Course Title: Aircraft Stability and Control Course Code: AAE014

Course Instructor: Dr. Yagya Dutta Dwivedi

Professor Department of Aeronautical Engineering

Institute of Aeronautical Engineering Dundigal, Hyderabad, India -500043

Mob-8555815261, email: yddwivedi@gmail.com

Course Outcomes

2

COs Course Outcome

CO1 Demonstrate concept of stability and application to dynamic systems like Aircraft, and the role of primary controls and secondary controls in longitudinal stability

CO2 Learn about the mathematical modeling of an aircraft in longitudinal, lateral and directional cases

CO3 Estimate the longitudinal and directional parameters with the help of the linearized equations of aircraft motion.

CO4 Analyze the different type of modes in longitudinal, lateral and directional motion of aircraft, and recovery from those modes

TEXT BOOKS:

1.Yechout, T.R. et al., “Introduction to Aircraft

Flight Mechanics”, AIAA education Series, 2003,

ISBN 1-56347-577-4.

2. Nelson, R.C., “Flight Stability and Automatic

Control”, 2nd Edn., Tata McGraw Hill, 2007,

ISBN 0- 07-066110.

3.Etkin, B and Reid, L.D., “Dynamics of Flight”,

3rd Edn., John Wiley, 1998, ISBN0-47103418-5

4

It consist of Five Units

Unit I: INTRODUCTION AND LONGITUDINAL STABILITY

CONSIST OF: Aircraft axis system, equilibrium, stability, controllability, static and dynamic stability, criteria and trim condition, contribution of Aircraft components on stability, static margin, neutral point, elevator hinge

Hinge moment, trim tabs, mass balancing etc

UNIT II: Lateral Directional static stability Consist of: Lateral-directional stability, forces and moments, rolling yawing due to side slip, component contribution on dirctional stability UNIT III- Aircraft Equation of Motion: Consist of: Reference frames, Euler angle, transformation, rotating system, liniear and angular acceleration, EOM of long and lat direction, UNIT IV: Linearization of EOM, aerodynamic forces and moments

Consist of: Perturbed EOMs, Linearization of EOMs, Different derivatives

UNIT V: Aircraft Dynamic Stability

UNIT-I

INTRODUCTION AND

LONGITUDINAL

SATABILITY

5

6

Course Learning Outcomes

CLOs Course Learning Outcome

CLO1 Apply concept of stability, controllability and

maneuverability in an aircraft.

CLO2 Use and interpret the basic mathematics, science

and engineering for solving problems of longitudinal,

lateral and directional static stability.

CLO3

Describe stick fixed and stick free conditions for

neutral point.

CLO4 Demonstrate different methods for finding static

margin, control force and CG limitation.

INTRODUCTION

7

AIRCRAFT STABILITY AND CONTROL

First we should know about the three words in this course i.e

AIRCRAFT : An aircraft is a vehicle that is able to fly by

gaining support from the air. It counters the force of gravity by

using either static lift or by using the dynamic lift of an airfoil,

or in a few cases the downward thrust from jet engines.

Fig an airplane, helicopter, or other machine capable of flight”

Stability and Control - Definitions

8

Definition of stability

1: the quality, state, or degree of being stable: such as

the strength to stand or endure : FIRMNESS

the property of a body that causes it when disturbed

from a condition of equilibrium or steady motion to

develop forces or moments that restore the original

condition

resistance to chemical change or to physical disintegration

Definition of Control

a:Power or authority to guide or manage

b:a device or mechanism used to regulate or guide the

operation of a machine, apparatus, or system.

Aircraft Stability

9

History and Growth Control and Stability of Aircraft • Write Brother’s achievement was to find the masterly of the

three main areas, which required by functional airplane, these are

1. Lift 2. Propulsion 3. Control • First two had been studies by many researchers like Sir

George Cayley, Otto Lilienthal, Octave Chanute and Samuel Langley

•The bigger innovation of Write Brother’s Flyer was control system, they installed in their airplane.

10

Aircraft Stability

Aircraft Stability and control- Aims of the Study

Suppose an aircraft in some state of steady flight. If it is disturbed, by a gust say, or by the pilot, it is regarded as stable if it returns to a sensibly steady state within a finite time.

We may be able to tolerate a small degree of instability. even deliberately design an aircraft to be quite unstable; in the latter case, however, a reliable automatic stabilization system will be required.

We normally require more than mere stability; the response to gusts must not make the pilot's task difficult,

produce an uncomfortable ride for passengers, impose excessive loads on the aircraft, or make the aircraft unsuitable as an aiming platform.

The pilot must be able to control the aircraft accurately without having to perform excessive feats of skill or strength.

Aims of this course

11

Our first aim then is to study the dynamics of the aircraft and its interaction with the aerodynamics in order to be able to assess and possibly improve the dynamic characteristics. A further aim is to understand the physics of the processes involved. We make approximations for better numerical results can generally be found using a computer, little real understanding follows its use alone. With a good understanding of the physics involved, solutions to design problems can be put forward.

Aircraft Axes System

12

An Aircraft in flight is free to rotate in three dimensions: Yaw, nose left or right about an axis running up and down. Pitch, nose up or down about an axis running from wing to wing. Roll, rotation about an axis running from nose to tail. The axes are alternatively designated as vertical, transverse,

and longitudinal respectively. These axes move with the vehicle and rotate relative to the

Earth along with the craft.

The position of all three axes, with the right hand

rules for its rotations

13

Axes Representation

14

Heading Rotation and angle Representation

15

Pitch Rotation Representation

16

Representation of Roll

17

Control surfaces

18

These rotations are produced by torques (or Moments) about the principal axes. On an aircraft, these are intentionally produced by means of moving control surfaces, which vary the distribution of the net Aerodynamic force about the vehicle's CG. a) Elevators (moving flaps on the horizontal tail) produce pitch, b) rudder on the vertical tail produces yaw, c) ailerons (flaps on the wings that move in opposing directions)

produce roll. On a spacecraft, the moments are usually produced by a a) Reaction control system consisting of small rocket thrusters used to apply asymmetrical thrust on the vehicle.

Stability

19

OVER VIEW of the class

Definitions: Equilibrium Stability Controllability Maneuverability Examples of stability by simple mechanical system Types of stability

Definitions

20

Equilibrium: If a system in an equilibrium state, returns to equilibrium

following a small disturbance, the state is said to be stable equilibrium Figure 1.

On the other hand, if the system diverges from equilibrium when slightly disturbed, the state is said to be an unstable equilibrium.

Strictly speaking, Figure 1(d) is also a case of stable equilibrium, because a very small disturbance from equilibrium would result in a force and moment imbalance that would return the ball to its original equilibrium state.

But a little extra disturbance, towards right could cause the ball to move past the apex, which would produce a force and moment imbalance that would cause the ball to move away from its original equilibrium state.

States of equilibrium

21

Figure 1: States of equilibrium

Fig. 1 (a), Shows the stable equilibrium as disturbing force removed, the ball will restore its original position. Fig 1. (b), Shows unstable equilibrium as once the ball is disturbed by some external force, the ball will never come back to original position again. Fig. 1 (c), Shows Neutral equilibrium as the ball is disturbed it will remain in new position. Fig1. (d), is also a case of stable equilibrium

Stability

22

Static Stability: If an airplane disturbed from equilibrium state has “Initial Tendency” to return to its equilibrium state, then the aircraft is assumed to have static stability. Dynamic Stability: Not only initial tendency, but also the amplitudes of the response due to disturbance decay in finite time to attain the equilibrium state.

There are two types of Stability as mentioned below

Controllability

23

Controllability: The response of an aircraft in steady flight, on pilot control inputs. For instance deflecting the ailerons: a high resulting roll rate means a fast response. The relationship between stability and controllability has been that greater stability means less controllability and vice versa. An aircraft becomes less controllable, especially at slow flight speeds, as the CG is moved further aft

Maneuverability

24

It is the ability to change the direction of motion of a body (normally a vehicle, aircraft) without any loss in speed with which the body moving. Maneuver is nothing but when a body moving at certain speed, which changes its direction and attains the same initial speed at which the body was moving before changing the direction. For example. Consider an aircraft traveling at 500 kmph towards north which change its direction towards south and again reaches 500 kmph with in 10 sec have high maneuverability than an aircraft traveling at 500 kmph towards north which change its direction towards south and again reaches 500 kmph with in 30 sec.

Controllability

25

Controllability: The response of an aircraft in steady flight, on pilot control inputs. For instance deflecting the ailerons: a high resulting roll rate means a fast response. The relationship between stability and controllability has been that greater stability means less controllability and vice versa. An aircraft becomes less controllable, especially at slow flight speeds, as the CG is moved further aft

Stability and types

26

Stability is the ability of an aircraft to correct for conditions that act on it, like turbulence or flight control inputs. For aircraft, there are two general types of stability: static and dynamic. Most aircraft are built with stability in mind, but that's not always the case. Some aircraft, like training airplanes, are built to be very stable. But others, like fighter jets, tend to be very unstable, and can even be unflyable without the help of computer controlled fly-by-wire systems. Static Stability Static stability is the initial tendency of an aircraft to return to its original position when it's disturbed. There are three kinds of static stability: a) Positive Static stability b) Neutral Static stability c) Negative Static stability

Types of stability-Contd-----

27

Positive Static Stability

An aircraft that has positive static stability tends to return to its

original attitude when it's disturbed. Let's say you're flying an

aircraft, you hit some turbulence, and the nose pitches up.

Immediately after that happens, the nose lowers and returns to

its original attitude. That's an example positive static stability,

and it's something you'd see flying an airplane like a Cessna

172.

Positive Static Stability

Contd-----

28

Neutral static stability An aircraft that has neutral static stability tends to stay in its new attitude when it's disturbed. For example, if you hit turbulence and your nose pitches up 5 degrees, and then immediately after that it stays at 5 degrees nose up, your airplane has neutral static stability.

Neutral static stability

Contd---

29

Negative static stability Finally, an aircraft that has negative static stability tends to continue moving away from its original attitude when it's disturbed. For example, if you hit turbulence and your nose pitches up, and then immediately continues pitching up, you're airplane has negative static stability. For most aircraft, this is a very undesirable thing.

Negative static stability

Simple Mechanical static stability

30

31

Dynamic Stability

Dynamic stability is how an airplane responds over time to a disturbance. And it's probably no surprise that there are three kinds of dynamic stability as well: They are a)Positive Static stability b)Neutral Static stability c) Negative Static stability

Positive Dynamic Stability

32

Aircraft with positive dynamic stability have oscillations that dampen out over time. The Cessna 172 is a great example. If your 172 is trimmed for level flight, and you pull back on the yoke and then let go, the nose will immediately start pitching down. Depending on how much you pitched up initially, the nose will pitch down slightly nose low, and then, over time, pitch nose up again, but less than your initial control input. Over time, the pitching will stop, and your 172 will be back to its original attitude.

Neutral dynamic stability

33

Aircraft with neutral dynamic stability have oscillations that never dampen out. As you can see in the diagram below, if you pitch up a trimmed, neutrally dynamic stable aircraft, it will pitch nose low, then nose high again, and the oscillations will continue, in theory, forever.

Negative dynamic stability

34

Aircraft with negative dynamic stability have oscillations that get worse over time. The diagram below pretty much sums it up. Over time, the pitch oscillations get more and more amplified.

Negative dynamic stability illustration

35

Contd– dynamic stability

36

Dynamic stability contd---

37

Some confusions on longitudinal stability

38

Contribution of the wing

39

Contd– wing contribution

40

Contd---

41

Wing contribution- stability

Combination of static and dynamic stability

42

Stable and Unstable view

43

CRITERIA FOR LONGITUDINAL STATIC STABILITY

44

Thrust Forces and Moments

45

Longitudinalforcesandmomentsresultingfromenginethrustmustalsobedefinedtocompletetheappliedforcesandmomentssideoftheaircraftequationsofmotion.

Wewillonlyconsiderdirectthrusteffectsontheaircraft.Indirectthrusteffects,suchasjetexhaustimpingingonliftingsurfaces,willbeignored.

Inaddition,theorientationofthethrustvectorproducedbytheengineorengineswillbeassumedtobeinthexzbodyaxisplane(no side force components).

Theseassumptionsleadtoasimplerepresentationofthethrustforcesandmomentsinthebodyandstabilityaxis

Longitudinal Static Stability

46

Staticstabilityreferstotheinitialtendencyofanairplane,followingadisturbancefromsteady-stateflight,todevelopaerodynamicforcesandmomentsthatareinadirectiontoreturntheaircrafttothesteady-stateflightcondition.

Forpurposesofthistext,longitudinalstaticstabilitywillprimarilyrefertoaircraftpitchingmomentcharacteristicsandwillbeanalyzedforthestickfixedcondition.

Therequirementtotrimtheaircraftatusableanglesofattackisalsodiscussedwiththelongitudinalstabilityrequirementbecausebotharegenerallynecessarytoachieveacceptableflightcharacteristics

Stability Requirements

47

Stability requirement Contd----

48

Neutral Point and Static Margin

49

For neutral static stability, Cm(alpha )will be equal to zero. This equates to a horizontal line on a Cm vs alpha graph.

The condition for neutral static stability is important because it represents the boundary between static stability and instability.

If the c.g. is located aft of the neutral point, the aircraft will

be statically unstable(longitudinally)and Cm(alpha)will be positive.

Aerodynamic balancing

50

The ways and means of reducing the magnitudes of Ch α t and Ch δe are called aerodynamic balancing.

The methods for aerodynamic balancing are:

Setback hinge,

Horn balance and

Internal balance

Set back hinge or over hang balance

51

In this case, the hinge line is shifted behind the leading edge of the control.

As the hinge line shifts, the area of the control surface ahead of

the hinge line increases.

Types of trailing edge

52

Horn balance

53

In this method of aerodynamic balancing, apart of the control surface near the tip, is ahead of the hinge line.

There are two types of horn balances– shielded and unshielded

Internal balance or internal seal

54

In this case, the portion of the control surface ahead of the hinge line, projects in the gap between the upper and lower surfaces of the stabilizer.

The upper and lower surfaces of the projected portion are vented to the upper and lower surface pressures respectively at a chosen chordwise position.

A seal at the leading edge of the projecting portion ensures that the pressures on the two sides of the projection do not equalize.

This method of aerodynamic balancing is complex but is reliable

Tabs

55

Tabs

The methods of aerodynamic balancing described earlier are sensitive to fabrication defects and surface curvature.

Hence, tabs are used for finer adjustment to make the hinge moment zero.

Tabs are also used for other purposes.

A brief description of different types of tabs is given in the following subsections

Trim tab

56

It is used to trim the stick or bring Ch to zero by tab deflection.

After the desired elevator deflection (δe) is achieved, the tab is deflected in a direction opposite to that of the elevator so that the hinge moment be comes zero.

Since the tab is located far from the hinge line, a small amount of tab deflection is adequate to bring Che to zero.

As the lift due to the tab is in a direction opposite to that of the elevator, a slight adjustment in elevator deflection would be needed after application of tab.

Though the pilot subsequently does not have to hold the stick all the time, the initial effort to move the control is not reduced when this tab is used

Link balance Tab

57

In this case the tab is linked to the main control surface. As the main surface moves up the tab deflects in the opposite

direction in a certain proportion. This way the tab reduces the hinge moment and hence it is

called “Balancetab”

Servo tab

58

Servo tab

In this case the pilot does not move the main surface which is free to rotate about the hinge.

Instead the pilot moves only the tab as a result of which the pressure distribution is altered on the main control surface and it attains a floating angle such that Ch is zero.

The action of the tab is like a servo action and hence it is called “Servotab”. This type of tab is used on the control surfaces of large air planes

Mass balancing

59

This ensures that the c.g. of the control surface lies ahead or on the hinge line.

All movable tail In some military and large civil airplanes the entire horizontal tail is

hinged and rotated to obtain larger longitudinal control.

Elevons In a tailless configuration (e.g. concorde airplane) the functions of

the elevator and the aileron are combined in control surfaces called elevons.

Like ailerons they are located near the wingtip but the movable surfaces on the two wing halves can move in the same direction or in different directions.

When they move in the same direction, they provide pitch control and when they move indifferent directions they provide control in roll

Contd----

60

V–tail In some older airplanes the functions of horizontal and vertical

tails were combined in a V-shaped tail. Though the area of the V-tail is less than the sum of the areas

of the horizontal and vertical tail, it leads to undesirable coupling of lateral and longitudinal motions and is seldom used.

Configuration with two vertical tails At supersonic speeds the slope of the lift curve (dCL/dα) is

proportional to Mach square, where M∞ is the free stream Mach number.

Thus, CL α and intern the tail effectiveness decreases significantly at high Mach numbers. Hence some military airplanes have two moderate sized vertical tails instead of one large tail

Definition of the longitudinal static stability

61

Contd-----

62

Flow field created by the Wing

63

Contribution to Stability

64

Static margin: Stability criteria

65

66

67

Wing contribution in moment

68

Wing contribution in moment

69

Contd---

70

Wing contribution: simplification

71

Other Approximations

72

Conclusions of wing contributions

73

About static stability

74

Static Longitudinal Stability: Conceptual Description

75

Equilibrium

Longitudinal Stability: Conceptual Description

76

Nose up Configuration

Longitudinal Stability: Conceptual Description

77

Stability of the Equilibrium

Numerical 1.2

78

solution

Slope of lift curve (CLα) and angle of zero lift (α0L) of the airplane:

79

Stick fixed Neutral point

80

Indirect contributions of power plant to Cmcg and Cmα

81

Angle of zero lift (α0L) forairplane:

82

UNIT- II

LATERAL-DIRECTIONAL

STATIC STABILITY

83

84

Course Learning Outcomes

CLOs Course Learning Outcome

CLO5 Organize total stability parameters in order of merit

of flight conditions.

CLO6 Locate the cause of instability in an aircraft and solve

the issue.

CLO7

Identify aircraft different types of stability for

different categories of aircraft.

CLO8 Demonstrate the aircraft component contribution for

different stability.

Introduction

85

Introduction contd----

86

87

Sideslip and yaw

88

Lateral-Directional Applied Forces and Moments

89

Because we have assumed that longitudinal and lateral-directional motion are independent of each other.

Lateral-directional motion is assumed to consist of roll and yaw rotation and y-axis translation.

These two rotations and the translation are typically coupled (that is, they occur together)

Aircraft Side Force

90

Aerodynamic side-force acts along the number two stability axis (positive out the right wing) and may be expressed using the side-force coefficient.

Aircraft Rolling Moment

91

Aircraft rolling moment acts about the x axis and may be expressed using the rolling moment coefficient as

Aircraft Yawing Moment

92

Aircraft yawing moment acts about the z axis and may be expressed using the yawing moment coefficient as

Lateral-Directional Static Stability

93

Static stability refers to the initial tendency of an airplane, following a disturbance from steady-state flight, to develop aerodynamic forces and moments that are in a direction to return the aircraft to the steady-state flight condition.

For purposes of this text, lateral-directional static stability will primarily refer to aircraft rolling moment and yawing moment characteristics.

Lateral and directional stability will be discussed separately, but it should be realized that rolling and yawing motions are inherently coupled.

This highly coupled behavior necessitates consideration of these motions together, especially when analyzing and designing lateral-directional handling qualities.

Stability Requirements

94

Lateral motion: The requirement for lateral (roll) static stability is

Contd----

95

Directional motion: The requirement for directional (yaw)

static stability is

Illustration of static directional stability

96

Illustration of static roll stability

97

Wing and fuselage contribution to dihedral effect

98

Contd----

99

Crosswind Landings

100

Landing approaches with a component of the wind across the runway can generally be handled in two ways by a pilot.

The first approach is to ‘‘crab in to the wind’’. The degree of crab is adjusted until the aircraft ground track aligns with the direction of the runway.

This approach works well until the aircraft is at the point of touch down on the runway.

Then the aircraft must generally align the x-body axis with the runway direction so that the landing gear wheels are aligned with the direction of touch down

Crabing

101

Generation of the side wing

102

Engine-Out Analysis

103

The lateral-directional force and moment equations can be used to analyze the case of an engine failure in flight that results in a yawing moment.

Consider a twin-engine aircraft that has experienced a right engine failure

Contd---

104

The rolling moment resulting from an asymmetric

thrust configuration should also be considered.

For a jet engine configuration with the engines

mounted forward and below the wing, a right engine

out configuration will probably result in a negative

rolling moment be cause at the lower pressure

generated below the left wing by the high velocity

exhaust from the operating engine.ne

Requirement for directional Control

105

Lateral and directional; stability

106

Definition of all stability

107

Lateral stability is roll stability: the tendency of the aircraft

to reduce its rolling and return to an upright position unless

continually maintained in position by e.g. the ailerons. (This

is usually only partial.)

Longitudinal stability is pitch stability: the tendency of the

aircraft to reduce its pitching and return to a level position

(relative to the direction it's traveling, at least) unless

countered by e.g. the elevators. Directional stability (also known as vertical stability) is yaw

stability: the tendency of the aircraft to reduce its yawing and

return to a straight position (relative to the direction it's

traveling, at least) unless countered by e.g. the rudder.

Yawing stability due to vertical tail

108

Directional stability

109

Aircraft directional stability and vertical tail

110

Dihedral and sweep back effect

111

Formula Directional stability

112

Lateral Stability- Main Sources

113

Lateral-Directional Applied Forces and Moments

114

Because we have assumed that longitudinal and lateral-directional motion are independent of each other, lateral-directional motion is assumed to consist of roll and yaw rotation and y-axis translation.

These two rotations and the translation are typically coupled (that is, they occur together).

We will now expand the applied lateral-directional aero force and moment terms with conventional aerodynamic coefficients

Determination of maneuver point

115

Aircraft Side Force

116

Aerodynamic side-force acts along the number two stability axis (positive out the right wing) and may be expressed using the side-force coefficient (Cy) as

Side force is a function of the angle of sideslip (beta or b), aileron deflection (da), rudder deflection (dr), angle of attack (a), Mach number, and Reynolds number. A positive sideslip angle (b) is defined in Fig. below. It can be easily remembered as positive b is ‘wind in the right ear’ for the pilot. Our Taylor series expansion of the side-force coefficient will include the first three terms.

Rolling moment because of sideslip with an aft-swept wing

117

118

A method to estimate Cyb based on aircraft configuration begins with the definition of the aero sideforce acting on the vertical tail using the sideforce coefficient.

The contribution of the vertical tail to Cyb

can be estimate

The contribution of the vertical tail to Cyb can be estimated

where s is the sidewash angle

Side force resulting from differential stabilator deflection

119

the aspect ratio of the vertical tail

The derivative Cydr is positive because a positive rudder deflection (trailing edge left) will generate a side force along the positive y axis.

Aircraft Rolling Moment

120

Aircraft rolling moment acts about the x axis and may be expressed using the rolling moment coefficient as

The rolling moment coefficient is a function of the same parameters we considered for side force; namely, sideslip angle, aileron deflection, rudder deflection, angle of attack, Mach number, and Reynolds number. We will again use sideslip angle, aileron deflection, and rudder deflection in our first-order Taylor series expansion

121

is the lateral (roll) static stability derivative. It is also sometimes called the dihedral effect. the sign of Clb must be negative if an

aircraft has roll static stability. A negative Clb simply implies that the aircraft generates a rolling moment that rolls the aircraft away from the direction of sideslip. Four aspects of an aircraft design primarily influence Clb : geometric dihedral, wing position, wing sweep, and the contribution of the vertical tail. In other words,

Geometric dihedral

122

Provides a significant negative contribution to Clb . The larger the dihedral angle, the more negative rolling moment will result from a positive sideslip angle and the more positive rolling moment will result from a negative sideslip angle. This occurs because the wing toward the relative wind (right wing for positive sideslip and left wing for negative sideslip) experiences a higher angle of attack than that experienced by the opposite wing

Effect of Wing position

123

A high wing position will provide a negative contribution to Clb , a low wing position will provide a positive contribution, and a mid-wing position will provide a fairly neutral contribution.

Effects of Wing sweep angle

124

• A sideslip angle results in a side velocity that can be broken into

• vector components normal and parallel to the leading edge of each wing.

• With aft sweep, the wing toward the velocity vector (the leading wing) has a larger normal velocity component than the wing opposite the velocity vector (the trailing wing).

• As a result, the upstream wing will produce more lift than the downstream wing (resulting in a rolling moment away from the sideslip direction), and a negative contribution will result for Clb

Effect of the vertical tail

125

• A positive sideslip angle will result in an aerodynamic force on the vertical tail in the negative y-axis direction.

• Because the vertical tail is normally above the x (or rolling) axis of the aircraft, this aerodynamic force produces a negative rolling moment that results in a negative contribution to Clb .

• A similar analysis holds for negative sideslip angles. The larger and higher the vertical tail, the more negative the contribution to Clb.

zv may be estimated

Illustration of vertical tail moment arm

126

Effects of

127

• Ailerons are typically the primary control surface for producing rolling moment in response to a pilot command.

• A positive aileron deflection results in a positive rolling moment about the x axis.

• Ailerons are generally not deflected symmetrically so that adverse yaw effects can be minimized. For example, in response to a right stick input, the right aileron may have a larger trailing edge up deflection that the left aileron has a trailing edge down deflection.

• we define the magnitude of aileron deflection (using the convention that trailing edge down is positive) as

Contd--- Clda

128

• The derivative Clda defines the change in rolling moment that results from aileron deflection.

• It is also called the aileron control power. • Clda is positive based on the definition of a positive aileron

deflection. • The magnitude of Clda depends on several factors. • The aileron chord to wing chord ratio is a measure of the

relative size of the aileron in terms of wing chord. • The larger the ratio, the larger Clda becomes. • The aileron span location on the wing determines the

moment arm and length of the ailerons. • The larger the moment arm (the further outboard) and the

longer the length, the larger Clda becomes. • The magnitude of aileron deflection is also a factor in

defining the magnitude of Clda .

Rolling moment wind tunnel data

129

YF 17 view on Clda

130

Roll control power is generally an important requirement in high-performance aircraft. As an example, the roll performance of the YF-17 was found to be unacceptable during initial Air Force flight evaluations because of the aeroelastic aileron reversal effect

Three-view

drawing of A-

7 corsair

Aileron control power

131

Clda is positive based on the definition of a positive aileron deflection. The magnitude of Clda depends on several factors. The aileron chord to wing chord ratio is a measure of the relative size of the aileron in terms of wing chord. The larger the ratio, the larger Clda becomes. The aileron span location on the wing determines the moment arm and length of the ailerons. The larger the moment arm (the further outboard) and the longer the length, the larger Clda becomes. The magnitude of aileron deflection is also a factor in defining the magnitude of Clda .

Aircraft Yawing Moment

132

Aircraft yawing moment acts about the z axis and may be expressed using the yawing moment coefficient as

These parameters are again sideslip angle, aileron deflection, rudder deflection, angle of attack, Mach number, Reynolds number, and center of gravity location. Sideslip angle, aileron deflection, and rudder deflection will again be used in our first-order Taylor series expansion

Directional (yaw) static stability derivative

133

It is sometimes called the weathercock stability derivative. The sign of Cnb must be positive if the aircraft has yaw static

stability. A positive Cnb implies that in response to a sideslip angle, the aircraft will generate an aerodynamic yawing moment, which tends to reduce or zero-out the sideslip.

For example, a positive Cnb will result in a positive yawing moment being generated in response to a positive sideslip angle.

This yawing moment will tend to yaw the aircraft toward the relative wind and reduce the sideslip angle. We can also think of this as the weathervane effect

Effect of aircraft components on yawing moment

134

Vertical tail

The vertical tail is the primary aircraft component that drives the magnitude of Cnb. The larger the vertical tail, the more positive Cnb will be. The x-axis distance between the c.g. and the a.c. of the tail is another design feature that influences Cnb .

The larger this distance, the more positive Cnb will be. Cnb vertical tail ay be estimated by again starting with Eq

derivative Cnda

135

The derivative Cnda defines how yawing moment changes with aileron deflection.

For aircraft equipped with conventional ailerons, Cnda is typically negative, indicating that adverse yaw is generated as a result of the control input. This means that a positive aileron input (right wing down) will have a nose left yawing moment result.

This yawing moment away from the direction of the turn results from the differential induced drag.

A TED aileron deflection reduces the lift on the wing being rolled into, while a TED aileron deflection increases the lift on the wing coming up

Proverse yaw

136

Cnda may also be positive. This is called a proverse yaw condition and results when roll control surfaces such as spoilers are used.

For example, many sailplanes use differential spoilers to generate a rolling moment.

Lift is decreased using spoiler deployment on the wing being rolled into. The spoiler deployment increases drag on the wing at the same time it is decreasing lift.

This increased drag generates a yawing moment in the direction of the turn.

The F-4 Phantom incorporated a combination of these

ideas to minimize adverse yaw.

The lateral control system incorporated both ailerons, spoilers, and an aileron to rudder interconnect

Illustration of adverse yaw

137

Yawing moment wind tunnel data

138

Fig. Yawing moment wind tunnel data for F-16 VISTA aircraft

Rudder control power

139

The rudder is typically the primary control surface for producing a yawing moment in response to a pilot command.

A positive rudder deflection is defined as trailing edge left. The derivative Cndr defines the change in yawing moment

that results from rudder deflection. It is also called the rudder control power. Cndr is negative because a positive rudder deflection results

in a negative yawing moment. The magnitude of Cndr depends on several factors. The rudder chord to vertical tail chord ratio is a measure of

the relative size of the rudder in terms of the vertical tail chord.

The larger the ratio, the larger the magnitude of Cndr

Lateral-Directional Static Stability

140

static stability refers to the initial tendency of an airplane, following a disturbance from steady-state flight, to develop aerodynamic forces and moments that are in a direction to return the aircraft to the steady-state flight condition.

lateral-directional static stability will primarily refer to aircraft rolling moment and yawing moment characteristics. it should be realized that rolling and yawing motions are inherently coupled. This highly coupled behavior necessitates consideration of these motions together, especially when analyzing and designing lateral-directional handling qualities

Trim Conditions

141

Lateral-directional trim requirements can be simply stated

as achieving a total aircraft rolling moment and yawing

moment of zero. In coefficient terms, trim equates to

Lateral-directional trim is typically the condition of a zero sideslip angle. This condition is more correctly referred to as coordinated flight (beta equal to zero). With the assumption of a symmetrical aircraft (Cl0 and Cn0 equal to zero) and coordinated flight (b equal to zero), zero roll and yaw coefficients are achieved simply with da and dr equal to zero

Numerical 2.1

142

Determine the aileron and rudder deflections required for an F-15 to maintain a þ1 degree ‘‘wings level’’ ideslip at 0.9 Mach and 6000 m. Determine the value of the sideforce coefficient under these conditions. Applicable derivatives follow

Solution

143

We know that

For trim, Cl =Cn= 0, and the two equations become

We have two equations and two un knowns by solving we get

Contd---

144

The result makes sense from a sign standpoint. Left rudder (positive) is needed to generate the sideslip and opposite aileron (positive) is needed to offset the rolling moment generated by the aircraft’s lateral stability. To calculate the sideforce coefficient, we use Eq. as given

the result makes sense from a direction standpoint. The

negative sign indicates a side force in the negative y

direction, which is what will result from a positive sideslip

angle.

Stability Requirements- lateral motion

145

The sign of the stability derivatives Clb and Cnb is key in determining the lateral and directional static stability of the aircraft. Lateral motion The requirement for lateral (roll) static stability is It is shown in figure of Cl vs beta

Fig. Rolling moment coefficient vs sideslip characteristics

146

Sideforce vector for an aircraft

in a bank

Geometric dihedral, wing position, wing sweep, and vertical tail size, must be balanced in an overall aircraft design to achieve an acceptable degree of lateral stability. Too much lateral stability typically results in unacceptable dutch roll and crosswind landing characteristics

Directional motion.

147

The requirement for directional (yaw) static stability is

and is illustrated in Fig below

This requirement results in an aircraft that generates a yawing moment that yaws the nose of the aircraft toward the direction of sideslip. An aircraft with a positive Cnb will generate a positive aerodynamic yawing moment in response to a positive sideslip angle. Directional stability attempts to keep the aircraft in a coordinated flight condition where the sideslip angle is equal to zero.

Fig. Yawing moment vs sideslip characteristics

GA aircrafts

148

Engine-Out Analysis

149

The lateral-directional force and moment equations can be used to analyze the case of an engine failure in flight that results in a yawing moment. Consider a twin-engine aircraft that has experienced a right engine failure as shown in Fig. below.

The left engine is still operating while the right engine is producing wind milling drag. The yawing moment that results is

150

Notice that for the engine out case presented, the asymmetrical thrust and ram drag moments are both in the positive direction. For trimmed flight and assuming Cn0 is zero, these terms are included in the directional Taylor series.

For a jet aircraft, DD for a windmilling engine can be estimated as 10 to 15% of the thrust normally produced by the engine. For propeller-powered aircraft, the windmilling drag is significantly higher than this

151

The following lateral Taylor series (assuming Cl0 is zero) is then appropriate for trimmed flight.

We will also present the sideforce Taylor series for trimmed flight, including the gravity term.

Two options for engine-out flight

152

Two options for engine-out flight will now b Option 1: The aileron and rudder control deflections remain at zero after engine failure, and the aircraft is allowed to attain a steady-state trim condition with asymmetric thrust considered. A 1-degree-of-freedom (DOF) estimate of the resulting steady-state sideslip angle can be obtained using only Eq. (5.105) with da and dr equal to zero

Option 2: The rudder is deflected to zero out the sideslip. For the right engine-out case, this would require left rudder. A 1-DOF estimate of the rudder required can be obtained using Eq. with b and da equal to zero

UNIT- III

EQUATIONS OF MOTION

153

154

Course Learning Outcomes

CLOs Course Learning Outcome

CLO9

CLO10

CLO11

CLO12

Aircraft Axis Systems

155

These include the body axis system fixed to the aircraft, the Earth axis system, which we will assume to be an inertial axis system fixed to the Earth, and the stability axis system, which is defined with respect to the relative wind.

Each of these systems is useful in that they provide a convenient system for defining a particular vector, such as, the aerodynamic forces, the weight vector, or the thrust vector.

Body Axis System

156

The body axis system is fixed to the air craft with its origin at the aircraft’s center of gravity.

The x axis is defined out the nose of the aircraft along some reference line.

The reference line may be chosen to be the chord line of the aircraft or may be along the floor of the aircraft, as is often the case in large transports.

The y axis is defined out the right wing of the aircraft, and the z axis is defined as down through the bottom of the aircraft in accordance with the right- hand rule,

Body axes system

157

Earth Axis System

158

The Earth axis system is fixed to the Earth with its

z axis pointing to the center of the Earth.

The x axis and y axis are orthogonal and lie in the

local horizontal plane with the origin at the aircraft

center of gravity.

Often, the x axis is defined as North and the y axis

defined as East.

The Earth axis system is assumed to be an inertial

axis system for aircraft problems.

This is important because Newton’s 2nd law is valid

only in an inertial system.

While this assumption is not total ly accurate, it

works well for aircraft problems where the aircraft

rotation rates are large compared to the rotation

rate of the Earth

Illustration of Eath’s Axis system

159

Stability Axis System

160

The stability axis system is rotated relative to the body

axis system through the angle of attack.

This means that the stability x axis points in the

direction of the projection of the relative wind on to

the x z plane of the aircraft.

The origin of the stability axis system is also at the

aircraft center of gravity.

The y axis is out the right wing and coincident with the

y axis of the body axis system.

The z axis is orthogonal and points down ward in

accordance with the right-handrule.

Stability Axis system

161

Combination of axis system

162

Illustration of mass of an airplane

163

Body and inertial axes system

164

Earth Axis to Body Axis Transformation

165

Transformation of Axes contd-----

166

Stability Axis to Body Axis Transformation

167

Aircraft Force Equations

168

Moment Equations

169

Longitudinal Equations of Motion

170

Lateral-Directional Equations of Motion

171

Kinematic Equations

172

In addition to the six force and moment EOM, additional equations are required in order to completely solve the aircraft problem.

These additional equations are necessary because there are more than six unknowns due to the presence of the Euler angles in the force equations.

Three equations are obtained by relating the three body axis system rates, P, Q, and R to the three Euler rates

Summary of Force and Moment Eq.

173

Velocity of aircraft in fixed frame of reference

174

Introduction to EOMs

175

Introduction: After being given a small disturbance, it has a tendency to return to the equilibrium position. To analyse the static stability, the moments brought about immediately after the disturbance are only to be considered. However, for a system to be dynamically stable it must finally return to the equilibrium position. Thus, to examine the dynamic stability, the motion following a disturbance or an intended control input needs to be analysed. This motion is called response. However, an airplane is a system with six degrees of freedom and obtaining the response is a difficult task. However, in this introductory course the equations of motion are derived and simplified forms are obtained. Subsequently, the conditions that ensure dynamic stability are deduced without solving the equations.

Equations of motion in vector and scalar forms

176

The equations of motion are obtained by applying the Newton’s second law to the motion of airplane. For this purpose the airplane is treated as a rigid body which is translating as well as rotating.

This motion is decomposed as: (a) translation of the c.g. of the airplane with reference to an inertial frame which is taken as a frame fixed at a point on the earth and

(b) rotation with respect to the inertial system of a body axes system, attached to the airplane. The linear velocity, vector and the angular velocity vector are resolved along the body axes system.

To apply Newton’s second law to the motion of an airplane, requires an expression for the acceleration of an elemental mass ′dm′ located at a point on the body

Acceleration of a particle on a rigid body

177

The velocity of an elemental mass ‘dm’ at point P is the time derivative of its position vector i.e

Derivation of acceleration contd--

178

Vector form of equations of motion

179

Applying Newton’s second law of motion, the equations of motion is vector form are

Scalar form of equations of motion

Contd---

180

Contd---

181

Forces acting on the airplane

182

The external forces acting on an airplane are the thrust ( T ), the aerodynamic forces (A) (lift, drag and side force) and the gravitational force (mg). In vector form

Following assumptions have been made during the above derivation. (a) The airplane is rigid. (b) The reference frame attached to the earth is a Newtonian frame. (c) Flat earth model is used for gravitational force. Before obtaining the scalar form of Eq.(7.37) the following points may be noted. (a) The thrust vector acts roughly along the fuselage reference line (FRL). (b) The aerodynamic forces are resolved so that the drag is parallel to the free stream direction and the lift and the side force are in mutually perpendicular directions to the free stream. (c) The gravitational force acts vertically downwards. (d) To obtain the scalar form of Eq. (7.37), T, A and mg must be expressed in a single coordinate system

Concept of Newton’s Laws

183

184

185

Problem one frame to other

186

Equations of motion

187

Contd----

188

Problem

189

Euler Angles

190

Contd----

191

Contd-----

192

Euler Anglel Contd-----

193

Aircraft Moment of inertia

194

Illustration of heading angular rate Earth axis system

195

Illustration of pitch attitude angular rate

196

Illustration of role attitude angular rate

197

UNIT- IV

LINEARIZATION OF

EQUATIONS OF MOTION

AND AERODYNAMIC

FORCES AND MOMENTS

DERIVATIVES

198

199

Course Learning Outcomes

CLOs Course Learning Outcome

CLO13

CLO14

CLO15

CLO16

u/U1 Derivatives

200

u/U1 Derivatives contd----

201

Mach Tuck Derivatives

202

U/U1 Derivatives

203

U/U1 Derivatives contd---

204

Mach Tuck Derivatives

205

Contd------

206

Mach Tuck illustration

207

Mach tuck is an aerodynamic effect whereby the nose of an aircraft tends to pitch downward as the airflow around the wing reaches supersonic speeds. This diving tendency is also known as "tuck under". The aircraft will first experience this effect at significantly below Mach 1

u/U1 derivatives

208

Contd------

209

210

Longitudinal Static Stability derivatives

211

212

213

214

Illustration of derivative parameters

215

Contd----

216

217

Contd----

218

Contd------

219

Contd----

220

Numerical 4.1

221

Solution of 4.1

222

Numerical 4.2

223

Solution of 4.2

224

Numerical 4.3

225

Solution 4.3

226

Numerical 4.4

227

Solution of 4.4

228

Pitch rate derivatives

229

Illustration of pitch rate derivatives

230

Contd---

231

Contd-----

232

233

234

Numerical 4.5

235

Summary of derivatives of flight dynamics

236

237

238

Contd-----

239

Numerical 4.6

240

Solution

241

242

Effect of roll rates

243

Effects of roll rates contd-----

244

Roll helix angle

245

The absolute value of this change in angle of attack at the wing tips due to roll rate (pb/2U1) is called the roll helix angle. It provides the basis for the form of the nondimensionalization approach used for angular rates.

The roll helix angle has physical meaning as well. It can be thought of generally as the angle that the wing tip light would make with the horizon for an aircraft undergoing a roll rate

Side force on vertical tail

246

Vertical tail contribution on damping derivatives

247

248

Cross Derivative

249

Contribution of the wing in Cnp

250

The wing contributes to Cnp in three ways that will be addressed qualitatively. The first contribution comes from the

Increase in drag that results from the increase in angle of attack on the wing being rolled into, and

Decrease on drag that results from the decrease in angle of attack on the wing being rolled a

The increased drag on that results on the right wing and decreased drag that results on the left wing will provide a positive yawing moment to the aircraft, resulting in a positive contribution to Cnp.

the increase in angle of attack on the right wing results in tilting of the lift vector forward, while the decrease in angle of attack on the left wing provides an aft tilting of the lift vector.

The net result is a negative contribution to yawing moment; thus, a negative contribution to Cnp. See Fig below. way from

Lift vector tilting

251

Side force effects

252

253

Wing Contribution to Cross derivative

254

Yaw rate increases effective velocity on one

wing and decreases on opposite wing.

Nose right yaw rate, increases the effective

velocity on left wing and decreases on right

wing.

This increase the lift in left wing and decrease

lift in right wing.

Net result is positive rolling moment (right

wing down)

Thus wing make a positive contribution to Clr

Vertical tail or Fin Contribution to Cross derivative

255

Vertical tail contribution contd------

256

Yaw Damping derivative

257

The derivative Cnr is called the yaw damping derivative. It represents the change in yawing moment coefficient with respect to non dimensional yaw rate and will always be negative (providing a moment which opposes the direction of the yaw rate). Cnr is also an important factor in lateral-directional stability characteristics. The wing and vertical tail are the primary components that contribute to Cnr.

Wing contribution on Cnr

258

The wing contribution to Cnr results from the yaw rate.

Which increases the effective velocity on one wing and decreasing the effective velocity on the opposite wing.

A positive ‘‘nose right’’ yaw rate will provide an angular rate that increases the effective velocity on the left wing and decreases the effective velocity on the right wing.

The increase in velocity results in increased lift and induced drag on the left wing.

The decrease in velocity results in decreased lift and decreased induced drag on the right wing.

The net result is a negative yawing moment (nose left). Thus, the wings make a negative contribution to Cnr

Vertical Tail Contribution on Cnr

259

Numerical 4.7

260

Solution

261

262

Summary of Lateral and Directional

Forces and moments

263

Summary of Lateral and Directional Derivatives

264

Regular perturbation

265

UNIT-V

DYNAMIC SATABILITY

266

267

Course Learning Outcomes

CLOs Course Learning Outcome

CLO17

CLO18

CLO19

CLO20

Aircraft Dynamic Stability

268

Aircraft dynamic stability focuses on the time history of

aircraft motion after the aircraft is disturbed from an

equilibrium or trim condition.

This motion may be first order (exponential response) or second

order (oscillatory Response).

Will have either positive dynamic stability (aircraft returns to the

trim condition as time goes to infinity), neutral dynamic stability

(aircraft neither returns to trim nor diverges further from the

disturbed condition), or

Dynamic instability (aircraft diverges from the trim condition and

the disturbed condition as time goes to infinity).

The study of dynamic stability is important to understanding

aircraft handling qualities and the design features that make an

airplane fly well or not as well while performing specific mission

tasks.

Aircraft Dynamic Modes

269

There are basically two modes of aircraft dynamic motion

1. Longitudinal Modes

a) Phugoid (longer period) oscillation

b) Short period oscillations

2. Longitudinal modes

Oscillating motions can be described by two parameters, the period

of time required for one complete oscillation, and the time required

to damp to half-amplitude, or the time to double the amplitude , a

dynamically unstable motion. The longitudinal motion consists of

two distinct oscillations, a long-period oscillation called a Phugoid

mode and a short-period oscillation referred to as the short-period

mode

Contd--

270

Phugoid mode is the one in which there is a large-amplitude variation of air-speed, pitch angle, and altitude, but almost no angle-of-attack variation.

The phugoid oscillation is really a slow interchange of kinetic energy (velocity) and potential energy (height) about some equilibrium energy level.

The motion is so slow that the effects of inertia forces and damping forces are very low.

Although the damping is weak, the period is so long that the pilot usually corrects for this motion without being aware that the oscillation even exists.

Typically the period is 20–60 seconds. This oscillation can generally be controlled by the pilot.

Phugoid

271

Phugoid is an aircraft motion in which the vehicle pitches up and climbs, and then pitches down and descends, accompanied by speeding up and slowing down as it goes "downhill" and "uphill”

Incidents: 1975 USAF C5 flight control damaged-153 died,

1985 Japan airlines, 520 death. 2009, Airbus 320-214

landed Hudson river

Short period oscillations

272

The short-period mode is a usually heavily damped

oscillation with a period of only a few seconds.

The motion is a rapid pitching of the aircraft about the

center of gravity.

The period is so short that the speed does not have

time to change, so the oscillation is essentially an

angle-of-attack variation.

The time to damp the amplitude to one-half of its value

is usually on the order of 1 second.

Ability to quickly self damp when the stick is briefly

displaced is one of the many criteria for general

aircraft certification

Lateral-directional" modes

273

"Lateral-directional" modes involve rolling and yawing motions.

Motions always couples into the other so the modes are generally

discussed as the "Lateral-Directional modes“. There are three types

of possible lateral-directional dynamic motion- roll subsidence

mode, Spiral mode, and Dutch roll mode.

1. Roll subsidence mode

This is simply the damping of rolling motion. There is no direct

aerodynamic moment created tending to directly restore wings-level,

i.e. there is no returning "spring force/moment" proportional to roll

angle. However, there is a damping moment (proportional to

roll rate) created by the slewing-about of long wings. This prevents

large roll rates from building up when roll-control inputs are made or

it damps the roll rate(not the angle) to zero when there are no roll-

control inputs.

Roll mode can be improved by dihedral effects coming from design

characteristics, such as high wings, dihedral angles or sweep angles

Dutch roll mode

274

• The second lateral motion is an oscillatory combined roll and

yaw motion called Dutch roll.

• The Dutch roll may be described as a yaw and roll to the right,

followed by a recovery towards the equilibrium condition, then

an overshooting of this condition and a yaw and roll to the left,

then back past the equilibrium attitude, and so on.

• The period is usually on the order of 3–15 seconds.

• Damping is increased by large directional stability and small

dihedral and decreased by small directional stability and large

dihedral.

• Although usually stable in a normal aircraft, the motion may be

so slightly damped that the effect is very unpleasant and

undesirable..

Contd---

275

In swept-back wing aircraft, the Dutch roll is solved by

installing a yaw damper,

in effect a special-purpose automatic pilot that damps out

any yawing oscillation by applying rudder corrections.

Some swept-wing aircraft have an unstable Dutch roll.

If the Dutch roll is very lightly damped or unstable, the yaw

damper becomes a safety requirement, rather than a pilot

and passenger convenience.

Dual yaw dampers are required and a failed yaw damper is

cause for limiting flight to low altitudes, and possibly

lower Mach numbers, where the Dutch roll stability is improved

Dutch Roll

276

Spiral Divergence

277

Types of Divergence

278

Stable and Natural Ditch Roll

279

Negative Dutch Roll and Manual Dutch Roll

280

Characteristics of Spiral Divergence

281

Types of Divergence

282

Dutch roll further explaination

283

Reasons for Dutch Roll

284

Illustration of Dutch Roll

285

Mass Spring Damper System

286

The mass–spring–damper system as in Fig. 7.1, provides a

starting point for analysis of system dynamics and aircraft

dynamic stability. This is an excellent model to begin the

understanding of dynamic response. Sum of the forces is

given

----7.1

287

There are two forces acting on the mass, the damping force (Ff), and the spring Force (Fs). This resistance force Ff can be expressed as Ff= CV where C is slope. The spring force (Fs) is directly proportional to the displacement (x) of the mass and can be represented as Fs = Kx, where K is the spring constant. If the mass is displaced in the positive x direction, both the damping and spring forces act in a direction opposite to this displacement and can be represented by Ff + Fs = -CV – Kx ---------------7.2

Contd----

288

First Order System

289

In a spring–mass–damper system where the mass is very small or

negligible compared to the size of the spring and damper. We will

call such a system a mass less or first-order (referring to the order of

the highest derivative) system. The following differential equation

results when the mass is set equal to zero.

To solve this differential equation, we will first describe the

method of differential operators where P is defined as the

differential operator, d/dt, so that

----7.6

Contd----

290

We will first attack the homogeneous form (forcing function equal to zero) of Eq 7.6

Substituting in the differential operator, P, Eq. (7.7) becomes

----7.7

Contd----

291

Numerical 5.1

292

Second Order systems

293

Contd---

294

Damping Ratio and natural Frequency

295

In Eq. (7.12), two new parameters are there: damping ratio (ζ) and natural frequency (ωN ). These parameters have physical meaning for Case 3 and lead directly to the time solution for common inputs such as steps and impulses.

The damping ratio (ζ) provides an indication of the system

damping and will fall between -1 and 1 for Case 3. For

stable systems, the damping ratio will be between 0 and 1.

For this case, the higher the damping ratio, the more

damping is present in the system. which show the

influence of damping ratio. Notice that the number of

overshoots/undershoots varies inversely with the damping

ratio.

------7.16

Unit step responses for different damping ratios

296

Types of frequencies

297

The natural frequency is the frequency (in rad/s) that the system would oscillate at if there were no damping. It represents the highest frequency that the system is capable of, but it is not the frequency that the system actually oscillates at if damping is present. For the mass–spring–damper system,

Natural Frequency

Damped Frequency

The damped frequency (ωD) represents the frequency (in rad/s) that the system actually oscillates at with damping present. we can use the quadratic formula to solve for the roots of the homogeneous form of the equation

Contd---

298

Contd----

299

Numerical 5.2

300

Dynamic Stability Criteria

301

Routh’s Criteria

302

Presently, the roots of the stability quartic are obtained by the

iterative procedure described above or by using packages like

Matlab. However, earlier the tendency was to look for elegant

analytical / approximate solutions. Routh’s criteria is a method

which indicates whether a system is stable without solving the

characteristic equation. The criteria is presented without giving

the mathematical proof.

A quartic Aλ 4 + Bλ 3 + Cλ 2 + Dλ + E = 0 will have roots

indicating stability i.e. real roots negative and complex roots

with negative real part when A>0 and the functions T1,T2, T3

and T4, given below, are positive.

Contd---

303

304

In the case of longitudinal stability quartic with A = 1, the criteria simplify to: B > 0 ; D > 0; E > 0 and R = T3 = BCD - B 2E - AD2 > 0 (8.26) The term ‘R’ is called Routh’s discriminant. The reader can verify that for the stability quartic given by Eq.(8.18), the value of R is positive

Damping and rate of divergence when roots are real

305

As mentioned earlier, when a root is real and non-zero, a negative root indicates subsidence and a positive root indicates divergence. Larger the magnitude of the negative root, faster will the system return to the undisturbed position. This is clear from Eq.(8.15), which shows that the response of the system corresponding to the root λ1 is 11 1λ t e . At t = 0, the amplitude of the response is 11 . Further, when 1 λ is negative, the term 1λ t e indicates that the amplitude would decrease exponentially with time (Fig 8.1b). The time when the amplitude decreases to half of its value at t = 0, is a measure of the damping. This time is denoted by t1/2. This quantity (t1/2) is obtained from the following equation.

For the sake of generality the root is denoted by λ instead of 1 λ .

306

e = 2 Or 1/2 t = (ln2) / λ = 0.693 / λ --------------(8.27) When the root is positive, the amplitude increases exponentially with time (Fig 8.1a). The time when the amplitude is twice the value at t = 0, is a measure of divergence. This time is denoted by t2. This quantity (t2) is obtained from the following equation. 2 t λ e = 2; ; Note λ is positive Or t2 = (ln 2) / λ = 0.693 / λ --------(8.28)

Damping, rate of divergence, period of oscillation

307

A complex root is usually written as: λ = η ± iω When η is negative, the response is a damped oscillation. The damping is characterized by the time when the quantities ηte becomes half. This time is denoted by 1 2 t . Consequently , 1 2 η t e = 0.5 Or (ln 2) 1/2 t = / η = 0.693/ η When η is positive, the response is a divergent oscillation. The time when the term ηt e equals two is a measure of the rate of divergence. This time is denoted by t2. It is easy to show that : t = ln 2 / 2 η = 0.693/η The time period of the oscillation (P) is given by P = 2π / ω (8.31) When η is negative, the number of cycles from t = 0 to t1/2 is denoted by N1/2 and equals: N = t / P 1/2 ½ Similarly, when η is positive, the number of cycles from t = 0 to t2 is denoted by N2 and equals: N = t / P 2 2

Linearized EOMs in Laplace form

308

these three EOM have five aircraft motion variables (u, θ, α, w,

and q) and δe. We have three equations with 5 variables and

one forcing function. We need to reduce in three motion

variables i.e α ,u and θ. assumption of initial trimmed flight

with the wings level condition.

The EOMs are repeated here

Contd---

309

Therefore, our aircraft motion variables are reduced to α ,u and θ. These should be thought of as the outputs for our system of differential equations. With zero initial conditions, the Laplace transform

310

311

Each of these transfer functions can be represented as

the ratio of two polynomials in the Laplace variables

Contd-----

312

There is a separate transfer function for each of our three longitudinal

motion variables (α ,u and θ). Also, each of these transfer functions has the

same characteristic equation.

Characteristic equation determines the dynamic stability characteristics of the response, and therefore all three transfer functions will have the same dynamic characteristics

Each motion variable will have a different magnitude of response but with the same dynamic characteristics.

Phugoid as slow interchange of kinetic energy and potential energy

313

(a)As the pitch angle goes through a cycle (Fig. 8.4 b), while the angle of attack remains nearly constant implies that the altitude of the airplane also changes in a periodic manner (Fig 8.5). (b)The damping of the phugoid is very light and the flight speed changes periodically. (c) Items (a) and (b) suggest that the motion, during one cycle, can be considered as an exchange between potential energy and kinetic energy of the airplane.The total energy (i.e. sum of potential and kinetic energies) remains nearly constant during the cycle

Standard notation for ASC

314

Effect of Re in Lift and drag

315

Elevator and trim tab representation

316

Complex conjugate mode

317

Short period dynamics

318

319

The phugoid mode consists of poorly damped, long-period oscillations of the point-mass aircraft. We can derive an approximation for the phugoid mode along the lines of the derivation for the SP mode We 5 start by assuming that ∆α = ∆q = 0. Furthermore, we normalize the perturbation ∆V by V 0 to keep it non-dimensional. The phugoid dynamics are then given by

Roll Mode

320

Dutch roll mode

321

Illustration of dutch role

322

Spiral Mode

323

Illustration of Spiral roll

324

Lateral-Directional Modes at Higher Angles of Attack

325

Basic Elements of a Control System

326

An automatic control system is used on aircraft for several reasons: • Reduce pilot workload by automating routine manoeuvres such as wings-level flight and turns. • Improve the handling qualities by augmenting the eigenvalues of the flight dynamics. • Eliminate pilot-induced oscillations by filtering the pilot inputs appropriately. • Help the aircraft achieve the desired dynamical response, including for the purpose of creating control system test beds.

Types of Flight Control

327

Depending on the nature of the problem and the sensed variables, control problems can be classified as follows

• State versus output feedback problems: when z = g(x) =

x, all of the aircraft states are available to the flight

computer in real time, and it can use them for control. This

is called full state feedback. In contrast, when z ⊂ x, i.e.,

only some of the state variables are measured, the

resulting control problem is called an output feedback

problem. Output feedback problems are generally harder

to solve, and the flight computer will usually make for the

limited information by using algorithms which help it predict

the state variables that are not measured. • Regulation

versus tracking problems: a tracking problem is one where

the output y(t) is required to track a reference signal r(t).

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In such cases, it suffices to ensure that the remaining state variables do

not assume unreasonable values; their exact values are not important

subject to the qualification of boundedness within a reasonable range. On

the other hand, if the control objective is to ensure that y(t) tends to zero

asymptotically, then the control problem is referred to as a regulation

problem. Regulation problems are generally easier to solve than tracking

problems, although it is usually possible to convert a tracking problem

into a regulation problem by appropriate coordinate transformations.

Sensor and Actuator Limitations

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Consider the system ˙x = Ax + Bu. When we design a control law for such systems by writing u = Kx, we implicitly assume that the control actuator can physically produce any signal given to it. This is usually not true owing to some physical constraints of the actuator:

Fig. Hysteresis. Green: expected

cyclic path; Red: actual

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