An estimate of the stable initial condition region based on the describing function.
Nonlinear control system, discussing method to obtain estimate of region of stable initial conditions
NETWORK SYNTHESIS↗
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A development of the describing functions for two nonlinearities separated by a linear function Final report, 1 Sep. 1965 - 1 Feb. 1966
Describing functions to analyze feedback control systems containing nonlinearities of dead zone and backlash
NONLINEARITY↗
Development of a nonlinear switching function and its application to static lift characteristics of straight wings
A mathematical modeling technique was developed for the lift characteristics of straight wings throughout a very wide angle of attack range. The technique employs a mathematical switching function that facilitates the representation of the nonlinear aerodynamic characteristics in the partially and fully stalled regions and permits matching empirical data within + or - 4 percent of maximum values. Although specifically developed for use in modeling the lift characteristics, the technique appears to have other applications in both aerodynamic and nonaerodynamic fields.
Hewes, D. E.↗
Two-dimensional parametric design of short pulse laser driven iron opacity experiments
High intensity short pulse lasers are commonly used to create high temperature and high density plasmas. These plasmas are used to study fundamental properties such as the equation of state and opacity. The targets often use small-diameter, thin buried layers of the materials of interest, surrounded by a tamper of low atomic number material such as plastic. Computational modeling is used to design and interpret experiments with short pulse lasers. Most of the modeling to date has assumed one-dimensional plane-parallel geometry. In this paper, the effects of radial gradients in the irradiation of thin planar targets are studied with one- and two-dimensional radiation/hydrodynamic simulations. It is found that the main effect of radial gradients is the averaging of plasma conditions and x-ray emission over the pattern of irradiation. Additionally, differences between one- and two-dimensional simulations arise because the plasma conditions and x-ray emission are, in general, nonlinear functions of the temperature of the plasma and thereby also nonlinear functions of the irradiation intensity. The differences increase with the ratio of the buried layer radius to the laser spot radius. The root mean square difference in the inferred iron L-shell opacity is less than 30% when the ratio is less than 1.0 and about 70% when the ratio is 1.5.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Marshall system for aerospace system simulation (MARSYAS), user's manual
The capabilities of the Marshall system for aerospace system simulation (MARSYAS) and how to use it are described. MARSYAS is a software system that allows easy setup and control of the simulation of the dynamics of large physical systems on a digital computer. The physical systems are modeled in the form of block diagrams or equations. The blocks can have multiple inputs and multiple outputs, and they can be nested to form hierarchies. The block diagrams can contain transfer functions, nonlinear and logical functions, equations, analog computer elements and FORTRAN programs. The input format of the equations can be combinations of nonlinear, time-varying differential equations and algebraic equations in their original format. MARSYAS could also serve as a storage and retrieval system for models as a basis for a model configuration control system on a central time-shared computer. The outputs of the simulation system can be not only time-responses but also other analysis data such as frequency response, power spectrum and stability parameters. The MARSYAS translator is written in FORTRAN running on the Univac 1108 computer under the EXEC 8 operating system.
Ventre, A.↗
Growth stage estimation
Of the three candidate approaches to adjustment of the crop calendar to account for year-to-year weather differences, the Robertson triquadratic unit, a function of a nonlinear function of maximum and minimum temperature and day length, best described the rate of phenological development of wheat. The adjustable crop calendar (ACC) as implemented for LACIE is used to calculate the daily increment of development through six physiological stages of growth. Topics covered include dormancy modeling, the spring restart model, spring wheat starter model, winter starter model, winter wheat starter model, inclusion of the moisture variable, and display of crop stage estimation results. Assessment of the ACC accuracy over the period of LACIE operation indicates that the adjustable crop calendars used provided more accurate information than would have been available using historical norms. The models performed best under the conditions from which they were derived (Canadian spring wheat) and most poorly for the dwarf varieties and Southern Hemisphere applications.
Whitehead, V. S.↗
Detection of nonlinear transfer functions by the use of Gaussian statistics
The possibility of using on-line signal statistics to detect electronic equipment nonlinearities is discussed. The results of an investigation using Gaussian statistics are presented, and a nonlinearity test that uses ratios of the moments of a Gaussian random variable is developed and discussed. An outline for further investigation is presented.
Sheppard, J. G.↗
Mathematical modeling of the aerodynamic characteristics in flight dynamics
Basic concepts involved in the mathematical modeling of the aerodynamic response of an aircraft to arbitrary maneuvers are reviewed. The original formulation of an aerodynamic response in terms of nonlinear functionals is shown to be compatible with a derivation based on the use of nonlinear functional expansions. Extensions of the analysis through its natural connection with ideas from bifurcation theory are indicated.
Tobak, M.↗
Analytical aerodynamic model of a high alpha research vehicle wind-tunnel model
A 6 DOF analytical aerodynamic model of a high alpha research vehicle is derived. The derivation is based on wind-tunnel model data valid in the altitude-Mach flight envelope centered at 15,000 ft altitude and 0.6 Mach number with Mach range between 0.3 and 0.9. The analytical models of the aerodynamics coefficients are nonlinear functions of alpha with all control variable and other states fixed. Interpolation is required between the parameterized nonlinear functions. The lift and pitching moment coefficients have unsteady flow parts due to the time range of change of angle-of-attack (alpha dot). The analytical models are plotted and compared with their corresponding wind-tunnel data. Piloted simulated maneuvers of the wind-tunnel model are used to evaluate the analytical model. The maneuvers considered are pitch-ups, 360 degree loaded and unloaded rolls, turn reversals, split S's, and level turns. The evaluation finds that (1) the analytical model is a good representation at Mach 0.6, (2) the longitudinal part is good for the Mach range 0.3 to 0.9, and (3) the lateral part is good for Mach numbers between 0.6 and 0.9. The computer simulations show that the storage requirement of the analytical model is about one tenth that of the wind-tunnel model and it runs twice as fast.
Cao, Jichang↗
Effect of Forcing Function on Nonlinear Acoustic Standing Waves
Nonlinear acoustic standing waves of high amplitude have been demonstrated by utilizing the effects of resonator shape to prevent the pressure waves from entering saturation. Experimentally, nonlinear acoustic standing waves have been generated by shaking an entire resonating cavity. While this promotes more efficient energy transfer than a piston-driven resonator, it also introduces complicated structural dynamics into the system. Experiments have shown that these dynamics result in resonator forcing functions comprised of a sum of several Fourier modes. However, previous numerical studies of the acoustics generated within the resonator assumed simple sinusoidal waves as the driving force. Using a previously developed numerical code, this paper demonstrates the effects of using a forcing function constructed with a series of harmonic sinusoidal waves on resonating cavities. From these results, a method will be demonstrated which allows the direct numerical analysis of experimentally generated nonlinear acoustic waves in resonators driven by harmonic forcing functions.
Finkheiner, Joshua R.↗
Curved Displacement Transfer Functions for Geometric Nonlinear Large Deformation Structure Shape Predictions
For shape predictions of structures under large geometrically nonlinear deformations, Curved Displacement Transfer Functions were formulated based on a curved displacement, traced by a material point from the undeformed position to deformed position. The embedded beam (depth-wise cross section of a structure along a surface strain-sensing line) was discretized into multiple small domains, with domain junctures matching the strain-sensing stations. Thus, the surface strain distribution could be described with a piecewise linear or a piecewise nonlinear function. The discretization approach enabled piecewise integrations of the embedded-beam curvature equations to yield the Curved Displacement Transfer Functions, expressed in terms of embedded beam geometrical parameters and surface strains. By entering the surface strain data into the Displacement Transfer Functions, deflections along each embedded beam can be calculated at multiple points for mapping the overall structural deformed shapes. Finite-element linear and nonlinear analyses of a tapered cantilever tubular beam were performed to generate linear and nonlinear surface strains and the associated deflections to be used for validation. The shape prediction accuracies were then determined by comparing the theoretical deflections with the finiteelement- generated deflections. The results show that the newly developed Curved Displacement Transfer Functions are very accurate for shape predictions of structures under large geometrically nonlinear deformations.
displacement theory↗
Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work
Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.
Johnson, Charles A.↗
Green's-function theory of nonlinear transport coefficients.
Green function theory of nonlinear transport coefficients
GREEN FUNCTION↗
Optimum sensitivity derivatives of objective functions in nonlinear programming
The feasibility of eliminating second derivatives from the input of optimum sensitivity analyses of optimization problems is demonstrated. This elimination restricts the sensitivity analysis to the first-order sensitivity derivatives of the objective function. It is also shown that when a complete first-order sensitivity analysis is performed, second-order sensitivity derivatives of the objective function are available at little additional cost. An expression is derived whose application to linear programming is presented.
Barthelemy, J.-F. M.↗
One type of functional converter with several inputs
Nonlinear functional converter with several inputs
CONVERTER↗
Connecting functions and nonlinear normal modes for dimension reduction in conservative structural dynamics models.
Abstract not provided.
Kuether, Robert↗
Improved Displacement Transfer Functions for Structure Deformed Shape Predictions Using Discretely Distributed Surface Strains
In the formulations of earlier Displacement Transfer Functions for structure shape predictions, the surface strain distributions, along a strain-sensing line, were represented with piecewise linear functions. To improve the shape-prediction accuracies, Improved Displacement Transfer Functions were formulated using piecewise nonlinear strain representations. Through discretization of an embedded beam (depth-wise cross section of a structure along a strain-sensing line) into multiple small domains, piecewise nonlinear functions were used to describe the surface strain distributions along the discretized embedded beam. Such piecewise approach enabled the piecewise integrations of the embedded beam curvature equations to yield slope and deflection equations in recursive forms. The resulting Improved Displacement Transfer Functions, written in summation forms, were expressed in terms of beam geometrical parameters and surface strains along the strain-sensing line. By feeding the surface strains into the Improved Displacement Transfer Functions, structural deflections could be calculated at multiple points for mapping out the overall structural deformed shapes for visual display. The shape-prediction accuracies of the Improved Displacement Transfer Functions were then examined in view of finite-element-calculated deflections using different tapered cantilever tubular beams. It was found that by using the piecewise nonlinear strain representations, the shape-prediction accuracies could be greatly improved, especially for highly-tapered cantilever tubular beams.
Ko, William L.↗
Inversion of multiwavelength radiometer measurements by three-dimensional filtering
Remote sensing data from satellites typically have three dimensions: scan position, spacecraft position, and wavelength. Inversion of the radiometric data to infer geophysical parameters is a filtering problem in which the dimension of wavelength (or channel number) is transformed into a dimension of geophysical parameters, and the most general solution is a three-dimensional filter. Linear filters have the advantages of computational speed and easily described transfer functions; but often the measurements are nonlinear functions of the parameters to be inferred. To the extent that the nonlinear inversion problem is overdetermined, it can be modeled by a critically determined linear problem. As an example, inversion of Scanning Multichannel Microwave Radiometer (SMMR) data by means of a three-dimensional Wiener Filter is described. Atmospheric water vapor content, rain liquid water content, surface wind speed and surface temperature are the parameters inferred from the measurements. Nonprecipitating liquid water and water vapor scale height are also modeled but not retrieved. The a priori statistics on which the filter is trained have the effect of governing the selection of a trade-off point of noise as a function of resolution (in all three retrieval dimensions).
Rosenkranz, P. W.↗