Two-dimensional dynamic simulation of a continuous foil bearing

The objective of this paper is to present an integrated procedure which will analyse the dynamics of a foil bearing on elastic supports. A complete time-dependent Navier-Stokes formulation is used to solve for the interaction between the hydrodynamic effects of the fluid lubricant, the motion of the journal, and the deformable foil boundary. The elastic deformation of the foil and its supports are simulated by a finite element model. The steady state, the quasi-transient and the full transient dynamics of the foil-fluid-journal interaction are examined. For the steady state simulation, for a particular journal position, the fluid lubricant pressures are evaluated by means of a joint iterative scheme until convergence is achieved in both the fluid pressure and the corresponding foil deformation. For the quasi-transient case, the transient motion of the journal is calculated using a numerical integration scheme for the velocity and displacement of the journal. The deformation of the foil is calculated through numerical iteration in feedback mode in combination with the fluid film pressure generated by the journal motion, until convergence at every time step is achieved. For the full transient simulation, a parallel real-time integration scheme is used, at each time step, to evaluate simultaneously the new journal position and the new deformed shape of the foil.     

Gas-dynamic foil bearing model

A method is developed to calculate the characteristics of gas-dynamic bearings of rotors of gas-turbine engines and gas-turbine units. The method takes into account the contact interactions between the shaft journal, the fluid film, and the elastic bearing elements. The problem of multidisciplinary mathematical simulation of the elastic gas-dynamic contact is formulated and solved to determine the parameters of lubrication and deformations of the shaft and bearing. The considerable nonlinearity of the problem is governed by the equations describing the fluid flow in the bearing and the features of the elastic contact during deformation of the bearing elements. The calculation of the fluid-film flow in the bearing is based on the solution of the nonlinear two-dimensional Reynolds equation for a compressible fluid. The method of consecutive loading with error correction that within the interval prompts to the linearized Reynolds equation solution for fluid film pressure increment is used. The results of calculation of the fluid-flow parameters in the clearance between the shaft and the bearing are compared with the results obtained by solving the fluid-flow problem in a bearing modeled with the Navier-Stokes equations with the STAR-CD software. The stress-strain state of the elastic bearing elements is studied with the finite element model taking into account the contact interaction between the foils themselves and with the bearing race. The pressure distribution and the clearance in the shaft are determined iteratively by the coupled solution of the fluid flow and bearing foil deformation problems. Bearing stiffness characteristics, its carrying force and attitude angle are determined versus shaft journal displacement value and direction. It is shown that the stiffness characteristics of the bearing depend on the direction of displacement of the shaft journal in the bearing. The influence of the bearing elastic element deformations on the support load carrying capacity and the stiffness characteristics are studied. The results yielded by the calculations with the developed method are compared with those when the fluid-layer thickness in the bearing was calculated using the analytical model proposed by H. Heshmat and co-authors.     

基于流固耦合的波箔动压气体轴承承载性能优化

基于有限元软件Ansys Workbench,建立波箔动压气体轴承在可压缩流体介质中运动的有限元模型,采用6DOF动网格计算方法对轴承的运动状态进行流固耦合数值模拟,探讨不同转速和波箔片结构参数（波箔的长度比、高度以及厚度）对轴承承载性能的影响规律。结果表明：轴承气膜压力大小的分布与平箔片变形量的大小成对应关系,说明提出的流固耦合方法能很好地反映波箔动压气体轴承的润滑状态;随着转速的增大,轴承动压效应不断增强,承载力增大,且平箔片的结构变形不断增大,致使气膜压力的收敛区发生波动;随着波箔的长度比和波箔片厚度的增加,轴承承载力先快速增大后趋于稳定,而波箔片高度对承载力影响不大,表明适当增加波箔的长度比和波箔片厚度可以提高承载力。     

The effect of Coulomb friction on the static performance of foil journal bearings

In foil bearings, the friction between bumps and their mating surfaces is the major factor which exerts great influence on the bearing performance. From this point of view, many efforts have been made to improve the understanding of the influence of the friction on the foil bearing performance by developing a number of analytical models. However, most of them did not consider the hysteretic behavior of the foil structure resulting from the friction. The present work developed the static structural model in which hysteretic behavior of the friction was considered. The foil structure was modeled using finite element method and the algorithm which determines the conditions of the contact nodes and the directions of the friction forces was used to take into account the friction. The developed model was integrated into the foil bearing prediction code to investigate the effects of the friction on the static performance of the bearing. The results of analysis show that multiple static equilibrium positions are presented for the one static load due to the friction, inferring its great effects on the dynamic performance. However, the effect of friction on the minimum film thickness which determines the load capacity of the bearing is negligible.     

Research of oil film thickness model and surface quality in cold rolling copper foil

Rolled copper foil is widely used in high frequency and speed transmission of fine line printed circuit board, because of its high strength, good toughness and high density. In this paper, a theoretical model for copper foil rolling in mixed lubrication regime was developed on the basis of the average volume flow model and asperity flattening model. A more accurate relation for the variation of the lubricant viscosity with pressure and temperature was considered. The cold rolled copper foil experiment was carried on with different viscosity of rolling oil and pass reduction. The effects of rolling oil viscosity and pass reduction on lubricant pressure, contact area ratio and film thickness ratio were studied. The calculation results agree well with the measured data from copper foil rolling experiment. For obtaining higher surface quality, the rolling oil viscosity is about 10 mm²/s, and the pass reduction is about 30%. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of gas foil bearings integrating FE top foil models

Gas foil bearings (GFBs) find widespread usage in oil-free turbo expanders, APUs, and micro gas turbines for distributed power due to their low drag friction and ability to tolerate high-level vibrations. The performance of GFBs depends largely on the support elastic structure, i.e. a smooth foil on top of bump strips. Conventional models include only the bumps as equivalent stiffnesses uniformly distributed around the bearing circumference. More complex finite element (FE) models couple the elastic deformations of the 2D shell or 1D beam-like top foil to the bump deflections as well as to the gas film hydrodynamics. Predictions of journal attitude angle and minimum film thickness for increasing static loads and two journal speeds are obtained for a GFB tested decades ago. For the GFB studied, 2D FE model predictions overestimate the minimum film thickness at the bearing centerline, while underestimating it at the bearing edges. Predictions from the 1D FE model compare best to the limited tests data, reproducing closely the experimental circumferential wavy-like film thickness profile. Predicted stiffness and damping coefficients versus excitation frequency show that the two FE models result in slightly lower direct stiffness and damping coefficients than those from the simple elastic foundation model.     

波箔式气体轴承气膜流场分析

模拟在压力远场条件下,考虑气体可压缩性、平箔变形和端泄效应的箔片轴承三维气膜流场,得到轴承气膜压力和速度分布。刚性气体轴承的压力分布结果与文献结果吻合很好。研究考虑平箔变形情况下的承载力、端部泄漏流量等随转速和偏心率的变化关系,结果发现:在轴承端部存在出气的正压区和进气的负压区,与之前的文献端部为大气压的模拟结果不同;轴承两端的进气、出气是独立的,中心截面的流量为0;平箔变形使箔片轴承端部泄漏质量流量变大,承载力变小。     

弹性箔片动压气体推力轴承承载性能研究

气弹耦合解是准确预测弹性箔片动压气体轴承承载性能的有效方法。通过引入箔片的弹性变形以及联立求解动压气体润滑Reynolds方程和弹性箔片的变形方程,给出了弹性箔片动压气体推力轴承的气弹耦合解。应用气弹耦合解理论,将顶层箔片的局部弹性变形纳入考虑范围,对弹性箔片动压气体推力轴承的承载性能进行了计算和分析。有限元数值仿真结果表明:顶层箔片在气膜压力作用下的局部弹性变形直接导致弹性箔片动压气体推力轴承承载能力的降低;根据轴承瓦块上气膜压力分布的特点合理设计支承拱箔的结构形式,可以减小顶层箔片的局部弹性变形,有效提高轴承的承载能力。得到了一种承载能力较高的弹性箔片动压气体推力轴承支承拱箔结构设计方案。     

Theoretical considerations of static and dynamic characteristics of air foil thrust bearing with tilt and slip flow

The thrust pad of the rotor is used to sustain the axial force generated due to the pressure difference between the compressor and turbine sides of turbomachinery such as gas turbines, compressors, and turbochargers. Furthermore, this thrust pad has a role to maintain and determines the attitude of the rotor. In a real system, it also helps reinforce the stiffness and damping of the journal bearing. This study was performed for the purpose of analyzing the characteristics of the air foil thrust bearing. The model for the air foil thrust bearing used in this study is composed of two parts: one is an inclined plane, which plays a role in increasing the load carrying capacity using the physical wedge effect, and the other is a flat plane. This study mainly consists of three parts. First, the static characteristics were obtained over the region of the thin air film using the finite-difference method (FDM) and the bump foil characteristics using the finite-element method (FEM). Second, the analysis of the dynamic characteristics was conducted by perturbation method. For more exact calculation, the rarefaction gas coefficients perturbed about the pressure and film thickness were taken into consideration. At last, the static and dynamic characteristics of the tilting condition of the thrust pad were obtained. Furthermore, the load carrying capacity and torque were calculated for both tilting and nontilting conditions. From this study, several results were presented: (1) the stiffness and damping of the bump foil under the condition of the various bump parameters, (2) the load carrying capacity and bearing torque at the tilting state, (3) the bearing performance for various bearing parameters, and (4) the effects considering the rarefaction gas coefficients.     

Effect of internal friction in the dynamic behavior of aerodynamic foil bearings

The development of aerodynamic foil bearings in the past decade is due, for a major part to environmental reasons because: first, air is a clean lubricant and second foil bearings can be used in very critical thermal environment: air has a poor sensitivity to high temperature changes. Foil bearings are often used for very high velocity turbines. Experimental studies have shown the capability of foil bearing to work under rather high load capacity with good dynamic behavior. Numerical simulations are now able to predict with a good accuracy foil bearing load capacity. They take into account foil deformation and dry friction between foils. For dynamic simulation, dry friction has been taken into account only through damping coefficient. As damping is not completely similar to dry friction, this paper is a first attempt in taking account its effect in the foil bearing dynamic behavior. A non-linear model, coupling a simplified equation for the rotor motion to both Reynolds equation and foil assembly model is described. Then the dynamic behavior, for a given unbalance is studied. For different values of friction coefficient, the rotor trajectory is studied when velocity increased. For low and high friction coefficient, the dynamic behavior shows critical speeds. For medium values (between 0.2 and 0.4), these critical speeds disappear. This work outlines that it is possible to optimize the friction between the foils in order to greatly improve the dynamic behavior of foil bearings. With a detailed analysis of these first results we propose primary physical explanation of this phenomena.     

Rigidity and damping characteristics of hydrodynamic sliding bearing with deformable working surfaces

The model of a hydrodynamic sliding bearing has been developed that takes into consideration the effect of the deformation of sliding surfaces on the bearing characteristics. The deformations of the sliding surfaces are determined when solving the problem of elastohydrodynamic contact of the journal and bearing with account for the pressure in the lubricating film. Variation in the clearance size at the deformation of the bearing and shaft surfaces is found by iterations when solving jointly the problems of lubricant flow and working surface deformation. Elastic deformations of the working surfaces are calculated using a two-dimensional boundary element model and a three-dimensional finite element model of the shaft and bearing. The method of finite elements is applied to calculate the parameters of lubricant flow in the bearing based on the solution of Reynolds equation in the disturbed form. The rigidity and damping characteristics of the sliding bearing with the deformable surfaces are compared to those of the bearing with the rigid surfaces; the results of the two-dimensional model of bearing deformation are compared to those of the three-dimensional one.     

Elasto‐aerodynamic lubrication analysis of a self‐acting air foil journal bearing

Nowadays, air foil bearings find widespread use in very high speed, lightly loaded oil‐free rotating turbomachineries such as compressors and microgas turbines because they have theoretically no speed limitations and they are environmentally benign. In the design of such bearings, it is of cardinal importance to enhance their steady‐state and dynamic performance characteristics for the safety operation, especially against the external dynamic excitations. Most of elasto‐aerodynamic approaches under dynamic conditions proposed in the technical literature include only the static pressure induced deformation of foils. This paper presents a theoretical investigation on the effects of both static and dynamic deformations of the foils on the dynamic performance characteristics and stability of a self‐acting air foil journal bearing operating under small harmonic vibrations. For the dynamic deformations of foils to be taken into account, the perturbation method is used for determining the gas‐film stiffness and damping coefficients for given values of excitation frequency, compressibility number and compliance factor of the bump foil. The rotor‐dynamic coefficients serve as input data for the linear stability analysis of rotor‐bearing system. The nonlinear stationary Reynolds' equation is solved by means of the Galerkin's finite element formulation, whereas the finite differences method are used to solve the first‐order complex dynamic equations resulting from the perturbation of the transient compressible Reynolds' equation. As a first approximation, the corrugated subfoil is modelled as a simple elastic foundation, i.e. the stiffness of a bump is uniformly distributed throughout the bearing surface. It was found that the dynamic properties and stability of the compliant finite length journal bearing are significantly affected by the compliance of foils especially when the dynamic deformation of foils is considered in addition to the static one by applying the principle of superposition. Copyright © 2012 John Wiley & Sons, Ltd.     

Steady-State and Transient Dynamic Simulation of a Continuous Foil Bearing

The objective of this paper is to present an integrated procedure which will analyze the dynamics of a foil hearing on elastic supports. A complete time-dependent Navier-Stokes formulation is used to solve for the interaction between the hydrodynamic effects of the fluid lubricant, the motion of the journal, and the deformable foil boundary. The elastic deformation of the foil and its supports are simulated by a finite-element model. The steady-state, quasi-transient and full transient dynamics of the foil-fluid-journal interaction are examined. For the steady-state simulation of a particular journal position, the fluid lubricant pressures are evaluated by means of a joint iterative scheme until convergence is achieved in both the fluid pressure and the corresponding foil deformation. For the quasi-transient case, the transient motion of the journal is calculated using a numerical integration scheme for the velocity and displacement of the journal. The deformation of the foil is calculated through numerical iteration in the feedback mode in combination with the fluid film pressure generated by the journal motion, until convergence at every time step is achieved. For the full transient simulation, a parallel real-time integration scheme is used, at each time step, to simultaneously evaluate the new journal position and the new deformed shape of the foil.     

LOADING CAPACITY ANALYSIS OF THE MISALIGNED CONICAL-CYLINDRICAL BEARING WITH NON NEWTONIAN LUBRICANTS

A novel numerical method to lubricate a conventional finite diameter conical-cylindrical bearing with a non-Newtonian lubricant, while adhering to the power-law model, is presented. The elastic deformation of bearing and varied viscosity of lubrication due to the pressure distribution of film thickness are also considered. Simulation results indicate that the normal load carrying capacity is more pronounced for higher values of flow behavior index n, higher eccentricity ratios and larger misalignment factors. It is found that the viscosity-pressure to the effect of lubricant viscosity is significant.     

Refrigerant foil bearing behavior—A Thermo-HydroDynamic study: (Application to rigid bearings)

Internal experiments at Liebherr-Aerospace FRANCE (LTS) on new refrigerant-lubricated compressor designs have shown that under specific operating conditions, a mixture of vapor and liquid appears in the compressor, instead of a single-phase vapor flow. Therefore, refrigerant-lubricated foil bearings behavior is studied, including the likelihood of two-phase flow in the lubricant. In this paper, we focus on the lubricant behavior only, in the operating conditions of foil bearings. The coupling with structural behavior is a complex process and will be the topic of another paper. The Thermo-HydroDynamic approach describes lubricant characteristics such as pressure, density, viscosity, and temperature. It involves the use of a Generalized Reynolds Equation for turbulent flow, a nonlinear cubic Equation of State for two-phase flow and a 3D turbulent thin-film energy equation. Journal bearing global parameters are calculated for steady-state conditions.     

Operating characteristics of the bump foil journal bearings with top foil bending phenomenon and correlation among bump foils

In many previous studies, the experimental results show the ripple traces of the post-test top foil due to the top foil deflection differences between the areas touched with and without bump foil. This phenomenon represents clearly the behaviour of the top foil motion. Therefore the top foil deflecting appearance should be taken into consideration for more reliable estimation of the bump foil bearing behaviour. However inherent top foil bending effect was ignored under assumption to the rigid body able to act only vertically. In this paper, a numerical analysis and its experimental investigation were studied with the intention to abstract the static and dynamic characteristics of the bump foil bearing considering the top foil bending effect and correlation among bumps. The bump and top foil inserted between the journal and the bearing housing have each mechanism as a shock absorber and journal supporter, and each mechanism was analysed using numerical method, respectively. In the case of the top foil, the deflection of that and Coulomb damping generated by relative friction was taken into consideration theoretically. In addition bump foil analysis was executed to be assumed to have a structure that flexible bumps with corrugated shape are connected with each other. This analysis verifies that the stiffness at the fixed end where the friction forces between the bearing housing and bump foil superpose is more than that at the free end. Using this structural analysis, the overall analysis of the bump foil bearing was executed through coupling process. In parallel, the experimental process was carried into execution with the bump foil bearing that L/D is 1. The rotating speed was from 10,000 up to 30,000 rpm and the load was 50 N. By comparing the analysis with the experimental results, the effects of the structural characteristics of bump foil were presented.     

以超临界二氧化碳为介质的径向箔片轴承静特性分析

以超临界二氧化碳(S-CO2)布雷顿循环所用压缩机中的径向箔片气体轴承为研究对象,考虑S-CO2在近临界区的实际物性,运用有限差分法数值求解变密度变黏度湍流雷诺方程,耦合弹性箔片变形方程,计算以S-CO2为润滑介质的径向气体轴承的气膜压力分布规律,并分析近临界区S-CO2物性变化对轴承承载力、摩擦力矩的影响.计算结果表...     

Minimum Film Thickness in a Gas Foil Journal Bearing with an Unbalanced Rotor

A gas-lubricated foil journal bearing consists of a compliant metal shell structure that supports a rigid journal or rotor by means of a gas film. The response of this system to the periodic forces of an unbalanced rotor supported by a single bearing is predicted using perturbation analysis. The foil structure and the gas film are modeled with an analytically perturbed finite element approach to predict the rotor dynamic coefficients. A dynamic model of the rotor is used to predict periodic journal motion. The perturbation analysis is then used with the periodic response of the rotor to calculate periodic changes in the gas film thickness. Other quantities such as the gas film pressure and the foil deflection can also be calculated. The model includes bending and membrane effects in the top foil, coupled radial and circumferential deflections in the corrugated sub-foil, and the equivalent viscous dissipation of Coulomb friction effects in the foil structure. The approach is used to investigate the effects of top-foil thickness on minimum film thickness in a bearing.     

Subfoil Stiffness Effects in Gas-Lubricated Foil Journal Bearings

A gas-lubricated foil journal bearing consists of a compliant foil structure that supports a rigid journal by means of a gas film. The foil structure consists of a top foil and a subfoil. The subfoil, which is constructed of one or more corrugated strips of sheet metal, supports the top foil and provides the primary resistance to radial deflection. As such, the radial stiffness of the subfoil has a direct influence on the gas film thickness and pressure distributions in the bearing. A finite element model incorporating the gas film and the foil structure will be used to investigate the effects of axial variations in the stiffness of the subfoil.     

Low-friction wear-resistant coatings for high-temperature foil bearings

Compliant foil bearings offer many advantages over rolling element bearings in high-speed and high-temperature applications. However, implementation of foil bearings in these applications requires development of solid lubricant coatings that can survive the severe operating conditions encountered at high speeds and high temperatures. The objective of this paper is to present results on development of an advanced coating system for use with compliant foil bearings that permits higher operating speeds and temperatures. In order to evaluate the coating performance and to select the best coating combination for implementation, tests were conducted using a high-temperature, high-speed tribometer. In these tests, Inconel test substrates, representative of a portion of a foil bearing, were coated with several different Korolon^TM coatings. The counterface disks were coated with a dense chrome, plasma sprayed PS304, hard chrome and Korolon^TM 1350B. Each test was conducted for 500 start–stop cycles up to 810 °C foil pad temperature under 13.8 kPa normal loading.The test results confirmed the excellent tribological behavior of Korolon^TM coatings for high-speed, high-temperature foil bearing applications. While the tribological behavior of Korolon^TM coatings were determined to be a function of temperature, in most cases a minimum coefficient of friction less than 0.1 was observed during startup/shutdown periods. Based on the measured coefficient of friction and post-test visual inspection of the mating surfaces, the hard chrome coating proved unacceptable for high-temperature applications due to extensive surface cracking. The other disk coatings exhibited excellent tribological performance.Following these tests, a foil journal bearing was designed and a composite coating consisting of Korolon^TM 1350A with an overcoat of Korolon^TM 800 was applied to the bearing top foil; and a dense chrome coating was applied to the journal surface. The foil bearing and journal were installed in a 240-lb thrust turbojet engine and operated successfully to 54,000 rpm for over 70 start–stop cycles and 14 h.