Application of the matrix form of the Reynolds equation to dynamic journal bearings.

This paper studies the bifurcation of a rigid rotor supported by a gas film bearing. A time-dependent mathematical model for gas journal bearings is presented. The finite differences method and the Successive Over Relation (S.O.R) method are employed to solve the Reynolds’ equation. The system state trajectory, Poincare´ maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor center in the horizontal and vertical directions under different operating conditions. The analysis shows how the existence of a complex dynamic behavior comprising periodic and subharmonic response of the rotor center. This paper shows how the dynamic behavior of this type of system varies with changes in rotor mass and rotational velocity. The results of this study contribute to a further understanding of the nonlinear dynamics of gas film rotor-bearing systems.

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The Effect of Journal Bearing Misalignment on Load and Cavitation for Non-Newtonian Lubricants

Journal of Tribology ◽

10.1115/1.3261295 ◽

1986 ◽

Vol 108 (4) ◽

pp. 645-654 ◽

Cited By ~ 40

Author(s):

R. H. Buckholz ◽

J. F. Lin

Keyword(s):

Reynolds Equation ◽

Numerical Solutions ◽

Journal Bearings ◽

Surface Boundary ◽

Film Rupture ◽

Aspect Ratios ◽

Free Surface Boundary Condition ◽

Load Carrying ◽

Boundary Location ◽

Surface Boundary Condition

An analysis for hydrodynamic, non-Newtonian lubrication of misaligned journal bearings is given. The hydrodynamic load-carrying capacity for partial arc journal bearings lubricated by power-law, non-Newtonian fluids is calculated for small valves of the bearing aspect ratios. These results are compared with: numerical solutions to the non-Newtonian modified Reynolds equation, with Ocvirk’s experimental results for misaligned bearings, and with other numerical simulations. The cavitation (i.e., film rupture) boundary location is calculated using the Reynolds’ free-surface, boundary condition.

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Theoretical Condition for the Cxy=Cyx Relation in Fluid Film Journal Bearings

Journal of Tribology ◽

10.1115/1.3261942 ◽

1989 ◽

Vol 111 (3) ◽

pp. 426-429 ◽

Cited By ~ 4

Author(s):

T. Kato ◽

Y. Hori

Keyword(s):

Reynolds Equation ◽

Journal Bearing ◽

Journal Bearings ◽

Fluid Film ◽

Finite Width ◽

Damping Coefficients ◽

Dynamic Coefficients ◽

Cross Terms ◽

Linear Damping ◽

Theoretical Condition

A computer program for calculating dynamic coefficients of journal bearings is necessary in designing fluid film journal bearings and an accuracy of the program is sometimes checked by the relation that the cross terms of linear damping coefficients of journal bearings are equal to each other, namely “Cxy = Cyx”. However, the condition for this relation has not been clear. This paper shows that the relation “Cxy = Cyx” holds in any type of finite width journal bearing when these are calculated under the following condition: (I) The governing Reynolds equation is linear in pressure or regarded as linear in numerical calculations; (II) Film thickness is given by h = c (1 + κcosθ); and (III) Boundary condition is homogeneous such as p=0 or dp/dn=0, where n denotes a normal to the boundary.

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The Effect of Lubricant Inertia in Journal-Bearing Lubrication

Journal of Applied Mechanics ◽

10.1115/1.4011588 ◽

1957 ◽

Vol 24 (4) ◽

pp. 494-496

Author(s):

J. F. Osterle ◽

Y. T. Chou ◽

E. A. Saibel

Keyword(s):

Reynolds Number ◽

Steady State ◽

Reynolds Equation ◽

Journal Bearing ◽

Journal Bearings ◽

Hydrodynamic Theory ◽

Simple Relationship ◽

Inertia Effect ◽

Laminar Regime ◽

Steady State Operation

Abstract The Reynolds equation of hydrodynamic theory, modified to take lubricant inertia into approximate account, is applied to the steady-state operation of journal bearings to determine the effect of lubricant inertia on the pressure developed in the lubricant. A simple relationship results, relating this “inertial” pressure to the Reynolds number of the flow. It is found that the inertia effect can be significant in the laminar regime.

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An Application of Fuzzy Clustering to Customer Portfolio Analysis in Automotive Industry

International Journal of Fuzzy System Applications ◽

10.4018/ijfsa.2016040102 ◽

2016 ◽

Vol 5 (2) ◽

pp. 13-25

Author(s):

Abdulkadir Hiziroglu ◽

Umit Dursun Senbas

Keyword(s):

Fuzzy Clustering ◽

Automotive Industry ◽

Quantitative Assessment ◽

Portfolio Analysis ◽

Matrix Form ◽

Automotive Supplier ◽

Qualitative And Quantitative ◽

Marketing Managers ◽

The Matrix

Having achieved an optimized customer portfolio has been of significant importance for companies. The literature provides several portfolio models and vast majority of them are in matrix form where several descriptors are used as dimensions of the matrix. These dimensions are characterized in ambiguity and require specific methods to tackle with it. The aim of this paper is to utilize fuzzy clustering in customer portfolio analysis to reduce this uncertainty and to make a comparison with a traditional customer portfolio model. A dataset of 130 customers of an automotive supplier in Turkey is used to perform the analyses and the results are compared with a conventional customer portfolio matrix. By making use of substantiality and balance of portfolio parameters, a qualitative and quantitative assessment of categorization generated by both approaches are evaluated. The use of fuzzy clustering gives more substantial clusters and a more balanced customer portfolio compared to the traditional matrix form of portfolio. Marketing managers can understand their overall customer portfolio better and reduce the effect of descriptive indicators via benefiting the fuzzy clustering results.

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Numerical and Experimental Investigations on Preload Effects in Air Foil Journal Bearings

Volume 7A: Structures and Dynamics ◽

10.1115/gt2017-63284 ◽

2017 ◽

Cited By ~ 1

Author(s):

Marcel Mahner ◽

Pu Li ◽

Andreas Lehn ◽

Bernhard Schweizer

Keyword(s):

Reynolds Equation ◽

Journal Bearing ◽

Journal Bearings ◽

Compressible Fluids ◽

Experimental Investigations ◽

Hysteresis Curves ◽

Numerical Predictions ◽

Bearing Stiffness ◽

Gasdynamic Model ◽

Good Agreement

A detailed elasto-gasdynamic model of a preloaded three-pad air foil journal bearing is presented. Bump and top foil deflections are herein calculated with a nonlinear beamshell theory according to Reissner. The 2D pressure distribution in each bearing pad is described by the Reynolds equation for compressible fluids. With this model, the influence of the assembly preload on the static bearing hysteresis as well as on the aerodynamic bearing performance is investigated. For the purpose of model validation, the predicted hysteresis curves are compared with measured curves. The numerically predicted and the measured hysteresis curves show a good agreement. The numerical predictions exhibit that the assembly preload increases the bearing stiffness (in particular for moderate shaft displacements) and the bearing damping.

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Analysis of the effect of oil inlet size on the static performance characteristics of non-Newtonian lubricated hole-entry hybrid journal bearings

Journal of Mechanics ◽

10.1093/jom/ufab020 ◽

2021 ◽

Vol 37 ◽

pp. 522-531

Author(s):

Haiyin Cao ◽

Yu Huang ◽

Youmin Rong ◽

Hao Wu ◽

Minghui Guo

Keyword(s):

Finite Element Method ◽

High Speed ◽

Reynolds Equation ◽

Journal Bearings ◽

Performance Characteristics ◽

Power Law Model ◽

The Finite Element Method ◽

Solution Strategy ◽

Nonuniform Mesh ◽

Static Performance

Abstract In this study, the influence of inlet pocket size on the static performance of non-Newtonian lubricated hole-entry hybrid journal bearings is theoretically analyzed. The oil film of the bearing is discretized into a nonuniform mesh containing the geometric characteristics of the oil inlet pocket, and the inlet pocket is treated as a micro-oil recess. The Reynolds equation is solved by the finite element method based on Galerkin's techniques, and a new solution strategy to solve the recess/pocket pressure is proposed. The power-law model is used to introduce the non-Newtonian effect. The results show that the static performance characteristics of this type of bearing are greatly affected by the pocket size at both zero speed and high speed.

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Matrix form of interval multivariable gray model and its application

Grey Systems Theory and Application ◽

10.1108/gs-09-2020-0120 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Sandang Guo ◽

Yaqian Jing ◽

Bingjun Li

Keyword(s):

Three Dimensional ◽

Original Data ◽

Upper Bounds ◽

Matrix Form ◽

Gray Model ◽

Interval Numbers ◽

Content Type ◽

The Matrix ◽

Number Sequences ◽

The Difference

PurposeThe purpose of this paper is to make multivariable gray model to be available for the application on interval gray number sequences directly, the matrix form of interval multivariable gray model (IMGM(1,m,k) model) is constructed to simulate and forecast original interval gray number sequences in this paper.Design/methodology/approachFirstly, the interval gray number is regarded as a three-dimensional column vector, and the parameters of multivariable gray model are expressed in matrix form. Based on the dynamic gray action and optimized background value, the interval multivariable gray model is constructed. Finally, two examples and comparisons are carried out to verify the effectiveness of IMGM(1,m,k) model.FindingsThe model is applied to simulate and predict expert value, foreign direct investment, automobile sales and steel output, respectively. The results show that the proposed model has better simulation and prediction performance than another two models.Practical implicationsDue to the uncertainty information and continuous changing of reality, the interval gray numbers are used to characterize full information of original data. And the IMGM(1,m,k) model not only considers the characteristics of parameters changing with time but also takes into account information on lower, middle and upper bounds of interval gray numbers simultaneously to make better suitable for practical application.Originality/valueThe main contribution of this paper is to propose a new interval multivariable gray model, which considers the interaction between the lower, middle and upper bounds of interval numbers and need not to transform interval gray number sequences into real sequences. According to combining different characteristics of each bound of interval gray numbers, the matrix form of interval multivariable gray model is established to simulate and forecast interval gray numbers. In addition, the model introduces dynamic gray action to reflect the changes of parameters over time. Instead of white equation of classic MGM(1,m), the difference equation is directly used to solve the simulated and predicted values.

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Symmetries of the Degeneracy of the Vertebrate Mitochondrial Genetic Code in the Matrix Form

Symmetrical Analysis Techniques for Genetic Systems and Bioinformatics ◽

10.4018/978-1-60566-124-7.ch002 ◽

2010 ◽

pp. 31-49

Author(s):

Sergey Petoukhov ◽

Matthew He

Keyword(s):

Mathematical Modeling ◽

Genetic Code ◽

Matrix Form ◽

Black And White ◽

Mitochondrial Genetic Code ◽

The Matrix

Symmetries of the degeneracy of the vertebrate mitochondrial genetic code in the mosaic matrix form of its presentation are described in this chapter. The initial black-and-white genomatrix of this code is reformed into a new mosaic matrix when internal positions in all triplets are permuted simultaneously. It is revealed unexpectedly that for all six variants of positional permutations in triplets (1-2-3, 2-3-1, 3-1-2, 1-3-2, 2-1-3, 3-2-1) the appropriate genetic matrices possess symmetrical mosaics of the code degeneracy. Moreover the six appropriate mosaic matrices in their binary presentation have the general non-trivial property of their “tetra-reproduction,” which can be utilized in particular for mathematical modeling of the phenomenon of the tetra-division of gametal cells in meiosis. Mutual interchanges of the genetic letters A, C, G, U in the genomatrices lead to new mosaic genomatrices, which possess similar symmetrical and tetra-reproduction properties as well.

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A Hybrid Mobility Solution Method for Dynamically Loaded Misaligned Journal Bearings

ASME/STLE 2012 International Joint Tribology Conference ◽

10.1115/ijtc2012-61225 ◽

2012 ◽

Cited By ~ 2

Author(s):

S. Boedo

Keyword(s):

Finite Element ◽

Reynolds Equation ◽

Solution Method ◽

Hybrid Approach ◽

Journal Bearings ◽

Finite Element Solution ◽

Computational Time ◽

Element Solution ◽

Dynamically Loaded ◽

Mobility Solution

This paper presents a hybrid mobility solution approach to the analysis of dynamically loaded misaligned journal bearings. Mobility data obtained for misaligned bearings (calculated from a finite element representation of the Reynolds equation) are compared with existing curve-fitted mobility maps representative of a perfectly aligned bearing. A relative error analysis of mobility magnitude and direction provides a set of misaligned journal bearing configurations (midplane eccentricity ratio and normalized misalignment angle) where existing curve-fitted mobility map components based on aligned bearings can be used to calculate the resulting journal motion. For bearing configurations where these mobility maps are not applicable, the numerical simulation process proceeds using a complete finite element solution of the Reynolds equation. A set of numerical examples representing misaligned main and connecting rod bearings in a four-stroke automotive engine illustrate the hybrid solution method. Substantial savings in computational time are obtained using the hybrid approach over the complete finite element solution method without loss of computational accuracy.

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