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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Asymptotic expansions in nonlinear rotordynamics

This paper is an examination of special nonlinearities of the Jeffcott equations in rotordynamics. The immediate application of this analysis is directed toward understanding the excessive vibrations recorded in the LOX pump of the SSME during hot-firing ground testing. Deadband, side force, and rubbing are three possible sources of inducing nonlinearity in the Jeffcott equations. The present analysis initially reduces these problems to the same mathematical description. A special frequency, named the nonlinear natural frequency, is defined and used to develop the solutions of the nonlinear Jeffcott equations as singular asymptotic expansions. This nonlinear natural frequency, which is the ratio of the cross-stiffness and the damping, plays a major role in determining response frequencies.

Day, William B.↗
NASA NTRS1986

A canonical expansion for nonlinear systems

The importance of differential geometry, particularly Lie brackets of vector fields, in the study of nonlinear systems is well established. Under very mild assumptions, it is shown that a real-analytic nonlinear system has an expansion in which the coefficients are computed in terms of Lie brackets. This expansion occurs in a special coordinate system. The concept of a pure feedback system is also explained. For control design involving a nonlinear system, one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths.

Su, R.↗
NASA NTRS1984

Singular asymptotic expansions in nonlinear rotordynamics

During hot firing ground testing of the Space Shuttle's Main Engine, vibrations of the liquid oxygen pump occur at frequencies which cannot be explained by the linear Jeffcott model of the rotor. The model becomes nonlinear after accounting for deadband, side forces, and rubbing. Two phenomena present in the numerical solutions of the differential equations are unexpected periodic orbits of the rotor and tracking of the nonlinear frequency. A multiple scale asymptotic expansion of the differential equations is used to give an analytic explanation of these characteristics.

Day, W. B.↗
NASA NTRS1985

Singular asymptotic expansions in nonlinear rotordynamics

During hot firing ground testing of the Space shuttle's Main Engine, vibrations of the liquid oxygen pump occur at frequencies which cannot be explained by the linear Jeffcott model of the rotor. The model becomes nonlinear after accounting for deadband, side forces, and rubbing. Two phenomena present in the numerical solutions of the differential equations are unexpected periodic orbits of the rotor and tracking of the nonlinear frequency. A multiple scale asymptotic expansion of the differential equations is used to give an analytic explanation of these characteristics.

Day, W. B.↗
NASA NTRS1974

Development of solution techniques for nonlinear structural analysis

Nonlinear structural solution methods in the current research literature are classified according to order of the solution scheme, and it is shown that the analytical tools for these methods are uniformly derivable by perturbation techniques. A new perturbation formulation is developed for treating an arbitrary nonlinear material, in terms of a finite-difference generated stress-strain expansion. Nonlinear geometric effects are included in an explicit manner by appropriate definition of an applicable strain tensor. A new finite-element pilot computer program PANES (Program for Analysis of Nonlinear Equilibrium and Stability) is presented for treatment of problems involving material and geometric nonlinearities, as well as certain forms on nonconservative loading.

Vos, R. G.↗
NASA NTRS1994

TEXCAD: Textile Composite Analysis for Design. Version 1.0: User's manual

The Textile Composite Analysis for Design (TEXCAD) code provides the materials/design engineer with a user-friendly desktop computer (IBM PC compatible or Apple Macintosh) tool for the analysis of a wide variety of fabric reinforced woven and braided composites. It can be used to calculate overall thermal and mechanical properties along with engineering estimates of damage progression and strength. TEXCAD also calculates laminate properties for stacked, oriented fabric constructions. It discretely models the yarn centerline paths within the textile repeating unit cell (RUC) by assuming sinusoidal undulations at yarn cross-over points and uses a yarn discretization scheme (which subdivides each yarn not smaller, piecewise straight yarn slices) together with a 3-D stress averaging procedure to compute overall stiffness properties. In the calculations for strength, it uses a curved beam-on-elastic foundation model for yarn undulating regions together with an incremental approach in which stiffness properties for the failed yarn slices are reduced based on the predicted yarn slice failure mode. Nonlinear shear effects and nonlinear geometric effects can be simulated. Input to TEXCAD consists of: (1) materials parameters like impregnated yarn and resin properties such moduli, Poisson's ratios, coefficients of thermal expansion, nonlinear parameters, axial failure strains and in-plane failure stresses; and (2) fabric parameters like yarn sizes, braid angle, yarn packing density, filament diameter and overall fiber volume fraction. Output consists of overall thermoelastic constants, yarn slice strains/stresses, yarn slice failure history, in-plane stress-strain response and ultimate failure strength. Strength can be computed under the combined action of thermal and mechanical loading (tension, compression and shear).

Naik, Rajiv A.↗
NASA NTRS1995

TEXCAD: TEXile Composite Analysis for Design

The Textile Composite Analysis for Design (TEXCAD) code provides the materials/design engineer with a user-friendly, desktop computer based tool for the analysis of a wide variety of fabric reinforced woven and braided composites. It can be used to calculate overall thermal and mechanical properties along with engineering estimates of damage progression and strength. TEXCAD also calculates laminate properties for stacked, oriented fabric constructions. It discretely models the yarn centerline paths within the textile repeating unit cell (RUC) by assuming sinusoidal undulations at yarn cross-over points and uses a year discretization scheme (which subdivides each yarn into smaller, piecewise straight yarn slices) together with a 3-D stress averaging procedure to compute overall stiffness properties. In the calculations for strength, it uses a curved beam-on-elastic foundation model for yarn undulating regions together with incremental approach in which stiffness properties for the failed yarn slices are reduced based on the predicted yarn slice failure mode. Nonlinear shear effects and nonlinear geometric effects can be stimulated. Input to TEXCAD consists of: (1) material parameters like impregnated yarn and resin properties such as moduli, Poisson's ratios, coefficients of thermal expansion, nonlinear shear parameters, axial failure strains, and in-plane failure stresses; and (2) fabric parameters like yarn sizes, braid angle, yarn packing density, filament diameter, and overall fiber volume fraction. Output consists of overall thermoelastic constants, yarn slice strains/stresses, yarn slice failure history, in-plane strain response, and ultimate failure strength. Strength can be computed under the combined action of thermal and mechanical loading (tension, compression, and shear). A brief overview of the analytical capabilities, program organization, and modules, input and output parameters, computer platforms, distribution, and modifications/extensions of the TEXCAD code are presented here.

Naik, Rajiv A.↗
NASA NTRS1980

Heat-pipe sensor for remote leveling

System gives level readings in inaccessible areas. Level sensor is equipped with three thermocouples used to measure temperature differences that arise when pipe is tilted. When platform on which pipe is resting is level, three thermocouple recordings are identical. When readings are unequal, platform is leveled by remote control. System can replace expensive optical equipment and can function in cold, vacuum, and hot humid environments that produce nonlinear expansion and contraction in conventional equipment. Other advantages include low cost, no moving parts, and operation in toxic environments.

Marshburn, J. P.↗
NASA NTRS2017

Thermal Expansion from Stochastic Nonlinear Acoustic Fields

The methods of stochastic mechanics are applied to nonlinear acoustic fields generated by random acoustic radiation sources associated with oscillating lattice sites in nonlinear lattices. The assumption of stochastically independent zero-point and temperature-dependent acoustic fields leads to expressions of the thermodynamic internal and Helmholtz free energies per unit mass in terms of modal energies per unit mass that account for the nonlinearity of the propagation modes. The thermodynamic state functions canonically transform to the familiar results of a system of quantized, simple harmonic oscillators in the linear field limit. The thermal expansion coefficient, derived from the Helmholtz free energy per unit mass, is obtained as a sum of the zero-point modal nonlinearity parameters, weighted by temperature-dependent, modal heat capacities. The relationship is fully anharmonic, in contrast to commonly used quasi-harmonic models, and predicts with excellent agreement the experimentally observed null thermal expansion of vitreous silica at the ‘cross-over’ temperature corresponding to the balance between long wavelength modes that contribute negative nonlinearity parameters to the modal sum and short wavelength modes that contribute positive nonlinearity parameters.

Cantrell, John H., Jr.↗
NASA NTRS1992

Nonlinear acoustics and the thermal expansivity of glass

The paper shows the intrinsic relationship between acoustical nonlinearity and the thermal expansivity of crystalline solids (a thermal measure of anharmonicity) from consideration of the 'static' radiation field generated by the vibrating lattice (atomic) sources of the crystal. A modification of the theory to account for long-range structural disorder in glass is proposed and applied to an explanation of the zero thermal expansivity at the cross-over temperature in silicate and titanium silicate glasses. Experimental evidence which validates the essential features of the theory is presented. An application of nonlinearity measurement, based on these results, to the processing of ULE glass, where the addition of titanium in various amounts is used to establish the temperature at which the zero thermal expansivity occurs, is suggested.

Cantrell, John H.↗
NASA NTRS1985

Nonlinear rotordynamics analysis

The special nonlinearities of the Jeffcott equations in rotordynamics are examined. The immediate application of this analysis is directed toward understanding the excessive vibrations recorded in the LOX pump of the SSME during hot firing ground testing. Deadband, side force and rubbing are three possible sources of inducing nonlinearity in the Jeffcott equations. The present analysis initially reduces these problems to the same mathematical description. A special frequency, named the nonlinear natural frequency is defined and used to develop the solutions of the nonlinear Jeffcott equations as asympotic expansions. This nonlinear natural frequency which is the ratio of the cross-stiffness and the damping, plays a major role in determining response frequencies. Numerical solutions are included for comparison with the analysis. Also, nonlinear frequency-response tables are made for a typical range of values.

Day, W. B.↗
NASA NTRS1974

Nonlinear acoustic propagation in rectangular ducts

The method of multiple scales is used to obtain a second-order uniformly valid expansion for nonlinear acoustic wave propagation in a rectangular 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 materials attenuate sound more than linear materials except at high acoustic frequencies. The nonlinear materials produce higher and combination tones which have higher attenuation rates than the fundamentals. Moreover, the attenuation rates of the fundamentals increase with increasing amplitude.

Nayfeh, A. H.↗
NASA NTRS1980

Theoretical prediction of nonlinear propagation effects on noise signatures generated by subsonic or supersonic propeller or rotor-blade tips

The nonlinear propagation equations for sound generated by a constant speed blade tip are presented. Propagation from a subsonic tip is treated as well as the various cases that can occur at supersonic speeds. Some computed examples indicate that the nonlinear theory correlates with experimental results better than linear theory for large amplitude waves. For swept tips that generate a wave with large amplitude leading expansion, the nonlinear theory predicts a cancellation effect that results in a significant reduction of both amplitude and impulse.

Barger, R. L.↗
NASA NTRS1980

A modified Barrett-Lampard expansion and its application to bandpass nonlinearities with both AM-AM and AM-PM conversion

A two-dimensional Barrett-Lampard expansion is developed to handle two-dimensional nonlinear systems with random Gaussian inputs. The theory is applied to the problem of evaluating the mean signal and the inphase and quadrature noise correlation functions at the output of a TWT nonlinearity exhibiting arbitrary AM-AM and AM-PM characteristics, when the input consists of a narrow-band signal and Gaussian noise. As a specialized application, the classical problem of determining the coherent and incoherent noise correlation functions at the output of a power-law bandpass nonlinearity is also considered. The results are of interest in assessing the performance of coherent satellite communication channels. The theory developed is accompanied by numerical examples of practical interest.

Chie, C. M.↗