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Two variable expansion method applied to ordinary differential equation and two wave propagation problems, comparing results with matched asymptotic expansions and coordinate stretching methods
A quantitative method for astrometrically detecting perturbations induced in a star's motion by the presence of a planetary object is described. A periodogram is defined, wherein signals observed from a star show exactly periodic variations, which can be extracted from observational data using purely statistical methods. A detection threshold is defined for the frequency of occurrence of some detectable signal, e.g., the Nyquist frequency. Possible effects of a stellar orbital eccentricity and multiple companions are discussed, noting that assumption of a circular orbit assures the spectral purity of the signal described. The periodogram technique was applied to 12 yr of astrometric data from the U.S. Naval Observatory for three stars with low mass stellar companions. Periodic perturbations were confirmed. A comparison of the accuracy of different astrometric systems shows that the detection accuracy of a system is determined by the measurement accuracy and the number of observations, although the detection efficiency can be maximized by minimizing the number of data points for the case when observational errors are proportional to the square root of the number of data points. It is suggested that a space-based astrometric telescope is best suited to take advantage of the method.
If a satellite has a mean daily motion nearly commensurable with the angular velocity of rotation of the earth, long-period perturbations will influence its motion. The eccentricity of such a commensurable orbit is not necessarily small, as the orbit of Explorer VI shows. This circumstance caused the author to develop a semianalytical method of treating these perturbations, which avoids the development of the disturbing function into powers of the eccentricity and consequently includes the cases of elongated orbits and orbits with moderate eccentricities.
Efforts are described to extend the averaged Lagrangian method of describing small signal wave propagation and nonlinear wave interaction, developed by earlier workers for cold plasmas, to the more general conditions of warm collisionless plasmas, and to demonstrate particularly the effectiveness of the method in analyzing wave-wave interactions. The theory is developed for both the microscopic description and the hydrodynamic approximation to plasma behavior. First, a microscopic Lagrangian is formulated rigorously, and expanded in terms of perturbations about equilibrium. Two methods are then described for deriving a hydrodynamic Lagrangian. In the first of these, the Lagrangian is obtained by velocity integration of the exact microscopic Lagrangian. In the second, the expanded hydrodynamic Lagrangian is obtained directly from the expanded microscopic Lagrangian. As applications of the microscopic Lagrangian, the small-signal dispersion relations and the coupled mode equations are derived for all possible waves in a warm infinite, weakly inhomogeneous magnetoplasma, and their interactions are examined.
A structural expansion for the static ground-state energy of a simple metal is derived. Two methods are presented, one an approach based on single-particle band structure which treats the electron gas as a nonlinear dielectric, the other a more general many-particle analysis using finite-temperature perturbation theory. The two methods are compared, and it is shown in detail how band-structure effects, Fermi-surface distortions, and chemical-potential shifts affect the total energy. These are of special interest in corrections to the total energy beyond third order in the electron-ion interaction and hence to systems where differences in energies for various crystal structures are exceptionally small. Preliminary calculations using these methods for the zero-temperature thermodynamic functions of atomic hydrogen are reported.
In the Probabilistic Finite Element Method (PFEM), finite element methods have been efficiently combined with second-order perturbation techniques to provide an effective method for informing the designer of the range of response which is likely in a given problem. The designer must provide as input the statistical character of the input variables, such as yield strength, load magnitude, and Young's modulus, by specifying their mean values and their variances. The output then consists of the mean response and the variance in the response. Thus the designer is given a much broader picture of the predicted performance than with simply a single response curve. These methods are applicable to a wide class of problems, provided that the scale of randomness is not too large and the probabilistic density functions possess decaying tails. By incorporating the computational techniques we have developed in the past 3 years for efficiency, the probabilistic finite element methods are capable of handling large systems with many sources of uncertainties. Sample results for an elastic-plastic ten-bar structure and an elastic-plastic plane continuum with a circular hole subject to cyclic loadings with the yield stress on the random field are given.
An operational stratospheric correction scheme used after the Mount Pinatubo (Phillipines) eruption (Jun. 1991) is presented. The stratospheric aerosol distribution is assumed to be only variable with latitude. Each 9 days the latitudinal distribution of the optical thickness is computed by inverting radiances observed in the NOAA AVHRR channel 1 (0.63 micrometers) and channel 2 (0.83 micrometers) over the Pacific Ocean. This radiance data set is used to check the validity of model used for inversion by checking consistency of the optical thickness deduced from each channel as well as optical thickness deduced from different scattering angles. Using the optical thickness profile previously computed and radiative transfer code assuming Lambertian boundary condition, each pixel of channel 1 and 2 are corrected prior to computation of NDVI (Normalized Difference Vegetation Index). Comparison between corrected, non corrected, and years prior to Pinatubo eruption (1989 to 1990) NDVI composite, shows the necessity and the accuracy of the operational correction scheme.
The coupled-cluster and Brueckner electron correlation methods are used to determine the equilibrium structures, dipole moments, harmonic vibrational frequencies, and IR intensities of NH3, FON, Be3, BeC2, and BeO2. The singles and doubles coupled-cluster and Brueckner doubles methods are employed, and the corresponding methods that include a perturbational estimate of connected triple excitations are also investigated. The T sub 1 diagnostic is found to provide a good indication of the magnitude of the difference between the results obtained with the coupled-cluster and Brueckner methods. For NH3, the T sub 1 diagnostic is small and so the differences between results obtained from coupled-cluster and Brueckner theories are quite small. It is found for all of the molecules under consideration that inclusion of the contribution from connected triple excitations is more important than the differences between the Brueckner and coupled-cluster correlation methods.
Halphen method and a resonance theory for determining long range effects of moon, sun and earth gravitational field irregularity in motion of artificial satellites
Determining higher order perturbations for lunar orbiter by Krylov-Bololiubov and Poincare methods
A method is presented of computing the dayside global earth magnetic field which is in equilibrium with the plasma pressure, based on satellite observations at a local region of the magnetosphere. The method, which utilizes a perturbation around a dipole magnetic field, involves computation of the global plasma pressure profile based on the equatorial (anisotropic) pressure data, derivation of the current profile which satisfies the equilibrium condition, and computation of the magnetic field using the current profile and the boundary current produced by the solar wind. The method is applied for the Active Magnetospheric Particle Tracer Explorers data, and the result of the computation is found to compare reasonably well with the observed magnetic field profile near the geomagnetic equator.
A method is presented for calculating the three orthogonal components of the velocity perturbations of satellite fragments, with a view to ascertaining the nature and intensity of the satellite breakup. The method employs three simultaneous equations furnished by changes in fragment specific energy, specific angular momentum, and plane orientation. Velocity perturbations are thereby calculated for fragments from 20 major satellite breakup events; these results, in conjunction with a technique for determining fragment masses, yield a description of the breakup process.
An implicit finite difference code using flux vector splitting has been developed for solving the two-dimensional inviscid gas dynamics equations. The method is spatially second-order acurate, fully conservative, and uses body-conforming generalized coordinates for treating complex geometries. Numerical results have been obtained for transonic flow over a circular cylinder and airfoils. Steady results for a half cylinder (top and bottom symmetry-imposed) range from critical flow to a strong shock case with rotationally induced flow separation. Full cylinder solutions at freestream Mach number values of 0.5, however, show unsteady oscillation. A perturbation form of the method has also been developed and used to compute both fore and aft inviscid flow separation about a cylinder for a nonuniform incoming stream.
A first order, semianalytical theory for the long term motion of resonant satellites is presented. The theory is valid for all eccentricities and inclinations and for all commensurability ratios. The method allows the inclusion of all the zonal and tesseral harmonics as well as luni solar perturbations and radiation pressure. The method is applied to a synchronous satellite including only the J sub 2 and J sub 22 harmonics. Global, long term solutions for this problem, eccentricity, argument of perigee, and inclination are obtained.
A small-scale wind tunnel previously used for turbulent boundary layer experiments was modified for two sets of boundary layer transition studies. The first study concerns a laminar separation/turbulent reattachment. The pressure gradient and unit Reynolds number are the same as the fully turbulent flow of Spalart and Watmuff. Without the trip wire, a laminar layer asymptotes to a Falkner & Skan similarity solution in the FPG. Application of the APG causes the layer to separate and a highly turbulent and approximately 2D mean flow reattachment occurs downstream. In an effort to gain some physical insight into the flow processes a small impulsive disturbance was introduced at the C(sub p) minimum. The facility is totally automated and phase-averaged data are measured on a point-by-point basis using unprecedently large grids. The evolution of the disturbance has been tracked all the way into the reattachment region and beyond into the fully turbulent boundary layer. At first, the amplitude decays exponentially with streamwise distance in the APG region, where the layer remains attached, i.e. the layer is viscously stable. After separation, the rate of decay slows, and a point of minimum amplitude is reached where the contours of the wave packet exhibit dispersive characteristics. From this point, exponential growth of the amplitude of the disturbance is observed in the detached shear layer, i.e. the dominant instability mechanism is inviscid. A group of large-scale 3D vortex loops emerges in the vicinity of the reattachment. Remarkably, the second loop retains its identify far downstream in the turbulent boundary layer. The results provide a level of detail usually associated with CFD. Substantial modifications were made to the facility for the second study concerning disturbances generated by Suction Holes for laminar flow Control (LFC). The test section incorporates suction through interchangeable porous test surfaces. Detailed studies have been made using isolated holes in the impervious test plate that used to establish the Blasius base flow. The suction is perturbed harmonically and data are averaged on the basis of the phase of the disturbance, for conditions corresponding to strong suction and without suction. The technique was enhanced by using up to nine multiple probes to reduce the experimental run-time. In both cases, 3D contour surfaces in the vicinity of the hole show highly 3D TS waves which fan out in the spanwise direction forming bow-shaped waves downstream. The case without suction has proved useful for evaluating calculation methods. With suction, the perturbations on the centerline are much stronger and decay less rapidly, while the TS waves in the far field are similar to the case without suction. Downstream, the contour surfaces of the TS waves develop spanwise irregularities which eventually form into clumps. The spanwise clumping is evidence of a secondary instability that could be associated with suction vortices. Designers of porous surfaces use Goldsmith's Criterion to minimize cross-stream interactions. It is shown that partial TS wave cancellation is possible, depending on the hole spacing, disturbance frequency and free-stream velocity. New high-performance Constant Temperature Hot-Wire Anemometers were designed and built, based on a linear system theory analysis that can be extended to arbitrary order. The motivation was to achieve the highest possible frequency reponse while ensuring overall system stability. The performance is equal to or superior to commercially available instruments at about 10% of the cost. Details, such as fabrication drawings and a parts list, have been published to enable the instrument to be construced by others.
Improved formulation reduces error and number of computational steps. Report describes method for calculating partial derivative of geopotential perturbation with respect to time. Useful in computations of orbit of lightweight satellite in which total orbital energy or related parameter one of dependent variables.
Techniques for improving the performance of robotic-manipulator path-control systems comprising independent SISO feedback controllers for each joint are discussed and illustrated with block diagrams, reviewing the results of recent analytical investigations. Topics examined include the servo design for a single link, the equations of motion for manipulators, SISO servo design for multiple links, inverse methods, pole placement with compensation of the gravity terms, linear state-feedback control based on the perturbation equations, and adaptive control methods. Consideration is given to variable-structure systems, suboptimal controllers, and the optimal-design-strategy approach.
The technique of implicit differentiation has been used in combination with linearized lifting-surface theory to derive analytical expressions for aerodynamic sensitivities (i.e., rates of change of lifting pressures with respect to general changes in aircraft geometry, including planform variations) for steady or oscillating planar or nonplanar lifting surfaces in subsonic, sonic, or supersonic flow. The geometric perturbation is defined in terms of a single variable, and the user need only provide simple expressions or similar means for defining the continuous or discontinuous global or local perturbation of interest. Example expressions are given for perturbations of the sweep, taper, and aspect ratio of a wing with trapezoidal semispan planform. In addition to direct computational use, the analytical method presented here should provide benchmark criteria for assessing the accuracy of aerodynamic sensitivities obtained by approximate methods such as finite geometry perturbation and differencing. The present process appears to be readily adaptable to more general surface-panel methods.