Transverse Oscillation Amplitude Research Articles

Parametric excitation of an immersed, vertical and slender beam using reduced-order models: influence of hydrodynamic coefficients

The paper addresses the effects of hydrodynamic parameters in the post-critical transversal vibration of an immersed, vertical, slender, extensible and flexible cylinder subjected to parametric excitation due to harmonic vertical displacements imposed at the upper extremity. The main objective of this research paper is to numerically investigate the influence of two hydrodynamic parameters, namely added mass and mean drag coefficients on the post-critical transversal oscillation amplitude. A single degree-of-freedom nonlinear reduced-order model is derived from the continuum beam equation of motion and numerically integrated using a Runge–Kutta scheme. Considering different values of added mass and mean drag coefficients, maps of post-critical amplitude are presented as functions of the amplitude and frequency of the parametric excitation. The results obtained are expected to be of importance to offshore engineering, particularly to riser and tether dynamics, since standard analysis approaches usually assume both hydrodynamic coefficients to be constant ab initio, no matter their values might vary significantly according to distinct design conditions.

Параметрические резонансы в задаче о продольном ударе по тонкому стержню

A longitudinal impact on a thin elastic rod generating a periodic system of longitudinal waves is addressed. Linear problem is considered and for certain values of parameters these waves are shown to generate parametric resonance accompanied by unbounded increase in the amplitude of transverse oscillations. In order to obtain the finite values of amplitudes we consider a quasi-linear system that takes into account the influence of transverse vibrations on the axial ones. In the earlier research this system was numerically solved by the Bubnov—Galerkin approach which resulted in beats displaying the exchange of energy between the axial and transverse oscillations. This paper presents an approximate analytic solution to this systems based on the two-scale expansions is constructed. A qualitative analysis is carried out, too. We obtained an estimate of the maximum transverse deflection depending upon the method of loading. Both the short-term and long-term axial loading are analysed. Intensive transverse vibrations are found out to appear in the case of a suddenly applied axial pulse with the magnitude below the critical Euler force. Refs 17. Figs 7. Tables 1.

Flow-induced vibrations of in-line cylinder arrangements at low Reynolds numbers

The present study numerically explores the limiting two degrees of freedom (streamwise and transverse) free oscillation response of three circular cylinders, placed in an in-line configuration subjected to a uniform cross flow at low Reynolds numbers. Three identical cylinders with a low mass ratio (m⁎=4/π) and zero damping are considered. The spacing between the cylinders L/D is equal to 4. The reduced velocity Ur is varied from 2 to 13 for two values of the Reynolds number (Re=100 and Re=150). For comparison purposes, the free oscillation responses of an isolated cylinder and a tandem cylinder pair under the same conditions are also evaluated. The results show that the dynamic behaviors of three in-line cylinders are significantly different from those of the tandem cylinder pair. While the maximum transverse oscillation amplitude increases by about 25%, there is now a very large streamwise oscillation amplitude (AX/D≈1.3), comparable to those in the transverse direction. They appear at Ur>9. The frequency responses of the triple cylinder case are much richer; in particular, at higher Ur. There is a clear low frequency component which is most evident in the streamwise direction. Many of the displacement trajectories of the triple cylinder case can almost be described as “bounded random movements”. The associated phase portraits and the Poincaré maps show that the dynamical characteristics of the triple cylinder configuration are considerably richer than those of the tandem cylinder pair. Even at such low Re, the free oscillations of three in-line cylinders already seem to approach a chaotic response. In particular, there is evidence that the fluid–structure system approaches to chaos via the quasi-periodic route. Due to such much more complex dynamical characteristics, it is therefore highly risky to predict the free oscillation behaviors of multiple in-line cylinders by extrapolating those of the tandem cylinder pair.

Flow-induced vibrations of two circular cylinders in tandem with shear flow at low Reynolds number

The flow-induced vibrations of two elastically mounted circular cylinders subjected to the planar shear flow in tandem arrangement are studied numerically at Re=160. A four-step semi-implicit Characteristic-based split (4-SICBS) finite element method is developed under the framework of the fractional step method to cope with the vortex-induced vibration (VIV) problem. For the computational code verification, two benchmark problems are examined in the laminar region: flow-induced vibration of an elastically mounted cylinder having two degrees of freedom and past two stationary ones in tandem arrangement. Regarding the two-cylinder VIVs in shear flow, the computation is conducted with the cylinder reduced mass Mr=2.5π and the structural damping ratio ξ=0.0. The effects of some key parameters, such as shear rate (k=0.0, 0.05, 0.1), reduced velocity (Ur=3.0–18.0) and spacing ratio (Lx/D=2.5, 3.5, 4.5, 8.0), are demonstrated. It is observed that the shear rate and reduced velocity play an important role in the VIVs of both cylinders at various center-to-center distances. Additionally, in comparison with the single cylinder case, a further study indicated that the gap flow has a significant impact on such a dynamic system, leading it to be more complex. The results show that, the performances of ‘dual-resonant’ are discovered in the shear flow. A valley is formed in transverse oscillation amplitude of DC for each spacing ratio when Ur is about 6.0. For the X–Y trajectories of the circular cylinders, figure-eight, figure-O and oval shape are obtained. Finally, the interactions between cylinders are revealed, together with the wake-induced vibration (WIV) mechanism underlying the oscillation characteristics of both cylinders exposed to shear flow. Besides, the “T+P” wake pattern is discovered herein.

FLOWS AND WAVES IN BRAIDED SOLAR CORONAL MAGNETIC STRUCTURES

We study the high frequency dynamics in the braided magnetic structure of an active region (AR 11520) moss as observed by High-Resolution Coronal Imager (Hi-C). We detect quasi periodic flows and waves in these structures. We search for high frequency dynamics while looking at power maps of the observed region. We find that shorter periodicites (30 - 60 s) are associated with small spatial scales which can be resolved by Hi-C only. We detect quasi periodic flows with wide range of velocities from 13 - 185 km/s associated with braided regions. This can be interpreted as plasma outflows from reconnection sites. We also find presence of short period and large amplitude transverse oscillations associated with braided magnetic region. Such oscillations could be triggered by reconnection or such oscillation may trigger reconnection.

Numerical Simulation of Three-Dimensional Flow Past a Cylinder Oscillating at the Strouhal Frequency

The paper presents computational results of 3D flow past a cylinder forced to oscillate: (a) transversely with respect to a uniform stream and (b) both transversely and in-line with respect to a uniform stream, following a figure-eight trajectory. For a flow from left to right the figure-eight is traversed counterclockwise in the upper half-plane. Direct numerical simulation (DNS) of the Navier–Stokes equations for 3D flow is performed using a spectral element code. Computations are carried out for a Reynolds number equal to 400, at a transverse oscillation frequency equal to the natural frequency of the Kármán vortex street. For both oscillation modes, the transverse oscillation amplitude is varied from 0 to 0.60 cylinder diameters. The forces on the cylinder are calculated and related to flow structure in the wake. The results indicate that, in general, the presence of in-line oscillation increases the magnitude of forces acting on the cylinder, as well as the power transfer from the flow to the structure. Flow visualizations indicate that, for the figure-eight mode, low-amplitude forcing tends to reduce the wake three-dimensionality. However, at high oscillation amplitudes, the wake structure is found to become more complex at increasing amplitude.

The appearance of P+S modes in the wake of a freely vibrating, highly flexible cylinder

We present visual evidence of a stable P+S (vortex pair and singlet) wake mode in the free vibration of a highly flexible, cantilevered cylinder. Our experiments cover a reduced velocity range of 17–91, and a Reynolds number range of 70–363, over which the flow remains laminar. The P+S wake appears at certain spanwise locations, coexisting with a 2S (2 singlet) wake at other locations. The presence of a co-existing P+S mode sometimes occurs with high amplitude transverse oscillations of the cylinder, but the magnitude of oscillations is not necessarily predictive of wake modes. Hysteresis in the transverse amplitudes occurs with increasing or decreasing reduced velocity. The hysteresis region indicates the lowest range of reduced velocity over which P+S wakes are found, though the P+S wake modes occur with the lower amplitude bound of the hysteresis region, rather than the larger amplitude bound.

Dynamics of three simultaneously stored beams in a storage ring

In this study, the observation of three simultaneously rotating beams in the SOLEIL storage ring is reported. This event occurred in November 2007 while operating in a low momentum compaction factor mode. The dynamics of the three beams is simulated using the longitudinal equations of motion and the longitudinal phase-space Hamiltonian which are extended to include higher-order terms of momentum compaction factor up to the third order. The effect of the transverse oscillation amplitude of the particles is also included in this work. It is shown that this term, which is experimentally very difficult to compensate for very low momentum compaction factor optics, can strongly affect the longitudinal beam dynamics. Finally, an extended formula of the variation of synchrotron frequency with respect to the relative variation of the rf frequency is derived and is used to deduce the higher-order terms of the momentum compaction factor from experimental data.

Free vibrations of a cylinder: 3-D computations at Re=1000

Vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m⁎=10.0) are investigated at Re=1000. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables in three dimensions. Computations are carried out for reduced velocities in the range 4.2≤U⁎≤6.2 for, both, increasing and, decreasing U⁎. Three branches are identified for the cylinder response: initial, upper and lower. The cylinder response and force coefficients exhibit beats in the initial branch. The initial–upper branch transition is hysteretic. The maximum amplitude of transverse oscillations is 0.7D. This value is larger than that observed in the laminar regime and smaller than that reported earlier for large Re. Intermittency is observed in the transition between upper and lower branches. During the transition, the fraction of time spent by the cylinder in lower branch increases with U⁎. Soft-lock-in is observed on the upper branch. Although the frequency of vortex shedding and the cylinder vibration match, they are different than the natural frequency of the spring-mass system. The mode of vortex shedding is 2S on the initial branch while it is 2P along the lower branch. Perhaps, for the first time, 2S mode of vortex shedding is found along the upper branch. This is confirmed via investigation of phase difference between the lift and cylinder response.

Damped large amplitude transverse oscillations in an EUV solar prominence, triggered by large-scale transient coronal waves

Aims. We investigate two successive trains of large amplitude transverse oscillations in an arched EUV prominence, observed with SoHO/EIT on the north-east solar limb on 30 July 2005. The oscillatory trains are triggered by two large scale coronal waves, associated with an X-class and a C-class flare occurring in the same remote active region. Methods. The oscillations are tracked within rectangular slits parallel to the solar limb at different heights, which are taken to move with the apparent height profile of the prominence to account for solar rotation. Time series for the two prominence arch legs are extracted using Gaussian fitting on the 195 angstrom absorption features, and fitted to a damped cosine curve to determine the oscillatory parameters. Results. Differing energies of the two triggering flares and associated waves are found to agree with the velocity amplitudes, of 50.6 +/- 3.2 and 15.9 +/- 8.0 km s(-1) at the apex, for the first and second oscillatory trains respectively, as estimated in the transverse direction. The period of oscillation is similar for both trains, with an average of 99 +/- 11 min, indicating a characteristic frequency as predicted by magnetohydrodynamics. Increasing velocity amplitude with height during the first oscillatory train, and in-phase starting motions of the two legs regardless of height, for each train, demonstrate that the prominence exhibits a global kink mode to a first approximation. However, discrepancies between the oscillatory characteristics of the two legs and an apparent dependence of period upon height, suggest that the prominence actually oscillates as a collection of separate but interacting threads. Damping times of around two to three cycles are observed. Combining our results with those of previously analysed loop oscillations, we find an approximately linear dependence of damping time upon period for kink oscillations, supporting resonant absorption as the damping mechanism despite limitations in testing this theory.

Controlling the betatron oscillations of a wakefield-accelerated electron beam by temporally asymmetric laser pulses

Based on two-dimensional particle-in-cell simulations, we investigated the electron beam’s transverse oscillations by temporally asymmetric laser pulses in laser wakefield acceleration. Of particular interest in this article are the effects of ultrashort laser pulses having sharp rising and slow falling time scales. In this situation, the accelerated electron beam interacts directly with the laser field and undergoes transverse oscillations due to a phase-slip with the laser field. This oscillation can be matched with the betatron oscillation due to the focusing force of the ions, which can lead to a large transverse oscillation amplitude due to the resonance between them. Furthermore, in this case, the electron beam can be microbunched at the laser wavelength, which may provide the possibility for generation of a coherent synchrotron radiation.

Vortex-induced vibration of two circular cylinders at low Reynolds number

Abstract Vortex-induced vibration of a pair of equal-sized circular cylinders in tandem and staggered arrangements in laminar flow regime is investigated. A stabilized finite element method is utilized to carry out the computations in two dimensions. Both cylinders are free to oscillate in transverse as well as in-line directions. The Reynolds number, based on the free-stream speed, U, and the diameter, D, of the cylinders is 100. To encourage high amplitude of oscillation the structural damping is set to zero and cylinders of low nondimensional mass are considered ( m * = 10 ) . The computations are carried out for various values of reduced speed of the oscillator ( 2 ⩽ U * ⩽ 15 ) . The cylinders are separated by 5.5 D in the streamwise direction. They are separated by 0.7 D in the cross-flow direction to study the effect of stagger. The downstream cylinder lies in the wake of the upstream one and experiences an unsteady inflow. The upstream cylinder in both tandem and staggered arrangement responds qualitatively similarly to a single cylinder. Compared to an isolated cylinder, a small increase in transverse oscillation amplitude of the upstream cylinder is observed due to the presence of the downstream cylinder. In both arrangements, the downstream cylinder shows very large amplitude transverse oscillations comparable to that of a single cylinder at higher Re. In the staggered arrangement, very large streamwise oscillations of the downstream cylinder are observed. Compared to an isolated cylinder, the synchronization range for the two-cylinder arrangement is larger. The downstream cylinder in the staggered arrangement undergoes two types of motion: an orbital motion at most of the U * studied, and figure-of-eight motion for a small range of U * . In the tandem arrangement, only the figure-of-eight motion is observed. The stagger in the arrangement of the two cylinders is found to have a significant effect on the flow.

Betatron radiation from density tailored plasmas

In laser wakefield accelerators, electron motion is driven by intense forces that depend on the plasma density. Transverse oscillations in the accelerated electron orbits produce betatron radiation. The electron motion and the resulting betatron radiation spectrum can therefore be controlled by shaping the plasma density along the orbit of the electrons. Here, a method based on the use of a plasma with a longitudinal density variation (density depression or step) is proposed to increase the transverse oscillation amplitude and the energy of the electrons accelerated in a wakefield cavity. For fixed laser parameters, by appropriately tailoring the plasma profile, the betatron radiation emitted by these electrons is significantly increased in both flux and energy.

Effect of blockage on free vibration of a circular cylinder at low Re

AbstractThe effect of the blockage on vortex‐induced vibrations of a circular cylinder of low non‐dimensional mass (m*=10) in the laminar flow regime is investigated in detail. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. The transverse response of the cylinder is found to be hysteretic at both ends of synchronization/lock‐in region for 5% blockage. However, for the 1% blockage hysteresis occurs only at the higher Re end of synchronization/lock‐in region. Computations are carried out at other blockages to understand its effect on the hysteretic behavior. The hysteresis loop at the lower Re end of the synchronization decreases with decrease in blockage and is completely eliminated for blockage of 2.5% and less. On the other hand, hysteresis persists for all values of blockage at the higher Re end of synchronization/lock‐in. Although the peak transverse oscillation amplitude is found to be same for all blockage (∼0.6D), the peak value of the aerodynamic coefficients vary significantly with blockage. The r.m.s. values show lesser variation with blockage. The effect of streamwise extent of computational domain on hysteretic behavior is also studied. The phase between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and is independent of blockage. Copyright © 2008 John Wiley & Sons, Ltd.

Thin crystal layers in superstrong laser fields: Dynamics and coherent x-ray generation

Coherent x-ray generation is investigated for a thin solid layer or any other periodic atomic structure with limited dimensions exposed to a short, superintense laser pulse. We single out a selective range of parameters at which the impact of nonlinear collective processes on ion dynamics is reduced. This allows us to consider an ionic structure that is quasiregular during the brief interaction, which we prove also by particle-in-cell simulations. In the relativistic regimes of the interaction, when the transversal oscillation amplitude of the electron in the laser field exceeds the layer transversal width, the multiphoton scattering of electrons dominates over recombination to bound states. Electrons accelerated by the laser field penetrate through the periodic ionic structure, emitting coherent bremsstrahlung. Hard x rays are shown to amplify along the interaction length due to stimulated coherent bremsstrahlung. The high gain of the scheme compensates for the short interaction length, yielding a gain-length product competitive with other coherent x-ray sources. Limitations on the gain due to the energy spread of the electrons and the short time of the interaction are estimated.

Vortex induced loads on marine risers

The Vortex-Induced Vibration on a circular cylinder is investigated as basis for applications on dynamics of risers used in offshore oil and gas industry. The numerical solution of the two-dimensional Reynolds Averaged Navier–Stokes equations is obtained for a circular cylinder laterally supported by a spring and a damper and free to oscillate in the transversal direction. The Beam and Warming implicit factorization scheme is used to solve the governing equations and the Baldwin–Lomax model is used to simulate the turbulent flow in the wake of the cylinder. The proposed numerical solution was able to provide a good picture of the real physics of the phenomenon including the Kármán vortex street effects on the lift and drag coefficients. The numerical results for the transversal oscillation amplitude are compared to experimental data showing a fairly precise agreement even at the difficult to simulate regime of the lock-in phenomenon.

Numerical Investigation of Vortex Induced Vibration on an Elastically Mounted Cylinder

The Vortex-Induced Vibration on a circular cylinder is investigated by the numerical solution of the two-dimensional Reynolds Averaged Navier-Stokes equations. Results are compared with experimental measurements obtained by different author to verify the quality of the numerical solution. The Beam and Warming implicit factorization scheme is used to solve the governing equations and the Baldwin - Lomax model is used to simulate the turbulent flow in the wake of the cylinder. The cylinder is laterally supported by a spring and a damper and free to oscillate in the transversal direction. The numerical results showed that the curve of transversal oscillation amplitude presents a quite different behaviour as the data is obtained with increasing or decreasing velocity, characterizing a hysteresis effect. The best agreement with experiment is obtained as the numerical results are obtained with decreasing velocity.

Hot coronal loop oscillations observed with SUMER: Examples and statistics

We give an extensive overview of Doppler shift oscillations in hot active region loops obtained with SUMER. The oscillations have been detected in loops sampled 50-100 arcsec off the limb of the Sun in ultraviolet lines, mainly and , with formation temperature greater than 6 MK. The spectra were recorded along a 300 arcsec slit placed at a fixed position in the corona above the active regions. Oscillations are usually seen along an extended section of the slit and often appear to be from several different portions of the loops (or from different loops). Different portions are sometimes in phase, sometimes out of phase and sometimes show phase shifts along the slit. We measure physical parameters of 54 Doppler shift oscillations in 27 flare-like events and give geometric parameters of the associated hot loops when soft X-ray (SXR) images are available. The oscillations have periods in the range 7-31 min, with decay times 5.7-36.8 min, and show an initial large Doppler shift pulse with peak velocities up to 200 km s-1. The oscillation periods are on average a factor of three longer than the TRACE transverse loop oscillations. The damping times and velocity amplitude are roughly the same, but the derived displacement amplitude is four or five times larger than the transverse oscillation amplitude measured in TRACE images. Unlike TRACE oscillations, only a small fraction of them are triggered by large flares, and they often recur 2-3 times within a couple of hours. All recurring events show initial shifts of the same sign. These data provide the following evidence to support the conclusion that these oscillations are slow magnetoacoustic standing waves in hot loops: (1) the phase speeds derived from observed periods and loop lengths roughly agree with the sound speed; (2) the intensity fluctuation lags the Doppler shifts by 1/4 period; (3) The scaling of the dissipation time of slow waves with period agrees with the observed scaling for 49 cases. They seem to be triggered by micro- or subflares near a footpoint, as revealed in one example with SXR image observations. However other mechanisms cannot as yet be ruled out. Some oscillations showed phase propagation along the slit in one or both directions with apparent speeds in the range of 8-102 km s-1, together with distinctly different intensity and line width distributions along the slit. These features can be explained by the excitation of the oscillation at a footpoint of an inhomogeneous coronal loop, e.g. a loop with fine structure.

Ion-hose instability in a long-pulse linear induction accelerator

The ion-hose instability is a transverse electrostatic instability which occurs on electron beams in the presence of a low-density ion channel. It is a phenomenon quite similar to the interaction between electron clouds and proton or positron beams in high-energy accelerators and storage rings. In the DARHT-2 accelerator, the 2-kA, $2\mathrm{\text{\ensuremath{-}}}\ensuremath{\mu}\mathrm{s}$ beam pulse produces an ion channel through impact ionization of the residual background gas (${10}^{\ensuremath{-}7}--{10}^{\ensuremath{-}6}\text{ }\mathrm{torr}$). A calculation of the linear growth by Briggs indicates that the instability could be strong enough to affect the radiographic application of DARHT, which requires that transverse oscillations be small compared to the beam radius. We present semianalytical theory and 3D particle-in-cell simulations (using the Lsp code) of the linear and nonlinear growth of the instability, including the effects of the temporal change in the ion density and spatially decreasing beam radius. We find that the number of $e$-foldings experienced by a given beam slice is given approximately by an analytic expression using the local channel density at the beam slice. Hence, in the linear regime, the number of $e$-foldings increases linearly from head to tail of the beam pulse since it is proportional to the ion density. We also find that growth is strongly suppressed by nonlinear effects at relatively small oscillation amplitudes of the electron beam. This is because the ion oscillation amplitude is several times larger than that of the beam, allowing nonlinear effects to come into play. An analogous effect has recently been noted in electron-proton instabilities in high-energy accelerators and storage rings. For DARHT-2 parameters, we find that a pressure of $\ensuremath{\le}1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\mathrm{torr}$ is needed to keep the transverse beam oscillation amplitude less than about 20% of the rms beam radius.

Remarks on the non-linear vibration of an axisymmetric circular disk near critical speed

We re-examine the large amplitude transverse oscillations of axisymmetric disks spinning near a critical speed resonance as considered by Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139). Averaged equations in traveling wave based coordinates are shown to provide a transparent explanation for certain bifurcations presented in Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139). The theoretical results in Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139) are proved analytically, and an enhanced interpretation of their results is presented.