Parallel Iterative Solution Schemes for the Analysis of Air Foil Bearings

2012 ◽  
Vol 28 (3) ◽  
pp. 413-422 ◽  
Author(s):  
N. Wang ◽  
S.-H. Chang
Keyword(s):  
Reynolds Equation ◽  
Iterative Solution ◽  
Solution Procedure ◽  
Sor Method ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Sor Methods ◽  
Numerical Damping ◽  
Parallel Iterative ◽  
Successive Over Relaxation

AbstractIn an air foil bearing analysis the model is usually solved iteratively due in part to the nonlinearity of the modeling Reynolds equation and the compliance of the bearing surface. The solution procedure requires a multiple-level-deep nested iteration, which involves extended solution time and convergence difficulty. In this study, a simple air foil bearing model is used and the compressible-fluid Reynolds equation for modeling gas lubrication is linearized by Newton's method. The discretized equation is solved by one of the two parallel iterative methods, red-black or strip partition successive-over-relaxation (SOR) method. The parallel programming is conducted using OpenMP programming in an eight-core work-station. Then, a numerical damping scheme for the film-profile convergence is presented. Finally, a root-finding process is conducted to iteratively attain the eccentricity of the bearing for a given load. It is found that the numerical damping step is crucial, which allows the use of a larger relaxation factor to have a fast rate of convergence. Both the parallel SOR methods are easy to implement and the red-black SOR method exhibits better efficiency in the studied cases. This study presents a parallel computing scheme for analyzing air foil bearing of bump-type by today's shared-memory multicore platforms.

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Nenzi Wang ◽  
Shih-Hung Chang ◽  
Hua-Chih Huang
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Compressible Fluid ◽  
Newton's Method ◽  
Reynolds Equation ◽  
Multigrid Method ◽  
Iterative Solution ◽  
Sor Method ◽  
Solution Methods ◽  
Iterative Solution Methods

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

The Effect of External Pressurization on Self-Acting Foil Bearings

1965 ◽  
Vol 87 (3) ◽  
pp. 631-638 ◽  
Author(s):  
M. Wildmann ◽  
A. Wright
Keyword(s):  
Pressure Distribution ◽  
Equilibrium Equation ◽  
Reynolds Equation ◽  
Foil Bearings ◽  
Foil Bearing

The effects of introducing a small amount of lubricant under pressure into a self-acting foil bearing film are investigated. Foil shape and pressure distribution under the foil are obtained by combining the equilibrium equation with the Reynolds’ equation and solving the resulting equation. The results show that the effect of even small external pressurization in a self-acting foil bearing is very important.

Thermo-elastohydrodynamic analysis of the inhomogeneous foil bearing with misalignment

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Hao Li ◽  
Haipeng Geng ◽  
Hao Lin
Keyword(s):  
Reynolds Equation ◽  
Energy Equation ◽  
Dynamic Performance ◽  
Gas Temperature ◽  
Content Type ◽  
Rotor Systems ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Deformation Equation ◽  
Small Perturbation Method

Purpose The misalignment is generally inevitable in the process of machining and assembly of rotor systems with gas foil bearings, but the exploration on this phenomenon is relatively less. Therefore, the purpose of this paper is to carry out the thermo-elastohydrodynamic analysis of the foil bearing with misalignment, especially the inhomogeneous foil bearing. Design/methodology/approach The rotor is allowed to misalign in two non-rotating directions. Then the static and dynamic performance of the inhomogeneous foil bearing is studied. The thermal-elastohydrodynamic analysis is realized by combining the Reynolds equation, foil deformation equation and energy equation. The small perturbation method is used to calculate the dynamic coefficients, then the critical whirl ratio is obtained. Findings The gas pressure, film thickness and temperature distribution distort when the misalignment appears. The rotor misalignment can improve the loading capacity but rise the gas temperature at the same time. Furthermore, the rotor misalignment can affect the critical whirl ratio which demonstrates that it is necessary to analyze the misalignment before the rotordynamic design. Originality/value The value of this paper is the exploration of the thermo-elastohydrodynamic performance of the inhomogeneous foil bearing with misalignment, the analysis procedure and the corresponding results are valuable for the design of turbo system with gas foil bearings.

Fast Convergence of Iterative Computation for Incompressible-Fluid Reynolds Equation

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Nenzi Wang ◽  
Kuo-Chiang Cha ◽  
Hua-Chih Huang
Keyword(s):  
Reynolds Equation ◽  
Truncation Error ◽  
Solution Method ◽  
Iterative Solution ◽  
Stopping Criterion ◽  
Iterative Computation ◽  
Grid Convergence ◽  
Grid Convergence Index ◽  
Successive Over Relaxation ◽  
Iterative Solution Method

When a discretized Reynolds equation is to be solved iteratively at least three subjects have to be determined first. These are the iterative solution method, the size of gridwork for the numerical model, and the stopping criterion for the iterative computing. The truncation error analysis of the Reynolds equation is used to provide the stopping criterion, as well as to estimate an adequate grid size based on a required relative precision or grid convergence index. In the simulated lubrication analyses, the convergent rate of the solution is further improved by combining a simple multilevel computing, the modified Chebyshev acceleration, and multithreaded computing. The best case is obtained by using the parallel three-level red-black successive-over-relaxation (SOR) with Chebyshev acceleration. The speedups of the best case relative to the case using sequential SOR with optimal relaxation factor are around 210 and 135, respectively, for the slider and journal bearing simulations.

Effects of wedge curvature on performances of foil thrust bearing and the profile design in compressor system

2020 ◽  
pp. 135065012097552
Author(s):  
Hao Li ◽  
Haipeng Geng ◽  
Lei Qi ◽  
Lu Gan
Keyword(s):  
Reynolds Equation ◽  
Thrust Bearing ◽  
Dynamic Performance ◽  
Testing System ◽  
Thrust Bearings ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Set Up ◽  
Foil Thrust Bearing ◽  
Excellent Stability

Foil thrust bearings have attracted considerable attention in small-sized turbo machines with its excellent stability, high compliance, temperature durability. Geometric structure play an important role on the performance of foil thrust bearings. However, the current research on the structure mainly focuses on the underlying foil type, such as bump foil, protuberant bump. In fact, the foil profile, especially in the convergent region has significant influence. In this paper, foil thrust bearings were classified into convex, slope and concave types according to the profile curvature. A numerical model of six pads foil thrust bearing was established by combining the shell model and Reynolds equation. The static and dynamic performance of thrust bearings with different curvature was calculated. The results showed that the convex convergent possessed higher capacity and was not sensitive to displacement disturbance. A stiffness testing system for thrust foil bearing was set up, and the results verified that the foil with convex wedge had higher stiffness. The experiment also indicated that all the thrust foil bearings had typical damping hysteresis. The axial force of a 10 kW on-board compressor was calculated. Based on the conclusion of this paper, the design scheme of curvature value β = 0.6 and gas thickness h2=15 µm was given in consideration of bearing capacity and machining robustness.

Calculation of Stiffness and Damping Coefficients for Elastically Supported Gas Foil Bearings

1993 ◽  
Vol 115 (1) ◽  
pp. 20-27 ◽  
Author(s):  
J.-P. Peng ◽  
M. Carpino
Keyword(s):  
Reynolds Equation ◽  
Structural Model ◽  
Structural Equations ◽  
Perfect Gas ◽  
Foil Bearings ◽  
Damping Coefficients ◽  
Dynamic Coefficients ◽  
Foil Bearing ◽  
Stiffness And Damping ◽  
Elastically Supported

The stiffness and damping coefficients of an elastically supported gas foil bearing are calculated. A perfect gas is used as the lubricant, and its behavior is described by the Reynolds equation. The structural model consists only of an elastic foundation. The fluid equations and the structural equations are coupled. A perturbation method is used to obtain the linearized dynamic coefficient equations. A finite difference formulation has been developed to solve for the four stiffness and the four damping coefficients. The effect of the bearing compliance on the dynamic coefficients is discussed in this paper.

Parallel iterative solution-based tabu search for the obnoxious p-median problem

2021 ◽  
Vol 127 ◽  
pp. 105155
Author(s):  
Jian Chang ◽  
Lifang Wang ◽  
Jin-Kao Hao ◽  
Yang Wang
Keyword(s):  
Tabu Search ◽  
Iterative Solution ◽  
Median Problem ◽  
Parallel Iterative

scholarly journals Self-updated four-node finite element using deep learning

2021 ◽  
Author(s):  
Jaeho Jung ◽  
Hyungmin Jun ◽  
Phill-Seung Lee
Keyword(s):  
Finite Element ◽  
Deep Learning ◽  
Iterative Solution ◽  
Solution Procedure ◽  
Element Formulation ◽  
Zero Energy ◽  
Energy Mode ◽  
Bending Modes ◽  
Solution Accuracy ◽  
Iterative Solution Procedure

AbstractThis paper introduces a new concept called self-updated finite element (SUFE). The finite element (FE) is activated through an iterative procedure to improve the solution accuracy without mesh refinement. A mode-based finite element formulation is devised for a four-node finite element and the assumed modal strain is employed for bending modes. A search procedure for optimal bending directions is implemented through deep learning for a given element deformation to minimize shear locking. The proposed element is called a self-updated four-node finite element, for which an iterative solution procedure is developed. The element passes the patch and zero-energy mode tests. As the number of iterations increases, the finite element solutions become more and more accurate, resulting in significantly accurate solutions with a few iterations. The SUFE concept is very effective, especially when the meshes are coarse and severely distorted. Its excellent performance is demonstrated through various numerical examples.

Foil Gas Bearing With Compression Springs: Analyses and Experiments

2007 ◽  
Vol 129 (3) ◽  
pp. 628-639 ◽  
Author(s):  
Ju-ho Song ◽  
Daejong Kim
Keyword(s):  
Loss Factor ◽  
Load Capacity ◽  
Underlying Structure ◽  
Equivalent Viscous Damping ◽  
Foil Bearings ◽  
Foil Bearing ◽  
Different Types ◽  
Structural Loss ◽  
Compression Springs ◽  
Gas Bearing

A new foil gas bearing with spring bumps was constructed, analyzed, and tested. The new foil gas bearing uses a series of compression springs as compliant underlying structures instead of corrugated bump foils. Experiments on the stiffness of the spring bumps show an excellent agreement with an analytical model developed for the spring bumps. Load capacity, structural stiffness, and equivalent viscous damping (and structural loss factor) were measured to demonstrate the feasibility of the new foil bearing. Orbit and coast-down simulations using the calculated stiffness and measured structural loss factor indicate that the damping of underlying structure can suppress the maximum peak at the critical speed very effectively but not the onset of hydrodynamic rotor-bearing instability. However, the damping plays an important role in suppressing the subsynchronous vibrations under limit cycles. The observation is believed to be true with any air foil bearings with different types of elastic foundations.

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