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NASA NTRS1983

Global transformations of nonlinear systems

Necessary and sufficient conditions for a nonlinear system of equations to be locally equivalent, in a neighborhood of the origin in the real number system, to a controllable linear system are combined with several versions of the global inverse function theorem to define sufficient conditions for transforming the nonlinear system into a linear system. Additionally, a technique is introduced for developing a transformation under the assumptions that the columns of a controllability matrix span an n-dimensional space. Finally, the n-l form of the controllability matrix columns is demonstrated to be involutive

Hunt, L. R.↗

NASA NTRS1983

An overview of two nonlinear supersonic wing design studies

The progress of two studies which apply nonlinear aerodynamics to supersonic wing design is reviewed. The first study employed a nonlinear potential flow code to design wings for high lift and low drag due to lift by employing a controlled leading-edge expansion in which the crossflow accelerates to supercritical conditions and decelerates through a weak shock. The second study utilized a modified linearized theory code to explore the concept of using 'attainable' leading-edge thrust as a guide for selecting a wing leading-edge shape (planform and radius) for maintaining attached flow and maximizing leading-edge thrust. Experimental and theoretical results obtained during the course of these two studies are discussed.

Miller, D. S.↗

NASA NTRS1983

Stratospheric sudden coolings and the role of nonlinear wave interactions in preconditioning the circumpolar flow

The mechanisms responsible for the transition of the circumpolar flow from its normal midwinter state to the preconditioned state that should evolve before a wavenumber-2 major warming are investigated, through a combination of observational, numerical and theoretical studies. Observations of Eliassen-Palm flux cross sections indicate that while wave zonal mean flow interaction theory could account for the qualitative evolution of the circumpolar flow during the warming, substantial nonlinear wave interactions were active during the cooling period, and these interactions significantly influenced the evolution of the circumpolar flow. Numerical experiments employing a truncated, semispectral model indicate that this cooling phenomenon is realistically reproducible in an idealized integration in which wave-wave interactions are present. Two different mechanisms are proposed to account for these nonlinearities.

Palmer, T. N.↗

NASA NTRS1982

Theory of design using nonlinear transformations

This paper is presenting an overview of the theory of transformations from nonlinear systems to linear systems. Topics covered include (1) necessary and sufficient conditions for transformations to exist, (2) a method of constructing transformations (3) robustness in design (based on transformations theory) and Liapunov functions, (4) estimation theory, and (5) the relationship between transformation theory and 'nonlinear zeros'. Application of these results to automatic flight control is presented in another paper at this session.

Su, R.↗

NASA NTRS1983

Convergence characteristics of nonlinear vortex-lattice methods for configuration aerodynamics

Nonlinear panel methods have no proof for the existence and uniqueness of their solutions. The convergence characteristics of an iterative, nonlinear vortex-lattice method are, therefore, carefully investigated. The effects of several parameters, including 1) the surface-paneling method, 2) the integration method of the trajectories of the wake vortices, 3) vortex-grid refinement , and 4) the initial conditions for the first iteration on the computed aerodynamic coefficients and on the flow field details are presented. The convergence of the iterative-solution procedure is usually rapid. The solution converges with grid refinement to a constant value, but the final value is not unique and varies with the wing surface-paneling and wake-discretization methods within some range in the vicinity of the experimental result.

Seginer, A.↗

NASA NTRS1983

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.↗

NASA NTRS1984

Computational aspects of the nonlinear normal mode initialization of the GLAS 4th order GCM

Using the normal modes of the GLAS 4th Order Model, a Machenhauer nonlinear normal mode initialization (NLNMI) was carried out for the external vertical mode using the GLAS 4th Order shallow water equations model for an equivalent depth corresponding to that associated with the external vertical mode. A simple procedure was devised which was directed at identifying computational modes by following the rate of increase of BAL sub M, the partial (with respect to the zonal wavenumber m) sum of squares of the time change of the normal mode coefficients (for fixed vertical mode index) varying over the latitude index L of symmetric or antisymmetric gravity waves. A working algorithm is presented which speeds up the convergence of the iterative Machenhauer NLNMI. A 24 h integration using the NLNMI state was carried out using both Matsuno and leap-frog time-integration schemes; these runs were then compared to a 24 h integration starting from a non-initialized state. The maximal impact of the nonlinear normal mode initialization was found to occur 6-10 hours after the initial time.

Navon, I. M.↗

NASA NTRS1983

Hybrid perturbation/Bubnov-Galerkin technique for nonlinear thermal analysis

A two step hybrid analysis technique to predict the nonlinear steady state temperature distribution in structures and solids is presented. The technique is based on the regular perturbation expansion and the classical Bubnov-Galerkin approximation. The functions are obtained by using the regular perturbation method. These functions are selected as coordinate functions and the classical Bubnov-Galerkin technique is used to compute their amplitudes. The potential of the proposed hybrid technique for the solution of nonlinear thermal problems is discussed. The effectiveness of this technique is demonstrated by the effects of conduction, convection, and radiation modes of heat transfer. It is indicated that the hybrid technique overcomes the two major drawbacks of the classical techniques: (1) the requirement of using a small parameter in the regular perturbation method; and (2) the arbitrariness in the choice of the coordinate functions in the Bubnov-Galerkin technique. The proposed technique extends the range of applicability of the regular perturbation method and enhances the effectiveness of the Bubnov-Galerkin technique.

Noor, A. K.↗

NASA NTRS1984

The nonlinear viscoelastic response of resin matrix composite laminates

Possible treatments of the nonlinear viscoelastic behavior of materials are reviewed. A thermodynamic based approach, developed by Schapery, is discussed and used to interpret the nonlinear viscoelastic response of a graphite epoxy laminate, T300/934. Test data to verify the analysis for Fiberite 934 neat resin as well as transverse and shear properties of the unidirectional T300/934 composited are presented. Long time creep characteristics as a function of stress level and temperature are generated. Favorable comparisons between the traditional, graphical, and the current analytical approaches are shown. A free energy based rupture criterion is proposed as a way to estimate the life that remains in a structure at any time.

Hiel, C.↗

NASA NTRS1984

A computer program for predicting nonlinear uniaxial material responses using viscoplastic models

A computer program was developed for predicting nonlinear uniaxial material responses using viscoplastic constitutive models. Four specific models, i.e., those due to Miller, Walker, Krieg-Swearengen-Rhode, and Robinson, are included. Any other unified model is easily implemented into the program in the form of subroutines. Analysis features include stress-strain cycling, creep response, stress relaxation, thermomechanical fatigue loop, or any combination of these responses. An outline is given on the theoretical background of uniaxial constitutive models, analysis procedure, and numerical integration methods for solving the nonlinear constitutive equations. In addition, a discussion on the computer program implementation is also given. Finally, seven numerical examples are included to demonstrate the versatility of the computer program developed.

Chang, T. Y.↗

NASA NTRS1983

Estimation of delays and other parameters in nonlinear functional differential equations

A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

Banks, H. T.↗

NASA NTRS1984

Nonlinear evolution of magnetopause tearing modes

Since the magnetosheath plasma is highly turbulent, reconnection at the dayside magnetopause is likely to be temporally unsteady. The tearing mode can be viewed as a model for the unsteady development of a reconnecting magnetic topology. Magnetopause tearing occurs in the guide-field limit and has a wave packet spatial structure in the east-west direction. This paper solves for the nonlinear evolution of a single wavelength guide-field tearing mode including the effects of finite transit time on the Landau resonant electrons. Short wavelength modes evolve algebraically in time with perturbation amplitudes proportional to t-squared. Long wavelength modes are fully nonlinear, and the amplitude grows linearly in time.

Coroniti, F. V.↗

NASA NTRS1984

Two dimensional nonlinear analysis of sound transmission through a near-sonic throat flow

A two-dimensional nonlinear theory of sound transmission through a nonuniform duct carrying a near-sonic throat flow is described. The mean flow in the duct is treated according to a generalized quasi-one dimensional model, and the analysis of the unsteady perturbations is carried out using the Method of Matched Asymptotic Expansions. The linearized acoustic field in the subsonic regions of the duct is approximated by a Wave Envelope model which is matched asymptotically with a nonlinear inner solution valid in the near-sonic throat region. Numerical results are presented to illustrate the predictions of the theory. It is found that, in general, shock waves develop in the acoustic field during transmission through the near-sonic flow. Unlike previous one-dimensional theories, the current study shows that dispersion can play a major role in the propagation process.

Myers, M. K.↗

NASA NTRS1982

Model reference adaptive control for linear time varying and nonlinear systems

Model reference adaptive control is applied to linear time varying systems and to nonlinear systems amenable to virtual linearization. Asymptotic stability is guaranteed even if the perfect model following conditions do not hold, provided that some sufficient conditions are satisfied. Simulations show the scheme to be capable of effectively controlling certain nonlinear systems.

Abida, L.↗

NASA NTRS1984

On the passage of high-level pulsed radio frequency interference through a nonlinear satellite transponder

Attention is given to the uncoded bit error rate (BER) performance of a satellite communications system whose modulation scheme is binary PSK and whose transponder contains an arbitrary amplitude nonlinearity, all in the presence of high level pulsed radio frequency interference (RFI). A general approach is presented for direct BER evaluations, in contrast to other approaches which may employ SNR suppression factors. The computed results are based on arbitrarily specified RFI scenarios, in the presence of hard limiter, clipper, or blanker amplitude nonlinearities. Performance curves demonstrate the superiority of an appropriately chosen blanker when the RFI environment is most severe. The results obtained also pertain to the sensitivity of performance to the information bit rate, signal power variations, and the ratio of CW to noise content. The CW effects are found to be the most severe.

Weinberg, A.↗

NASA NTRS1983

Design of nonlinear discrete-time controllers using a parameter space sampling procedure

The design of nonlinear discrete-time controllers is investigated where the control algorithm assumes a special form. State-dependent control actions are obtained from tables whose values are the design parameters. A new design methodology capable of dealing with nonlinear systems containing parameter uncertainty is used to obtain the controller design. Various controller strategies are presented and illustrated through an example.

Young, G. E.↗

NASA NTRS1984

Application of nonlinear transformations to automatic flight control

The theory of transformations of nonlinear systems to linear ones is applied to the design of an automatic flight controller for the UH-1H helicopter. The helicopter mathematical model is described and it is shown to satisfy the necessary and sufficient conditions for transformability. The mapping is constructed, taking the nonlinear model to canonical form. The performance of the automatic control system in a detailed simulation on the flight computer is summarized.

Meyer, G.↗

NASA NTRS1984

Nonlinear mode coupling theory of the lower-hybrid-drift instability

A nonlinear mode coupling theory of the lower-hybrid-drift instability is presented. A two-dimensional nonlinear wave equation is derived which describes lower-hybrid drift wave turbulence in the plane transverse to B (k.B = 0), and which is valid for finite beta, collisional and collisionless plasmas. The instability saturates by transferring energy from growing, long wavelength modes to damped, short wavelength modes. Detailed numerical results are presented which compare favorably to both recent computer simulations and experimental observations. Applications of this theory to space plasmas, the earth's magnetotail and the equatorial F region ionosphere, are discussed. Previously announced in STAR as N84-17734

Drake, J. F.↗