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Flight dynamics-I Prof. E.G. Tulapurkara Chapter-1

Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1

Chapter 1 Introduction

(Lectures 1, 2 and 3)

Keywords: Definition and importance of flight dynamics; forces acting on an airplane; degrees of freedom for a rigid airplane; subdivisions of flight dynamics;

simplified treatment of performance analysis; course outline.

Topics 1.1 Opening remarks

1.1.1 Definition and importance of the subject

1.1.2 Recapitulation of the names of the major components of the airplane

1.1.3 Approach in flight dynamics

1.1.4 Forces acting on an airplane in flight

1.1.5 Body axes system for an airplane

1.1.6 Special features of flight dynamics

1.2 A note on gravitational force 1.2.1 Flat earth and spherical earth models

1.3 Frames of reference 1.3.1 Frame of reference attached to earth

1.4 Equilibrium of airplane 1.5 Number of equations of motion for airplane in flight 1.5.1 Degrees of freedom

1.5.2 Degrees of freedom for a rigid airplane

1.6 Subdivisions of flight dynamics 1.6.1 Performance analysis

1.6.2 Stability and control analysis

1.7 Additional definitions 1.7.1 Attitude of the airplane

1.7.2 Flight path

1.7.3 Angle of attack and side slip



1.8 Simplified treatment of performance analysis 1.9 Course outline 1.10 Background expected References Exercises



Chapter 1 Lecture 1 Introduction 1 Topics 1.1 Opening remarks


1.1.2 Recapitulation of the names of the major components of the airplane

1.1.3 Approach in flight dynamics

1.1.4 Forces acting on an airplane in flight

1.1.5 Body axes system for an airplane

1.1.6 Special features of flight dynamics

1.2 A note on gravitational force 1.2.1 Flat earth and spherical earth models

1.3 Frames of reference 1.3.1 Frame of reference attached to earth

1.1 Opening remarks

At the beginning of the study of any subject, it is helpful to know its definition,

scope and special features. It is also useful to know the benefits of the study of

the subject, background expected, approach, which also indicates the limitations,

and the way the subject is being developed. In this chapter these aspects are

dealt with.


The normal operation of a civil transport airplane involves take-off, climb to

cruise altitude, cruising, descent, loiter and landing (Fig.1.1). In addition, the

airplane may also carry out glide (which is descent with power off), turning

motion in horizontal and vertical planes and other motions involving

accelerations.



Fig.1.1 Typical flight path of a passenger airplane

Apart from the motion during controlled operations, an airplane may also

be subjected to disturbances which may cause changes in its flight path and

produce rotations about its axes.

The study of these motions of the airplane either intended by the pilot or

those following a disturbance forms the subject of Flight dynamics.

Flight dynamics: It is a branch of dynamics dealing with the motion of an object moving in the earths atmosphere.

The study of flight dynamics will enable us to (a) obtain the performance of the

airplane which is described by items like maximum speed, minimum speed,

maximum rate of climb, distance covered with a given amount of fuel, radius of

turn, take-off distance, landing distance etc., (b) estimate the loads on the

airplane, (c) estimate the power required or thrust required for desired

performance, (d) determine the stability of the airplane i.e. whether the airplane

returns to steady flight conditions after being disturbed and (e) examine the

control of the airplane.

Flight dynamics is a basic subject for an aerospace engineer and its

knowledge is essential for proper design of an airplane.

Some basic ideas regarding this subject are presented in this chapter. The topics

covered herein are listed in the beginning of this chapter.



In this course, attention is focused on the motion of the airplane. Helicopters,

rockets and missiles are not covered. 1.1.2 Recapitulation of the names of the major components of the airplane

At this stage it may be helpful to recapitulate the names of the major

components of the airplane. Figures 1.2a, b and c show the three-view drawings

of three different airplanes.

Fig.1.2a Major components of a piston engined airplane

(Based on drawing of HANSA-3 supplied by

National Aerospace Laboratories, Bangalore, India)



Fig.1.2b Major components of an airplane with turboprop engine

(Based on drawing of SARAS airplane supplied by

National Aerospace Laboratories, Bangalore, India)



Fig.1.2c Major components of an airplane with jet engine

(Note: The airplane shown has many features, all of which may not be there in a single airplane).

1.1.3 Approach The approach used in flight mechanics is to apply Newtons laws to the

motion of objects in flight. Let us recall these laws:

Newtons first law states that every object at rest or in uniform motion

continues to be in that state unless acted upon by an external force.

The second law states that the force acting on a body is equal to the time

rate of change of its linear momentum.

The third law states that to every action, there is an equal and opposite

reaction.

Newtons second law can be written as:

F = ma ; a = dV / dt ; V = dr / dt (1.1)



Where F = sum of all forces acting on the body, m = mass, a = acceleration, V = velocity, r = the position vector of the object and t = time (Note: quantities in bold are vectors).

Acceleration is the rate of change of velocity and velocity is the rate of

change of position vector.

To prescribe the position vector, requires a co-ordinate system with

reference to which the position vector/displacement is measured.

1.1.4 Forces acting on an airplane During the analysis of its motion the airplane will be considered as a rigid

body. The forces acting on an object in flight are:

Gravitational force

Aerodynamic forces and

Propulsive force.

The gravitational force is the weight (W) of the airplane.

The aerodynamic forces and moments arise due to the motion of the

airplane relative to air. Figure 1.3 shows the aerodynamic forces viz. the drag

(D), the lift (L) and the side force (Y).

The propulsive force is the thrust(T) produced by the engine or the engine-

propeller combination.



Fig.1.3 Forces on an airplane

1.1.5 Body axes system of an airplane

To formulate and solve a problem in dynamics requires a system of axes.

To define such a system it is noted that an airplane is nearly symmetric, in

geometry and mass distribution, about a plane which is called the Plane of

symmetry (Fig.1.4a). This plane is used for defining the body axes system.

Figure 1.4b shows a system of axes (OXbYbZb) fixed on the airplane which

moves with the airplane and hence is called Body axes system. The origin O of

the body axes system is the center of gravity (c.g.) of the body which, by

assumption of symmetry, lies in the plane of symmetry. The axis OXb is taken

positive in the forward direction. The axis OZb is perpendicular to OXb in the

plane of symmetry, positive downwards. The axis OYb is perpendicular to the

plane of symmetry such that OXbYbZb is a right handed system.



Fig.1.4a Plane of symmetry and body axis system

Fig.1.4b The forces and moments acting on an airplane and the components of

linear and angular velocities with reference to the body axes system



Figure 1.4b also shows the forces and moments acting on the airplane

and the components of linear and angular velocities. The quantity V is the velocity vector. The quantities X, Y, Z are the components of the resultant

aerodynamic force, along OXb, OYb and OZb axes respectively. L, M, N are the

rolling moment, pitching moment and yawing moment respectively about OXb,

OYb and OZb axes; the rolling moment is denoted by L to distinguish it from lift

(L). u,v,w are respectively the components, along OXb, OYb and OZb, of the

velocity vector (V). The angular velocity components are indicated by p, q, and r. 1.1.6 Special features of Flight Dynamics The features that make flight dynamics a separate subject are:

i)During its motion an airplane in flight, can move along three axes and can

rotate about three axes. This is more complicated than the motions of machinery

and mechanisms which are restrained by kinematic constraints, or those of land

based or water based vehicles which are confined to move on a surface.

ii)The special nature of the forces, like aerodynamic forces, acting on the

airplane(Fig.1.3). The magnitude and direction of these forces change with the

orientation of the airplane, relative to its flight path.

iii)The system of aerodynamic controls used in flight (aileron, elevator, rudder).

1.2 A note on gravitational force In the case of an airplane, the gravitational force is mainly due to the

attraction of the earth. The magnitude of the gravitational force is the weight of

the airplane (in Newtons).

W = mg; where W is the gravitational force, m is the mass of the airplane and g is the acceleration due to gravity.

The line of action of the gravitational force is along the line joining the

centre of gravity (c.g.) of the airplane and the center of the earth. It is directed

towards the center of earth.

The magnitude of the acceleration due to gravity (g) decreases with

increase in altitude (h). It can be calculated based on its value at sea level (go),

and using the following formula.

(g / g0) = [R / (R + h)]2 (1.2)



where R is the radius of the earth,

R = 6400 km (approx.) and g0 = 9.81ms-2

However, for typical airplane flights (h < 20 km), g is generally taken to be

constant. 1.2.1 Flat earth and spherical earth models

In flight mechanics, there are two ways of dealing with the gravitational

force, namely the flat earth model and the spherical earth model.

In the flat earth model, the gravitational acceleration is taken to act

vertically downwards (Fig 1.5).

When the distance over which the flight takes place is small, the flat earth

model is adequate. Reference 1.1, chapter 4 may be referred to for details.

Fig.1.5 Flat earth model

In the spherical earth model, the gravitational force is taken to act along

the line joining the center of earth and the c.g. of the airplane. It is directed

towards the center of the earth (Fig.1.6).

The spherical earth model is used for accurate analysis of flights involving

very long distances.



Fig.1.6. Spherical earth model

Remarks: In this course the flat earth model is used. This is adequate for the

following reasons.

i) The distances involved in flights with acceleration are small and the

gravitational force can be considered in the vertical direction by proper choice of

axes.

ii) In unaccelerated flights like level flight, the forces at the chosen instant of time

are considered and the distance covered etc. are obtained by integration. This

procedure is accurate as long as it is understood that the altitude means height

of the airplane above the surface of the earth and the distance is measured on a

sphere of radius equal to the sum of the radius of earth plus the altitude of

airplane.

iii) As mentioned in section 1.1.4, the forces acting on the airplane are the

gravitational force, the aerodynamic forces and the propulsive force. The first one

has been discussed in this section.The discussion on aerodynamic forces will be

covered in chapter 3 and that on propulsive force in chapter 4.

1.3 Frame of reference A frame of reference (coordinate system) in which Newtons laws of

motion are valid is known as a Newtonian frame of reference.



Since Newtons laws deal with acceleration, a frame of reference moving

with uniform velocity with respect to a Newtonian frame is also a Newtonian

frame or inertial frame.

However, if the reference frame is rotating with an angular velocity (), then, additional accelerations like centripetal acceleration { x ( x r)} and Coriolis acceleration (V x ) will come into picture. Reference 1.2,chapter 13 may be referred to for further details on non-Newtonian

reference frame.

1.3.1 Frame of reference attached to earth In flight dynamics, a co-ordinate system attached to the earth is taken to

approximate a Newtonian frame (Fig.1.7).

The effects of the rotation of earth around itself and around the sun on this

approximation can be estimated as follows.

It is noted that the earth rotates around itself once per day. Hence

= 2 / (3600x24) = 7.27x10-5 s-1; Since r roughly equals 6400 km; the maximum centripetal acceleration (2r) equals 0.034 ms-2.

The earth also goes around the sun and completes one orbit in approximately

365 days. Hence in this case,

= 2 / (365 x 3600 x 24) = 1.99x10-7s-1; Further, in this case, the radius would be roughly the mean distance between the

sun and the earth which is 1.5x1011m. Consequently, 2 r = 0.006 ms-2.



Fig.1.7 Earth fixed and body fixed co-ordinate systems

Thus, it is observed that the centripetal accelerations due to rotation of earth

about itself and around the sun are small as compared to the acceleration due to

gravity.

These rotational motions would also bring about Coriolis acceleration

(V x ). However, its magnitude, which depends on the flight velocity, would be much smaller than the acceleration due to gravity in flights up to Mach number of

3. Hence, the influence can be neglected.

Thus, taking a reference frame attached to the surface of the earth as a

Newtonian frame is adequate for the analysis of airplane flight. Figure 1.7 shows

such a coordinate system.



Chapter 1 Lecture 2 Introduction 2 Topics 1.4 Equilibrium of airplane 1.5 Number of equations of motion for airplane in flight 1.5.1 Degrees of freedom

1.5.2 Degrees of freedom for a rigid airplane

1.6 Subdivisions of flight dynamics 1.6.1 Performance analysis

1.6.2 Stability and control analysis

1.7 Additional definitions 1.7.1 Attitude of the airplane

1.7.2 Flight path

1.7.3 Angle of attack and side slip



1.4 Equilibrium The above three types of forces (aerodynamic, propulsive and

gravitational) and the moments due to them govern the motion of an airplane in

flight.

If the sums of all these forces and moments are zero, then the airplane is

said to be in equilibrium and will move along a straight line with constant velocity

(see Newton's first law). If any of the forces is unbalanced, then the airplane will

have a linear acceleration in the direction of the unbalanced force. If any of the

moments is unbalanced, then the airplane will have an angular acceleration

about the axis of the unbalanced moment.

The relationship between the unbalanced forces and the linear

accelerations and those between unbalanced moments and angular

accelerations are provided by Newtons second law of motion. These

relationships are called equations of motion.

1.5 Number of equations of motion for an airplane in flight To derive the equations of motion, the acceleration of a particle on the

body needs to be known. The acceleration is the rate of change of velocity and

the velocity is the rate of change of position vector with respect to the chosen

frame of reference.

1.5.1 Degrees of freedom The minimum number of coordinates required to prescribe the motion is

called the number of degrees of freedom. The number of equations governing

the motion equals the degrees of freedom. As an example, it may be recalled

that the motion of a particle moving in a plane is prescribed by the x- and y-

coordinates of the particle at various instants of time and this motion is described

by two equations.

Similarly, the position of any point on a rigid pendulum is describe by just

one coordinate namely the angular position () of the pendulum (Fig.1.8). In this case only one equation is sufficient to describe the motion. In yet another



example, if a particle is constrained to move on a sphere, then its position is

completely prescribed by the longitude and the latitude. Hence, this motion has

only two degrees of freedom.

From the discussion in this subsection it is clear that the coordinates needed to

prescribe the motion could be lengths and/or angles.

Note : The bobs in the figure are circular in shape. Please adjust the resolution of

your monitor so that they look circular.

Fig.1.8 Motion of a single degree of freedom system

1.5.2 Degrees of freedom for a rigid airplane To describe its motion, the airplane is treated as a rigid body. It may be

recalled that in a rigid body the distance between any two points is fixed. Thus

the distance r in Fig. 1.9 does not change during the motion. To decide the minimum number of coordinates needed to prescribe the position of a point on a

rigid body which is translating and rotating, one may proceed as follows.



Fig 1.9 Position of a point on a rigid airplane

A rigid body with N particles may appear to have 3N degrees of freedom,

but the constraint of rigidity reduces this number. To arrive at the minimum

number of coordinates, let us approach the problem in a different way. Following

Ref.1.3, it can be stated that to fix the location of a point on a rigid body one does

not need to prescribe its distance from all the points, but only needs to prescribe

its distance from three points which do not lie on the same line (points 1, 2 and 3

in Fig.1.10a). Thus, if the positions of these three points are prescribed with

respect to a reference frame, then the position of any point on the body is known.

This may indicate nine degrees of freedom. This number is reduced to six

because the distances s12, s23 and s13 in Fig.1.10a are constants.



Fig 1.10a Position of a point with respect to three reference points

Another way of looking at the problem is to consider that the three

coordinates of point 1 with respect to the reference frame are prescribed. Now

the point 2 is constrained, because of rigid body assumption, to move on a

sphere centered on point 1 and needs only two coordinates to prescribe its

motion. Once the points 1 and 2 are determined, the point 3 is constrained, again

due to rigid body assumption, to move on a circle about the axis joining points 1

and 2. Hence, only one independent coordinate is needed to prescribe the

position of point 3. Thus, the number of independent coordinates is six (3+2+1).

Or a rigid airplane has six degrees of freedom.

In dynamics the six degrees of freedom associated with a rigid body,

consist of the three coordinates of the origin of the body with respect to the

chosen frame of reference and the three angles which describe the angular

position of a coordinate system fixed on the body (OXbYbZb) with respect to the



fixed frame of reference (EXeYeZe) as shown in Fig.1.10b. These angles are

known as Eulerian angles. These are discussed in ch.7 of flight dynamics- II. See

also Ch.4 of Ref.1.3.

Fig 1.10b Coordinates of a point (P) on a rigid body

Remarks: i) The derivation of the equations of motions in a general case with six degrees of

freedom (see chapter 7 of Flight dynamics-II or Ref 1.4 chapter 10, pt.3 or

Ref.1.5, chapter 10) is rather involved and would be out of place here.

ii) Here, various cases are considered separately and the equations of motion

are written down in each case.

1.6 Subdivisions of flight dynamics The subject of flight dynamics is generally divided into two main branches viz.

(i) Performance analysis and (ii) Stability and control

1.6.1 Performance Analysis In performance analysis, only the equilibrium of forces is generally

considered. It is assumed that by proper deflections of the controls, the moments



can be made zero and that the changes in aerodynamic forces due to deflection

of controls are small. The motions considered in performance analysis are steady

and accelerations, when involved, do not change rapidly with time.

The following motions are considered in performance analysis

- Unaccelerated flights,

Steady level flight

Climb, glide and descent

- Accelerated flights,

Accelerated level flight and climb

Loop, turn, and other motions along curved paths which are

called manoeuvres

Take-off and landing.

1.6.2 Stability and control analyses Roughly speaking, the stability analysis is concerned with the motion of

the airplane, from the equilibrium position, following a disturbance. Stability

analysis tells us whether an airplane, after being disturbed, will return to its

original flight path or not.

Control analysis deals with the forces that the deflection of the controls

must produce to bring to zero the three moments (rolling, pitching and yawing)

and achieve a desired flight condition. It also deals with design of control

surfaces and the forces on control wheel/stick /pedals. Stability and control are

linked together and are generally studied under a common heading.

Flight dynamics - I deals with performance analysis. By carrying out this

analysis one can obtain various performance characteristics such as maximum

level speed, minimum level speed, rate of climb, angle of climb, distance covered

with a given amount of fuel called Range, time elapsed during flight called

Endurance, minimum radius of turn, maximum rate of turn, take-off distance,

landing distance etc. The effect of flight conditions namely the weight, altitude

and flight velocity of the airplane can also be examined. This study would also

help in solving design problems of deciding the power required, thrust required,



fuel required etc. for given design specifications like maximum speed, maximum

rate of climb, range, endurance etc.

Remark: Alternatively, the performance analysis can be considered as the analysis

of the motion of flight vehicle considered as a point mass, moving under the

influence of applied forces (aerodynamic, propulsive and gravitational forces).

The stability analysis similarly can be considered as motion of a vehicle of finite

size, under the influence of applied forces and moments.

1.7 Additional definitions 1.7.1 Attitude: As mentioned in section 1.5.2 the instantaneous position of the airplane,

with respect to the earth fixed axes system (EXeYeZe), is given by the

coordinates of the c.g. at that instant of time. The attitude of the airplane is

described by the angular orientation of the OXbY

bZ

b system with respect to

OXeYeZe system or the Euler angles. Reference 1.4, chapter 10 may be referred

to for details. Let us consider simpler cases. When an airplane climbs along a

straight line its attitude is given by the angle between the axis OXb and the

horizontal (Fig.1.11a). When an airplane executes a turn, the projection of OXb

axis, in the horizontal plane, makes an angle with reference to a fixed horizontal axis (Fig.1.11b). When an airplane is banked the axis OYb makes an

angle with respect to the horizontal (Fig.1.11c) and the axis OZb makes an angle with respect to the vertical.

Fig 1.11a Airplane in a climb



Note : The flight path is circular. Please adjust the resolution of your monitor

so that the flight path looks circular

Fig 1.11b Airplane in a turn - view from top

Fig 1.11c Angle of bank ( )



1.7.2 Flight path: In the subsequent sections, the flight path, also called the trajectory,

means the path or the line along which the c.g. of the airplane moves. The

tangent to this curve at a point gives the direction of flight velocity at that point on

the flight path. The relative wind is in a direction opposite to that of the flight

velocity.

1.7.3. Angle of attack and side slip While discussing the forces acting on an airfoil, the chord of the airfoil is

taken as the reference line and the angle between the chord line and the relative

wind is the angle of attack (). The aerodynamic forces viz. lift (L) and drag (D) , produced by the airfoil, depend on the angle of attack () and are respectively perpendicular and parallel to relative wind direction (Fig.1.11 d).

Fig 1.11d Angle of attack and forces on a airfoil

In the case of an airplane the flight path, as mentioned earlier, is the line along

which c.g. of the airplane moves. The tangent to the flight path is the direction of

flight velocity (V). The relative wind is in a direction opposite to the flight velocity. If the flight path is confined to the plane of symmetry, then the angle of attack

would be the angle between the relative wind direction and the fuselage

reference line (FRL) or OXb axis (see Fig.1.11e). However, in a general case the

velocity vector (V) will have components both along and perpendicular to the



plane of symmetry. The component perpendicular to the plane of symmetry is

denoted by v. The projection of the velocity vector in the plane of symmetry

would have components u and w along OXb and OZb axes (Fig.1.11f). With this

background the angle of sideslip and the angle of attack are defined as follows.

Fig 1.11e Flight path in the plane of symmetry



Fig 1.11f Velocity components in a general case and definition of angle of attack

and sideslip

The angle of sideslip () is the angle between the velocity vector (V) and the plane of symmetry i.e.

= sin-1 (v/ |V|); where |V| is the magnitude of V. The angle of attack () is the angle between the projection of velocity vector (V) in the Xb - Zb plane and the OXb axis or

-1 -1 -12 2 2 2

w w w = tan = sin = sinu | | -v u +wV

Remarks: i) It is easy to show that, if V denotes magnitude of velocity (V), then u = V cos cos , v = V sin ; w = V sin cos . ii) By definition, the drag (D) is parallel to the relative wind direction. The lift force

lies in the plane of symmetry of the airplane and is perpendicular to the direction

of flight velocity.



Chapter 1 Lecture 3 Introduction 3 Topics 1.8 Simplified treatment of performance analysis 1.9 Course outline 1.10 Background expected

1.8 Simplified treatment in performance analysis In a steady flight, there is no acceleration along the flight path and in a

level flight; the altitude of the flight remains constant. A steady, straight and level

flight generally means a flight along a straight line at a constant velocity and

constant altitude.

Sometimes, this flight is also referred to as unaccelerated level flight. To illustrate

the simplified treatment in performance analysis, the case of unaccelerated level

flight is considered below.

The forces acting on an airplane in unaccelerated level flight are shown in the

Fig.1.12.

They are: Lift (L), Thrust (T), Drag (D) and Weight (W) of the airplane.

It may be noted that the point of action of the thrust and its direction depend on

the engine location. However, the direction of the thrust can be taken parallel to

the airplane reference axis.



Fig.1.12 Forces acting in steady level flight

The lift and drag, being perpendicular to the relative wind, are in the

vertical and horizontal directions respectively, in this flight. The weight acts at the

c.g. in a vertically downward direction.

In an unaccelerated level flight, the components of acceleration in the

horizontal and vertical directions are zero.

Hence, the sums of the components of all the forces in these directions

are zero. Resolving the forces along and perpendicular to the flight path (see

Fig.1.12.), gives the following equations of force equilibrium.

T cos D = 0 (1.3) T sin + L W = 0 (1.4)

Apart from these equations, equilibrium demands that the moment about

the y-axis to be zero, i.e.,

Mcg = 0

Unless the moment condition is satisfied, the airplane will begin to rotate

about the c.g.

Let us now examine how the moment is balanced in an airplane. The

contributions to Mcg come from all the components of the airplane. As regards the

wing, the point where the resultant vector of the lift and drag intersects the plane

of symmetry is known as the centre of pressure. This resultant force produces a

moment about the c.g. However, the location of the center of pressure depends

on the lift coefficient and hence the moment contribution of wing changes with

the angle of attack as the lift coefficient depends on the angle of attack. For



convenience, the lift and the drag are transferred to the aerodynamic center

along with a moment (Mac). Recall, that moment coefficient about the a.c. (Cmac)

is, by definition, constant with change in angle of attack.

Similarly, the moment contributions of the fuselage and the horizontal tail

change with the angle of attack. The engine thrust also produces a moment

about the c.g. which depends on the thrust required.

Hence, the sum of the moments about the c.g. contributed by the wing,

fuselage, horizontal tail and engine changes with the angle of attack. By

appropriate choice of the horizontal tail setting (i.e. incidence of horizontal tail

with respect to fuselage central line), one may be able to make the sum of these

moments to be zero in a certain flight condition, which is generally the cruise

flight condition. Under other flight conditions, generation of corrective

aerodynamic moment is facilitated by suitable deflection of elevator (See

Fig.1.2a, b and c for location of elevator). By deflecting the elevator, the lift on the

horizontal tail surface can be varied and the moment produced by the horizontal

tail balances the moments produced by all other components.

The above points are illustrated with the help of an example.

Example 1.1 A jet aircraft weighing 60,000 N has its line of thrust 0.15 m below the line

of drag. When flying at a certain speed, the thrust required is 6000 N and the

center of pressure of the wing lift is 0.45 m aft of the airplane c.g. What is the lift

on the wing and the load on the tail plane whose center of pressure is 7.5 m

behind the c.g.? Assume unaccelerated level flight and the angle of attack to be

small during the flight.

Solution: The various forces and dimensions are presented in Fig.1.13. The lift on

the wing is LW and the lift on the tail is LT. Since the angle of attack () is small, it may be considered that cos = 1 and sin = 0. Thus, the force equilibrium (Eqs. 1.3 and 1.4), yields :

T D = 0

LW + LT W = 0



i.e. D = T = 6000 N and LT + LW = 60000 N

From Fig. 1.13., the moment equilibrium about the c.g. gives:

Mcg = T (zd + 0.15) D.zd 0.45.LW 7.5.LT = 0 where zd is the distance of drag

below the c.g; not shown in figure as it is of no significance in the present

context.

Fig.1.13 Forces acting on an airplane in steady level flight

Solving these equations, gives :

LW = 63702.13 N and LT = -3702.13 N

Following observations can be made. A) The lift on the wing is about 63.7 kN. The lift on the tail is only 3.7 kN and is in

the downward direction.

B) The contribution of tail to the total lift is thus small, in this case, about 6% and

negative. This negative contribution necessitates the wing lift to be more than the

weight of the airplane. This increase in the lift results in additional drag called trim

drag.

C) The distance zd is of no significance in this problem as the drag and thrust

form a couple whose moment is equal to the thrust multiplied by the distance

between them.

D) Generally, the angle of attack () is small. Hence, sin is small and cos is nearly equal to unity. Thus, the equations of force equilibrium reduce to

T D = 0 and L W = 0.

E) It is assumed that the pitching moment equilibrium i.e. Mcg = 0 is achieved by appropriate deflection of the elevator. The changes in the lift and drag due to



elevator deflections are generally small and in performance analysis, as stated

earlier, these changes are ignored and the simplified picture as shown in Fig.1.14

is considered adequate.

Fig.1.14. Simplified picture of the forces acting on an airplane in level flight.

1.9 Course outline Let us consider the background material required to carry-out the

performance analysis. It is known that :

L = (1/2) V2 S CL

D = (1/2) V2 S CD where CL and CD are the lift and drag coefficients; S is the area of the wing.

The quantities CL and CD depend on , Mach number (M = V / a) and Reynolds number (Re = V l /); where l is the reference length. Thus CD = f (CL, M, Re) (1.6)

The relation between CL and CD at given M and Re is known as the drag

polar of the airplane. This has to be known for carrying the performance analysis. The density of air () depends on the flight altitude. Further the Mach number depends on the speed of sound, which in turn depends on the ambient

air temperature. Thus, performance analysis requires the knowledge of the



variations of pressure, temperature, density, viscosity etc. with altitude in earths

atmosphere.

The evaluation of performance also requires the knowledge of the engine

characteristics such as, variations of thrust (or power) and fuel consumption with

the flight speed and altitude.

Keeping these aspects in view, following will be the contents of this course.

Earths atmosphere (chapter 2)

Drag polar (chapter 3)

Engine characteristics (chapter 4)

Performance analysis. ( chapters 5 to 10)

These topics will be taken up in the subsequent chapters.

The Appendices A and B present the performance analyses of piston-engined

and jet airplane respectively.

1.10 Back ground expected The student is expected to have undergone courses on (a) Vectors (b)

Rigid body dynamics (c) Aerodynamics and (d) Aircraft engines.

Remark: References 1.5 to 1.14 are some of the books dealing with airplane performance. They can be consulted for additional information.



Chapter 1 References

1.1 Miele, A. Flight mechanics Vol I Addison Wesley (1962).

1.2 Shames, I.H. and Krishna Mohana Rao, G. Engineering mechanics statics

and dynamics, 4th Edition, Dorling Kindersley (India), licensees of Pearson

Education (2006).

1.3 Goldstein H. Classical mechanics Second edition Addison Wesley (1980).

1.4 Davies, M. (Editor) The standard handbook for aeronautical and

astronautical engineers McGraw Hill (2003).

1.5 Perkins, C.D. and Hage, R. E. Airplance performance, stability and

control John Wiley (1963).

1.6 Dommasch, D.O. Sherby, S.S. and Connolly, T.F. Airplane

aerodynamics Pitman (1967).

1.7 Houghton E.L. and Carruthers N.B. Aerodynamics for engineering

students, Edward Arnold (1982).

1.8 Hale, F.J. Introduction to aircraft performance, selection and design,

John Wiley (1984).

1.9 McCormick B.W. Aerodynamics, aeronautics and flight mechanics, John

Wiley (1995).

1.10 Anderson, Jr. J.D. Aircraft performance and design McGraw Hill

International edition (1999).

1.11 Eshelby, M.E. Aircraft performance-theory and practice, Butterworth-

Heinemann, Oxford, U.K., (2001).

1.12 Pamadi, B. Performance, stability, dynamics and control of an

airplane, AIAA (2004).

1.13 Anderson, Jr. J.D. Introduction to flight Fifth edition, McGraw-Hill,

(2005).

1.14 Phillips, W.F. Mechanics of flight 2nd Edition John Wiley (2010).



1.15 Jackson, P. (Editor) Janes all the worlds aircraft Published annually

by Janes information group Ltd., Surrey, U.K..



Chapter 1 Exercises 1. Sketch the three views of an airplane and show its axes systems.

2. Define, with neat sketches, the following terms.

(a) flight path

(b) flight velocity

(c) body axes system

(d) angle of attack

(e) angle of slide slip and

(f) bank angle.

3.Janes All the World Aircraft (Ref.1.15) is a book published annually and

contains details of airplanes currently in production in various countries. Refer to

this book and study the three view drawings, geometrical details and

performance parameters of different types of airplanes.



Chapter 2 Earths atmosphere (Lectures 4 and 5)

Keywords: Earths atmosphere; International standard atmosphere; geopotential altitude; stability of atmosphere. Topics 2.1 Introduction 2.2 Earths atmosphere 2.2.1 The troposphere

2.2.2 The stratosphere

2.2.3 The mesosphere

2.2.4 The ionosphere or thermosphere

2.2.5 The exosphere

2.3 International standard atmosphere (ISA) 2.3.1 Need for ISA and agency prescribing it.

2.3.2 Features of ISA

2.4 Variations of properties with altitude in ISA 2.4.1 Variations of pressure and density with altitude

2.4.2 Variations with altitude of pressure ratio, density ratio speed of

sound, coefficient of viscosity and kinematic viscosity.

2.5 Geopotential altitude 2.6 General remarks

2.6.1 Atmospheric properties in cases other than ISA

2.6.2 Stability of atmosphere

References Exercises



Chapter 2 Lecture 4 Earths atmosphere 1 Topics 2.1 Introduction 2.2 Earths atmosphere 2.2.1 The troposphere

2.2.2 The stratosphere

2.2.3 The mesosphere

2.2.4 The ionosphere or thermosphere

2.2.5 The exosphere

2.3 International standard atmosphere (ISA) 2.3.1 Need for ISA and agency prescribing it.

2.3.2 Features of ISA

2.1 Introduction

Airplanes fly in the earths atmosphere and therefore, it is necessary to

know the properties of this atmosphere.

This chapter, deals with the average characteristics of the earths

atmosphere in various regions and the International Standard Atmosphere (ISA)

which is used for calculation of airplane performance.

2.2 Earths atmosphere The earths atmosphere is a gaseous blanket around the earth which is

divided into the five regions based on certain intrinsic features (see Fig.2.1).

These five regions are: (i) Troposphere, (ii) Stratosphere, (iii) Mesosphere,

(iv) Ionosphere or Thermosphere and (v) Exosphere. There is no sharp

distinction between these regions and each region gradually merges with the

neighbouring regions.



Fig.2.1 Typical variations of temperature and pressure in the earths atmosphere

2.2.1 The troposphere

This is the region closest to the earths surface. It is characterized by

turbulent conditions of air. The temperature decreases linearly at an approximate

rate of 6.5 K / km. The highest point of the troposphere is called tropopause. The

height of the tropopause varies from about 9 km at the poles to about 16 km at

the equator.

2.2.2 The stratosphere This extends from the tropopause to about 50 km. High velocity winds

may be encountered in this region, but they are not gusty. Temperature remains

constant up to about 25 km and then increases. The highest point of the

stratosphere is called the stratopause.

2.2.3 The mesosphere The mesosphere extends from the stratopause to about 80 km. The

temperature decreases to about -900C in this region. In the mesosphere, the



pressure and density of air are very low, but the air still retains its composition as

at sea level. The highest point of the mesosphere is called the mesopause.

2.2.4 The ionosphere or thermosphere This region extends from the mesopause to about 1000 km. It is

characterized by the presence of ions and free electrons. The temperature

increases to about 00C at 110 km, to about 10000C at 150 km and peak of about

17800C at 700 km (Ref.2.1). Some electrical phenomena like the aurora borealis

occur in this region.

2.2.5 The exosphere This is the outer fringe of the earths atmosphere. Very few molecules are

found in this region. The region gradually merges into the interplanetary space.

2.3 International Standard Atmosphere (ISA) 2.3.1 Need for ISA and agency prescribing it

The properties of earths atmosphere like pressure, temperature and

density vary not only with height above the earths surface but also with the

location on earth, from day to day and even during the day. As mentioned in

section 1.9, the performance of an airplane is dependent on the physical

properties of the earths atmosphere. Hence, for the purpose of comparing

(a) the performance of different airplanes and (b) the performance of the same

airplane measured in flight tests on different days, a set of values for atmospheric

properties have been agreed upon, which represent average conditions

prevailing for most of the year, in Europe and North America. Though the agreed

values do not represent the actual conditions anywhere at any given time, they

are useful as a reference. This set of values called the International Standard

Atmosphere (ISA) is prescribed by ICAO (International Civil Aviation

Organization). It is defined by the pressure and temperature at mean sea level,

and the variation of temperature with altitude up to 32 km (Ref.1.11, chapter 2).

With these values being prescribed, it is possible to find the required physical

characteristics (pressure, temperature, density etc) at any chosen altitude.



Remark: The actual performance of an airplane is measured in flight tests under

prevailing conditions of temperature, pressure and density. Methods are

available to deduce, from the flight test data, the performance of the airplane

under ISA conditions. When this procedure is applied to various airplanes and

performance presented under ISA conditions, then comparison among different

airplanes is possible.

2.3.2 Features of ISA The main features of the ISA are the standard sea level values and the

variation of temperature with altitude. The air is assumed as dry perfect gas.

The standard sea level conditions are as follows:

Temperature (T0) = 288.15 K = 150C

Pressure (p0) = 101325 N/m2 = 760 mm of Hg

Rate of change of temperature:

= - 6.5 K/km upto 11 km

= 0 K/km from 11 to 20 km

= 1 K/km from 20 to 32 km

The region of ISA from 0 to 11 km is referred to as troposphere. That

between 11 to 20 km is the lower stratosphere and between 20 to 32 km is the

middle stratosphere (Ref.1.11, chapter 2).

Note: Using the values of T0 and p0 , and the equation of state, p = RT, gives the sea level density (0) as 1.225 kg/m3.



Chapter 2 Lecture 5 Earths atmosphere 2 Topics 2.4 Variations of properties with altitude in ISA

2.4.1 Variations of pressure and density with altitude

2.4.2 Variations with altitude of pressure ratio, density ratio speed of

sound, coefficient of viscosity and kinematic viscosity.

2.5 Geopotential altitude 2.6 General remarks

2.6.1 Atmospheric properties in cases other than ISA

2.6.2 Stability of atmosphere

Atmospheric properties of ISA (Table 2.1)

2.4 Variations of properties with altitude in ISA For calculation of the variations of pressure, temperature and density with

altitude, the following equations are used.

The equation of state p = R T (2.1) The hydrostatic equation dp/dh = - g (2.2)

Remark: The hydrostatic equation can be easily derived by considering the balance of

forces on a small fluid element.

Consider a cylindrical fluid element of area A and height h as shown in Fig.2.2.



Fig.2.2 Equilibrium of a fluid element.

The forces acting in the vertical direction on the element are the pressure forces

and the weight of the element.

For vertical equilibrium of the element,

pA {p + (dp /dh) h} A g A h = 0 Simplifying, dp /dh = - g 2.4.1 Variations of pressure and density with altitude Substituting for from the Eq.(2.1) in Eq.(2.2) gives: dp / dh = -(p/RT) g

Or (dp/p) = -g dh/RT (2.3)

Equation (2.3) is solved separately in troposphere and stratosphere, taking into

account the temperature variations in each region. For example, in the

troposphere, the variation of temperature with altitude is given by the equation

T = T0 h (2.4) where T0 is the sea level temperature, T is the temperature at the altitude h and is the temperature lapse rate in the troposphere.

Substituting from Eq.(2.4) in Eq.(2.3) gives:

(dp /p) = - gdh /R (T0 h) (2.5)



Taking g as constant, Eq.(2.5) can be integrated between two altitudes h1 and

h2. Taking h1 as sea level and h2 as the desired altitude (h), the integration gives

the following equation, the intermediate steps are left as an exercise.

(p/p0) = (T/T0)(g/R) (2.6)

where T is the temperature at the desired altitude (h) given by Eq.(2.4).

Equation (2.6) gives the variation of pressure with altitude.

The variation of density with altitude can be obtained using Eq.(2.6) and

the equation of state. The resulting variation of density with temperature in the

troposphere is given by:

(/0) = (T/T0)(g/R)-1 (2.7) Thus, both the pressure and density variations are obtained once the

temperature variation is known.

As per the ISA, R = 287.05287 m2sec-2 K and g = 9.80665 m/s2.

Using these and = 0.0065 K/m in the troposphere yields (g/R) as 5.25588. Thus, in the troposphere, the pressure and density variations are :

(p/p0) = (T/T0)5.25588 (2.8)

(/0) = (T/T0)4.25588 (2.9) Note: T= 288.15 - 0.0065 h; h in m and T in K.

In order to obtain the variations of properties in the lower stratosphere (11

to 20 km altitude), the previous analysis needs to be carried-out afresh with = 0 i.e., T having a constant value equal to the temperature at 11 km (T = 216.65 K).

From this analysis the pressure and density variations in the lower stratosphere

are obtained as :

(p / p11) = ( / 11) = exp { -g (h - 11000) / RT11 } (2.10) where p11, 11 and T11 are the pressure, density and temperature respectively at 11 km altitude.

In the middle stratosphere (20 to 32 km altitude), it can be shown that (note in

this case = -0.001 K / m): (p / p20) = (T / T20)- 34.1632 (2.11)

( / 20) = (T/ T20)- 35.1632 (2.12)



where p20, 20 and T20 are pressure, density and temperature respectively at 20 km altitude.

Thus, the pressure and density variations have been worked out in the

troposphere and the stratosphere of ISA. Table 2.1 presents these values.

Remark: Using Eqs.(2.1) and (2.2) the variations of pressure and density can be worked

out for other variations of temperature with height (see exercise 2.1).

2.4.2. Variations with altitude of pressure ratio, density ratio, speed of sound, coefficient of viscosity and kinematic viscosity The ratio (p/p0) is called pressure ratio and is denoted by . Its value in ISA can be obtained by using Eqs.(2.8),(2.10) and (2.11). Table 2.1 includes these

values.

The ratio ( / 0) is called density ratio and is denoted by . Its values in ISA can be obtained using Eqs.(2.9),(2.10) and (2.12). Table 2.1 includes these values.

The speed of sound in air, denoted by a, depends only on the temperature and

is given by:

a = ( RT)0.5 (2.13) where is the ratio of specific heats; for air = 1.4. The values of a in ISA can be obtained by using appropriate values of temperature. Table 2.1 includes these

values.

The kinematic viscosity ( ) is given by: = / where is the coefficient of viscosity.

The coefficient of viscosity of air () depends only on temperature. Its variation with temperature is given by the following Sutherland formula.

3/2-6 T = 1.458X10 [ ]

T+110.4, where T is in Kelvin and is in kg m-1 s-1 (2.14)

Table 2.1 includes the variation of kinematic viscosity with altitude.

Example 2.1 Calculate the temperature (T), pressure (p), density ( ), pressure ratio ( ) , density ratio ( ), speed of sound (a) , coefficient of viscosity ( ) and kinematic viscosity ( ) in ISA at altitudes of 8 km, 16 km and 24 km.



Solution: It may be noted that the three altitudes specified in this example, viz.

8 km, 16 km and 24 km, lie in troposphere, lower stratosphere and middle

stratosphere regions of ISA respectively.

(a) h = 8 km

Let the quantities at 8 km altitude be denoted by the suffix 8.

In troposphere: 0T = T -h where, T0 = 288.15 K, = 0.0065 K /m Hence, 8T = 288.15 - 0.0065 8000 = 236.15K From Eq.(2.8)

5.25588 5.255888 8 00

p = = T/T = 236.15/288.15 = 0.35134p

Or 28p = 0.35134 101325 = 35599.5 N/m

38 8 8 35599.5 = p / RT = = 0.52516 kg/m287.05287236.15 8 8 0 = / = 0.52516/1.225 = 0.42870 a8 = ( RT8)0.5 0.5= 1.4287.05287236.15 = 308.06 m/s From Eq.(2.14):

1.5 1.5-6 -6 -5 -1 -18

88

T 236.15 = 1.45810 = 1.45810 = 1.526810 kg m sT +110.4 236.15+110.4

-5 -5 2

8 8 8= / = 1.526810 / 0.52516 = 2.907210 m /s Remarks: (i) The values calculated above and those in Table 2.1 may differ from each

other in the last significant digit. This is due to the round-off errors in the

calculations.

(ii) Consider an airplane flying at 8 km altitude at a flight speed of 220 m/s.

The Mach number of this flight would be: 220/308.06 = 0.714



(iii) Further if the reference chord of the wing (cref) of this airplane be 3.9 m,

the Reynolds number in this flight, based on cref, would be:

6refe -5V c 2203.9R = = = 29.5110

2.907210

(iv) For calculation of values at 16 km altitude, the values of temperature,

pressure and density are needed at the tropopause viz. at h=11 km.

Now 11T = 288.15-0.006511000 = 216.65 K

5.25588 211p = 101325 216.65/288.15 = 22632 N/m 311 = 22632/ 287.05287216.65 = 0.36392 kg/m

(b) h = 16 km

In lower stratosphere Eq.(2.10) gives :

1111 11

p = = exp -g h-11000 /RTp

Consequently,

16 1611 11

p = = exp -9.80665 16000-11000 / 287.05287216.65 = 0.45455p

Or 216p = 226320.45455 = 10287 N/m

316 = 0.363920.45455 = 0.16541kg/m 16 = 10287 /101325 = 0.10153 16 = 0.16541/1.225 = 0.13503 0.516a = 1.4287.05287216.65 = 295.07m/s

1.5-6 -5 -1 -1

16216.65 = 1.45810 = 1.421610 kg m s

216.65+110.4

-5 -5 216 = 1.421610 / 0.16541 = 8.59410 m /s Remark : To calculate the required values at 24 km altitude, the values of T and p are

needed at h = 20 km. These values are :

T20 = 216.65



2011

p = exp -9.80665 20000-11000 / 287.05287216.65 = 0.24191p

Or 220p = 22632 0.24191 = 5474.9 N/m

(c) h = 24 km

24T = 216.65+0.001 24000-20000 = 220.65K From Eq.(2.11):

-34.163224 24 2020

p = T /Tp

Or -34.1632 224p = 5474.9 220.65/216.65 = 2930.5N/m 24 = 2930.5/ 287.05287220.65 = 0.04627 Hence, 24 = 2930.5/101325 = 0.02892 and 24 = 0.04627/1.225 = 0.03777

0.524a = 1.4287.05287220.65 = 297.78 m/s 1.5

-6 -5 -1 -124

220.65 = 1.45810 = 1.443510 kg m s220.65+110.4

-5 -4 224 = 1.443510 / 0.04627 = 3.1210 m /s

Answers:

h (km) 8 16 24

T (K) 236.15 216.65 220.65

p (N/m2) 35599.5 10287.0 2930.5

0 = p/p 0.35134 0.10153 0.02892 3 kg/m 0.52516 0.16541 0.04627 0 = / 0.42870 0.13503 0.03777 a (m/s) 308.06 295.07 297.78

-1 -1 kg m s 1.5268 x 10-5 1.4216 x 10-5 1.4435 x 10-5 2m /s 2.9072 x 10-5 8.594 x 10-5 3.12 x 10-4



2.5 Geopotential altitude The variations of pressure, temperature and density in the atmosphere

were obtained by using the hydrostatic equation (Eq.2.2). In this equation g is

assumed to be constant. However, it is known that g decreases with altitude.

Equation (1.1) gives the variation as:

0

G

Rg = g ( )R+h

where R is the radius of earth and hG is the geometric altitude above earths

surface.

Thus, the values of p and obtained by assuming g = 0

g are at an

altitude slightly different from the geometrical altitude (hG). This altitude is called

geopotential altitude, which for convenience is denoted by h. Following Ref.1,

the geopotential altitude can be defined as the height above earths surface in

units, proportional to the potential energy of unit mass (geopotential), relative to

sea level. It can be shown that the geopotential altitude (h) is given, in terms of

geometric altitude (hG), by the following relation. Reference 1.13, chapter 3 may

be referred to for derivation.

GRh = h

R-h

It may be remarked that the actual difference between h and hG is small

for altitudes involved in flight dynamics; for h of 20 km, hG would be 20.0627 km.

Hence, the difference is ignored in performance analysis.

2.6 General remarks: 2.6.1 Atmospheric properties in cases other than ISA It will be evident from chapters 4 to 10 that the engine characteristics and

the airplane performance depend on atmospheric characteristics. Noting that ISA

only represents average atmospheric conditions, other atmospheric models have

been proposed as guidelines for extreme conditions in arctic and tropical regions.

Figure 2.3 shows the temperature variations with altitude in arctic and tropical

atmospheres along with ISA. It is seen that the arctic minimum atmosphere has

the following features. (a) The sea level temperature is -500C (b) The



temperature increases at the rate of 10 K per km up to 1500 m altitude. (c) The

temperature remains constant at -350C up to 3000 m altitude. (d) Then the

temperature decreases at the rate of 4.72 K per km up to 15.5 km altitude (e)

The tropopause in this case is at 15.5 km and the temperature there is -940c.

The features of the tropical maximum atmosphere are as follows.

(a) Sea level temperature is 450 C.

(b) The temperature decreases at the rate of 6.5 K per km up to 11.54 km

and then remains constant at -300 C.

Fig.2.3 Temperature variations in arctic minimum, ISA and tropical maximum

atmospheres (Reproduced from Ref.1.7, Chapter 3 with permission of author)

Note: (a) The local temperature varies with latitude but the sea level pressure (p0)

depends on the weight of air above and is taken same at all the places i.e.

101325 N/m2. Knowing p0 and T0, and the temperature lapse rates, the pressure,

temperature and density in tropospheres of arctic minimum and tropical



maximum can be obtained using Eqs. (2.4), (2.6) and (2.7). (see also exercise

2.1).

(b) Some airlines/ air forces may prescribe intermediate values of sea level

temperature e.g. ISA +150C or ISA +200C. The variations of pressure,

temperature and density with altitude in these cases can also be worked out from

the aforesaid equations.

2.6.2 Stability of atmosphere It is generally assumed that the air mass is stationary. However, some

packets of air mass may acquire motion due to local changes. For example, due

to absorption of solar radiation by the earths surface, an air mass adjacent to the

surface may become lighter and buoyancy may cause it to rise. If the

atmosphere is stable, a rising packet of air must come back to its original

position. On the other hand, if the air packet remains in the disturbed position,

then the atmosphere is neutrally stable. If the rising packet continues to move up

then the atmosphere is unstable.

Reference 1.7, chapter 3 analyses the problem of atmospheric stability

and concludes that if the temperature lapse rate is less than 9.75 K per km, then

the atmosphere is stable. It is seen that the three atmospheres, representing

different conditions, shown in Fig.2.3 are stable.



Altit-

ude

(m)

Tempe-

rature

(K)

Pressure

(N/m2)

(p/po)

Density

(kg/m3)

(/o)

speed

of

sound

(m/s)

Kinematic

viscosity

(m2/s)

0 288.15 101325.0 1.00000 1.22500 1.00000 340.29 1.4607E-005

200 286.85 98945.3 0.97651 1.20165 0.98094 339.53 1.4839E-005

400 285.55 96611.0 0.95348 1.17864 0.96216 338.76 1.5075E-005

600 284.25 94321.6 0.93088 1.15598 0.94365 337.98 1.5316E-005

800 282.95 92076.3 0.90872 1.13364 0.92542 337.21 1.5562E-005

1000 281.65 89874.4 0.88699 1.11164 0.90746 336.43 1.5813E-005

1200 280.35 87715.4 0.86568 1.08997 0.88977 335.66 1.6069E-005

1400 279.05 85598.6 0.84479 1.06862 0.87234 334.88 1.6331E-005

1600 277.75 83523.3 0.82431 1.04759 0.85518 334.10 1.6598E-005

1800 276.45 81489.0 0.80423 1.02688 0.83827 333.31 1.6870E-005

2000 275.15 79494.9 0.78455 1.00649 0.82162 332.53 1.7148E-005

2200 273.85 77540.6 0.76527 0.98640 0.80523 331.74 1.7432E-005

2400 272.55 75625.4 0.74636 0.96663 0.78908 330.95 1.7723E-005

2600 271.25 73748.6 0.72784 0.94716 0.77319 330.16 1.8019E-005

2800 269.95 71909.7 0.70969 0.92799 0.75754 329.37 1.8321E-005

3000 268.65 70108.2 0.69191 0.90912 0.74214 328.58 1.8630E-005

3200 267.35 68343.3 0.67450 0.89054 0.72697 327.78 1.8946E-005

3400 266.05 66614.6 0.65744 0.87226 0.71205 326.98 1.9269E-005

3600 264.75 64921.5 0.64073 0.85426 0.69736 326.18 1.9598E-005

3800 263.45 63263.4 0.62436 0.83655 0.68290 325.38 1.9935E-005

4000 262.15 61639.8 0.60834 0.81912 0.66867 324.58 2.0279E-005

4200 260.85 60050.0 0.59265 0.80197 0.65467 323.77 2.0631E-005

4400 259.55 58493.7 0.57729 0.78510 0.64090 322.97 2.0990E-005

4600 258.25 56970.1 0.56225 0.76850 0.62735 322.16 2.1358E-005

4800 256.95 55478.9 0.54753 0.75217 0.61402 321.34 2.1734E-005

Table 2.1 Atmospheric properties in ISA (Cont..)



5000 255.65 54019.4 0.53313 0.73611 0.60091 320.53 2.2118E-005

5200 254.35 52591.2 0.51903 0.72031 0.58801 319.71 2.2511E-005

5400 253.05 51193.7 0.50524 0.70477 0.57532 318.90 2.2913E-005

5600 251.75 49826.4 0.49175 0.68949 0.56285 318.08 2.3324E-005

5800 250.45 48488.8 0.47855 0.67446 0.55058 317.25 2.3744E-005

6000 249.15 47180.5 0.46564 0.65969 0.53852 316.43 2.4174E-005

6200 247.85 45900.9 0.45301 0.64516 0.52666 315.60 2.4614E-005

6400 246.55 44649.5 0.44066 0.63088 0.51501 314.77 2.5064E-005

6600 245.25 43425.9 0.42858 0.61685 0.50355 313.94 2.5525E-005

6800 243.95 42229.6 0.41677 0.60305 0.49229 313.11 2.5997E-005

7000 242.65 41060.2 0.40523 0.58949 0.48122 312.27 2.6480E-005

7200 241.35 39917.1 0.39395 0.57617 0.47034 311.44 2.6974E-005

7400 240.05 38799.9 0.38292 0.56308 0.45965 310.60 2.7480E-005

7600 238.75 37708.1 0.37215 0.55021 0.44915 309.75 2.7998E-005

7800 237.45 36641.4 0.36162 0.53757 0.43884 308.91 2.8529E-005

8000 236.15 35599.2 0.35134 0.52516 0.42870 308.06 2.9073E-005

8200 234.85 34581.2 0.34129 0.51296 0.41875 307.21 2.9629E-005

8400 233.55 33586.9 0.33148 0.50099 0.40897 306.36 3.0200E-005

8600 232.25 32615.8 0.32189 0.48923 0.39937 305.51 3.0784E-005

8800 230.95 31667.6 0.31254 0.47768 0.38994 304.65 3.1383E-005

9000 229.65 30741.9 0.30340 0.46634 0.38069 303.79 3.1997E-005

9200 228.35 29838.2 0.29448 0.45521 0.37160 302.93 3.2627E-005

9400 227.05 28956.1 0.28577 0.44428 0.36268 302.07 3.3272E-005

9600 225.75 28095.2 0.27728 0.43355 0.35392 301.20 3.3933E-005

9800 224.45 27255.2 0.26899 0.42303 0.34533 300.33 3.4611E-005

10000 223.15 26435.7 0.26090 0.41270 0.33690 299.46 3.5307E-005

10200 221.85 25636.2 0.25301 0.40256 0.32862 298.59 3.6020E-005

10400 220.55 24856.4 0.24531 0.39262 0.32050 297.71 3.6752E-005

10600 219.25 24096.0 0.23781 0.38286 0.31254 296.83 3.7503E-005




10800 217.95 23354.4 0.23049 0.37329 0.30473 295.95 3.8274E-005

11000 216.65 22631.5 0.22336 0.36391 0.29707 295.07 3.9065E-005

11200 216.65 21929.4 0.21643 0.35262 0.28785 295.07 4.0316E-005

11400 216.65 21248.6 0.20971 0.34167 0.27892 295.07 4.1608E-005

11600 216.65 20588.9 0.20320 0.33106 0.27026 295.07 4.2941E-005

11800 216.65 19949.7 0.19689 0.32079 0.26187 295.07 4.4317E-005

12000 216.65 19330.4 0.19078 0.31083 0.25374 295.07 4.5736E-005

12200 216.65 18730.2 0.18485 0.30118 0.24586 295.07 4.7202E-005

12400 216.65 18148.7 0.17911 0.29183 0.23823 295.07 4.8714E-005

12600 216.65 17585.3 0.17355 0.28277 0.23083 295.07 5.0275E-005

12800 216.65 17039.4 0.16817 0.27399 0.22366 295.07 5.1886E-005

13000 216.65 16510.4 0.16294 0.26548 0.21672 295.07 5.3548E-005

13200 216.65 15997.8 0.15789 0.25724 0.20999 295.07 5.5264E-005

13400 216.65 15501.1 0.15298 0.24925 0.20347 295.07 5.7035E-005

13600 216.65 15019.9 0.14823 0.24152 0.19716 295.07 5.8862E-005

13800 216.65 14553.6 0.14363 0.23402 0.19104 295.07 6.0748E-005

14000 216.65 14101.8 0.13917 0.22675 0.18510 295.07 6.2694E-005

14200 216.65 13664.0 0.13485 0.21971 0.17936 295.07 6.4703E-005

14400 216.65 13239.8 0.13067 0.21289 0.17379 295.07 6.6776E-005

14600 216.65 12828.7 0.12661 0.20628 0.16839 295.07 6.8916E-005

14800 216.65 12430.5 0.12268 0.19988 0.16317 295.07 7.1124E-005

15000 216.65 12044.6 0.11887 0.19367 0.15810 295.07 7.3403E-005

15200 216.65 11670.6 0.11518 0.18766 0.15319 295.07 7.5754E-005

15400 216.65 11308.3 0.11160 0.18183 0.14844 295.07 7.8182E-005

15600 216.65 10957.2 0.10814 0.17619 0.14383 295.07 8.0687E-005

15800 216.65 10617.1 0.10478 0.17072 0.13936 295.07 8.3272E-005

16000 216.65 10287.5 0.10153 0.16542 0.13504 295.07 8.5940E-005

16200 216.65 9968.1 0.09838 0.16028 0.13084 295.07 8.8693E-005

16400 216.65 9658.6 0.09532 0.15531 0.12678 295.07 9.1535E-005




16600 216.65 9358.8 0.09236 0.15049 0.12285 295.07 9.4468E-005

16800 216.65 9068.2 0.08950 0.14581 0.11903 295.07 9.7495E-005

17000 216.65 8786.7 0.08672 0.14129 0.11534 295.07 1.0062E-004

17200 216.65 8513.9 0.08403 0.13690 0.11176 295.07 1.0384E-004

17400 216.65 8249.6 0.08142 0.13265 0.10829 295.07 1.0717E-004

17600 216.65 7993.5 0.07889 0.12853 0.10492 295.07 1.1060E-004

17800 216.65 7745.3 0.07644 0.12454 0.10167 295.07 1.1415E-004

18000 216.65 7504.8 0.07407 0.12068 0.09851 295.07 1.1780E-004

18200 216.65 7271.9 0.07177 0.11693 0.09545 295.07 1.2158E-004

18400 216.65 7046.1 0.06954 0.11330 0.09249 295.07 1.2547E-004

18600 216.65 6827.3 0.06738 0.10978 0.08962 295.07 1.2949E-004

18800 216.65 6615.4 0.06529 0.10637 0.08684 295.07 1.3364E-004

19000 216.65 6410.0 0.06326 0.10307 0.08414 295.07 1.3793E-004

19200 216.65 6211.0 0.06130 0.09987 0.08153 295.07 1.4234E-004

19400 216.65 6018.2 0.05939 0.09677 0.07900 295.07 1.4690E-004

19600 216.65 5831.3 0.05755 0.09377 0.07654 295.07 1.5161E-004

19800 216.65 5650.3 0.05576 0.09086 0.07417 295.07 1.5647E-004

20000 216.65 5474.9 0.05403 0.08803 0.07187 295.07 1.6148E-004

20200 216.85 5305.0 0.05236 0.08522 0.06957 295.21 1.6694E-004

20400 217.05 5140.5 0.05073 0.08251 0.06735 295.34 1.7257E-004

20600 217.25 4981.3 0.04916 0.07988 0.06521 295.48 1.7839E-004

20800 217.45 4827.1 0.04764 0.07733 0.06313 295.61 1.8440E-004

21000 217.65 4677.9 0.04617 0.07487 0.06112 295.75 1.9060E-004

21200 217.85 4533.3 0.04474 0.07249 0.05918 295.89 1.9701E-004

21400 218.05 4393.4 0.04336 0.07019 0.05730 296.02 2.0363E-004

21600 218.25 4257.9 0.04202 0.06796 0.05548 296.16 2.1046E-004

21800 218.45 4126.8 0.04073 0.06581 0.05372 296.29 2.1752E-004

22000 218.65 3999.7 0.03947 0.06373 0.05202 296.43 2.2480E-004

22200 218.85 3876.7 0.03826 0.06171 0.05038 296.56 2.3232E-004




22400 219.05 3757.6 0.03708 0.05976 0.04878 296.70 2.4009E-004

22600 219.25 3642.3 0.03595 0.05787 0.04724 296.83 2.4811E-004

22800 219.45 3530.5 0.03484 0.05605 0.04575 296.97 2.5639E-004

23000 219.65 3422.4 0.03378 0.05428 0.04431 297.11 2.6494E-004

23200 219.85 3317.6 0.03274 0.05257 0.04291 297.24 2.7376E-004

23400 220.05 3216.1 0.03174 0.05091 0.04156 297.38 2.8287E-004

23600 220.25 3117.8 0.03077 0.04931 0.04026 297.51 2.9228E-004

23800 220.45 3022.6 0.02983 0.04776 0.03899 297.65 3.0198E-004

24000 220.65 2930.4 0.02892 0.04627 0.03777 297.78 3.1200E-004

24200 220.85 2841.1 0.02804 0.04482 0.03658 297.92 3.2235E-004

24400 221.05 2754.6 0.02719 0.04341 0.03544 298.05 3.3302E-004

24600 221.25 2670.8 0.02636 0.04205 0.03433 298.19 3.4404E-004

24800 221.45 2589.6 0.02556 0.04074 0.03325 298.32 3.5542E-004

25000 221.65 2510.9 0.02478 0.03946 0.03222 298.45 3.6716E-004

25200 221.85 2434.7 0.02403 0.03823 0.03121 298.59 3.7927E-004

25400 222.05 2360.9 0.02330 0.03704 0.03024 298.72 3.9178E-004

25600 222.25 2289.4 0.02259 0.03589 0.02929 298.86 4.0468E-004

25800 222.45 2220.1 0.02191 0.03477 0.02838 298.99 4.1800E-004

26000 222.65 2153.0 0.02125 0.03369 0.02750 299.13 4.3174E-004

26200 222.85 2087.9 0.02061 0.03264 0.02664 299.26 4.4593E-004

26400 223.05 2024.9 0.01998 0.03163 0.02582 299.40 4.6056E-004

26600 223.25 1963.9 0.01938 0.03064 0.02502 299.53 4.7566E-004

26800 223.45 1904.7 0.01880 0.02969 0.02424 299.66 4.9124E-004

27000 223.65 1847.3 0.01823 0.02878 0.02349 299.80 5.0732E-004

27200 223.85 1791.8 0.01768 0.02788 0.02276 299.93 5.2391E-004

27400 224.05 1737.9 0.01715 0.02702 0.02206 300.07 5.4102E-004

27600 224.25 1685.8 0.01664 0.02619 0.02138 300.20 5.5868E-004

27800 224.45 1635.2 0.01614 0.02538 0.02072 300.33 5.7690E-004

28000 224.65 1586.2 0.01565 0.02460 0.02008 300.47 5.9569E-004

Table 2.1 Atmospheric properties in ISA (Cont)



28200 224.85 1538.7 0.01519 0.02384 0.01946 300.60 6.1508E-004

28400 225.05 1492.6 0.01473 0.02311 0.01886 300.74 6.3508E-004

28600 225.25 1448.0 0.01429 0.02239 0.01828 300.87 6.5572E-004

28800 225.45 1404.8 0.01386 0.02171 0.01772 301.00 6.7700E-004

29000 225.65 1362.9 0.01345 0.02104 0.01718 301.14 6.9896E-004

29200 225.85 1322.2 0.01305 0.02040 0.01665 301.27 7.2161E-004

29400 226.05 1282.8 0.01266 0.01977 0.01614 301.40 7.4497E-004

29600 226.25 1244.7 0.01228 0.01916 0.01564 301.54 7.6906E-004

29800 226.45 1207.6 0.01192 0.01858 0.01517 301.67 7.9391E-004

30000 226.65 1171.8 0.01156 0.01801 0.01470 301.80 8.1954E-004

30200 226.85 1137.0 0.01122 0.01746 0.01425 301.94 8.4598E-004

30400 227.05 1103.3 0.01089 0.01693 0.01382 302.07 8.7324E-004

30600 227.25 1070.6 0.01057 0.01641 0.01340 302.20 9.0136E-004

30800 227.45 1038.9 0.01025 0.01591 0.01299 302.33 9.3035E-004

31000 227.65 1008.1 0.00995 0.01543 0.01259 302.47 9.6026E-004

31200 227.85 978.3 0.00966 0.01496 0.01221 302.60 9.9109E-004

31400 228.05 949.5 0.00937 0.01450 0.01184 302.73 1.0229E-003

31600 228.25 921.4 0.00909 0.01406 0.01148 302.87 1.0557E-003

31800 228.45 894.3 0.00883 0.01364 0.01113 303.00 1.0895E-003

32000 228.65 867.9 0.00857 0.01322 0.01079 303.13 1.1243E-003

Table 2.1 Atmospheric properties in ISA

Note: Following values / expressions have been used while preparing ISA table. 2 -2

2

R=287.05287m sec Kg= 9.80665m/s

Sutherland formula for viscosity: 3/2

-6 T = 1.458X10 [ ]T+110.4



In troposphere (h = 0 to 11000 m): T= 288.15 - 0.0065 h. p = 101325 [1-0.000022588h] 5.25588

= 1.225 [1-0.000022588h]4.25588 . In lower stratosphere (h = 11000 to 20000 km): T=216.65 K. p = 22632 exp {-0.000157688 (h-11000)} = 0.36391 exp {-0.000157688 (h-11000)} In middle stratosphere (h = 20000 to 32000 km): T = 216.65 + 0.001h p = 5474.9 [1+0.000004616(h-20000)]-34.1632 = 0.08803 [1+0.000004616(h-20000)]-35.1632



Chapter 2 Table 2.1 Atmospheric properties in ISA



Altit-

ude

(m)

Tempe-

rature

(K)

Pressure

(N/m2)

(p/po)

Density

(kg/m3)

(/o)

speed

of

sound

(m/s)

Kinematic

viscosity

(m2/s)

0 288.15 101325.0 1.00000 1.22500 1.00000 340.29 1.4607E-005

200 286.85 98945.3 0.97651 1.20165 0.98094 339.53 1.4839E-005

400 285.55 96611.0 0.95348 1.17864 0.96216 338.76 1.5075E-005

600 284.25 94321.6 0.93088 1.15598 0.94365 337.98 1.5316E-005

800 282.95 92076.3 0.90872 1.13364 0.92542 337.21 1.5562E-005

1000 281.65 89874.4 0.88699 1.11164 0.90746 336.43 1.5813E-005

1200 280.35 87715.4 0.86568 1.08997 0.88977 335.66 1.6069E-005

1400 279.05 85598.6 0.84479 1.06862 0.87234 334.88 1.6331E-005

1600 277.75 83523.3 0.82431 1.04759 0.85518 334.10 1.6598E-005

1800 276.45 81489.0 0.80423 1.02688 0.83827 333.31 1.6870E-005

2000 275.15 79494.9 0.78455 1.00649 0.82162 332.53 1.7148E-005

2200 273.85 77540.6 0.76527 0.98640 0.80523 331.74 1.7432E-005

2400 272.55 75625.4 0.74636 0.96663 0.78908 330.95 1.7723E-005

2600 271.25 73748.6 0.72784 0.94716 0.77319 330.16 1.8019E-005

2800 269.95 71909.7 0.70969 0.92799 0.75754 329.37 1.8321E-005

3000 268.65 70108.2 0.69191 0.90912 0.74214 328.58 1.8630E-005

3200 267.35 68343.3 0.67450 0.89054 0.72697 327.78 1.8946E-005

3400 266.05 66614.6 0.65744 0.87226 0.71205 326.98 1.9269E-005

3600 264.75 64921.5 0.64073 0.85426 0.69736 326.18 1.9598E-005

3800 263.45 63263.4 0.62436 0.83655 0.68290 325.38 1.9935E-005

4000 262.15 61639.8 0.60834 0.81912 0.66867 324.58 2.0279E-005

4200 260.85 60050.0 0.59265 0.80197 0.65467 323.77 2.0631E-005

4400 259.55 58493.7 0.57729 0.78510 0.64090 322.97 2.0990E-005

4600 258.25 56970.1 0.56225 0.76850 0.62735 322.16 2.1358E-005




4800 256.95 55478.9 0.54753 0.75217 0.61402 321.34 2.1734E-005

5000 255.65 54019.4 0.53313 0.73611 0.60091 320.53 2.2118E-005

5200 254.35 52591.2 0.51903 0.72031 0.58801 319.71 2.2511E-005

5400 253.05 51193.7 0.50524 0.70477 0.57532 318.90 2.2913E-005

5600 251.75 49826.4 0.49175 0.68949 0.56285 318.08 2.3324E-005

5800 250.45 48488.8 0.47855 0.67446 0.55058 317.25 2.3744E-005

6000 249.15 47180.5 0.46564 0.65969 0.53852 316.43 2.4174E-005

6200 247.85 45900.9 0.45301 0.64516 0.52666 315.60 2.4614E-005

6400 246.55 44649.5 0.44066 0.63088 0.51501 314.77 2.5064E-005

6600 245.25 43425.9 0.42858 0.61685 0.50355 313.94 2.5525E-005

6800 243.95 42229.6 0.41677 0.60305 0.49229 313.11 2.5997E-005

7000 242.65 41060.2 0.40523 0.58949 0.48122 312.27 2.6480E-005

7200 241.35 39917.1 0.39395 0.57617 0.47034 311.44 2.6974E-005

7400 240.05 38799.9 0.38292 0.56308 0.45965 310.60 2.7480E-005

7600 238.75 37708.1 0.37215 0.55021 0.44915 309.75 2.7998E-005

7800 237.45 36641.4 0.36162 0.53757 0.43884 308.91 2.8529E-005

8000 236.15 35599.2 0.35134 0.52516 0.42870 308.06 2.9073E-005

8200 234.85 34581.2 0.34129 0.51296 0.41875 307.21 2.9629E-005

8400 233.55 33586.9 0.33148 0.50099 0.40897 306.36 3.0200E-005

8600 232.25 32615.8 0.32189 0.48923 0.39937 305.51 3.0784E-005

8800 230.95 31667.6 0.31254 0.47768 0.38994 304.65 3.1383E-005

9000 229.65 30741.9 0.30340 0.46634 0.38069 303.79 3.1997E-005

9200 228.35 29838.2 0.29448 0.45521 0.37160 302.93 3.2627E-005

9400 227.05 28956.1 0.28577 0.44428 0.36268 302.07 3.3272E-005

9600 225.75 28095.2 0.27728 0.43355 0.35392 301.20 3.3933E-005

9800 224.45 27255.2 0.26899 0.42303 0.34533 300.33 3.4611E-005

10000 223.15 26435.7 0.26090 0.41270 0.33690 299.46 3.5307E-005

10200 221.85 25636.2 0.25301 0.40256 0.32862 298.59 3.6020E-005

10400 220.55 24856.4 0.24531 0.39262 0.32050 297.71 3.6752E-005




10600 219.25 24096.0 0.23781 0.38286 0.31254 296.83 3.7503E-005

10800 217.95 23354.4 0.23049 0.37329 0.30473 295.95 3.8274E-005

11000 216.65 22631.5 0.22336 0.36391 0.29707 295.07 3.9065E-005

11200 216.65 21929.4 0.21643 0.35262 0.28785 295.07 4.0316E-005

11400 216.65 21248.6 0.20971 0.34167 0.27892 295.07 4.1608E-005

11600 216.65 20588.9 0.20320 0.33106 0.27026 295.07 4.2941E-005

11800 216.65 19949.7 0.19689 0.32079 0.26187 295.07 4.4317E-005

12000 216.65 19330.4 0.19078 0.31083 0.25374 295.07 4.5736E-005

12200 216.65 18730.2 0.18485 0.30118 0.24586 295.07 4.7202E-005

12400 216.65 18148.7 0.17911 0.29183 0.23823 295.07 4.8714E-005

12600 216.65 17585.3 0.17355 0.28277 0.23083 295.07 5.0275E-005

12800 216.65 17039.4 0.16817 0.27399 0.22366 295.07 5.1886E-005

13000 216.65 16510.4 0.16294 0.26548 0.21672 295.07 5.3548E-005

13200 216.65 15997.8 0.15789 0.25724 0.20999 295.07 5.5264E-005

13400 216.65 15501.1 0.15298 0.24925 0.20347 295.07 5.7035E-005

13600 216.65 15019.9 0.14823 0.24152 0.19716 295.07 5.8862E-005

13800 216.65 14553.6 0.14363 0.23402 0.19104 295.07 6.0748E-005

14000 216.65 14101.8 0.13917 0.22675 0.18510 295.07 6.2694E-005

14200 216.65 13664.0 0.13485 0.21971 0.17936 295.07 6.4703E-005

14400 216.65 13239.8 0.13067 0.21289 0.17379 295.07 6.6776E-005

14600 216.65 12828.7 0.12661 0.20628 0.16839 295.07 6.8916E-005

14800 216.65 12430.5 0.12268 0.19988 0.16317 295.07 7.1124E-005

15000 216.65 12044.6 0.11887 0.19367 0.15810 295.07 7.3403E-005

15200 216.65 11670.6 0.11518 0.18766 0.15319 295.07 7.5754E-005

15400 216.65 11308.3 0.11160 0.18183 0.14844 295.07 7.8182E-005

15600 216.65 10957.2 0.10814 0.17619 0.14383 295.07 8.0687E-005

15800 216.65 10617.1 0.10478 0.17072 0.13936 295.07 8.3272E-005

16000 216.65 10287.5 0.10153 0.16542 0.13504 295.07 8.5940E-005

16200 216.65 9968.1 0.09838 0.16028 0.13084 295.07 8.8693E-005




16400 216.65 9658.6 0.09532 0.15531 0.12678 295.07 9.1535E-005

16600 216.65 9358.8 0.09236 0.15049 0.12285 295.07 9.4468E-005

16800 216.65 9068.2 0.08950 0.14581 0.11903 295.07 9.7495E-005

17000 216.65 8786.7 0.08672 0.14129 0.11534 295.07 1.0062E-004

17200 216.65 8513.9 0.08403 0.13690 0.11176 295.07 1.0384E-004

17400 216.65 8249.6 0.08142 0.13265 0.10829 295.07 1.0717E-004

17600 216.65 7993.5 0.07889 0.12853 0.10492 295.07 1.1060E-004

17800 216.65 7745.3 0.07644 0.12454 0.10167 295.07 1.1415E-004

18000 216.65 7504.8 0.07407 0.12068 0.09851 295.07 1.1780E-004

18200 216.65 7271.9 0.07177 0.11693 0.09545 295.07 1.2158E-004

18400 216.65 7046.1 0.06954 0

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