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.     

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.     

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 analysis of an aerodynamic journal bearing system considering the effect of stationary herringbone grooves

This paper investigates the bifurcation and nonlinear behavior of an aerodynamic journal bearing system taking into account the effect of stationary herringbone grooves. A finite difference method based on the successive over relation approach is employed to solve the Reynolds’ equation. The analysis reveals a complex dynamical behavior comprising periodic and quasi-periodic responses of the rotor center. The dynamic behavior of the bearing system varies with changes in the bearing number and rotor mass. The results of this study provide a better understanding of the nonlinear dynamics of aerodynamic grooved journal bearing systems.     

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.     

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.     

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.     

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)     

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)     

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．     

Dynamic analysis and experiment investigation of a cracked dual-disc bearing-rotor system based on orbit morphological characteristics

The paper is aimed to examine dynamic behaviors of a dual-disc bearing-rotor system in multi-fault state, and the crack detection based on the orbit morphological characteristics and vibration responses is proposed. Dynamic response and vibration signal analysis are two significant studies in rotor system. Most researchers have simulated the nonlinear dynamics and analyzed the fault signal using various methods separately. However, the fault feature from vibration signal is tightly connected with the dynamic mechanism in the rotor system, especially in rotor system with coupling multi-fault. In the paper, the dynamic model of the dual-disc bearing-rotor system is established, which takes into account the effects of crack, rub-impact and nonlinear oil-film forces. The vibration responses and the effect of crack on dual-disc rotor system with multi faults are investigated. The existence of crack and the coupling effect of multi faults enrich dynamic behavior of the dual-disc bearing-rotor system, and the response near the 1/2 subcritical speed provides a criterion for crack detection. Experiment investigation is attempted for the first time, which is based on the changes of crack depth and rotation speed for multi-fault dual-disc rotor system. The analysis of the dynamic response and the orbit morphological characteristics from experiment can effectively detect the crack information.     

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)     

On the dynamics of a nonlinear rotor-floating ring bearing system

Today, in high speed applications the rotors are commonly supported by hydrodynamic journal bearings. One typical configuration of journal bearings incorporated in automotive turbochargers is the floating ring bearing. Rotors supported by floating ring bearings have many advantages, regarding costs and power consumption for example. However, they might become unstable with increasing speed of rotation. At the onset of instability both the perfectly balanced and unbalanced rotor undergo self-excited vibrations which could cause the mechanical breakdown of the system. The “oil whip”-phenomenon, very well known from the investigations of the plain journal bearing occurs here in a different form. At the stability limit the rotor begins either oscillating with about the half of the ring speed or the half of the ring speed plus the half of the journal speed depending on the system parameters. For this reason a rotor-floating ring bearing model is presented showing the mentioned characteristics. By applying the nonlinear equations of motion the limit cycles of the system are determined and its loss of stability is investigated. (© 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.     

Theoretical analysis of high speed spindle air bearings by a hybrid numerical method

To study the behavior of the high speed spindle air bearing (HSSAB) system, we conduct the research by means of a hybrid numerical method which combines the differential transformation method and the finite difference method in this paper. According to the results of the research, the flexible rotor center is found to include a complex dynamic behavior that comprises periodic, sub-harmonic and quasi-periodic responses. In addition, as the rotor mass and the bearing number are increased, there will be some changes taking place in the dynamic behavior of the bearing system. The results are proven to have no conflict with those of the other numerical methods, which enables an effective means in gaining insights into the nonlinear dynamics of HSSAB systems.     

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)     

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves under the cavitation boundary condition is proposed. The pressure distribution of the Reynolds equation of a finitely long journal bearing with axial grooves is expressed as a particular solution and a homogeneous solution. The particular solution and the homogeneous solution are separated, respectively, into an additive form and a multiplicative form by the method of separation of variables. The circumferential separable function of the homogeneous solution can be expanded on the basis of the infinite series of trigonometric functions. The pressure distribution of the particular solution is obtained by the Sommerfeld transformation. The termination positions of the fluid film are determined by the continuity condition. The analytical expressions for the nonlinear fluid film forces of a finitely long journal bearing with two axial grooves are obtained. The fluid film forces calculated by the proposed method agree well with the results obtained by the finite-difference method. The effects of the bearing parameters on the nonlinear fluid film forces are analyzed.     

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.     

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.