Introduction

Electric circuits composed of discrete elements in which the voltages and currents are described in continuous time are sometimes called analog circuits, to distinguish them from circuits used in digital signal processors. We now model some analog circuits by writing ordinary differential equations in the state-variable form just as we modeled mechanical systems in Chapter 2. We apply the basic principles of circuit analysis, which are Kirchhoff's voltage and current laws, Ohm's law, Faraday's law, and the law relating the voltage across a capacitor to the current through it. We consider only circuits composed of the following elements: voltage sources, current sources, resistors, capacitors, inductors, and operational amplifiers. Our treatment here is limited to the type of circuits commonly encountered in the signal-processing parts of control systems, as opposed to those in power supplies or in other highpowered devices.

In Section 3.2 we confine our attention to passive circuits. These circuits contain no energy sources (such as operational amplifiers or batteries) except for the voltage or current sources that drive them. We can apply nearly all the basic circuit laws in this simple setting, and we can demonstrate an important principle of control engineering pertaining to interconnected elements, called the loading effect.

In Section 3.3 we model active circuits having operational amplifiers. This extends significantly our capability to design dynamic circuits for control system purposes. We also study simple instrumentation circuits for measuring dynamic variables.

Introduction

We now develop methods for calculating the dynamic response of a linear system that is modeled by its transfer functions. The response depends upon the physical parameters of the system, the input function, and any nonzero initial conditions. We are particularly interested in relating the dynamic features of the response to the physical properties of the system. If the input is a step function, the response may fluctuate temporarily and eventually reach a constant value. The nature of the fluctuations – the length of time during which they persist, and whether they cause the response to overshoot its final value excessively or to oscillate with both positive and negative values – are dynamic features of vital importance. The initial values of the response and its derivatives and the final value of the response are also important characteristics that depend on the physical parameters. In simple systems with simple inputs – for example, a firstor second-order system with a step input – the dynamic features of the response are directly related to simple combinations of the parameter values. But in higher-order systems these important relationships are less obvious because the significant dynamic features of the response depend on complicated combinations of the parameter values. We must now employ a mixture of analysis tools, computer calculations, and approximation techniques to determine which of our system parameters have the most influence on the significant dynamic features of the response.

The thrust F produced by the engines is of great importance in almost every phase of flight because it counteracts the drag and enables the aircraft to climb if required. The maximum available thrust Fm depends on the height and speed of the aircraft and is limited by the approved ‘rating’ for the appropriate phase of the flight. The three ratings that are important in relation to aircraft performance calculations are those specified for take-off, climb and cruise, and the rated thrust for each of these is the maximum available. In any phase of flight the thrust can of course be reduced by the pilot below the rated value, usually by moving a single control lever which is commonly known as a ‘throttle’ lever, even though it may act on a complex engine control system.

For almost all aspects of aircraft performance calculation it is necessary to know how Fm varies with the speed and height of the aircraft. In addition, for calculations of range, endurance and operating cost, a knowledge is required of the rate of consumption of fuel and the way in which this varies with flight speed, height and engine throttle setting. In this chapter the principles governing these variations will be discussed and approximate equations will be introduced for representing the variations in calculations of aircraft performance. For this purpose the rate of consumption of fuel will be expressed as the ratio of the rate of consumption to either the thrust or the shaft power of the engine.

The performance of an aircraft is essentially a statement of its capabilities and a different selection of these will normally be specified for the various categories such as transport, military and light aircraft, even though several common performance factors will feature in every such selection. For the engineer involved in the creation of a new design, these performance features serve as design criteria or at least desirable objectives, whereas late in the design and development stages the sales staff will quote the performance features as the basis for the commercial strength of the emerging aircraft. For either reason the performance will be stated in terms of quantities such as direct operating cost (DOC), maximum range for various payloads and fuel loads, cruising speed and airport requirements for landing and take-off. While the sales and design attitudes will be distinct, although related, this book addresses the early stages of the design process which must also bear considerable allegiance to performance as viewed by a potential customer.

The estimation of performance proceeds in stages, starting with parametric studies based on simple assumptions and progressing to more refined calculations as the main features of the design become established and the confidence in data grows. Estimation techniques are important not only because they allow the engineering team to proceed while data are crude or speculative, but also because construction of the new aeroplane will begin well in advance of the engineering refinements, and if there is accuracy in the early estimations this will be rewarded by a reduction in modifications as the fabrication and assembly effort progresses toward regular production.

In the design of a civil aircraft the condition of steady level cruise is of prime importance because improved fuel economy in this flight regime makes a direct and valuable contribution to the reduction of operating costs. Performance in the climb is often less important, but it cannot be ignored because a climb is always needed to reach the required cruising height after take-off and Air Traffic Control may also require the aircraft to change height during the cruise. For military aircraft, performance in the climb may be a primary design requirement because there is often a need to reach a specified height and speed in the shortest possible time, either from take-off or from some other prescribed initial conditions of height and speed.

The quantities that are of most interest in calculations of climbing performance are the rate of climb VC = V sin γ and the time required and fuel used in climbing from one specified height to another. In many cases there is a change of speed during the climb, so that the aircraft is accelerating, but it will be shown that a correction can easily be made for the effect of the acceleration on the rate of climb. The angle of climb is also of some interest, although it is important mainly at low altitudes where there may be obstacles to be cleared or where a large angle of climb may be required for reasons of noise abatement.

In earlier chapters, except in the discussions of the landing flare and the take-off transition immediately after lift-off, it has been assumed that the flight path is straight, so that there is no component of acceleration normal to the flight path. In this chapter flight in a curved path will be considered, concentrating on the usual form of banked turn as shown in Figure 8.1, in which the angle of bank is adjusted so that there is no sideslip and therefore no component of aerodynamic force normal to the plane of symmetry of the aircraft. In such a turn the required lift is greater than the weight, thus CL is greater than it would be in straight and level flight at the same speed and consequently the drag is also greater. This raises the requirement for thrust, even to maintain level flight, and thus the rate of climb obtainable with the maximum available thrust is reduced and may become negative. As the turn becomes tighter and the normal acceleration V2/R is increased, due to either a high speed or a small turn radius, or both, there will be increased demands for CL and for thrust to maintain height, with the consequence that limitations may be imposed by stalling or buffeting or by the engine rating.

This chapter addresses the interdependences among speed, rate of turn, rate of climb and additional ‘g-load’ on the pilot, as well as the limitations on one or other of these when some are fixed.

The estimation of the performance of an aircraft requires calculations of quantities such as rate of climb, maximum speed, distance travelled while burning a given mass of fuel and length of runway required for take-off or landing. The aim of this book is to explain the principles governing the relations between quantities of this kind and the properties of the aircraft and its power plant. Thus the emphasis is on the development of simple analytical expressions which depend only on the basic aircraft properties such as mass, lift and drag coefficients and engine thrust characteristics. Although extensive numerical data are required for the most accurate estimates of performance in the later stages of a design, the use of such data is not considered here and the data required for use in the simple expressions to be derived are of the kind that would be readily available at the preliminary design stage of an aircraft. Only fixed wing aircraft are considered and the measurement of performance in flight is not discussed.

One of the authors (WAM) has given for many years a short course of lectures on aircraft performance to engineering students at the University of Cambridge. Experience with these lectures has drawn attention to the shortcomings of existing books and to the need for a new book with the aim stated above. The book follows the same approach as the lectures, although it covers a greater range of topics and these are examined in much greater detail. Little previous knowledge of aircraft is assumed and the level of mathematics required should be well within the capabilities of engineering students, even in their first year.

The first chapter has given an introduction to the characteristics of atmospheric air and has provided a valid basis for the expression of the aerodynamic force developed during flight in that air, but there has not yet been any attempt to consider the balance of forces necessary to satisfy the laws of mechanics. Except for Chapters 6 and 8 and parts of Chapters 4 and 10, this book is directed mainly towards flight with zero or negligible acceleration so that the equations to be developed are those of statics, not dynamics.

Consideration of the effects of varying speed and altitude on the aerodynamic force on an aircraft can be greatly simplified by examining the dependence of the lift and drag coefficients on the Reynolds and Mach numbers. This dependence has already been mentioned briefly and is discussed further in this chapter, where it is shown that for a given aircraft the variations of Reynolds number caused by changing speed and altitude are likely to have only small effects. With increasing Mach number in the high subsonic range there is usually a large increase of drag coefficient and this important effect is introduced briefly, deferring a more detailed account until Chapter 10.

An important measure of the aerodynamic efficiency of an aircraft is the ratio L/D of lift to drag, since there is always a desire to create lift with as little cost in drag as possible. In this chapter the effects of this ratio (or its reciprocal D/L) on some important performance parameters are examined and it is shown that there is a minimum value of D/L which is especially important.

An aircraft with vectored thrust is defined here as one in which the pilot is able to vary the direction of the engine thrust over a wide range, usually at least 90°. The main advantage of this facility is that if the maximum available thrust is greater than the weight, the aircraft is able to take-off and land vertically, i.e. with zero ground run. If the thrust is large but still less than the take-off weight it may be possible to use thrust vectoring to give a substantial reduction in the distance required for take-off and in this case the landing weight may be less than the available thrust so that a vertical landing may be possible. Aircraft with vectored thrust are commonly known as V/STOL aircraft because they are capable of vertical or short take-off and landing.

V/STOL aircraft have been designed with several distinct configurations, the best known being that used in the Harrier as shown in Figure 9.1 and described by Fozard (1986), where the propulsive jets can be deflected downward by movable nozzles. Other forms have been reviewed by Poisson-Quinton (1968) and include the tilt rotor, where lifting rotors of the kind used in helicopters are tilted forward to operate like normal propellers in forward flight, and the tilt wing where the rotor axes remain fixed in the wing and the whole wing–rotor assembly is rotated relative to the fuselage. An example of a tilt-rotor aircraft is the Bell Textron Osprey shown in Figure 9.2.