1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Simulation and analysis of the Earth’s polar oscillations

The fundamental model considered in [2–9] is used to interpolate and forecast the polar motion by the least-squares method applied to IERS data [1]. On the basis of the fundamental model, we consider an extended model of the Earth’s polar motion taking into account oscillations due to lunar influence. It is established that high-frequency lunar oscillations can occur on the beats period and one can interpolate them on a short time interval. A comparative analysis of models taking into account the month and two-week frequencies is performed.     

Nonlinear oscillations of a charged drop levitated in gravity and electrostatic fields

Analytically, on the basis of asymptotic methods, the problem of the nonlinear oscillations of a charged ideal incompressible electroconductive fluid drop levitated at rest in gravity and homogeneous electrostatic fields is solved in the quadratic approximation in two small parameters: the initial drop shape deformation amplitude and the stationary eccentricity of the equilibrium drop shape in the electrostatic field. The calculations are performed in fractional powers of the nonlinear oscillation amplitude. The nonlinear corrections to the oscillation frequencies are always negative and already present in the second-order approximation due to the stationary deformation of the drop in the external fields rather than nonlinear interaction between the modes. In the case considered, in contrast to the nonlinear oscillations of a free charged drop, the expression for the generator of the nonlinearly oscillating drop shape contains terms proportional to the oscillation amplitude to the power 3/2.     

CHARACTERISTIC OF DROPLET OSCILLATION ON THE SURFACE OF RECTANGULAR HYDROPHOBIC GROOVES

液滴振荡行为是液滴运动中的重要伴随现象,具有重要科研价值.由于液滴撞击疏水沟槽板时运动行为与光滑表面明显不同,可以推测疏水沟槽表面液滴振荡特性也将会呈现与众不同的行为特点.采用高速摄像技术,研究了矩形疏水沟槽表面上水滴高度和接触线振荡行为随沟槽尺寸和撞击速度的变化规律.结果发现,矩形疏水沟槽造成的各向润湿异性使得振荡过程中水滴在平行沟槽方向上的接触线长度大于垂直方向,但并不影响水滴高度方向上衰减振荡的周期,即水滴振荡周期与沟槽间距无关;同时由于疏水沟槽表面上存在能垒束缚效应,致使水滴振荡过程中接触线的铺展和回缩运动不服从典型阻尼振荡规律,而呈现振荡数次后直接趋稳的特点.如水滴以0.61 m/s撞击时,接触线经历2次振荡后即维持稳定,但此时水滴仍在持续振荡中.另外,还初步分析了水滴振荡周期与沟槽间距无关的原因.     

Decay of longitudinal plasma oscillations to ion-sound oscillations

Recently, the question of the nonlinear relation between various plasma oscillations has been the subject of much attention as a result of a series of circumstances. The most important of these is the fact that in the majority of experiments on beam instabilities [1, 2] the intensity of the oscillations excited is very large, so that nonlinear effects in the interaction of oscillations must be significant. It should be noted that beam instability is not the only method of exciting highfrequency plasma oscillations. As was shown in [3], very intense oscillations may also be excited by beams of transverse waves of various frequency ranges, among which are powerful light beams [4]. Finally, excitation is possible by means of shock waves [5] and large-amplitude waves propagating through a plasma.Nonlinear coupling of plasma and low-frequency ion-sound oscillations leads, in particular, to the generation of the latter [6]. On the one hand, this is of interest as regards the problem of turbulent heating of a plasma, since the absorption of ion-sound oscillations in a plasma is usually stronger than the absorption of plasma oscillations. On the other hand, ion-sound oscillations may bring about the acceleration of low-energy ions due to the effects of induced erenkov absorption and radiation of waves by ions, as considered in the work of one of the authors [7], Although plasma oscillations accelerate particles more effectively [8], the injection conditions in the configuration for acceleration by plasma oscillations are very stringent v > v_e. The number of ions with such velocity for small ion temperatures T_i is small. Thus, the acceleration of ions will arise in this case as a result of the interaction of ion-sound oscillations until such time as their velocity reaches values of the order v_e. This question is of interest not only for the acceleration of ions (heating) in the presence of high-frequency turbulence created by beams of charged particles or as a result of the action of powerful radiation on a plasma, but also for the problem of neutron radiation from powerful impulse discharges in a plasma and for a series of astrophysical problems.In what follows we consider a number of one-dimensional self-consistent problems regarding the interaction (decay and fusion) of plasma and ion-sound oscillations resulting from the induced Raman scattering of the former by the latter. It is shown that the development of instability in a turbulent plasma with a high level of excited plasma oscillations leads both to the excitation of ion-sound oscillations, and also to the appearance in the plasma oscillation spectrum of satellites differing from the basic frequency _0e by a frequency of the order _0i and with greater intensities for the lower frequencies. The qualitative change of the plasma oscillation spectrum may serve as an immediate indication of the excitation of ion-sound oscillations in the system. The results obtained allow one to trace the process of development of instabilities. It is shown that in a plasma with a high level of ion-sound oscillations violet satellites are excited in the plasma oscillation spectrum, while the intensities of the violet satellites have a tendency to level out and form a satellite plateau if the level of ion-sound waves is high enough.     

Modelling of a resonant inertial oscillation within a torus

Summary  We consider the air contained in a pneumatic tyre with the purpose of investigating its inertial oscillations. We model the tyre as a torus limited by a membrane in contact with the ground. According to this model, we prove that the flow within this torus may be considered as one at low Mach number and that it is ruled by oscillations of incompressible rotating fluid. Investigating such inertial oscillations, we show that the geostrophic oscillation is resonant, and we study the resonance phenomenon. Received 6 June 2000; accepted for publication 22 November 2000     

Hyperchaotic beats and their collapse to the quasiperiodic oscillations

The letter shows the possibility of generation of hyperchaotic beats characterized by four, three or two positive Lyapunov exponents. The beats are a result of linear coupling of two identical nonlinear subsystems describing second-harmonic generation of light (SHG). The rapid transition from highly chaotic beats to quasiperiodic oscillations is studied.     

Wavelet decomposition method decoupled boiling/evaporation oscillation mechanisms over two to three timescales: A study for a microchannel with pin fin structure

Boiling/evaporation heat transfer in a microchannel with pin fin structure was performed with water as the working fluid. Simultaneous measurements of various parameters were performed. The chip wall temperatures were measured by a high spatial-time resolution IR image system, having a sensitivity of 0.02 °C. The flow pattern variations synchronously changed wall temperatures due to ultra-small Bi number. The wavelet decomposition method successfully identified the noise signal and decoupled various temperature oscillations with different amplitudes and frequencies. Three types of temperature oscillations were identified according to heat flux q and mass flux G. The first type of oscillation occurred at q/G < 0.62 kJ/kg. The approximation coefficient of wavelet decomposition decided the dominant cycle period which was ∼3 times of the fluid residence time in the microchannel, behaving the density wave oscillation characteristic. The detail coefficients of wavelet decomposition decided the dominant cycle period, which matched the flow pattern transition determined value well, representing the flow pattern transition induced oscillation. For the second type of oscillation, the wavelet decomposition decoupled the three oscillation mechanisms. The pressure drop oscillation caused the temperature oscillation amplitudes of 5–10 °C and cycle periods of 10–15 s. The density wave oscillation and flow pattern transition induced oscillation are embedded with both the pressure rise and decrease stages of the pressure drop oscillation. The third type of oscillation happened at q/G > 1.13 kJ/kg, having the density wave oscillation coupled with the varied liquid film evaporation induced oscillation. The liquid island, retention bubble induced nucleation sites and cone-shape two-phase developing region are unique features of microchannel boiling with pin fin structure. This study illustrated that pressure drop oscillation and density wave oscillation, usually happened in large size channels, also take place in microchannels. The flow pattern transition and varied liquid film evaporation induced oscillations are specific to microchannel boiling/evaporation flow.     

The Saddle-Node of Nearly Homogeneous Wave Trains in Reaction–Diffusion Systems

We study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. Beyond the stability of the primary homogeneous oscillations created in the bifurcation, we investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.     

Nonlinear resonances and self-excited oscillations of a rotor caused by radial clearance and collision

This paper investigates oscillations in a flexible rotor system with radial clearance between an outer ring of the bearing and a casing by experiments and numerical simulations. The mathematical model considers the collisions of the bearing with the casing. The following phenomena are found: (1) Nonlinear resonances of subharmonic, super-subharmonic and combination oscillation occur. (2) Self-excited oscillation of a forward whirling mode occurs in a wide range above the major critical speed. (3) Entrainment phenomena from self-excited oscillation to nonlinear forced oscillation occur at these nonlinear resonance ranges. Moreover, this study analyzes periodic solutions of the mathematical model by the Harmonic Balance Method (HBM). As the results, the nonlinear resonances of subharmonic oscillation and its entrainment phenomenon can be explained theoretically by investigating the stability of the periodic solutions. The influence of the static force and the bearing damping on these oscillation are also clarified.     

VORTEX SHEDDING RESONANCE FROM A ROTATIONALLY OSCILLATING CYLINDER

Vortex shedding resonance of a circular cylinder wake to a forced rotational oscillation has been investigated experimentally by measuring the velocity fluctuations in the wake, pressure distributions over the cylinder surface, and visualizing the flow field with respect to cylinder oscillations. The vortex shedding resonance occurs near the natural shedding frequency at small amplitude of cylinder oscillations, while the peak resonance frequency shifts to a lower value with an increase in oscillation amplitude. The drag and lift forces acting on the cylinder at fixed forcing Strouhal number indicate that the phase lag of fluid forces to the cylinder oscillations increases with an increase in oscillation amplitude, supporting the variation of resonance frequency with oscillation amplitude. The comparative study of the measured pressure distributions and the simultaneous flow visualizations with respect to cylinder rotation shows the mechanisms of phase lag, which is due to the strengthened vortex formation and the modification of the surface pressure distributions.     

Electron oscillations in a plasma

The conditions for periodicity of the spatial distribution of plasma oscillation intensity upon excitation by an electron beam are investigated.It is established that periodicity is observed in the presence of a boundary reflecting primary or emitting secondary electrons which form a reflected wave. If there is no such boundary, then the oscillations build up in accordance with the principles of the theory of convective plasma instability.Merrill and Webb [1] first discovered that in gas discharge tubes at low pressure, regions of intense plasma oscillation and regions scattering of primary electrons emitted from the cathode have a spatially periodic structure. A theory to account for these results was developed in [2] on the basis of the notion of velocity modulation of primary electrons on passage through an oscillating double layer at the boundary of the plasma and bunching near the focal plane (in much the same way as in a klystron). Later [3, 4] it was noted that periodicity of oscillation intensity distribution is observed in some cases, but not in others.In [5] it was shown that the presence of a second boundary electrode affects the nature of intensity distribution (the first boundary is the cathode) if it reflects primary electrons or emits secondary electrons and thus leads to the formation of reflected waves. When the interelectrode distance exceeded the mean free path of beam electrons in the gas, the oscillation distribution had the form of a smooth curve with a maximum somewhere in the central region between the electrodes. However, when the distance between electrodes is less than the mean free path, periodicity appears, as described in [1]. Theory [2] does not reflect this aspect of the phenomenon, since it assumes that the medium is bounded only on one side (semibounded) in the direction of motion of the beam.It is of great interest from the theoretical viewpoint to have information on the spatial distribution of the oscillations, since this makes it possible to draw conclusions concerning the nature of plasma instability as a result of which the oscillations are excited. In this article, experiments are described that confirm the role of reflected electrons in the formation of a periodic oscillation intensity. Moreover, data on the buildup of oscillations are presented and a comparison is made with the conclusions of convective instability theory.     

Sub/superharmonic oscillations and the perturbation procedure

The problem of sub/super and harmonic oscillations of non-linear systems is considered in this paper. This is made possible by the development of a modified version of the constrained Linstedt-Poincare perturbation procedure. The generality of the procedure is such that the existence of sub/super and harmonic oscillations can be simultaneously handled. Furthermore, the methodology developed is such that no a priori knowledge of solution form is necessary to establish the existence of such oscillation modes. Based on the approach, the steady response of Duffing's equation wherein sub/super and harmonic oscillations exist is studied in detail.     

Optimal control of the deployment (and retrieval) of a tethered satellite under small initial disturbances

The deployment and retrieval processes of satellites from a space station are demanding tasks during the operations of tethered satellite systems. The satellite should be steered into its working state within a reasonable amount of time and without too much control efforts. For the pure in-plane oscillation we have found time-optimal solutions with bang–bang control strategy for the deployment and retrieval process. In our working group we have also investigated different stabilization methods of the vertical equilibrium configuration, for example parametric swing control and chaotic control. In this article we concentrate on the final stage of the operation, when the oscillations around the vertical configuration should be brought to halt. While this task is quite simple for a motion of the satellite in the orbital plane, it is considerably more difficult, if the satellite has been perturbed out of that plane. We first analyze the control for a purely out-of-plane oscillation, which is governed by a Hamiltonian Hopf bifurcation, and then investigate the combined control for the spatial dynamics. Using a center manifold ansatz for the in-plane oscillations, we can show, that it is possible to diminish the oscillations of the tethered satellite in both directions, but the decay is extremely slow.     

Hydrodynamics in weak gravitational fields small oscillations of an ideal liquid in a cylindrical vessel

The problem of determining the frequencies and forms of small natural oscillations of an ideal liquid in a cylindrical vessel under conditions close to weightlessness is examined. It is assumed that a weak homogeneous gravitational field acts parallel to the vertical generatrix forming the cylinder. In contrast to [1], where only the first antisymmetric oscillation frequency is found for a semiinfinite cylindrical vessel, the frequencies of several axiosymmetric, antisymmetric, etc. oscillations are obtained as functions of the gravitational-field intensity and other parameters of the problem. The Ritz method is employed for two different variations of the problem, equivalent to that of oscillations of an ideal liquid under conditions of weightlessness [1–5].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–13, March–April, 1973.     

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

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High-speed imaging of an ultrasound-driven bubble in contact with a wall: “Narcissus” effect and resolved acoustic streaming

We report microscopic observations of the primary flow oscillation of an acoustically driven bubble in contact with a wall, captured with the ultra high-speed camera Brandaris 128 (Chin et al. 2003). The driving frequency is up to 200 kHz, and the imaging frequency is up to 25 MHz. The details of the bubble motion during an ultrasound cycle are thus resolved, showing a combination of two modes of oscillations: a radius oscillation and a translation oscillation, perpendicular to the wall. This motion is interpreted using the theory of acoustic images to account for the presence of the wall. We conclude that the bubble is subjected to a periodic succession of attractive and repulsive forces, exerted by its own image. Fast-framing recordings of a tracer particle embedded in the liquid around the particle are performed. They fully resolve the acoustic streaming flow induced by the bubble oscillations. This non-linear secondary flow appears as a tiny drift of the particle position cycle after cycle, on top of the primary back and forth oscillation. The high oscillation frequency accounts for a fast average particle velocity, with characteristic timescales in the millisecond range at the lengthscale of the bubble. The features of the bubble motion being resolved, we can apply the acoustic streaming theory near a wall, which provides predictions in agreement with the observed streaming velocity.     

Generation of Chaotic Beats in a Modified Chua's Circuit Part II: Circuit Design

This paper and the companion one [Cafagna, D., and Grassi, G., Nonlinear Dynamics, this issue] describe the new phenomenon of chaotic beats in a modified version of the Chua's circuit, driven by two sinusoidal inputs with slightly different frequencies. In particular, this paper presents the circuit design, including a novel implementation of the Chua diode, and investigates the beats phenomenon generated by the considered circuit.     

Stability of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure

Conclusions 1. The coupling of the vehicle oscillation equation and the equations for the currents in the guideway structures must be taken into account in the stability analysis of vertical oscillations in an electrodynamic suspension system with a discrete guideway structure.2. The reported calculations indicate that vertical oscillations are unstable over a wide range of travel velocities. Consequently, the execution of steady-state vehicle motion requires vertical stabilization of the system.3. Effective stabilization can be achieved by means of a dynamic vibration damper. Even for a small relative mass =0.025 of the optimally tuned damper the perturbed motion decays rapidly.Physicotechnical Problems Branch of the Institute of Geotechnical Mechanics, National Academy of Sciences of Ukraine, Dnepropetrovsk. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 88–93, July, 1995.