Compliant Foil Research Articles

Starved Hydrodynamic Gas Foil Bearings—Experiment, Micromechanical Phenomenon, and Hypotheses

In this paper, a hypothesis for the operating tribological mechanisms and phenomena occurring in compliant surface gas foil bearings subjected to low ambient pressure conditions, such as occur at high altitude or in soft vacuum, will be presented and discussed. Both theoretical and experimental evidence supporting the proposed hypothesis will be presented to show that, under low ambient pressure conditions (i.e., something akin to starved fluid film lubrication), the shaft is supported by a combination of hydrodynamic and morphological elements. The theoretical treatment of the compressible fluid film in a simple gas bearing is highly nonlinear in-and-of-itself, and especially more so when combined with a compliant surface supported on a frictional-elastic foil foundation. Adding a “molecularly starved gas film” to this highly nonlinear system, one encounters a very interesting and complex system that, heretofore, has not been considered. When operating compliant foil gas bearings in a near or soft vacuum, the term hydrodynamic may be considered oxymoronic in that there is little or no apparent fluid/gas to provide “a full hydrodynamic” action. However, theoretical and experimental evidence of compliant surface foil gas bearings operating at low ambient pressures show that they do continue to work and, in fact, can do so quite well given the appropriate compliancy and other factors, as yet to be discussed. In this paper, the situation will be addressed based upon the experimental evidence that resulted in the essential hypothesis that there are elements at work that go above and beyond purely hydrodynamic phenomenon or so-called solid lubrication. These elements include both tribological and morphological interactions, which are at work at all times and it is the respective ratios of hydrodynamic and morphological elements that characterize operation. Evidence is presented to the effect that, even when hydrodynamic effects dominate, morphological interactions contribute to bearing performance and load-carrying capacity and that, when morphological effects dominate, third body and surface elements impart to the interface many of the characteristics and effects of a hydrodynamic film. Thus, by combining classical Reynolds equation modified for compressible media with the quasi-hydrodynamic/continuum equations and the appropriate rheological and morphological parameters, meaningful solutions for foil bearing operating with extreme low-pressure boundary conditions are possible, and which result in increased load-carrying capacity contrary to classical hydrodynamic theory.

Structural Stiffness of X-750 Alloy Bump Foil Strips for Compliant Foil Bearings With Different Heat Treatments

This paper focuses on the structural stiffness of bump foils which are used for compliant foil bearings with different heat treatments. After heat treatments in vacuum environments, the mechanical properties of the foil strips were tested, and the structural stiffness was estimated from the static load versus displacement curves obtained from the experiments. High cycle dynamic load tests were also applied to the bump foil under different cycle loads, and the shape of the foil was scanned after the tests to measure the height variation of the bumps. The results show that the modulus of elasticity and strength of Inconel X-750 strip with thickness of 0.1 mm after different treatments are lower than that with the thickness of 0.18 mm at room temperature. Moreover, the sample foil strips which have been treated with a lower solution anneal temperature at 980 °C (2 hrs) and precipitation heat treatment at 732 °C (16 hrs) have the largest modulus of elasticity and strength at room temperature. Therefore, heat treatments have a great influence on the structure stiffness of the bump foil. At last, the results of the high cycle dynamic load tests show that the bump foil with suitable heat treatment will have a good load capacity and stress-relaxation property.

Foil Bearing Stiffness Estimation with Pseudospectral Scheme

Compliant foil gas lubricated bearings are used for the support of light loads in the order of few kilograms at high speeds, in the order of 50,000 RPM. The stiffness of the foil bearings depends both on the stiffness of the compliant foil and on the lubricating gas film. The stiffness of the bearings plays a crucial role in the stable operation of the supported rotor over a range of speeds. This paper describes a numerical approach to estimate the stiffness of the bearings using pseudo spectral scheme. Methodology to obtain the stiffness of the foil bearing as a function of weight of the shaft is given and the results are presented. Keywords—Foil bearing, simulation, numerical, stiffness estimation.

Foil Bearing Stiffness Estimation with Pseudospectral Scheme

Compliant foil gas lubricated bearings are used for the support of light loads in the order of few kilograms at high speeds, in the order of 50,000 RPM. The stiffness of the foil bearings depends both on the stiffness of the compliant foil and on the lubricating gas film. The stiffness of the bearings plays a crucial role in the stable operation of the supported rotor over a range of speeds. This paper describes a numerical approach to estimate the stiffness of the bearings using pseudo spectral scheme. Methodology to obtain the stiffness of the foil bearing as a function of weight of the shaft is given and the results are presented.

Application of a compliant foil bearing for the thrust force estimation in the single stage radial blower

AbstractThe paper presents the application of a compliant foil bearing for estimation of the thrust force in a single stage radial blower under operational conditions. The bump foil of the thrust bearing behaves as a nonlinear spring. The knowledge of the spring deflection curve allows estimation of the actual thrust force for a measured bump deflection at the given rotational speed. To acquire the deflection curve, static calibration of the axial shaft displacement sensor was performed. During the calibration, the information about voltage signals of the sensor for the given loading force was collected. The measured voltage values at different speeds and loadswere then converted into the thrust force. The results were verified by comparison to the thrust force resulting from the pressure distribution on the impeller.

Modeling of Gas Lubricated Compliant Foil Bearings Using Pseudo Spectral Scheme

Modeling of Gas Lubricated Compliant Foil Bearings Using Pseudo Spectral Scheme

A fully coupled 3D thermo-elastohydrodynamics model for a bump-type compliant foil journal bearing

Coupled thermo-elastohydrodynamics model has been developed to predict three-dimensional thermal, structural and hydrodynamic performance of foil bearings. Bearing deformation and film pressure has been coupled in FEM considering thermal/centrifugal growths, and thermo-mechanical material properties. Energy equation has been solved for film temperature using FDM, which is also coupled to FEM via successive iterations. Augmented-Lagrangian method and thermal contact models have been applied to solve for mechanical and thermal contacts. The predicted values have been benchmarked against published measurements, which indicated reasonable correlation. A parametric study has been conducted for various shaft speeds and load conditions to visualize bearing performance.

Experimental Identification of Dynamic Force Coefficients for a 110 MM Compliantly Damped Hybrid Gas Bearing

Compliant hybrid gas bearings (HGBs) combine key enabling features from both fixed geometry externally pressurized gas bearings and compliant foil bearings. The compliant hybrid bearing relies on both hydrostatic and hydrodynamic film pressures to generate load capacity and stiffness to the rotor system, while providing damping through integrally mounted metal mesh bearing support dampers. This paper presents experimentally identified force coefficients for a 110 mm compliantly damped gas bearing using a controlled-motion test rig. Test parameters include hydrostatic inlet pressure, excitation frequency, and rotor speed. The experiments were structured to evaluate the feasibility of implementing these bearings in large size turbomachinery. Dynamic test results indicate weak dependency of equivalent direct stiffness coefficients to most test parameters except for frequency and speed, where higher speeds and excitation frequency decreased equivalent bearing stiffness values. The bearing system equivalent direct damping was negatively impacted by increased inlet pressure and excitation frequency, while the cross-coupled force coefficients showed values an order of magnitude lower than the direct coefficients. The experiments also include orbital excitations to simulate unbalance response representative of a target machine while synchronously traversing a critical speed. The results indicate the gas bearing can accommodate vibration levels larger than the set bore clearance while maintaining satisfactory damping levels.

In-situ characterization of strain in lithium battery working electrodes

During lithiation and detlithiation, substantial volumetric changes occur within the electrode materials used for rechargeable lithium batteries. The magnitude of these deformations is inherently linked to the electrical capacity of the battery electrical capacity, which tends to degrade with repeated cycling. In this work, the relationship between electrical discharge capacity and mechanical deformation state is examined using in-situ imaging of the working electrode surface within a custom CR2032 coin cell lithium battery. Digital image correlation is used to quantify electrode strains throughout the discharge–charge process. The effect of constraint due to substrate stiffness is investigated for two film materials: traditional graphite and a carbon nanotube based composite. Results for all cases show that as discharge capacity decreases with repeated cycling, increasing residual electrode strains are observed. The thin, compliant foil substrates allowed over double the bi-axial strain state to be induced within electrodes, compared to that found for the thick copper disk substrates under the same electrical cycling conditions.

Efficient solution of the non-linear Reynolds equation for compressible fluid using the finite element method

An efficient finite element scheme for solving the non-linear Reynolds equation for compressible fluid coupled to compliant structures is presented. The method is general and fast and can be used in the analysis of airfoil bearings with simplified or complex foil structure models. To illustrate the computational performance, it is applied to the analysis of a compliant foil bearing modelled using the simple elastic foundation model. The model is derived and perturbed using complex notation. Top foil sagging effect is added to the bump foil compliance in terms of a close-form periodic function. For a foil bearing utilized in an industrial turbo compressor, the influence of boundary conditions and sagging on the pressure profile, shaft equilibrium position and dynamic coefficients is numerically simulated. The proposed scheme is faster, leading to the conclusion that it is suitable, not only for steady-state analysis, but also for non-linear time domain analysis of rotors supported by airfoil bearings.

A Thermohydrodynamic Sparse Mesh Model of Bump-Type Foil Bearings

A numerical model for 3D thermohydrodynamic analysis of bump-type foil bearings with a sparse mesh across the air film is described. The model accounts for heat convection into cooling air, thermal expansion of the bearing components, and material property variations due to temperature rise. Deflection of the compliant foil strip, described as a link-spring structure, is coupled to the solution of the generalized Reynolds equation and the energy equation to account for the effect of foil deformation on the film thickness. The variation in bump stiffness with the thermal growth of bumps is also considered in the model. The unique airflow in foil bearings created by the top foil detachment in the subambient region is analyzed for use in modifying the thermal boundary condition. The Lobatto point quadrature algorithm is used to represent the model on a sparse mesh and thereby reduce the computational effort. The calculated bearing temperatures are in remarkable agreement with both the published test data with the use of cooling air and that without the use of cooling air. The change of bearing radial clearance due to thermal growth of the bearing components was found to significantly affect the bearing load and to be a likely cause of the obvious drop in load capacity with a rise in ambient temperature.

Key Problem of Solving Nonlinear Reynolds Equation

Finite element equation of the nonlinear dimensionless Reynolds equation, based on the Galerkin finite element method, was derived. Three key points of solving the equation was studied in detail, i.e. Boolean matrix was calculated under the nonlinear conditions, and a method of integrating discrete element equations was provided; Nonlinear algebraic equations set, resulted from integrated finite element equations, was obtained and a method how to substitute boundary conditions into the algebraic equations was presented; A method of calculating the Jacobi matrix of the equations set were described in this paper. All of them are crucial to solve the nonlinear Reynolds equation and helpful for promoting the further research on compliant foil gas bearing.

Simulation of Compliant Bump Foil Journal Bearing Using Coupled Reynolds Equation and Finite Element Model Method

Based on a “1st generation” compliant foil journal bearing, the performance of foil bearing was simulated according to the model base on Reynolds equation coupled the finite element method. Model considers the bending of top foil, elastic deformation of bump foil, the friction between top foil and bump foil and the interaction between bumps. After compared analysis results to test data, the result of simulation was more accurate than other models, thus this model establish a basis for further study of the compliant foil bearing.

Experimental Investigation of Prototype Water-Lubricated Compliant Foil Bearings

First gas-lubricated compliant foil bearings (CFBs) were built in the 1950s. Due to their significant advantages, such as oil-free operation, good tolerance to bearing misalignment and very low maintenance, they have been penetrating the bearing applications for high speed compressors, air-cycle machines and gas turbines. The work presented here investigates a novel idea of water-lubricated compliant foil bearings, which could be used in applications where environmentally friendly lubrication is desired, for example in hydroelectric turbines or water pumps. Experimental results collected for three prototype water-lubricated foil journal bearings are presented. The tests were conducted under steady radial load and with the sliding speed varied incrementally. A sequence of design improvements is presented, with the best bearing demonstrating friction coefficient of about 0.01 at the sliding speed of about 4 m/s and the radial load of about 300 kPa. Encountered difficulties, research methodology and the testing equipment are also described.

Dynamic Properties Analysis of Compliant Foil Aerodynamic Bearings Based on Spring Model

The dynamic coefficients are the essential parameters, which could be used to determine the critical rotational speed, an unbalance response and the stability analysis. A plural mathematic model of the coefficients calculation is developed by using the numerical perturbation method. A spring model is applied to get the foil deformation. The dynamic coefficients of the gas foil bearings are obtained through the application of the finite difference method and the finite element method. The results show that the dynamic coefficients are closely related to the journal eccentricity and whirl ratio. With the results compared to the published data, the model is proved to be accurate. And the coefficients could be used as a reference for the design of a gas foil bearing.

Prediction of Dynamic Coefficients of Bump-Type Foil Bearings with Bumps Considered as Link-Spring Structures

Bump-type foil bearings have demonstrated excellent performance in service experiences in recent 30 years. Due to their special structures, the performance prediction models must couple the air pressure with the elastic deflection of the compliant foil structure. The solution of air pressure has been well settled in the hydrodynamic lubrication theory, while the deflection of foils now still cannot be accurately solved due to the mechanical complexity of the corrugated bump structure. A theoretical model, link-spring model, was developed to predict the structure characteristics of foil strips in a previous work. Effects of four factors, i.e., the elasticity of the bump foil, the interaction forces between bumps, the friction forces at contact surfaces, and the local deflection in the top foil, had been taken into consideration. In this investigation, based on the link-spring model, a perturbation approach is used to determine the dynamic characteristics of bump-type foil bearings. The calculations have been performed with different meshes. And the accordant results show that the calculation is independent on the mesh. The effects on the dynamic coefficients of the friction forces and radial clearance are also evaluated using the perturbation method.

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.

Structural Properties of Foil Bearings: A Closed-Form Solution Validated with Finite Element Analysis

Fluid film thickness in a compliant foil bearing is greatly influenced by the deflection of the bearing structure. Therefore, in order to properly model performance of a foil bearing, it is mandatory that the deflection of the compliant bearing structure due to the generated hydrodynamic pressure is determined accurately. This article proposes an easy-to-use two-dimensional model, which takes into account detailed geometry of the bump foil-top foil assembly and the interaction between bumps and which can predict the bearing deflection and stresses due to an arbitrary pressure load. The proposed model is first validated using a finite element analysis and the results available in the literature and then used to conduct a parametric study investigating the influence of bump foil geometry, the coefficient of friction between the bearing components, and the type of loading on the structural properties of the bearing. The most important parameters are also identified. The proposed analytical model is completely algebraic and can be easily implemented using any programming language, a spreadsheet, or even a calculator. The resulting solution can also be coupled with the appropriate hydrodynamic model to predict static performance of compliant foil bearings.

Compliant foil fluid film radial bearing or seal

Compliant foil fluid film radial bearing or seal

Dynamic stability experiments of compliant foil thrust bearing with viscoelastic support

In order to reduce the complexity of machine construction and improve the mechanical efficiency, high speed rotation machineries usually implement self-acting gas bearings to substitute the traditional oil-lubricated bearings. This paper presents test results of a gas thrust bearing with viscoelastic support which is designed for high speed turbo-machinery. The gas bearing, which belongs to compliant foil bearings, consists of a top thin metal foil and a bottom thin rubber foil. Static and stability experiments are conducted on a high speed gas turbine test rig. The static results indicate that the structural stiffness of test bearing generally increases with the increase in axial load and the decrease in thickness of bottom foil. In the rotation tests, rotor runs stably with small vibration amplitude, which is dominant in waterfall plot during whole speed up procedure. It shows that test bearing has preferable stability characteristics for high speed gas turbines.