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An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.
The development by JPL of a four band differential radiometer (FBDR) which is capable of providing a fast rate of response, accurate measurements of methane, ethane, and propane concentrations on the periphery of a dispersing LNG cloud. The FBDR is a small, low power, lightweight, portable instrument system that uses differential absorption of near infrared radiation by the LNG cloud as a technique for the determination of concentration of the three gases as the LNG cloud passes the instrument position. Instrument design and data analysis approaches are described. The data obtained from the FBDR prototype instrument system deployed in an instrument array during two 40 cubic meter spill tests are discussed.
The use of pressure differentials in a flight control system was evaluated. The pressure profile around the test surface was determined using two techniques: (1) windtunnel data (actual); and (2) NASA/Langley Single Element Airfoil Computer Program (theoretical). The system designed to evaluate the concept of using pressure differentials is composed of a sensor drive and power amplifiers, actuator, position potentiometer, and a control surface. The characteristics (both desired and actual) of the system and each individual component were analyzed. The desired characteristics of the system as a whole are given. The flight control system developed, the testing procedures and data reduction methods used, and theoretical frequency response analysis are described.
The extent and mechanisms of global differentiation and the early thermal and tectonic histories of the terrestrial planets are surveyed in order to provide constraints on the first billion years of earth history. Indirect and direct seismic evidence for crusts on the moon, Mars and Venus is presented, and it is pointed out that substantial portions of these crusts have been in place since the cessation of heavy bombardment of the inner solar system four billion years ago. Evidence for sizable cores on Mars and Mercury and a small core on the moon is also discussed, and the heat involved in core formation is pointed out. Examination of the volcanic and tectonic histories of planets lacking plate tectonics indicates that core formation was not closely linked to crust formation on the moon or Mars, with chemical differentiation restricted to shallow regions, and was much more extensive on Mercury. Extension of these considerations to the earth results in a model of a hot and vigorously convecting mantle with an easily deformable crust immediately following core formation, and the gradual development of a lithosphere and plates.
This paper examines the effect that solar differential rotation would have on a hypothetical large-scale equatorial dipole field. The evolving large-scale field pattern is expressed as a series of non-axisymmetric moments. As time increases, power is transferred to progressively higher order moments. In the 27d rotating coordinate system, each moment undergoes a small retrograde drift which remains nearly uniform until that mode begins to fade. The synodic rotation periods of the first few moments are comparable to the observed 28.5d period of the sun's large-scale field near sunspot maximum. Differential rotation may be the source of this 28.5d period, but the eruption of new flux is necessary to keep the pattern going.
Barrier surface construction in the joint space of the differential turning game for thrust-vectored vs. conventional aircraft is discussed. Differential-turn studies are based on modifications of existing computer programs including an energy-turn program, and one which generates hodograph data. Optimal turning flight in energy approximation is discussed for the conventional aircraft configurations. It is concluded that any advantages realized from thrust-vectoring are minor, unless hover is possible, where advantages would be major at low energies, and affect tactics at high energies as well.
The effects of time dependent phenomena, such as flare energization and decay, on the temperature and density structure of the transition region and, in particular, on the form of the differential emission measure are studied. It is found that unlike the case of the static models, the form of the differential emission measure can be used to determine the important physical mechanisms in the dynamic models.
Subsystems aboard the Geostationary Operational Environmental Satellites 4 and 5 showed instances of anomalous changes in state corresponding to false commands. Evidence linking the anomalous changes to geomagnetic activity, and presumably static discharges generated by spacecraft differential charging induced by substorm particle injection events is presented. The anomalies are shown to be correlated with individual substorms as monitored by stations of the North American Magnetometer Chain. The relative frequency of the anomalies is shown to be a function of geomagnetic activity. Finally a least squares fit to the time delay between substorm initiation and spacecraft anomaly as a function of spacecraft local time is shown to be consistent with injected electron populations with energy in the range 10 keV to 15 keV, in agreement with present understanding of the spacecraft charging mechanism. The spacecraft elements responsible for the differential charging were not satisfactorily identified. That question is currently under investigation.
A class of nonlinear, zero-sum differential games, exhibiting time-scale separation properties, can be analyzed by singular-perturbation techniques. The merits of such an analysis, leading to an approximate game solution, as well as the 'well-posedness' of the formulation, are discussed. This approach is shown to be attractive for investigating pursuit-evasion problems; the original multidimensional differential game is decomposed to a 'simple pursuit' (free-stream) game and two independent (boundary-layer) optimal-control problems. Using multiple time-scale boundary-layer models results in a pair of uniformly valid zero-order composite feedback strategies. The dependence of suboptimal strategies on relative geometry and own-state measurements is demonstrated by a three dimensional, constant-speed example. For game analysis with realistic vehicle dynamics, the technique of forced singular perturbations and a variable modeling approach is proposed. Accuracy of the analysis is evaluated by comparison with the numerical solution of a time-optimal, variable-speed 'game of two cars' in the horizontal plane.
The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.
The NAVSTAR Global Positioning System (GPS), currently being developed by the DOD, is a space based navigation system that will provide the user with precise position, velocity, and time information on a 24 hour basis, in all weather conditions and at any point on the globe. The baseline GPS system will provide guaranteed high accuracy to only a limited number of users, mostly the military. The civilian community has to devise a variation of this system to allow for an assured, uninterrupted level of accuracy. Differential GPS provides such a capability. In connection with the conceived possibility of the use of GPS by an enemy, it is found to be necessary to implement a selective availability technical capability. Differential GPS provides an approach for the civilian community to have a guaranteed level of accuracy better than the 250 meters presently planned for GPS.
The results reported here are based on applying the method of parametric differentiation (MPD) to transform the nonlinear differential equation governing small-disturbance transonic flow to a linear equation. Implicit approximate factorization and monotone methods were used to accelerate convergence of the linear problem by an order of magnitude over successive line over-relaxation. The relative merits of using MPD are discussed in comparison to conventional small-disturbance applications. Several MPD analyses are performed on an array of airfoils. A design procedure utilizing MPD is discussed and demonstrated for two nonlifting cases.
A study was made on the cure kinetics of an epoxy neat-resin (Narmco 5208) using Differential Scanning Calorimetry (DSC). Two interrelated analytical methods were applied to dynamic DSC data for evaluating the kinetic parameters, such as activation energy, E, the order of reaction, n, and the total heat of polymerization (or crosslinking), delta H sub t. The first method was proposed by Ellerstein (1968), and uses a thorough differential-integral analysis of a single DSC curve to evaluate the kinetic parameters. The second method was proposed by Kissinger (1957), and uses multiple DSC curves obtained at various heating rates to evaluate E regardless of n. Kinetic analysis of Narmco 5208 epoxy resin showed that the reaction order, n, is substantially affected by the rate of heating; i.e., n is approximately 2 at slow scan rates but is reduced to 1.5 at higher scan rates. The activation energy, E, is not affected by the scan rate, and the average value of E is 25.6 + or - 1.8 kcal/mole.
The propagation of errors in the solutions of the differential equations for the orbital elements of perturbed two-body motion is investigated. It is shown that the error in the time-element grows linearly for differential equations for orbital elements when only perturbations are present on the right-hand side, cubically for formulations which have a two-body term on the right-hand side, and linearly for formulations based upon extended phase space Hamiltonians.
Four approaches to radio interferometric tracking of geosynchronous satellites are analyzed and compared. Quasar-based differential very-long-baseline interferometry, which requires a very sensitive receiver, can achieve meter-level position accuracy with a two-baseline system. Satellite-based differential VLBI gives somewhat lower accuracy with a compact, inexpensive receiver. Nondifferential VLBI, using less precise media and clock calibrations obtained by observing the GPS satellites, still gives 5-10 m position accuracy with two baselines. For a sufficiently inclined orbit, all interferometric approaches can yield six-component satellite state from a single baseline.
An investigation has been conducted in the Langley 8 foot transonic pressure tunnel to determine the effects of differential and symmetrical aileron deflection on the longitudinal and lateral directional aerodynamic characteristics of an 0.087 scale model of an NASA supercritical wing research airplane (TF-8A). Tests were conducted at Mach numbers from 0.25 to 0.99 in order to determine the effects of differential aileron deflection and at Mach numbers of 0.25 and 0.50 to determine the effects of symmetrical aileron (flap) deflection. The angle of attack range for all tests varied from approximately -12 deg to 20 deg.
A compact differentiation technique (without using indexes) is developed for scalar functions that depend on complex matrix arguments which are combined by operations of complex conjugation, transposition, addition, multiplication, matrix inversion and taking the direct product. The differentiation apparatus is developed in order to simplify the solution of extremum problems of scalar functions of matrix arguments.
A broadband differential optical interferometer that uses an optical-fiber waveguide to transmit coherent light to a surface under inspection is discussed. Input laser light is coupled into two single-mode optical fibers in a stationary input optics enclosure. The fibers transmit the light to a small, remote detection head where it is used in a wide-band differential interferometer to detect continuous and pulsed surface-particle displacements. The effective separation between the input optics and the detection head allows simple surface scanning and probe alignment. Detection of simulated acoustic emission pulses by a prototype test unit is reported.