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)     

Chaotic response and bifurcation analysis of a flexible rotor supported by porous and non-porous bearings with nonlinear suspension

This study presents numerical work investigating the dynamic responses of a flexible rotor supported by porous journal bearings. Both porous and non-porous bearing types are taken into consideration in this study. The rotating speed ratios and imbalance parameters are also presented and proved to be important control parameters. Many non-periodic responses to chaotic and quasi-periodic motions are found, too. From the bifurcation diagrams in this paper, it is also evidenced that the vibration behaviors would be improved by porous bearings. The modeling result obtained here can be employed to predict the dynamics of bearing–rotor systems, and undesirable behavior of the rotor and bearing orbits can be avoided. Also, this could help engineers and researchers in designing and studying bearing–rotor systems or some turbo-machinery in the future.     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Nonlinear stability analysis of rotor-bearing systems

High-speed rotors supported by floating ring bearings exhibit beside self-excited vibrations various nonlinear vibration effects, which may cause the damage of the rotor. After deriving the equations of motion of a perfectly balanced turbocharger rotor supported by floating ring bearings, bifurcation analyses are carried out with both rigid and flexible model by applying numerical continuation methods. Thereby, the main focus of the investigation is on the critical bifurcations emanating destructive limit-cycle oscillations of higher amplitudes. Finally, the influence of the shaft elasticity on the critical limit-cycle oscillations is discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sensitivity of Computational Rotor Dynamics Towards the Empirically Estimated Lubrication Gap Clearance of Foil Air Journal Bearings

This contribution is concerned with the computational analysis of a rigid rotor supported by means of two self-acting foil air journal bearings. Even though the overall equation system is thereby typically written in a nondimensional form, prior knowledge about realistic value ranges of occurring dimensionless numbers is required in order to parameterize and interpret such simulations correctly. Unlike all other quantities, the nominal lubrication gap clearance between the rotating journal and the undeformed foil structure is reported to be only poorly known. Thus, even in the light of an advanced understanding of the bearing rotor system's fundamental behavior, the quantitative reproduction and prediction of experimental results by means of computational analysis need to be viewed critically. In this study, the sensitivity of numerical results towards the assumed nominal lubrication gap clearance will be investigated. To this end, the stability of the system is considered and the characteristics of occasionally observed equilibrium points and limit cycles are addressed. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

非线性转子系统稳定性量化分析方法

转子轴承系统是一类多自由度非线性动力系统,广泛应用于工程实际．设计观念和维修体制的变革提出了稳定性量化分析的要求．本文利用轨线保稳降维方法提出了转子系统稳定性的量化分析方法．首先,对高维非线性非自治转子系统进行数值积分,将n维空间的轨线映射为一系列一维的映象轨线,并将各自由度的运动方程中除该自由度外的所有状态变量用积分结果代换,得到n个互相解耦,含有多个时变参数的单自由度方程．然后,在一维观察空间的外力位移扩展相平面上定义了动态中心点,研究转子系统中常见的几种运动的动态中心点动能差序列的特点,给出了上述典型运动形式的轨线稳定裕度的定量评估指标,应用灵敏度分析技术快速有效地预测周期运动的倍周期分岔点和Hopf分岔点．以一个具有非线性支承的滑动轴承柔性转子模型为例,证明了该方法的有效性．     

Approximation of quasi-periodic solutions of a rotor in two-lobe bearings with time-varying geometry

Improving the dynamic behaviour of rotor systems in journal bearings represents an ongoing topic of research. The pressure distribution within journal bearings is described by the Reynolds equation, whereby unwanted oscillations can be caused by the fluid-solid interaction within the bearings. An approach of a two-lobe bearing with time-varying geometry is suggested to suppress or at least to reduce occurring oscillations. In order to systematically analyse the system, a spectral reduction is performed, allowing to handle also quasi-periodic behaviour by means of numerical continuation algorithms. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrations of Rotors Supported on Nonlinear Bearings

Rotors in electrical machines are supported by various types of bearings. In general, the rotor bearings have nonlinear stiffness properties and they influence the rotor vibrations significantly. In this work, this influence of these nonlinearities is investigated. A simplified finite element model using Timoshenko beam elements is set up for the heterogeneous structure of the rotor. A transversally isotropic material model is adopted for the rotor core stack. Imposing the nonlinear bearing stiffnesses on the model, the Newton-Raphson procedure is used to carry out a run up simulation. The spectral content of these results shows nonlinear effects due to the bearings. The rotor vibrations are further investigated in detail for various constant speeds. These results show non-harmonic vibrations of the rotor in a section of the investigated speed range. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical analysis of the rotor system

This paper aims at creating a mathematical model of a bending oscillation rotor system which enables to execute a dynamical analysis of its vibration including the influence of nonlinear bearing characteristics. More specifically, using the finite element method the model of rotating system supported by four hydrodynamic bearings was created. The basic dynamical analysis of the rotor system was performed and the eigenvalues, eigenvectors and stability conditions were evaluated. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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。     

滑动轴承转子系统动力学稳定域的数值分析方法

根据Floquet理论定义了非线性非自治系统周期解的稳定度.从动力系统流的概念出发,给出利用非线性非自治系统稳态周期解受扰后的瞬态响应,计算周期解稳定度的数值计算方法.以稳定度等于零为临界判据,分析计算了滑动轴承平衡和不平衡刚性转子系统的稳定吸引域.研究发现,平衡转子随着转速的升高稳定域减小;不平衡转子随着不平衡量的增大稳定域减小;且工频周期解的稳定域比同样系统条件下平衡点的稳定域小.     

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.     

The influences of longitudinal surface roughness on sub-critical and super-critical limit cycles of short journal bearings

By applying the stochastic model of rough surfaces by Christensen (1969–1970, 1971)  and  together with the Hopf bifurcation theory by Hassard et al. (1981) [3], the present study is mainly concerned with the influences of longitudinal roughness patterns on the linear stability regions, Hopf bifurcation regions, sub-critical and super-critical limit cycles of short journal bearings. It is found that the longitudinal rough-surface bearings can exhibit Hopf bifurcation behaviors in the vicinity of bifurcation points. For fixed bearing parameter, the effects of longitudinal roughness structures provide an increase in the linear stability region, as well as a reduction in the size of sub-critical and super-critical limit cycles as compared to the smooth-bearing case.     

Application of a hybrid method to the nonlinear dynamic analysis of a flexible rotor supported by a spherical gas-lubricated bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the nonlinear dynamic behavior of a flexible rotor supported by a spherical gas-lubricated bearing system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and quasi-periodic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those from other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of spherical gas film rotor–bearing systems.     

Dynamic analysis of short and long journal bearings in laminar and turbulent regimes,application in critical shaft stiffness determination

Linear and non-linear stability of a flexible rotor-bearing system supported on short and long journal bearings is studied for both laminar and turbulent operating conditions. The turbulent pressure distribution and forces are calculated analytically from the modified Reynolds equation based on two turbulent models; Constantinescu's and Ng–Pan–Elrod. Hopf bifurcation theory was utilized to estimate the local stability of periodic solutions near bifurcating operating points. The shaft stiffness was found to play an important role in bifurcating regions on the stable boundaries. It was found that for shafts supported on short journal bearings with shaft stiffness above a critical value, the dangerous subcritical region can be eliminated from a range of operating conditions with high static load. The results presented have been verified by published results in the open literature.     

Non-linear dynamic analysis of a flexible rotor supported on porous oil journal bearings

In the present paper, the non-linear dynamic analysis of a flexible rotor with a rigid disk under unbalance excitation mounted on porous oil journal bearings at the two ends is carried out. The system equation of motion is obtained by finite element formulation of Timoshenko beam and the disk. The non-linear oil-film forces are calculated from the solution of the modified Reynolds equation simultaneously with Darcy’s equation. The system equation of motion is then solved by the Wilson-θ method. Bifurcation diagrams, Poincaré maps, time response, journal trajectories, FFT-spectrum, etc. are obtained to study the non-linear dynamics of the rotor-bearing system. The effect of various non-dimensional rotor-bearing parameters on the bifurcation characteristics of the system is studied. It is shown that the system undergoes Hopf bifurcation as the speed increases. Further, slenderness ratio, material properties of the rotor, ratio of disk mass to shaft mass and permeability of the porous bush are shown to have profound effect on the bifurcation characteristics of the rotor-bearing system.     

Bifurcation and nonlinear analysis of a flexible rotor supported by a relative short spherical gas bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the bifurcation and nonlinear dynamic behavior of a flexible rotor supported by a relative short spherical gas bearing (RSSGB) system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic, and chaotic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of RSSGB systems.     

Identification and Self Tuning Control for Active Magnetic Bearing Systems: Levitation of Unknown Rotors

The present paper deals with the problem of levitating rotors with unknown characteristics by means of active magnetic bearings whose properties are known. This problem is of interest in a technical setting to shorten the development time of AMB systems, in particular for controller design. Theoretical interest arises from the fact that several issues in the area of identification and self tuning control are addressed for an unstable system. Our aim is to identify the flexible rotor including gyroscopic effects and to automatically design a robustly stabilizing controller for this system that can be used for running the system under regular operating conditions. To this end, a rigid body model of the rotor is identified based on measured step responses from the plant. Then, the bearings are adjusted to have very low stiffness, and a controller with steep roll‐off is designed in order to avoid excitation of the unknown flexible modes of the system. Once the rotor is floating, the identification algorithm from [1] is applied to obtain information on the flexible modes of the system. Based on this extended model, a robust controller allowing for slow rotation of the rotor is designed. With the rotor rotating at a moderate speed, the frequency response functions are measured, and based on these measurements, the gyroscopic matrixof the system is identified, completing the system model and allowing for design of the desired controller for normal operation. The present contribution focusses on identification of the rigid body model of the flexible rotor.     

Numerical analysis of a rigid rotor supported by aerodynamic four-lobe journal bearing system with mass unbalance

This paper presents the effect of rotor mass on the nonlinear dynamic behavior of a rigid rotor-bearing system excited by mass unbalance. Aerodynamic four-lobe journal bearing is used to support a rigid rotor. A finite element method is employed to solve the Reynolds equation in static and dynamical states and the dynamical equations are solved using Runge-Kutta method. To analyze the behavior of the rotor center in the horizontal and vertical directions under different operating conditions, the dynamic trajectory, the power spectra, the Poincare maps and the bifurcation diagrams are used. From this study, results show how the complex dynamic behavior of this type of system comprising periodic, KT-periodic and quasi-periodic responses of the rotor center varies with changes in rotor mass values by considering two bearing aspect ratios. Results of this study contribute a better understanding of the nonlinear dynamics of an aerodynamic four-lobe journal bearing system.