Optimal guidance from hyperbolic to circular orbits.
Minimum fuel guidance from hyperbolic into specified circular orbit
Edelbaum, T. N.↗
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Optimum transfer between hyperbolic asymptotes about a planet of finite size.
Optimal transfer between hyperbolic asymptotes about finite size planet, discussing twelve types of analytical solutions in terms of arrival, departure and escape velocities
Culp, R. D.↗
Hyperbolic position finding with synchronous satellites
Hyperbolic position finding with synchronous satellites having frequency conversion unit for operations at higher frequency satellite signals
Casserly, G. W.↗
Perturbation solution of a hyperbolic equation governing longitudinal wave propagation in certain nonuniform bars
Longitudinal wave propagation in variable section bars, solving hyperbolic equation of motion by perturbation method
Davis, R. T.↗
Formulas for nth order derivatives of hyperbolic and trigonometric functions
Formulas for nth order derivatives of hyperbolic and trigonometric functions used in evaluating Fourier sine and cosine integrals
Wintucky, E. G.↗
Use of orbit-to-orbit shuttles for hyperbolic rendezvous with returning planetary spacecraft
Use of orbit-to-orbit shuttles for hyperbolic rendezvous with returning interplanetary spacecraft
Brooks, D. R.↗
Experimental study of magnetic flux transfer at the hyperbolic neutral point
Magnetic flux transfer measurement at hyperbolic neutral point resulting in plasma compression
Bratenahl, A.↗
Optimum transfer between hyperbolic asymptotes with turn angle less than the maximum natural turn angle.
The impulsive transfer of minimum characteristic velocity between two given hyperbolic asymptotes associated with a real planet is considered. The constraint of remaining above the surface of the planet introduces a maximum natural turn angle. This divides the problem naively into two categories. The first is the case when the turn angle required by the given asymptotes is less than or equal to the maximum natural turn angle. This is the more reasonable situation for a swingby maneuver, and is the case considered in this paper. The resulting optimal transfers are of five basic types. In boundary conditions space, by far the largest area of transfers are either single impulse transfers or transfers through the parabolic level requiring two finite impulses. In no case has an optimal transfer been found which uses more than two finite impulses and three infinitesimal impulses. Complete results are presented for the case in which the given turn angle is less than or equal to the maximum natural turn angle.
Walton, J. M.↗
A source for hyperbolic cosmic dust particles
Earlier analyses of the Pioneer 8 and 9 experimental meteoroid data have shown that the detectors on these two spacecraft are intercepting meteoroids with hyperbolic orbital parameters. It is shown in this paper that these results are entirely consistent with and, indeed, to be expected from other observations of the interplanetary meteoroid complex. Collisional breakup of meteoroids and post-collision radiation pressure modification of their orbits is found to be a sufficient cause for the observed results. Details of the calculations as well as of the results are presented.
Zook, H. A.↗
A generalized hyperbolic marching method for chemically reacting 3-D supersonic flow using a splitting technique
A generalized hyperbolic marching method employing a nonorthogonal coordinate system and using a split differencing scheme for calculating steady supersonic flow over aerodynamic shapes is presented. It is a second-order-accurate mixed explicit-implicit procedure that solves the inviscid adiabatic and nondiffusive equations for chemically reacting flow in integral conservation-law form. The relationship between the integral and differential forms of the equations are examined and the relative merits of each discussed. The method admits initial Cauchy data situated on any arbitrary surface and integrates them forward along a general curvilinear coordinate, distorting and deforming the surface as it advances. The chemical kinetics term is split from the convective terms which are themselves dimensionally split, thereby freeing the fluid operators from the restricted step-size imposed by the chemical reactions and increasing the computational efficiency. The accuracy of this splitting technique is analyzed, a sufficient stability criterion is established, and comparisons are made with another method.
Rizzi, A. W.↗
Nonlinear hyperbolic theory of thermal waves in metals
A closed-form solution for cylindrical thermal waves in metals is given based on the nonlinear hyperbolic system of energy-conservation and heat-flux relaxation equations. It is shown that heat released from a line source propagates radially outward with finite speed in the form of a thermal wave which exhibits a discontinuous wave front. Unique nonlinear thermal-wave solutions exist up to a critical amount of driving energy, i.e., for larger energy releases, the thermal flow becomes multivalued (occurrence of shock waves). By comparison, it is demonstrated that the parabolic thermal-wave theory gives, in general, a misleading picture of the profile and propagation of thermal waves and leads to physical (infinite speed of heat propagation) and mathematical (divergent energy integrals) difficulties. Attention is drawn to the importance of temporal heat-flux relaxation for the physical understanding of fast transient processes such as thermal waves and more general explosions and implosions.
Wilhelm, H. E.↗
Highly efficient antenna system using a corrugated horn and scanning hyperbolic reflector
In a horn-reflector antenna system for producing a spherical aperture phase front, a corrugated conical horn illuminates a section of a hyperbolic reflector to produce a spherical aperture phase front. This front produces a far field beam with low sidelobes and high beam efficiency. The system is insensitive to frequency and polarization changes, and is also insensitive to orientation about the axis of the conical horn for beam scanning.
Green, K. A.↗
An implicit finite-difference algorithm for hyperbolic systems in conservation-law form
An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.
Beam, R. M.↗
Composite methods for hyperbolic equations
A composite approximation procedure combining the properties of the Lax-Wendroff and leapfrog algorithms is proposed for solving hyperbolic equations. For a one-dimensional equation, a three-step approximation consisting of a two-step Richtmeyer method followed by a leapfrog step is considered. This is a two-level scheme, so all difficulties, including storage requirements, associated with the three-level leapfrog are eliminated. For two-dimensional problems a generalization of the preceding method is used consisting of a rotated Richtmeyer method followed by a modified leapfrog step. It is found that the composite schemes are effective in reducing oscillations and nonlinear instabilities that affect the leapfrog method. The dissipation in the composite schemes is much less than in the Richtmeyer algorithm, and hence can be used for long term integrations.
Turkel, E.↗
Energy conserving norms for the solution of hyperbolic systems of partial differential equations
The hyperbolic system of partial differential equations with a real constant square coefficient matrix A is considered. The problem of finding an energy conserving norm for the solution of the system is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm. The conditions serve as criteria on a given system which determine whether or not the solution will have its energy conserved in some norm. Examples of specific systems and boundary conditions are also provided.
Gunzburger, M. D.↗
Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems
Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.
Skollermo, G.↗
The Split Coefficient Matrix method for hyperbolic systems of gasdynamic equations
The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.
Chakravarthy, S. R.↗
Central configurations and hyperbolic-elliptic motion in the three-body problem
A refined classification of motion for the planar three-body problem with zero-sum total energy is presented. The structure and dimensions of the sets of initial conditions leading to parabolic expansion or hyperbolic-elliptic motion are found. Attention is given to the use of the Hamiltonian of the system and to the two- and three-dimensional problem. It is verified that the new coordinates of position and linear momentum approaches limits as t (time) approaches infinity, and that these limits constitute equilibrium solutions to the three-body problem representing control configurations.
Hulkower, N. D.↗