Problem Of Parametric Excitation Research Articles

Development of Coupled Program fast-simdyn and Study of Non-Linear Hydrodynamics, Coupled With Negative Damping and Aerodynamics of Floating Offshore Wind Turbines

Abstract Non-linear hydrostatic and wave forces on floating structures are very important during large amplitude waves. The computer program simdyn is a blended time domain program developed by Marine Dynamics Laboratory at TAMU and is capable of capturing the role of non-linear fluid forces. simdyn has previously been used to demonstrate that nonlinear hydrostatics have become very important in the problem of parametric excitation. In the current work simdyn is coupled with the computer program fast developed by U.S. National Renewable Energy Laboratory (NREL) for numerical simulation of floating offshore wind turbines (FOWTs). fast-simdyn is now a tool that is capable of studying large amplitude motions of FOWTs in extreme seas. fast-simdyn was then used to study the classic instability of negative damping that occurs in FOWTs that use conventional land-based control. The development of platform pitch and platform surge instability are studied in relation to different wave and wind scenarios. The intent was to do an analysis to see if the non-linear forces do play a significant role in large amplitude motions induced by negative damping. This study gives an indication of whether the development and application of higher fidelity hydrodynamic modules are justified.

Asynchronous parametric excitation, total instability and its occurrence in engineering structures

In mechanical engineering systems self-excited and parametrically excited vibrations are in general unwanted and sometimes dangerous. There are many systems exhibiting such vibrations which up to this day cannot be completely avoided, such as brake squeal, the galloping vibrations of overhead transmission lines, the ground resonance in helicopters and others. In general, problems of parametric excitation are studied for the case in which all the time-periodic terms are synchronous. In this case the stability behavior is well understood. However, if the time-periodic terms are asynchronous, an “atypical” behavior may occur: The linear system may then be unstable for all frequencies of the parametric excitation, and not only in the neighborhood of certain discrete frequencies (total instability). Until recently it was believed that such “atypical” behavior would not appear in mechanical systems. The present paper discusses some recent insights and results obtained for linear and nonlinear systems with asynchronous parametric excitation. The method of normal forms is used to prove total instability and to calculate limit cycles of a generalized nonlinear system. Further, a mechanical example of a minimal disk brake model featuring such out of phase parametric excitation is presented. The example outlines the importance of the observed effects from the engineering point of view, since similar terms are also expected in the equations of motion of disk brakes with disks with ventilation channels and most likely also in other physical systems.

Atypical parametric instability in linear and nonlinear systems

The linear parts of the equations of motion of parametrically excited mechanical systems are characterized by the M, D, G, K, N matrices which may all be time-periodic (mass, damping, gyroscopic, stiffness and circulatory matrices, respectively). The stability of these systems can be studied via Floquet theory. A typical property of parametric instability behavior is the existence of combination resonances. It has been known for a long time that the type of parametric resonance depends very much on whether the excitation is in the in the K or in the N matrices, or simultaneously in both of them, the other matrices being constant. In general, problems of parametric excitation are studied for the case in which all the excitation terms are in phase. If this is not the case, an atypical behavior may occur: The linear system may then be unstable for all frequencies of the parametric excitation, and not only in the neighborhood of certain discrete frequencies. Examples of differential equations of this type were first given about 70 years ago by Lamberto Cesari, but seem largely to have fallen into oblivion since then. It was recently observed that the linearized equations of motion for a minimal model of a squealing disk brake have such out of phase parametric excitation. Additional nonlinearities are introduced in the equations and the corresponding limit cycles are calculated using normal form theory.

Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation

The non-linear equations and boundary conditions of non-planar (two bending and one torsional) vibrations of inextensional isotropic geometrically imperfect beams (i.e. slightly curved and twisted beams) are derived using the extended Hamilton's principle. The assumptions of Euler–Bernoulli beam theory are used. The order of magnitude of the natural geometric imperfection is assumed to be the same as the first order of vibrations amplitude. Although the natural imperfection is small, in contrast to the case of straight beams (i.e. geometrically perfect beams), this study shows that the vibration equations are linearly coupled and have linear and quadratic terms in addition to cubic terms. Also, in the case of near-square or near-circular beams, coupling terms between lateral and torsional vibrations exist. Furthermore, a problem of parametric excitation in the case of perfect beams changes to a problem of mixed parametric and external excitation in the case of imperfect beams. The validity of the model is investigated using the existing experimental data.

Parametric excitation of high-mode oscillations for a non-linear telegraph equation

The problem of parametric excitation of high-mode oscillations is solved for a non-linear telegraph equation with a parametric external excitation and small diffusion. The equation is considered on a finite (spatial) interval with Neumann boundary conditions. It is shown that under a proper choice of parameters of the external excitation this boundary-value problem can have arbitrarily many exponentially stable solutions that are periodic in time and rapidly oscillate with respect to the spatial variable.

Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.

Nonlinear oscillating structures in the earthquake and seaquake dynamics.

The article deals with the problem of parametric excitation of oscillating structures during an earthquake and a seaquake associated with external forces. Tidal forces and their linear and nonlinear components are considered as possible causes of earthquakes. A seaquake is regarded as a typical large-scale structural disturbance of the water surface resulting from the ocean bottom earthquake. There are given results of original laboratory seaquake modeling where wave structures with hexagonal and square cells appeared. The received wave lattices were similar to Faraday ripples, but with the size of cells from 15 to 120 mm. These experiments provided parameters on transition from a wave structure to chaos. Comparison of laboratory experimental data with descriptions of full-scale seaquakes and parametric wave theory has confirmed the submitted interpretation of the phenomenon. (c) 1996 American Institute of Physics.

A quantum approach to the parametric excitation problem in solids

This report is a review of recent results in the quantum treatment of the parametric interactions in electron-phonon systems subjected to the action of high-power electromagnetic radiation fields and described by Hamiltonians depending on time explicitly. The main content of the work that follows after a short introduction includes two parts. In the first one (sections 2–3) a general quantum formalism is developed and then applied to the problem of parametric excitation of waves in homogeneous isotropic electron-phonon systems and in real solids where different effects of lattice symmetry and electronic band structure are taken into account. The second part (section 4) is devoted t the theory of anomalous fluctuations in connection with the possibility of experimental observation of parametric instability phenomena in solids. In the concluding part some remarks about the possible directions of further development of the problem are made.

Parametric decay of magneto-acoustic oscillations in a cylindrical plasma

Parametric decay of magneto-acoustic oscillations in a cylindrical plasma column is considered, including the Hall effect and the effects of finite resistivity. Growth rates for axisymmetric modes in decay instabilities are derived and applied to the problems of parametric excitation of Alfvén waves and supplementary heating of plasmas. It is shown that both the slow (ion cyclotron) and fast (compressional) modes may be parametrically excited. The threshold &litudes for nonlinear dissipation of pump field energy are calculated and briefly discussed.

On the theory of parametric excitation in quantum electron–phonon systems

AbstractThe problem of parametric excitation of electromagnetic vibrations of a quantum electron‐phonon system is studied using a formalism based on the equation of motion for the quantum distribution function of an electron gas interacting with phonons in the presence of high‐intensity radiation field and the complete system of Maxwell equations. Calculations of the growth rates and the threshold values of the external electric field E0 are presented for both, uncoupled electron and electron‐phonon systems in different regions of wave number κ for κ ‖ E0 as well as for κ ⊥ E/0.

Nonlinear theory of the excitation of spin waves in ferrites by a longitudinal ultrahigh-frequency magnetic field

The problem of parametric excitation of spin waves in ferrites due to the action of a perturbing super high-frequency (SHF) magnetic field polarized along the direction of the constant magnetic field H0 is considered in a nonlinear approximation. Such an examination enables one to determine the conditions for parametric excitation of spin waves and to find the amplitude and phase of the oscillations of the system's magnetization in the steady state. The frequency interval, within whose limits parametric excitation of spin waves is possible, is determined. Stability of the steady state under the conditions of parametric resonance is investigated. The results obtained are compared with existing experimental data.

Energy methods in parametric excitation of rings

Hamilton's principle is employed to derive equations that may be taken as the basis for nonlinear problems of parametric excitation of rings. Such problems may arise in ring reinforced missile or submarine structures or wherever rings are employed in dynamical situations. Both normally directed and centrally directed pressures are considered. Analogous to the static stability problem, for the linear case, by means of a simple calculation, the stability regions are shown to be sensitive to the nature of the pressures.