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Showing papers in "Journal of Aircraft in 1975"



Journal ArticleDOI
TL;DR: In this paper, the shape factor of the boundary layer, d*/0 £ = plate length L = lift m = exponent in Cp=x flows, also lift magnification factor (5.1) M = Mach number p = pressure q = dynamic pressure Q = flow rate R = Reynolds number (= u Ox/v in Stratford flows) R6 = Reynolds Number based on momentum thickness uee/v S = Stratford's separation constant (4.10)
Abstract: c. f = chord fraction, see Eq. (5.1) H = shape factor of the boundary layer, d*/0 £ = plate length L = lift m = exponent in Cp=x flows, also lift magnification factor (5.1) M = Mach number p = pressure q = dynamic pressure Q = flow rate R = Reynolds number (= u Ox/v in Stratford flows) R6 = Reynolds number based on momentum thickness uee/v S = Stratford's separation constant (4.10); also peripheral distance around a body or wing area / = blowing slot gap, also thickness ratio of a body u = velocity in x-direction u0 = initial velocity at start of deceleration in canonical and Stratford flows v = velocity normal to the wall V = a general velocity x = length in flow direction, or around surface of a body measured from stagnation point if used in connection with boundary-layer flow

478 citations



Journal ArticleDOI
TL;DR: A new method of providing motion cues to a moving base six-degree-of-freedom flight simulator utilizing nonlinear filters utilizing Coordinated adaptive filters derived based on the method of continuous steepest descent.
Abstract: This paper introduces a new method of providing motion cues to a moving base six-degree-of-freedom flight simulator utilizing nonlinear filters. Coordinated adaptive filters, used to coordinate translational and rotational motion, are derived based on the method of continuous steepest descent, and the basic concept of the digital controllers used for the uncoordinated heave and yaw cues is also presented. The coordinated adaptive washout method is illustrated by an application in a six-degree-of-freedom fixed-base environment.

153 citations


Journal ArticleDOI
TL;DR: In this paper, the concept of circulation control by tangential upper surface blowing over a circular trailing edge has been investigated for application to fixed wing STOL aircraft, and two-dimensional airfoils employing nominal blowing have demonstrated lift gains double to triple that of the conventional flapped airfoil, and associated large increases in drag.
Abstract: The concept of circulation control by tangential upper surface blowing over a circular trailing edge has been investigated for application to fixed wing STOL aircraft. Experimental investigations on both two- and threedimensional airfoils employing nominal blowing have demonstrated lift gains double to triple that of the conventional flapped airfoil, and associated large increases in drag (which further serve to reduce landing velocities and distances). An additional two-dimensional investigation into the basic fluid mechanics of the concept has shown that jet Mach numbers considerably above choked produced no adverse effects on the mechanism of the trailing edge Coanda flow, but instead yielded additional lift gains. These results appear quite promising where high lift generation is desired for a STOL aircraft having a nominal amount of auxiliary bleed air available, but where substitution of increased pressure ratios can produce added jet velocity to obtain the required momentum (blowing) coefficient.

109 citations


Journal ArticleDOI
TL;DR: In this paper, the authors identify two characteristic flow regions for the dependence of vortex maximum tangential velocity on downstream distance, the first is a region with little, if any, change in velocity, extending from wake rollup to downstream distances as great as 100 span lengths depending on span loading and angle of attack.
Abstract: Velocity measurements have been made in the wake of wings that were being towed underwater. As the wake aged, measurements of the tangential and axial velocity profiles were made with a two-dimensional scanning laser velocimeter at downstream distances of 5 to 200 span lengths behind wings with different span loadings. The results identify two characteristic flow regions for the dependence of vortex maximum tangential velocity on downstream distance. The first, a region with little, if any, change in maximum tangential velocity, extends from wake rollup to downstream distances as great as 100 span lengths, depending on span loading and angle of attack. This is followed by a decay region in which the maximum tangential velocity decreases with downstream distance at rates nominally proportional to the inverse one-half power.

94 citations


Journal ArticleDOI
TL;DR: The first flight test demonstration of active flutter suppression has been successfully completed and Comparisons between flight test and theoretical results are presented.
Abstract: The first flight test demonstration of active flutter suppression has been successfully completed. The Control Configured Vehicles (CCV) B-52 test airplane was twice flown 10 knots faster than its flutter speed relying solely on an automatic control system for adequate damping. The design, safety considerations, mechanization, ground testing, and flight testing of the flutter mode control system are reported. Comparisons between flight test and theoretical results are presented. The system was tested at heavy and light airplane weights and tested for compatibility with simultaneous ride control, maneuver load control, fatigue reduction, and augmented stability.

88 citations


Journal ArticleDOI
TL;DR: In this paper, the aeromechanical stability of the blade-disk system is expressed in terms of a stability parameter which measures the amount of unsteady work done by the air on the system, when the system is vibrating in one of its natural modes.
Abstract: A unified approach to flutter prediction has been developed at Pratt & Whitney Aircraft (P&WA). The aeromechanical stability of the blade-disk system is expressed in terms of a stability parameter which measures the amount of unsteady work done by the air on the system, when the system is vibrating in one of its natural modes. In neutrally stable systems, the unsteady work done by the air on the blades will balance the work dissipated by friction and by material damping. An accurate prediction of the vibrational deflections and of the unsteady aerodynamic forces is required at every spanwise location on each blade, so that the work done by the unsteady aerodynamic forces may be calculated. Recent progress is described in the prediction of unsteady aerodynamic forces and the determination of mode shapes. The stability model is applied to the prediction of supersonic flutter, chord wise bending flutter, and stall flutter. Recommendations are made for additional work necessary to improve the prediction model.

83 citations


Journal ArticleDOI
TL;DR: It has been demonstrated that the use of CSOR b results in material savings in computational times over theuse of Jh, GSh, andSORh.
Abstract: techniques QN x 10~ -9.3, 10.5, 20.9, for CSORb, GSb, and Jb, respectively and QNA0 >9.3 for SO/?/, due to computational times incurred in determining the optimum value of a( = 1.1 in this case). Testing of the four techniques for other configurations and Mach numbers, subsonic and supersonic, have been performed within the methods of Ref. 1 with essentially equivalent results. Thus, it has been demonstrated that the use of CSOR b results in material savings in computational times over the use of Jh, GSh, andSORh.

83 citations


Journal ArticleDOI
TL;DR: In this paper, an investigation has been conducted to evaluate the aerodynamic effects associated with blowing a jet spanwise over a wing's upper surface in a direction parallel to the leading edge.
Abstract: An investigation has been conducted to evaluate the aerodynamic effects associated with blowing a jet spanwise over a wing's upper surface in a direction parallel to the leading edge. Experimental pressure and force data were obtained on wings with sweep angles of 30 and 45 degrees and showed that spanwise blowing aids in the formation and control of the leading-edge vortex and, hence, significantly improves the aerodynamic characteristics at high angles of attack. Full vortex section lift is achieved at the inboard span station with a small blowing rate, but successively higher blowing rates are necessary to attain the full vortex-lift level at increased span distances. Spanwise blowing generates large increases in lift at high angles of attack, improves the drag polars, and extends the linear pitching moment to high lifts.

77 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of optimizing a sailplane flight path to achieve maximum cross-country speeds with zero net altitude loss is considered, and a variational formulation is chosen that includes three modes of cross country soaring: thermalling, essing, and straight dolphining.
Abstract: The problem of optimizing sailplane flight paths to achieve maximum cross-country speeds with zero net altitude loss is considered. A variational formulation is chosen that includes three modes of cross-country soaring: thermalling, essing, and straight dolphining. Optimal solutions are obtained numerically for various atmospheric vertical velocity distributions using quadratic approximations to the polars of two sailplanes representing current high performance Standard and Open Class designs. Implications of the solutions are discussed; especially the optimality of any one mode when more than one mode is possible and the advance knowledge of the atmosphere required in order to choose an optimal speeds-to-fly policy. Particularly important is the result that the maximum cross-country speed through an element of the atmosphere capable of sustaining straight dolphin flight is attained with a speeds-to-fly policy identical to that of some equivalent interthermal flight.

Journal ArticleDOI
TL;DR: In this paper, a full-scale system of this type should be able to detect wingtip vortices over a runway, and the laser wavelength should be selected such that the light beams would not transmit through the aircraft windshields.
Abstract: light reaching the photomultiplier. If one beam is unaffected but the other is deflected in the vertical plane, a smaller net change in the photomultiplier current would result. In full-scale operation over a runway, the laser beams should propagate above and parallel to the runway. Existing runway approach light towers on both ends of the runway could be used to house the necessary system of mirrors. This would not, therefore, add any new protruding structures adjacent to the runway. The beam separation and height above ground are variables which must be determined for best system response to a vortex. For safety, the laser wavelength should be selected such that the light beams would not transmit through the aircraft windshields. The laboratory tests described above indicate that, in principle, a full-scale system of this type should be able to detect wingtip vortices over a runway.

Journal ArticleDOI
TL;DR: In this article, Curwen, P. W., Nypan, L. C., and Hanrock, B. C. presented a series-Hybrid Bearing for small high performance aircraft gas turbine.
Abstract: Thomson, W. T., Vibration Theory and Applications, Prentice-Hall, Englewood Cliffs, N.J., 1965, pp. 84-86. Greene, R. B., "Gyroscopic Effects on the Critical Speeds of Flexible Rotors," ASME Transactions, Journal of Applied Mechanics, Vol. 70, 1948, pp. 369-376. Huang, T. C. and Huang, F. C. C., "On Precession and Critical Speeds of Two-Bearing Machines with Overhung Weight," ASME Paper 76-VIBR-19, 1967 ASME Vibrations Conference, Boston, Mass., March 1967. (Also published in the ASME Transactions, Ser. B: Journal of Engineering for Industry, Vol. 89, Sec. 1, pp. 713-718). Curwen, P. W., "Feasibility of Gas Bearings for Small High Performance Aircraft Gas Turbines," USAAVLABS Tech. Rept. 68-87, 1968, U.S. Army Aviation Material Labs., Fort Eustis, Va. Nypan, L. J., Scibbe, H. W., and Hanrock, B. J., "Optimal Speed Sharing Characteristics of a Series-Hybrid Bearing," ASME Transactions, Ser. F: Journal of Lubrication Technology, Vol. 95, Sec. 4, Jan. 1973, pp. 76-81.

Journal ArticleDOI
TL;DR: In this article, an end-fire microphone array that utilizes a digital time delay system has been designed and evaluated for measuring noise in wind tunnels, and it is estimated that four and eight-element arrays reject 6 and 9 dB, respectively, of microphone wind noise, as compared with a conventional omnidirectional microphone with nose cone.
Abstract: An end-fire microphone array that utilizes a digital time delay system has been designed and evaluated for measuring noise in wind tunnels. The directional response of both a four- and eight-element linear array of microphones has enabled substantial rejection of background noise and reverberations in the NASA Ames 40- by 80-foot wind tunnel. In addition, it is estimated that four- and eight-element arrays reject 6 and 9 dB, respectively, of microphone wind noise, as compared with a conventional omnidirectional microphone with nose cone. Array response to two types of jet engine models in the wind tunnel is presented. Comparisons of array response to loudspeakers in the wind tunnel and in free field are made.


Journal ArticleDOI
TL;DR: In this paper, an analysis of the steady and unsteady aerodynamics of sharp-edged slender wings has been performed, and the results indicate that the effects of delta planform lifting surfaces can be included in a simple manner when determining the aeroelastic characteristics of the space shuttle lift-off configuration.
Abstract: An analysis of the steady and unsteady aerodynamics of sharp-edged slender wings has been performed. The results show that slender wing theory can be modified to give the potential flow static and dynamic characteristics in incompressible flow. A semiempirical approximation is developed for the vortex-induced loads, and it is shown that the analytic approximation for sharp-edged slender wings gives good prediction of experimentall y determined steady and unsteady aerodynamics. The results indicate that the effects of delta planform lifting surfaces can be included in a simple manner when determining the aeroelastic characteristics of the space shuttle lift-off configuration.

Journal ArticleDOI
TL;DR: In this article, an experimental investigation was conducted in the UARL Acoustic Research Tunnel to define the noise characteristics associated with the interaction of a stationary tip vortex and a downstream stationary airfoil.
Abstract: An experimental investigation was conducted in the UARL Acoustic Research Tunnel to define the noise characteristics associated with the interaction of a stationary tip vortex and a downstream stationary airfoil. This model test geometry simulated, in its simplest form, the tip vortexblade interaction which occurs on single rotor helicopters during hover. For moderate to high lift test conditions, the vortex-airfoil interaction was found to cause local blade stall with an attendant increase in the blade far-field noise. These results indicated that this interaction may be an important source of helicopter broadband noise during hover. Cross-correlation measurements conducted amongst surface-mounted and far-field microphones demonstrated that the operative noise mechanism was "trailing edge noise" arising from the interaction of stall generated eddies with the airfoil trailing edge. This mechanism would be expected to be responsible for increased noise at stall conditions in other, nonrotary wing, applications.

Journal ArticleDOI
TL;DR: In this article, a nonlinear model is developed which determines the swirling and axial velocities in an aircraft vortex wake, given wing lift and drag distributions, and the model is shown to reduce to that given by Betz when the axial velocity is the freestream value.
Abstract: A nonlinear model is developed which determines the swirling and axial velocities in an aircraft vortex wake, given wing lift and drag distributions. The model is shown to reduce to that given by Betz when the axial velocity is the freestream value. The nonlinear interaction of swirling and axial velocities may lead to velocity distributions which are different from those previously calculated. Qualitatively, drag reduces the axial velocity in the vortex and results in an enlarged vortex radius and, therefore, a reduction in swirl velocity. The inviscid model that predicts that significant changes in the structure of the vortex wake, brought about solely by modification of the drag distribution, may require prohibitively large drag penalties. Theoretical results compare favorably with measurements made by Orloff and Grant. A model is developed to estimate the time to roll up a two-dimensional vortex sheet. Results are presented for the cases of linear, parabolic, and elliptic wing loading.

Journal ArticleDOI
TL;DR: A comprehensive engine/airframe screening methodology has been developed based on surface fitting and nonlinear optimization procedures, and a gradient-based method for nonlinear constrained minimization.
Abstract: A comprehensive engine/airframe screening methodology has been developed based on surface fitting and nonlinear optimization procedures. These procedures include the use of experimental design techniques, and a gradient-based method for nonlinear constrained minimization. The methodology has been programed for use on the CDC 6600 computer and has been successfully demonstrated on extensive test cases. One test case involved selection of an optimum airplane design using performance data for only 28 designs. A total of 256 designs were required to locate this optimum graphically.


Journal ArticleDOI
TL;DR: In this paper, a combined numerical-experimental study of the effects of varying tip twist and increasing centrifugal loading on the resonant characteristics of cantilevered plates is presented.
Abstract: A combined numerical-experimental study of the effects of varying tip twist and increasing centrifugal loading on the resonant characteristics of cantilevered plates is presented. The finite element computer program, NASTRAN, is used to compute the natural frequencies, mode shapes, and normalized shear stress distribution for each mode of vibration for cantilevered plates having, a), varying degrees of tip twist and, b) increasing centrifugal loading. For the case of zero centrifugal load, the resulting mode shapes are compared to those obtained experimentally using holographic iiiterferometry. The agreement between the two is found to be quite good. For increasing centrifugal loading, it is found that the nodal lines of the flexural modes of vibration shift toward the plate root. Increased centrifugal loading is also found to strongly effect the character of some of the plate-like vibration modes of the cantilevered plate.

Journal ArticleDOI
TL;DR: In this article, an experimental investigation was conducted to examine an airfoil durability problem in the first fan rotor of the F100 engine, and the results of this investigation's initial testing showed that rotor failure at high-flight Mach numbers and low altitudes was caused by torsional stall flutter instability.
Abstract: An experimental investigation was conducted to examine an airfoil durability problem in the first fan rotor of the F100 engine. This study incorporated laboratory and simulated engine flight tests, an empirical correlation of aeroelastic stability parameters from engine test data, and substantiation testing of the redesign. The results of this investigation's initial testing showed that rotor failure at high-flight Mach numbers and low altitudes was caused by torsional stall flutter instability. The results of the empirical correlation indicated that a design free of flutter required a decrease in both normalized incidence and reduced velocity. Further, the correlation indicated that the flutter was affected by inlet pressure, a heretofore undocumented phenomenon. The results of the substantiation testing confirmed that the redesign made the rotor flutter-free throughout the entire aircraft flight envelope. It was concluded that an improved stall flutter analysis was required to ensure stable fan and compressor rotor designs. It was further concluded that the effect of changes in inlet pressure level on rotor stability was, in part, the result of the accompanying changes in air density and steady-state aerodynamic loading.

Journal ArticleDOI
TL;DR: In this article, the unsteady Kutta condition is discussed in the light of some recent experimental measurements made near the trailing edge of a long flat plate and a 10C4 airfoil.
Abstract: The unsteady Kutta condition is discussed in the light of some recent experimental measurements made near the trailing edge of a long flat plate and a 10C4 airfoil. The hierachy of disagreement from the theoretically predicted zero trailing edge loading caused by viscous instabilities is found to be acoustically correlated vortex shedding, natural vortex shedding, Tollmien-Schlichting waves, and, by implication, turbulent boundary-layer eddies. The region of significant chordwise disagreement scales with the wake perturbation wavelength of the corresponding instability. Coordinating the vorticity of the turbulent boundary layer shed from the profile airfoil with a transverse acoustic resonance produced a distinct disagreement of the Kutta condition at high reduced frequency parameters (\ = wc/U). In this case and for vortex shedding, the extrapolated loading coefficient at the trailing edge increased with the nondimensional acoustic amplitude. I. Introduction T^HE Kutta condition as applied in unsteady JL potential airfoil analyses is essentially an extension of steady theory. Kutta ! postulated that a value of circulation should be chosen in his steady potential model to avoid a velocity singularity at the sharp trailing edge of an airfoil. This condition can be established if the trailing edge is also the rear stagnation point. The resulting modeled flow pattern agrees with that observed in steady flow and also predicts the lift and its chordwise distribution well at low angles of attack. The theoretical consequences of this hypothesis are that the lift loading or chordwise vorticity jump approaches zero at the trailing edge. An alternative statement is that the surface velocities on either side of the airfoil approach a common value at the rear stagnation point. For rounded trailing edges, the position of the rear stagnation point is indeterminate , as there is no velocity singularity to be avoided and so fix its location. In this case and for the situation of real flows with viscosity, Taylor2 proposed the condition of zero net vorticity discharge to establish the steady lift value. Preston3 explained the deviation of the lift of an airfoil at low angles of incidence from the potential theory value as due to the profile alteration from the boundary-layer growth. His calculations incorporated Taylor's vorticity discharge condition. Various approximate steady lift calculation methods for the rounded trailing edge geometry have been proposed by Gostello 4 and others. These extend the upper and lower lift distributions, at a selected chordwise position, to the trailing edge to give zero loading and thereby remove the stagnation point indeterminacy. In the unsteady case there are all the previous theoretical difficulties and, in addition, the unsteady effects on the viscous boundary layer and the shed vorticity. The latter complicates the airfoil response, making it a function of the airfoil's vorticity history. However, same theoretical assumption for the Kutta condition, of no unsteady loading at the

Journal ArticleDOI
TL;DR: In this paper, the mutual instability of a trailing vortex pair has been studied in a large wind tunnel and the vortices were visualized using heliumfilled soap bubbles and the cores probed with a hot-wire anemometer.
Abstract: The mutual instability of a trailing vortex pair has been studied in a large wind tunnel. The vortices were visualized using helium-filled soap bubbles and the cores probed with a hot-wire anemometer. Measurements were made to permit calculation of the wing lift, the circulation of the trailing vortices, the diameter of the vortex cores, and the wavelength and plane of oscillation of the unstable vortices. The results show that the linear theories of either Crow or Parks closely predict the characteristics of the instability, the agreement depending on the definition of the vortex core diameter.

Journal ArticleDOI
TL;DR: In this paper, it has been shown by a new static airflow visualization method that a drogue device properly positioned downstream of the wing tip causes vortex breakdown, and this same result has been obtained by mounting a jet engine simulator at the wingtip and directing the high-energy jet blast downstream into the vortex.
Abstract: It has been shown by a new static airflow visualization method that a drogue device properly positioned downstream of the wing tip causes vortex breakdown. This same result has been obtained by mounting a jet engine simulator at the wing tip and directing the high-energy jet blast downstream into the vortex. These configurations, among others, are now under intensive investigation in the new Langley Vortex Research Facility. In this facility, a balance mounted vortex generating model is propelled along the 1800-ft track while a second model trailed at 160 ft (scale distance of 1 mile) measures the far-field rolling moment induced by the vortex of the generating model.

Journal ArticleDOI
TL;DR: In this paper, it was found that for any practical biplane configuration, it is possible to design a wing system which has a much lower weight per unit area and which has essentially the same maximum useable lift coefficient as that of a comparable, well-designed, monoplane wing, in either clean or flapped configurations.
Abstract: LTHOUGH biplanes were quite popular during the early days of aviation, they had virtually disappeared from service by the mid-1930's. The biplane seemed to be plagued by inherently high drag and low maximum lift coefficient. The purpose of the present study was to re-examine the reasons for the biplane's decline. It was found that for any practical biplane configuration, it is possible to design a wing system which has a much lower weight per unit area and which has essentially the same maximum useable lift coefficient as that of a comparable, well-designed, monoplane wing, in either clean or flapped configurations. Improvements in the design of fairings permit large drag reductions, relative to the earlier designs. Because of these improvements, it is possible to design a biplane whose performance is superior, for some applications, to that of a well-designed monoplane. These applications are those for which excellent low-speed maneuverability, good short-field performance, good loadcarrying ability, low cost, and rugged construction are of primary importance.! Content If a biplane wing system is to develop a high CLmax, both wings must stall nearly together. A method presented by Fuchs was used to identify those configurations for which a good stall match could be obtained. Each wing is idealized in this method as a single horseshoe vortex, with the bound part of each vortex located at the center of pressure. Satisfactory stall match was defined as a ratio of leading wing to trailing wing Coequal to unity at an assumed CLmax for the combination. Some results of this analysis are given in Fig. 1, and some nomenclature is defined in the sketch. The airfoil chord lines are parallel; results of this study showed no advantage to be gained from the use of decalage. The geometric stagger angle a for best stall match is quite insensitive to the gap/chord ratio, and it decreases as the aspect ratio increases. The best stall match always corresponds to small values of 0, the aerodynamic stagger angle. The chordwise variation in vertical velocity induced by one airfoil upon the other corresponds to a local curvature of the flow. Since within the framework of thin airfoil theory, this effect is identical to the effect of a change in the mean camber line, it will be referred to as induced camber. Data for NACA airfoils of the four- and five-digit series show increases in the maximum section lift coefficient Cfmax


Journal ArticleDOI
TL;DR: In this paper, a simple analysis is presented which, using static experimental data as an input, can predict these two buffet-components for a wing in high Mach number subsonic flow.
Abstract: Shock-induced flow separation is the flow mechanism usually responsible for what the structural dynamicist terms 'buffet.' The shock-induced flow separation affects the aeroelastic response via two different mechanisms: (1) the flow separation generates fluctuating pressures, i.e., a forcing function that is independent of the motion of the aerodynamic surface, e.g., an aircraft wing, and (2) the flow separation affects the motion-dependent forces and can in some cases generate negative aerodynamic damping. A simple analysis is presented which, using static experimental data as an input, can predict these two buffet-components for a wing in high Mach number subsonic flow.

Journal ArticleDOI
TL;DR: It is foreseen that automation will be introduced to replace the human operator in several major areas such as surveillance, much decision making, and most communication.
Abstract: The steady growth of civil aviation has produced a corresponding increase in the size and complexity of the system for air traffic surveillance and control. To meet the operational demands forecast for the end of the century, it is anticipated that the system must not only continue to grow but also undergo a change in character, making a transition from a man-intensive to a machine-intensive system. It is foreseen that automation will be introduced to replace the human operator in several major areas such as surveillance, much decision making, and most communication. The role of man is expected to evolve into that of a manager of automatd processes, involving matching the capacity of system resources to the total workload imposed by demand, assigning of resources, assuring quality, and maintaining the operation of the system in the event of automated resource failure or malfunction.

Journal ArticleDOI
TL;DR: The optimal control problem is converted to a parameter optimization problem by chosing a form for the control which contains unknown parameters, and a gradient method is used to obtain approximate values for the parameters and Lagrange multipliers associated with the constraints.
Abstract: The purpose of this paper is to demonstrate the applicability of parameter optimization methods to the computation of optimal aircraft trajectories. First, the optimal control problem is converted to a parameter optimization problem by chosing a form for the control which contains unknown parameters. Then, a gradient method is used to obtain approximate values for the parameters and Lagrange multipliers associated with the constraints. Finally, a second-order method is used to obtain the converged trajectory. The test problem is that of finding the angle of attack history which minimizes the time to climb of a supersonic aircraft operating at full power. The angle of attack history is approximated by a fifth-order series of Chebyshev polynomials. An inequality constraint prevents the aircraft from descending below the takeoff altitude, and this constraint is handled in both the penalty function manner and in the hard constraint manner. The latter produces a lower climb time because all the parameters can be used for optimization; also, the shape of the trajectory in the altitudeMach number plane closely resembles that obtained from an energy-state analysis, with the transitions occurring at load factors ranging between 0.5 and 1.8.