Three-Dimensional Semiperiodic Line Contact–Periodic in Contact Length Direction

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Shuangbiao Liu ◽  
Diann Y. Hua
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Three Dimensional ◽  
Contact Problems ◽  
Contact Length ◽  
Computational Domain ◽  
Line Contact ◽  
Discrete Convolution ◽  
Length Direction ◽  
Contact Domain

Line contact problems, such as those seen in spur gears and cam-roller follower systems, are often simplified with the plane-strain assumption and thus modeled by two-dimensional equations. However, in order to address the effects of roughness and textured surfaces, three-dimensional modeling is necessary. The challenge arises when the contact domain is several orders of magnitude greater than the grid size needed to properly describe the surface roughness or texture. Considering the surface geometry of a so-called “line contact,” the contact domain is nonperiodic in contact width direction, but it can be treated as periodic in the contact length direction–semiperiodic line contact problem. Thus, only a section of the entire contact domain is used as the computational domain with a much-reduced size. Based on an in-depth investigation of available algorithms, DC-FFTS and DC-CC-FFT algorithms are proposed. The DC-FFTS algorithm is a modified discrete convolution and fast Fourier transform algorithm with superposition of influence coefficients. The DC-CC-FFT algorithm is a hybrid fast Fourier transform based algorithm, which combines the discrete convolution–FFT and the continuous convolution–FFT methods. The proposed algorithms are used to solve three-dimensional displacement, contact pressure, and stresses for line contact problems. The results are compared with the other available algorithms from literature. The accuracy and efficiency of different algorithms are discussed.

scholarly journals FFT-ASSISTED ALGORITHMS FOR 3D LINE-CONTACT PROBLEMS

2021 ◽  
Vol 13 (2) ◽  
pp. 124-129
Author(s):  
Sergiu Spinu ◽  
◽  
Keyword(s):  
Fourier Transform ◽  
Contact Problem ◽  
Fast Fourier Transform ◽  
Contact Area ◽  
Contact Problems ◽  
Contact Length ◽  
The Other ◽  
Line Contact ◽  
Contact Width ◽  
Length Direction

The line-contact is a particular type of contact with a contact length much greater than its width. Such contact scenarios can be treated in the frame of a two-dimensional plane-strain problem if the contacting surfaces can be considered nominally smooth. However, surface irregularities inherent to any manufacturing technique lead to a discontinuous contact area that differs from the one derived on the basis of the smooth profile assumption. It is therefore tantalizing to pursue the solution of a line-contact problem using an intrinsically three-dimensional (3D) model, which can only be numerical due to lack of general analytical solutions in contact mechanics. Considering the geometry of the line-contact, a major challenge in its numerical modelling is that the expected contact area is orders of magnitude larger in one direction compared to the other. This may lead to an unreasonably large number of grids in the contact length direction, which translates to a prohibitive computational burden. An alternative approach, employed in this paper, is to treat the line-contact as non-periodic in the contact width direction, but periodic in the contact length direction, with a period equal to the window required to capture and replicate the surface specific texture. This periodicity encourages the contact problem solution by spectral methods based on the fast Fourier transform (FFT) algorithm. Based on this idea, two methods are derived in this paper from the existing Discrete Convolution Fast Fourier Transform (DCFFT) technique, which was previously developed for purely non-periodic contact problems. A first algorithm variant employs a special padding technique for pressure, whereas a second one mimics the contribution of multiple pressure periods by summation of the influence coefficients over a domain a few times larger than the target domain. Both techniques are validated against the existing analytical Hertz solution for the line-contact and a good agreement is found. The advanced methods seem well adapted to the simulation of contact problems that can be approximated as periodic in one direction and non-periodic in the other.

A Fast Method for Solving Contact Plasticity in a Half Space

2011 ◽  
Author(s):  
Zhanjiang Wang ◽  
Xiaoqing Jin ◽  
Shuangbiao Liu ◽  
Leon M. Keer ◽  
Jian Cao ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Residual Stresses ◽  
Fast Fourier Transform ◽  
Half Space ◽  
Three Dimensional ◽  
New Method ◽  
Fast Method ◽  
Discrete Convolution ◽  
Influence Coefficients ◽  
The One

This paper presents a new method of contact plasticity analysis based on Galerkin vectors to solve the eigenstresses due to eigenstrain. The influence coefficients relating eigenstrains to eigenstresses thus can be divided into four terms the one due to the eigenstrains in the full space, others due to the mirrored eigenstrains in the mirror half space. Each term can be solved fast and efficient by using the three-dimensional discrete convolution and fast Fourier transform (DC-FFT) or the three-dimensional discrete correlation and fast Fourier transform (DCR-FFT). The new method is used to analyze the contact plastic residual stresses in half space.

Note on the FFT Based Computational Code and Its Application

2008 ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Numerical Simulation ◽  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Toeplitz Matrix ◽  
Contact Problems ◽  
Point Of View ◽  
Fast Fourier Transform Algorithm ◽  
Discrete Convolution ◽  
Mathematical Point ◽  
Computational Code

The discrete convolution based Fast Fourier Transform algorithm (DC-FFT) has been successfully applied in numerical simulation of contact problems. The algorithm is revisited from a mathematical point of view, equivalent to a Toeplitz matrix multiplied by a vector. The nature of the convolution property permits one to implement the algorithm with fewer constraints in choosing the computational domains. This advantageous feature is explored in the present work, and is expected to be beneficial to many tribological studies.

A Three-Dimensional Deterministic Model for Rough Surface Line-Contact EHL Problems

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Ning Ren ◽  
Dong Zhu ◽  
W. Wayne Chen ◽  
Yuchuan Liu ◽  
Q. Jane Wang
Keyword(s):  
Fourier Transform ◽  
Film Thickness ◽  
Fast Fourier Transform ◽  
Rough Surface ◽  
Deterministic Model ◽  
Three Dimensional ◽  
Elastohydrodynamic Lubrication ◽  
Line Contact ◽  
Point Contacts ◽  
Transverse Roughness

This paper reports the development of a novel three-dimensional (3D) deterministic model (3D L-EHL) for rough surface line-contact mixed-elastohydrodynamic lubrication (EHL) problems. This model is highly demanded because line contacts are found between many mechanical components, such as various gears, roller and needle bearings, cams and followers, and work rolls and backup rolls in metal-forming equipment. The macro aspects of a line-contact problem can be simplified into a two-dimensional (2D) model; however, the topography of contacting rough surfaces, microasperity contacts, and lubricant flows around asperities are often three-dimensional. The present model is based on Hu and Zhu’s unified 3D mixed-EHL model (Hu and Zhu, 2000, “Full Numerical Solution to the Mixed Lubrication in Point Contacts,” ASME J. Tribol., 122(1), pp. 1–9) originally developed for point contacts and the mixed fast Fourier transform (FFT)-based approach for deformation calculation formulated by Chen et al. (2008, “Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts With Normally Flat Surface,” ASME J. Appl. Mech., 75(1), 011022-1-11). It is numerically verified through comparisons with results from the line-contact Hertzian theory and the conventional 2D line-contact smooth-surface EHL formulas. Numerical examples involving 3D sinusoidal and digitized machined surfaces are also analyzed. Sample cases indicate that transverse roughness may yield greater film thickness than longitudinal roughness. This observation is qualitatively in agreement with the trend predicted by Patir and Cheng’s stochastic model (1978, “Effect of Surface Roughness on the Central Film Thickness in EHL Contacts,” Proceedings of the Fifth Leeds-Lyon Symposium on Tribology, London, pp. 15–21). However, the roughness orientation effect does not appear to be quantitatively as great as that shown in the work of Patir and Cheng for the same range of λ ratio.

Simulation of Plasto-Elastohydrodynamic Lubrication in Line Contacts of Infinite and Finite Length

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Tao He ◽  
Jiaxu Wang ◽  
Zhanjiang Wang ◽  
Dong Zhu
Keyword(s):  
Surface Roughness ◽  
Finite Length ◽  
3D Model ◽  
Three Dimensional ◽  
Elastohydrodynamic Lubrication ◽  
Axial Direction ◽  
Contact Length ◽  
Line Contact ◽  
Line Contacts ◽  
3D Surface Topography

Line contact is common in many machine components, such as various gears, roller and needle bearings, and cams and followers. Traditionally, line contact is modeled as a two-dimensional (2D) problem when the surfaces are assumed to be smooth or treated stochastically. In reality, however, surface roughness is usually three-dimensional (3D) in nature, so that a 3D model is needed when analyzing contact and lubrication deterministically. Moreover, contact length is often finite, and realistic geometry may possibly include a crowning in the axial direction and round corners or chamfers at two ends. In the present study, plasto-elastohydrodynamic lubrication (PEHL) simulations for line contacts of both infinite and finite length have been conducted, taking into account the effects of surface roughness and possible plastic deformation, with a 3D model that is needed when taking into account the realistic contact geometry and the 3D surface topography. With this newly developed PEHL model, numerical cases are analyzed in order to reveal the PEHL characteristics in different types of line contact.

Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts of Nominally Flat Surfaces

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
W. Wayne Chen ◽  
Shuangbiao Liu ◽  
Q. Jane Wang
Keyword(s):  
Fourier Transform ◽  
Numerical Methods ◽  
Fast Fourier Transform ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Stress And Strain ◽  
Two Dimensional ◽  
Response Functions ◽  
Strain Fields ◽  
Flat Surfaces

This paper presents a three-dimensional numerical elasto-plastic model for the contact of nominally flat surfaces based on the periodic expandability of surface topography. This model is built on two algorithms: the continuous convolution and Fourier transform (CC-FT) and discrete convolution and fast Fourier transform (DC-FFT), modified with duplicated padding. This model considers the effect of asperity interactions and gives a detailed description of subsurface stress and strain fields caused by the contact of elasto-plastic solids with rough surfaces. Formulas of the frequency response functions (FRF) for elastic/plastic stresses and residual displacement are given in this paper. The model is verified by comparing the numerical results to several analytical solutions. The model is utilized to simulate the contacts involving a two-dimensional wavy surface and an engineering rough surface in order to examine its capability of evaluating the elasto-plastic contact behaviors of nominally flat surfaces.

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