Monotone Iterations for Nonlinear Equations with Application to Gauss-seidel Methods
Monotone iterations for nonlinear equations with application to Gauss-Seidel methods
GAUSS FUNCTION↗
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Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces
The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.
Tiffany, Sherwood H.↗
Construction of the Yapunov Function for a case of stability with constantly acting disturbances in nonlinear spaces
Construction of Lyapunov function for case of stability with constantly acting disturbances in nonlinear spaces
BANACH SPACE↗
The choice of trial functions in the perturbation-variation method.
Nonlinear variational trial function for atomic hydrogen dynamic polarizability, discussing first order perturbed wave function and perturbation- variation method
HYDROGEN ATOM↗
Study of a nonlinear lead compensator
Design and functioning of split path nonlinear compensator compared with conventional controllers
Jenney, W. P.↗
Aeroelastic Modeling and Behavior of Lifting Surfaces Incorporating Aerodynamic and Structural Nonlinearities: Volterra Series and Indicial Function Approach
Within this NASA Grant, the following points should be emphasized: 1) All the objectives stated in the proposal of the grant have been accomplished. Moreover, the activity within the project has addressed additional issues, of the linear and nonlinear aeroelasticity, not included in the objectives of the grant. 2) During the activities within the grant, we have been in a permanent contact with Dr. Walter A. Silva, the monitor of the NASA Project, to whom we have reported continuously our achievements. 3) As a result of the activities within the grant a number of papers: a. have been submitted for publication to the AIAA Journals, namely the AIAA Journal and the Journal of Guidance, Control, and Dynamics, b. have been presented at the specialized National Conferences and an International Congress, and have appeared in the proceedings of these Conferences, 4) A list of papers submitted for publication and presented at Conferences is appended herewith. 5) In all these papers, an acknowledgment to NASA Langley Research Center was included.
Librescu, Liviu↗
Aeroelastic Modeling and Behavior of Lifting Surfaces Incorporating Aerodynamic and Structural Nonlinearities: Volterra Series and Indicial Function Approach
Within this NASA Grant, the following points should be emphasized: 1) All the objectives stated in the proposal of the grant have been accomplished. Moreover. the activity within the project has addressed additional issues, of the linear and nonlinear aeroelasticity, not included in the objectives of the grant. 2) During the activities within the grant, we have been in a permanent contact with Dr. Walter A. Silva, the monitor of the NASA Project, to whom we have reported continuously our achievements. 3) As a result of the activities within the grant a number of papers: a. have been submitted for publication to the AIAA Journals, namely the AIAA Journal and the Journal of Guidance, Control, and Dynamics, and b. have been presented at the specialized National Conferences and an International Congress, and have appeared in the proceedings of these Conferences. 4) A list of papers submitted for publication and presented at Conferences is appended herewith. 5) In all these papers, an acknowledgment to NASA Langley Research Center was included.
Librescu, Liviu↗
Design of fuzzy system by NNs and realization of adaptability
The issue of designing and tuning fuzzy membership functions by neural networks (NN's) was started by NN-driven Fuzzy Reasoning in 1988. NN-driven fuzzy reasoning involves a NN embedded in the fuzzy system which generates membership values. In conventional fuzzy system design, the membership functions are hand-crafted by trial and error for each input variable. In contrast, NN-driven fuzzy reasoning considers several variables simultaneously and can design a multidimensional, nonlinear membership function for the entire subspace.
Takagi, Hideyuki↗
Parallel processors and nonlinear structural dynamics algorithms and software
A nonlinear structural dynamics finite element program was developed to run on a shared memory multiprocessor with pipeline processors. The program, WHAMS, was used as a framework for this work. The program employs explicit time integration and has the capability to handle both the nonlinear material behavior and large displacement response of 3-D structures. The elasto-plastic material model uses an isotropic strain hardening law which is input as a piecewise linear function. Geometric nonlinearities are handled by a corotational formulation in which a coordinate system is embedded at the integration point of each element. Currently, the program has an element library consisting of a beam element based on Euler-Bernoulli theory and trianglar and quadrilateral plate element based on Mindlin theory.
Belytschko, Ted↗
Programming the gradient projection algorithm
The gradient projection method of numerical optimization which is applied to problems having linear constraints but nonlinear objective functions is described and analyzed. The algorithm is found to be efficient and thorough for small systems, but requires the addition of auxiliary methods and programming for large scale systems with severe nonlinearities. In order to verify the theoretical results a digital computer is used to simulate the algorithm.
Hargrove, A.↗
Vibration Control in Turbomachinery Using Active Magnetic Journal Bearings
The effective use of active magnetic bearings for vibration control in turbomachinery depends on an understanding of the forces available from a magnetic bearing actuator. The purpose of this project was to characterize the forces as functions shaft position. Both numerical and experimental studies were done to determine the characteristics of the forces exerted on a stationary shaft by a magnetic bearing actuator. The numerical studies were based on finite element computations and included both linear and nonlinear magnetization functions. Measurements of the force versus position of a nonrotating shaft were made using two separate measurement rigs, one based on strain gage measurement of forces, the other based on deflections of a calibrated beam. The general trends of the measured principal forces agree with the predictions of the theory while the magnitudes of forces are somewhat smaller than those predicted. Other aspects of theory are not confirmed by the measurements. The measured forces in the normal direction are larger than those predicted by theory when the rotor has a normal eccentricity. Over the ranges of position examined, the data indicate an approximately linear relationship between the normal eccentricity of the shaft and the ratio of normal to principal force. The constant of proportionality seems to be larger at lower currents, but for all cases examined its value is between 0.14 and 0.17. The nonlinear theory predicts the existence of normal forces, but has not predicted such a large constant of proportionality for the ratio. The type of coupling illustrated by these measurements would not tend to cause whirl, because the coupling coefficients have the same sign, unlike the case of a fluid film bearing, where the normal stiffness coefficients often have opposite signs. They might, however, tend to cause other self-excited behavior. This possibility must be considered when designing magnetic bearings for flexible rotor applications, such as gas turbines and other turbomachinery.
Knight, Josiah D.↗
Interactive orbital proximity operations planning system instruction and training guide
This guide instructs users in the operation of a Proximity Operations Planning System. This system uses an interactive graphical method for planning fuel-efficient rendezvous trajectories in the multi-spacecraft environment of the space station and allows the operator to compose a multi-burn transfer trajectory between orbit initial chaser and target trajectories. The available task time (window) of the mission is predetermined and the maneuver is subject to various operational constraints, such as departure, arrival, spatial, plume impingement, and en route passage constraints. The maneuvers are described in terms of the relative motion experienced in a space station centered coordinate system. Both in-orbital plane as well as out-of-orbital plane maneuvering is considered. A number of visual optimization aids are used for assisting the operator in reaching fuel-efficient solutions. These optimization aids are based on the Primer Vector theory. The visual feedback of trajectory shapes, operational constraints, and optimization functions, provided by user-transparent and continuously active background computations, allows the operator to make fast, iterative design changes that rapidly converge to fuel-efficient solutions. The planning tool is an example of operator-assisted optimization of nonlinear cost functions.
Grunwald, Arthur J.↗
Improved Fine Sun Sensor Field of View Calibration
The fine Sun sensor used on many spacecraft consists of two independent single-axis sensor heads, nominally mounted perpendicularly. These detect the Sun angle over a field of view typically of +32 deg. (There is a trade-off between accuracy and size of the field of view that allows for much latitude in any numbers quoted.) The nonlinear "transfer" function that maps the telemetered counts into observed angles consists of 9 adjustable parameters for each axis (1 8 total). An augmented transfer function has previously been reported that achieves a significant accuracy improvement across the entire field of view. That function expands the parameter set to 12 coefficients per axis (24 total) and includes cross terms combining counts from both axes. To make the best use of the Sun sensor for attitude determination, it must be calibrated after launch. However, the large number of parameters and the nonlinearity of the problem make this a challenging task. The purpose of this paper is to examine ways to improve convergence of the parameter search algorithm. In particular, experience has shown that the problem should be broken down into several steps, solving for a selected subset of the parameters at each step. This approach has now been incorporated as an option in the calibration utility.
Sedlak, J.↗
Error rate information in attention allocation pilot models
The Northrop urgency decision pilot model was used in a command tracking task to compare the optimized performance of multiaxis attention allocation pilot models whose urgency functions were (1) based on tracking error alone, and (2) based on both tracking error and error rate. A matrix of system dynamics and command inputs was employed, to create both symmetric and asymmetric two axis compensatory tracking tasks. All tasks were single loop on each axis. Analysis showed that a model that allocates control attention through nonlinear urgency functions using only error information could not achieve performance of the full model whose attention shifting algorithm included both error and error rate terms. Subsequent to this analysis, tracking performance predictions for the full model were verified by piloted flight simulation. Complete model and simulation data are presented.
Faulkner, W. H.↗
A class of neutral functional differential equations.
Formulation and study of the initial value problem for neutral functional differential equations. The existence, uniqueness, and continuation of solutions to this problem are investigated, and an analysis is made of the dependence of the solutions on the initial conditions and parameters, resulting in the derivation of a continuous dependence theorem in which the fundamental mathematical principles underlying the continuous dependence problem for a very general system of nonlinear neutral functional differential equations are separated out.
Melvin, W. R.↗
Electrical Transfer Function and Poling Mechanisms for Nonlinear Optical Polymer Modulators
Electro-Optic Polymers hold great promise in increased electro-optic coefficients as compared to their inorganic corollaries. Many researchers have focused on quantum chemistry to describe how the dipoles respond to temperature and electric fields. Much work has also been done for single layer films to confirm these results. For optical applications, waveguide structures are utilized to guide the optical waves in 3 layer stacks. Electrode poling is the only practical poling method for these structures. This research takes an electrical engineering approach to develop poling models and electrical and optical transfer functions of the waveguide structure. The key aspect of the poling model is the large boundary charge density deposited during the poling process. The boundary charge density also has a large effect on the electrical transfer function which is used to explain the transient response of the system. These models are experimentally verified. Exploratory experiment design is used to study poling parameters including time, temperature, and voltage. These studies verify the poling conditions for CLDX/APC and CLDZ/APEC guest host electro optic polymer films in waveguide stacks predicted by the theoretical developments.
Watson, Michael Dale↗
Nonlinear Dynamic Models in Advanced Life Support
To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.
Jones, Harry↗
An expert system based software sizing tool, phase 2
A software tool was developed for predicting the size of a future computer program at an early stage in its development. The system is intended to enable a user who is not expert in Software Engineering to estimate software size in lines of source code with an accuracy similar to that of an expert, based on the program's functional specifications. The project was planned as a knowledge based system with a field prototype as the goal of Phase 2 and a commercial system planned for Phase 3. The researchers used techniques from Artificial Intelligence and knowledge from human experts and existing software from NASA's COSMIC database. They devised a classification scheme for the software specifications, and a small set of generic software components that represent complexity and apply to large classes of programs. The specifications are converted to generic components by a set of rules and the generic components are input to a nonlinear sizing function which makes the final prediction. The system developed for this project predicted code sizes from the database with a bias factor of 1.06 and a fluctuation factor of 1.77, an accuracy similar to that of human experts but without their significant optimistic bias.
Friedlander, David↗