A high-order perturbation theory using rectangular coordinates.
Runge-kutta integration to approximate system of nonlinear equations by series of linear equations
RUNGE-KUTTA INTEGRATION↗
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Stability of sequences generated by nonlinear differential systems
A local stability analysis is presented for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. It is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and the one-leg k-step numerical solution can be approximated. Attention is given to the example of the two-dimensional problem involving the stability of the longitudinal equations of motion of a gliding jet aircraft.
Brown, R. L.↗
A three-dimensional model of corotating streams in the solar wind. I - Theoretical foundations
The paper is concerned with the development of the theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 Rs where the flow is supersonic) is approximately described by the nonlinear, single-fluid, polytropic magnetohydrodynamic or hydrodynamic equations. Efficient numerical techniques are outlined for solving this complex system of coupled, hyperbolic partial differential equations. The present formulation is inviscid and nonmagnetic, but the methods used allow for the potential inclusion of both features with only modest modifications. A simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution. It is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction. Comparison with simpler models demonstrates the essential nonlinear, multidimensional nature of the interplanetary dynamics.
Pizzo, V.↗
A three-dimensional model of corotating streams in the solar wind. 1: Theoretical foundations
The theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions (3-D) is discussed. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 solar radii where the flow is supersonic) is approximately described by the nonlinear, single fluid, polytropic (magneto-) hydrodynamic equations. Efficient numerical techniques for solving this complex system of coupled, hyperbolic partial differential equations are outlined. The formulation is inviscid and nonmagnetic, but methods allow for the potential inclusion of both features with only modest modifications. One simple, highly idealized, hydrodynamic model stream is examined to illustrate the fundamental processes involved in the 3-D dynamics of stream evolution. Spatial variations in the rotational stream interaction mechanism were found to produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction.
Pizzo, V. J.↗
A Comparative Study of Some Dynamic Stall Models
Three semi-empirical aerodynamic stall models are compared with respect to their lift and moment hysteresis loop prediction, limit cycle behavior, easy implementation, and feasibility in developing the parameters required for stall flutter prediction of advanced turbines. For the comparison of aeroelastic response prediction including stall, a typical section model and a plate structural model are considered. The response analysis includes both plunging and pitching motions of the blades. In model A, a correction to the angle of attack is applied when the angle of attack exceeds the static stall angle. In model B, a synthesis procedure is used for angles of attack above static stall angles and the time history effects are accounted through the Wagner function. In both models the life and moment coefficients for angle of attack below stall are obtained from tabular data for a given Mach number and angle of attack. In model C, referred to an the ONERA model, the life and moment coefficients are given in the form of two differential equations, one for angles below stall, and the other for angles above stall. The parameters of those equations are nonlinear functions of the angle of attack.
Reddy, T. S. R.↗
Approximate Solutions Of Equations Of Steady Diffusion
Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.
Edmonds, Larry D.↗
An application of a linear programing technique to nonlinear minimax problems
A differential correction technique for solving nonlinear minimax problems is presented. The basis of the technique is a linear programing algorithm which solves the linear minimax problem. By linearizing the original nonlinear equations about a nominal solution, both nonlinear approximation and estimation problems using the minimax norm may be solved iteratively. Some consideration is also given to improving convergence and to the treatment of problems with more than one measured quantity. A sample problem is treated with this technique and with the least-squares differential correction method to illustrate the properties of the minimax solution. The results indicate that for the sample approximation problem, the minimax technique provides better estimates than the least-squares method if a sufficient amount of data is used. For the sample estimation problem, the minimax estimates are better if the mathematical model is incomplete.
Schiess, J. R.↗
Linear and Nonlinear Aspects of Rotordynamics
Excessive vibrations of the liquid oxygen pump in the Space Shuttle's Main Engine have been recorded during hot firing ground testing. In order to determine mathematical explanations of this possibility, destructive phenomenon differential equations have been examined which describe the rotordynamics of the pump. Modeling the rotor as a random eigenvalue problem was considered. Analytical expressions were derived for the solution in the case of symmetric damping and stiffness. This enables one to determine accuracy estimates when testing numerical techniques to solve both asymmetric and nonlinear problems. Finally, the rotor model has had nonlinear elements incorporated to improve its simulation of the pump and to expand the corresponding mathematical theory.
Day, W. B.↗
The role of latent heat release in baroclinic waves - Without beta-effect
An attempt is made to understand the role of latent heat release in the development of baroclinic waves by developing the analytical theory of two-level, quasi-geostrophic baroclinic waves proposed by Phillips (1954) without beta effect, for the case of an atmosphere initially saturated with water vapor and subject to pseudo-adiabatic lifting and dry adiabatic subsistence. The saturated atmosphere condition precludes consideration of conditional convective instability. A consideration of disturbances at each level which depends only on the zonal coordinates and time results in a set of differential equations which is linear in all respects except for the latent heating term in the thermodynamic energy equation. A fundamental nonlinearity is present in this model which is due to heating via the release of latent heat when vertical motion is upward.
Tang, C.-M.↗
Eclipse-Free-Time Assessment Tool for IRIS
IRIS_EFT is a scientific simulation that can be used to perform an Eclipse-Free- Time (EFT) assessment of IRIS (Infrared Imaging Surveyor) mission orbits. EFT is defined to be those time intervals longer than one day during which the IRIS spacecraft is not in the Earth s shadow. Program IRIS_EFT implements a special perturbation of orbital motion to numerically integrate Cowell's form of the system of differential equations. Shadow conditions are predicted by embedding this integrator within Brent s method for finding the root of a nonlinear equation. The IRIS_EFT software models the effects of the following types of orbit perturbations on the long-term evolution and shadow characteristics of IRIS mission orbits. (1) Non-spherical Earth gravity, (2) Atmospheric drag, (3) Point-mass gravity of the Sun, and (4) Point-mass gravity of the Moon. The objective of this effort was to create an in-house computer program that would perform eclipse-free-time analysis. of candidate IRIS spacecraft mission orbits in an accurate and timely fashion. The software is a suite of Fortran subroutines and data files organized as a "computational" engine that is used to accurately predict the long-term orbit evolution of IRIS mission orbits while searching for Earth shadow conditions.
Eagle, David↗
Non-Gaussian approach for parametric random vibration of non-linear structures
The dynamic response of a nonlinear, single degree of freedom structural system subjected to a physically white noise parametric excitation is investigated. The Ito stochastic calculus is employed to derive a general differential equation for the moments of the response coordinates. The differential equations of moments of any order are found to be coupled with higher order moments. A non-Gaussian closure scheme is developed to truncate the moment equations up to fourth order. The statistical of the stationary response are computed numerically and compared with analytical solutions predicted by a Gaussian closure scheme and the stochastic averaging method. It is found that the computed results exhibit the jump phenomenon which is typical of the characteristics of deterministic nonlinear systems. In addition, the numerical algorithm leads to multiple solutions all of which give positive mean squares. However, two of these solutions are found to violate the properties of high order moments. One solution preserves the moments properties and demonstrates that the system achieves a stationary response.
Ibrahim, R. A.↗
Bifurcation and stability of low-order steady flows in horizontally and vertically forced convection
A nonlinear spectral model of two-dimensional, shallow Boussinesq convection which responds to heating in both the horizontal and vertical directions is examined. The governing partial differential system is converted to an infinite set of ordinary differential equations and truncated to a small set to permit detailed study of the number and types of transitions from one flow configuration to another. The Hadley number and the Rayleigh number are defined as the horizontal and vertical thermal forcing mechanisms, respectively, for inclusion in the nonlinear spectral model, which is composed of three equations. The model is then used to describe steady states, linearly stable solutions, and balancing factors in unstable stratification. The number and the distribution of the steady states are found to be qualitatively independent of the aspect ratio and the Prandtl number.
Yost, D. A.↗
Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations
The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.
Lee, Sang Soo↗
Bounded state space
This investigation is divided functionally into three different areas: (1) study of bounded state space, (2) nonlinear smoothing theory, and (3) system identification. (1) Study of bounded state space: necessary and sufficient conditions for an optimal control are obtained for a bounded state space optimal control problem. The difficulty of determining the so-called jump conditions is eliminated; however, the problem of determining the points where the response either enters or leaves the boundary still remains unsolved. (2) Nonlinear smoothing theory: nonlinear fixed-interval, fixed-point and fixed-lag smoothing of a random signal generated by a stochastic differential equation are investigated. Results on the asymptotic stability of a linear constant-parameter fixed-interval smoothing filter are obtained. (3) System identification: a particular stochastic modelling problem is solved. An Ito stochastic integral equation is used to mathematically model a black box having multiple inputs and multiple outputs. A new method for identifying system parameters is presented.
Eyman, E. D.↗
Parameter estimation of nonlinear nonautonomous distributed systems
We present an abstract approximation framework for estimation of parameters in nonlinear nonautonomous distributed systems. Specific examples involving linear spline approximations for retarded delay equations and cubic spline approximations for parabolic partial differential equations are discussed and shown to be included as special cases of our general framework.
Banks, H. T.↗
A note concerning the onset of three dimensionality and time dependence in Goertler vortices
Recently Hall and Seddougui (1989) considered the secondary instability of large amplitude Goertler vortices in a growing boundary layer evolving into a three-dimensional flow with wavy vortex boundaries. They obtained a pair of coupled, linear ordinary differential equations for this instability which constituted an eigenproblem for the wavelength and frequency of this wavy mode. Investigations into the nonlinear version of this problem by Seddougui and Bassom have revealed several omissions in the numerical work of Hall and Seddougui. These issues are addressed in this note. In particular, it is found that many neutrally stable modes are possible. The properties of such modes are derived in a high wavenumber limit and it is shown that the combination of the results of Hall and Seddougui and the modifications made here lead to conclusions which are consistent with the available experimental observations.
Bassom, Andrew P.↗
Parameter identification for nonlinear aerodynamic systems
Parameter identification for nonlinear aerodynamic systems is examined. It is presumed that the underlying model can be arranged into an input/output (I/O) differential operator equation of a generic form. The algorithm estimation is especially efficient since the equation error can be integrated exactly given any I/O pair to obtain an algebraic function of the parameters. The algorithm for parameter identification was extended to the order determination problem for linear differential system. The degeneracy in a least squares estimate caused by feedback was addressed. A method of frequency analysis for determining the transfer function G(j omega) from transient I/O data was formulated using complex valued Fourier based modulating functions in contrast with the trigonometric modulating functions for the parameter estimation problem. A simulation result of applying the algorithm is given under noise-free conditions for a system with a low pass transfer function.
Pearson, Allan E.↗
A Thermodynamic Theory of Solid Viscoelasticity: Nonlinear Thermo-viscoelasticity - Part II:
This paper, second in the series of three papers, develops a general, nonlinear, non-isothermal, compressible theory for finite rubber viscoelasticity and specifies it in a form convenient for solving problems important to the rubber, tire, automobile, and air-space industries, among others. Based on the quasi-linear approach of non-equilibrium thermodynamics, a general nonlinear theory of differential type has been developed for arbitrary non-isothermal deformations of viscoelastic solids. In this theory, the constitutive equations were presented as the sum of a rubber elastic (equilibrium) and a liquid type viscoelastic (non-equilibrium) terms. These equations have then been simplified using several modeling and simplicity arguments.
Freed, Alan D.↗