PERFORMANCE analysis of an airplane consists

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1 WESTERN JOINT COMPUTER CONFERENCE is presented in Fig. 7, shoing the forces hich act upon it, as ell as a numerical verification that the equilibrium conditions are satisfied. It is perhaps superfluous to add that the final test of results obtained ith any such mathematical model must be their reasonableness, based upon experience ith the behavior of similar structures and upon experimental verification henever this is possible. CONCLUSIONS A general digital-computer program for static stress analysis has been described. This program is applicable to a large class of structures ithout additional programming time, and relieves the stress analyst of most of the computational burden. After the analyst has idealized the structure, and has made it statically determinate, his task consists of a straightforard tabulation of elastic and geometric properties. Once a unit load solution has been obtained, multiple load conditions can be investigated at little additional cost, and the effect of varied elastic properties can be studied by rerepeating only the solution of the continuity equations. Extensive use of magnetic tapes makes possible the large-order matrix operations characteristic of the stress analysis problem. The program is designed throughout for economy of computation. Aircraft Performance Studied on an Electronic Analog Computer L. B. WADEL AND C. C. WANt INTRODUCTION PERFORMANCE analysis of an airplane consists of the study of various operating characteristics and operating limits. These characteristics may be divided essentially into three major groups as follos: 1. Performance at Constant A ltitudes. These include the conventional performance characteristics such as maximum speed, rate of climb, and equilibrium turn performance. All of these items may be determined by comparison of "thrust available" and "thrust required" for steady-state operations, based on constant airplane eight and constant altitude. 2. Performance at Varying Altitudes. For certain portions of a typical flight program, fuel economy and elapsed time become factors' of major importance. It is necessary to establish a theoretical optimum operating schedule hich ould serve as an upper bound to the attainable performance. Optimum-climb programs for an Interceptor and optimum-cruise profiles for a long range bomber belong to this category. Although these problems may be rigorously resolved by means of the calculus of variations, equivalent methods requiring only steady-state considerations can be formulated to yield the required solutions. 3. Range Performance. The maximum range, or radius of action, of an airplane for a specific mission profile is one of the most important items in performance analysis. Here, the variation of operating altitudes and airplane eight must be properly accounted for. Generally, this can be achieved through appropriate combination of results obtained from the first to groups. t Chance Vought Aircraft, Inc., Dallas, Tex. During the preliminary design of a prototype aircraft, a fairly exhaustive study of range performance is usually carried out for a range of design parameters. The proper selection of compromises among these design parameters is made on the basis of the results obtained from such a study. Similar performance calculations ould be made also for guided missiles, but different properties ill be emphasized. For production aircraft, performance data must be presented in the pilot's handbook. A ide range of variations in atmospheric conditions and mission types is usually included in this presentation. It may readily be realized that a large amount of computational effort is necessary to provide a complete assessment of the pertinent performance characteristics of an aircraft. It is, therefore, desirable to mechanize performance calculations as much as possible. Several exploratory analog computer studies have been made, ith promising results, of certain performance characteristics of a high speed fighter aircraft. The method of approach and the nature of the results obtained during these studies are described in this paper. THE ELECTRONIC ANALOG COMPUTER The general-purpose electronic analog computer (electronic differential analyzer) as e kno it today as developed in the years immediately folloing World War II; its original mission as to solve the ordinary differential equations hich describe the dynamic behavior of guided missiles. The key to such computers is the operational or computing amplifier. This is a highgain dc amplifier hich, hen supplied ith proper precision input and feedback impedances (usually resistors and capacitors), performs summation, integration, signchanging, or other more specialized linear operations.

2 Wadel and Wan: Aircraft Performance Studied on an Electronic A nalog Computer 79 Other computer components are potentiometers, used to establish constant coefficients and additive constants, and such nonlinear devices as multipliers and arbitrary function generators. For a given problem, these basic "building blocks" are interconnected such that the mathematical equations of the electrical behavior of the circuits so established are identical ith the equations corresponding to the aerodynamic, mechanical, chemical, pneumatic, hydraulic, economic, biological, electrical, or electronic system or process that is under investigation. At any time the output of a given unit in volts represents the value of a variable, or a function of a variable or combination of variables in the system being studied. Direct-recording oscillographs provide time-histories of selected outputs. For most problems solved on electronic analog computers, nonlinearities, although often important, are fe in number. Attention is focused upon relatively small dynamic oscillations about a predetermined set of equilibrium conditions. The order of the system of differential equations being solved determines the number of integrators. For example, the linearized equations for the control-fixed longitudinal motion of an aircraft require four integrators. The number of summers and signchangers, as ell as potentiometers, depends upon the number and signs of the terms in the equations. Perhaps 3 per cent of the amplifiers employed ill be in tegra tors. As noted earlier in the paper, e are concerned here ith the solution of problems of special concern to the aircraft designer. Obviously, e must kno hat equilibrium flight conditions are possible or desirable for an aircraft or missile before it makes sense to discuss dynamic performance about a certain flight condition. As far as the computer is concerned, hat is special about performance equations? Qualitatively,,nothing. The same type of computer components are required, but the emphasis is drastically shifted. The percentage of integrators required may drop to 5 or less, but multipliers and function generators are required in profusion. The many nonlinearities in the equations assume fundamental importance, and their accurate representation in the computer becomes the prime difficulty in solving the equations. INPUT DATA Basic data required for performance analysis may be grouped into four general categories as follos: 1. Aerodynamic Data. These include the lift and drag characteristics. The drag coefficients, being directly related to thrust required, require careful simulation. These coefficients are functions of both Mach number and the lift coefficient. 2. Poer Plant Data. These include the installed thrust and corrected fuel rate for a specific poer plant installation. These coefficients are concerned ith available thrust and fuel rate hich are functions of altitude, Mach number and poer setting. A very elaborate simulation setup is usually required for these coefficients. 3. Atmospheric Data. These include the air density and the speed of sound; they are functions of altitude only. 4. Geometrical and Weight Data. These are essentially constant factors for a particular design. Hoever, for problems involving the consideration of the entire flight profile, the fuel rate and other step variations due to jettisoning of bombs or external stores must be taken into account. The general approach to the simulation problem is to establish a polynomial approximation to the basic data in terms of the most significant independent variables. By this means, functions of more than one variable may be simulated by a combination of function generators and multiplying servomechanisms. For example, the drag coefficien t CD of an airplane may be expressed in terms of the lift coefficient C L as CD = do + d 2CL 2 + d4cl 4, here do, d 2 and d 4 are all functions of Mach number and may be stored on such function devices as photoformers, diode function generators, or nonlinear potentiometers. CONSTANT ALTITUDE PERFORMANCE Performance characteristics of an airplane at constant altitude is related to the difference beteen the available thrust (T) and the total airplane drag (D). It is, therefore, necessary to generate by appropriate function devices these to quantities for various operating conditions. The range of simulation, of course, should be sufficiently extensive to include all conceivable altitudes and speeds. The available thrust is a function of both altitude and speed for a specific airplane-poer plant combination. The airplane drag maybe expressed in terms of the airplane drag coefficient (CD) as here q* is the dynamic pressure at sonic speed and a function of altitude, S is the ing area, M is the Mach number and CD is further related to the airplane lift coefficien t (C L) by The airplane lift coefficient is related to the airplane eight (W) and the airplane load factor (n) by the equation For constant altitude performance calculations, the airplane eight is usually treated as a constant. 1. Equilibrium Turn Performance. During equilibrium turns, there is to be no loss in speed and altitude. The controlling equation is therefore (1) (2) (3) T - D = o. (4)

3 WESTERN JOINT COMPUTER CONFERENCE The airplane load factor (n) corresponding to this condition is of considerable interest for fighter type aircraft, as it provides an index to the maneuverability of the aircraft. Furthermore, the maximum and the minimum speeds for level flight may be defined by means of (4) by setting the airplane load factor equal to one. A feedback circuit may be used for equilibrium turn computation by treating the quantity (T-D) as an error signal input to the C L channel. An integrator may be used to provide a variable Mach number input so that the final turning performance curve may be recorded directly on a plotter. Typical results of this calcula tion are displayed in Fig. 1. c:: 6 t- o ~4 « MACH NUMBER Fig. i-equilibrium turn performance of a typical fighter airplane. 2. Acceleration Characteristics. The time required to reach a given speed and the variation of speed during a turning maneuver are required in the study of combat situations. Depending on the magnitude of the airplane load factor during the turn, the quantity (T-D) may be either positive or negative. The equilibrium turn discussed above defines a limiting case here this quantity (T-D) is zero. When the airplane load factor is higher than the equilibrium turn load factor, at the same Mach number, the airplane ould decelerate. Acceleration is only possible hen the airplane load factor is loer than the equilibrium turn load factor. For operations at constant altitudes, it is convenient to study only turn at constant load factors. The basic equation for this case is (Wa/g)M = T - D, (5) here a is the speed of sound at the given altitude and g is the gravitational acceleration'. The constant load factor under consideration may be used as a limiter for the generation of the CL signal. The time history may be obtained through integration of (5). Typical results of this problem are displayed in Fig Rate oj Climb. The classical treatment of rate of climb is based on constant speed operations at fixed poer setting. Under these conditions the rate of climb (it) maybe expressed as it = (T - D)Ma/W, (6) hen a is the speed of sound and a function of altitude. The airplane drag term in (6) must be evaluated for a lift coefficient (C L) defined by CL = CLJW - T(ao + e)]/[t + q*sm2cla J. (7) In (7), CLa is the slope of the trimmed lift coefficient curve ith respect to the angle of attack, CXo is the angle of the zero lift line and e is the thrust angle. Both C La and CXo are usually functions of Mach number hile the thrust angle is a constant for a specific airplane. 14 c:r: en ~ 12 z J: o «1 ~ = LOULA rred V~LUES V n=15 / -- / V ~ } /' n-2 f/ n'", -,.!!!J...8 o 1 2 ~ 4 TIME IN SECONDS Fig. 2-Acceleration characteristics of a typical fighter airplane. The rate of climb may be obtained by direct computation by means of only function devices and appropriate circuitry. An integrator may be included to provide a variable Mach number input and the final rate of climb at constant altitude may be recorded directly on a plotter. The best-climb speed and the best-climb Mach number may be obtained from this plot. Typical examples of rate of climb calculations are displayed in Fig frl ~18 t-= 1L ~ 16 ::i u 1L 14 LLJ ~ 12 :: 1 ~ MACH NUMBER Fig. 3-Rate of climb characteristics of a typical fighter airplane. PERFORMANCE AT VARYING ALTITUDES In addition to the constant altitude performance discussed above, it is frequently desirable to establish optimum operating schedules for various portions of a flight plan. Typical among these are the optimum climb profile and the optimum cruise profile. The optimization in these cases may be made on the basis of either fuel economy or elapsed time. Although these

4 Wadel and Wan: Aircraft Performance Studied on an Electronic Analog Computer 81 problems could be formulated rigorously by means of the calculus of variations, solutions are not alays attainable due to mathematical complexities. Equivalent methods requiring only steady-state considerations are available, hereby the solutions may be obtained by graphical means. Certain parametric curves must be constructed, hoever, before proceeding ith the graphical solution. The types of parametric curves required are listed in Table I for both optimum climb and optimum cruise calculations. A discussion of the significance of these curves may be found,l but is not included here for the sake of brevity. Additional function devices are required to generate the fuel rate input (Wf) for these calculations. Optimum Climb Optimum Cruise TABLE I PARAMETRIC CURVES FOR PERFORMANCE CALCULATIONS Minimum Time Minimum Fuel h=(t-d)ma/w = constant k/j = constant h-mplane h-mplane Maximum h=constant f-m plane Time Minimum h=constant Ma/f-M plane Fuel '. For optimum climb calculations, the pertinent factors may be maintained at a constant value by using the deviation from the constant as a feedback to the altitude channel. For optimum cruise calculations, direct computation is possible and no special technique is required. RANGE PERFORMANCE The calculation of maximum range, or radius of action, of an aircraft requires the consideration of the complete profile necessary for the accomplishment of a specific mission. Fig. 4 shos schematically the various hile others may be typical of the particular aircraft, regardless of mission. An initial guide to the other phases can be obtained from the results of previous performance calculations. Because of the complexity of the equations and the number of phases of flight, radius of action may be computed most easily by an iterative procedure. This method may be illustrated by the folloing much-simplified example: Consider a flight to consist only of a straight-line cruise-to-target phase plus an immediate return-cruise phase, all at a constant speed and altitude (no ind), ith an instantaneous discharge of a eight W over the target. The folloing equations then apply: (for a typical airplane) W = 1 5 C L lb. T = 1 5 C D = 1 5 ( C.&2) lb. TV = - Wf = T lb/hr. x = Vt. Given: W(o) =3, lb. W = bomb eight = 2, lb. Wee) = minimum landing eight=2, lb. V = speed = 5 knots (8) (9) (1) (11) A possible iteration scheme is as follos: Choose a trial radius of action R o, corresponding to a time-to-target, to. Carry out the computations outlined above, stopping hen t =to. Reduce W by, then continue until the allotted fuel is exhausted. The total distance covered ill be Xo=Ro+ro, here ro is the return distance traveled from the target. We seek a radius of action R such that r = R, or X = 2R; i.e., fuel exhausted exactly upon return to base. To find this radius of action, e choose Rl =Xo/c (in general, Rk+l =Xk/c), and repeat the solu,tions until sufficient convergence is achieved. It is possible to make such iteration automatically handled by an electronic analog computer.2 Fig. 5 shos the <V Q) Q) TAKE - RETURN CRUISE CRUISE (i) COMBAT Fig. 4-Typical mission profile. phases of flight: take-off, climb, cruise, combat, return cruise, descent, and landing. The characteristics of certain phases ill be determined by the specific mission, 1 E. S. Rutoski, "Energy approach to the general aircraft performance problem," Jour. Aeronaut. Sci., vol. 21, pp ; March, ~ 35,------,---,---,----, ; 3 k !t J: RUN 3 RUN 2 RUN 1 ~ tlof-7,..., j 3: ~ 2 ~= Lj ii a:: 196 N,M-+---~ «15 I--.L.-..l--.l--_---l o 1 2 DISTANCE IN NAUTICAL MILES Fig. 5-Iterative solution of a range problem. 2 L. B. Wadel, "Automatic Iteration on an Electronic Analog Computer." 1954 Western Electronics Sho and Convention; August 25, 1954.

5 WESTERN JOINT COMPUTER CONFERENCE cessive results obtained for the sample problem just discussed. In this case an analytical solution exists, yielding R = 1,25 nautical miles. To handle actual radius of action problems, it ould be desirable to incorporate in the computer automatic sitching beteen successive phases of a flight plan, as ell as a more sophisticated automatic iteration technique. CONCLUSION AND FUTURE PLANS Several exploratory electronic analog computer studies have been made of certain performance capabilities of a modern, high-speed fighter aircraft; the method of approach and the nature of the results obtained for a theoretical aircraft have been described. A longer-range program is under study hich ould make available sufficient computer equipment, - particularly function generators, and develop applicable techniques such that complete missions may be studied at once. A compromise is thus sought beteen special-purpose computing equipment that cannot easily be used for other purposes, and the general-purpose electronic analog computer hose normal complement of computer elements is not chosen for maximum efficiency in solving performance problems. It is believed that considerable advantages of speed and convenience may thus be achieved in the preliminary design of aircraft or missiles, and in the later tabulation of data for the pilot's handbook. ACKNOWLEDGMENT I t is a pleasure to acknoledge the general assistance of C. C. Calvin, B. B. Mackey, Mrs. Olive Warram and Mrs. Evan Cotten, as ell as the assistance of Mmes. Wilma Drake and Martha Cohen in typing the manuscript and Mrs. Elizabeth Bird in executing the figures. Coding a General-Purpose Digital Computer to Operate as a Differential Analyzer R. G. SELFRIDGEt Summary-This paper describes a system of coding a generalpurpose digital computer so that differential equations may be solved easily and rapidly. While originally developed for an IBM 71, the system is applicable to any of several large computers, in that at no stage is any reference to a particular machine needed. The process is such that any engineer ho has learned ho to set up an analog computer should be able to code ithout hesitation, and should have very little difficulty even if no prior computer experience exists. No attempt has been made to solve partial differential equations. The present system could be used, but ould be inefficient in all but the simplest problems. INTRODUCTION THE SYSTEM TO BE described in this paper as set up specifically to enable problems hich ere to be run on a REAC analog computer to be run instead on an IBM 71 computer. The reason for this as tofold. First, that larger problems could be handled by a digital machine than as possible on the available analog computer, and secondly, that greater accuracy as possible. To some extent, therefore, there is a bias toards these to machines. Nevertheless, the basic ideas can easily be used on any COIllPuter ith consequent advantages. Nearly all the tricks available to an analog user can be duplicated ith extreme ease, and in many cases the desired result is available ithout resorting to such tricks. t USNOTS, Inyokern, Calif. Since presently available digital computers operate serially, that is, only one operation is performed at a time, the solution of the equations ill proceed step by step, one function at a time. Each equation must therefore be broken don into all its parts, and each part developed as needed. In the course of this breakdon each function is assigned a location, an amplifier number for an analog computer, or storage address for the digital computer. For the present coding there is no need or advantage in knoing here the actual storage takes place in the computer. After assigning the storage locations (amplifier numbers), the coding for each location must be set up. For the digital computer this consists of a series of numbers; for the analog computer it is a iring diagram and actual iring of a patch board. What follos is a brief description of the coding that the digital computer ill accept and then a comparison of the advantages and disadvantages ith an analog computer. The system as presented here has been changed slightly to eliminate a fe details that are tied to the specific machine and dependent primarily on the method by hich the system is constructed. One such case is the permissible range of multiplication constants. DESCRIPTION The operations that are available consist of addition, multiplication, division, integration, and a relay operation. Each of these operations is assigned a number, and