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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.
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.
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.
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.
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.
Green function theory of nonlinear transport coefficients
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.
Nonlinear functional converter with several inputs
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.
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).
Classes of n dimensional functions nonlinear matrix mapping, emphasizing block-Jacobi and block-Gauss-Seidel process convergence for continuous surjective M functions
High-temperature creep behavior of ceramics is characterized by nonlinear time-dependent responses, asymmetric behavior in tension and compression, and nucleation and coalescence of voids leading to creep rupture. Moreover, creep rupture experiments show considerable scatter or randomness in fatigue lives of nominally equal specimens. To capture the nonlinear, asymmetric time-dependent behavior, the standard linear viscoelastic solid model is modified. Nonlinearity and asymmetry are introduced in the volumetric components by using a nonlinear function similar to a hyperbolic sine function but modified to model asymmetry. The nonlinear viscoelastic model is implemented in an ABAQUS user material subroutine. To model the random formation and coalescence of voids, each element is assigned a failure strain sampled from a lognormal distribution. An element is deleted when its volumetric strain exceeds its failure strain. Element deletion has been implemented within ABAQUS. Temporal increases in strains produce a sequential loss of elements (a model for void nucleation and growth), which in turn leads to failure. Nonlinear viscoelastic model parameters are determined from uniaxial tensile and compressive creep experiments on silicon nitride. The model is then used to predict the deformation of four-point bending and ball-on-ring specimens. Simulation is used to predict statistical moments of creep rupture lives. Numerical simulation results compare well with results of experiments of four-point bending specimens. The analytical model is intended to be used to predict the creep rupture lives of ceramic parts in arbitrary stress conditions.
The augmented Lagrangian (ALAG) Penalty Function Algorithm for optimizing nonlinear mathematical models is discussed. The mathematical models of interest are deterministic in nature and finite dimensional optimization is assumed. A detailed review of penalty function techniques in general and the ALAG technique in particular is presented. Numerical experiments are conducted utilizing a number of nonlinear optimization problems to identify an efficient ALAG Penalty Function Technique for computer implementation.
A large number of roll rate stimuli, covering rates from zero to plus or minus 25 deg/sec, were presented to subjects in random order at 2 sec intervals. Subjects were to make estimates of magnitude of perceived roll rate stimuli presented on either a central display, on displays in the peripheral ield of vision, or on all displays simultaneously. Response was by way of a digital keyboard device, stimulus exposition times were varied. The present experiment differs from earlier perception tasks by the same authors in that mean rate perception error (and standard deviation) was obtained as a function of rate stimulus magnitude, whereas the earlier experiments only yielded mean absolute error magnitude. Moreover, in the present experiment, all stimulus rates had an equal probability of occurrence, whereas the earlier tests featured a Gaussian stimulus probability density function. Results yield a ood illustration of the nonlinear functions relating rate presented to rate perceived by human observers or operators.
Partial pressure of CO2 is a nonlinear function of several seawater properties. Due to the nonlinearity in this relationship, the partial pressure of a uniform ocean would be different from that of a nonuniform ocean with the same bulk seawater properties. Assuming uniformity of seawater properties at some temporal and spatial scales in carbon models leads to systematic errors in partial pressure of CO2. In this paper we evaluate the magnitude of these errors. We partition the Geochemical Ocean Sections Study and Transient Tracers in the Oceans data according to the horizontal structure of several box models from the literature. Our results suggest that assumption of uniformity at large scales leads to understanding of underestimation of global surface ocean partial pressure of CO2 by at least 3 - 12 microatm. Nonlinear effects also introduce systematic errors in the buffer factor estimated from bulk seawater properties. We find the standard deviation of partial pressure of CO2 to be an indicator of the magnitude of the nonlinear effects. We discuss the implications of these errors for some conclusions drawn from carbon models. Biogeochemical processes, such as mixing, gas exchange, or biological activity, influence the distribution of the seawater properties. A shift in spatial or temporal patterns of these processes can modify the nonuniformity of the seawater properties and thus alter the partial pressure of the surface waters, even if the mean intensities of the processes remain constant.
The results of this investigation indicate that the penetration of projectiles into quasi-infinite targets can be correlated as a function of the maximum momentum per unit area possessed by the projectiles. The penetration of projectiles into aluminum, copper, and steel targets was found to be a linear function while the penetration into lead targets was a nonlinear function of the momentum per unit area of the impacting projectiles. Penetration varied inversely as the projectile density and the elastic modulus of the target material for a given projectile momentum per unit area. Crater volumes were found to be a linear function of the kinetic energy of the projectile, the greater volumes being obtained in the target materials which had the lowest yield strength and the lowest speed of sound.
The objective of this paper is to develop a nonlinear viscoelastic–viscoplastic constitutive model for epoxy polymers. The classic nonlinear viscoelasticity model is reformulated to yield a closed-form incremental constitutive relation, which relates the stress increments to the viscoelastic strain increments. A viscoplasticity model, which consists of the Drucker–Prager yield function, nonlinear isotropic and kinematic hardening laws, and the Perzyna viscosity function, is subsequently developed. The nonlinear viscoelasticity and the vis-coplasticity models are then implemented in a radial return algorithm. A closed-form algorithmic tangent operator is derived to facilitate model implementation. The nonlinear viscoelasticity model is validated by reproducing the test data for polymethyl methacrylate(PMMA). The present constitutive model’s capabilities are demonstrated through modeling viscoelastic–viscoplastic PMMA loaded at different strain rates. The mechanics of structure genome-based micromechanics approach, embedded with the present constitutive model, is used to homogenize a unidirectional fiber-reinforced composite with a PMMA matrix, subjected to uniaxial and shear loading at different strain rates. The viscoelastic characterization of PMT-F7 epoxy is evaluated by comparing the predicted creep–recovery responses by the present constitutive model with the experimental ones.
Sufficient conditions derived for absolute stability of dynamic systems containing nonlinear functions of several state variables