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NASA NTRS1992

Eleven-Species Thermochemical Model Of Air

SPECIES program computes thermodynamic and transport properties and equilibrium constants. Uses least-squares polynomial curve fits of most accurate data available to provide requested values more quickly than table-lookup methods. Subroutines computing transport coefficients and collision cross sections include additional code to correct for any electron pressure when working with ionic species. Species represented: diatomic nitrogen, diatomic oxygen, atomic nitrogen, atomic oxygen, nitric oxide, ionized nitric oxide, free electron, ionized atomic nitrogen, ionized atomic oxygen, ionized diatomic nitrogen, and ionized diatomic oxygen. Written in FORTRAN 77.

Thompson, Richard A.↗

NASA NTRS2012

Efficient Computation of Separation-Compliant Speed Advisories for Air Traffic Arriving in Terminal Airspace

A class of problems in air traffic management asks for a scheduling algorithm that supplies the air traffic services authority not only with a schedule of arrivals and departures, but also with speed advisories. Since advisories must be finite, a scheduling algorithm must ultimately produce a finite data set, hence must either start with a purely discrete model or involve a discretization of a continuous one. The former choice, often preferred for intuitive clarity, naturally leads to mixed-integer programs, hindering proofs of correctness and computational cost bounds (crucial for real-time operations). In this paper, a hybrid control system is used to model air traffic scheduling, capturing both the discrete and continuous aspects. This framework is applied to a class of problems, called the Fully Routed Nominal Problem. We prove a number of geometric results on feasible schedules and use these results to formulate an algorithm that attempts to compute a collective speed advisory, effectively finite, and has computational cost polynomial in the number of aircraft. This work is a first step toward optimization and models refined with more realistic detail.

Sadovsky, Alexander V.↗

NASA NTRS1979

Diagrammatic theory of transition of pendulum like systems

Orbit-orbit and spin-orbit gravitational resonances are analyzed using the model of a rigid pendulum subject to both a time-dependent periodic torque and a constant applied torque. First, a descriptive model of passage through resonance is developed from an examination of the polynomial equation that determines the extremes of the momentum variable. From this study, a probability estimate for capture into libration is derived. Second, a lowest order solution is constructed and compared with the solution obtained from numerical integration. The steps necessary to systematically improve this solution are also discussed. Finally, the effect of a dissipative term in the pendulum equation is analyzed.

Yoder, C. F.↗

NASA NTRS1989

Estimation of aircraft aerodynamic parameters from flight data

Several ways for obtaining aerodynamic parameters of an aircraft from flight data are presented. A brief description of data analysis from steady measurements is given, and a concept of system identification is introduced. This is followed by a mathematical model of an aircraft with aerodynamic forces and moments approximated either by polynomials or splines and a detailed treatment of two often-used techniques for parameter estimation. Because of renewed interest in frequency-domain analysis, one section of the paper is devoted to this problem. The methods are demonstrated in several examples using real flight data.

Klein, Vladislav↗

NASA NTRS1991

Hierarchic Extensions in the Static and Dynamic Analysis of Elastic Beams

Approximate solutions of static and dynamic beam problems by the p-version of the finite element method are investigated. Within a hierarchy of engineering beam idealizations, rigorous formulations of the strain and kinetic energies for straight and circular beam elements are presented. These formulations include rotating coordinate system effects and geometric nonlinearities to allow for the evaluation of vertical axis wind turbines, the motivating problem for this research. Hierarchic finite element spaces, based on extensions of the polynomial orders used to approximate the displacement variables, are constructed. The developed models are implemented into a general purpose computer program for evaluation. Quality control procedures are examined for a diverse set of sample problems. These procedures include estimating discretization errors in energy norm and natural frequencies, performing static and dynamic equilibrium checks, observing convergence for qualities of interest, and comparison with more exacting theories and experimental data. It is demonstrated that p-extensions produce exponential rates of convergence in the approximation of strain energy and natural frequencies for the class of problems investigated.

Watson, Robert A.↗

NASA NTRS1991

Analysis of IRAS solar system dust data

Data in the Infrared Astronomical Satellite (IRAS) Zodiacal History File were analyzed to extract dust band locations and peak brightness measurements from approximately 1,000 individual IRAS scans. The study had three goals. One was to show that the prominent solar system dust bands are associated with Hirayama asteroid families and thus that collisions between asteroids account for a significant fraction of the particles in the zodiacal cloud. Recent work suggests that while the Hirayama families are a major source of the dust in the bands, there may also be contributions from two or three smaller, more recently recognized asteroid families. A second goal was to show that there is evidence in the IRAS dust data for the transport of particles from asteroid belt to the Earth by Poynting-Robertson light drag and thus account for the fact that asteroid particles are collected in the Earth's stratosphere. Results of the study will confirm the location of the dust bands within the inner asteroid belt, and show conclusively that the material seen by IRAS is now spread over a wide range of distances from the sun. The third goal was to construct a model of the background zodiacal cloud that satisfies the proper dynamical constraints. Figures are provided to show the scans processed to remove zodiacal background and Galactic signals, and the resulting polynomial fits to the 25 micron scan. The latter provided objective estimates of band widths, peak locations, and peak fluxes. Modelling and analysis of the resulting band data has been presented at several conferences and is the subject of a number of forthcoming papers.

Dermott, S. F.↗

NASA NTRS2010

Estimation of Supersonic Stage Separation Aerodynamics of Winged-Body Launch Vehicles Using Response Surface Methods

Response surface methodology was used to estimate the longitudinal stage separation aerodynamic characteristics of a generic, bimese, winged multi-stage launch vehicle configuration at supersonic speeds in the NASA LaRC Unitary Plan Wind Tunnel. The Mach 3 staging was dominated by shock wave interactions between the orbiter and booster vehicles throughout the relative spatial locations of interest. The inference space was partitioned into several contiguous regions within which the separation aerodynamics were presumed to be well-behaved and estimable using central composite designs capable of fitting full second-order response functions. The underlying aerodynamic response surfaces of the booster vehicle in belly-to-belly proximity to the orbiter vehicle were estimated using piecewise-continuous lower-order polynomial functions. The quality of fit and prediction capabilities of the empirical models were assessed in detail, and the issue of subspace boundary discontinuities was addressed. Augmenting the central composite designs to full third-order using computer-generated D-optimality criteria was evaluated. The usefulness of central composite designs, the subspace sizing, and the practicality of fitting lower-order response functions over a partitioned inference space dominated by highly nonlinear and possibly discontinuous shock-induced aerodynamics are discussed.

Erickson, Gary E.↗

NASA NTRS1993

Microwave Scattering Model for Grass Blade Structures

In this paper, the electromagnetic scattering solution for a grass blade with complex cross-section geometry is considered. It is assumed that the blade cross section is electrically small, but its length is large compared to the incident wavelength. In a recent study it has been shown that the scattering solution for such problems, in the form of a polarizability tensor, can be obtained using the low-frequency approximation in conjunction with the method of moments. In addition, the study shows that the relationship between the polarizability tensor of a dielectric cylinder and its dielectric constant can be approximated by a simple algebraic expression. The results of this study are used to show that this algebraic approximation is valid also for cylinders with cross sections the shape of grass blades, providing that proper values am selected for each of three constants appearing in the expression. These constants are dependent on cylinder shape, and if the relationship between the constants and the three parameters describing a grass blade shape can be determined, an algebraic approximation relating polarizability tensor to blade shape, as well as dielectric constant, can be formed. Since the elements of the polarizability tensor are dependent on only these parameters, this algebraic approximation can replace the cumbersome method of moments model. A conjugate gradient method is then implemented to correctly determine the three constants of the algebraic approximation for each blade shape. A third-order polynomial fit to the data is then determined for each constant, thus providing a complete analytic replacement to the numerical (moment method) scattering model. Comparisons of this approximation to the numerical model show an average error of less than 3%.

Stiles, James M.↗

NASA NTRS2006

Low-Velocity Impact Response of Sandwich Beams with Functionally Graded Core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young's modulus is represented by a polynomial in the thickness coordinate, but the Poisson's ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel - the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of thc beam with graded core Increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load arc reduced considerably due to grading of thc core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

Apetre, N. A.↗

NASA NTRS1989

The effect of an unsteady three-dimensional wake on elastic blade-flapping eigenvalues in hover

This paper describes the formulation of a finite-state inflow model based on an acceleration potential and a helical wake geometry. The states of the model are coefficients of an inflow expansion in terms of a Fourier series (azimuthally) and of special polynomials (radially). The integrals over the wake are done in closed-form to obtain a set of ordinary differential equations for the inflow coefficients. The forcing functions for these equations are generalized forces which are integrals of the blade loading exactly as in structural dynamics. This model implicitly includes (for the hover case) Prandtl-Goldstein tip losses, dynamic inflow, and Theodorsen/Loewy lift deficiency. Thus, it is a fully three-dimensional unsteady wake model. Here, this model is coupled with elastic-blade equations in hover and eigenvalues are found. The results show that the three-dimensional wake has a large effect on the flap damping of all modes.

Peters, David A.↗

NASA NTRS1993

Geophysical Model Functions for the Sea State Bias in Satellite Altimetry

The sea state bias in altimeter measurements of sea surface height has been investigated by many authors, resulting in several proposed algorithms of the form SSB = epsilonH where H is the significant wave height and epsilon is a nondimensional function of wind and wave parameters. All known algorithms are presently rated. By analyzing the performance of different models we conclude that further progress can hardly be achieved by raising the degree of the polynomial for epsilon(U,H). Physically based approaches employing a small number of adjustable parameters and theoretically justified non-dimensional combinations of external wind and wave factors appear to be more promising.

geophysical model functions sea state bias in sate↗

NASA NTRS2015

Random Predictor Models for Rigorous Uncertainty Quantification: Part 1

This and a companion paper propose techniques for constructing parametric mathematical models describing key features of the distribution of an output variable given input-output data. By contrast to standard models, which yield a single output value at each value of the input, Random Predictors Models (RPMs) yield a random variable at each value of the input. Optimization-based strategies for calculating RPMs having a polynomial dependency on the input and a linear dependency on the parameters are proposed. These formulations yield RPMs having various levels of fidelity in which the mean and the variance of the model's parameters, thus of the predicted output, are prescribed. As such they encompass all RPMs conforming to these prescriptions. The RPMs are optimal in the sense that they yield the tightest predictions for which all (or, depending on the formulation, most) of the observations are less than a fixed number of standard deviations from the mean prediction. When the data satisfies mild stochastic assumptions, and the optimization problem(s) used to calculate the RPM is convex (or, when its solution coincides with the solution to an auxiliary convex problem), the model's reliability, which is the probability that a future observation would be within the predicted ranges, can be bounded tightly and rigorously.

Crespo, Luis G.↗

NASA NTRS1989

Report on the multicolor pyrometry session at the Second NASA Non-Contact Temperature-Measurement Workshop

The session met jointly with the other special sessions. Due to time constraints, the concerns of multicolor pyrometry were not addressed in depth. Multicolor pyrometry attempts to determine an object's temperature and emittance as a function of wavelength from simultaneous measurement of the spectral radiance of the object at multiple wavelengths. The main issues are those of the most appropriate model for the wavelength dependence of the emittance, the number of undetermined parameters, and the error in the measured temperature. It has long been known that it is inappropriate to attempt to model the emittance with a functional form that contains as many unknowns as wavelengths. Such attempts will lead to gross errors in the measured temperature. However, more recent work by Hunter and Khan has indicated that emittance can be modeled by a function with many fewer degrees of freedom than sampled wavelengths (e.g., it may be appropriate to use a second order polynomial to fit data from a forty-color pyrometer). In a platform lecture, Khan described algorithms for determination of the parameters by minimization of the errors between the model and the data in a least-squares sense. The first was a linear method and the second was a non-linear method where the emittance could assume any functional form. Both methods were shown to be very sensitive to random noise and non-linearity in the measurement system.

Spjut, R. Erik↗

NASA NTRS1993

Robust algebraic image enhancement for intelligent control systems

Robust vision capability for intelligent control systems has been an elusive goal in image processing. The computationally intensive techniques a necessary for conventional image processing make real-time applications, such as object tracking and collision avoidance difficult. In order to endow an intelligent control system with the needed vision robustness, an adequate image enhancement subsystem capable of compensating for the wide variety of real-world degradations, must exist between the image capturing and the object recognition subsystems. This enhancement stage must be adaptive and must operate with consistency in the presence of both statistical and shape-based noise. To deal with this problem, we have developed an innovative algebraic approach which provides a sound mathematical framework for image representation and manipulation. Our image model provides a natural platform from which to pursue dynamic scene analysis, and its incorporation into a vision system would serve as the front-end to an intelligent control system. We have developed a unique polynomial representation of gray level imagery and applied this representation to develop polynomial operators on complex gray level scenes. This approach is highly advantageous since polynomials can be manipulated very easily, and are readily understood, thus providing a very convenient environment for image processing. Our model presents a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets.

Lerner, Bao-Ting↗

NASA NTRS2015

Random Predictor Models for Rigorous Uncertainty Quantification: Part 2

This and a companion paper propose techniques for constructing parametric mathematical models describing key features of the distribution of an output variable given input-output data. By contrast to standard models, which yield a single output value at each value of the input, Random Predictors Models (RPMs) yield a random variable at each value of the input. Optimization-based strategies for calculating RPMs having a polynomial dependency on the input and a linear dependency on the parameters are proposed. These formulations yield RPMs having various levels of fidelity in which the mean, the variance, and the range of the model's parameter, thus of the output, are prescribed. As such they encompass all RPMs conforming to these prescriptions. The RPMs are optimal in the sense that they yield the tightest predictions for which all (or, depending on the formulation, most) of the observations are less than a fixed number of standard deviations from the mean prediction. When the data satisfies mild stochastic assumptions, and the optimization problem(s) used to calculate the RPM is convex (or, when its solution coincides with the solution to an auxiliary convex problem), the model's reliability, which is the probability that a future observation would be within the predicted ranges, is bounded rigorously.

Crespo, Luis G.↗

NASA NTRS1973

Parametric analysis of ATM solar array.

The paper discusses the methods used for the calculation of ATM solar array performance characteristics and provides the parametric analysis of solar panels used in SKYLAB. To predict the solar array performance under conditions other than test conditions, a mathematical model has been developed. Four computer programs have been used to convert the solar simulator test data to the parametric curves. The first performs module summations, the second determines average solar cell characteristics which will cause a mathematical model to generate a curve matching the test data, the third is a polynomial fit program which determines the polynomial equations for the solar cell characteristics versus temperature, and the fourth program uses the polynomial coefficients generated by the polynomial curve fit program to generate the parametric data.

Singh, B. K.↗

NASA NTRS1997

LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation

A PDE-based feedback control method for thin cylindrical shells with surface-mounted piezoceramic actuators is presented. Donnell-Mushtari equations modified to incorporate both passive and active piezoceramic patch contributions are used to model the system dynamics. The well-posedness of this model and the associated LQR problem with an unbounded input operator are established through analytic semigroup theory. The model is discretized using a Galerkin expansion with basis functions constructed from Fourier polynomials tensored with cubic splines, and convergence criteria for the associated approximate LQR problem are established. The effectiveness of the method for attenuating the coupled longitudinal, circumferential and transverse shell displacements is illustrated through a set of numerical examples.

delRosario, R. C. H.↗

NASA NTRS1975

Recovery of atmospheric refractivity profiles from simulated satellite-to-satellite tracking data

Techniques for recovering atmospheric refractivity profiles from simulated satellite-to-satellite tracking data are documented. Examples are given using the geometric configuration of the ATS-6/NIMBUS-6 Tracking Experiment. The underlying refractivity model for the lower atmosphere has the spherically symmetric form N = exp P(s) where P(s) is a polynomial in the normalized height s. For the simulation used, the Herglotz-Wiechert technique recovered values which were 0.4% and 40% different from the input values at the surface and at a height of 33 kilometers, respectively. Using the same input data, the model fitting technique recovered refractivity values 0.05% and 1% different from the input values at the surface and at a height of 50 kilometers, respectively. It is also shown that if ionospheric and water vapor effects can be properly modelled or effectively removed from the data, pressure and temperature distributions can be obtained.

Murray, C. W., Jr.↗