Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “nonlinear normal modes”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

Nonlinear behavior of acoustic waves in combustion chambers

The nonlinear growth and limiting amplitude of acoustic waves in a combustion chamber are considered. A formal framework is provided within which practical problems can be treated with a minimum of effort and expense. The general conservation equations were expanded in two small parameters, one characterizing the mean flow field and one measuring the amplitude of oscillations, and then combined to yield a nonlinear inhomogeneous wave equation. The unsteady pressure and velocity fields were expressed as syntheses of the normal modes of the chamber, but with unknown time-varying amplitudes. This procedure yielded a representation of a general unsteady field as a system of coupled nonlinear oscillators. The system of nonlinear equations was treated by the method of averaging to produce a set of coupled nonlinear first order differential equations for the amplitudes and phases of the modes. The analysis is applicable to any combustion chamber. The most interesting applications are probably to solid rockets, liquid rockets, or thrust augmentors on jet engines.

Culick, F. E. C.↗
NASA NTRS1977

Solar oscillations

The main observational results and related theoretical investigations concerning solar oscillations are reviewed. The normal modes of oscillation of the sun are classified according to their angular eigenvalues and the number of their radial nodes. Observations of excited normal modes are discussed, particularly in relation to five-minute oscillations, nonlinearly coupled unstable g-modes, oscillations of the sun's apparent diameter, and reported oscillations of the solar surface with a period of 2 hr 40 min. Linear stability calculations are briefly examined, and stochastic excitation of p-modes by turbulent convection is considered. The five-minute oscillations are described as the superposition of many excited nonradial p-modes.

Goldreich, P.↗
NASA NTRS1994

Gravitational instabilities in protostellar disks

The nonaxisymmetric stability of self-gravitating, geometrically thick accretion disks has been studied for protostellar systems having a wide range of disk-to-central object mass ratios. Global eigenmodes with four distinctly different characters were identified using numerical, nonlinear hydrodynamic techniques. The mode that appears most likely to arise in normal star formation settings, however, resembles the 'eccentric instability' that was identified earlier in thin, nearly Keplerian disks: It presents an open, one-armed spiral pattern that sweeps continuously in a trailing direction through more than 2-pi radians, smoothly connecting the inner and outer edges of the disk, and requires cooperative motion of the point mass for effective amplification. This particular instability promotes the development of a single, self-gravitating clump of material in orbit about the point mass, so its routine appearance in our simulations supports the conjecture that the eccentric instability provides a primary route to the formation of short-period binaries in protostellar systems.

Tohline, J. E.↗
NASA NTRS1975

Analysis of nonlinear structures via mode synthesis

An effective procedure for NASTRAN was developed that permits any number of substructures of any size to be synthesized for the purpose of developing normal modes of vibration of the complete structural system. The technique is extended to permit modal transient analysis of the subdivided system. This latter procedure permits the use of NASTRAN's ability to include nonlinear forces in the problem. The five-phase process is accomplished using standard NASTRAN rigid formats with problem-independent alter packages and DMAP sequences.

Gieseke, R. K.↗
NASA NTRS1993

The barotropic normal modes in certain shear flows and the traveling waves in the atmosphere

It is shown analytically and numerically that in certain shear flows the linearized nondivergent barotropic vorticity equation has a limited number of neutral normal modes. The latitudinal structures of these shear flows can be expressed as polynomials of the sine of latitude. The first few such shear flows resemble the gross features of the zonal winds in the atmosphere of the earth at different times and altitudes. The spatial structures of the neutral normal modes in these shear flows are spherical harmonics, and, as a consequence, these modes are also the exact solutions of the fully nonlinear equation because the nonlinear interaction term vanishes identically. The spatial structures of the observed 5-, 4-, 2-, and 16-day free traveling waves in the atmosphere are often identified with the spherical harmonics with indices of (m, n) = ( 1, 2), (2, 3), (3, 3), and ( 1, 4), which are known previously as the neutral normal modes of the nondivergent barotropic vorticity equation in a motionless background state. Our results could explain why these free traveling waves can survive the shearing effects of zonal flows that are far different from rest because these spherical harmonics are also normal modes in certain shear flows that resemble the observations of the atmosphere.

Chen, Ping↗
NASA NTRS1983

On error detection in the dynamics part of primitive equation models

Rossby-Haurwitz waves have been commonly employed as initial conditions for testing the dynamics part of global atmospheric models. Difficulties arise in connection with the primitive equation (PE) model. Chao and Geller (1982) have suggested that it would be better if normal mode solutions of the linearized primitive equations would be used as initial conditions. However, tests involving normal mode conditions are not adequate for the detection of errors in the nonlinear terms. The present investigation has the objective to review the study conducted by Chao and Geller and to provide tests for the nonlinear terms.

Chao, W. C.↗
NASA NTRS1994

Numerical simulation of baroclinic Jovian vortices

We examine the evolution of baroclinic vortices in a time-dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and first two baroclinic modes. Results for the stability of baroclinic vortices on an f plane in the absence of a mean zonal flow are similar to results of Earth vortex models, although the presence of a fluid interior on the Jovian planets shifts the stability boundaries to smaller length scales. The presence of a barotropic mean zonal flow in the interior stabilizes vortices against instability and significantly modifies the finite amplitude form of baroclinic instabilities. The effect of a zonal flow on a form of barotropic instability produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable vortices in a zonal shear flow are observed in this baroclinic model, as in simpler two-dimensional models. Such oscillations are also observed in the atmospheres of Jupiter and Neptune. The meridional propagation and decay of vortices on a beta plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model assumption for the flow in the deep interior of the Jovian planets.

Achterberg, Richard K.↗
NASA NTRS1988

Dynamical criterion for a marginally unstable, quasi-linear behavior in a two-layer model

A two-layer quasi-geostrophic flow forced by meridional variations in heating can be in regimes ranging from radiative equilibrium to forced geostrophic turbulence. Between these extremes is a regime where the time-mean (zonal) flow is marginally unstable. Using scaling arguments, it is concluded that such a marginally unstable state should occur when a certain parameter, measuring the strength of wave-wave interactions relative to the beta effect and advection by the thermal wind, is small. Numerical simulations support this proposal. A transition from the marginally unstable regime to a more nonlinear regime is then examined through numerical simulations with different radiative forcings. It is found that transition is not caused by secondary instability of waves in the marginally unstable regime. Instead, the time-mean flow can support a number of marginally unstable normal modes. These normal modes interact with each other, and if they are of sufficient amplitude, the flow enters a more nonlinear regime.

Ebisuzaki, W.↗
NASA NTRS1986

Differential stiffness effects

Differential stiffness as developed in NASTRAN is a linear change in stiffness caused by applied loads. Examples of differential stiffness are the stiffening effects of gravity forces in a pendulum, centrifugal forces in rotor blades and pressure loading of shell structures. In cases wherein this stiffness caused by a load is destabilizing, the differential stiffness concept lends itself to nonlinear structural analysis. Rigid Formats 4 (static analysis with differential stiffness) and 13 (normal modes with differential stiffness) are specifically designed to account for such stiffness changes. How pressure loading may be treated in these rigid formats is clarified. This clarification results from modal correlation of Ground Vibration Test (GVT) results from the empty and pressurized Filament Wound Case (FWC) quarter-scale Space Shuttle solid rocket booster (QSSRB). A sketch of the QSSRB cantilevered to the floor by its external tank attachments is shown.

Ujihara, B. H.↗
NASA NTRS1993

Instabilities of two-dimensional inviscid compressible vortices

An investigation of the linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is presented. The effects of the entropy gradient are investigated, and for the particular entropy profile chosen, the positive average entropy in the vortex core is destabilizing, while the opposite is true for the negative average entropy gradient. Finite-difference methods are used to study the linear initial value problem. These methods are found to be capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behavior is found to correspond to the most unstable normal mode in all cases. The numerical solution of the Euler equations is used to study the nonlinear development of an elliptic-mode perturbation.

Chan, W. M.↗
NASA NTRS2002

Nonlinear Pressurization and Modal Analysis Procedure for Dynamic Modeling of Inflatable Structures

An introduction and set of guidelines for finite element dynamic modeling of nonrigidized inflatable structures is provided. A two-step approach is presented, involving 1) nonlinear static pressurization of the structure and updating of the stiffness matrix and 2) hear normal modes analysis using the updated stiffness. Advantages of this approach are that it provides physical realism in modeling of pressure stiffening, and it maintains the analytical convenience of a standard bear eigensolution once the stiffness has been modified. Demonstration of the approach is accomplished through the creation and test verification of an inflated cylinder model using a large commercial finite element code. Good frequency and mode shape comparisons are obtained with test data and previous modeling efforts, verifying the accuracy of the technique. Problems encountered in the application of the approach, as well as their solutions, are discussed in detail.

Smalley, Kurt B.↗
NASA NTRS1986

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree geometric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗
NASA NTRS1987

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree goemetric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of Pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment.

Subrahmanyam, K. B.↗
NASA NTRS2012

Nonlinear Reduced-Order Simulation Using An Experimentally Guided Modal Basis

A procedure is developed for using nonlinear experimental response data to guide the modal basis selection in a nonlinear reduced-order simulation. The procedure entails using nonlinear acceleration response data to first identify proper orthogonal modes. Special consideration is given to cases in which some of the desired response data is unavailable. Bases consisting of linear normal modes are then selected to best represent the experimentally determined transverse proper orthogonal modes and either experimentally determined inplane proper orthogonal modes or the special case of numerically computed in-plane companions. The bases are subsequently used in nonlinear modal reduction and dynamic response simulations. The experimental data used in this work is simulated to allow some practical considerations, such as the availability of in-plane response data and non-idealized test conditions, to be explored. Comparisons of the nonlinear reduced-order simulations are made with the surrogate experimental data to demonstrate the effectiveness of the approach.

Rizzi, Stephen A.↗
NASA NTRS1980

Generalized Grueneisen tensor from solid nonlinearity parameters

Anharmonic effects in solids are often described in terms of generalized Grueneisen parameters which measure the strain dependence of the lattice vibrational frequencies. The relationship between these parameters and the solid nonlinearity parameters measured directly in ultrasonic harmonic generation experiments is derived using an approach valid for normal-mode elastic wave propagation in any crystalline direction. The resulting generalized Grueneisen parameters are purely isentropic in contrast to the Brugger-Grueneisen parameters which are of a mixed thermodynamic state. Experimental data comparing the isentropic generalized Grueneisen parameters and the Brugger-Grueneisen parameters are presented.

Cantrell, J. H., Jr.↗
NASA NTRS1982

Nonlinear longitudinal resonance interaction of energetic charged particles and VLF waves in the magnetosphere

The longitudinal resonance of waves and energetic electrons in the Earth's magnetosphere, and the possible role this resonance may play in generating various magnetospheric phenomena are studied. The derivation of time-averaged nonlinear equations of motion for energetic particles longitudinally resonant with a whistler mode wave propagating with nonzero wave normal is considered. It is shown that the wave magnetic forces can be neglected at lower particle pitch angles, while they become equal to or larger than the wave electric forces for alpha 20 deg. The time-averaged equations of motion were used in test particle simulation which were done for a wide range of wave amplitudes, wave normals, particle pitch angles, particle parallel velocities, and in an inhomogeneous medium such as the magnetosphere. It was found that there are two classes of particles, trapped and untrapped, and that the scattering and energy exchange for those two groups exhibit significantly different behavior.

Tkalcevic, S.↗
NASA NTRS1993

Upper Atmosphere Research Satellite (UARS) onboard attitude determination using a Kalman filter

The Upper Atmospheric Research Satellite (UARS) requires a highly accurate knowledge of its attitude to accomplish its mission. Propagation of the attitude state using gyro measurements is not sufficient to meet the accuracy requirements, and must be supplemented by a observer/compensation process to correct for dynamics and observation anomalies. The process of amending the attitude state utilizes a well known method, the discrete Kalman Filter. This study is a sensitivity analysis of the discrete Kalman Filter as implemented in the UARS Onboard Computer (OBC). The stability of the Kalman Filter used in the normal on-orbit control mode within the OBC, is investigated for the effects of corrupted observations and nonlinear errors. Also, a statistical analysis on the residuals of the Kalman Filter is performed. These analysis is based on simulations using the UARS Dynamics Simulator (UARSDSIM) and compared against attitude requirements as defined by General Electric (GE). An independent verification of expected accuracies is performed using the Attitude Determination Error Analysis System (ADEAS).

Garrick, Joseph↗
NASA NTRS1979

A theory of stationary long waves. III - Quasi-normal modes in a singular waveguide

The present paper deals with the fundamental issue of whether one can treat waves as normal modes when critical surface, where the phase speed of the wave matches the zonal wind speed, are present. In particular the question of whether a Rossby critical level (such as the zero-wind line for stationary waves) is absorbing or reflecting is raised and subsequently addressed. It is found that the critical level is never totally absorbing; Rossby waves are partially reflected even if the critical layer is dominated by dissipative processes. The relevance of nonlinearity in planetary-scale Rossby wave critical layers is also discussed and it is found to be the dominant mechanism. With the relative magnitudes of nonlinearity versus viscosity relevant to the earth's atmosphere it is found that the steady-state critical level should be almost perfectly reflecting to incident Rossby waves. Consequently, normal-mode solutions can be found; the quantization condition for these waves is also derived.

Tung, K. K.↗