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Research Papers: Contact Mechanics

# Modeling of Finite-Length Line Contact Problem With Consideration of Two Free-End Surfaces

[+] Author and Article Information
Haibo Zhang, Shengguang Zhang, Ziqiang Zhao

School of Mechanical Engineering,
Beijing Institute of Technology,
Beijing 10008, China

Wenzhong Wang

School of Mechanical Engineering,
Beijing Institute of Technology,
Beijing 10008, China
e-mail: wangwzhong@bit.edu.cn

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received May 15, 2015; final manuscript received July 20, 2015; published online October 15, 2015. Assoc. Editor: Sinan Muftu.

J. Tribol 138(2), 021402 (Oct 15, 2015) (10 pages) Paper No: TRIB-15-1159; doi: 10.1115/1.4031403 History: Received May 15, 2015; Revised July 20, 2015

## Abstract

Finite-length line contact conditions, existing in applications such as gears or roller bearings, lead to subsurface stress distribution influenced by the free boundaries. This paper presents a semi-analytical method (SAM) for the finite-length line contact problem, based on the overlapping concept and matrix formation, to consider the effect of two free-end surfaces. In order to obtain two free surfaces, three half-spaces with mirrored loads to be solved are overlapped to cancel out the stresses at expected surfaces. The error introduced by this method is analyzed and proven to be negligible. The conjugate gradient method (CGM) is used to solve the pressure distribution, and the fast Fourier transform (FFT) is used to speed up the elastic deformation and stress-related calculation. The model is verified by finite element method (FEM) and shows a high conformity and efficiency. Besides, the line contact situations are discussed to explore the effect of free surfaces.

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## References

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## Figures

Fig. 1

Schematic plot for finite-length space problem

Fig. 2

Fig. 3

The load condition of the vertical (V) half-space

Fig. 4

The load condition of the reversed vertical (V) half-space

Fig. 5

Flowchart for solving finite-length space contact problem

Fig. 6

Schematic diagram of contact model in present paper (y = 0 plane)

Fig. 7

Three-dimensional (3D) surface of overlapped σxx, τxy, and τxz on surface Ι when Lf = Lc = 5 mm and the variation of R41 and R51 with increasing γ (Lf = Lc = 1, 3, 5, 7, 9, and 11 mm): (a) the overlapped normal stress σxx, (b) the overlapped shear stress τxy, (c) the overlapped shear stress τxz, and (d) the variation of R41 and R51 with γ

Fig. 8

Comparison between the present method and the FEM. (The simulation domain of the present method and FEM is both (0, 17.243a) × (−10a, 10a) × (0, 10a) along the x, y, and z directions.) (a) FEM model with mesh design and (b) pressures along the x-axis.

Fig. 9

Contour plots of the results obtained by FEM and present method. (FEM solutions are shown in left plots and solutions based on the present method are shown in right plots.) (a) Contour plots of P/p0 in the plane z = 0, (b) contour plots of the σvon/p0 in the plane y = 0, and (c) contour plots of the σxx/p0 in the plane y = 0.

Fig. 10

Grid effect and comparisons of the CPU time between the present method and FEM for different mesh densities. (The simulation domain of the present method and FEM is both (0, 17.243a) × (−10a, 10a) × (0, 10a) along the x, y, and z directions. “Matrix calculation” stands for the CPU time consumed for the calculation of three matrices MA, MB, and MC; “call the matrices” represents the CPU time consumed by calling the matrices MA, MB, and MC.) (a) σvon along the x-axis (half-domain is plotted) and (b) CPU time consumption.

Fig. 11

Effect of free surface distance d when a cylindrical roller contacts with a finite-length space and half-space: (a) pressure profile along x-axis and (b) σvon contour plots in y = 0 plane

Fig. 12

Pressure profile along x-axis for different r0 and e (Lf = Lc = 5 mm, R1 = R2): (a) pressure profile along x-axis when r0 = 9a and (b) pressure profile along x-axis when r0 = 39a

Fig. 13

The σvon distribution for different e when r0 = 34a

Fig. 14

The contact situation of tapered roller with coincident end: (a) the pressure along x-axis when R1 increases while R2 remains 19.05 mm and (b) the contour plots of pressure (in z = 0 plane) and σvon (in y = 0 plane)

## Errata

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