Vertically forced surface wave in weakly viscous fluids bounded in a circular cylindrical vessel

Two-time scale perturbation expansions were developed in weakly viscous fluids to investigate surface wave motions by linearizing the Navier－Stokes equation in a circular cylindrical vessel which is subject to a vertical oscillation. The fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates a damping term and external excitation, was derived for the weakly viscid fluids. The condition for the appearance of stable surface waves was obtained and the critical curve was determined. In addition, an analytical expression for the damping coefficient was determined and the relationship between damping and other related parameters (such as viscosity, forced amplitude, forced frequency and the depth of fluid, etc.) was presented. Finally, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent.     

Resonant and nonresonant patterns in forced oscillators

Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillator's frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.     

Breakdown of Stability of Motion in Superquadratic Potentials

Based on the KAM theory, investigation of the equation of motion of a classical particle in a one-dimensional superquadratic potential well, under the influence of an external time-periodic forcing, raised a hope that all the solutions are bounded. Indeed, due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to infinity as the amplitude increases. Therefore, because of this relationship between the frequency and the amplitude, intuitively one might expect that all resonances that could cause the accumulation of energy would be destroyed, and thus all solutions would stay bounded for all time. More formally, according to Moser's twist theorem, this could mean the existence of invariant tubes in the extended phase space and therefore would result in the boundedness of the solutions. Actually, the boundedness results have been established for a large class of superquadratic potentials, but in general, the above intuition turns out to be wrong. Littlewood showed it by creating a superquadratic potential in which an unbounded motion occurs in the presence of some particular piecewise constant forcing. Moreover, Littlewood's result holds for a larger class of forcings. Here it is proven for the continuous time-periodic forcing. For this purpose a new averaging technique for the forced motions in superquadratic potentials with rather weak assumptions on the differentiability of the potentials has been developed. Received: 1 February 1995 / Accepted: 15 March 1997     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

Some comments on the Mindlin-Goodman method of solving vibrations problems with time dependent boundary conditions

Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.     

Bifurcation structures of periodically forced oscillators

A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned.     

Dynamic stability of weakly damped oscillators with elastic impacts and wear

The dynamics of non-linear oscillators comprising of a single-degree-of-freedom system and beams with elastic two-sided amplitude constraints subject to harmonic loads is analyzed. The beams are clamped at one end, and constrained against unilateral contact sites near the other end. The structures are modelled by a Bernoulli-type beam supported by springs using the finite element method. Rayleigh damping is assumed. Symmetric and elastic double-impact motions, both harmonic and sub-harmonic, are studied by way of a Poincaré mapping that relates the states at subsequent impacts. Stability and bifurcation analyses are performed for these motions, and domains of instability are delineated. Impact work rate, which is the rate of energy dissipation to the impacting surfaces, is evaluated and discussed. In addition, an experiment conducted by Moon and Shaw on the vibration of a cantilevered beam with one-sided amplitude constraining stop is modelled. Bifurcation observed in the experiment could be captured.     

Motion of spiral tip driven by local forcing in excitable media

We study the motion of a spiral wave controlled by a local periodic forcing imposed on a region around the spiral tip in an excitable medium. Three types of trajectories of spiral tip are observed： the epicycloid-like meandering, the resonant drift, and the hypocycloid-like meandering. The frequency of the spiral is sensitive to the local periodic forcing. The dependency of spiral frequency on the amplitude and size of local periodic forcing are presented. In addition, we show how the drift speed and direction are adjusted by the amplitude and phase of local periodic forcing, which is consistent with a theoretical analysis based on the weak deformation approximation.     

Hexagonal Standing-Wave Patterns in Periodically Forced Reaction-Diffusion Systems

The periodically forced spatially extended Brusselator is investigated in the oscillating regime. The temporal response and pattern formation within the 2：1 frequency-locking band where the system oscillates at one half of the forcing frequency are examined. An hexagonal standing-wave pattern and other resonant patterns are observed. The detailed phase diagram of resonance structure in the forcing frequency and forcing amplitude parameter space is calculated. The transitions between the resonant standing-wave patterns are of hysteresis when control parameters are varied, and the presence of multiplicity is demonstrated. Analysis in the framework of amplitude equation reveals that the spatial patterns of the standing waves come out as a result of Turing bifurcation in the amplitude equation.     

Efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading

A method is developed to compute low-level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing. Emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters. This gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure. The method exploits the First-Order Reliability Method (FORM) in which the failure surface is constructed via the Harmonic Balance Method (HBM). This combined approach is tested on a Duffing oscillator with harmonic forcing and up to three uncertain parameters, for which the frequency of multiple-solution-maximum-amplitude is found directly, and the probability computed via the Hasofer-Lind reliability index. The accuracy of the proposed HBM-FORM, in the tails of the amplitude exceedance probability, is shown for the Duffing example to be acceptably accurate, whereas the efficiency is shown to be around 1000 times faster than Direct Integration and around 200 times faster than Monte Carlo simulation.     

Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation

The magneto-rheological damper(MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom(2-DOF) MR suspension system was established first, by employing the modified Bouc–Wen force–velocity(F –v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface.The largest Lyapunov exponent(LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density(PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy(K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system.     

NON-LINEAR AEROELASTIC ANALYSIS USING THE POINT TRANSFORMATION METHOD, PART 1: FREEPLAY MODEL

A point transformation technique is developed to investigate the non-linear behavior of a two-dimensional aeroelastic system with freeplay models. Two formulations of the point transformation method are presented, which can be applied to accurately predict the frequency and amplitude of limit cycle oscillations. Moreover, it is demonstrated that the developed formulations are capable of detecting complex aeroelastic responses such as periodic motions with harmonics, period doubling, chaotic motions and the coexistence of stable limit cycles. Applications of the point transformation method to several test examples are presented. It is concluded that the formulations developed in this paper are efficient and effective.     

Nonlinear effects in the dynamics of clouds of bubbles

This paper presents a spectral analysis of the response of a fluid containing bubbles to the motions of a wall oscillating normal to itself. First, a Fourier analysis of the Rayleigh-Plesset equation is used to obtain an approximate solution for the nonlinear effects in the oscillation of a single bubble in an infinite fluid. This is used in the approximate solution of the oscillating wall problem, and the resulting expressions are evaluated numerically in order to examine the nonlinear effects. Harmonic generation results from the nonlinearity. It is observed that the bubble natural frequency remains the dominant natural frequency in the volume oscillations of the bubbles near the wall. On the other hand, the pressure perturbations near the wall are dominated by the first and second harmonics present at twice the natural frequency while the pressure perturbation at the natural frequency of the bubble is inhibited. The response at the forcing frequency and its harmonics is explored along with the variation with amplitude of wall oscillation, void fraction, and viscous and surface tension effects. Splitting and cancellation of frequencies of maximum and minimum response due to enhanced nonlinear effects are also observed.     

Large variations in NLS bi-soliton wave groups

The nonlinear Schrödinger (NLS) equation describes the spatial–temporal evolution of the complex amplitude of wave groups in beams and pulses in both second and third order nonlinear material. In this paper we investigate in detail the wave group that has the exact two-soliton solution as amplitude, and show that large variations in the amplitude appear to form a pattern that, at the peak interaction, resembles quite well the linear superposition. The complexity of the phenomenon is a combination of nonlinear effects and linear interference of the carrier waves: the characteristic parameter is the quotient of wave amplitude and frequency difference of the carrier waves, which is also proportional to the quotient of the modulation period of the carrier waves during interaction and the interaction period of the soliton envelopes.     

Flexomagnetoelectric interaction in multiferroics

We present a theoretical analysis of phase separation in the presence of a spatially periodic forcing of wavenumber q traveling with a velocity v. By an analytical and numerical study of a suitably generalized 2d-Cahn-Hilliard model we find as a function of the forcing amplitude and the velocity three different regimes of phase separation. For a sufficiently large forcing amplitude a spatially periodic phase separation of the forcing wavenumber takes place, which is dragged by the forcing with some phase delay. These locked solutions are only stable in a subrange of their existence and beyond their existence range the solutions are dragged irregularly during the initial transient period and otherwise rather regular. In the range of unstable locked solutions a coarsening dynamics similar to the unforced case takes place. For small and large values of the forcing wavenumber analytical approximations of the nonlinear solutions as well as for the range of existence and stability have been derived.     

洛仑兹脉冲光束的复振幅包络解和复解析信号解的比较研究

使用复振幅包络表示式和复解析信号表示式推导了洛仑兹脉冲光束的传输公式. 通过具体数 值计算对脉冲光束的复振幅包络解和复解析信号解在不同带宽时的传输进行了对比研究. 数 值结果表明脉冲光束为窄带时，在传输方向轴中心的一定范围内两种解是一致的，而对于宽 带脉冲光束，复振幅包络解在轴中心较近的距离即表现出奇异性，复解析信号解才是符合物 理意义的表示式. 由数值计算得出了选择脉冲光束研究方法的条件，并从所得公式对复振幅 包络解出现奇异性的现象进行了解释. 关键词： 复振幅包络 复解析信号 窄带脉冲 宽带脉冲     

Cross-focusing of two laser beams in a plasma

Cross-focusing of two copropagating laser beams in a plasma is investigated using paraxial ray theory. If the lasers have a frequency difference equal to the electron plasma frequency, they can drive a large amplitude plasma wave. The ponderomotive force due to the plasma wave forces the plasma electrons outwards thereby generating a parabolic density profile giving rise to cross-focusing. The results show a decrease in threshold for focusing by two orders of magnitude as compared to focusing due to the ponderomotive force of the laser beams.     

Enhanced Beat Wave Saturation Amplitude in an Ionizing Plasma

The excitation of a plasma wave by two laser beams, whose frequency difference is near the plasma frequency, is studied in a plasma with a density that is slowly increasing with time due to ongoing ionization as appropriate for experiments done in laser breakdown plasmas. Numerical integration of the relativistic equation for the evolution of the wave amplitude reveals that for a rate of increase of the plasma density of approximately 1017 cm-3/ns at a laser intensity I = 1014 W/cm2, the wave amplitude can rise considerably above the relativistic saturation limit of Rosenbluth and Liu which was obtained for a plasma of constant density. This increase in plasma density compensates the reduction in plasma frequency caused by the relativistic electron mass increase when the wave amplitude is large. The frequency and phase excursions of the plasma wave are reduced for an optimum time increasing density. We find that moderate damping can stabilize both the amplitude and the phase of the plasma wave with respect to the pump.     

Terahertz radiation generation by beating of two chirped laser pulses in a warm collisional magnetized plasma

Analytical equations of terahertz(THz) radiation generation based on beating of two laser beams in a warm collisional magnetized plasma with a ripple density profile are developed. In this regard, the effects of frequency chirp on the field amplitude of the terahertz radiation as well as the temperature and collision parameters are investigated. The ponderomotive force is generated in the frequency chirp of beams. Resonant excitation depends on tuning of the plasma beat frequency,magnetic field frequency, thermal velocity, collisional frequency, and effect of the frequency chirp with the plasma density.For optimum parameters of frequency and temperature the maximum THz amplitude is obtained.     

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.