Incompressible Reynolds Equation Research Articles

Nonlinear dynamic analysis of worn gas foil bearings

Highly nonlinear vibration is one of important factors influencing the failure of gas foil bearing-rotor systems. In this paper, an integrated nonlinear dynamic model is constructed to study the dynamic vibration of worn gas foil bearings. In the wear model, the deformation of foil is considered. The Iordanoff model is used to solve the stiffness of foil, and equivalent viscous damping is introduced to amend the friction factor at the interface between the foil and bearing bracket. The incompressible Reynolds equation is solved by using ADI method, and the Newmark-β method is applied to obtain the trajectory of axis center. In numerical examples, the effects of worn zone on the trajectory of axis center, the Poincare map and bifurcation diagram are discussed. The nonlinear vibration resulting from the worn zone is revealed. Comparison with existing results validates this model.

Dynamic characteristics of high-speed gasoline engine turbocharger based on thermo-elasto-hydrodynamic lubrication bearing model and flexible multibody dynamics method.

A flexible multibody dynamic calculation model based on thermo-elasto-hydrodynamic lubrication bearing model was established. This numerical simulation method provided a more realistic turbocharger calculation model and a more reliable theoretical support for studying the dynamic vibration characteristics of the floating ring bearing turbocharger system. In order to fully consider the dynamic characteristics of each component, the behavior of the floating ring bearing was described by generalized incompressible Reynolds equation in thermo-elasto-hydrodynamic lubrication model. The flexible body substructure models were established by the modal synthesis method. Based on this model, the direct mathematical model of the relationship between the eccentricity of the rotor and the oil film clearance on the turbocharger's surface vibration was established. The influence of eccentricity and oil film thickness on the surface vibration of the turbocharger body was calculated by transient dynamics method. The results showed that the eccentricity of the rotor and the vibration of turbocharger housing were monotonic functions, but the interaction between the whirl of internal and external oil films made the mechanism of the influence of the oil film thickness on the turbocharger body's vibration complicated. The research provided a new idea for the structural vibration and synchronous noise control of the supercharger.

Prediction and Measurement of Sealing Properties of Joints Between Wavy Metal Surfaces

The transmissivity of metal-metal sealing joints is investigated experimentally and compared to predictions obtained by modeling. The focus is laid upon a wavy surface contacting a flat rigid part, representative of a seat-to-plug contact in an internal sealing valve encountered in nuclear power plants for instance. Experimental transmissivities are obtained from water leak-rate and pressure drop measurements carried out on a model ring-shape sample seat holding a controlled wavy defect and pressed against a rigid flat plug with a controlled normal load. The sample seat surface is manufactured by face turning a tubular part under radial stress and waviness is obtained after elastic relaxation. Modeling is performed on a three-dimensional finite element model of the assembly, composed of the plug, the sample seat, and its holder. The upper sample seat surface, in which topography is recorded by confocal microscopy, is reconstructed using a modal decomposition on the basis of vibrational eigenmodes. Its lower surface, in contact with the holder, is considered as perfectly flat or with its own defects. The contact aperture field between the seat and the plug is computed for a given normal load and is used to solve the incompressible Reynolds equation with a boundary element method, yielding the transmissivity. Predicted transmissivities reveal to be in good agreement with experimental data at low clamping loads and are overestimated for larger ones. Defects on the lower surface of the seat are shown to have a significant impact on the seat-to plug contact transmissivity.

Performance of Balancing Wedge Action in Textured Hydrodynamic Pad Bearings

This study investigates a mechanism of textured features taking into account the balance of moment termed “balancing wedge action.” The principle of the suggested mechanism is that a change in moment applied to the lubricated area by incorporating textured features promotes the entire wedge action over the lubricated area. In the current study, multiple dimples are created on the stationary surface of an infinite pad bearing. A one-dimensional incompressible Reynolds equation is solved numerically to determine the load-carrying capacity of infinite pad bearings with a centrally located pivot. Numerical results show the importance of the balancing wedge action. When multiple dimples are created at the inlet side or outlet side of the lubricated area, positive load-carrying capacity is realized. When multiple dimples are located around the central area, no balance solution is obtained for the pad. The dimple depth, width, and distribution are varied to investigate the behavior of the load-carrying capacity realized by the action of the balancing wedge.

Workstation Computing of Discretized Reynolds Equations

In this study, the solution methods suitable for use in a workstation to solve the discretized incompressible and compressible fluid Reynolds equations are examined. The workstation used for computing consists of dual six-core central processing units (CPUs) and a 256-core graphics processing unit (GPU) dedicated for computing. To compare the computational performance, the Reynolds equations are solved by a parallel iterative method using multithreaded and GPU computing. Multithreaded computing is conducted by OpenMP directives and GPU computing is executed using either NVIDIA's compute unified device architecture (CUDA) programming or the accelerator programming model (APM). The lubrication models used are an inclined surface slider with or without a central recess and an air journal bearing. In GPU computing, both the CUDA and APM are subjected to communication latency to achieve efficient fine-grained computations. The results show that the performance of the less expensive GPU computing is close to that of multithreaded computing in which the grid sizes are large in the analyses. The performance of the OpenMP-like APM programming is compatible with CUDA computing in the slider cases but is significantly worse for air bearings. This study illustrates the workstation computing for lubrication analyses in which the computing devices can be multi-core CPUs or many-core GPUs.

Numerical Simulation of Gasoline Blending in a Cylindrical Tank Agitated by Side-Entering Agitators

Side-entering agitators are the main mixing system in the process of petroleum blending. Based on the three-dimensional gasoline components mixing model, the incompressible Reynolds equation is selected as the momentum equation and the algorithm of SIMPLE is employed to simulate the flow field of three side-entering agitators. In the unsteady multiphase flow, moving mesh and the standard k-ε turbulent model has been employed to get the mixing time. The results show that the model of side-entering agitators exists low-velocity zones, which are like structures of vortex in the central regions of blending tank, and the mixing time is 33.7s.

Numerical Simulation of Gasoline Blending with Two Different Mixing Systems

Based on two different mixing systems: Rotary Jet Mixing (RJM) system and side-entering agitator, two kinds of three-dimensional gasoline components mixing models are established. The incompressible Reynolds equation is selected as the momentum equation and the algorithm of SIMPLE is used to simulate the jet facility. To get the mixing time, moving mesh and the standard k-ε turbulent model has been employed in the multiphase unsteady flow. The results show that the dead areas of RJM are less than side-entering agitator, and the mixing effects are much better. Furthermore, the mixing time of RJM is only 58.2s, which is 69.7% of Side-entering Agitator.

Improved identification of squeeze-film damper models for aeroengine vibration analysis

Numerical solution of the Reynolds equation imposes a prohibitive computational cost on the dynamic analysis of practical squeeze film damped turbomachinery. To surmount this problem, the present paper develops the use of Chebyshev polynomial fits to identify finite difference (FD) solution of the incompressible Reynolds equation. The proposed method manipulates the Reynolds equation to allow efficient and accurate identification in the presence of cavitation, the feed-groove, feed-ports, end-plate seals and supply pressure. The ability of Chebyshev polynomials to rapidly reproduce FD routines is demonstrated. The bearing models developed are experimentally proven to give more accurate results than alternative analytical bearing models.

Numerical modeling of wave overtopping a levee during Hurricane Katrina

In this paper, a numerical wave model based on the incompressible Reynolds equations and k– ε equations has been applied to estimate the impact of overtopping on levee during storm surge. The free surface locations are represented by volume of fluid function (VOF). The model was satisfactorily tested against empirical equation of overflow discharge at a vertical seawall, and experimental data of overtopping discharge at a sloping seawall. The validated model was used to simulate wave overtopping of the levee system during storm surge of Hurricane Katrina. The time history of wave profiles and velocity magnitude field in the vicinity of the levee are demonstrated and analyzed. It is concluded that the failure of parts of the levee system was caused by erosion during wave overtopping.

Homogenization of Reynolds Equation by Two-Scale Convergence

To increase the hydrodynamic performance in different machine elements, as e.g. journal bearings and thrust bearings, during lubrication it is important to understand the influence of surface roughness. In this connection one encounters homogenization of the incompressible Reynolds equation, where the roughness of the lubricated surface is assumed to be periodic. This problem has recently been studied in more engineering-oriented papers by using the formal method of multiple scale expansion. In this paper, we rigorously prove both homogenization and corrector results by using two-scale convergence, which may be regarded as a justification of the formal multiple scale expansion method described above. Moreover, some numerical illustrations and results are presented.

Homogenization of the unstationary incompressible Reynolds equation

This paper is devoted to the effects of surface roughness during hydrodynamic lubrication. In the numerical analysis a very fine mesh is needed to resolve the surface roughness, suggesting some type of averaging. A rigorous way to do this is to use the general theory of homogenization. In most works about the influence of surface roughness, it is assumed that only the stationary surface is rough. This means that the governing Reynolds type equation does not involve time. However, recently, homogenization was successfully applied to analyze a situation where both surfaces are rough and the lubricant is assumed to have constant bulk modulus. In this paper we will consider a case where both surfaces are assumed to be rough, but the lubricant is incompressible. It is also clearly demonstrated, in this case that homogenization is an efficient approach. Moreover, several numerical results are presented and compared with those corresponding to where a constant bulk modulus is assumed to govern the lubricant compressibility. In particular, the result shows a significant difference in the asymptotic behavior between the incompressible case and that with constant bulk modulus.

109 液体メニスカスが局在する固体面間に生ずる液体反力の解析 : 平行平面間の場合(情報機器コンピュータメカニクス1 : ヘッド・ディスク・インタフェース)

The dynamic force between solid plane surface linked by a liquid meniscus bridge was analysed assuming small vibration of the spacing. The infinite width meniscus and the finite meniscus ring were considered as analytical models. The incompressible Reynolds equation was solved under the Laplace pressure boundary condition and the solution was linearlized by the small vibration assumption. The derivation of the solutions and the differences by the models were discussed. It was found that the pressure and the load carrying capacity in the linearlized analysis had three terms, i. e., squeeze term by the viscosity of liquid, spring term by the dynamic Laplace pressure and static meniscus force term.

136 HDD用スピンドルモータの流体軸受の潤滑解析 : ラジアル軸受とスラスト軸受の相互影響(情報機器コンピュータメカニクス2 : 機構)

Generally, fluid dynamic bearing of the spindle motor for HDD is constituted combining two or more radial bearings and thrust bearings. It is filled with lubricating oil between these bearings, and it is thought that these influence each other through lubricating oil during operation. This paper describes the lubricating analysis with consideration to the interaction between radial bearings and thrust bearings which constitute fluid dynamic bearing of the spindle motor for HDD. Lubricating analysis is solving the incompressible Reynolds equation by the finite element method using the model which combined radial bearings and thrust bearings. The influence which the pump-up value of radial bearing has on the characteristics of thrust bearing is discussed.

Development of a Numerical Procedure to Predict the Flowfield with Sand Erosion.

This paper presents a numerical procedure to predict sand erosion and the interaction between the flowfield and the eroded surface. To simulate this phenomenon, the turbulent flowfield, the particle trajectory and the amount removed from the eroded wall are calculated iteratively. In computations of the flowfield, incompressible Reynolds equations and a high-Reynolds-number type k-e turbulence model are adopted. Assuming that the particle suspension is dilute, the collision of particles and the influence of particle motions to the flowfield are ignored. The erosion model proposed by Neilson and Gilchrist is used to calculate the weight loss. To verify this method, a 90°curved channel and a channel with a bump are computed. The results show that the present procedure can reproduce the sand erosion process and the temporal change of the flowfield qualitatively.

Effects of Membrane Stresses in the Prediction of Foil Bearing Performance©

The effects of membrane stresses which result from both viscous drag and the three-dimensional structure of an incompressible journal foil bearing are discussed. The structural model includes combined bending, membrane, and elastic foundation effects. The pressure in the lubricant film is predicted using an incompressible Reynolds equation. Finite element methods are used to predict both the structural deflections and the pressures in the lubricant film. Results will demonstrate that membrane effects are significant in an elastically supported foil bearing where elastic foundation effects are normally considered to provide the primary resistance to deflection. The effects of viscous drag are small but not insignificant. Presented at the 48th Annual Meeting in Calgary, Alberta, Canada May 17–20, 1993

Theoretical Performance of a Hydrostatic Foil Bearing

The performance of a hydrostatic foil journal bearing operating with an incompressible fluid is discussed. In the configuration considered here, pressurized lubricant fluid is supplied through capillaries to recessed pockets on the surface of the journal. The foil is treated as a perfectly flexible, inextensible shell by a three dimensional finite element model. The pressure distribution is predicted by a finite element formulation of the incompressible Reynolds equation. Results are presented which demonstrate that the foil structure enhances load support while increasing lubricant film thickness. In addition, results indicate that membrane effects are essential in the structural model.

The effects of variable permeability and roughness of porous bearings

The effects of surface roughness and variable permeability on the hydrodynamic characteristics of squeeze-film porous bearings have been studied by analyzing the ensemble-averaged incompressible Reynolds equation. Film thickness averages are derived for a roughness distribution given by a polynomial fit of a Gaussian distribution. The numerical results are obtained and compared with the anisotropic case having a different constant permeability in the respective directions. These findings are also compared with those for bearings having a precise description of roughness. It is shown that the load-carrying capacity decreases for higher values of the permeability parameter and that the optimum load shifts for off-squared plates. The squeeze-film time for the anisotropic case is found to be quite large compared to that for the isotropic case.

Investigation of a Flexible Disk Rotating Near a Rigid Surface

A flexible disk rotating in close proximity to a flat rigid surface has been studied experimentally and theoretically. A bump is located on the surface to create a non-axisymmetric disturbance. The gas film between the disk and the rigid surface is represented by an incompressible Reynolds equation with inertial effects included. The general equations of a disk including both membrane and bending effects with displacement dependent tension distributions are presented. Numerical results are developed where the effects of displacement on the tension distributions and of bending in the disk are not included.

Axisymmetric Solution of a Flexible Disk Rotating Near a Rigid Surface

A rotating flexible disk separated from a rigid flat surface by a gas film is addressed. The gas film between the disk and the plate is represented by an incompressible Reynolds equation. Inertial effects are included. The disk is treated as a membrane where the tension is found from the plane stress solution for a spinning disk. Two different methods for the axisymmetric solution of this system are developed. The first uses the method of matched asymptotic expansions. The second method is a mixed numerical/perturbation procedure.

Analytic Solution of a Finite-Width Rough Surface Hydrodynamic Bearing

An analytic solution of the full incompressible unaveraged Reynolds equation is obtained by regular perturbation expansion for the case of a finite width bearing with transverse roughness. Results show that the unaveraged pressure is first influenced by the roughness statistics at the 0 (ε2) level and is negligibly affected by the details of the roughness distribution. Through second order effects, it is shown that the closure assumption utilized by Christensen and To̸ender in developing their theory for striated roughness effects on incompressible lubrication is correct. That is, for transverse roughness [h = h(x)], h3 and ∂P/∂y are statistically independent. Due to the linearity of the incompressible Reynolds equation, it is likely that a similar approach can be used for the study of fully three-dimensional roughness effects on lubrication in order to reveal the detailed relationship between clearance and pressure.