Showing papers in "AIAA Journal in 1974"

Some Approximation Concepts for Structural Synthesis

Abstract: An efficient automated minimum weight design procedure is presented which is applicable to sizing structural systems that can be idealized by truss, shear panel, and constant strain triangles. Static stress and displacement constraints under alternative loading conditions are considered. The optimization algorithm is an adaptation of the method of inscribed hyperspheres and high efficiency is achieved by using several approximation concepts including temporary deletion of noncritical constraints, design variable linking, and Taylor series expansions for response variables in terms of design variables. Optimum designs for several planar and space truss examples problems are presented. The results reported support the contention that the innovative use of approximation concepts in structural synthesis can produce significant improvements in efficiency.

Statistical Identification of Structures

Abstract: A method is formulated for systematically using experimental measurements of the natural frequencies and mode shapes of a structure to modify stiffness and mass characteristics of a finite element model. Throughout the modification process, which does not require complete data, the finite element model remains consistent. An additional feature is that the engineer's confidence in the modeling of the various finite elements is quantified and incorporated into the revision procedure. Examples demonstrate the convergence and versatility of the method.

Subsonic Potential Aerodynamics for Complex Configurations: A General Theory

Abstract: A general theory of subsonic potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented. By using the Green function method, an integral representation for the velocity potential is obtained for both supersonic and subsonic flow. Under the small perturbation assumption, the potential at any point in the field depends only upon the values of the potential and its normal derivative on the surface of the body. On the surface of the body, this representation reduces to an integro-differential equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface. The theory is applied to finite-thickness wings in subsonic steady and oscillatory flows.

A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems

Abstract: 1 Hoffman, W. C, Zvara, J., and Bryson, A. E., Jr., "A Landing Approach Guidance Scheme for Unpowered Lifting Vehicles," Journal of Spacecraft and Rockets, Vol. 7, No. 2, Feb. 1970, pp. 196-202. 2 Duiven, E. M., "Quadratic Cost-Re-Entry Guidance Laws," Internal Memo 2-5482-0100-020, Dec. 1969, Aerospace Group, Boeing Co., Seattle, Wash. 3 Kovatch, G., "Optimal Guidance and Control Synthesis for Maneuverable Lifting Space Vehicles," Transactions of the 9th Symposium on Ballistic Missile and Space Technology, Vol. 1, Air Force Command, San Diego, Calif., 1964, pp. 327-379. 4 Glover, L. A. and Hagan, J. C, "The Motion of Ballistic Missiles," TG1164, July 1971, The Johns Hopkins University, Applied Physics Lab., Baltimore, Md. 5 White, J. S. and Lee, H. Q., "Users Manual for the Variable Dimension Automatic Synthesis Program (VASP)," TM X-2417, Oct. 1971, NASA. 6 Anderson, B. D. O. and Moore, J. B., Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N.J., 1971. 7 Bryson, A. E. and Ho, H. C, Applied Optimal Control, Blaisdell, Waltham, Mass., 1969. 8 Kwakernaak, H. and Si van, R., Linear Optimal Control Systems, Wiley, New York, 1972.

Vorticity Associated with a Jet in a Cross Flow

Abstract: An extensive wind-tunnel test of a round turbulent jet directed normally through a flat plate into a subsonic cross flow has been conducted. The results of the velocity field measurements are presented in a concise and usable form through the use of simple models to relate the velocity field to empirical values for the strength and location of the pair of contrarotating vortices associated with the jet.

Centerline Velocity Decay of Compressible Free Jets

Abstract: Theme A empirically determined formulation for the eddy viscosity is introduced into the turbulent axially symmetric compressible free jet theory of Kleinstein. * The easily evaluated algebraic equation that results shows excellent agreement with an extensive compilation of experimental data for the centerline velocity decay in the main region. It is found that the eddy viscosity can be treated as constant, with functional dependence on only the exit Mach number and the jet-to-freestream density ratio. The theory presented is applicable to the main region of most classes of free jets, including heterogeneous, nonisothermal, and subsonic and properly-expanded supersonic flows.

Calculation of the Flow on a Cone at High Angle of Attack

Abstract: 2 Kreith, F. and Sonju, O. K., "The Decay of a Turbulent Swirl in a Pipe," Journal of Fluid Mechanics, Vol. 22, No. 2, 1965, pp.257-271. 3 Menis, M., "Effect of Vortex Decay on Pipe Flow," Rept. 61, Nov. 1960, Gas Turbine Lab., MIT, Cambridge', Mass. 4 Youssef, T. E. A., "Some Investigations on Rotating Flow with a Recirculation Core in Straight Pipes," Paper 66-WA-FE-36, Nov. 27, 1966, American Society of Mechanical Engineers, New York. 5 Schlichting, H., Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960, pp. 53-54. 6 Thompson, J. R., Jr., "The Structure of Free and Confined Turbulent Vortices," Research Note 44, May 1963, Aero Physics Dept, Mississippi State University, State College, Miss. 7 Yeh, H., "Boundary Layer Along Annular Walls in a Swirling Flow," and "Discussion," Transactions of the ASME, May 1958, pp. 767-776. 8 Kinney, R. B., "Universal Velocity Similarity in Fully Turbulent Rotating Flows," ASME Paper 67-APM-23, June 1967, Applied Mechanic Conference, Pasadena, Calif. 9 Baker, D. W., "Decay of Swirling, Turbulent Flow of Incompressible Fluids in Long Pipes," Ph.D. thesis, 1967, Univ. of Maryland, College Park, Md. 10 Schlichting, H., Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960, pp. 441-144. 11 Schlichting, H., Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960, p. 503. 12 Schlichting, H., Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960, pp. 502-508.

Effect of the Adverse Pressure Gradient on Vortex Breakdown

Abstract: of the Influence of the Turbulent Boundary Layer on the Pressure Distribution over a Rigid Two-Dimensional Wavy Wall," TN D-6477, Aug. 1971, NASA. 4 Lighthill, M. J., "On Boundary Layers and Upstream Influence II. Supersonic Flows without Separation," Proceedings of the Royal Society, Vol. A217, 1953, pp. 478 and 504; see also Quarterly Journal of Mechanics, Vol. 3, 1950, p. 303. 5 Benjamin, T. B., "Shearing Flow over a Wavy Boundary," Journal of Fluid Mechanics, Vol. 6, 1959, p. 161. 6 Miles, J. W., "On Panel Flutter in the Presence of a Boundary Layer," Journal of Aerospace Sciences, Vol. 26, No. 2, Feb. 1959, pp. 81-93. 7 McClure, J. D., "On Perturbed Boundary Layer Flows," Rept. 62-2, June 1962, M.I.T. Fluid Dynamic Research Lab., Cambridge, Mass. 8 Anderson, W. J. and Fung, Y. C, "The Effect of an Idealized Boundary Layer on the Flutter of Cylindrical Shells in Supersonic Flow," GALCIT Structural Dynamics Rept. SM62-49, Dec. 1962, Graduate Aeronautical Lab., California Institute of Technology, Pasadena, Calif. 9 Zeydel, E. F. E., "Study of the Pressure Distribution on Oscillating Panels in Low Supersonic Flow with Turbulent Boundary Layer," NASA CR-691, Feb. 1967, Georgia Institute of Technology, Atlanta, Ga. 10 Do well, E. H., "Generalized Aerodynamic Forces on a Flexible Plate Undergoing Transient Motion in a Shear Flow with an Application to Panel Flutter," AIAA Journal, Vol. 9, No. 5, May 1971, pp. 834-841. 11 Ventres, C. S., "Transient Panel Motion in a Shear Flow," AMS Rept. 1062, Aug. 1972, Princeton Univ., Princeton, N.J. 12 Garrick, I. E. and Rubinow, S. I., "Theoretical Study of Air Forces on an Oscillating or Steady Thin Wing in a Supersonic Main Stream," TN 1383, July 1947, NACA. 13 Yates, J. E., "A Study of Panel Flutter with the Exact Method of Zeydel," NASA CR-1721, Dec. 1970, Aeronautical Research Associates of Princeton, Princeton, N.J. 14 Do well, E. H. and Ventres, C. S., "Derivation of Aerodynamic Kernel Functions," AIAA Journal, Vol. 11, No. 11, Nov. 1973, pp. 1586-1588.

Turbulence-Model Predictions for Turbulent Boundary Layers

Abstract: A set of turbulence model equations, originally postulated by Saffman, forms the basis of this three-part study of steady turbulent-boundary-layer structure above a flat plate In one part of the study the turbulence equations are integrated through the viscous sublayer by means of time-marching numerical integration techniques and the constant in the law of the wall is predicted as a function of wall roughness In a second part, the Van Driest compressible law of the wall is deduced by classical mathematical methods Then, using the Van Driest law as a wall boundary condition, the model equations are integrated through the entire boundary layer, again by time-marching methods, for a freestream Mach number of 296 NCREASINGLY, turbulent-flow studies have concentrated on phenomenologi cal-model equations describing turbulent motion The most promising approach seeks rate equations which describe evolution of the Reynolds stresses; development of equations of this type has progressed for both low- and highspeed flows1"5 This paper focuses upon one such set of turbulence-mo del equations originally devised by Saffman 2 and subsequently modified by Wilcox and Alber3 to account for the effects of compressibility; the model equations are based on the hypothesis that transfer of momentum and heat by turbulence can be described by an eddy viscosity which is a function of two turbulence densities, viz, an energy density, e, and a pseudovorticity density, Q Previous work 2'3 has concentrated mostly on formulation of the nonlinear diffusion equations satisfied by the turbulence densities; included in the present study is some model development which is directed primarily to the boundary conditions imposed at or near a solid boundary As a result, uncertainties regarding boundary conditions have been eliminated A concurrent study has been made of the predictions of the model equations for boundary layers on a flat plate under both compressible and incompressible flow conditions Past studies have left at least two areas of boundary-condition uncertainty On the one hand, Saffman2 indicated that the most natural boundary conditions for the mean velocity and the turbulence densities follow from matching to the law of the wall Matching is valid for the incompressible Saffman model because law-of-the-wall behavior near a solid boundary is contained within the equations; however, Wilcox and Alber3 did not generalize the matching concept for compressible flows On the other hand, for some applications, integration through the sublayer might be necessary (because of, eg, numerical reasons or sublayer-structure uncertainties) Thus, the question of boundary conditions for the turbulence densities appropriate to a solid boundary was considered briefly by Saffman and in a little more detail by Wilcox and Alber Although the boundary condition on e is straightforward (it vanishes at solid boundaries), both studies fall short of specifying a boundary condition on the vorticity density

Reliability Analysis of Aircraft Structures under Random Loading and Periodic Inspection

Abstract: A reliability analysis of fatigue-sensi tive aircraft structures, based on the application of the approach developed in the "random vibration theory," is presented. Operational service loads, composed of ground loads, ground-air-ground loads, and gust loads, are all random in nature. The fatigue process involved here consists of crack initiation, crack propagation, and strength degradation. The time to crack initiation and the ultimate strength are random variables. After a fatigue crack is initiated, fracture mechanics is applied to predict crack propagation under random loading. While the fatigue crack is propagating, the residual strength of the structure decreases progressively, thus increasing the failure rate with time. The aircraft structure is subjected to periodic inspection in service. When a fatigue crack is detected during inspection, the implicated component is either repaired or replaced, so that both the static and the fatigue strength are renewed. Such a renewal process is taken into account in the present analysis. The detection of an existing fatigue crack during inspection is also a random variable which depends on the resolution capability of the particular technique employed and the size of the existing crack. Taking into account all the random variables as well as all the random loadings, the solution for the probability of first failure in a fleet of aircraft is derived. Finally, numerical examples are given to demonstrate the effect of inspection and fleet size on the fleet reliability.

Derivatives of Eigenvalues and Eigenvectors for a General Matrix

Abstract: Expressions are obtained for the derivatives of the eigenvalues and eigenvectors which are expressions of only one left-hand and one right-hand eigenvector. The approach described makes use of a Choleski decomposition or some other decomposition method. The method may be extended to find any order of derivative of the eigenvalue and eigenvector. The expressions obtained for finding the derivatives of eigenvalues and eigenvectors for nonself-adjoint systems may be applied to self-adjoint systems.

Role of Binders in Solid Propellant Combustion

Abstract: : The objective of this program was to investigate and define the effects of inert binder properties on composite solid propellant burning rate. Experimental pyrolysis data were obtained for many binders of practical interest over a wide range of heating rates and pressures, in several environmental gases, with and without 10-percent ammonium perchlorate (AP) contained in the sample, and in some cases with catalysts. These data were used to extract kinetics constants from Arrhenius plots, and heat of decomposition. In addition, motion pictures were taken of the pyrolyzing surface and gas samples were extracted for analysis. Pyrolysis kinetics varied between binders, but were found to be independent of pressure, the presence of AP, and the presence of burn rate catalysts; however, a chlorine gas environment had a material effect upon the results. All of the binders exhibited molten, boiling surfaces mingled with char, to varying degrees; the amount of char increased with AP present, and in chlorine. Relevant data were input to the Derr-Beckstead-Price combustion model in order to associate binder properties with known binder effects on burning rate. Although the effects were predictable, they stemmed from properties other than pyrolysis kinetics; however, the binder data as applied to the model revealed possible deficiencies in the model, which are discussed. It appears that the approach of combustion tailoring by binder modification would have to involve the gas phase combustion processes rather than surface pyrolysis. Therefore, future work concerning the role of binder should be directed toward the gas phase.

Initially Stressed Mindlin Plates

Abstract: Equations of motion for a transversely isotropic plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived by two methods. The first method is to perturb the nonlinear equations of elasticity by an incremental deformation. The resulting equations are linearized and integrated through the thickness of the plate to obtain the plate equations. The second method is to derive nonlinear equations of motion for a thick plate variationally by Hamilton's principle. These equations are then perturbed and suitably linearized to obtain the same equations as were obtained by the first method. A reduced set of equations for a thin plate are also given. Finally, the thick plate equations are solved for a simply supported rectangular plate in a state of uniform compressive stress plus a uniform bending stress both acting in the same direction.

Composite Material Mechanics: Structural Mechanics

Abstract: Introduction T purpose of this Survey is to review and bring together in an orderly fashion some of the principal contributions to the field of structural mechanics of structures containing composite materials. The topics of micromechanics and fracture, while quite important, are not considered in this Survey. Emphasis is given to the macromechanical structural analysis of various structural elements, including response under conditions of stable static loading, buckling, and dynamics. The Survey unfolds in the following sequence: Straight and Curved Laminated Bars, Laminated Plates, Laminated Shells, Sandwich Structures, Applications to Practical Structural Systems, and Future Trends. The authors hope that this contribution will be a useful reference tool for researchers and engineers already involved in structural aspects of advanced composites, as well as for those who are just entering the field. No Survey can do full justice to such a wide field as compositematerial structural mechanics. The references cited give only a glimpse of the extensive literature in this field. The authors apologize for not citing a number of important contributions in the field.

Propulsion by Absorption of Laser Radiation

Abstract: The results of an experimental study of the physics of propulsion by absorption of radiation from a remotely stationed high-power laser are presented. Both pulsed and steady-state simulation experiments were performed utilizing a megawatt peak power pulsed CO2 laser. The steady-state simulation experiments were carried out in a vacuum chamber with graphite, Lucite and lithium fluoride as solid propellants. The pulsed experiments consisted of ballistic pendulum measurements of thrust, wall pressure traces, and streak camera records of plasma velocity to deduce the efficiency of a laser powered pulsejet, where air is utilized as fuel. Results indicate that for steady-state propulsion a high ratio of thrust to laser power (~ 10-100 dynes/w) can be obtained by using the laser to simply vaporize a suitable solid propellant. However, for high specific impulse (~1000 sec) it is necessary to add energy to the vapor in a stable manner. This may require gaseous propellants with external control over the absorption characteristics for stable heating in a throatless nozzle. The pulsejet experiments resulted in a ratio of thrust to laser power ~ 25 dynes/w. In order to increase the efficiency of propellant mass utilization, shorter laser pulses with higher power are desirable.

Theoretical Analysis of the Aerodynamic Stability of Multiple, Interdigitated Helical Vortices

Abstract: A small perturbation stability analysis of a doubly infinite array of interdigitated, right circular helical vortices has been formulated. This array corresponds to the vortices trailed from the tips of the blades of a helicopter rotor or propeller in static thrust or axial flight condition and at great distance from the plane of rotation of the blades. A continuum of instability modes has been found associated with all values of wave numbers; only modes with wave numbers 0 and 1 are so much as neutrally stable, and for the case of a single helix. The most unstable modes involve the most axial motion of adjacent vortex segments relative to each other. Maximum divergence rates increase as the helix pitch decreases, increase as the number of helices increase and decrease as the number of cycles of deformations in one turn of the helix (i.e., wave number) increases. The helix filament core diameter can have substantial effect on the stability of a single helix, but not for multiple arrays. The larger the core diameter, the more sensitive the analysis is to the means by which the singularities in the self-induction integrals are eliminated. Increasing core diameters reduces the maximum divergence rates in all cases.

Compressibility Effects in Unsteady Thin-Airfoil Theory

Abstract: This is the same equation as obtained by Goodman, for the melting rate, using the heat balance integral method Results for surface temperature-time history

Dynamic Transformation Method for Modal Synthesis

Abstract: This paper presents a condensation method for large discrete parameter vibration analysis of complex structures that greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range. A dynamic transformation is obtained from the partitioned equations of motion that relates modes not explicitly in the condensed solution to the retained modes at a selected system frequency. The generalized mass and stiffness matrices, obtained with existing modal synthesis methods, are reduced using this transformation and solved. Revised solutions are then obtained using new transformations at the calculated eigenvalues and are also used to assess the accuracy of the results. If all the modes of interest have not been obtained, the results are used to select a new set of retained coordinates and a new transformation frequency and the procedure repeated for another group of modes. Computations are made tractable by simplified forms of the transformation that result with various modal synthesis methods. Three examples using the dynamic transformation in conjunction with a General Electric stiffness coupling method and the method of Craig and Bampton indicate large reductions in truncation errors and demonstrate the method for sequential groups of modes. Comparisons with truncated results using current methods indicate that two to three times as many accurate modes are obtained from solutions keeping less than half the component modes. Nomenclature [/c] = stiffness matrix for total structure in {x} physical coordinates [X] = generalized stiffness matrix for total structure in {q} modal coordinates [/CCPL] = stiffness matrix in {x} coordinates for coupling substructures [m] = mass matrix for total structure in {x} physical coordinates [M] = generalized mass matrix for total structure in {q} modal coordinates [Am] = incremental mass matrix of coupling structures in {x} coordinates

Gas Concentration Measurement by Coherent Raman Anti-Stokes Scattering

Abstract: A novel method for gas concentration measurement is described. This method is based on a four wave mixing process: two collinear light beams of frequencies o>1 and a>2 generate a collinear (anti-Stokes) wave at frequency 2(Dl—co2 when traversing a gas containing a Raman active molecular species with vibrational frequency wv = coj — co2. The intensity of the new wave is proportional to the square of the number density of resonant molecules. This scattering is much more intense than spontaneous Raman scattering. Possible experimental set-ups are reviewed. Typical results obtained with H2 gas are also presented: the concentration in a hydrocarbon flame was mapped, and small jets through the atmosphere were photographed in anti-Stokes light. The limitations of the method are discussed, and its field of application is assessed.

Stability of Nonequilibrium MHD Plasma in the Regime of Fully Ionized Seed

Abstract: The stability of nonequilibriu m MHD plasmas in the regime of fully ionized seed, taking into account the ionization of the noble gas, is studied theoretically and experimentally. It is shown that the complete ionization of the seed is attainable with the discharge current and the corresponding electric field able to be achieved in actual MHD generators by reducing the seed fraction down to 10~ 5. The experimental data show that in the regime of fully ionized seed the plasma is stable in agreement with the theoretical predictions. Effective Hall parameters as high as 6 were obtained, while the effective conductivity remained at the value of the laminar conductivity. The application of this stabilization method to actual MHD generators is discussed.

Numerical-Perturbation Method for the Nonlinear Analysis of Structural Vibrations

Abstract: A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is

On the Accuracy of the Taylor Approximation for Structure Resizing

Abstract: P2 ~ P2 + Pi 2 are the probabilities of occurrence of x^t) or x 1(t)H-x2(f) and x2(r) or xl(t)+x2(t\ respectively. Further development of this case is left to the interested reader. It is important to note, however, that setting p1>2 = 0 implies that xl (t) and x2(f) are mutually exclusive 100% of the time, and setting p1 2 = 1 implies x^t) and x2(t) act concurrently 100% of the time. The former case illustrates how to resolve x(t) into mutually exclusive components, and the latter case leads to the principle of linear superposition. Finally, it should be noted that the p and p values are discrete and can also be determined by counting occurrences.

Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves

Abstract: We report results of an investigation into the roles played by thermal and radiative transport in the propagation of Laser Supported Combustion (LSC) waves We replace the volume radiation losses assumed by Raizer with detailed treatment of radiation in 19 frequency groups, the flux of each being calculated from the temperature in discrete cells along the LSC wave axis and allowed to deposit in other cells according to temperature dependent absorption coefficients for air Calculations of LSC wave structure and propagation velocity as a function of laser intensity are compatible with available experimental data a = c = CP = f = h = h = hi(a) = K = k = KV =

Electric Probes in Stationary and Flowing Plasmas: Part 1. Collisionless and Transitional Probes

Abstract: The operation and theory for collisionless and transitional electric probes are reviewed. The following topics are discussed: (1) orbital motion limit, (2) cold ion approximation, (3) exact theories for current collection by spherical and cylindrical probes in the collisionless limit, (4) collisional effects on probe response, and (5) the effect of flow on aligned cylindrical probes under collisionless conditions. (MOW)

Inclusion of Transverse Shear Deformation in Finite Element Displacement Formulations

Abstract: A stiffness matrix is derived for a beam element with transverse shear deformation. It is shown that straightforward energy minimization yields the correct stiffness matrix in displacement formulations when transverse shear effects are considered. Since the TIM4 beam element does not represent the geometric boundary conditions for a cantilever beam the rotation of the normal must be retained as a grid point degree of freedom.

Minimum Impulse Three-Body Trajectories

Abstract: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1973.

The Homicidal Chauffeur

Abstract: P differential games had their beginning in the pioneering work of Isaacs. These differential games are a generalization of the optimal guidance problem, in the sense that guidance strategies are to be found for each of two players having conflicting objectives. The players' strategies, respectively, are found to minimize and to maximize a specific criterion, which is often taken to be the time to termination, or "capture." The second-order differential equations describing the dynamics of the homicidal chauffeur game include two parameters, which are the speed ratio and the ratio of capture-radius to pursuer's minimum-turn radius. It is the purpose of this paper to show that this apparently simple, physically motivated differential game can lead to novel and remarkably complex solutions. It is shown that the classical necessary conditions often fall far short of providing the complete solution to the problem. On the contrary, a variety of "exceptional" lines are found to occur in the state space of the game. These lines are not predicted by the theory and they are of fundamental importance, since they separate regions of differing optimal strategies for either or both of the players. Isaacs discovered the barrier, universal line (singular arc), equivocal line, and a pursuer's dispersal line in his study of the problem. Other types of lines are found to include new dispersal lines, switch lines, switch envelopes, focal lines, and safe-contact motion. The switch envelope and focal line phenomena are believed to be new to the general theory of differential games.