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)     

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.     

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)     

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.     

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)     

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.     

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.     

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.     

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．     

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.     

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)     

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.     

Application of CFRP as a rotor shaft material

Conclusions Theoretical analysis and tests performed on rotors with composite shaft show that there is a sufficiently wide rotation stability region in the rotor parameter space despite comparatively high damping of a polymeric composite with respect to steel. Optimum parameters of the shaft (lay-up, thickness) and bearing (radial stiffness, damping) can be found within this region for each given rotor ensuring a low vibration level at critical frequencies.If rotor system parameters are far enough from the instability threshold, maximum vibration level is observed when rotor passes the first eigenfrequency zone. Further increase of rotation frequency leads to a rotor self-centering, and vibration level does not change passing the second eigenfrequency zone. The rotor response is not sensitive to small changes in rotor system parameters. If rotor system parameters are close to the instability threshold, vibration level at the second eigenfrequency dominates, and a small variation of bearing parameters causes significant changes in the vibration level.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 227–240, March–April, 1995.     

Wear and fluid-solid-thermal transient coupled model for journal bearings

In this study, the transient interactions between the sliding wear behaviour and fluid–solid–thermal (FST) characteristics of journal bearings are revealed using an established mathematical model. The calculated temperature distribution is validated by a comparison with experimental results from the literature. Furthermore, a wear test for lubricated journal bearings is conducted to verify the predicted wear rate. The time-varying wear and FST performances of the journal bearing, including the wear rate, wear depth, fluid pressure, contact pressure, and maximum temperature are calculated numerically. Through numerical simulations, the effects of the boundary friction coefficient and surface roughness on the wear and FST performances are evaluated. To demonstrate the importance of considering the three-dimensional (3D) thermal effect during the wear analysis of lubricated journal bearings, the numerical results predicted by the isothermal model and the thermal model are compared systematically within a wide range of operating conditions. The numerical results reveal that the worn surface profile slightly decreases the maximum temperature. Additionally, the worn region is primarily located at both edges of the bearing, and the time-varying worn surface profile may be beneficial for improving the hydrodynamic effect. Furthermore, the effect of the 3D thermal characteristics on the wear prediction of journal bearings cannot be ignored when the external load, boundary friction coefficient, surface roughness are relatively large.     

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.     

Active magnetic damper in a power transmission system

In rotor dynamics, the bearing characteristics exerts a decisive influence on dynamics of the rotating shaft. The research and application experience have led to active magnetic bearings (AMBs), which allow for unique applications in rotating systems. The paper presents the investigations concerning optimization of the magnetic bearing construction. An active magnetic bearing operates as a radial, auxiliary damper, which cooperates with the long, flexible shaft line (aircraft industry applications) and modifies its dynamic properties. In the developed concept of AMBs for aviation purposes, a necessity of increasing its bearing load capacity and damping has occurred. The second important criterion is a weight reduction. This advanced problem leads to specific requirements on the design and materials for the AMB. To achieve these goals, some simulations have been performed. The experimental results are presented as well.     

Lubrication analysis of externally pressurized circular porous bearings

The present study introduces a mathematical formulation for externally pressurized circular porous bearings. A porous layer is used to cover one of the bearing surfaces. An empirical boundary condition with a nonzero tangential velocity, which is known as the velocity slip at the interface, is incorporated into the analysis. The effect of pressure on lubricant viscosity is also considered. The mathematical model consists of two coupled partial differential equations; the first governs the pressure distribution in the film and the second governs the pressure distribution in the porous layer. A simultaneous numerical solution of these equations with the boundary conditions is presented. The effects of porous layer permeability parameter, lubricant viscosity parameter, recess radius, and film thickness on pressure distribution and load-carrying capacity are presented and discussed.     

Forced oscillations of an unbalanced rotor in nonisotropic bearings

The influence of anisotropy of elastic bearings on forced oscillations of a rotor with the static and moment unbalance is studied for the cases of its fastening on a rigid shaft and on a flexible one. The rotor with four degrees of freedom is considered. It is suggested that the shaft is fixed in linear elastic nonisotropic bearings. The differential equations of rotation of the rotor are written in complex variables, and an exact solution to the equation system is found that corresponds to the elliptical synchronous precession. The exact solution is a sum of two vectors, one of which parameterizes a forward precession, while another parameterizes a reverse precession. Amplitude-frequency characteristics of forward and reverse precessions and elliptical trajectories of the rotor axis ends are constructed. It is shown that, in case of nonisotropic bearings, both the forward and reverse precession, as well as the axis motion of nonsimple type (when its one end is moving forward, while another is moving in the reverse direction), can take place. The influence of anisotropy of elastic bearings also manifests itself by change in critical frequencies towards their reduction and by arising of additional critical frequencies in the bottom part of the spectrum, which significantly complicates dynamics of the high-speed rotor at the moment when it reaches the working angular speed.     

Dynamic modeling and analysis of wind turbine drivetrain considering the effects of non-torque loads

With the increase of the rotor diameter and the deterioration of operating conditions, modern wind turbines suffer from more and more significant time-varying non-torque loads, which increases the burden of turbine structures especially the gearbox. Based on an aeroelastic loose coupling approach and assembly of the finite element method, an integrated drivetrain coupling analysis model including blade module, aerodynamic module, and gearbox module is established in this study. This proposed model is validated by comparing the calculation results with previous literature. Taking National Renewable Energy Laboratory 5-MW wind turbine as the research object, the gearbox vibration responses, gear meshing forces and bearing forces under non-torque loads caused by blade gravity, wind shear (WS), tower shadow (TS) and yawed inflow are studied in detail. Results show that the y-direction displacements of the gearbox, sun gear 1, sun gear 2 and gear are larger than those in x-direction because F_y or M_x generated by blade gravity, WS and TS dominates the non-torque loads. The non-torque loads lead to a non-uniform planet load sharing especially for the planetary gear stage 1. Because of the fluctuations of non-torque loads, not only the rotation frequencies of the corresponding carrier but also the multiple frequencies of the carrier 1 are observed in the frequency spectrums. The non-torque loads are mainly borne by carrier 1 bearings. Except for the blade gravity, the bearing forces caused by other unsteady inflows have obvious fluctuations.