Showing papers on "Nonlinear resonance published in 1990"

Nonlinear motions of beam-mass structure

Abstract: We study the planar dynamic response of a flexible L-shaped beam-mass structure with a two-to-one internal resonance to a primary resonance. The structure is subjected to low excitation (mili g) levels and the resulting nonlinear motions are examined. The Lagrangian for weakly nonlinear motions of the undamped structure is formulated and time averaged over the period of the primary oscillation, leading to an autonomous system of equations governing the amplitudes and phases of the modes involved in the internal resonance. Later, modal damping is assumed and modal-damping coefficients, determined from experiments, are included in the analytical model. The locations of the saddle-node and Hopf bifurcations predicted by the analysis are in good agreement, respectively, with the jumps and transitions from periodic to quasi-periodic motions observed in the experiments. The current study is relevant to the dynamics and modeling of other structural systems as well.

Phase space bifurcation structure and the generalized local-to-normal transition in resonantly coupled vibrations

Abstract: The generalization of the local‐to‐normal transition seen in symmetric triatomics is considered for nonsymmetric molecules and 2:1 Fermi resonance systems. A straightforward generalization based on a division of phase space into local and normal regions is not possible. Instead, classification of the phase space bifurcation structure is presented as the complete generalization of the local–normal concept for all spectroscopically relevant systems of two vibrations interacting via a single nonlinear resonance. The polyad phase sphere (PPS) is shown to be the natural arena to analyze the bifurcation structure for resonances of arbitrary order. For 1:1 and 2:1 resonances, the bifurcation problem is reduced to one or two great circles on the phase sphere. All bifurcations are shown to be examples of elementary bifurcations of vector fields in one dimension. The classification of the bifurcation structure is therefore governed and greatly simplified by the theory of the universal unfolding and codimension of e...

A resonance broadening kinetic theory of the modified two-stream instability: Implications for radar auroral backscatter experiments

Abstract: The kinetic theory of the modified two-stream instability (MTSI) is usually considered more accurate than the corresponding fluid theory, for the purpose of interpreting VHF and UHF coherent radar backscatter measurements. However, recent developments in the nonlinear theory of the MTSI have retained the fluid theory formalism and consequently may not be entirely valid in the short-wavelength regime where VHF and UHF radars operate. In this paper, a nonlinear kinetic theory dispersion relation which takes account of the nonlinear resonance broadening effects is developed. With the aid of this dispersion relation, the phase speeds of the short wavelength plasma waves are calculated, as functions of wavelength, aspect angle, and flow angle. The results indicate that phase speeds tend to increase with increasing drift speed, at all wavelengths. Furthermore, under given flow conditions, the phase speeds are relatively insensitive to the flow angle and aspect angle but vary considerably with altitude. However, unlike long-wavelength fluid type waves, short-wavelength MTSI waves are moderately dispersive, the shorter wavelengths having the larger phase speeds. Finally, these kinetic theory calculations are also used to estimate the form of the k spectrum of saturated MTSI waves, and the results are compared with previously published fluid theorymore » predictions of spectral density.« less

Weinberg's nonlinear wave mechanics.

Abstract: From Weinberg's nonlinear wave equation we derive the nonlinear generalization of the optical Bloch equations with radiative damping. We compare these equations with Fermi's nonlinear radiation-reaction theory and with Jaynes's neoclassical theory. The nonlinear constant \ensuremath{\epsilon}, which is small if compared to the binding energy per nucleon, can be comparable to the radiative lifetime of an excited state of an ion stored in an ion trap. We show that in such a case the nonlinear modification of the optical Bloch equations can lead to large-scale effects in coherent transients, in atomic quantum jumps, and in resonance fluorescence and can modify the emission line shape.

Stochastic resonance in the linear and nonlinear responses of a bistable system to a periodic field.

Abstract: The validity of the theory of the linear response (the satisfaction of the standard spectral relations) during a stochastic resonance is demonstrated A stochastic resonance in relatively strong fields is studied The reason for the appearance of a stochastic resonance and the region of parameter values in which it is seen are pointed out

Wind-driven nonlinear oscillations of cables

Abstract: The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.

Behavior of a nonlinear resonator driven at subharmonic frequencies

Abstract: We have experimentally investigated the behavior of a driven nonlinear electrical resonator over a large region of its control parameter space. If one regards the various responses of the resonator as different ``phases'' and constructs a ``phase diagram'' in the system control parameter space, many intriguing regularities become apparent. At drive frequencies far below the system's resonant frequency, there exists a series of regions which are bounded by contours that mark the successive bifurcations in a period-doubling route to chaos. There are geometrical regularities in the size and location of these regions, and we suggest empirical scaling laws to describe these features. The appearance of period doubling and chaos in nonlinear systems that are driven far below resonance can have considerable practical significance and, in the empirical observations that are given in this paper, are a step in understanding the global parameter-space behavior of nonlinear systems.

Nonlinear energy exchange among harmonic modes and its applications to nonlinear imaging

Abstract: Nonlinear energy exchange and redistribution among the harmonic modes have been investigated. A problem with the pre‐bias boundary condition, which means that in addition to the fundamental harmonic, nonzero higher harmonics with adjustable amplitudes and phases are also applied to the input simultaneously, has been considered. The results show that the pre‐bias technique that is proposed in this paper may offer a better means for detecting the nonlinear properties of the medium. The feasibility of the approach was validated by computer simulation and experimental results.

Whither Nonlinear Acoustics

Abstract: If we include consideration of wave propagation in an ideal gas, we can trace the origin of theoretical nonlinear acoustics, at least as far back as Poisson’s work in 1808 [1]. The first experiments in air were done 1.27 centuries later [2]. With liquids Fox and Wallace [3] tried to explain experimental results on sound attenuation with a correct, but somewhat inadequate nonlinear theory. Keck and Beyer [4] used the equations of hydrodynamics and showed that nonlinear considerations lead to the prediction of nonlinear distortion, something that had been observed in fluids by optical techniques [5] and showed that the nonlinear equation for wave propagation in fluids has the same form as that for an ideal gas. Different thermodynamical quantities appeared in the nonlinear equations, however. It is significant that Keck and Beyer were able to perceive the inherent similarity of the two descriptions without becoming confused by the dissimilarities of certain details of equations describing ideal gases compared with those describing liquids.

Nonlinear Waves in Mechanics and Gas Dynamics

Abstract: : The author has studied the qualitative behaviour of nonlinear waves for hyperbolic conservation laws with or without the effects of dissipations, discretization, or nonlinear resonance. The fundamental problem of well-posedness theory for hyperbolic conservation laws is being resolved. It is shown that no physical law, beyond the second law of thermodynamics, is needed. The shock waves for finite difference schemes are shown to have slow decaying tails due to the effect of small divisor. Physical degenerate dissipation matrix is shown to give rise to rich nonlinear wave phenomena. Nonlinear waves for non-strictly hyperbolic system are shown to behave sensitively as a functional of the dissipation matrix. The ideas of wave tracing and pointwise estimates introduced by the author play the central role in the analysis of these problems.

Self-similarity in quantum dynamics

Abstract: We have developed a renormalization transformation, based on the existence of higher-order nonlinear resonances in the double-resonance model, that gives good predictions for the extension of the wave function in that system due to nonlinear resonance overlap. The double-resonance model describes the qualitative behavior, in local regions of the Hilbert space, of many quantum systems with two degrees of freedom whose dynamics is described by a nonlinear Hamiltonian but a linear Schr\"odinger equation.

Nonlinear oscillations in the convective zone

Abstract: Radiation-hydrodynamics for stellar oscillations in convective zone (Xiong 1989 is reformulated into a form suitable for non-linear dynamical interpretation of semi-regular and irregular variability of red super-giant stars

Third-harmonic studies of nonlinear anelasticity and determination of higher-order elastic constants in Mn-7 at.% Cu

Abstract: A nonlinear equation of motion relevant to a vibrating reed with one end clamped and the other free has been derived in order to analyse nonlinear anelasticity due to motion of twin boundaries. Application of the iteration method of Duffing to solve the equation has yielded an expression describing the shape of a resonance curve and an approximate solution that contains the third harmonic, the amplitude of which is a function of strain and higher-order elastic constants. Nonlinear resonance curves and third harmonics were measured for the martensite of a Mn-7 at.% Cu shape-memory alloy in an effort to test the theory. The experimental results have been found to be consistent with the theory, and the even-order elastic constants have been determined as high as the eighth order.

Theory of resonance dechanneling in strained-layer superlattices

Abstract: The general theory of resonance dechanneling of ions, positrons and electrons in crystals with superlattices has been developed. The application of the multiple scale method in the framework of the nonlinear oscillation theory gives the opportunity to investigate various types of spatially-periodic channel potential distortions. The resonance conditions are obtained. They take the form of relations between the transverse oscillation frequency of a particle in the channel and superlattice period. The occurrence or the absence of certain resonances directly correlates with the character of the lattice distortions. A set of two nonlinear partial differential equations describing the time dependences of the amplitude and phase of particle oscillations in the channel was obtained. Unlike the modified harmonic approximation used for a description of catastrophic dechanneling in strained-layer superlattices, the present theory is also valid for the case of weak channel potential perturbations when the anharmonicity of transverse particle oscillations plays an important role.

Mathematics of nonlinear science : proceedings of an AMS special session held January 11-14, 1989

Abstract: Multiple steady states in tubular chemical reactors by R. K. Alexander Two new approaches to large amplitude quasi-periodic motions of certain nonlinear Hamiltonian systems by M. S. Berger Vortices for the Ginzburg-Landau equations-the nonsymmetric case in bounded domain by Y. Y. Chen Nonlinear stability and bifurcation in Hamiltonian systems with symmetry by A. Szeri and P. Holmes Nonlinear resonance in inhomogeneous systems of conservation laws by E. Isaacson and B. Temple Bifurcation and stability in rotating, plane Couette-Poiseuille flow by G. H. Knightly and D. Sather Bifurcations of central configurations in the $N$-body problem by K. R. Meyer and D. S. Schmidt Leapfrogging of vortex filaments in an ideal fluid by M. S. Berger and J. Nee Calculation of sharp shocks using Sobolev gradients by J. W. Neuberger Monodromy preserving deformation of the Dirac operator acting on the hyperbolic plane by J. Palmer and C. A. Tracy Bifurcation from equilibria for certain infinite-dimensional dynamical systems by M. S. Berger and M. Schechter On dynamics of discrete and continuous $\sigma$-models (chiral fields) with values in Riemannian manifolds by V. Shubov Direct study for some nonlinear elliptic control problems by S. Stojanovic.

Effect of wavefront curvature within the cavity on the shift of the saturated-absorption resonance

Abstract: The shift of a nonlinear resonance due to wavefront curvature is shown to be proportional to the difference of the intensities of the counterpropagating waves with in a cavity. The depends on the direction of observation shift is calculated to be, i.e. from the left or to the right of the resonator.

Resonance Behavior of a Two-Level Atom Driven by Amplitude- or Frequency-Modulated Fields

Abstract: Recently, we have commented on equivalent magnetic and optical resonance experiments involving modulated fields.1 Also, we have shown how the concepts of multiple quantum resonances and Bloch-Siegert shifts apply to the interaction of a fully-amplitude-modulated (FAM) field with a two-level atom.2 Elaborating on work by Agarwal and Nayak,3 we apply the same concepts here to the interaction of not- fully amplitude-modulated (AM) or frequency-modulated (FM) fields (strictly speaking, phase-modulated fields) with a two-level atom. Specifically, we calculate the resonance behavior in the limit of no damping. Among the new features that we have found are the curious role of Haroche-like resonances4 and the appearance of interference effects in modulated components of the interaction.

Forced systems with multiple steady states

Abstract: The design and analysis of electronic systems often require determination of the steady state response of nonlinear circuits. The purpose of the paper is to reexamine some of the well known facts about nonlinear systems and their analysis. For nonlinear systems there may be two or more different steady-state responses, as in the case of jump resonance; there can be responses of different periods, as in the case of subharmonics; or there can be a nonperiodic response, as in the case of systems that exhibit chaos. This work deals with the application of iteration and Volterra series analysis methods to certain types of problems with interesting solutions. In particular, the jump resonance phenomenon and the initial conditions leading to different harmonic solutions are considered. >