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Research Papers: Hydrodynamic Lubrication

A New Singularity Treatment Approach for Journal-Bearing Mixed Lubrication Modeled by the Finite Difference Method With a Herringbone Mesh

[+] Author and Article Information
Yanfeng Han, Jiaxu Wang

State Key Laboratory
of Mechanical Transmission,
Chongqing University,
174 Shazhengjie, Shapingba District,
Chongqing 40044, China

Shangwu Xiong

Department of Mechanical Engineering, Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208

Q. Jane Wang

Department of Mechanical Engineering,
Northwestern University,
2145 Sheridan Road,
Evanston, IL 60208;
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
174 Shazhengjie, Shapingba District,
Chongqing 40044, China
e-mail: qwang@northwestern.edu

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received March 14, 2015; final manuscript received July 21, 2015; published online August 31, 2015. Assoc. Editor: Mihai Arghir.

J. Tribol 138(1), 011704 (Aug 31, 2015) (10 pages) Paper No: TRIB-15-1079; doi: 10.1115/1.4031138 History: Received March 14, 2015; Revised July 21, 2015

Steady-state mixed hydrodynamic lubrication of rigid journal bearing is investigated by using a finite difference form of the Patir–Cheng average Reynolds equation under the Reynolds boundary condition. Two sets of discretization meshes, i.e., the rectangular and nonorthogonal herringbone meshes, are considered. A virtual-mesh approach is suggested to resolve the problem due to the singularities of pressure derivatives at the turning point of the herringbone mesh. The effectiveness of the new approach is examined by comparing the predicted load with that found in the literature for a smooth-surface case solved in the conventional rectangular mesh. The effects of the skewness angles of symmetric and asymmetric herringbone meshes on the predicted parameters, such as load, friction coefficient, attitude angle, and maximum pressure, are investigated for smooth, rough, and herringbone-grooved bearing surfaces. It is found that the new approach helps to improve the computational accuracy significantly, as demonstrated by comparing the results with and without the treatment of the pressure derivative discontinuity although the latter costs slightly less computational time.

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Figures

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Fig. 1

(a) Illustration of a skewed discretization mesh and (b) the neighboring nodes of node (i, j) when β = 0 deg. x-y are the Cartesian coordinates, β indicates the skewness angle of the herringbone meshes, A is a node in λ1−φ1 skewed coordinate, B is a node in λ2−φ2 skewed coordinate, and C is the turning location.

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Fig. 2

Node (i, j) and its neighbors. (a) Common approach without any singularity treatment and (b) new approach using virtual nodes, marked by cross, to treat the singularity. L and R mark the points neighboring virtual node (i ̃, J + 1) on the line of y=yJ+1 in the φ − λ2 grids.

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Fig. 4

Total bearing load and attitude angle obtained in the rectangular mesh and herringbone meshes without singularity treatment at the turning points. β indicates the skewness angle of the herringbone meshes, with β = 0 for the rectangular mesh, and yc is for the turning location. (a) yC = 0.5L, symmetric, using 80 × 41 and 160 × 41 nodes and (b) yC = 0.25L and yC = 0.125L using 160 × 41 nodes.

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Fig. 5

Predicted total bearing load and attitude angle using the rectangular mesh (β = 0) and the herringbone meshes with the first virtual-node interpolation technique. (a) yC = 0.5L, symmetric, using 160 × 41 nodes; (b) yC = 0.25L using 160 × 41 nodes; and (c) yC = 0.125L using 160 × 41 nodes.

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Fig. 6

Predicted total bearing load and attitude angle using the rectangular mesh (β = 0) and herringbone meshes with the second and third virtual-node interpolation technique (yC = 0.5L). (a) Second virtual-node interpolation technique using 160 × 41 nodes and (b) third virtual-node interpolation technique using 160 × 41 nodes.

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Fig. 7

Total load error using the rectangular mesh (β = 0) and herringbone meshes with 160 × 41 nodes (yC = 0.5L). (a) Without singularity treatment; (b) first virtual-node interpolation technique; and (c) comparison between without singularity treatment and virtual-node interpolation technique.

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Fig. 8

Comparison of the total bearing load obtained by using the rectangular mesh (β = 0) and the symmetric herringbone mesh (yC = 0.5L) with the first virtual-node interpolation technique

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Fig. 9

Predicted pressures for smooth bearing surface by using the rectangular and herringbone meshes (eccentricity ratios at 0.75 and 0.99)

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Fig. 10

Numerical results for mixed lubrication obtained by using the rectangular and symmetric herringbone meshes, withthe new method enforced by the first singularity treatment technique for the latter. (a) Attitude angle and the total load; (b) asperity contact load and the average friction coefficient; and (c) comparison of total load error for without singularity treatment and first singularity treatment technique.

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Fig. 11

Comparison of load capacity with Refs. [7] and [10]. The geometry and lubricant parameters are the same with Ref.[10].

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Fig. 12

Herringbone grooves in the bearing surface

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Fig. 13

Numerical results for the herringbone-grooved bearing surface (groove angle = mesh angle = 60 deg) obtained by using the rectangular and symmetric herringbone meshes, with the new method enforced by the first singularity treatment technique for the latter. (a) Maximum lubricant pressure and theminimum film thickness; (b) asperity contact load and the average friction coefficient; and (c) attitude angle and the total load. The β = 0 (rectangular mesh) and β = 60 deg (herringbone mesh) cases with yC = 0.5L are directly comparable, showing good agreement.

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Fig. 14

Predicted film thickness and pressure distributions for the herringbone-grooved surface (groove angle = 60 deg) at eccentricity ratio = 0.75. (a) Rectangular mesh, β = 0 deg and yC=0.5L and (b) herringbone meshes with the first singularity treatment, β = 60 deg and yC=0.5L.

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