Stiffness and damping coefficients for journal bearing using the 3D transient flow calculation

The dynamic coefficients of journal bearing are necessary components in the analysis of linear stability and response of rotating dynamic systems. We propose a new method for the numerical identification of bearing support force coefficients in flexible rotor-bearing systems based on the 3D transient flow calculation. The CFD commercial software FLUENT is mainly used in this simulation, which employs a finite volume method for the discretization of the Navier-Stokes equations. To determine the dynamic coefficients, a new mesh movement approach is presented to update the volume mesh when the journal moves during the 3D transient flow calculation of a journal bearing. Existing dynamic mesh models provided by FLUENT are not suitable for the transient oil flow in journal bearings. Measurements and identification are performed on a test rotor supported on a pair of identical two-lobe fluid film bearings, and the results obtained from the CFD methods agree well with experimental results. The results indicate that the methods proposed in this paper can predict the dynamic coefficients of the journal bearing in a rotor-bearing system effectively, and provide a further tool for stability analysis.     

动压气体轴承的动态刚度和动态阻尼系数

采用偏导数法求解动压气体润滑Reynolds方程,给出动压气体轴承动态刚度和动态阻尼系数普遍适应的计算方法.进行有限元数值仿真,计算动压气体轴承的动态刚度和动态阻尼系数,并重点研究轴颈扰动频率和不同轴颈扰动频率下的轴承静态载荷及轴承数对这些系数的影响.从理论上解释动压气体轴承的动态刚度和动态阻尼系数与轴颈扰动频率的密切相关性.     

Application of computational fluid dynamics and fluid structure interaction techniques for calculating the 3D transient flow of journal bearings coupled with rotor systems

Journal bearings are important parts to keep the high dynamic performance of rotor machinery. Some methods have already been proposed to analysis the flow field of journal bearings, and in most of these methods simplified physical model and classic Reynolds equation are always applied. While the application of the general computational fluid dynamics (CFD)-fluid structure interaction (FSI) techniques is more beneficial for analysis of the fluid field in a journal bearing when more detailed solutions are needed. This paper deals with the quasi-coupling calculation of transient fluid dynamics of oil film in journal bearings and rotor dynamics with CFD-FSI techniques. The fluid dynamics of oil film is calculated by applying the so-called ’’dynamic mesh’’ technique. A new mesh movement approach is presented while the dynamic mesh models provided by FLUENT are not suitable for the transient oil flow in journal bearings. The proposed mesh movement approach is based on the structured mesh. When the journal moves, the movement distance of every grid in the flow field of bearing can be calculated, and then the update of the volume mesh can be handled automatically by user defined function (UDF). The journal displacement at each time step is obtained by solving the moving equations of the rotor-bearing system under the known oil film force condition. A case study is carried out to calculate the locus of the journal center and pressure distribution of the journal in order to prove the feasibility of this method. The calculating results indicate that the proposed method can predict the transient flow field of a journal bearing in a rotor-bearing system where more realistic models are involved. The presented calculation method provides a basis for studying the nonlinear dynamic behavior of a general rotor-bearing system.     

Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method

This paper proposes a method to calculate the stiffness and the damping coefficients of the coupled journal and thrust bearings. Reynolds equations and their perturbation equations of journal and thrust bearings are transformed to the finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. It also includes the Reynolds boundary condition in the numerical analysis to simulate the cavitation phenomenon. The stiffness and damping coefficients of the proposed mathematical method are compared with those of the numerical differentiation of the bearing force with respect to finite displacements and finite velocities of bearing center. It shows that the proposed method can calculate the dynamic coefficients of a coupled journal and thrust bearing more numerically stable and computationally efficient than the differentiation method. It also investigates the coupling effect of the coupled journal and thrust bearing and it shows that the proposed method makes it possible to calculate the cross-coupled dynamic coefficients in the radial–axial direction of the coupled journal and thrust bearing.     

Lateral Vibration Analysis of Flexible Shafts Supported on Elliptical Journal Bearings

This paper presents the development of a finite element procedure for the dynamic analysis of flexible rotors supported on fluid-film elliptical journal bearings operating under several operating conditions. The rotating shaft is modeled by using Timoshenko beam theory and the coupled rotating components, such as disks and impellers, are modeled by using lumped masses. The modeling of the elliptical journal bearings is performed by solving the lubrication equations generated from the application of a linearized perturbation method on the classical Reynolds equation. The bearing carrying-load capacity and the linearized dynamic force coefficients can be predicted for elliptical bearings with different preloads and journal eccentricities. The rotor transient whirling unbalance response is estimated by performing the time integration of the finite element equations using Newmark method. Experimental whirling unbalance response of a rotating shaft supported at two identical journal bearings is used to validate the finite element procedure. A comparative analysis of the dynamic response of flexible rotors supported on both cylindrical and elliptical journal bearings is performed to show that some elliptical bearings are capable of attenuating the rotor unbalance response more efficiently than cylindrical bearings are.     

Dynamic stiffness and damping coefficients of aerodynamic tilting-pad journal bearings

The dynamic gas–film forces of aerodynamic bearing often can be characterized by eight linear stiffness and damping coefficients. How to theoretically predict these coefficients is a very difficult issue for tilting-pad gas bearing design because of its structural complexity. The current study presents a novel and universal theoretical analysis method for calculating the dynamic stiffness and damping coefficients of aerodynamic tilting-pad bearing. The gas–film pressure within the bearing is expressed in the form of dimensionless compressible gas-lubricated Reynolds equation, which is solved by means of the finite element method. With the assumption that the journal and the pads are disturbed with the same frequency, the dynamic coefficients of tilting-pad gas bearing are computed by using the partial derivative method and the equivalent coefficient method. Finally, the investigations are conducted about the effects of bearing number, perturbation frequency of the journal and the pads, eccentricity ratios, preload and length-to-diameter ratio of the bearing on the dynamic coefficients of aerodynamic tilting-pad journal bearing. The numerical results indicate that the dynamic stiffness and damping coefficients of tilting-pad gas bearing are closely related with these factors. The proposed analytical method provides a valuable means of predicting dynamic performances of tilting-pad gas bearing. The solution can be used for the purpose of prediction of dynamic behavior of the rotor systems supported by aerodynamic tilting-pad bearings.     

Numerical prediction of slip flow effect on gas-lubricated journal bearings for MEMS/MST-based micro-rotating machinery

The fluid dynamics of gas-lubricated journal bearings in micro-rotating machinery is different from those of larger size. One point to be considered is a slip flow effect. In this paper, the slip flow effect is considered in order to estimate load-carrying capacity and dynamic coefficients of micro gas-lubricated journal bearings. Based on a modified compressible Reynolds equation including slip flow effect, the first slip approximation was applied. To extract dynamic coefficients, the linearized dynamic equations were formulated by the perturbation method. Numerical predictions compared the static and dynamic characteristics considering slip flow at room-to-high temperature in a range of bearing numbers. The results including load-carrying capacity and dynamic coefficients demonstrate that the slip flow effect becomes significant with temperature increase as well as in the lower range of bearing numbers.     

Performance Characteristics of Worn Journal Bearings in Both Laminar and Turbulent Regimes. Part II: Dynamic Characteristics

The effects of geometric change due to wear on the dynamic characteristics of journal bearings are determined theoretically in both laminar and turbulent regimes. The dynamic characteristics such as spring and damping coefficients and whirl onset speed of a rigid rotor supported by two identical symmetrically aligned bearings are analyzed by a semianalytical finite element method, and the numerical results for various wear depth parameters are indicated in graphical form.

The geometric change due to wear has significant effects on the principal spring coefficients and on the cross-coupled damping coefficients. The whirl onset speed for a worn journal bearing whose wear depth parameter is larger than 0.3 becomes higher than the speed for a nonworn bearing.     

滑动轴承动特性系数新算法

轴承的动特性系数与轴颈在轴承中的位置有关。对于给定工况的转子-轴承系统,必须先求出轴颈的平衡位置,然后再解得轴承动力特性系数。用Newton法迭代求轴颈平衡位置,轴承的动特性系数已在获得轴颈平衡位置的同时求得,无需额外增加求解Reynolds方程的次数。     

Computation of dynamic force coefficients for hybrid (hydrostatic/hydrodynamic) journal bearings by the finite disturbance and perturbation techniques

This paper presents non-dimensional linearized bearing coefficients for oil lubricated plain line-entry (slot) hybrid journal bearings. This represents the first stage of a programme of work on the dynamic characteristics of plain line-entry hybrid journal bearings. A full-scale dynamic analysis of this type of bearing, both theoretical and experimental is currently underway. Two techniques for predicting the linearized bearing coefficients are presented and compared: (i) the finite disturbance technique; (ii) the perturbation technique. Both analyses are based on a finite-difference approximation of the classical Reynolds equation, with provision for source flow and fluid-film rupture, although an isoviscous fluid is assumed.     

动压气体轴承-转子系统的分岔和混沌

基于非线性动力学理论,研究了气体动压轴承-转子系统的不平衡响应及分岔行为。建立了与时间相关的非线性气体动压轴承的压力分布模型和气体动压轴承-刚性Jeffcott转子系统的动力学模型。运用有限差分法和逐次超松弛迭代法求解动压气体润滑雷诺方程;运用轨迹图、Poincaré映射图、时间历程图、频谱图和分岔图研究了有限宽气体轴承支承的非线性转子系统的不平衡响应及分岔;数值模拟结果揭示了系统存在复杂多样的非线性现象,这对气体润滑轴承支承的实际轴承—转子系统的设计提供了理论依据。     

Novel approach to solve the dynamical porous journal bearing problem

Analytical pressure solutions are a simple and robust way to model plain journal bearings in rotordynamics, but cannot be extended to porous journal bearings. Using the method of weighted residuals—viz. Galerkin's method with global shape functions—a novel and fast approach to solve the dynamical porous journal bearing problem is proposed. This approach allows for the influence of rough surfaces on the hydrodynamic pressure implemented through flow-factors, journal misalignment and calculation of stiffness and damping coefficients for finite bearings. The proposed method will be verified using analytical expressions and new results will be shown for porous journal bearings including the influence of rough surfaces.     

Controllable high speed journal bearings, lubricated with electro-rheological fluids. An analytical and experimental approach

The problem of journal bearings control is of great importance in mechanical engineering. A very recent method for doing this is the creation of ‘smart' journal bearings using electro-rheological (ER) fluids. If such a fluid is used to lubricate a journal bearing system, it is expected that the imposition of an electric field between the rotor and the stator will cause an alteration in the dynamic properties of the journal bearing. In this paper an experiment in a high speed journal bearing (16 000 to 35 000 s⁻¹), with small radial clearance is presented. The alternation of the attitude locus (eccentricity and attitude angle) and the stiffness coefficients in a loaded journal bearing lubricated with ER fluid is investigated and presented. The Reynolds equation is solved using the finite element method in order to get the dynamic characteristics of the ER bearings vs the electric field and to simulate its dynamic behavior. The Bingham plastic model of non-Newtonian fluid flow behavior is used to described the ER lubricant. The accuracy of the algorithm is obtained by comparing the results published by previous investigators and the experimental data described in this paper. It is concluded that ER fluids can be used to create ‘smart' journal bearings. and vibration controllers can be constructed to control the dynamics and stability of the ER fluid lubricated bearings.     

Improvement in Dynamic Characteristics of Circular Journal Bearings by Means of Longitudinal Microgrooves

Longitudinal microgrooves were assumed on the circular journal bearings and static and dynamic characteristics were investigated by solving the modified Reynolds equation for a rough bearing surface. It was found that the dynamic characteristics of the journal bearings were improved by longitudinal microgrooves or truncated micro-grooves on the bearing surface while the static characteristics, such as load-carrying capacity and friction coefficients, were not changed by the microgrooves. Calculations regarding the linear stability of a symmetrical rotor supported by two journal bearings were also carried out and it was found that the stable region was expanded on the stability chart by microgrooves.     

一种典型的静压轴承动态特性分析

用有限差分方法求出线性化的四腔静压轴承的8项动特性系数,并由此分析了典型的毛细管节流静压轴承的动态性能.对静压轴承的稳定性进行了讨论,提出了判定静压轴承稳定性的简明方法.该分析方法及所得数据在实际工作中很有意义.     

轴质量对用液体油膜轴承支承转子动特性的影响

本文研究了轴质量对支承于液体油膜轴承上的转子的影响.转子假定是位于弹性轴中部的薄盘,它考虑非零弹性轴质量由液体油膜轴承支承.该径向轴承假定为短简易型,用纵向的和相互作用的刚度及阻尼系数表示,这些系数与系统的角速度有关.假定转于由一电机加速,假定电机速度由一两阶传递函数控制.仿真结果表示质量比的小值,它由轴质量对盘质量比...     

Stiffness and damping coefficients of an inclined journal bearingSteifigkeit und dämpfungskoeffiziente eines schrägen gleitlagers

In the determination of the dynamic behaviour of a rotating shaft, the fluid film stiffness and damping coefficients of the bearings play an important role. The general practice is to ignore the rotational stiffnesses and damping coefficients due to the tilt of the journal in the bearing. This paper presents the stiffness and damping coefficients of such journal bearings. Using the expression for film thickness, the modified Reynolds' Equation for the tilted finite journal bearing is set up. The solution of this equation for the film pressure is obtained by using Fedor's proportionality hypothesis. The results obtained are presented in the form of non-dimensional charts.     

Bifurcation and chaos analysis of nonlinear rotor system with axial-grooved gas-lubricated journal bearing support

Axial-grooved gas-lubricated journal bearings have been widely applied to precision instrument due to their high accuracy, low friction, low noise and high stability. The rotor system with axial-grooved gas-lubricated journal bearing support is a typical nonlinear dynamic system. The nonlinear analysis measures have to be adopted to analyze the behaviors of the axial-grooved gas-lubricated journal bearing-rotor nonlinear system as the linear analysis measures fail. The bifurcation and chaos of nonlinear rotor system with three axial-grooved gas-lubricated journal bearing support are investigated by nonlinear dynamics theory. A time-dependent mathematical model is established to describe the pressure distribution in the axial-grooved compressible gas-lubricated journal bearing. The time-dependent compressible gas-lubricated Reynolds equation is solved by the differential transformation method. The gyroscopic effect of the rotor supported by gas-lubricated journal bearing with three axial grooves is taken into consideration in the model of the system, and the dynamic equation of motion is calculated by the modified Wilson-0-based method. To analyze the unbalanced responses of the rotor system supported by finite length gas-lubricated journal bearings, such as bifurcation and chaos, the bifurcation diagram, the orbit diagram, the Poincar6 map, the time series and the frequency spectrum are employed. The numerical results reveal that the nonlinear gas film forces have a significant influence on the stability of rotor system and there are the rich nonlinear phenomena, such as the periodic, period-doubling, quasi-periodic, period-4 and chaotic motion, and so on. The proposed models and numerical results can provide a theoretical direction to the design of axial-grooved gas-lubricated journal bearing-rotor system.     

Effect of Fluid Inertia on Journal Bearing Parameters

In the modeling and analysis of rotordynamic systems with journal bearings, the stiffness and damping parameters are usually obtained from the Reynolds equation of hydrodynamic lubrication. The Reynolds equation is derived from the continuity and momentum equations with several assumptions; the principal one among them being that the inertia terms are negligible since the lubricant flow is viscosity-dominated. Some previous work has shown, however, that the effect of fluid inertia on the static and dynamic properties of a bearing is not negligible in many circumstances.

This paper uses a perturbation approach to present a rigorous derivation of the correction terms to be added to account for the effect of inertia in the case of a journal on a short bearing. The governing equation for pressure correction is derived and the corrected stiffness, damping, and inertia coefficients to the first order are displayed as a function of the equilibrium position.     

混合轴承柔性转子系统稳定性研究

用数值解方法分析了多腔混合轴承的动态特性。对于典型的毛细管节流４腔轴承计算了在各种设计参数下的线性化的刚度系数和阻尼系数,从而得到了轴承相应的刚度和稳定性速度阈值,指出了动压效应对这两者的负面影响。讨论了转子柔性对自激涡动的作用,提出了在设计转子混合轴承系统中实用的确定稳定性阈值的方法。