Long-wave convection in a two-layer system

Periodic regimes of large-scale convection in a two-layer system with a deformable interface are considered. In the limiting case of strong surface tension, an amplitude equation describing the interface deformation is derived. Two-dimensional periodic flow regimes are studied. It is shown that, depending on the governing parameters, both hard and soft excitation of convection is possible. In the presence of heating from above, the finite-amplitude excitation of convection was observed.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 7–20. Original Russian Text Copyright © 2004 by Lyubimov and Shklyaev.     

Chaos in Acoustic Subspace Raised by the Sommerfeld–Kononenko Effect

This paper deals with vibrations of an infinite plate in contact with an acoustic medium where the plate is subjected to a point excitation by an electric motor of limited power-supply. The whole system is divided into two “exciter - foundation” and “foundation-plate-medium”. In the system “motor-foundation” three classes of steady state regimes are determined: stationary, periodic and chaotic. The vibrations of the plate and the pressure in the acoustic fluid are described for each of these regimes of excitation. For the first class they are periodic functions of time, for the second they are modulated periodic functions, in general with an infinite number of carrying frequencies, the difference between which is constant. For the last class they correspond to chaotic functions. In another mathematical model where the exciter stands directly on an infinite plate (without foundation) it was shown that chaos might occur in the system due to the feedback influence of waves in the infinite hydro-elastic subsystem in the regime of motor shaft rotation. In this case the process of rotation can be approximately described as a solution of the fourth order nonlinear differential equation and may have the same three classes of steady state regimes as the first model. That is the electric motor may generate periodic acoustic waves, modulated waves with an infinite number of frequencies or chaotic acoustic waves in a fluid.     

Bifurcation and chaos of biochemical reaction model with impulsive perturbations

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem for the impulsive equation and small-amplitude perturbation skills, we see that the boundary-periodic solution ([(x)\tilde](t),0)(\tilde{x}(t),0) is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. By numerical simulation, we can show that the system presents rich dynamics, including periodic solutions, quasi-periodic oscillations, period doubling cascades, periodic halving cascades, symmetry bifurcations, and chaos.     

Dynamic analysis of mathematical model of ethanol fermentation with gas stripping

In this paper, a mathematical model for ethanol fermentation with gas stripping is investigated. Firstly, the model with continuous substrate input is taken. We study the existence and local stability of two equilibrium points. According to Poincare–Bendixson Theorem, the sufficient condition for the globally asymptotical stability of positive equilibrium point is obtained, which implies that we can get stable ethanol product. Secondly, we study the model with impulsive substrate input and obtain the sufficient condition for the local stability of cell-free periodic solution by using the Floquet’s theory of impulsive differential equation and small-amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical (subcritical) bifurcation. Finally, our results are confirmed by means of numerical simulation.     

Dynamical behavior of a class of prey-predator system with impulsive state feedback control and Beddington–DeAngelis functional response

This paper studies systematically a Bedd-ington?CDeAngelis prey?Cpredator system with harvesting and impulsive state feedback control. Conditions for existence and stability of predator-free periodic solution are obtained. When the predator-free periodic solution loses its stability, the existence and stability of nontrivial period solution are also established. Furthermore, computer simulations show that this impulsive system displays a series of complex phenomena, including period-doubling bifurcation and cascade, period window, and chaotic bands. Through numerical simulation, it is also observed that capture capability can influence the amount of predator released and the interval of the stability for nontrivial period-1 solution. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.     

Exploiting the subharmonic parametric resonances of a buckled beam for vibratory energy harvesting

The potential of harvesting vibratory energy via a bistable beam subjected to subharmonic parametric excitations is investigated. The vibrating structure is a buckled beam with two stable equilibria separated by a potential barrier. The beam is subjected to a superposition of a static axial load beyond its buckling load and a harmonic axial excitation whose frequency is around twice the frequency of the buckled beam’s first vibration mode. A macro-fiber composite patch is attached to one side of the beam to convert the strain energy resulting from the beam’s oscillation into electricity. The study considers two regimes of excitations: an amplitude sweep and a frequency sweep. In the first regime, the amplitude of excitation is quasi-statically varied while the excitation frequency is tuned at twice the natural frequency of the first vibration mode. In the second regime, the excitation frequency is swept forward and backward around the subharmonic resonant frequency while the amplitude of excitation is kept constant. A theoretical model which governs the electromechanical coupling of the transverse vibrations of the beam and the output voltage is used to monitor the response as the excitation parameters are changed. An experimental setup is also built and a series of tests is performed to validate the theoretical findings. It is shown that, depending on the amplitude and frequency of excitation, the harvester can perform small-amplitude periodic intra-well motion, intra- and inter-well chaotic motions, as well as periodic inter-well motions. Experimental results also show that, as compared to the classical linear resonance, utilizing the sub-harmonic resonance of a bistable energy harvesters can result in a broadband frequency response.     

多自由度参激系统稳定性的数值分析

提出多自由度周期参激系统稳定性的数值直接法。通过将扰动方程表示成状态方程形式,再根据Flo-quet理论将扰动解表示成指数特征分量与周期分量之积,并将其周期分量与系统周期系数展成Fourier级数,导出一系列代数方程,建立矩阵特征值问题,从而由数值求解特征值可直接确定参激系统的稳定性。该方法可用于一般周期参激阻尼系统,特征值矩阵不含逆子阵。应用于斜拉索在支座周期运动激励下的参激振动不稳定性分析,数值结果表明该方法的有效性。     

Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

Analyse von Systemen mit Impuls-Parametererregung

Übersicht Unter Impuls-Parametererregung versteht man eine Erregungsart, die im mathematischen Modell durch periodische Koeffizienten in Form periodischer Folgen von Dirac-Funktionen beschrieben werden kann. Einige Beispiele für diesen Erregertyp sind angegeben. Ziel der vorliegenden Arbeit ist die Analyse von periodischen Systemen mit Impuls-Parametererregung unter Berücksichtigung deterministischer und stochastischer Störungen. Neben den üblichen Simulationsmethoden dienen dazu die auf der Floquetschen Theorie basierenden Verfahren, die so erweitert werden, daß sie auf den gegebenen Systemtyp anwendbar sind. Der Vorteil der letztgenannten Methoden liegt in einer Verkürzung der Integrationszeit auf nur eine Periodendauer.

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Summary By impulsive parametric excitation we mean those excitations which may be represented in a mathematical model by periodic coefficients consisting of periodic sequences of Dirac functions. Some examples for this typ of excitation are given. The aim of the paper is the analysis of periodic systems with impulsive parametric excitation regarding deterministic and stochastic disturbances. For that the common simulation techniques are used beside the methods based on Floquet's theory, which are extended to the given system typ. The benefit of the latter methods is a reduction of integration time to only one period.

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Herrn Professor Dr. Fritz Weidenhammer zum 60. Geburtstag gewidmet     

Non-resonant response, bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control

The maglev system with delayed position feedback control is excitated by the deflection of flexible guideway and resonant response may take place. This paper concerns the non-resonant response of the system by employing centre manifold reduction and method of multiple time scales. The dynamical model is presented and expanded to the third-order Taylor series. Taking time delay as its bifurcation parameter, the condition with which the Hopf bifurcation may occur is investigated. Centre manifold reduction is applied to get the Poincaré normal form of the nonlinear system so that we can study the relationship between periodic solution and system parameter. At first, the non-resonant periodic solution of the normal form is calculated based on the method of multiple time scales. Then the bifurcation condition of the free oscillation in the solution is analyzed, and we get the conditions with which the free oscillation has maximum and minimum values. The relationship between external excitation and the periodic solution is also discussed in this paper. Finally, numerical simulation results show how system and excitation parameters affect the system response. It is shown that the existence of the free oscillation and the amplitude of the forced oscillation can be determined by time delay and control parameters. So felicitously selecting them can suppress the oscillation effectively.     

Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations

A linear oscillator (LO) coupled with two vibro-impact (VI) nonlinear energy sinks (NES) in parallel is studied under periodic and transient excitations, respectively. The objective is to study response regimes and to compare their efficiency of vibration control. Through the analytical study with multiple scales method, two slow invariant manifolds (SIM) are obtained for two VI NES, and different SIM that result from different clearances analytically supports the principle of separate activation. In addition, fixed points are calculated and their positions are applied to judge response regimes. Transient responses and modulated responses can be further explained. By this way, all analysis is around the most efficient response regime. Then, numerical results demonstrate two typical responses and validate the effectiveness of analytical prediction. Finally, basic response regimes are experimentally observed and analyzed, and they can well explain the complicated variation of responses and their corresponding efficiency, not only for periodic excitations with a fixed frequency or a range of frequency, but also for transient excitation. Generally, vibration control is more effective when VI NES is activated with two impacts per cycle, whatever the types of excitation and the combinations of clearances. This observation is also well reflected by the separate activation of two VI NES with two different clearances, but at different levels of displacement amplitude of LO.     

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.     

Non-linear response of a string to random excitation

The mean square deflection of a non-linear string subjected to nonplanar Gaussian white noise excitation is determined by the perturbation method. It is shown that increase in tension due to stretching, and transverse transverse mode coupling tend to reduce the mean square deflection; while longitudinal-transverse mode coupling tends to counter this effect to some extent. These results are in conformity with the trend observed in the case of periodic excitation.     

一类双自由度碰振系统运动分析

基于Poincare映射方法对一类两自由度碰撞系统进行了分析。经过详细的理论演算得到单碰周期n的次谐运动的存在性判据和稳定性条件,给出计算Jacobi矩阵特征值的公式。数值模拟表明,该方法具有令人满意的结果。此外,还讨论了当不满足所提出的单碰周期n次谐运动的存在性条件时,可能会出现的运动形式。     

Periodic viscous flow: A bench-mark problem

A double-transform technique provides a semi-analytic solution in the form of a series expansion for unsteady axisymmetric Stokes flow in the entrance region of a semi-infinite rigid cylindrical tube. This in turn offers an appropriate bench-mark problem for evaluating the quality of numerical approximations. To illustrate this, periodic axial flow in a circular cylinder is considered. Some aspects of the bench-mark problem that are of interest include the reverse flow in the wall layers, the accuracy of the approximate method in different flow regimes and the mesh grading. This bench-mark problem and the numerical study provide some insight into practical issues pertinent to the approximate solution of unsteady and periodic flows.     

Parametric excitation of waves on the surface of an inviscid liquid

A study is made of the stability of the equilibrium of the free surface of an infinite layer of inviscid incompressible liquid executing oscillations along the vertical axis. The problem is solved in the nonlinear formulation by series expansion with respect to the amplitude of the excitation. Soft and hard excitation regimes of the surface waves are obtained. The stability of the regimes is investigated. It is shown that the plane wave formed on the surface of the liquid is unstable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–75, September–October, 1982.I thank V. A. Briskman for suggesting the problem and for constant interest in the work and also A. A. Nepomnyashchii for discussing the results.     

Neimark-Sacker （N-S） in bifurcation of oscillator with dry friction 1：4 strong resonance

An oscillator with dry friction under external excitation is considered.The Poincaré map can be established according to the series solution near equilibrium in the case of 1:4 resonance.Based on the theory of normal forms,the map is reduced into its normal form.It is shown that the Neimark-Sacker(N-S) bifurcations may occour.The theoretical results are verified with the numerical simulations.     

Numerical modeling of vortex/shock wave interaction and its transformation by localized energy deposition

Three-dimensional unsteady Euler simulations are presented for the interaction of a streamwise vortex with an oblique shock of angle β = 23.3° at Mach 3 and 5. The flowfield features are analyzed for weak, moderate and strong interaction regimes. The details of the free recirculation zone at conditions of subsonic and supersonic flow on the vortex axis are considered. The vortex breakdown under conditions of a subsonic vortex core is characterized by a continuous growth and gradual degeneration of the region, unlike the supersonic core condition wherein a steady recirculation zone is achieved. The possibility of using a localized steady and pulsed periodic energy deposition on the vortex axis for stimulating the breakdown is demonstrated for various interaction regimes. It is shown that the formation of a subsonic wake downstream of an energy source lying on the vortex axis contributes to a more significant growth of the dimensions of the recirculation zone compared to the case when the vortex core remains supersonic. The possibility of achieving the effects similar to the steady case is demonstrated by the effect of a pulsed periodic energy source on the flow under consideration for corresponding equivalence parameters.      

Periodic and quasiperiodic galloping of a wind-excited tower under external excitation

The galloping of tall structures excited by steady and unsteady wind may be periodic or quasiperiodic (QP) with amplitudes having the same order of magnitude. While the onset of periodic and QP galloping was studied, their control on the other hand has received less attention. In this paper, we conduct analytical study on the effect of a fast harmonic excitation on the onset of periodic and QP galloping in the presence of steady and unsteady wind. We consider the cases where the unsteady wind activates either external excitation, parametric one or both. A perturbation analysis is performed to obtain close expressions of QP solution and the corresponding modulation envelopes. We show that at various loading situations, the periodic and QP galloping onset is significantly influenced by the amplitude of the fast external excitation. In the case where the unsteady wind activates parametric excitation, the QP galloping occurs with higher frequency modulation compared to the case where the unsteady wind activates external excitation. In the case where external and parametric excitations are activated simultaneously, fast harmonic excitation eliminates bistability in the amplitude response and gives rise to a new small QP modulation envelope.     

Convergent analytic solutions for homoclinic orbits in reversible and non-reversible systems

In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important non-linear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic settings. We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE.