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NASA NTRS1972

Buckling of circular cylindrical shells under axial compression.

The infinitely long cylindrical shell under axial compression with axisymmetric sinusoidal imperfections is considered. The bifurcation problem is formulated exactly, and a complete family of buckling modes is identified. Koiter's (1963) upper bound pertains to the critical stress associated with one restricted set of modes, and the critical stress for a set of long wavelength modes reproduces some numerical results obtained by Almroth (1966). The initial postbuckling analysis is also formulated exactly. An exact analytical solution is obtained for the limiting case of modes with infinitely long wavelengths, and numerical analysis is used to solve the equations for the other cases. An appendix gives the details of the analysis.

Budiansky, B.↗

NASA NTRS1972

Response of convectively controlled burning to nonlinear disturbances.

A process where energy release rate is proportional to the square root of the Reynolds number of a burning particle or surface is used to demonstrate some distinctive dynamic properties of a disturbed convectively controlled burning process. The instantaneous energy release rates caused by assumed periodic flow field disturbances are numerically evaluated and examined. Correlation coefficients which express the energy released in-phase with the pressure disturbance are evaluated and analytical solutions for these coefficients are derived. The response of the process is shown to be highly sensitive to the harmonic distortion of the disturbance. For some disturbances the energy released in-phase with the pressure disturbance is an order of magnitude larger than that for linear (sinusoidal) disturbances. The results show that harmonic distortion increases the coupling between the burning process and the flow field. This amplifying effect of harmonic distortion is suppressed when a disturbance becomes steep-fronted; therefore, steepening can act to limit the equilibrium amplitudes in disturbed systems. Amplification is also suppressed when the velocity disturbance is phase shifted with respect to the pressure disturbance or when the process is exposed to a high steady velocity.

Heidmann, M. F.↗

NASA NTRS1973

Use of multivariable asymptotic expansions in a satellite theory

Initial conditions and perturbative force of satellite are restricted to yield motion of equatorial satellite about oblate body. In this manner, exact analytic solution exists and can be used as standard of comparison in numerical accuracy comparisons. Detailed numerical accuracy studies of uniformly valid asymptotic expansions were made.

Dallas, S. S.↗

NASA NTRS1972

Monte Carlo inversion of seismic data

The analytic solution to the linear inverse problem provides estimates of the uncertainty of the solution in terms of standard deviations of corrections to a particular solution, resolution of parameter adjustments, and information distribution among the observations. It is shown that Monte Carlo inversion, when properly executed, can provide all the same kinds of information for nonlinear problems. Proper execution requires a relatively uniform sampling of all possible models. The expense of performing Monte Carlo inversion generally requires strategies to improve the probability of finding passing models. Such strategies can lead to a very strong bias in the distribution of models examined unless great care is taken in their application.

Wiggins, R. A.↗

NASA NTRS1973

The induced magnetosphere of the moon. 1: Theory

An analytic solution for the magnetic field in the space defined by a spherical moon and its downstream cylindrical cavity formed by the solar wind is derived for interplanetary magnetic fields both parallel and perpendicular to the cavity axis. By superposition, the solution is obtained for arbitrary orientations of the interplanetary field. The theory is quasi-static and is formulated in terms of a scalar magnetic potential. Thus the moon model consists of a core of arbitrary size and infinite electrical conductivity surrounded by a nonconducting shell; the cavity volume is assumed to be nonconducting. The variation of the magnetic field on the lunar surface, both on the sunlit and on the dark side hemispheres, and on the cavity boundary is presented for various values of core radius. The solution yields the distribution of currents on the lunar sunlit surface and the surface of the cavity. Theoretical transfer functions are presented and their variations with position on the lunar surface and with core size are discussed.

Schubert, G.↗

NASA NTRS1972

Models for the interaction of rarefied gas with a surface

Various theoretical models are discussed for the interaction of gaseous atoms with solids at energies of approximately 10 eV. The surface is modeled by a one- and three-dimensional aggregate of atoms with different initial conditions and lattice parameters. In calculating the accommodation coefficients, the direct integration of the equations of motion of gaseous molecules and atoms in the crystal lattice of the solid is used. An analysis of the advantages and the drawbacks of the models, as well as the possibilities of the analytical solutions in a one-dimensional case, is given. The dependence of the energy and the normal and the tangential momenta accommodation coefficients on the collision parameters is obtained.

Pyarnpuu, A. A.↗

NASA NTRS1973

Alfven waves in spiral interplanetary field

A theoretical study is presented of the Alfven waves in the spiral interplanetary magnetic field. The Alfven waves under consideration are arbitrary, large amplitude, non-monochromatic, microscale waves of any polarization. They superpose on a mesoscale background flow of thermally anisotropic plasma. Using WKB approximation, an analytical solution for the amplitude vectors is obtained as a function of the background flow properties: density, velocity, Alfven speed, thermal anisotropy, and the spiral angel. The necessary condition for the validity of the WKB solution is discussed. The intensity of fluctuations is calculated as a function of heliocentric distance. Relative intensity of fluctuations as compared with the magnitude of the background field has its maximum in the region near l au. Thus outside of this region, the solar wind is less turbulent.

Whang, Y. C.↗

NASA NTRS1973

High speed flow past wings

The analytical solution to the transonic small perturbation equation which describes steady compressible flow past finite wings at subsonic speeds can be expressed as a nonlinear integral equation with the perturbation velocity potential as the unknown function. This known formulation is substituted by a system of nonlinear algebraic equations to which various methods are applicable for its solution. Due to the presence of mathematical discontinuities in the flow solutions, however, a main computational difficulty was to ensure uniqueness of the solutions when local velocities on the wing exceeded the speed of sound. For continuous solutions this was achieved by embedding the algebraic system in an one-parameter operator homotopy in order to apply the method of parametric differentiation. The solution to the initial system of equations appears then as a solution to a Cauchy problem where the initial condition is related to the accompanying incompressible flow solution. In using this technique, however, a continuous dependence of the solution development on the initial data is lost when the solution reaches the minimum bifurcation point. A steepest descent iteration technique was therefore, added to the computational scheme for the calculation of discontinuous flow solutions. Results for purely subsonic flows and supersonic flows with and without compression shocks are given and compared with other available theoretical solutions.

Norstrud, H.↗

NASA NTRS1973

Study of the far wake vortex field generated by a rectangular airfoil in a water tank

Underwater towing experiments were carried out with a rectangular airfoil of aspect ratio 5.3 at 4 and 8 deg angles of attack and at chord-based Reynolds numbers between 2 x 100,000 and 7.5 x 100,000. Quantitative measurements by means of the hydrogen bubble technique indicated lower peak swirl velocities in the range of 100 to 1000 lenghts downstream than have been measured in wind tunnel of flight tests. The maximum circumferential velocity decayed whereas the turbulent eddy viscosity increased. This behavior and other known rates of vortex decay are explained in terms of an analytical solution for the vortex problem with time varying eddy viscosity. It is shown that this case corresponds to nonequilibrium turbulent vortex flow.

Lezius, D. K.↗

NASA NTRS1973

Optical potential approach to the electron-atom impact ionization threshold problem

The problem of the threshold law for electron-atom impact ionization is reconsidered as an extrapolation of inelastic cross sections through the ionization threshold. The cross sections are evaluated from a distorted wave matrix element, the final state of which describes the scattering from the Nth excited state of the target atom. The actual calculation is carried for the e-H system, and a model is introduced which is shown to preserve the essential properties of the problem while at the same time reducing the dimensionability of the Schrodinger equation. Nevertheless, the scattering equation is still very complex. It is dominated by the optical potential which is expanded in terms of eigen-spectrum of QHQ. It is shown by actual calculation that the lower eigenvalues of this spectrum descend below the relevant inelastic thresholds; it follows rigorously that the optical potential contains repulsive terms. Analytical solutions of the final state wave function are obtained with several approximations of the optical potential.

Temkin, A.↗

NASA NTRS1973

The application of far IR lasers to the investigation of the optical properties of materials

The effect of diffusion on the saturation and absorption properties of molecular gases is considered. The analysis starts with a simple derivation of the multi-level rate equations. Incorporating the diffusion term, calculations are carried out both for the homogeneous and inhomogeneous broadening cases. Both graphical and analytical solutions are obtained for the absorption coefficient and the saturation intensity. It is shown that the effect of diffusion is characterized by a diffusion function, which varies inversely with the radius of the beam at low pressures and small beam sizes. The analysis is done in terms of two parameters k1a and k2a, which are simple functions of the intensity of the beam I, the effective relaxation rate w, and the effective diffusion constant D the a is designated as the radius of the beam.

Stafsudd, O. M.↗

NASA NTRS1973

Optimum TPS Design with Rei-mullite

It has been shown that a ceramic mullite fiber/mullite insulation has adequate margins-of-safety based on minimum strengths for the critical thermostructural conditions of the shuttle mission. Introduction into the design of the new lower density foam bond material provides cold soak and cold entry capability at 116 K (-250 F). Finally, design solutions to the coating cracking tendencies have been identified and the solution analytically verified with three dimensional thermostructural analysis.

Hess, T. E.↗

NASA NTRS1972

Constrained optimal design of columns against buckling.

The buckling loads of straight, simply supported columns, loaded by axial thrust are maximized, subject to the restriction that the available volume of structural material is specified. In addition, a constraint is placed upon the maximum allowable prebuckling stress. Analytic solutions are presented for a variety of different cross-sectional geometries.

Frauenthal, J. C.↗

NASA NTRS1973

Analysis of sublimation-cooled coated mirrors in convective and radiative environments.

Analytical solutions were obtained for the thermal response of a transpiration- or sublimation-cooled spherical mirror coating exposed to convective and radiative heating. The solutions allow unlimited spectral detail to be accounted for. Results indicate that transpiration-cooled thick coatings (1 cm) may withstand up to 10 kW/sq cm on a steady basis without excessive temperature rise for quartzlike materials with an internal absorption coefficient of 0.01 per cm. On a transient basis, fluxes up to 20 kW/sq cm can be accommodated for a second (cW laser exposure time), 4 kW/sq cm for 5 sec (planetary entry heating time), and of the order of MW/sq cm for millisecond times (short-duration laser bursts) without transpiration cooling for a material with an absorption coefficient of 0.1 per cm. Proportionately higher fluxes can be accommodated with lower absorption coefficients. Thermal stresses produced by the heat pulse are found to be high but within the strength of the materials. The regime in which meaningful solutions may be obtained is mapped in detail.

Howe, J. T.↗

NASA NTRS1973

Dynamics of moisture diffusion through a partially liquid filled porous matrix.

Models for unsteady water transport in fuel cell configurations of Bacon-Pratt and Whitney type are derived. Tests with a specially constructed isothermal cell producing no water and order of magnitude analysis show that diffusion through the liquid alone cannot account for the observed behavior. An analytical solution which includes diffusion in both the liquid and the gas spaces of the matrix is in better agreement with the data than models which consider diffusion in either the liquid or in the gas phase alone.

Murty, B. G. K.↗

NASA NTRS1973

Induced magnetosphere of the moon. I - Theory.

An analytic solution for the magnetic field in the space defined by a spherical moon and its downstream cylindrical cavity formed by the solar wind is derived for interplanetary magnetic fields both parallel and perpendicular to the cavity axis. By superposition, the solution is obtained for arbitrary orientations of the interplanetary field. The theory is quasi-static and is formulated in terms of a scalar magnetic potential. Thus, the moon model consists of a core of arbitrary size and infinite electrical conductivity surrounded by a nonconducting shell; the cavity volume is also assumed to be nonconducting. The variation of the magnetic field on the lunar surface (both sunlit and dark hemispheres) and on the cavity boundary is presented for various values of core radius.

Schubert, G.↗

NASA NTRS1973

On the structure of the magnetotail current sheet.

A self-consistent tail current sheet model described by an exact analytic solution of the time-independent Vlasov-Maxwell equations is presented. The model has a 'slingshot' field configuration with field lines outside the plasma sheet slightly flared in the antisolar direction. It is pointed out that when the model parameters are adjusted to agree with the spatial variation along the tail the required thickness of the neutral sheet must be about 2.5 earth radii, instead of less than or equal to 1 earth radius, as indicated by observations. Furthermore, it is shown qualitatively that a considerable velocity shear must be present in the tail current sheet if the plasma sheet is indeed much thicker than the neutral sheet.

Kan, J. R.↗

NASA NTRS1973

Quasi-periodic orbits about the translunar libration point.

Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincare and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32,379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a 'halo orbit' of the moon.

Farquhar, R. W.↗