机械化数学在转子动力学研究中的应用

基于机械化数学-吴文俊消去法,分别采用短轴承油膜力模型和Muszynska转子力学模型,对转子轴承系统中的动力学行为与稳定性进行了分析研究.具体分析时,采用吴文俊特征列概念和基于Maple软件的符号计算平台,对短轴承涡动参数进行了解析分析,以及试算构造出了Liapunov函数,并给出了转子系统运动稳定性条件.     

悬臂柔轴的干摩擦失稳

本文讨论柔轴高速自转时由于摩擦激发的弓状回转现象.摩擦力作用面不与集聚质量面重合.导出了回转频率和回转形态的计算公式, 分析了它的运动稳定性.结果指出,存在着稳定的弓状回转,其频率大于转轴的临界转速.     

Instability of Unbalance Excited Synchronous Forward Whirl at Rotor‐Stator‐Contact

Normally rotor unbalance causes synchronous forward whirl of rotor‐stator systems, even if rub occurs due to rotorstatorcontact. This synchronous forward whirl has to be stable in order to avoid destructive self‐excited dry friction backward whirl, chaotic motions or sub‐ and superharmonic vibrations. However, friction between rotor and stator can cause the synchronous forward whirl to become unstable within certain rotor speed ranges. In the present paper the stability of the synchronous forward whirl caused by unbalance is investigated for rotor motions under contact with the stator. To analyse the stability of synchronous forward whirl the equations of motion are linearised around the stationary synchronous motion. The characteristic polynomial of the perturbations is calculated and the stability is checked by the Hurwitz criterion.     

A nonlinear model for rotor-shaft joints of high speed rotor systems with internal damping

Viscous internal damping in joints of high speed rotor systems causes instabilities above a certain frequency of revolution. In the majority of cases a nonlinearity adjusts the stability margin towards higher frequencies. In this paper an analytical solution of a nonlinear four degrees of freedom rotor model with internal damping is proposed, which enables to clearly analyse the influence of shaft stiffness, connection stiffness, rotor mass and shaft mass. The steady state solution of the unbalance case and the stability boundaries are deduced analytically. The stabilizing effect of the nonlinearity is shown. The analytical solutions are in good agreement with numerical results obtained by FERAN, a rotor dynamic simulation tool. A model, representing the rotor–shaft connection with an o–ring has been analyzed by a hydro pulse rig. Beneath the linear way, two further approaches to describe the measured hysteresis, a cubic and a bilinear force law are shown in the paper. The different analytical and numerical results for the whole rotor system with these three approaches are compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

迷宫密封不平衡转子动力系统的稳定性与分岔

研究迷宫密封对不平衡转子系统动力稳定性的影响.存在不平衡量的转子在旋转过程中受到周期激励,低转速时,转子作与激励同频率的周期运动,随着转速的提高,达到一定阈值时周期运动开始失稳.对迷宫密封的气动力采用Muszynska非线性力学模型,用打靶法求解转子运动周期解,并根据Floquet理论分析了周期解的稳定性及失稳后的动力学特性.     

Persistence and global stability of Bazykin predator–prey model with Beddington–DeAngelis response function

In this article, a predator–prey model of Beddington–DeAngelis type with discrete delay is proposed and analyzed. The essential mathematical features of the proposed model are investigated in terms of local, global analysis and bifurcation theory. By analyzing the associated characteristic equation, it is found that the Hopf bifurcation occurs when the delay parameter τ crosses some critical values. In this article, the classical Bazykin’s model is modified with Beddington–DeAngelis functional response. The parametric space under which the system enters into Hopf bifurcation for both delay and non-delay cases are investigated. Global stability results are obtained by constructing suitable Lyapunov functions for both the cases. We also derive the explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Our analytical findings are supported by numerical simulations. Biological implication of the analytical findings are discussed in the conclusion section.     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

Synchronization effects in rotors partially filled with fluid-influence on propulsion

In fluid-filled Rotors occur self-excited vibrations induced by surface waves of the fluid. A characteristic property is the instability over an interval of angular velocity above the natural frequency of the system. One explanation is the occurrence of synchronization effects between fluid waves and the critical rotor speed. 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 for observer-based real-time control. This paper analyses a model based on a laval rotor and shows synchronization effects between fluid waves and rotor model and its influence on the rotor propulsion. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and Bifurcations of Rotors in Fluid Film Bearings

To study the nonlinear phenomena of rotors in the sense of bifurcation theory, the mechanical model of a symmetric flexible rotor is investigated which is supported by two identical journal bearings. Two types of journal bearings are considered. While the oil whirl and oil whip oscillations of rotors in plain journal bearings are widely examined, the floating ring bearings cause a quite different vibration behavior with several mode interactions and an area of so-called critical limit cycles leading to a rotor damage. For both types a Hopf bifurcation marks the beginning of the self-excited oscillations in the case of a perfectly balanced rotor. By applying the methods of numerical continuation the occurring limit cycles as well as their stability are determined. The different nonlinear effects with the corresponding bifurcations are explained by describing the global solution behavior of the rotor-bearing systems. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability study and control of helicopter blade flapping vibrations

The response of a two-degree-of-freedom, controlled, autoparametric system to harmonic excitations is studied and solved. The objective of this research is to investigate the effect of linear absorber on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to obtain the periodic response equation near the primary resonance in the presence of internal resonance of the system. The stability of the obtained numerical solution is investigated using both phase plane methods and frequency response equations. Variation of some parameters leads to the bending of the frequency response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

磁轴承失灵后坠落转子瞬态振动灾变机理研究

研究一个带磁轴承的转子系统,在磁轴承失灵后转子坠入备用轴承引起的非线性瞬态振动。通过严格建立运动方程和数值仿真计算,详尽地分析了坠落转子转动角速度变化和轴颈与备用轴承接触点法向力变化的时间历程及备用轴承振动位移的频谱,发现系统发生灾变破坏的原因是由于高速不平衡阻尼转子减速通过临界速度时引起的强烈非稳态受迫弯曲振动加上轴颈与备用轴承接触点碰摩的非线性引起的高频颤振。     

考虑径向内间隙的滚动轴承平衡转子系统的非线性动力稳定性

研究滚动轴承平衡转子系统在不同轴承内间隙量,不同转速下系统的稳定性及其分岔特性和混沌．考虑Hertz接触力、 滚动体通过振动和轴承径向内间隙等非线性因素建立数学模型,根据Floquet理论分析不同间隙量下滚动轴承转子系统的周期解稳定性, 找到了3种导致周期解失稳的方式:倍周期分岔失稳、拟周期分岔失稳和边界激变导致混沌失稳．通过对各间隙量下转子系统拓扑特性变化和失稳区域的研究,表明滚动轴承间隙量是影响转子系统动力稳定性的一个重要因素．     

Stability Analysis of Full Annular Rub in Rotor‐to‐Stator Systems

This paper outlines the analytical study of the stability of a more realistic rotor‐to‐stator contact model, which considers the stator as an elastic body having finite mass and being elastically supported. For this model, no report on the stability analysis is available in the literature by our knowledge. In this study, the analytical solution, corresponding to the synchronous full annular rub motion, is first solved, then the stability condition of it is derived. Based on the systematic parametric studies, one gains a deeper insight into the nature of the dynamical behavior of the rotor‐to‐stator contact system. Since the previously investigated simpler rotor‐to‐stator contact models are only the special cases of the present model, the limitation of the simpler models and the influence of the system parameters, which were omitted or assigned to specific values in the simpler models, have been revealed through the present investigation.     

Kink–antikink density wave of an extended car-following model in a cooperative driving system

We propose an extended optimal velocity model applicable to cooperative driving control system by considering the headway of arbitrary number of cars that precede and the relative velocity. The stability condition of the extended model is obtained by using the linear stability theory. The modified Korteweg-de Vries (mKdV) equation is derived to describe the traffic behavior near the critical point by applying the nonlinear analysis. Thus the traffic jams can be described by the kink–antikink density wave which is the solution of the mKdV equation. The simulation results confirm the analytical results and show that the traffic jams are suppressed more efficiently with considering not only the headway of more vehicles ahead but also the relative velocity.     

Whirling prediction with geometrical shaft profiling

Distributed parameter shaft–rotor models are considered. The multivariable irrational, hyperbolic and circular function, input–output relationship for the system, is derived. Arbitrary, geometrical shaft profiling may be accommodated within the analytical techniques outlined. Conventional frequency response methods are employed in the determination of the critical speed condition. Specific studies, incorporating cantilevered rotors with non-linear shaft length–diameter configurations are detailed. The general applicability of the procedures outlined is emphasised.     

Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

Auto-balancing of a rotor with an orthotropic elastic shaft

The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor.     

The effects of ridge-top and lee-slope fires upon rotor motions in the lee of a steep ridge

Soon after the outbreak of the South Australian Ash Wednesday I bushfire on a ridge in the Adelaide Hills, fires were observed burning up its lee slope in a direction opposite to that of the main fire. To investigate mathematically why this happened, a simple two-dimensional analytical model of a lee rotor (which occurs when airflow over a ridge separates from the lee slope) is developed by applying the first circulation theorem of physical hydrodynamics to a triangular material curve of air parcels in the lee of a ridge. The resultant equation describes how fire-generated buoyancy, frictional driving from above and lee-slope friction affect the average airspeed around a lee rotor.

A comparison between steady-state solutions of the rotor equation with and without a ridge-top fire indicates that the presence of the latter slightly retards any lee rotor which may have existed under neutral pre-fire conditions. The solution of an initial-value problem simulating the growth and decay of a lee-slope fire is next found and its dependence upon the ratio of the frictional relaxation time to the growth time of maximum fire-induced instability across the rotor is investigated and graphed. For growth times greater than 2.5 minutes and reasonable friction coefficients, it turns out that the response of the rotor to the slope fire is almost instantaneous and its maximum speed is more than twice its pre-fire speed. A halving of the friction coefficients even results in the maximum rotor speed being more than the airspeed above it. It is concluded that buoyant enhancement of lee rotors by fires greatly helps the latter to burn up lee slopes and that lee rotors containing intense fires may sometimes even drive the airflow above them.     

Nonlinear response and dynamic stability of a cracked rotor

Establishment of a new approach for analyzing the nonlinear behavior of a cracked rotor system is the main goal of the present research. Nonlinear governing equations of motion are developed for the cracked rotor system with asymmetrical viscoelastic supports. In establishing the approach, the masses of the rotational shaft and a disc mounted on the shaft, geometric nonlinearity of the shaft, and the rotor’s extra displacements due to the existence of the crack are all taken into account. On the basis of the governing equations, the nonlinear behavior of the rotor system is analyzed numerically with considerations of the effects of the crack depth, the crack location, the locations of the disc, and the shaft’s rotational speed. The effects of the crack and the other system parameters on the dynamic stability of the rotor system are also investigated.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.