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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Structural/control system optimization with variable actuator masses

A method is presented to integrate the design space for structural/control system optimization problems in the case of linear state feedback control. Nonstructural lumped masses and control system design variables as well as structural sizing variables are all treated equally as independent design variables in the optimization process. Structural and control design variable linking schemes are used in order to avoid a prohibitively large increase in the total number of independent design variables. When actuator masses are treated as nonstructural lumped mass design variables, special consideration is given to the relation between the transient peak responses and the required actuator masses which is formulated as a behavior constraint form. The original nonlinear mathematical programming problem based on a finite element formulation and linear state feedback is replaced by a sequence of explicit approximate problems exploiting various approximation concepts such as design variable linkings, temporary constraint deletion and first order Taylor series expansion of nonlinear behavior constraints in terms of intermediate design variables. Examples which involve a variety of dynamic behavior constraints (including constraints on closed-loop eigenvalues, peak transient displacements, peak actuator forces, and relations between the peak responses and the actuator masses) are effectively solved by using the method presented.

Jin, Ik M.↗
NASA NTRS1983

Temperature dependence of mechanical and thermal expansion properties of T300/5208 graphite/epoxy

The 116-394 K temperature range elastic constants, ultimate strengths and thermal expansion behavior of T300/5208 graphite/epoxy laminates of various different lay-up types have been determined. It is noted that most properties exhibit nearly linear temperature dependencies, although the transverse modulus and axial thermal expansion vary nonlinearly with temperature, and change in value by an order of magnitude over the present temperature range.

Hyer, M. W.↗
NASA NTRS1984

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.↗
NASA NTRS1984

Predictions of nucleation theory applied to Ehrenfest thermodynamic transitions

A modified nucleation theory is used to determine a critical nucleus size and a critical activation-energy barrier for second-order Ehrenfest thermodynamic transitions as functions of the degree of undercooling, the interfacial energy, the heat-capacity difference, the specific volume of the transformed phase, and the equilibrium transition temperature. The customary approximations of nucleation theory are avoided by expanding the Gibbs free energy in a Maclaurin series and applying analytical thermodynamic expressions to evaluate the expansion coefficients. Nonlinear correction terms for first-order-transition calculations are derived, and numerical results are presented graphically for water and polystyrene as examples of first-order and quasi-second-order transitions, respectively.

Barker, R. E., Jr.↗
NASA NTRS1989

A space-time discretization procedure for wave propagation problems

Higher order compact algorithms are developed for the numerical simulation of wave propagation by using the concept of a discrete dispersion relation. The dispersion relation is the imprint of any linear operator in space-time. The discrete dispersion relation is derived from the continuous dispersion relation by examining the process by which locally plane waves propagate through a chosen grid. The exponential structure of the discrete dispersion relation suggests an efficient splitting of convective and diffusive terms for dissipative waves. Fourth- and eighth-order convection schemes are examined that involve only three or five spatial grid points. These algorithms are subject to the same restrictions that govern the use of dispersion relations in the constructions of asymptotic expansions to nonlinear evolution equations. A new eighth-order scheme is developed that is exact for Courant numbers of 1, 2, 3, and 4. Examples are given of a pulse and step wave with a small amount of physical diffusion.

Davis, Sanford↗
NASA NTRS1979

Nonlinear effects on composite laminate thermal expansion

Analyses of Graphite/Polyimide laminates shown that the thermomechanical strains cannot be separated into mechanical strain and free thermal expansion strain. Elastic properties and thermal expansion coefficients of unidirectional Graphite/Polyimide specimens were measured as a function of temperature to provide inputs for the analysis. The + or - 45 degrees symmetric Graphite/Polyimide laminates were tested to obtain free thermal expansion coefficients and thermal expansion coefficients under various uniaxial loads. The experimental results demonstrated the effects predicted by the analysis, namely dependence of thermal expansion coefficients on load, and anisotropy of thermal expansion under load. The significance of time dependence on thermal expansion was demonstrated by comparison of measured laminate free expansion coefficients with and without 15 day delay at intermediate temperature.

Hashin, Z.↗
NASA NTRS1999

Quasi-Static Viscoelastic Finite Element Model of an Aircraft Tire

An elastic large displacement thick-shell mixed finite element is modified to allow for the calculation of viscoelastic stresses. Internal strain variables are introduced at the element's stress nodes and are employed to construct a viscous material model. First order ordinary differential equations relate the internal strain variables to the corresponding elastic strains at the stress nodes. The viscous stresses are computed from the internal strain variables using viscous moduli which are a fraction of the elastic moduli. The energy dissipated by the action of the viscous stresses is included in the mixed variational functional. The nonlinear quasi-static viscous equilibrium equations are then obtained. Previously developed Taylor expansions of the nonlinear elastic equilibrium equations are modified to include the viscous terms. A predictor-corrector time marching solution algorithm is employed to solve the algebraic-differential equations. The viscous shell element is employed to computationally simulate a stair-step loading and unloading of an aircraft tire in contact with a frictionless surface.

Johnson, Arthur R.↗
NASA NTRS1987

Interactive application of quadratic expansion of chi-square statistic to nonlinear curve fitting

This report contains a detailed theoretical description of an all-purpose, interactive curve-fitting routine that is based on P. R. Bevington's description of the quadratic expansion of the Chi-Square statistic. The method is implemented in the associated interactive, graphics-based computer program. Taylor's expansion of Chi-Square is first introduced, and justifications for retaining only the first term are presented. From the expansion, a set of n simultaneous linear equations is derived, then solved by matrix algebra. A brief description of the code is presented along with a limited number of changes that are required to customize the program of a particular task. To evaluate the performance of the method and the goodness of nonlinear curve fitting, two typical engineering problems are examined and the graphical and tabular output of each is discussed. A complete listing of the entire package is included as an appendix.

Badavi, F. F.↗
NASA NTRS1974

Nonlinear acoustic propagation in two-dimensional ducts

The method of multiple scales is used to obtain a second-order uniformly valid expansion for the nonlinear acoustic wave propagation in a two-dimensional duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear effects tend to flatten and broaden the absorption versus frequency curve, in qualitative agreement with the experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.

Nayfeh, A. H.↗
NASA NTRS1987

A computational code for two-dimensional unsteady magnetohydrodynamics by the method of characteristics

A computational code for solving two-dimensional, time-dependent MHD equations by the method of characteristics is presented. Its capabilities are demonstrated by solving two very different problems for which analytical solutions exist: linearized, standing MHD wave motions in a magnetized cylindrical plasma, and nonlinear self-similar expansion of a magnetized plasma ball. The nonlinear development of standing MHD wave solutions in a cylindrical plasma is also studied. The method can be naturally embedded in the computational architecture of massively parallel processors.

Lou, Y. Q.↗
NASA NTRS1974

Stability analysis of nonlinear systems by multiple time scaling

The asymptotic solution for the transient analysis of a general nonlinear system in the neighborhood of the stability boundary was obtained by using the multiple-time-scaling asymptotic-expansion method. The nonlinearities are assumed to be of algebraic nature. Terms of order epsilon to the 3rd power (where epsilon is the order of amplitude of the unknown) are included in the solution. The solution indicates that there is always a limit cycle which is stable (unstable) and exists above (below) the stability boundary if the nonlinear terms are stabilizing (destabilizing). Extension of the solution to include fifth order nonlinear terms is also presented. Comparisons with harmonic balance and with multiple-time-scaling solution of panel flutter equations are also included.

Morino, L.↗
NASA NTRS1995

Contact and pressure balance structures in two-fluid cosmic-ray hydrodynamics

The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Webb, G. M.↗
NASA NTRS1982

Computer modeling of multiple-channel input signals and intermodulation losses caused by nonlinear traveling wave tube amplifiers

The multiple channel input signal to a soft limiter amplifier as a traveling wave tube is represented as a finite, linear sum of Gaussian functions in the frequency domain. Linear regression is used to fit the channel shapes to a least squares residual error. Distortions in output signal, namely intermodulation products, are produced by the nonlinear gain characteristic of the amplifier and constitute the principal noise analyzed in this study. The signal to noise ratios are calculated for various input powers from saturation to 10 dB below saturation for two specific distributions of channels. A criterion for the truncation of the series expansion of the nonlinear transfer characteristic is given. It is found that he signal to noise ratios are very sensitive to the coefficients used in this expansion. Improper or incorrect truncation of the series leads to ambiguous results in the signal to noise ratios.

Stankiewicz, N.↗
NASA NTRS1976

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that was averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. An expansion is derived for the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest nontrivial order. The validity of this expansion is seen to be the same as that of the standard quasilinear expansion.

Jones, F. C.↗
NASA NTRS1978

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that has been averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. In this note we derive an expansion of the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest non-trivial order. The validity of this expansion is seen to be the same as that of the standard quasi-linear expansion.

Jones, F. C.↗
NASA NTRS1991

A geometrically nonlinear analysis of interlaminar stresses in unsymmetrically laminated plates subjected to uniform thermal loading

An analytical study of interlaminar stresses in unsymmetrically laminated plates is presented. The study examines the linear elastic, large deflection response of square laminated composite plates subjected to uniform thermal loading. Both cross-ply and angle-ply, symmetric and unsymmetric, laminates are examined to evaluate the effects of mismatch between adjacent layers in elastic constants and coefficients of thermal expansion. A geometrically nonlinear kinematic description is used to predict the large out-of-plane (transverse) deflections. The nonlinear, three-dimensional boundary value problems are formulated from elasticity theory and approximate solutions are determined using the finite element method. A global/local analysis procedure is used to obtain improved free edge stress predictions. For the laminates and loading conditions considered, the results indicate that the out-of-plane deflections of the unsymmetric laminates reduce interlaminar shear stresses. These deflections also reduce interlaminar normal stresses in some laminates and increase these stresses for others. The results indicate that as the out-of-plane deflections become large, the differences in interlaminar stress predictions between linear and nonlinear theory can become quite large.

Norwood, D. S.↗
NASA NTRS1976

Effects of high subsonic flow on sound propagation in a variable-area duct

The propagation of sound in a converging-diverging duct containing a quasi-one-dimensional steady flow with a high subsonic throat Mach number was studied. The behavior of linearized acoustic theory at the throat of the duct was shown to be singular. This singularity implies that linearized acoustic theory is invalid. The explicit singular behavior was determined and used to sketch the development (by the method of matched asymptotic expansions) of a nonlinear theory for sound propagation in a sonic throat region.

Callegari, A. J.↗