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We consider an oscillator model with nonpolynomial interaction. The model admits exact solutions for two situations: for energy eigenvalues in terms of zeros of Bessel functions, that were considered as functions of the continuous index; and for the corresponding eigenstates in terms of Lommel polynomials.
The stability of steady equilibrium amplitudes of magnetic fields generated by convection flows is investigated. Two cases are considered in detail, for which steady solutions are available from the previous work of Busse (1973) and Busse (1977). In the first case instability occurs primarily because of magnetic flux expulsion at high magnetic Reynolds numbers. In the second case the change in the velocity field caused by the Lorentz force enhances dynamo action. This subcritical finite amplitude dynamo is potentially unstable. In typical cases the nonlinear dynamo oscillations that replace the steady equilibrium solutions are investigated by numerical integration.
It is shown how some basic ideas from system theory and differential geometry can be used to establish new results concerning the existance of oscillations for autonomous feedback systems. The conditions obtained are expressed in terms of the frequency response characteristic of the open-loop system and certain general properties of the nonlinearity.
Nonlinear mixing of oscillations in a dusty plasma due to the harmonic time varying modulation of a nonlinear compressional oscillation is analyzed using a simple mathematical model consisting of a forced Korteweg-de Vries equation. An exact analytic solution of this equation is found to exhibit nonlinear mixing in the system. Here, the model solution can be usefully employed to predict the existence of nonlinear mixing of oscillations in a two-dimensional dusty plasma system of a particular experimental configuration.
In this second paper on the evolution of magnetic flux ropes we study the effects of gas pressure. We assume that the energy transport is described by a polytropic relationship and reduce the set of ideal MHD equations to a single, second-order, nonlinear, ordinary differential equation for the evolution function. For this conservative system we obtain a first integral of motion. To analyze the possible motions, we use a mechanical analogue -- a one-dimensional, nonlinear oscillator. We find that the effective potential for such an oscillator depends on two parameters: the polytropic index gamma and a dimensionless quantity kappa the latter being a function of the plasma beta, the strength of the azimuthal magnetic field relative to the axial field of the flux rope, and gamma. Through a study of this effective potential we classify all possible modes of evolution of the system. In the main body of the paper, we focus on magnetic flux ropes whose field and gas pressure increase steadily towards the symmetry axis. In this case, for gamma greater than 1 and all values of kappa, only oscillations are possible. For gamma less than 1, however, both oscillations and expansion are allowed. For gamma less than 1 and kappa below a critical value, the energy of the nonlinear oscillator determines whether the flux rope will oscillate or expand to infinity. For gamma less than 1 and kappa above critical, however, only expansion occurs. Thus by increasing kappa while keeping gamma fixed (less than 1), a phase transition occurs at kappa = kappa(sub critical) and the oscillatory mode disappears. We illustrate the above theoretical considerations by the example of a flux rope of constant field line twist evolving self-similarly. For this example, we present the full numerical MHD solution. In an appendix to the paper we catalogue all possible evolutions when (1) either the magnetic field or (2) the gas pressure decreases monotonically toward the axis. We find that in these cases critical conditions can occur for gamma greater than 1. While in most cases the flux rope collapses, there are notable exceptions when, for certain ranges of kappa and gamma, collapse may be averted.
The problem of forced acoustic oscillations in a pipe was experimentally investigated, taking into account the response of both open and closed tubes to near-resonant excitation by large amplitude oscillations of a piston at one end of the tube. Attention was given to the effect of the orifice area on shock waves. By comparing the experimental results with nonlinear theory, wave reflection coefficients of the orifice plates were determined at both closed-tube and open-tube resonant frequencies. This approach can even be used when the terminating elements are subjected to intense periodic pressure pulses.
The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. The linear theory predicts large amplitudes near resonance and that nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and waveform of the oscillation are predicted there. In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.
Optimizing resonator shapes for maximizing the ratio of maximum to minimum gas pressure at an end of the resonator is investigated numerically. It is well known that the resonant frequencies and the nonlinear standing waveform in an acoustical resonator strongly depend on the resonator geometry. A quasi-Newton type scheme was used to find optimized axisymmetric resonator shapes achieving the maximum pressure compression ratio with an acceleration of constant amplitude. The acoustical field was solved using a one-dimensional model, and the resonance frequency shift and hysteresis effects were obtained through an automation scheme based on continuation method. Results are presented for optimizing three types of geometry: a cone, a horn-cone and a half cosine-shape. For each type, different optimized shapes were found when starting with different initial guesses. Further, the one-dimensional model was modified to study the effect of an axisymmetric central blockage on the nonlinear standing wave.
Optimizing resonator shapes for maximizing the ratio of maximum to minimum gas pressure at an end of the resonator is investigated numerically. It is well known that the resonant frequencies and the nonlinear standing waveform in an acoustical resonator strongly depend on the resonator geometry. A quasi-Newton type scheme was used to find optimized axisymmetric resonator shapes achieving the maximum pressure compression ratio with an acceleration of constant amplitude. The acoustical field was solved using a one-dimensional model, and the resonance frequency shift and hysteresis effects were obtained through an automation scheme based on continuation method. Results are presented for optimizing three types of geometry: a cone, a horn-cone and a half cosine- shape. For each type, different optimized shapes were found when starting with different initial guesses. Further, the one-dimensional model was modified to study the effect of an axisymmetric central blockage on the nonlinear standing wave.
In the rapid directional solidification of a dilute binary alloy, analysis reveals that, in addition to the cellular mode of Mullins and Sekerka (1964), there is an oscillatory instability. For the model analyzed by Merchant and Davis (1990), the preferred wavenumber is zero; the mode is one of pulsation. Two strongly nonlinear analyses are performed that describe this pulsatile mode. In the first case, nonequilibrium effects that alter solute rejection at the interface are taken asymptotically small. A nonlinear oscillator equation governs the position of the solid-liquid interface at leading order, and amplitude and phase evolution equations are derived for the uniformly pulsating interface. The analysis provides a uniform description of both subcritical and supercritical bifurcation and the transition between the two. In the second case, nonequilibrium effects that alter solute rejection are taken asymptotically large, and a different nonlinear oscillator equation governs the location of the interface to leading order. A similar analysis allows for the derivation of an amplitude evolution equation for the uniformly pulsating interface. In this case, the bifurcation is always supercritical. The results are used to make predictions about the characteristics of solute bands that would be frozen into the solid.
The Frequency-Amplitude (F-A) curve on power system oscillation under a large disturbance characterizes how a natural oscillation mode transitions to nonlinear oscillations with growing amplitudes and decaying frequencies. The existing formulation of the F-A curve is derived by solving elliptical integrals on oscillation of a single-machine-infinite-bus equivalent about the targeted oscillation mode. The formula is in a form of infinite series and needs to sum a large number of terms for satisfactory accuracy. This paper introduces an explicit, approximate expression obtained from the Homotopy Analysis Method on the F-A curve. The proposed F-A curve expression is derived from an SMIB system and verified on the IEEE 3-machine 9-bus system to show how the oscillation frequency of a dominant mode varies with oscillation amplitude under large disturbances.
Steady state planar motion of fluid with free surface in container undergoing lateral harmonic vibration
Finite-amplitude free oscillations of inviscid incompressible fluid in cylindrical sector container
Solution for finite-amplitude free oscillations of fluid in partially filled axisymmetric container
Transverse and longitudinal combustion chamber oscillations in rocket motors with distributed mass and energy sources
Some results of bifurcation theory were used to study the existence of small-amplitude periodic behavior in launch vehicle dynamics, assuming that nonlinearity exists as a cubic term in the rudder response. The analysis follows closely Sattinger's (1973) approach to the theory of periodic bifurcations. The conditions under which a bifurcating branch of orbitally stable periodic solutions will exist are determined. It is shown that in more complicated cases, the conditions under which the system matrix has a pair of simple purely imaginary eigenvalues can be determined with the aid of linear stability techniques.
Consideration is given to synthesis methods for feedback systems with nonlinear dynamic compensation, which permits increased feedback while preserving robustness, global stability, good transient responses, and stability of the output processes. Material needed for the realization of design goals from the areas of linear systems, nonlinear oscillation, and stability of nonlinear systems is covered. The three main parts of the book are: (1) linear feedback systems, (2) nonlinear feedback system analysis, and (3) synthesis methods for globally stable Nyquist-stable feedback systems with increased feedback.
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