A New Active Tilting-Pad Bearing: Nonlinear Modeling and Feedback Control

This article introduces a new type of active fluid film bearing and its feedback control. In particular, the active adjustment of the angular velocity of the pads of a tilting-pad bearing in response to changes in the operating conditions of the rotating machine is proposed. This is motivated by the observation that there is more control authority in the pad tilting motion than in its radial translation. To this end, a dynamic model for the bearing system is first developed, inclusive of the nonlinear hydrodynamic force for the infinitely short bearing case. A model-based controller is then constructed, based on measurements of the journal position and velocity and pad tilting angles, to ensure that the journal is asymptotically regulated to the bearing center. Numerical simulations illustrate the performance of the active bearing under the proposed control in comparison with the bearing's standard passive mode of operation.     

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.     

A Three-Dimensional Foil Bearing Performance Map Applied to Oil-Free Turbomachinery

To effectively apply compliant foil gas bearings to increasingly larger and more challenging turbomachinery, a comprehensive method that compares a foil bearing's capabilities with the application's operating requirements is needed. Extensive laboratory and field experience suggests that foil bearing failure is generally due to thermal stress brought on by excessive viscous power loss; therefore, a map that graphically relates component- and system-level parameters (bearing size, applied loads, and shaft rotational speeds) directly to bearing power loss is more elucidating than a map based on a lumped speed/load parameter like the Sommerfeld number. In this article we describe a performance map featuring a three-dimensional contour plot that illustrates the expected power loss in a foil bearing as a function of applied load and shaft speed. Using this performance map, bearing capabilities can be examined at the anticipated system operating conditions and safety margins between an operating point and incipient bearing failure can be ascertained. To demonstrate the concept's features and usefulness, we present a performance map generated from foil bearing power loss test data. We expect that these maps, combined with other predictive tools, will help evaluate a foil bearing's general suitability for a candidate rotor system and will lead to more robust and successful oil-free turbomachinery designs.     

Transient analysis of misaligned elastic tilting-pad journal bearing

For several reasons, almost all bearings operate in a misaligned condition, the present research work deals with analyzing the performance of a misaligned tilting-pad journal bearing under transient loading condition. The elastic and thermal distortions of the pad are considered and finite element analysis is used to calculate the pad’s elastic deformation. Using finite difference method, the Reynolds equation is simultaneously solved with the energy equation to calculate the pressure distribution and hence the other bearing performance characteristics. A modified fluid film thickness equation is used to take the effect of shaft misalignment and bearing elastic and thermal distortion into consideration.The results have shown that considering the thermo-elasto-hydrodynamic distortion improves the bearing performance in the case of misalignment shaft. And, at low values of shaft misalignment, the decrease in oil film thickness due to shaft misalignment is compensated by the increase in oil film thickness due to elastic and thermal distortions.     

Wave Journal Bearing with Compressible Lubricant—Part I: The Wave Bearing Concept and a Comparison to the Plain Circular Bearing

To improve hydrodynamic journal bearing steady-stale and dynamic performance, a new bearing concept, the wave journal bearing, was developed at the author's lab. This concept features a waved inner bearing diameter. Compared to other alternative bearing geometries used to improve bearing performance such as spiral or herringbone grooves, steps, etc., the wave bearing's design is relatively simple and allows the shaft to rotate in either direction. A three-wave bearing operating with a compressible lubricant; i.e., gas, is analyzed using a numerical code. Its performance is compared to a plain (truly) circular bearing over a broad range of bearing working parameters, e.g., bearing numbers from 0.01 to 100. The geometry of the wave bearing gives the bearing its high load; i.e., stiffness, and stability characteristics. The wave bearing's performance is dependent upon the amplitude of the wave and the position of the waves relative to the applied load. To maximize wave bearing performance, the waves' position relative to the applied load should be carefully selected. The wave journal bearing offers better stability than the plain circular bearing' under all operating conditions and all wave-load orientations. Specifically, an unloaded journal bearing can be made to run stably in any operating regime by incorporating the wave geometry.     

The effect solid particle lubricant contamination on the dynamic behavior of compliant journal bearings

The proposed work concerns a theoretical and numerical investigation of the effect of solid particle contamination of lubricant oils on the static and dynamic characteristics of a finite length compliant journal bearing operating under isothermal conditions with laminar flow. In the present investigation, we use simple models based on the Einstein's mixture theory, which is characterized by the presence of suspended rigid particles in a fluid. Using the classical assumptions of lubrication, a Reynolds equation is derived and solved numerically by the finite difference method. The displacement field at the fluid film bearing liner interface due to pressure forces is determined using the elastic thin layer model. The results obtained show that the presence of suspended rigid particles in the lubricating oil (solid contamination) has significant effects on the hydrodynamic performance characteristics such as the pressure field, friction force, flow rate, elastic surface deformation as well as stability maps of the rotor‐bearing system (critical mass and whirl frequency) especially at high volumetric concentration.     

TEHD Analysis for Tilting-Pad Journal Bearings Using Upwind Finite Element Method

A general approach for incorporating heat transfer and elastic deformation effects into a tilting-pad journal bearing simulation model is presented. A global analysis method is used, which includes variable viscosity and heat transfer effects in the fluid film, elastic deformation and heat conduction effects in the pads, and elastic deformation effect in the pivots. The two-dimensional variable viscosity. Reynolds equation produces pressure distributions in the axial and circumferential directions. The energy equation is two-dimensional, assuming that the temperature variation in the axial direction is negligible. The elasticity and heat conduction models are also two-dimensional, being in the midline cross-section of the bearing, including the circumferential and cross-film directions. An upwind technique is used in the finite element formulation of the energy equation to remove numerical instability due to the convective term. Simulation results are compared with the test and predicted values of previous researchers.     

可倾瓦气体动压轴承承载能力的数值分析

应用MATLAB的偏微分方程工具箱,采用有限元法求解气体润滑Reynolds方程,通过完全装配分析法计算了可倾瓦动压气体轴承的承载能力,研究了轴承偏心和瓦块预负荷对可倾瓦轴承的承载能力、最小气膜厚度和最大气膜压力以及瓦块摆角的影响。计算结果表明,随轴承偏心或瓦块预负荷增大,各瓦块的承载能力和摆角发生明显变化,轴承的最小气膜厚度减小,最大气膜压力增大,承载能力增大。从理论上解释了可倾瓦轴承的承载能力与轴承偏心和瓦块预负荷的密切相关性。     

Nonlinear Dynamic Analysis of a Rigid Rotor Supported By a Spiral-Grooved Opposed-Hemisphere Gas Bearing

The nonlinear dynamic behavior of a rigid rotor supported by a spiral-grooved opposed-hemisphere gas bearing is investigated in this article, focusing particular attention on its whirl motion. The finite element method combined with the finite difference method is employed to solve the time-dependent Reynolds equation that is coupled with the rotor motion considering five degrees of freedom. The rotor responses to the initial disturbance and synchronous and nonsynchronous excitations are investigated. To analyze the complicated dynamic behavior of the rotor–bearing system, the trajectories of the rotor centerline, time responses, phase portraits, power spectra, Poincare maps, and bifurcation diagrams are obtained from the numerical procedure. The results show that the conical whirl instability appears earlier than the cylindrical whirl instability with increasing rotational speed for the rotor–bearing system with no unbalance mass. Moreover, it reveals that the complex dynamic behavior of the system excited by unbalance mass varies with rotational speed and rotor mass. In addition, bifurcation diagrams employing the rotating speed and rotor mass as bifurcation parameters are obtained. Finally, the nonsynchronous excitation responses are presented, which behave in a different way than the synchronous excitation responses. The results of this study offer a further understanding of the nonlinear characteristics of spiral-grooved opposed-hemisphere gas bearings.     

An analytical model for complete dynamical coefficients of a tilting-pad journal bearing

An analytical model as well as calculation method is presented for the complete dynamic characteristics of tilting-pad journal bearing. Using this model, the global oil-film forces, stiffness and damping coefficients acting on the journal and all pads can be calculated in a highly concise expression. To improve the computational efficiency, a fast algorithm for calculating the oil-film force and Jacobian of the local pad system is proposed and the Newton–Raphson method is used for solving the equilibrium positions of the journal and all pads. Taking a rigid rotor symmetrically supported on two identical five-shoe tilting-pad journal bearing as an example, the complete dynamic characteristics, damping natural frequency and the stability are calculated. Compared with the traditional reduced model, the numerical results show that the dynamic characteristics can be calculated efficiently and succinctly. The stability can also be overestimated by using the reduced model.     

Analysis of heavy duty tilting-pad journal bearing taking into account pad distortion and possible adoption of rubber pad segments

In this work a comparative study has been made between the thermo-hydrodynamic performance of a three shoe tilting-pad journal bearing with rigid and elastic pads subjected to unbalance load. A case study of a bearing adopting three rubber pad segments has also been studied. The distortion of the elastic pad is introduced into the distribution of the film thickness through an iterative scheme to assess its effect on the load carrying capacity of the bearing. A finite element mesh is used to calculate the distortion of the elastic pad while a finite difference mesh is used to calculate the pressure field in the lubricant film. Results have shown a number of interesting conclusions regarding the adoption of rubber pad segments instead of tilting-pads. There is an increase in minimum oil film thickness when using elastic pad or even rubber pad segments compared with rigid pads. Also the maximum pressure and load carrying capacity are not significantly affected.     

STOCHASTIC PROGNOSTICS FOR ROLLING ELEMENT BEARINGS

The capability to accurately predict the remaining life of a rolling element bearing is prerequisite to the optimal maintenance of rotating machinery performance in terms of cost and productivity. Due to the probabilistic nature of bearing integrity and operation condition, reliable estimation of a bearing's remaining life presents a challenging aspect in the area of maintenance optimisation and catastrophic failure avoidance. Previous study has developed an adaptive prognostic methodology to estimate the rate of bearing defect growth based on a deterministic defect-propagation model. However, deterministic models are inadequate in addressing the stochastic nature of defect-propagation. In this paper, a stochastic defect-propagation model is established by instituting a lognormal random variable in a deterministic defect-propagation rate model. The resulting stochastic model is calibrated on-line by a recursive least-squares (RLS) approach without the requirement of a priori knowledge on bearing characteristics. An augmented stochastic differential equation vector is developed with the consideration of model uncertainties, parameter estimation errors, and diagnostic model inaccuracies. It involves two ordinary differential equations for the first and second moments of its random variables. Solving the two equations gives the mean path of defect propagation and its dispersion at any instance. This approach is suitable for on-line monitoring, remaining life prediction, and decision making for optimal maintenance scheduling. The methodology has been verified by numerical simulations and the experimental testing of bearing fatigue life.     

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.     

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.     

Rotordynamic Characteristics of Flexure-Pivot Tilting-Pad Journal Bearings

Many of today's modern turbomachines, especially those running at high speeds and high power ratings, require the superior stability characteristics of tilting-pad journal bearings to prevent rotor-dynamic instabilities. Until now, the design complexity of tilting-pad bearings has precluded their use in many small, high-volume applications where cost and size are important. This paper introduces a new one-piece journal bearing design, the flexure-pivot bearing, that offers many of the beneficial rotodynamic advantages of tilting-pad bearings, without the complexities of a multi-piece design. Performance data for a flexure-pivot bearing is shown for an application requiring a highly stable design, illustrating the effectiveness of the flexure-pivot bearing in offering rotordynamic stability approaching that of a tilting-pad bearing.     

Design and parameter estimation of hybrid magnetic bearings for blood pump applications

This paper discusses the design and parameter estimation of the dynamics characteristics of a high-speed hybrid magnetic bearings (HMBs) system for axial flow blood pump applications. The rotor/impeller of the pump is driven by a three-phase permanent magnet (PM) brushless and sensorless DC motor. It is levitated by two HMBs at both ends in five-degree-of-freedom with proportional-integral-derivative (PID) controllers; among which four radial directions are actively controlled and one axial direction is passively controlled. Test results show that the rotor can be stably supported to speeds of 14,000 rpm. The frequency domain parameter estimation technique with statistical analysis is adopted to validate the stiffness and damping coefficients of the HMBs system. A specially designed test rig facilitated the estimation of the bearing's coefficients in air—in both the radial and axial directions. The radial stiffness of the HMBs is compared to the Ansoft's Maxwell 2D/3D finite element magnetostatic results. Experimental estimation showed that the dynamics characteristics of the HMBs system are dominated by the frequency-dependent stiffness coefficients. The actuator gain was also successfully calibrated and may potentially extend the parameter estimation technique developed in the study of identification and monitoring of the pump's dynamics properties under normal operating conditions with fluid.     

Wave Journal Bearing with Compressible Lubricant—Part II: A Comparison of the Wave Bearing with a Wave-Groove Bearing and a Lobe Bearing

To identify the potential advantages of the wave journal bearing, a three-wave journal bearing was compared to both a three-wave-groove bearing (a wave bearing with axial grooves that isolate each wave) and a three-lobe bearing. The lobe bearing's profile was selected to approximate the wave journal bearing's profile. The lubricant was assumed to be compressible (gas). The bearing number, A, was parameterized from 0.01 to 100, and the eccentricity ratio, ε, was varied from 0 to 0.4. Data at bearing numbers 0.1, 1, and 50, and eccentricity ratios of 0.1 and 0.4, were selected as representative of the bearing performance. The calculated load capacity and the critical mass are presented for the three bearings. The wave bearing shows a better load capacity than the other bearings at any applied load and running regime. However, at high bearing numbers the lubricant compressibility effect is predominant and all three analyzed bearings show similar load capacity. The critical masses of the wave-groove and lobe bearing are greater than the critical mass of the wave bearing if the applied load is small. For low and intermediate bearing numbers the wave-groove bearing is more stable than the other bearings especially at low wave's amplitude ratio. The lobe bearing is more stable than the other analyzed bearings at high bearing numbers or at large preload ratios. If the applied load increases, the wave bearing dynamic performance is competitive with both wave-groove and lobe bearings. In addition, at high bearing numbers, the wave bearing could run stably for any allocated rotor mass over a wide range of wave position angle. Three wave bearings are more sensitive to the direction of the applied load than the other bearings especially at low and intermediate bearing numbers. Therefore, a careful selection of the waves position angle has to be done to maximize the wave bearing performance.     

Numerical Calculation of Rotation Effects on Hybrid Air Journal Bearings

Hybrid air journal bearings are of great importance in the precision engineering. Despite much progress, the influence of the aerostatic effect and the aerodynamic effect on the bearings is still not clear. Numerical calculation is a useful technique to evaluate bearing performance. Many theoretical problems related to Reynolds equation have been figured out by numerical simulation. The present study analyzes the effects of rotational speed—that is, the bearing speed number—on the performance of hybrid bearings. The behaviors of the pure aerostatic bearing and the pure aerodynamic bearing are investigated for comparison. The second-order finite difference method (FDM) and an iterative procedure are proposed to solve the Reynolds equation and derive the air film pressure distribution. The bearing characteristics such as load capacity, stiffness, friction coefficient, attitude angle, and mass inflow rate are taken into consideration. The research reveals the very dependence of the hybrid bearing's performance on the journal rotation and eccentricity ratio. The numerical results indicate that at a small bearing speed number of 0.223 and eccentricity ratio of 0.15, about 99.8% of the load capacity and 99.7% of the stiffness are determined by the aerostatic effect, whereas at a large bearing speed number of 2.229 and eccentricity ratio of 0.55, about 63.2% of the load capacity and 83.3% of the stiffness are determined by the aerodynamic effect.     

Static and dynamic performance characteristics of an orifice compensated hydrostatic journal bearing with non-Newtonian lubricants

This paper presents a theoretical study of the performance characteristics of hydrostatic rigid orifice compensated multirecess journal bearings using non-Newtonian lubricants. The generalized Reynolds equation governing the flow of lubricant having variable viscosity has been solved using the finite element method and iterative procedure. The static and dynamic performance characteristics are presented for non-Newtonian lubricants of which constitutive equation has been represented by the cubic shear stress law. The non-linearity factor () in the cubic shear stress law significantly influences the bearing performance characteristics, particularly the dynamic characteristics.     

Stability of multi orifice active tilting-pad journal bearings

The stability properties of actively lubricated tilting-pad journal bearings are investigated theoretically. The bearing preload factor and control system gains are varied, and stable and unstable regions are identified. It is seen, that the control system influences bearing stability, and that the nature and magnitude of this influence depends on the rotor mass, preload factor and rotational speed. Furthermore, it is shown that assuming the bearing pads to be rigid can produce a substantial error. A rigid pad model will overpredict the stable range of the bearing, thus it may lead to failure if trusted.