Synchronization effects in rotors partly filled with fluid

In fluid-filled rotors self-excited vibrations occur induced by a surface wave of the fluid. A characteristic property is the instability over the full range of angular velocity above the Eigenfrequency of the system. A possible explanation is the occurrence of synchronization effects between fluid and rotor. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable due to the complexity of fluid modelling. This paper analyses a simple minimal model and shows synchronization effects between fluid and rotor model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical model and optimum design approach of sinusoidal pressure wave generator for downhole drilling tool

Mud pulse generators have been widely used for the real-time transmission of valuable directional and formation data from downholes with depths of thousands of meters. There have been numerous studies on the design of mud pulse generators in which the pressure waves were typically nonsinusoidal. Sinusoidal waves provide improved long-distance data transmission and signal noise suppression compared with nonsinusoidal waves. Although sinusoidal pressure wave generators have been studied in the published literature, the influence of the risks of clogging on the design of the generator for producing sinusoidal pressure waves has rarely been considered. To generate sinusoidal pressure waves and to reduce the risks of clogging, a mathematical model for the design of a sinusoidal pressure wave generator is developed in this paper. The effects of the axial and radial clearances between the rotor and stator on the design of the generator are considered in the model. An optimum design method for the generator is provided by combining the developed model and a computational fluid dynamics analysis. Finally, an experimental platform was built and experiments at frequencies 2 Hz and 10 Hz were conducted to validate the design result. The simulation and experimental results show that the optimized pressure waves closely approximate sine waves. Therefore, the developed mathematical model and optimization approach can be used to design a sinusoidal pressure wave generator.     

Stability and bifurcation behaviour of a Laval-Rotor considering fluid forces in compliant liquid seals

This contribution discusses the influence of fluid forces, stemming from compliant, contact-free annular rotor seals, on the steady state stability and bifurcation behaviour of a rotor. The model used in this work consists of a Laval-Rotor where the disc runs in a turbulently streamed seal. The compliance of the seal is reduced to a visco-elastically supported outer seal ring. In order to account for the fluid seal forces the Childs-Hirs-model is used. An investigation of the eigenvalues shows that the compliance of the seal support may lead to a significant increase in the stable operating range. A stability-loss via Hopf-, Hopf-Hopf or secondary Hopf-Bifurcations can occur depending on the system parameters. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation behaviour and limit cycles of a compliant seal-rotor system

This contribution discusses the non-linear dynamic behaviour of a rotor system equipped with a compliant seal. The investigated model consists of a Laval-Rotor and a stiff seal ring which is visco-elastically supported. The fluid forces stemming from the turbulent incompressible lubricant film are accounted for by the non-linear Muszynska model. The added compliance leads to an improved stability behaviour of the steady state. Within the post-critical regime the additional compliance gives rise to bifurcations of the stationary vibration: Hopf bifurcations lead to limit cycles which can loose their stability via Neimark-Sacker bifurcations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blowout bifurcations in a model for chaos in spin-wave dynamics

Blowout bifurcations are investigated numerically in a model for chaos in the coincidence regime of high-power ferromagnetic resonance, based on interactions between the uniform mode and two pairs of parametric spin waves. This model possess two orthogonal invariant manifolds corresponding to the excitation of only one spin–wave pair above the first-order Suhl instability threshold. Marginal synchronization of the amplitudes of spin–wave pairs, the exchange of stability of the invariant manifolds, as well as both supercritical and subcritical blowout bifurcations are observed as the system parameters are varied, with the accompanying on–off intermittency, attractor bubbling and intermingled basins of attraction.     

Rotor dynamics behavior of turbo-expander involving droplet impact

ABSTRACT

This paper dedicates on the rotor dynamics behaviour research on the turbo-expander rotor system involving droplet impact. A stochastic model based on Beta distribution and Bernoulli distribution of droplet generation is established and the formulations of droplet impact forces are deduced, which is applied on the rotor dynamics equations of the tilting pad bearing supported turbo-expander considering the temperature gradient for a further analysis. A time domain research is carried out and a conclusion that continuously droplet impact will perturb the steady vibration of the turbo-expander rotor system is obtained. Monte Carlo method is implemented for a statistics dynamics research and the results suggests that in the design of expander impellers, in order to decrease the uncertainty brought by droplet impact, the number of channels should be as few as possible, the droplet impact should be controlled to occur uniformly, and the collision, entrainment of the primary droplets and the stripping of the liquid film on the blade should be strictly restrained.     

Dynamics of a rotor partially filled with a viscous incompressible fluid

A rotor partially filled with a viscous incompressible fluid is modeled as planar system. Its structural part, i. e. the rotor, is assumed to be rigid, circular, elastically supported and running with a prescribed time-dependent angular velocity. Both parts, structure and fluid, interact via the no-slip condition and the pressure. The point of departure for the mathematical formulation of the fluid filling is the Navier-Stokes equation, which is complemented by an additional equation for the evolution of its free inner boundary. Further, rotor and fluid are subjected to volume forces, namely gravitation. Trial functions are chosen for the fluid velocity field, the pressure field and the moving boundary, which fulfill the incompressibility constraint as well as the boundary conditions. Inserting these trial functions into the partial differential equations of the fluid motion, and applying the method of weighted residuals yields equations with time derivatives only. Finally, in combination with the rotor equations, a nonlinear system of 12 differential-algebraic equations results, which sufficiently describes solutions near the circular symmetric state and which may indicate the loss of its stability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries

The mathematical modelling and numerical simulation of the human cardiovascular system is playing nowadays an important role in the comprehension of the genesis and development of cardiovascular diseases. In this paper we deal with two problems of 3D modelling and simulation in this field, which are very often neglected in the literature. On the one hand blood flow in arteries is characterized by travelling pressure waves due to the interaction of blood with the vessel wall. On the other hand, blood exhibits non-Newtonian properties, like shear-thinning, viscoelasticity and thixotropy. The present work is concerned with the coupling of a generalized Newtonian fluid, accounting for the shear-thinning behaviour of blood, with an elastic structure describing the vessel wall, to capture the pulse wave due to the interaction between blood and the vessel wall. We provide an energy estimate for the coupling and compare the numerical results with those obtained with an equivalent fluid-structure interaction model using a Newtonian fluid.     

Whirl characteristics of a flexible liquid-filled rotor under thermal shock

In this paper, the whirl characteristics of a flexible liquid-filled rotor subjected to thermal shock are investigated. On the basis of the Hamilton principle, the whirl frequency equation of the rotor system is derived. Using Laplace transform, the analytical model of the temperature field of the rotor is obtained. The validity of the developed temperature model is demonstrated by comparing with the finite element results. Then, the thermal axial force exerted on the rotor is calculated and the influence factors are studied. The system stability is analyzed in terms of the whirl frequency equation. The reasonability of the predict model for system stability is verified, and a good agreement can be seen in the comparison of the obtained results based on the presented analytical method with published data. Finally, the critical spinning speed of the rotor system is analyzed, and the effects of some main parameters on system critical speed are investigated.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Chaotic synchronization of two mutually coupled semiconductor lasers for optoelectronic logic gates

A physical model of the fundamental configuration of two mutually coupled semiconductor lasers is presented for logic-gate applications, and the principles of optoelectronic logic computing based on chaotic synchronization or chaotic de-synchronization are defined. Two laser diodes were coupled via injection of each into the opposite laser and became chaotic; our analysis showed that the oscillation derives from chaotic fluctuations after a progression from stability to period-doubling by varying the coupling factor, delay time or detuning. Chaotic synchronization is achieved between the two lasers through the coupling, where we found chaotic and quasi-periodic synchronization regions. Based on the chaotic synchronization system, three optoelectronic logic gates can be implemented by modulating the laser diode current to synchronize or de-synchronize the two chaotic states. Finally, we studied the effects of resynchronization time on logic gate function in a practical implementation of the system. Numerical results show the validity and feasibility of the method.     

Nonlinear dynamic analysis of a hybrid squeeze-film damper-mounted rigid rotor lubricated with couple stress fluid and active control

The hybrid squeeze-film damper bearing with active control is proposed in this paper and the lubricating with couple stress fluid is also taken into consideration. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincaré maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the non-dimensional speed ratio s = 3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action. Besides, the rotor dynamic responses of the system will be more stable by using couple stress fluid.     

The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields

The effects of electric fields on the reaction fronts that arisein a system governed by an autocatalytic reaction and a complexationreaction between the autocatalyst and a complexing agent areconsidered. The complexation reaction is assumed to be fastrelative to the autocatalytic reaction and the equations forthis limit are derived. The corresponding travelling waves arediscussed, the case of quadratic autocatalysis being treatedin detail. The existence of minimum speed waves is examined,being dependent on the ratio of diffusion coefficients D, theconcentration S₀ and equilibrium constant K of the complexationreaction as well as the electric field strength E. It is seenthat, for some parameter values, minimum speed waves have negativeautocatalayst concentrations, and waves which have the lowestspeed consistent with non-negative concentrations are also obtained.Numerical integrations of the initial-value problem are performedfor representative parameter values. These show the developmentof the appropriate travelling wave (when it exists) as the largetime behaviour of the system, and, in cases where no travellingwave exists, the numerical integrations show the electrophoreticseparation of substrate and autocatalyst.     

Hydromagnetic stability of rotating stratified compressible fluid flows

The hydromagnetic stability of a radially stratified compressible fluid rotating between two coaxial cylinders is investigated. The stability with respect to axisymmetric disturbances is examined. The fluid system is found to be thoroughly stable to axisymmetric disturbances provided the fluid rotates very rapidly. The system is shown to be unstable to non-axisymmetric disturbances, and the slow amplifying hydromagnetic wave modes propagate against the basic rotation. The lower and upper bounds of the azimuthal phase speeds of the amplifying waves are determined. A quadrant theorem on the slow waves characteristic of a rapidly rotating fluid is derived. Special attention is given to the effects of compressibility of the fluid. Some results concerning the stabilitiy of an incompressible fluid system are obtained as special cases of the present analysis.

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Zusammenfassung Es wird die hydromagnetische Stabilität eines radial geschichteten kompressiblen Fluids untersucht, das zwischen zwei koaxialen Zylindern rotiert. Die Untersuchung der Stabilität in Bezug auf axisymmetrischen Störungen gibt volle Stabilität bei rascher Rotation. Für nicht axisymmetrische Störungen zeigt sich das System instabil, und die langsam angefachten hydromagnetischen Wellen pflanzen sich entgegen der Rotationsrichtung des Fluids fort. Es wird die untere und obere Grenze der azimutalen Phasengeschwindigkeit für diese Wellen bestimmt. Für die langsamen Wellen, charakteristisch für rasche Rotation des Fluids, wird ein Quadrant-Theorem hergeleitet. Die Kompressibilitätseffekte werden diskutiert, und Resultate für inkompressible Systeme ergeben sich als Sonderfälle.

     

Statistical investigation of the nonlinear dynamical behavior of offshore structures

The increasing amount of reports concerning damages of ships, structures and loss of cargo due to wave structure interaction demand the development of systems to predict critical situations in the offshore environment. Within the last years, research has been done to predict encounter with critical wave or wave groups such as “rogue waves” or the so–called “three sisters”. The aim of such research is to develop programs that predict dangerous incidents and possibly alert the crew in time. The dangers of such extreme wave situations are severe, but occurrence of them is rather rare. Additionally and more often, structures are endangered because of fluid–structure–interaction leading to critical dynamical system behaviour in a wave environment that shows no extreme wave heights. The severeness of these incidents then depends on the experience and correctness of decision concerning the evasive actions of the master and his crew. Taking nonlinear effects into account, the statistical investigation of structures in waves shows critical behaviour of ships and structures without the absolute necessity of heavy sea conditions or the occurrence of dangerous sea phenomena. This paper describes the development of a program that uses a Monte–Carlo–Simulation technique based on a common panel–method for the creation of added masses and added dampings to predict the behaviour of the structure in several wave conditions. It shows possible ways to prevent the occurrence using similar early warning systems to those in development for critical wavegroups. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Couple stress fluid improve rub-impact rotor-bearing system – Nonlinear dynamic analysis

This study performs a dynamic analysis of the rub-impact rotor supported by two couple stress fluid film journal bearings. The strong nonlinear couple stress fluid film force, nonlinear rub-impact force and nonlinear suspension (hard spring) are presented and coupled together in this study. The displacements in the horizontal and vertical directions are considered for various non-dimensional speed ratios. The numerical results show that the dynamic behaviors of the system vary with the dimensionless speed ratios, the dimensionless unbalance parameters and the dimensionless parameter, l^∗. Inclusive of the periodic, sub-harmonic, quasi-periodic and chaotic motions are found in this analysis. The results of this study contribute to a further understanding of the nonlinear dynamics of a rotor-bearing system considering rub-impact force existing between rotor and stator, nonlinear couple stress fluid film force and nonlinear suspension. We also prove that couple stress fluid used to be lubricant do improve dynamics of rotor-bearing system.     

Efficient non-hydrostatic modelling of surface waves interacting with structures

An efficient three-dimensional non-hydrostatic model is applied to simulate free-surface waves interacting with structures. The model employs an implicit Crank–Nicholson scheme to discretize the Navier–Stokes equations under a Cartesian staggered grid framework. An integration method is introduced to account for the full effects of non-hydrostatic pressure at the free-surface layer. A domain decomposition method is proposed to effectively solve the resulting matrix system. The model is first validated by simulating three-dimensional sloshing waves in a container. The model is then applied to simulate waves propagating over two-dimensional and three-dimensional submerged structures, in which the effects of non-linearity and dispersion are important. The model results show that the model using only two vertical layers are in all favorable agreements with experimental data, demonstrating the efficiency and accuracy of the model on simulating surface waves interacting with structures.