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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.

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At least 19 records

Force Density Function Relationships in 2-D Granular Media

An integral transform relationship is developed to convert between two important probability density functions (distributions) used in the study of contact forces in granular physics. Developing this transform has now made it possible to compare and relate various theoretical approaches with one another and with the experimental data despite the fact that one may predict the Cartesian probability density and another the force magnitude probability density. Also, the transforms identify which functional forms are relevant to describe the probability density observed in nature, and so the modified Bessel function of the second kind has been identified as the relevant form for the Cartesian probability density corresponding to exponential forms in the force magnitude distribution. Furthermore, it is shown that this transform pair supplies a sufficient mathematical framework to describe the evolution of the force magnitude distribution under shearing. Apart from the choice of several coefficients, whose evolution of values must be explained in the physics, this framework successfully reproduces the features of the distribution that are taken to be an indicator of jamming and unjamming in a granular packing. Key words. Granular Physics, Probability Density Functions, Fourier Transforms

Youngquist, Robert C.↗

Bayesian estimation of life parameters in the Weibull distribution.

Development of a Bayesian analysis of the scale and shape parameters in the Weibull distribution and the corresponding reliability function with respect to the usual life-testing procedures. For the scale parameter theta, Bayesian estimates of theta and reliability are obtained for the uniform, exponential, and inverted gamma prior probability densities. Bhattacharya's results (1967) for the one-parameter exponential life-testing distribution are reduced to a special case of these results. A fully Bayesian analysis of both the scale and shape parameters is developed by assuming independent prior distributions; since in the latter case, analytical tractability is not possible, Bayesian estimates are obtained through a conjunction of Monte Carlo simulation and numerical-integration techniques. In both cases, a computer simulation is carried out, and a comparison is made between the Bayesian and the corresponding minimum-variance unbiased, or maximum likelihood, estimates. As expected, the Bayesian estimates are superior.

Canavos, G. C.↗

Optimal estimation for the satellite attitude using star tracker measurements

An optimal estimation scheme is presented, which determines the satellite attitude using the gyro readings and the star tracker measurements of a commonly used satellite attitude measuring unit. The scheme is mainly based on the exponential Fourier densities that have the desirable closure property under conditioning. By updating a finite and fixed number of parameters, the conditional probability density, which is an exponential Fourier density, is recursively determined. Simulation results indicate that the scheme is more accurate and robust than extended Kalman filtering. It is believed that this approach is applicable to many other attitude measuring units. As no linearization and approximation are necessary in the approach, it is ideal for systems involving high levels of randomness and/or low levels of observability and systems for which accuracy is of overriding importance.

Lo, J. T.-H.↗

Numerical simulations of soft and hard turbulence - Preliminary results for two-dimensional convection

Results on the transition from soft to hard turbulence in simulations of two-dimensional Boussinesq convection are reported. The computed probability densities for temperature fluctuations are exponential in form in both soft and hard turbulence, unlike what is observed in experiments. In contrast, a change is obtained in the Nusselt number scaling on Rayleigh number in good agreement with the three-dimensional experiments.

Deluca, E. E.↗

Focusing of waves in turbulent inhomogeneous media

A stochastic method using geometrical acoustics is employed to investigate the growth of a large fluctuation in the amplitude of high-frequency waves or shocks propagating through turbulent, inhomogeneous media. Nonlinear terms are retained in the analysis to correctly model focusing and growth of singular fluctuations in the amplitude. A two-dimensional analysis reveals that fluctuations in the ray-tube area grow exponentially and every ray displays caustics. Probability densities for the appearance of caustics are provided, and moments of the ray-tube area distribution and amplitude-related statistics are formulated for distances far into the region of caustic formation. Finally, a relationship is defined between theoretical predictions and measurements on an image plane.

Kulkarny, V. A.↗

Atmospheric attenuation for correlated satellite communication ground sites

Link margin estimates are calculated for 30/20 GHz satellite communication systems employing closely-spaced (4 to 100 km) 'dual diversity' switched ground sites. The link margin estimates are based on a new analysis in which the bivariate rain attenuation density function for two correlated ground sites is modelled by an exponential density function. The results of the exponential density function analysis enable derivation of a direct relation between rain margin estimates and probability of exceedance (link availability). Margins typically in the range 2 to 12 dB are calculated for various ground site separations and summarized for seven city locations and five satellite orbit positions over the U.S. The results can be easily extended to other EHF satellite frequencies and other locations.

Christopher, P.↗

Quantiles, parametric-select density estimation, and bi-information parameter estimators

A quantile-based approach to statistical analysis and probability modeling of data is presented which formulates statistical inference problems as functional inference problems in which the parameters to be estimated are density functions. Density estimators can be non-parametric (computed independently of model identified) or parametric-select (approximated by finite parametric models that can provide standard models whose fit can be tested). Exponential models and autoregressive models are approximating densities which can be justified as maximum entropy for respectively the entropy of a probability density and the entropy of a quantile density. Applications of these ideas are outlined to the problems of modeling: (1) univariate data; (2) bivariate data and tests for independence; and (3) two samples and likelihood ratios. It is proposed that bi-information estimation of a density function can be developed by analogy to the problem of identification of regression models.

Parzen, E.↗

Turbulence and the Stabilization Principle

Further results of research, reported in several previous NASA Tech Briefs articles, were obtained on a mathematical formalism for postinstability motions of a dynamical system characterized by exponential divergences of trajectories leading to chaos (including turbulence). To recapitulate: Fictitious control forces are introduced to couple the dynamical equations with a Liouville equation that describes the evolution of the probability density of errors in initial conditions. These forces create a powerful terminal attractor in probability space that corresponds to occurrence of a target trajectory with probability one. The effect in ordinary perceived three-dimensional space is to suppress exponential divergences of neighboring trajectories without affecting the target trajectory. Con sequently, the postinstability motion is represented by a set of functions describing the evolution of such statistical quantities as expectations and higher moments, and this representation is stable. The previously reported findings are analyzed from the perspective of the authors Stabilization Principle, according to which (1) stability is recognized as an attribute of mathematical formalism rather than of underlying physics and (2) a dynamical system that appears unstable when modeled by differentiable functions only can be rendered stable by modifying the dynamical equations to incorporate intrinsic stochasticity.

Zak, Michail↗

Radar scattering laws and wavelength dependence of the lunar surface

Data from Apollo lunar bistatic radar experiments have been processed to give probability density functions for surface slopes. These show best agreement with a Hagfors scattering law, though data having both gaussian and exponential characteristics also exist. Surface roughness estimates range from 4 deg in maria to at least 8 deg in highlands, values which are appropriate to 25 m horizontal scales and which are areal averages over tens of square kilometers. Roughness varies with wavelength, most strongly in maria.

Simpson, R. A.↗

Extending Newtonian Dynamics to Include Stochastic Processes

A paper presents further results of continuing research reported in several previous NASA Tech Briefs articles, the two most recent being Stochastic Representations of Chaos Using Terminal Attractors (NPO-41519), [Vol. 30, No. 5 (May 2006), page 57] and Physical Principle for Generation of Randomness (NPO-43822) [Vol. 33, No. 5 (May 2009), page 56]. This research focuses upon a mathematical formalism for describing post-instability motions of a dynamical system characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism involves fictitious control forces that couple the equations of motion of the system with a Liouville equation that describes the evolution of the probability density of errors in initial conditions. These stabilizing forces create a powerful terminal attractor in probability space that corresponds to occurrence of a target trajectory with probability one. The effect in configuration space (ordinary three-dimensional space as commonly perceived) is to suppress exponential divergences of neighboring trajectories without affecting the target trajectory. As a result, the post-instability motion is represented by a set of functions describing the evolution of such statistical quantities as expectations and higher moments, and this representation is stable.

Zak, Michail↗

On recontamination and directional-bias problems in Monte Carlo simulation of PDF turbulence models

Turbulent combustion can not be simulated adequately by conventional moment closure turbulence models. The difficulty lies in the fact that the reaction rate is in general an exponential function of the temperature, and the higher order correlations in the conventional moment closure models of the chemical source term can not be neglected, making the applications of such models impractical. The probability density function (pdf) method offers an attractive alternative: in a pdf model, the chemical source terms are closed and do not require additional models. A grid dependent Monte Carlo scheme was studied, since it is a logical alternative, wherein the number of computer operations increases only linearly with the increase of number of independent variables, as compared to the exponential increase in a conventional finite difference scheme. A new algorithm was devised that satisfies a restriction in the case of pure diffusion or uniform flow problems. Although for nonuniform flows absolute conservation seems impossible, the present scheme has reduced the error considerably.

Hsu, Andrew T.↗

A Numerical Approach to Estimate the Ballistic Coefficient of Space Debris from TLE Orbital Data

Low Earth Orbit (LEO) is full of space debris, which consist of spent rocket stages, old satellites and fragments from explosions and collisions. As of 2009, more than 21,000 orbital debris larger than 10 cm are known to exist], and while it is hard to track anything smaller than that, the estimated population of particles between 1 and 10 cm in diameter is approximately 500,000, whereas small as 1 cm exceeds 100 million. These objects orbit Earth with huge kinetic energies speeds usually exceed 7 kms. The shape of their orbit varies from almost circular to highly elliptical and covers all LEO, a region in space between 160 and 2,000 km above sea level. Unfortunately, LEO is also the place where most of our active satellites are situated, as well as, International Space Station (ISS) and Hubble Space Telescope, whose orbits are around 400 and 550 km above sea level, respectively.This poses a real threat as debris can collide with satellites and deal substantial damage or even destroy them.Collisions between two or more debris create clouds of smaller debris, which are harder to track and increase overall object density and collision probability. At some point, the debris density couldthen reach a critical value, which would start a chain reaction and the number of space debris would grow exponentially. This phenomenon was first described by Kessler in 1978 and he concluded that it would lead to creation of debris belt, which would vastly complicate satellite operations in LEO. The debris density is already relatively high, as seen from several necessary debris avoidance maneuvers done by Shuttle, before it was discontinued, and ISS. But not all satellites have a propulsion system to avoid collision, hence different methods need to be applied. One of the proposed collision avoidance concepts is called LightForce and it suggests using photon pressure to induce small orbital corrections to deflect debris from colliding. This method is very efficient as seen from theoretical simulations, even few continuous mode 10 kW ground-based lasers, focused by 1.5 m telescopes with adaptive optics, were enough to prevent significant amount of the debris collisions. Simulations were done by propagating all space objects in LEO by 1 year into the future and checking whether the probability of collision was high. For those space objects different ground-based lasers were used to divert them, afterwards collision probabilities were reevaluated. However, the actual accuracy of the LightForce software, which has been developed at NASA AmesResearch Center, depends on the veracity of the input parameters, one of which is the objects ballistic coefficient. It is a measure of bodys ability to overcome air resistance, which has a significant impact on the debris in LEO, and thus it is responsible for the shape of the trajectory of the debris. Having the exact values of the ballistic coefficient would make significantly better collision predictions, unfortunately, we do not know what are the values for most of the objects.In this research, we were working with part of LightForce code, which estimates the ballistic coefficient from ephemerides. Previously used method gave highly inaccurate values, when compared to known objects, and it needed to be changed. The goal of this work was to try out a different method of estimating the ballistic coefficient and to check whether or not it gives noticeable improvements.

Collisions↗

Radar scattering laws for the lunar surface

Bistatic radar observations of quasi-specular scatter from the lunar surface at 13 and 116 cm wavelengths have been expressed in terms of probability densities of the underlying surface slope distributions. These show best agreement with a Hagfors scattering function, although other forms having predominantly Gaussian or exponential characteristics also occur. Estimates of root mean square (rms) surface slope derived from these function range from 4 deg rms in maria to at least 8 deg rms in highlands. These are values appropriate to 25 m horizontal scales and represent averages over tens of square kilometers of surface area. The effective scattering roughness varies with wavelength, most strongly in maria, where its behavior is consistent with that expected from an immature, i.e., unsaturated cratered surface. There is very little variation in scattering roughness with wavelength in highlands area.

Simpson, R. A.↗

Modeling gap probability in discontinuous vegetation canopies

In the present model for the gap probability of a discontinuous vegetation canopy, the assumption of a negative exponential attenuation within individual plant canopies will yield a problem involving the distribution distances within canopies through which a ray will pass. If, however, the canopies intersect and/or overlap, so that foliage density remains constant within the overlap area, the problem can be approached with two types of approximations. Attention is presently given to the case of a comparison of modeled gap probabilities with those observed for a stand of Maryland pine, which shows good agreement for zenith angles of illumination up to about 45 deg.

Li, Xiaowen↗

Interrelated structure of high altitude atmospheric profiles

A preliminary development of a mathematical model to compute probabilities of thermodynamic profiles is presented. The model assumes an exponential expression for pressure and utilizes the hydrostatic law and equation of state in the determination of density and temperature. It is shown that each thermodynamic variable can be factored into the produce of steady state and perturbation functions. The steady state functions have profiles similar to those of the 1962 standard atmosphere while the perturbation functions oscillate about 1. Limitations of the model and recommendations for future work are presented.

Engler, N. A.↗

A Simple, Velocity Dependent, Collision Probability Algorithm for Small Combined Hard Body Radii Close Approach Events

Over the last few years progress has been made in the velocity dependent collision probability problem. Recent progress in this problem has been approached via integration of the probability flux into the surface of the combined hard body object. The algorithm presented uses the surface flux approach and is designed to compute the collision probability rate between two objects given their time dependent states and state error covariance matrices. With regards to the computation of the collision probability (Pc) rate, two major differences exist between the development of this algorithm and the usual Pc algorithm. First, the shape of the at-risk volume is assumed to be a cube rather than a sphere. The size of the cube is chosen so that it circumscribes the usual hard body sphere chosen for the spherical Pc problem. This will result in half the length of a side of the cube being equal to the hard body radius (HBR) of the sphere. Second, it is assumed that the HBR of the cube is much smaller than the smallest combined position uncertainty (σ(sub min) > 5*HBR) at each time point of evaluation. This is necessary as the actual collision probability calculation is based on a first order, small variable expansion in the position components of the Gaussian probability density function. This leads to the collision probability rate being to second order in the combined HBR. The collision probability rate is in a concise, closed form containing exponential and error functions.

Frisbee, Joe↗

Simulation of flight maneuver-load distributions by utilizing stationary, non-Gaussian random load histories

Random numbers were generated with the aid of a digital computer and transformed such that the probability density function of a discrete random load history composed of these random numbers had one of the following non-Gaussian distributions: Poisson, binomial, log-normal, Weibull, and exponential. The resulting random load histories were analyzed to determine their peak statistics and were compared with cumulative peak maneuver-load distributions for fighter and transport aircraft in flight.

Leybold, H. A.↗