stability is its capability to return to its initial flight condition after

the cause of deviation from the initial condition ceased to act.

The aircraft moves in space by Newtons laws of motion.

According to these laws the aircraft is the body of a variable mass, for the aircraft

mass decreases in flight in the course of time. In the general case the aircraft is also the

body of a variable shape, because it is subjected to deformations caused by the

aerodynamic load; besides, the controls deflections are necessary for the flight control.

It follows that the aircraft should be considered as the body with a great number of

degrees of freedom. In this case setting up the equation systems is extraordinarily

difficult, and solving the equation systems is also difficult. Therefore a number of

simplifying assumptions are taken for solving the problem. The most important of them

are:

1. The environment in which the flying aircraft is considered to be invariable in

time, the characteristics of this environment are uniquely considered the known

flight altitude functions.

2. We shall consider the aircraft as the body of variable composition (mass) with a

stiff fixed external shell. In other words, we shall neglect structural elastic

deformations connected with external loads and the aircraft surface kinetic

heating.

3. The no stationary of processes which occur inside the shell confining the variable

composition body will be neglected. Thus, we shall not take into account, e.g.,

the rapid fuel movement inside the aircraft.

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4. We shall take into account gravity forces, acting on the aircraft, caused only by

the Earth neglecting the gravity of other celestial bodies.

Since in the general case the aircraft motion can be represented in the form of

forward motions of the centre of mass and the system rotation relative to the centre of

mass, the flight dynamics problems can be divided into two classes:

- Problems of determining the path of the centre of mass.

- Problems of determining the aircraft necessary or possible angular

positions relative to the path of the centre of mass.

The first class of problems is described by force equations and moment equations

in projections on the coordinate axis, moment equations and some constraint equations.

The second class of problems is described by force equations in projections on

the coordinate axis and some constraint equations.

The first class of problems is solved in the first part of the flight dynamics

course, the aircraft being considered as a material point having a variable mass in the

general case.

The second class of problems is solved in the second part of the course, the

aircraft being considered as a system of material points.

In the first part of the course, considering only the equations for forces in the

projections on the coordinate axis, we assume that a quite definite angular position

corresponding to specified or necessary forces is given to the aircraft at any moment

with the help of controls deflection performed by the autopilot or the pilot.

The amount of the controls deflection and the possibility of performing

these deflections are defined in the second part of the flight dynamics course, in which

stability and controllability characteristics are determined.

External forces acting on the aircraft

To solve equations describing the aircraft motion it is necessary first of al to

know what external forces act on the aircraft considered as a body of variable

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composition.

The aircraft is acted upon by such external forces:

1) mass external forces caused by the suns, earths, moons and other planets

gravity;

2) aerodynamic forces (in the case of flight in sufficiently dense atmospheric

layers);

3) reactive force of engine thrust installed on the aircraft (in the case when the

engine is operating).

1. Mass external forces

During the aircraft flight in the space it is moving in the gravitational field of

different celestial bodies.

Relative position of the aircraft and celestial bodies in space continuously

changes. Therefore the problem of calculating the flight paths, known as the problem

of several bodies being attracted in celestial mechanics, is found to be complicated

enough.

However, if we limit ourselves to the study of the aircraft flight at comparatively

small distances from the Earths surface, then as shown by calculations, in consequence

of very great distances from the Earth even to the nearest celestial bodies the attraction

by these bodies is found to be essentially less than the Earths attraction.

Thus, e.g. the Suns attraction the mass of which is 33200 times that of the Earth,

and the distance from the Earth (the smallest) is about 147 million kilometers, even at

the flight altitude of 1000 kilometers is only about 0,1% of the Earths attraction.

The Moons attraction at the same flight altitude is approximately equal to

6105 the Earths attraction, because the Moon mass is about 80 times less than that

of the Earth, and the distance from Earth to the Moon ( the smallest) is 363000

kilometers.

In that way, considering low flight altitudes (up to H=1000 2000 kilometers),

we may limit the problem taking into account only mass forces of the Earths attraction.

Acceleration of gravity g on the surface of the Earth, in general, depends on

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the geographical location of the point at which g is determined. It is explained, on the

one hand, by the fact that masses inside the Earths volume are not quite uniformly

distributed; on the other hand, the Earth has not exactly spherical shape. Both these

reasons, however, do not have a great influence on g. Therefore it is quite

permissible to take some average acceleration of gravity, constant for all points of the

Earths surface.

Let us assume, as usual in calculations

g0=9.81 m/sec2

(1.1)

Thus, we shall consider further that the Earth has a spherical shape and the center

of its attraction is a geometrical center of this sphere. In other words, we shall proceed

from Keplers gravitational field.

According to Newtons law acceleration of gravity is inversely proportional to

the square of the distance from the point under consideration:

2

E

E0

Hr

rgg

==== , (1.2)

where Er =6371 km, Earths radius and H the altitude of the point under

consideration above the Earths surface that will be taken as sea level.

At not very high altitudes g little differs from g0 at sea level. E.g., at H=50

kilometers g=9,7 m/sec2.

Therefore for the problems related to the study of aircraft flight at low altitudes,

up to about 50 100 kilometers, for the purpose of simplification g is assumed to be

constant, independent on the flight altitude.

Thus, if low flight altitudes are studied, it is possible, to a sufficient degree of

accuracy, to consider the relation between the mass force and the force of weight as a

linear relation:

)H(mgG ==== . (1.3)

We agreed to consider the Earth as a spherical body. When the aircraft moves

along the path equidistant to the Earths surface (and also along other paths, except the

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flight along the path that is the continuation of the Earths radius), we should take into

account a centripetal force caused by the curvature of the Earths surface and acting on

the aircraft. Centripetal acceleration depends on the flight speed and increases with its

increase. The centripetal acceleration becomes comparatively not high as well for

comparatively not high flight speed. E.g., during the aircraft flight at the speed of

V=1000 m/sec parallel to the Earths surface the centripetal acceleration

2

E

2c sec/m16.0

r

Vj ======== , (1.4)

i.e. about 1.6% of acceleration of gravity.

Therefore when solving the problems related to the aircraft flight at

comparatively low speed, up to about 1000 1500 m/sec, it is permitted to neglect the

centripetal acceleration caused by the curvature of the Earths surface, i.e. to assume

the Earth as the flat one.

The role of the centripetal acceleration increases with the increase of the flight

speed, and when reaching the escape velocity (the escape velocity at sea level is

Vkr=7.9 km/sec), it becomes determining. When investigating the motion of ballistic

missiles or spaceships, it is not permitted to neglect the centripetal acceleration for

setting up the equations of the aircraft motion, as well as in some cases of winged

aircraft flight (the spaceships intended for reuse and others). In these cases the Earth

will be considered to be spherical, and the centripetal acceleration caused by the

curvature of the Earths surface will be taken into account.