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NASA NTRS1973

Modeling a particular class of multiple-input/multiple-output black boxes with stochastic integral equations and identifying the required parameters

A method is given for obtaining a mathematical model of a class of black boxes having multiple inputs and multiple outputs in terms of Ito stochastic integral equations. This method is applicable to the class of black boxes having ergodic correlation functions when there is zero applied input. The point of view adopted in this paper is phenomenological in that it is desired that calculations made using the mathematical model should be close to what is actually observed at the output of the black box. How close is defined in the problem statement.

Eyman, E. D.↗

NASA NTRS1977

Calibration of a universal indicated turbulence system

Theoretical and experimental work on a Universal Indicated Turbulence Meter is described. A mathematical transfer function from turbulence input to output indication was developed. A random ergodic process and a Gaussian turbulence distribution were assumed. A calibration technique based on this transfer function was developed. The computer contains a variable gain amplifier to make the system output independent of average velocity. The range over which this independence holds was determined. An optimum dynamic response was obtained for the tubulation between the system pitot tube and pressure transducer by making dynamic response measurements for orifices of various lengths and diameters at the source end.

Chapin, W. G.↗

NASA NTRS1977

A classification of large amplitude oscillations of a spring-pendulum system

We present a detailed classification of large amplitude oscillations of a non-integrable autonomous system with two degrees of freedom: the spring pendulum system. The classification is made with the method of invariant curves. The results show the importance of three types of motion: periodic, quasi-periodic and semi-ergodic. The numerical results are given for nine different values of the energy constant.

Broucke, R.↗

NASA NTRS1981

Current research activities at the Texas Center for Orbital Mechanics

The development of new techniques and their applications to problems in satellite geodesy, oceanography, orbit determination, long-time prediction, regularization, potential determination, stability of natural and artificial satellites and planets, etc. are described. Applications to polar motion, earth rotation and to tectonic plate motion are discussed as related to the Lageos, Starlette and Seasat missions. Numerical averaging techniques are described for the determination of the long term motion of near-resonant gravitational systems with application to Jupiter's Galilean satellites. Regions of long-time stability and ergodicity are established in the restricted problem.

Szebehely, V. G.↗

NASA NTRS1980

Nonlinear dynamics; Proceedings of the International Conference, New York, NY, December 17-21, 1979

Papers were presented on turbulence, ergodic and integrable behavior, chaotic maps and flows, chemical and fully developed turbulence, and strange attractors. Specific attention was given to measures describing a turbulent flow, stochastization and collapse of vortex systems, a subharmonic route to turbulent convection, and weakly nonlinear turbulence in a rotating convection layer. The Korteweg-de Vries and Hill equations, plasma transport in three dimensions, a horseshoe in the dynamics of a forced beam, and the explosion of strange attractors exhibited by Duffing's equation were also considered.

Helleman, R. H. G.↗

NASA NTRS1982

Scale-free models of galaxies. II - A complete survey of orbits

A complete set of orbits starting at over 400 distinct points spread out in the phase space of an oblate scale-free potential is investigated. Each orbit is followed for a time that corresponds to a Hubble time in a realistic galaxy potential suitable for an E5 or E6 galaxy. It is noted that none of the orbits in the survey is ergodic. All of the survey orbits are regular, visiting a region at least one dimension smaller than expected from the classical integrals of motion. Thus, for all practical purposes, they have an extra nonclassical isolating integral. Approximately 95% of the survey orbits are box orbits, the rest being pipes (formerly tubes). The survey exposes an ambiguity in the original classification scheme for orbits, which, it is noted, can be resolved on the basis of the topology of an orbit's surface of section. Nevertheless, the distinction between high order very convoluted pipes and boxes is probably artificial for practical purposes.

Richstone, D. O.↗

NASA NTRS1982

Stationarity of magnetohydrodynamic fluctuations in the solar wind

Solar wind research and studies of charged particle propagation often assume that the interplanetary magnetic field represents a stationary random process. The extent to which ensemble averages of the solar wind magnetic fields follow the asymptotic behavior predicted by the ergodic theorem was investigated. Several time periods, including a span of nearly two years, are analyzed. Data intervals which span many solar rotations satisfy the conditions of weak stationarity if the effects of solar rotation are included in the asymptotic analysis. Shorter intervals which include a small integral number of interplanetary sectors also satisfy weak stationarity. The results are illustrated using magnetometer data from the ISEE-3, Voyager and IMP spacecraft.

Matthaeus, W. H.↗

NASA NTRS1984

On the topological stability of magnetostatic equilibria

The topological stability of MHD equilibria is investigated by exploring the formal analogy, in the ideal MHD limit, between the topology of magnetic lines of force in coordinate space and the topology of integral surfaces of one- and two-dimensional Hamiltonian systems in phase space. It is demonstrated that in an astrophysical setting, symmetric magnetostatic equilibria satisfying the ideal MHD equations are exceptional. The principal result of the study is that previous infinitesimal perturbation theory calculations can be generalized to include finite-amplitude and symmetry-breaking effects. The effect of the ergodicity of perturbed symmetric equilibria on heat dispersal in magnetically dominated plasmas is discussed.

Tsinganos, K. C.↗

NASA NTRS1987

Modeling of controlled flexible structures with impulsive loads

The characteristic wave approach is developed as an alternative to modal methods which may lead to significant errors in the presence of impulsive or concentrated loads. The method is applied to periodic structures. Some special phenomena like cumulation effects and transitions to ergodicity are analyzed.

Zak, M.↗

NASA NTRS1987

Dynamical response to pulse excitations in large space structures

Finite dimensional approximations of large space structures as distributed parameter systems may lead to a loss of contribution of high frequencies to the dynamic response in case of impulsive or concentrated loads. It is shown that the unmodeled part of this response can be represented by a system of thin pulses which propagate as characteristic waves. It is demonstrated that the dynamical response to such a system of pulses can be modeled by a system of equations with delay argument. Fundamental dynamical properties of this system such as Liapunov stability, structural stability, loss of periodicity and transition to ergodicity are analyzed in this study. The results are illustrated by examples.

Zak, Michail↗

NASA NTRS1987

Effects of intensity modulations on the power spectra of random processes

Intensity-modulated random processes (IMRPs), defined as the products of (1) deterministic modulating functions or processes and (2) stationary modulated processes statistically independent of (1), are investigated analytically. Instantaneous power spectra are derived for IMRPs with different classes of (1), and a spectrum series expansion with a known locally stationary approximation as its first term is obtained. Numerical results for sample IMRPs with known (deterministic), stationary, nonstationary, ergodic, onset, and bell-shaped types of (1) are presented graphically and briefly characterized.

Mark, W. D.↗

NASA NTRS1989

A finite element formulation for the large deflection random response of thermally buckled beams

The effects of temperature and acoustic loading are included in a theoretical finite element large deflection formulation for thin, isotropic beams. Thermal loads are applied as steady-state temperature distributions, and acoustic loads are taken to be ergodic and Gaussian with zero mean and uniform magnitude and phase along the length of the beam. Material properties are considered presently to be independent of temperature. Also, inplane and rotary inertia terms are assumed to be negligible, and all inplane edge conditions are taken to be immovable. For the random response analysis, both auto- and cross-correlation terms are included. The nature of the loads leads to the solution of two separate problems. First, the problem of thermal postbuckling is solved to determine the deflections and stresses due to the thermal load only. These deflections and stresses are then used as initial deflections and stresses for the random vibration analysis. Root-mean-square (RMS) maximum deflections and strains are obtained and compared with previous classical equivalent linearization results.

Locke, James↗

NASA NTRS1990

An investigation of chaotic Kolmogorov flows

A two dimensional flow governed by the incompressible Navier-Stokes equations with a steady spatially periodic forcing (known as the Kolmogorov flow) is numerically simulated. The behavior of the flow and its transition states as the Reynolds number (Re) varies is investigated in detail, as well as a number of the flow features. A sequence of bifurcations is shown to take place in the flow as Re varied. Two main regimes of the flow were observed: small and large scale structure regimes corresponding to different ranges of Re. Each of the regimes includes a number of quasiperiodic, chaotic, and relaminarization windows. In addition, each range contains a chaotic window with non-ergodic chaotic attractors. Spatially disordered, but temporally steady states were discovered in large scale structure regime. Features of the diverse cases are displayed in terms of the temporal power spectrum, Poincare sections and, where possible, Lyapunov exponents and Kaplan-Yorke dimension.

Platt, N.↗

NASA NTRS1990

Spatial variability of rain rate and slant path attenuation distributions at 28 GHz in the mid-Atlantic coast region of the United States

Rain rate and corresponding estimated slant-path attenuation distributions derived from two years of measurements are presented for a network of nine rain-gage sites located in a region with extent 70 km N-S and 47 km E-W in the mid-Atlantic coast of the U.S. near the NASA Wallops Flight Facility, VA. The network average rain-rate distribution was compared with a previously measured 6-year average at one of the site locations. Agreement in rain rates was found to be within approximately 1 mm/h over the percentage range of 1-0.01 percent of the year, signifying the constancy of the rain-rate climatology for the region, and implying an ergodic rainfall process. Slant-path attenuation statistics were estimated at 28.56 GHz by interfacing the measured rain-rate distributions at each of nine site locations with Crane's (1980) global model.

Goldhirsh, Julius↗

NASA NTRS1991

An investigation of chaotic Kolmogorov flows

Platt, N.↗

NASA NTRS1991

Limits on the isolation of stochastic vibration for microgravity space experiments

The limitations on the isolation of stochastic vibrations for microgravity space experiments are explored. These limitations result from the restricted interior space available for vibration isolation. A one-degree-of-freedom representation of the experiment-spacecraft system is used, and an ideal vibration actuator is assumed. A kinematic representation results, and the problem becomes one of finding the minimum acceleration trajectory within a pair of stochastic walls. The wall motion is characterized by an ergodic, stationary, zero-mean, Gaussian random process with known power spectral density. The geometry of the wall trajectories is defined in terms of their significant extrema and zero crossings. This geomemtry is used in defining a composite trajectory that has a mean square acceleratiaon lower than that on the optimal path satisfying the stochastic wall inequality constraints. The optimal control problem is solved on a return path yielding the mean square acceleration in terms of the distributions of significant maxima and first-passage time of the wall process. The methodology is applied to a microgravity isolation problem to find the lower bounds on root-mean-square acceleration given the disturbance power spectral density.

Knospe, C. R.↗

NASA NTRS1991

Nonlinear dynamics and predictability in the atmospheric sciences

Systematic applications of nonlinear dynamics to studies of the atmosphere and climate are reviewed for the period 1987-1990. Problems discussed include paleoclimatic applications, low-frequency atmospheric variability, and interannual variability of the ocean-atmosphere system. Emphasis is placed on applications of the successive bifurcation approach and the ergodic theory of dynamical systems to understanding and prediction of intraseasonal, interannual, and Quaternary climate changes.

Ghil, M.↗

NASA NTRS1992

Turbulent fluid motion IV-averages, Reynolds decomposition, and the closure problem

Ensemble, time, and space averages as applied to turbulent quantities are discussed, and pertinent properties of the averages are obtained. Those properties, together with Reynolds decomposition, are used to derive the averaged equations of motion and the one- and two-point moment or correlation equations. The terms in the various equations are interpreted. The closure problem of the averaged equations is discussed, and possible closure schemes are considered. Those schemes usually require an input of supplemental information unless the averaged equations are closed by calculating their terms by a numerical solution of the original unaveraged equations. The law of the wall for velocities and temperatures, the velocity- and temperature-defect laws, and the logarithmic laws for velocities and temperatures are derived. Various notions of randomness and their relation to turbulence are considered in light of ergodic theory.

Deissler, Robert G.↗