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Ranjan Vepa
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To my teachers, Hans Wagner, Horst Leipholz, Holt Ashley, Art Bryson and Geoff Hancock
5
5.3
5.5
5.5.1
5.5.2
5.6
5.7
5.9
5.10
6.1
6.6.4
6.6.5
8.6 Application to the Design of Stability Augmentation Systems and Autopilots ....................................................................
8.6.1 Design of a Pitch Attitude Autopilot Using PID Feedback and the Root Locus Method .............................
8.6.2 Example of Pitch Attitude Autopilot Design for the Lockheed F104 by the Root Locus Method ........................
8.6.3 Example of Pitch Attitude Autopilot Design, Including a Stability Augmentation Inner Loop, by the Root Locus Method
8.6.4
8.6.5
8.6.7
8.6.8
8.6.9
8.7
8.8
8.11
8.9.1
8.11.4
8.11.5
List of Acronyms
AC Aerodynamic centre
ADF Automatic direction finding
amc Aerodynamic mean chord
AR Aspect ratio
BDF Backward difference formula
CG Centre of gravity
CH Cooper–Harper (rating)
CM Centre of mass
CP Centre of pressure
DME Distance measuring equipment
EFIS Electronic flight information system
EIS Electronic information system
EPR Engine pressure ratio
FBW Fly by wire
FCU Flight control unit
FDAU Flight data acquisition unit
FMGS Flight management and guidance system
GPS Global positioning system
HSI Horizontal situation indicator
HUD Head-up displays
IAS Indicated airspeed
IFS In-flight simulation
ILS Instrument landing system
INS Inertial navigation system
NDF Numerical differentiation formula
NP Neutral point
PD Proportion derivative
PID Proportional, integral, derivative
PIO Pilot-induced oscillation
psfc Power-specific fuel consumption
RMI Radio magnetic indicator
SISO Single input, singe output
TCAS Traffic collision avoidance system
TR Trapezoidal rule
Tsfc Thrust-specific fuel consumption
VHF Very high frequency
VOR VHF omni-range or vestibulo-ocular reflex
Preface
In the last decade, we have seen a phenomenal increase in air travel to phenomenal levels. A plethora of low-cost airlines have made it possible for the common man to travel between continents at relatively reasonable fares. This has also led to the design of newer energy-efficient aircraft incorporating the principles of feedback control. These aircraft have generally tended to be lighter and more flexible because of the use of composite structures and other smart materials. It therefore becomes important to consider the aircraft not as a rigid body, as has been done traditionally in the past, but as an inherently flexible body. Such considerations will require a revision of a number of traditional concepts, although many of them can be easily adapted to the flexible aircraft.
This book addresses the core issues involved in the dynamic modelling, simulation and control of a selection of aircraft. The principles of modelling and control could be applied both to traditional rigid aircraft as well as more modern flexible aircraft. A primary feature of this book is that it brings together a range of diverse topics relevant to the understanding of flight dynamics, its regulation and control and the design of flight control systems and flight simulators.
This book will help the reader understand the methods of modelling both rigid and flexible aircraft for controller design application as well as gain a basic understanding of the processes involved in the design of control systems and regulators. It will also serve as a useful guide to study the simulation of flight dynamics for implementing monitoring systems based on the estimation of internal system variables from measurements of observable system variables.
The book brings together diverse topics in flight mechanics, aeroelasticity and automatic controls. It would be useful to designers of hybrid flight control systems that involve advanced composite structure–based components in the wings, fuselage and control surfaces. The distinctive feature of this book is that it introduces case studies of practical control laws for several modern aircraft and deals with the use of non-linear model-based techniques and their applications to flight control.
Chapter 1 begins with an introduction and reviews the configuration of a typical aircraft and its components. Chapter 2 deals with the basic principles governing aerodynamic flows. Chapter 3 covers the mechanics of equilibrium flight and describes static equilibrium, trimmed steady level flight, the analysis of the static stability of an aircraft, static margins stick-fixed and stick-free, modelling of control-surface hinge moments and the estimation of the elevator angle for trim. Basic concepts of stability based on disturbances to one parameter alone are discussed. The effects of a change in the angle of
attack on the pitching moment and its application to stability assessment are discussed. Also considered are steady flight at an angle to the horizontal and the definition of flight path, incidence and pitch angles and the heading, yaw and sideslip angles. The assessment of manoeuvrability and the application of margins required for a steady pull-out from a dive are also introduced.
Chapter 4 is dedicated to the development of the non-linear equations of motion of an aircraft, including simple two-dimensional dynamic models, and the development of the aircraft’s equations of motion in three dimensions. The general Euler equations of rigid body and the definition and estimation of moments of inertia matrix are discussed. The definitions of motion-induced aerodynamic forces and moments and the need for various reference axes that are fixed in space, fixed to the body and fixed in the wind as well as the definition of stability axes are clearly explained. The non-linear dynamics of aircraft motion in the stability axes is derived both in terms of body axis degrees of freedom and wind axis variables. The concept of non-linear reduced order modelling is introduced, and the short period approximation is discussed. Finally, the trimmed equations of motion as well as the non-linear perturbation equations of motion are derived. The concept of linearisation is also introduced, and the linear equations of aircraft motion are briefly discussed. In Chapter 5, the small perturbation equations of motion are described in detail, and the equations are expressed as two sets of decoupled equations representing the longitudinal and lateral dynamics. Chapter 6 introduces the methodology of linear stability analysis and provides a modal description of aircraft dynamics. The application of small perturbation equations in determining the control setting angles for executing typical manoeuvres is also discussed in this chapter.
Chapter 7 covers the evaluation of aircraft dynamic response and the application of MATLAB®/Simulink® in determining the aircraft’s response to typical control inputs. A basic introduction to aircraft non-linear dynamic phenomenon is also presented in this chapter. Chapter 8 deals with aircraft flight control, the design of control laws, stability augmentation, autopilots and the optimal design of feedback controllers. Chapter 9 describes flight simulators and the principles governing their design. Finally, Chapter 10 is dedicated to the flight dynamics of elastic aircraft, including the principles of aeroelasticity from an aircraft perspective.
I thank my colleagues and present and former students at the School of Engineering and Material Science, Queen Mary University of London, for their support in this endeavour.
I thank my wife Sudha for her love, understanding and patience. Her encouragement and support provided me the motivation to complete this project. I also thank our children Lullu, Satvi and Abhinav for their understanding during the course of this project.
Ranjan Vepa London, United Kingdom
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Author
Dr. Ranjan Vepa earned his PhD in applied mechanics from Stanford University, Stanford, California, specialising in the area of aeroelasticity under the guidance of the late Prof. Holt Ashley. He currently serves as a lecturer in the School of Engineering and Material Science, Queen Mary University of London, where he has also been the programme director of the Avionics Programme since 2001. Prior to joining Queen Mary, he was with the NASA Langley Research Center, where he was awarded a National Research Council Fellowship and conducted research in the area of unsteady aerodynamic modelling for active control applications. Subsequently, he was with the Structures Division of the National Aeronautical Laboratory, Bengaluru, India, and the Indian Institute of Technology, Chennai, India.
Dr. Vepa’s research interests include the design of flight control systems and the aerodynamics of morphing wings and bodies with applications in smart structures, robotics and biomedical engineering and energy systems, including wind turbines. He is particularly interested in the dynamics and in the robust adaptive estimation and the control of linear and non-linear aerospace, energy and biological systems with uncertainties. The research in the area of the aerodynamics of morphing wings and bodies is dedicated to the study of aerodynamics and its control. This includes the use of smart structures and their applications to the control of aerospace vehicles, jet engines, robotics and biomedical systems. Other applications of this work are to wind turbine and compressor control, maximum power point tracking, flow control over smart flaps and the control of biodynamic systems. Dr. Vepa currently conducts research on biomimetic morphing and aerodynamic shape control and their applications, which include feedback control of aerofoil section shape in subsonic and transonic flow for unmanned aerial vehicles (UAV), airship and turbomachine applications and integration of computational aeroelasticity (CFD, computational fluid dynamics/ CSD, computational structural dynamics) with deforming grids as well as their applications to active flow control. Of particular interest are the boundary layer instabilities in laminar flow arising due to various morphinginduced disturbances. Dr. Vepa has also been studying the optimal use and regulation of alternate power sources such as fuel cells in hybrid electric vehicle power trains, modelling of fuel cell degradation and health monitoring of aircraft structures and systems. With regard to structural health monitoring and control, observer and Kalman filter–based crack detection filters are being designed and applied to crack detection and isolation in aeroelastic aircraft structures such as nacelles, casings, turbine rotors and rotor blades. Feedback control of crack propagation and compliance compensation in cracked vibrating structures is also being investigated. Another issue is the
modelling of damage in laminated composite plates, non-linear flutter analysis of their plates and their interaction with unsteady aerodynamics. These research studies are contributing to the holistic design of vision-guided autonomous UAVs, which are expected to be extensively used in the future.
Dr. Vepa is the author of three books: Biomimetic Robotics (Cambridge University Press, 2009), Dynamics of Smart Structures (Wiley, 2010) and Dynamic Modeling, Simulation and Control of Energy Generation (Springer, 2013). He is a member of the Royal Aeronautical Society, London; the Institution of Electrical and Electronic Engineers (IEEE), New York; a fellow of the Higher Education Academy; a member of the Royal Institute of Navigation, London; and a chartered engineer.
1 Introduction to Flight Vehicles
1.1 Introduction
While aerodynamics is the study of flows past and over bodies, the principles of flight are governed by the dynamics and aerodynamics of flight vehicles. The focus of this chapter is on the general principles of flight and the primary features of aircraft. Further details may be found in Anderson [1] and Shevell [2]. As the aerodynamics of bodies is greatly influenced by their external geometry, the aerodynamics of flight vehicles is entirely determined by their external geometry. The external geometry is in turn completely influenced by the entire complement of components external to the vehicle. The basic architecture of a typical aeroplane, the simplest of flight vehicles, is well known to any cursory observer of aeroplanes. It can be considered to be the assemblage of a number of individual components. The principal external components are the fuselage, the left and right wings, the power plant pods or nacelles, the tail plane unit comprising of the horizontal and vertical stabilisers, the various control flaps and control surfaces and the landing gear. When the components are assembled or integrated together, a complete external picture of a typical aeroplane emerges. A typical planform or top-down view of an aeroplane is shown in Figure 1.1.
1.2 Components of an Aeroplane
The primary components of an aeroplane are the fuselage, the wing, the tail surfaces which are collectively referred to as the empennage, the power plant, the various control surfaces used to control the flight of the aeroplane and the landing gear.
1.2.1 Fuselage
The fuselage is the main body of any aeroplane, housing the crew and passengers or the cargo or payload and the like.
1.1
Typical planform view of an aeroplane.
1.2.2 Wings
The wings are the main lifting element of the aeroplane. They comprise of the wing leading and trailing edges, flaps and slats that are used to augment the lift on the wing, ailerons to enable the aeroplane to bank while turning and spoilers that are capable of reducing the wing lift during landing and act as speed brakes. The high-lift devices controlled and operated below the wing permit the wing to develop the necessary lift during take-off when a large passenger jet attains speeds of the order of 320 km/h after accelerating down a runway of length 3–4 km. The controls and drive mechanisms linking these devices are usually shrouded in canoe-shaped fairings attached to the underside of the wing. The wing essentially carries the entire aeroplane and all other associated systems. The wing is essentially a single aerodynamic element although it extends symmetrically on either side of the fuselage.
1.2.3 Tail Surfaces or Empennage
The tail surfaces are the basic elements that stabilise and control the aeroplane. Normally, both the vertical and horizontal tail surfaces have a fixed forward portion and a hinged rearward portion. The forward portion of the horizontal tail surface is known as the stabiliser, while the rearward hinged portion on the same surface is known as the elevator. On many long-haul airliners, the horizontal stabiliser is an all movable unit. On the vertical tail, the fixed forward portion is known as the fin, while the hinged rearward portion is known as the rudder. Both on the rudder and on the elevator are additional hinged surfaces known as the trim tabs which are used to adjust the forces on
FIGURE
the pilot’s control column (which controls the movement of the elevator) and rudder pedals so that these are force free. Together, the entire horizontal and vertical tail surface assembly is known as the empennage.
1.2.4 Landing Gear
To enable an aeroplane to operate from land, aeroplanes are provided with landing gear comprising of wheels with types mounted on axles. Brakes are integral elements while the axles are attached via supporting struts and shock absorbers to the fuselage. To minimise drag during take-off and in steady flight, cowlings and retractable mechanisms are provided. The latter permit the retraction of the entire landing gear to an enclosed housing within the fuselage once the aeroplane is airborne.
1.3 Basic Principles of Flight
1.3.1 Forces Acting on an
Aeroplane
Consider the equilibrium of an aeroplane on the ground. Its weight may be regarded as acting vertically downwards through the aeroplane’s centre of gravity (CG) and this is balanced by two sets of reactions acting vertically upwards, one at the points of contact of the main undercarriage and the ground surface and the other either at the nose wheel or tail skid depending on the type of aeroplane. To maintain an aeroplane in vertical equilibrium during flight, the vertical reactions at the main undercarriage and nose wheels must be replaced by equivalent upward forces: the lift components acting on the main wing and tail plane surface. In the days of the lighter than air balloons, which were axially symmetric about the CG axis, the reaction was a single lift force due to the buoyancy. This force was due to the difference in the weight of the air displaced by the balloon and the gas contained within and acted in the vicinity of the CG. However, with the arrival of the airship, the forces were no longer acting in a single vertical line. Typically, a steady level flight is held in balance or equilibrium by a combination of forces (Figure 1.2a). The forces comprise
1. The lift on the aeroplane with the principal contributions being due to the wing and horizontal tail
2. The drag which consists of two main components the profile drag and the induced drag
3. The thrust produced by the power plants
4. The weight of the aeroplane
FIGURE 1.2
(a) Forces acting on aeroplane in steady, level, equilibrium flight and (b) pressure distribution on a wing: front and side view of a typical wing section.
In addition to the equilibrium of forces, the forces on the tail plane contribute principally towards rotational moments acting on the aeroplane. All the rotational moments acting on the aeroplane must cancel each other to ensure that the aeroplane is in rotational equilibrium. Rotational equilibrium is essential so the aeroplane can maintain steady orientation during a long and sustained flight. Thus, the attitude of the aeroplane must remain steady during extended periods of flight.
The principal phenomenon that is responsible for holding the aeroplane in flight is the wing lift which is caused as a result of the generation of a lowpressure or suction region over the top surface of the wing and high-pressure region below the lower surface of the wing (Figure 1.2b). The region of low pressure on the top surface of the wing is caused by the flow of air over the curved surface of the wing with a resultant increase in flow velocity and consequent decrease in pressure relative to the rest of the atmosphere. Similarly, the region of high pressure below the lower surface of the wing represents a region where the pressure is relatively greater than in the surrounding air. The result of these two complementary effects on the two surfaces of the
Relative wind Wing lift, LW
Induced drag, Di Pro le drag, Dp Tail lift, LT
rust, T
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CELANDINE: Cheledonium majus.
FUMITORY FAMILY.
FUMARIACEÆ.
Dutchman’s Breeches. DicentraCucullaria.
Found in rich, moist woodlands in April.
The flower-stems and leaves (from 5 to 8 inches in height) spring from the root.
The large feathery leaf is many times divided and sub-divided in groups of threes, the margin being entire. In texture it is thin and delicate, the surface being noticeably cool to the touch; in color, gray-green, bluish underneath. It is borne on a pale, juicy stem, which is tinged with pink or reddish at the foot. Several leaves spring up enclosed by 2 or 3 dry, reddish sheaths.
The petals of this curious flower are 4 in number, one pair being joined together to form a 2-spurred heart-shaped bag, with its spurs spread widely apart; the other 2 petals held within the narrow mouth of the bag are very small, and join their tips over the slightly protruding stamens; the texture is very thin and delicate and slightly ribbed; the color of the petals is a pure white, the spurs being tipped with pure yellow. The 2 divisions of the small calyx hug the bag betwixt the spurs,—it is white, a trifle tinted with green. The footstem on which the bag hangs is small and pale; the flowers are set in a nodding row upon the long curving or upright stem, which is pale or tinted red.
In New England this is a less common variety than its sister Squirrel Corn, D. Canadensis, which is very like, though smaller and pink-tinted instead of yellow.
DUTCHMAN’S BREECHES: Dicentra Cucullaria.
Pale Corydalis. Corydalisglauca.
Found in woodsy, rocky places during June and July.
The erect and leafy stalk grows from 6 to 15 inches in height, and is slender and smooth, with a slight bloom, which makes its color a pale or whitish-green.
The pretty leaf is not large; it is compound, its 3 leaflets being sub-divided and cut into deep scallops on the entire edges; the ribs are very delicate, the texture thin and fine and smooth; in color, a cool green, the underside whitened by a pale bloom. The lower leaves are on slender, smooth stems, and the upper clasp the stalk; they are placed alternately.
The flower is small, fragile and smooth in texture, and hung upside down; the corolla, like a one-sided flat bag with a round bottom and a 2-lipped mouth, is pale pink, the lips being golden; the 2-parted calyx is pinkish. Three or four flowers hang in terminal groups on slender reddish stems.
The seed-pod is out of all expectation long, frequently over an inch in length, but very slender. The whole plant has a smooth daintiness, and especially is the leaf pretty.
PALE CORYDALIS: Corydalis glauca.
MUSTARD FAMILY.
CRUCIFERÆ.
Toothwort. Dentarialaciniata.
Crinkle-root.
Pepper-root.
Found in flower, on the edge of thickets, in May.
A single stalk grows to the height of 6 or 7 inches; it is smooth, round, and juicy; light green.
The compound leaf is very deeply 3 times cut into long, narrow parts, with sharply notched edges; the texture is common (not to say coarse), and the color a full juicy green. Two or three leaves, on short foot-stems, grow in a whorl-like cluster about the stalk, a little below the flower-cluster.
The flower has 4 rounded petals spreading at the top; in texture rather thick, in color white, faintly tinged with violet; the 4-parted calyx is pale green, and the 6 stamens, 2 noticeably shorter than the others, are a dull greenish-yellow.
With the true vigor of its family this plant grows in small communities. It stays in bud a long time before the blossoms finally open. A number of small tubers are strung together on the roots, like beads on a necklace; pungent and peppery to the taste.
TOOTHWORT: Dentaria laciniata.
Herb of St. Barbara. Barbareavulgaris.
Yellow Rocket.
Winter Cress.
Found in sunny places, by clear water and in moist meadows during May.
The single stalk, from 12 to 16 inches in height, is branching only for the flowers; it is large, fibrous, and strong, grooved, but very smooth, and of a shining, pure, green color.
The lower leaves are 3 or 4 inches long, lyre-shaped, and cut nearly to the midrib into 5 or 7 irregular lobes, the middle lobe being very round; the upper leaves are cut less deeply, and are small; the margins are entire, the texture strong, and the lower surface rough, the upper being smooth and shining; in color, dark full green. The lowest leaves are on clasping stems, all the rest clasp the stalk with a pair of wings, alternately.
The 4 petals of the small flower are rounding, and arranged in pairs within the small 4-parted green calyx; they are of a charming light yellow color, and so are the 6 stamens. These stamens group themselves oddly about the central pistil,—two pairs stand in front of the pairs of petals, and the single shorter stamens fill the more open spaces left in the opposite angles. The flowers form loose terminal groups.
The stalk springs from a foot-rosette of the rich green leaves, and is one of the earliest risers of the spring, in its chosen home—a marshy meadow.
YELLOW ROCKET: Barbarea vulgaris.
Field Mustard. BrassicaSinapistrum.
Charlock.
Crowd Weed (W. Va.).
Found in grain fields, and along cultivated lands, from July to September.
The large and branching stalk is zigzag or curved in habit of growth, about 2 or 3 feet high, and ribbed; with occasional hairs; it is bright green in color.
The lower leaves are lyre-shaped and large, the middle lobe wide and curving to a long pointed tip, the side lobes being narrow; the upper leaves are irregularly cut; the margins are notched (the points of the notches often turn toward the stem), the edge curling or wavy; the midribs and netted veins often pucker the surface, which is shining; the color is a clear full green. They are set on short stems, or clasp the stalk, and are alternately arranged.
The flower has 4 shell-shaped petals, with very long and erect bases, of a fine texture, and yellow color with a tinge of green; the calyx is 4-divided, its parts slender and wide-spreading, also yellow but with a marked tendency to green. The flowers form close leafy terminal clusters.
As the petals fall early and only two or three flowers are open at once, the seed-pods, green and shining, form a distinctive feature of the plant; when they ripen, and in their turn drop, their little footstems are left bristling along the elongating branch. The leaf suggests Gothic ornament with its quaint curves and lines.
