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NASA NTRS1977

Mathematical and computational studies of equilibrium capillary free surfaces

The results of several independent studies are presented. The general question is considered of whether a wetting liquid always rises higher in a small capillary tube than in a larger one, when both are dipped vertically into an infinite reservoir. An analytical investigation is initiated to determine the qualitative behavior of the family of solutions of the equilibrium capillary free-surface equation that correspond to rotationally symmetric pendent liquid drops and the relationship of these solutions to the singular solution, which corresponds to an infinite spike of liquid extending downward to infinity. The block successive overrelaxation-Newton method and the generalized conjugate gradient method are investigated for solving the capillary equation on a uniform square mesh in a square domain, including the case for which the solution is unbounded at the corners. Capillary surfaces are calculated on the ellipse, on a circle with reentrant notches, and on other irregularly shaped domains using JASON, a general purpose program for solving nonlinear elliptic equations on a nonuniform quadrilaterial mesh. Analytical estimates for the nonexistence of solutions of the equilibrium capillary free-surface equation on the ellipse in zero gravity are evaluated.

Albright, N.↗

NASA NTRS1977

TDRSS performance evaluation for TDRSS services using analytical simulation

An analytical simulation package which can predict bit error rate and tracking performance for nonlinear, bandlimited satellite links was developed for the Tracking and Data Relay Satellite System (TDRSS). It is based on analytical simulation, i.e., a combination of simulation techniques and analytical performance evaluation. This allows the use of a more realistic signal format than is possible with a purely analytical approach and maintains the advantage of short computation time on bandlimited, nonlinear channels with a noisy uplink. The link models and the main features of the program are described and an overview of the mathematical approach is given. Sample results of computer runs are provided for current power budgets and system design parameters for both forward and return links.

Braun, W. R.↗

NASA NTRS1980

Focal axis resolver for offset reflector antennas

Described are electrical means for determining the focal axis of an offset reflector antenna whose physical rim is not coincident with the boundary of the electrical aperture. Even and odd sensing functions are employed in the focal region, leading to both amplitude and phase criteria for resolving a focal axis generally inclined with respect to the system axis. The analytical aspects of the problem are discussed, and an example related to a 4-meter Large-Antenna Multiple-Frequency Microwave Radiometer (LAMMR) is included. The technique is useful for focal axis determination in mathematical simulations and in the physical world.

Schmidt, R. F.↗

NASA NTRS1988

The optical field angle distortion calibration feasibility study for the Hubble Space Telescope fine guidance sensors

The results of an analytical study to investigate the feasibility of calibrating the Hubble Space Telescope's (HST's) fine guidance sensors (FGSs) within HST mission accuracy limits are presented. The study has two purposes: (1) to determine the mathematical feasibility of the optical field angle distortion (OFAD) calibration algorithm and (2) to confirm that the OFAD, plate scale, and FGS-to-FGS alignment calibration algorithms produced a calibration of the FGSs that satisfied mission requirements. The study concluded that the mathematical specification of the OFAD algorithm is adequate and permits a determination of the FGS calibration parameters (accurate to better than 0.003 arc-second) sufficient to meet the mission requirements. The algorithms implemented, the characteristics of the simulated data and procedures for data analysis, and the study's results are discussed. In addition, several useful techniques for improving the stability and accuracy of the OFAD solution are outlined.

Luchetti, K.↗

NASA NTRS1965

Earth-Moon trajectories, 1965-70

Analytical model for generation of earth-moon trajectory analysis data

MATHEMATICAL MODEL↗

NASA NTRS1980

Acoustoelasticity - Review and extension

Research on the interaction between the sound pressure field and the flexible wall of an enclosure is reviewed. The topics discussed include: a mathematical model, simplified solutions for sinusoidal excitation, damping, random excitation, numerical results, including comparisons with representative experimental data, and design analysis methodology. Results of various authors using finite element techniques and modal analysis suggest that effective analytical models are available.

Dowell, E. H.↗

NASA NTRS1994

Dynamics of a split torque helicopter transmission

Split torque designs, proposed as alternatives to traditional planetary designs for helicopter main rotor transmissions, can save weight and be more reliable than traditional designs. This report presents the results of an analytical study of the system dynamics and performance of a split torque gearbox that uses a balance beam mechanism for load sharing. The Lagrange method was applied to develop a system of equations of motion. The mathematical model includes time-varying gear mesh stiffness, friction, and manufacturing errors. Cornell's method for calculating the stiffness of spur gear teeth was extended and applied to helical gears. The phenomenon of sidebands spaced at shaft frequencies about gear mesh fundamental frequencies was simulated by modeling total composite gear errors as sinusoid functions. Although the gearbox has symmetric geometry, the loads and motions of the two power paths differ. Friction must be considered to properly evaluate the balance beam mechanism. For the design studied, the balance beam is not an effective device for load sharing unless the coefficient of friction is less than 0.003. The complete system stiffness as represented by the stiffness matrix used in this analysis must be considered to precisely determine the optimal tooth indexing position.

Krantz, Timothy L.↗

NASA NTRS1996

Gravity Induced Formation of Concentration Gradients in Supersaturated Binary Solutions

Experimental and theoretical studies of the formation of solute concentration gradient in supersaturated binary solutions in a gravitational field were carried out. The formation of solute concentration gradient was associated with the gravity induced redistribution of subcritical solute clusters. The birth-death process of the new solute-rich phase domains (subcritical solute clusters) was described in terms of the time-dependent Ginzburg Landau model developed for metastable state relaxation in binary (solute + solvent) non-critical solutions in the presence of a gravitational field. A new mathematical Ansatz was developed for solution of the model equations. This Ansatz has allowed to approach for the first time the following important problems: (1) Microstructure of solute distribution inside of the subcritical solute clusters. The analytical results obtained demonstrate that solute inside of the subcritical solute clusters is heterogeneously distributed with a spatially periodic structure. (2) Macrostructure of the solute subcritical clusters distribution in a gravitational field. The subcritical solute clusters are found to be distributed heterogeneously in a gravitational field. This heterogeneity, which is due to the heterogeneous birth-death process of the subcritical solute clusters in a gravitational field, initiates a noticeable solute concentration gradient in vertical columns of supersaturated binary solutions. An analysis and comparison of theoretical results and experimental data related to the solute concentration gradient formation in a gravitational field are presented. It is also demonstrated that the critical radius of solute clusters (radius of nucleation) and induction time are gravity-dependent.

Izmailov, Alexander F.↗

NASA NTRS1997

Mathematical and Computational Challenges in Population Biology and Ecosystems Science

Mathematical and computational approaches provide powerful tools in the study of problems in population biology and ecosystems science. The subject has a rich history intertwined with the development of statistics and dynamical systems theory, but recent analytical advances, coupled with the enhanced potential of high-speed computation, have opened up new vistas and presented new challenges. Key challenges involve ways to deal with the collective dynamics of heterogeneous ensembles of individuals, and to scale from small spatial regions to large ones. The central issues-understanding how detail at one scale makes its signature felt at other scales, and how to relate phenomena across scales-cut across scientific disciplines and go to the heart of algorithmic development of approaches to high-speed computation. Examples are given from ecology, genetics, epidemiology, and immunology.

Levin, Simon A.↗

NASA NTRS1973

Ion time-of-flight determinations of doubly to singly ionized mercury ion ratios from a mercury electron bombardment discharge

Doubly to singly charged mercury ion ratios in electron bombardment ion thruster exhaust beams have been determined as functions of bombardment discharge potential, thrust beam current, thrust beam radial position, acceleration-deceleration voltage ratio, and propellant utilization fraction. A mathematical model for two-step ionization processes has been derived, and calculated ion ratios are compared to observed ratios. Production of Hg(++) appears to result primarily from sequential ionization of Hg(+) in the discharge. Experimental and analytical results are presented, and design, construction, and operation features of an electrostatic deflection ion time-of-flight analyzer for the determination of the above-mentioned ratios are reviewed.

Sellen, J. M., Jr.↗

NASA NTRS1975

Dynamic heave-pitch analysis of air cushion landing systems

A program to develop analytical tools for evaluating the dynamic performance of Air Cushion Landing Systems (ACLS) is described. The heave (vertical) motion of the ACLS was analyzed, and the analysis was extended to cover coupled heave-pitch motions. The mathematical models developed are based on a fundamental analysis of the body dynamics and fluid mechanics of the aircraft-cushion-runway interaction. The air source characteristics, flow losses in the feeding ducts, trunk and cushion, the effects of fluid compressibility, and dynamic trunk deflections, including ground contact are considered. A computer program, based on the heave-pitch analysis, was developed to simulate the dynamic behavior of an ACLS during landing impact and taxi over an irregular runway. The program outputs include ACLS motions, loadings, pressures, and flows as a function of time. To illustrate program use, three basic types of simulations were carried out. The results provide an initial indication of ACLS performance during (1) a static drop, (2) landing impact, and (3) taxi over a runway irregularity.

Captain, K. M.↗

NASA NTRS1985

Identifying approximate linear models for simple nonlinear systems

This paper addresses the identification (realization) of approximate linear models from response data for certain nonlinear dynamic systems. Response characteristics for several typical nonlinear joints are analyzed mathematically and represented by series expansions. The parameters of the series expansion are then compared with the modal parameters of a linear model identified by the Eigensystem Realization Algorithm. The agreement of the identified model and the analytically derived representation is excellent for the cases studied. Also laboratory data from a model which exhibited stiffening behavior was analyzed using the Eigensystem Realization algorithm and Fast Fourier Transform. The laboratory experiment demonstrated the ability of the technique to recover the model characteristics using real data.

Horta, L. G.↗

NASA NTRS1983

Mathematical developments regarding the general theory of the Earth magnetism

A literature survey on the Earth's magnetic field, citing the works of Gauss, Erman-Petersen, Quintus Icilius and Neumayer is presented. The general formulas for the representation of the potential and components of the Earth's magnetic force are presented. An analytical representation of magnetic condition of the Earth based on observations is also made.

Schmidt, A.↗

NASA NTRS1987

Geometrical analysis of Cys-Cys bridges in proteins and their prediction from incomplete structural information

Analysis of C-alpha atom positions from cysteines involved in disulphide bridges in protein crystals shows that their geometric characteristics are unique with respect to other Cys-Cys, non-bridging pairs. They may be used for predicting disulphide connections in incompletely determined protein structures, such as low resolution crystallography or theoretical folding experiments. The basic unit for analysis and prediction is the 3 x 3 distance matrix for Cx positions of residues (i - 1), Cys(i), (i +1) with (j - 1), Cys(j), (j + 1). In each of its columns, row and diagonal vector--outer distances are larger than the central distance. This analysis is compared with some analytical models.

NASA Discipline Exobiology↗

NASA NTRS1974

Aeroelastic-stability characteristics of a V/STOL tilt-rotor aircraft with hingeless blades - Correlation of analysis and test

The potential aeroelastic problems of a tilt rotor aircraft are described. The type of mathematical model required to predict such behavior is discussed and results of a design study on a specific configuration are presented. The sensitivity of the results to variation of the structural parameters is evaluated. The methodology used for analytical studies is shown to predict accurately the behavior of a full-scale rotor-pylon-wing system tested in the NASA-Ames 40- by 80-foot wind tunnel. It is concluded that there is no impediment from an aeroelastic viewpoint in designing and flying a tilt-rotor aircraft using a soft-in-plane hingeless rotor.

Alexander, H. R.↗

NASA NTRS1986

Analytical determination of space station response to crew motion and design of suspension system for microgravity experiments

The objective of this investigation is to make analytical determination of the acceleration produced by crew motion in an orbiting space station and define design parameters for the suspension system of microgravity experiments. A simple structural model for simulation of the IOC space station is proposed. Mathematical formulation of this model provides the engineers a simple and direct tool for designing an effective suspension system.

Liu, F. C.↗

NASA NTRS1987

Analytical determination of space station response to crew motion and design of suspension system for microgravity experiments

Liu, Frank C.↗

NASA NTRS1988

Recursive forward dynamics for multiple robot arms moving a common task object

Recursive forward dynamics algorithms are developed for an arbitrary number of robot arms moving a commonly held object. The multiarm forward dynamics problem is to find the angular accelerations at the joints and the contact forces that the arms impart to the task object. The problem also involves finding the acceleration of this object. The multiarm forward dynamics solutions provide a thorough physical and mathematical understanding of the way several arms behave in response to a set of applied joint moments. Such an understanding simplifies and guides the subsequent control design and experimentation process. The forward dynamics algorithms also provide the necessary analytical foundation for conducting analysis and simulation studies. The multiarm algorithms are based on the filtering and smoothing approach recently advanced for single-arm dynamics, and they can be built up modularly from the single-arm algorithms. The algorithms compute recursively the joint-angle accelerations, the contact forces, and the task-object accelerations. Algorithms are also developed to evaluate in closed form the linear transformations from the active joint moments to the joint-angle accelerations, to the task-object accelerations., and to the task-object contact forces. A possible computing architecture is presented as a precursor to a more complete investigation of the computational performance of the dynamics algorithms.

Rodriguez, G.↗