An infinite class of periodic solutions of d super 2x/dt super 2 plus 2x super 3 equal p/t/.
Second order forced nonlinear oscillation differential equation with time-dependent term not even, having infinity of periodic solutions
PERIODIC FUNCTION↗
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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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Numerical integration of nonlinear differential equations by use of rational approximation Final report, 15 Jan. 1965 - 14 Jul. 1967
Numerical integration of nonlinear differential equations by use of rational approximation
NUMERICAL INTEGRATION↗
Luminescent screen composition for cathode ray tubes
Screen composition for cathode ray tubes exhibits differential color of emission as a function of beam current variation at a constant accelerating voltage. The screen consists of a mixture of phosphors which emit different hues, have different current saturation values and exhibit a nonlinear current-brightness characteristic.
Hilborn, E. H.↗
The value of improved (ERS) information based on domestic distribution effects of U.S. agriculture crops
The value of improving information for forecasting future crop harvests was investigated. Emphasis was placed upon establishing practical evaluation procedures firmly based in economic theory. The analysis was applied to the case of U.S. domestic wheat consumption. Estimates for a cost of storage function and a demand function for wheat were calculated. A model of market determinations of wheat inventories was developed for inventory adjustment. The carry-over horizon is computed by the solution of a nonlinear programming problem, and related variables such as spot and future price at each stage are determined. The model is adaptable to other markets. Results are shown to depend critically on the accuracy of current and proposed measurement techniques. The quantitative results are presented parametrically, in terms of various possible values of current and future accuracies.
Bradford, D. F.↗
Shuttle Experiment Pointing Mount /EPM/ Systems
Shuttle Experiment Pointing Mount System concepts and technology have been investigated and assessed with regard to payload requirements having a wide range of stability, accuracy, and control functions. Pointing systems were analyzed with end-mounted and center of gravity-mounted payloads viewing stellar and solar targets. Major error sources are identified with dynamical, stochastic, and nonlinear characteristics of structures, isolators, sensors, bearings, actuators, and controller. Results are presented which place a perspective on the potential of advanced technology to satisfy the most stringent sub-arc second pointing requirements.
Mettler, E.↗
Bifilar analysis study, volume 1
A coupled rotor/bifilar/airframe analysis was developed and utilized to study the dynamic characteristics of the centrifugally tuned, rotor-hub-mounted, bifilar vibration absorber. The analysis contains the major components that impact the bifilar absorber performance, namely, an elastic rotor with hover aerodynamics, a flexible fuselage, and nonlinear individual degrees of freedom for each bifilar mass. Airspeed, rotor speed, bifilar mass and tuning variations are considered. The performance of the bifilar absorber is shown to be a function of its basic parameters: dynamic mass, damping and tuning, as well as the impedance of the rotor hub. The effect of the dissimilar responses of the individual bifilar masses which are caused by tolerance induced mass, damping and tuning variations is also examined.
Miao, W.↗
Optimality criteria solution strategies in multiple constraint design optimization
Procedures and solution strategies are described to solve the conventional structural optimization problem using the Lagrange multiplier technique. The multipliers, obtained through solution of an auxiliary nonlinear optimization problem, lead to optimality criteria to determine the design variables. It is shown that this procedure is essentially equivalent to an alternative formulation using a dual method Lagrangian function objective. Although mathematical formulations are straight-forward, successful applications and computational efficiency depend upon execution procedure strategies. Strategies examined, with application examples, include selection of active constraints, move limits, line search procedures, and side constraint boundaries.
Levy, R.↗
Analysis of a longitudinal pilot-induced oscillation experienced on the approach and landing test of the space shuttle
During the final free flight (FF-5) of the shuttle's approach and landing tests, the vehicle experienced pilot-induced oscillations near touchdown. The light test data showed that pilot inputs to the hand controller reached peak-to-peak amplitudes of 20 deg at a frequency between 3 and 3.5 radians per second. The controller inputs were sufficient to exceed the priority rate limit set in the pitch axis. A nonlinear analytical study was conducted to investigate the combined effects of pilot input, rate limiting, and time delays. The frequency response of the total system is presented parametrically as a function of the three variables. In general, with no dead time, for controller inputs of 5 deg or less, the total system behaves in a linear fashion. For 10 deg of controller input, independent of the delay time, the elevon loop will be rate saturated above a frequency of 4 radians per second.
Smith, J. W.↗
Coupling of radiation into thin film modes by means of localized plasma resonances
The interaction between the surface plasmon mode that propagates at a metal dielectric interface and the localized plasma resonances (LPR) is investigated experimentally in Ag-island films. A stair-stepped sample geometry comprising a glass substrate, a continuous 50-nm Ag film, an LiF spacer film of thickness d = 5-60 nm, and an Ag-island film of mass thickness 3 nm is used in near-normal-reflectivity and plasmon-propagation-constant (k) determinations. The results are presented graphically and discussed. The overall shape of the reflectivity curves is found to be characteristic of Ag films, but with a dip at about 400 nm (corresponding to the absorption resonance of the island film) which is most pronounced with d = 25 nm. It is inferred that the island resonances are strongly coupled to a continuous-film dissipative mechanism at this d value. This inference is supported by the fact that the variation in k, correctd for LiF effects and plotted as a function of d, is greatest at around d = 25 nm. The implications of this finding for broad-band coupling into a thin-film mode, LPR enhancement of waveguide nonlinear effects, and new surface-enhanced-Raman-scattering geometries are indicated.
Holland, W. R.↗
Simulation of non-linear bearing forces for post-stability investigation
Different types of bearing designs were developed to improve dynamic properties of rotor-bearing systems. Elliptical bearings, multisleeve bearings, tilting pad and other designs such as herringbone groove were utilized to increase resistance to the onset of self excited vibrations. Experimental trials are costly, two alternative methods are used to gain a qualitative insight. The first one creates mathematical model and applies both a digital or an analog computer simulation. The second one investigates phenomena occurring on the laboratory rig with the bearing replaced by an electronic simulating device, working in a feedback loop, which produces forces,which are functions of journal displacement and velocity. The simulated hydrodynamic forces are produced according to assumed characteristics matched to the bearing type. The principal benefit of the analog simulation is that nonlinear characteristics of a subsystem are precisely identified and mathematical methods applied for a wide class of problems are checked on the experimental installation.
Parszewski, Z. A.↗
The Chinchilla's vestibulo-ocular reflex
The horizontal vestibulo-ocular reflex (VOR) was measured and characterized in seven adult chinchillas using 0.01 to 1.0 Hz angular velocity sinusoids. Gains were less than compensatory, and were variable from day to day, but phases were highly repeatable both within and between animals. The best fitting transfer function to the average data of all animals had a dominant time constant of 7.5 sec, and an adaptation operator with a time constant of 24.0 sec. There were certain nonlinearities in the horizontal VOR of this animal, and it was difficult to elicit a robust optokinetic response. Results are discussed in relation to similar measurements in other species.
Merwin, W. H., Jr.↗
Comparison of dynamical approximation schemes for non-linear gravitational clustering
We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of approximation by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(exp 2, sub G)) where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich approximation created by adding forces, were in fact generally worse by this measure. By explicitly checking, we verified that the success of our best-choice was a result of the best treatment of the phases of nonlinear Fourier components. Of all schemes tested, the adhesion approximation produced the most accurate nonlinear power spectrum and density distribution, but its phase errors suggest mass condensations were moved to slightly the wrong location. Due to its better reproduction of the mass density distribution function and power spectrum, it might be preferred for some uses. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even when subcondensations are present.
Melott, Adrian L.↗
Test particle study of minor ions in solar wind turbulence
We perform a parameter study of the temporal evolution of a test particle distribution function in MHD turbulence. The turbulent fields are calculated using a pseudo-spectral method and periodic boundary conditions on a regular grid of 180(exp 3) points, appropriate for incompressible, homogeneous and isotropic turbulence. Initially, the kinetic and the magnetic energy are equal on the average. Both, deterministic and random initial conditions are used, in the former case with zeros of the magnetic field located at grid points, in the latter case located by interpolation between grid points. The evolution of the minor ion distribution function is studied in detail as these turbulent fields evolve, developing strong current and vorticity sheets. Using the full collisionless equation of motion for the test particles, the efficiency of nonlinear interactions can be studied. The results are compared to theoretical predictions and are then discussed in connection with the observations of the dynamical properties of solar wind minor ions derived from in situ observations.
Zurbuchen, Th.↗
The Dynamics of Some Iterative Implicit Schemes
The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations is analyzed using the theory of dynamical systems. With the aid of parallel Connection Machines (CM-2 and CM-5), the associated bifurcation diagrams as a function of the time step, and the complex behavior of the associated 'numerical basins of attraction' of these iterative implicit schemes are revealed and compared. Studies showed that all of the four implicit LMMs exhibit a drastic distortion and segmentation but less shrinkage of the basin of attraction of the true solution than standard explicit methods. The numerical basins of attraction of a noniterative implicit procedure mimic more closely the basins of attraction of the differential equations than the iterative implicit procedures for the four implicit LMMs.
Yee, H. C.↗
Adaptive Performance Seeking Control Using Fuzzy Model Reference Learning Control and Positive Gradient Control
Performance Seeking Control attempts to find the operating condition that will generate optimal performance and control the plant at that operating condition. In this paper a nonlinear multivariable Adaptive Performance Seeking Control (APSC) methodology will be developed and it will be demonstrated on a nonlinear system. The APSC is comprised of the Positive Gradient Control (PGC) and the Fuzzy Model Reference Learning Control (FMRLC). The PGC computes the positive gradients of the desired performance function with respect to the control inputs in order to drive the plant set points to the operating point that will produce optimal performance. The PGC approach will be derived in this paper. The feedback control of the plant is performed by the FMRLC. For the FMRLC, the conventional fuzzy model reference learning control methodology is utilized, with guidelines generated here for the effective tuning of the FMRLC controller.
Kopasakis, George↗
Nonlinear Thermal Analyses of a Liquid Hydrogen Tank Wall
A thermal evaluation of a composite tank wall design for a liquid hydrogen tank was performed in the present study. The primary focus of the current effort was to perform one-dimensional, temperature nonlinear, transient thermal analyses to determine the through-the-thickness temperature profiles. These profiles were used to identify critical points within the flight envelope that could have detrimental effects on the adhesive bondlines used in the construction of the tank wall. Additionally, this paper presents the finite element models, analysis strategies, and thermal analysis results that were determined for several vehicle flight conditions. The basic tank wall configuration used to perform the thermal analyses consisted of carbon-epoxy facesheets and a Korex honeycomb core sandwich that was insulated with an Airex cryogenic foam and an Alumina Enhanced Thermal Barrier (AETB-12). Nonlinear, transient thermal analyses were conducted using the ABAQUS finite element code. Tank wall models at a windward side location on the fuel tank were analyzed for three basic flight conditions: cold-soak (ground-hold), ascent, and re-entry. Additionally, three ambient temperature boundary conditions were applied to the tank wall for the cold-soak condition, which simulated the launch pad cooldown process. Time-dependent heating rates were used in the analyses of the ascent and reentry segments of the flight history along with temperature dependent material properties. The steady-state through-the-thickness temperature profile from the cold-soak condition was used as the initial condition for the ascent analyses. Results from the nonlinear thermal analyses demonstrated very good correlation with results from similar models evaluated by Northrop- Grumman using a different analysis tool. Wall through-the-thickness temperature gradients as a function of flight time were obtained for future incorporation into a full-scale thermostructural analysis to evaluate the adhesive bondlines. As a result of the thermal analyses conducted, a sufficient level of confidence was demonstrated in the thermal modeling and analysis capabilities of ABAQUS to warrant future use as a thermo-structural analysis tool to evaluate cryogenic tank wall designs.
Smeltzer, Stanley S., III↗
On Certain Theoretical Developments Underlying the Hilbert-Huang Transform
One of the main traditional tools used in scientific and engineering data spectral analysis is the Fourier Integral Transform and its high performance digital equivalent - the Fast Fourier Transform (FFT). Both carry strong a-priori assumptions about the source data, such as being linear and stationary, and of satisfying the Dirichlet conditions. A recent development at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC), known as the Hilbert-Huang Transform (HHT), proposes a novel approach to the solution for the nonlinear class of spectral analysis problems. Using a-posteriori data processing based on the Empirical Mode Decomposition (EMD) sifting process (algorithm), followed by the normalized Hilbert Transform of the decomposed data, the HHT allows spectral analysis of nonlinear and nonstationary data. The EMD sifting process results in a non-constrained decomposition of a source real-value data vector into a finite set of Intrinsic Mode Functions (IMF). These functions form a nearly orthogonal derived from the data (adaptive) basis. The IMFs can be further analyzed for spectrum content by using the classical Hilbert Transform. A new engineering spectral analysis tool using HHT has been developed at NASA GSFC, the HHT Data Processing System (HHT-DPS). As the HHT-DPS has been successfully used and commercialized, new applications pose additional questions about the theoretical basis behind the HHT and EMD algorithms. Why is the fastest changing component of a composite signal being sifted out first in the EMD sifting process? Why does the EMD sifting process seemingly converge and why does it converge rapidly? Does an IMF have a distinctive structure? Why are the IMFs nearly orthogonal? We address these questions and develop the initial theoretical background for the HHT. This will contribute to the development of new HHT processing options, such as real-time and 2-D processing using Field Programmable Gate Array (FPGA) computational resources,
Kizhner, Semion↗
Discharge Oscillations in a Permanent Magnet Cylindrical Hall-Effect Thruster
Measurements of the discharge current in a cylindrical Hall thruster are presented to quantify plasma oscillations and instabilities without introducing an intrusive probe into the plasma. The time-varying component of the discharge current is measured using a current monitor that possesses a wide frequency bandwidth and the signal is Fourier transformed to yield the frequency spectra present, allowing for the identification of plasma oscillations. The data show that the discharge current oscillations become generally greater in amplitude and complexity as the voltage is increased, and are reduced in severity with increasing flow rate. The breathing mode ionization instability is identified, with frequency as a function of discharge voltage not increasing with discharge voltage as has been observed in some traditional Hall thruster geometries, but instead following a scaling similar to a large-amplitude, nonlinear oscillation mode recently predicted in for annular Hall thrusters. A transition from lower amplitude oscillations to large relative fluctuations in the oscillating discharge current is observed at low flow rates and is suppressed as the mass flow rate is increased. A second set of peaks in the frequency spectra are observed at the highest propellant flow rate tested. Possible mechanisms that might give rise to these peaks include ionization instabilities and interactions between various oscillatory modes.
Polzin, K. A.↗