On the notion of a rotating fluid force induced by swirling flow

The Bently/Muszynska (B/M) model shows that oil whirl and oil whip are both self-sustained vibrations associated with two unstable modes of a rotor–fluid system. The model includes a rotating fluid damping and inertia force. In certain configurations, the rotating damping force overcomes the frictional internal damping of the rotor and pushes the rotor into a stable limit cycle of circular orbiting. Such a notion of a rotating fluid force is based on bulk-flow models of fluid-filled clearances that could be approximated as narrow since the tangential velocity of the fluid then translates to one angular velocity at a certain radial distance defined by an average radius. This paper scrutinizes the assumption of a rotating fluid inertia force and pinpoints the additional inertial effects of the swirling flow as the gap width increases. These effects are clarified by deriving the equation of motion of a body with a mass subjected to motion-induced fluid forces of a confined swirling flow. We show that the inertial effects of the swirling flow counteract the destabilizing effect of the rotating damping force. However, if the body mass is larger than the displaced fluid mass, instability follows. The frequency of the unstable mode is unchanged by the additional inertial effects and is always equal to the frequency of the damping that induces the instability.     

Stability in parametric resonance of axially accelerating beams constituted by Boltzmann's superposition principle

Stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated. The governing equation is derived from Newton's second law, Boltzmann's superposition principle, and the geometrical relation. When the axial speed is a constant mean speed with small harmonic variations, the governing equation can be treated as a continuous gyroscopic system with small periodically parametric excitations and a damping term. The method of multiple scales is applied directly to the governing equation without discretization. The stability conditions are obtained for combination and principal parametric resonance. Numerical examples demonstrate that the increase of the viscosity coefficient causes the lager instability threshold of speed fluctuation amplitude for given detuning parameter and smaller instability range of the detuning parameter for given speed fluctuation amplitude. The instability region is much bigger in lower order principal resonance than that in the higher order.     

Parametric instability of a cantilevered column with end mass

Parametric instability regions in the first spatial and temporal modes of cantilevered columns having longitudinal inertia and end mass have been determined experimentally and analytically in the parametric space. Experiments have been conducted with three different columns, each having a set of five end masses. Analytical investigation has been carried out using small deflection beam theory. Under certain assumptions, the governing differential equation reduces to a modified Mathieu-Hill type equation, which gives the stability criteria. Effects of disturbances on parametric resonance have been observed experimentally and explained analytically.     

Parametric resonance of truncated conical shells rotating at periodically varying angular speed

Parametric resonance of a truncated conical shell rotating at periodically varying angular speed is studied in this paper. Based upon the Love?s thin shell theory and generalized differential quadrature (GDQ) method, the equations of motion of a rotating conical shell are derived. The time-dependent rotating speed is assumed to be a small and sinusoidal perturbation superimposed upon a constant speed. Considering the periodically rotating speed, the conical shell system is a parametric excited system of the Mathieu–Hill type. The improved Hill?s method is utilized for parametric instability analysis. Both the primary and combination instability regions for various natural modes and boundary conditions are obtained numerically. The effects of relative amplitude and constant part of periodically rotating speed and cone angle on the instability regions are discussed in detail. It is shown that for the natural mode with lower circumferential wavenumber, only the primary instability regions exist. With the increasing circumferential wavenumber, the instability widths are reduced significantly and the combination instability region might appear. The results for different boundary conditions are substantially similar. Increasing the constant rotating speed (or cone angle) all lead to the movements of instability regions and the appearance of combination instability region. The former will cause the instability width increasing, while the latter will reduce the instability width. The variation of length-to-radius ratio only causes the movements of instability regions.     

A SIMPLE MODEL TO ESTIMATE THE IMPACT FORCE INDUCED BY PISTON SLAP

The dynamics of piston's secondary motion (lateral and rotational motion) across the clearance between piston and cylinder inner wall of reciprocating machines are analyzed. This paper presents an analytical model, which can predict the impact forces and vibratory response of engine block surface induced by the piston slap of an internal combustion engine. A piston is modelled on a three-degree-of-freedom system to represent its planar motion. When slap occurs, the impact point between piston skirt and cylinder inner wall is modelled on a two-degree-of-freedom vibratory system. The equivalent parameters such as mass, spring constant, and damping constant of piston and cylinder inner wall are estimated by using measured (driving) point mobility. Those parameters are used to calculate the impact force and for estimating the vibration level of engine block surfaces. The predicted results are compared with experimental results to verify the model.     

Parametric response of viscoelastically supported beams

The stability of viscoelastic beams with an attached mass and viscoelastic end supports under axial and tangential periodic loads is investigated. Viscoelastic end supports are substituted for translational and rotational springs with viscoelastic damping. The regions of instability for simple and combination resonances are obtained from the ordinary Mathieu equation which is obtained from the equation of motion by application of Galerkin's method. In numerical computations, the influences of the direction of loading, the attached mass, the support stiffness, and the damping on the regions of instability for simple and combination resonances are clarified.     

Analytical investigation on unstable vibrations of single-mesh gear systems with velocity-modulated time-varying stiffness

Time-varying mesh stiffness parametrically excites gear systems and causes severe vibrations and instabilities. Taking speed fluctuations into account, the time-varying mesh stiffness is frequency modulated, and more complex instabilities might arise. Considering two different speed fluctuation models, parametric instability associated with velocity-modulated time-varying stiffness is analytically investigated using a typical single-mesh gear system model. Closed-form approximations are obtained by perturbation analysis, and verified by numerical analysis. The effects of the amplitude of the mesh stiffness variation, the characteristics of speed fluctuations and damping on parametric instability are systematically examined.     

Dynamic stability of thin walled beams under traveling follower load systems

It is found that the perturbation equation of motion of a thin walled beam under a traveling follower load system becomes Hill's equation and that parametrically excited unstable coupled vibration occurs. The boundary frequency equations of the simple parametric resonance, from which the unstable regions are estimated, are obtained by Bolotin's method. Stability maps of a simply supported beam are shown, with account taken of the effects of load mass and damping.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

逃逸电子和不稳定性波反常多普勒共振实验研究

利用电子回旋辐射诊断系统并结合其他相关诊断研究了HL-2A托卡马克中逃逸电子与波间的反常多普勒共振作用.结果显示:欧姆放电下提高等离子密度能抑制逃逸电子束的不稳定性,但等离子密度的再次降低导致逃逸电子又会激发不稳定性波,并耦合不稳定性波发生二次反常多普勒共振作用.利用统计方法分析了HL-2A上不同放电阶段逃逸电子反常多普勒共振阈值(ωpe/ωce)区间大致都在0.17-0.54范围内.此共振机制导致逃逸电子在速度空间被波散射,平行能量转化到垂直能量,pitch角增加,同步辐射功率增强,逃逸电子能量限制在反常多普勒效应的阈值能量附近.基于反常多普勒共振的逃逸抑制能有效减轻逃逸电子对装置第一壁的损坏.     

Internal resonance in a plane system of rods

The transverse vibrations of a plane system of rods is considered. The analysis of internal resonance in the system is a primary purpose of the paper. The internal resonance analyzed has an autoparametric nature. The couplings of the elements of the system through internal longitudinal forces, which are transverse forces at the ends of neighbouring rods, are taken into account. The amplitudes of the vibrations in the stationary states of internal resonance are investigated. Non-linear terms appear in the equations of motion. These terms are non-linear damping and non-linear inertia, and have a geometrical nature. The approximate method of calculation gives formulae for the vibration amplitudes of the rods. Plots of the amplitudes against frequency are presented. The stabilizing effect of masses placed at the articulated joints of the system is shown. The influences of the inertia and damping values on the character of the curves is considered. The results obtained are of a qualitative character.     

The effects of fluid motion on oscillatory and chaotic fronts

We study oscillatory and chaotic reaction fronts described by the Kuramoto-Sivashinsky equation coupled to different types of fluid motion. We first apply a Poiseuille flow on the fronts inside a two-dimensional slab. We show regions of period doubling transition to chaos for different values of the average speed of Poiseuille flow. We also analyze the effects of a convective flow due to a Rayleigh-Taylor instability. Here the front is a thin interface separating two fluids of different densities inside a two-dimensional vertical slab. Convection is caused by buoyancy forces across the front as the lighter fluid is under a heavier fluid. We first obtain oscillatory and chaotic solutions arising from instabilities intrinsic to the front. Then, we determine the changes on the solutions due to fluid motion.     

Dynamic stability of rotating flexible disk perturbed by the reciprocating angular movement of suspension-slider system

To simulate the dynamic process of a magnetic head reading/writing data in a hard disk drive, a rotating flexible thin disk perturbed by the reciprocating angular movement of a suspension-slider system is modelled, where the suspension-slider system is considered as a mass-damping-spring loading system. A system dynamic model is formulated as a parametrically excited system, and its dynamic stability is studied by Hill’s method involving harmonic balance. The reciprocating angular movement of the suspension-slider system causes system parametric instability at some angular movement frequencies. The large-amplitude angular movement is especially dangerous, and angular movement frequency must be reduced when the slider works at large radii of the disk. The parametric instability can be avoided or suppressed by operating at: low-frequency and small-amplitude reciprocating angular movement, small mass, large natural frequency and damping of the suspension-slider system, and low-speed rotation of the disk.     

Random excitation of a vibratory system with autoparametric interaction

The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.     

Stability analysis of a rotor–bearing system with time-varying bearing stiffness due to finite number of balls and unbalanced force

Previous investigations have indicated that the finite number of balls can cause the bearing stiffness to vary periodically. However, effects of unbalanced force in a rotor–bearing system on the bearing stiffness have not received sufficient attention. The present work reveals that the unbalanced force can also make the bearing stiffness vary periodically. The parametric excitations from the time-varying bearing stiffness can cause instability and severe vibration under certain operating conditions. Therefore, the determination of the operating conditions of parametric instability is crucial to the design of high speed rotating machinery. In this paper, an extended Jones–Harris stiffness model is presented to ascertain the stiffness of the angular contact ball bearing considering five degrees of freedom. Stability analysis of a rigid rotor–bearing system is performed utilizing the discrete state transition matrix (DSTM) method. The effects of unbalanced force, bearing loads and damping on the instability regions are discussed thoroughly. Investigations mainly show that the time-varying bearing stiffness fluctuates sinusoidally due to finite number of balls and unbalanced force. The locations and widths of the instability regions caused by these two parametric excitations differ distinctly. Unbalanced force could change the widths of the instability regions, but without altering their central positions. The axial and radial loads of the bearing only change the positions of the instability regions, without affecting their widths. Besides, damping can reduce the widths of the instability regions.     

Dynamics of a single particle in a Paul trap in the presence of the damping force

The motion of a single charged particle in a Paul trap in the presence of the damping force is investigated theoretically and the modified stability diagrams in the parameter space are calculated. The results show that the stable regions in thea–q parameter plane are not only enlarged but also shifted. Consequently, the damping force causes instability in some cases, contrary to intuition. As a by-product of the calculation, we derive new theoretical approximate expressions for the secular-oscillation frequency. In the limiting case of no damping, these formulas are in good agreement with early measurements done by Wuerker et al.     

Dynamic stability of a slender beam under horizontal–vertical excitations

The dynamic stability of a vertically standing cantilevered beam simultaneously excited in both horizontal and vertical directions at its base is studied theoretically. The beam is assumed to be an inextensible Euler–Bernoulli beam. The governing equation of motion is derived using Hamilton's principle and has a nonlinear elastic term and a nonlinear inertia term. A forced horizontal external term is added to the parametrically excited system. Applying Galerkin's method for the first bending mode, the forced Mathieu–Duffing equation is derived. The frequency response is obtained by the harmonic balance method, and its stability is investigated using the phase plane method. Excitation frequencies in the horizontal and vertical directions are taken as 1:2, from which we can investigate the influence of the forced response under horizontal excitation on the parametric instability region under vertical excitation. Three criteria for the instability boundary are proposed. The influences of nonlinearities and damping of the beam on the frequency response and parametric instability region are also investigated. The present analytical results for instability boundaries are compared with those of experiments carried out by one of the authors.     

Dynamic instability of supercritical driveshafts mounted on dissipative supports—Effects of viscous and hysteretic internal damping

The case of a rotating shaft with internal damping mounted either on elastic dissipative bearings or on infinitely rigid bearings with viscoelastic suspensions is investigated in order to obtain the stability region. A Euler-Bernoulli shaft model is adopted, in which the transverse shear effects are neglected and the effects of translational and rotatory inertia, gyroscopic moments, and internal viscous or hysteretic damping are taken into account. The hysteretic damping is incorporated with an equivalent viscous damping coefficient. Free motion analysis yields critical speeds and threshold speeds for each damping model in analytical form. In the case of elastic dissipative bearings, the present results are compared with the results of previous studies on finite element models. In the case of infinitely rigid bearings with viscoelastic suspensions, it is established that viscoelastic supports increase the stability of long shafts, thus compensating for the loss of efficiency which occurs with classical bearings. The instability criteria also show that the effect of the coupling which occured between rigid modes introducing external damping and shaft modes are almost more important than damping factor. Lastly, comparisons between viscous and hysteretic damping conditions lead to the conclusion that an appropriate material damping model is essential to be able to assess these instabilities.     

Atomistic spin dynamic method with both damping and moment of inertia effects included from first principles

We consider spin dynamics for implementation in an atomistic framework and we address the feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the spirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized equation of motion for the magnetization dynamics in the semiclassical limit, which is nonlocal in both space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation, also including the moment of inertia, and demonstrate how the exchange interaction, damping, and moment of inertia, all can be calculated from first principles.     

A Pair of Zakharov Equations with Collisional Damping and the Motion of Langmuir Soliton

The effects of collisional damping on high frequency Langmuir wave and low frequency ion-acoustic wave have been investigated. It is found that the governing equations for the waves are a pair of Zakharov equations with a damping term in each equation. By using the treatment which consists of approximate solutions of the balance equations, a set of first order ordinary differential equations have been derived for the solution parameters in order to study the motion of Zakharov solitons in presence of damping. It has been shown that the width of the soliton remains constant throughout the motion.