On the use of approximate analytical solutions in solving optimum trajectory problems
Method for numerical solutions of nonlinear optimum trajectory problems using approximate analytic method
APPROXIMATION METHOD↗
SEARCH · Engineering Papers
Results for “analytical solution”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Analytic Solutions Of Equations For Q-Switched Lasers
Approximate closed-form, analytic equations for characteristics of Q-switched, continuous-wave-pumped laser express relationships among characteristics of output, pumping rate above threshold pumping rate, and rate of repetition of switch. Equations describe performance in terms of peak output power, energy in each pulse, average output power, and duration of each pulse as functions of pulse-repetition rate, actual pump power, threshold pump power, volume occupied by laser electromagnetic mode, and reflectivity of output coupler. Useful for analyzing performances of diode-pumped solid-state lasers.
Wilson, Keith E.↗
A complete analytical solution for the inverse instantaneous kinematics of a spherical-revolute-spherical (7R) redundant manipulator
Using a method based upon resolving joint velocities using reciprocal screw quantities, compact analytical expressions are generated for the inverse solution of the joint rates of a seven revolute (spherical-revolute-spherical) manipulator. The method uses a sequential decomposition of screw coordinates to identify reciprocal screw quantities used in the resolution of a particular joint rate solution, and also to identify a Jacobian null-space basis used for the direct solution of optimal joint rates. The results of the screw decomposition are used to study special configurations of the manipulator, generating expressions for the inverse velocity solution for all non-singular configurations of the manipulator, and identifying singular configurations and their characteristics. Two functions are therefore served: a new general method for the solution of the inverse velocity problem is presented; and complete analytical expressions are derived for the resolution of the joint rates of a seven degree of freedom manipulator useful for telerobotic and industrial robotic application.
Podhorodeski, R. P.↗
Analytical solution for boundary heat fluxes from a radiating rectangular medium
Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.
Siegel, R.↗
Analytic solutions of the radial pulsation equation for rotating and magnetic star models
Eddington's (1926) form of wave equation for small-amplitude, radial, adiabatic stellar pulsations of spherically symmetric, gaseous stars is generalized to include the effects of axial rotation and tangled magnetic fields. Equilibrium quantities possessing dimensions are affected, and the relative importance of rotation and magnetism in affecting pulsation characteristics of the models depends on the choices of gamma and the type of model. Solutions are obtained in closed form for adiabatic pulsation periods of fine analytic stellar models, and nonadiabatic stability criteria are determined by means of the one-zone stellar model. Results are discussed for a range of physical parameters such as rotational angular momenta, central condensations, and magnetic energies; and applications are made to the case of classical Cepheids and other variable giant stars.
Stothers, R.↗
An approximate analytic solution of the steady- state thermal design problem of space radiators.
Space radiator steady state thermal design using approximate analytic method
Chapman, A. J.↗
Analytical solution for the wind-driven circulation in a lake containing an island
An analysis was carried out to determine analytically the effect of an island on the wind driven currents in a shallow lake (or sea). A general analysis is developed that can be applied to a large class of lake and island geometries and bottom topographies. Detailed numerical results are obtained for a circular island located eccentrically or concentrically in a circular lake with a logarithmic bottom topography. It is shown that an island can produce volume flow (vertically integrated velocities) gyres that are completely different from those produced by a normal basin without an island. These gyres in the neighborhood of the island will produce different velocity patterns, which include the acceleration of flow near the island shore.
Goldstein, M. E.↗
Construction of approximate analytical solutions to a new class of non-linear oscillator equation
The principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire. By assuming that harmonic balance will hold, solutions are devised for a steady state limit cycle and/or limit point motion. A method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective Fourier expansions. The latter technique is actually a generalization of the method proposed by Kryloff and Bogoliuboff (1943).
Mickens, R. E.↗
A simplified analytical solution for thermal response of a one-dimensional, steady state transpiration cooling system in radiative and convective environment
A simplified analytical method for calculation of thermal response within a transpiration-cooled porous heat shield material in an intense radiative-convective heating environment is presented. The essential assumptions of the radiative and convective transfer processes in the heat shield matrix are the two-temperature approximation and the specified radiative-convective heatings of the front surface. Sample calculations for porous silica with CO2 injection are presented for some typical parameters of mass injection rate, porosity, and material thickness. The effect of these parameters on the cooling system is discussed.
Kubota, H.↗
A canopy reflectance model based on an analytical solution to the multiple scattering equation
An approximation to the radiative transfer equation for solar radiation in relatively full, homogeneous plant canopies is presented and solved analytically for solar zenith angles less than 60 deg. The model predicts reflectance at any depth in the canopy and in any direction and may be inverted with bidirectional reflectance measurements. The model was fit to data at two sun angles and two wavebands (visible and NIR) to within the assumed errors on the reflectance data. The calculated albedos are insensitive to achievable measurement errors. Some of the parameter values themselves found by the inversion agree reasonably well with independent measurements, but the uncertainties introduced by the data noise are rather large. However, the agreement is good enough to demonstrate that the model is physically realistic.
Camillo, Peter↗
Analytical solutions with Generalized Impedance Boundary Conditions (GIBC)
The diffraction by a material discontinuity in a thick dielectric/ferrite layer is considered by modeling the layer as a distributed current sheet obeying generalized sheet transition conditions (GSTC's). The sheet currents are then formulated and solved via the standard dual integral equation approach. This yields the diffracted field in terms of unknown constants which underscore the non-uniqueness of the GSTC current sheet representation. The constants are dependent on the geometry and properties of the discontinuity and are determined by enforcing field continuity across the material junction. This requires the field internal to the slab which are determined from the external ones via analytic continuity. Results are given which validate the solution and demonstrate the importance of the constants.
Ricoy, Mark A.↗
A class of analytic solutions for the thermally balanced magnetostatic prominence sheet
A theoretical study is presented for the nonlinear interplay between magnetostatic equilibrium and energy balance in a Kippenhahn-Schlueter type solar prominence sheet. A class of theoretical models is presented, expressed in closed analytic forms, thus facilitating the direct illustration of the nonlinear physical properties. The model couples the equilibrium between magnetic field, plasma pressure, and weight on the one hand, with the balance between a rho-squared T radiative loss, a rho wave heating (where rho equals plasma density, and T equals plasma temperature), and thermal conduction channeled along magnetic field lines on the other. The steady solutions are divided into three classes, and are characterized by the total wave heating in the prominence sheet which is greater than, equal to, or less than the total radiative loss. The compaction of the plasma along the field lines, under its own weight, and the energy transport effects determine which of the three basic behaviors obtains in a particular situation. A discussion is presented of the implications of the steady solutions for the formation of prominences.
Low, B. C.↗
Numerical and Analytical Solutions of Hypersonic Interactions Involving Surface Property Discontinuities
The local viscous-inviscid interaction field generated by a wall temperature jump on a flat plate in supersonic flow and on the windside of a Reusable Launch Vehicle in hypersonic flow is studied in detail by both a Navier-Stokes numerical code and an analytical triple-deck model. Treatment of the rapid heat transfer changes both upstream and downstream of the jump is included. Closed form relationships derived from the triple-deck theory are presented. The analytically predicted pressure and heating variations including upstream influence are found to be in generally good agreement with the Computational Fluid Dynamic (CFD) predictions. These analyses not only clarify the interactive physics involved but also are useful in preliminary design of thermal protection systems and as an insertable module to improve CFD code efficiency when applied to such small-scale interaction problems. The analyses only require conditions at the wall and boundary-layer edge which are easily extracted from a baseline, constant wall temperature, CFD solution.
Gnoffo, Peter A.↗
On the energy dependence of the radial diffusion coefficient and spectra of inner radiation belt particles - Analytic solutions and comparison with numerical results
A theoretical method by which the energy dependence of the radial diffusion coefficient may be deduced from spectral observations of the particle population at the inner edge of the earth's radiation belts is presented. This region has previously been analyzed with numerical techniques; in this report an analytical treatment that illustrates characteristic limiting cases in the L shell range where the time scale of Coulomb losses is substantially shorter than that of radial diffusion (L approximately 1-2) is given. It is demonstrated both analytically and numerically that the particle spectra there are shaped by the energy dependence of the radial diffusion coefficient regardless of the spectral shapes of the particle populations diffusing inward from the outer radiation zone, so that from observed spectra the energy dependence of the diffusion coefficient can be determined. To insure realistic simulations, inner zone data obtained from experiments on the DIAL, AZUR, and ESRO 2 spacecraft have been used as boundary conditions. Excellent agreement between analytic and numerical results is reported.
Westphalen, H.↗
The analysis of non-linear dynamic behavior (including snap-through) of postbuckled plates by simple analytical solution
Static postbuckling and nonlinear dynamic analysis of plates are usually accomplished by multimode analyses, although the methods are complicated and do not give straightforward understanding of the nonlinear behavior. Assuming single-mode transverse displacement, a simple formula is derived for the transverse load displacement relationship of a plate under in-plane compression. The formula is used to derive a simple analytical expression for the static postbuckling displacement and nonlinear dynamic responses of postbuckled plates under sinusoidal or random excitation. Regions with softening and hardening spring behavior are identified. Also, the highly nonlinear motion of snap-through and its effects on the overall dynamic response can be easily interpreted using the single-mode formula. Theoretical results are compared with experimental results obtained using a buckled aluminum panel, using discrete frequency and broadband point excitation. Some important effects of the snap-through motion on the dynamic response of the postbuckled plates are found.
Ng, C. F.↗
A singularity free analytical solution of artificial satellite motion with drag
The connection between the existing Delaunay-Similar and Poincare-Similar satellite theories in the true anomaly version is outlined for the J(2) perturbation and the new drag approach. An overall description of the concept of the approach is given while the necessary expansions and the procedure to arrive at the computer program for the canonical forces is delineated. The procedure for the analytical integration of these developed equations is described. In addition, some numerical results are given. The computer program for the algebraic multiplication of the Fourier series which creates the FORTRAN coding in an automatic manner is described and documented.
Scheifele, G.↗
Analytical solution for the drift behavior of Dynamics Explorer-B
The Dynamics Explorer (DE) mission was designed to explore the earth's magnetosphere, ionosphere, and plasmasphere. The DE mission employs two spacecraft, including DE-A and DE-B. The spacecraft were launched from the Western Test Range on Aug. 3, 1981, onboard the same Delta launch vehicle. The two spacecraft were placed in a coplanar polar orbit, with DE-A in a high altitude orbit (perigee = 570 km and apogee = 23174 km) and DE-B in a low altitude orbit (perigee = 309 km and apogee = 1013 km). In the present investigation, an attempt has been made to validate the flight data for DE-B with analytic and simulation results. The external torques acting on the spacecraft are represented in terms of tractable mathematical functions. A piecewise linear model of the superrotation of the upper atmosphere is assumed. The effect of individual torques on the long-term pitch axis motion is investigated using analytic and simulation methods. The results are found to be in very good agreement with the available flight data.
Sellappan, R. G.↗
Analytic Solution to the Problem of Aircraft Electric Field Mill Calibration
It is by no means a simple task to retrieve storm electric fields from an aircraft instrumented with electric field mill sensors. The presence of the aircraft distorts the ambient field in a complicated way. Before retrievals of the storm field can be made, the field mill measurement system must be "calibrated". In other words, a relationship between impressed (i.e., ambient) electric field and mill output must be established. If this relationship can be determined, it is mathematically inverted so that ambient field can be inferred from the mill outputs. Previous studies have primarily focused on linear theories where the relationship between ambient field and mill output is described by a "calibration matrix" M. Each element of the matrix describes how a particular component of the ambient field is enhanced by the aircraft. For example the product M(sub ix), E(sub x), is the contribution of the E(sub x) field to the i(th) mill output. Similarly, net aircraft charge (described by a "charge field component" E(sub q)) contributes an amount M(sub iq)E(sub q) to the output of the i(th) sensor. The central difficulty in obtaining M stems from the fact that the impressed field (E(sub x), E(sub y), E(sub z), E(sub q) is not known but is instead estimated. Typically, the aircraft is flown through a series of roll and pitch maneuvers in fair weather, and the values of the fair weather field and aircraft charge are estimated at each point along the aircraft trajectory. These initial estimates are often highly inadequate, but several investigators have improved the estimates by implementing various (ad hoc) iterative methods. Unfortunately, none of the iterative methods guarantee absolute convergence to correct values (i.e., absolute convergence to correct values has not been rigorously proven). In this work, the mathematical problem is solved directly by analytic means. For m mills installed on an arbitrary aircraft, it is shown that it is possible to solve for a single 2m-vector that provides all other needed variables (i.e., the unknown fair weather field, the unknown aircraft charge, and the unknown matrix M). Numerical tests of the solution, effects of measurement errors, and studies of solution non-uniqueness are ongoing as of this writing.
Koshak, William↗