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Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years. The finite element method has been widely used to solve contact problems with various grades of complexity. Great progress has been made on both theoretical studies and engineering applications. This paper reviews some of the main developments in contact theories and finite element solution techniques for static contact problems. Classical and variational formulations of the problem are first given and then finite element solution techniques are reviewed. Available constraint methods, friction laws and contact searching algorithms are also briefly described. At the end of the paper, a bibliography is included, listing about seven hundred papers which are related to static contact problems and have been published in various journals and conference proceedings from 1976.

This paper aims to present a comparison of numerical methods for determining the contact pattern of Gleason-type bevel gears. The mathematical model of tooth contact analysis and the finite element method were taken into consideration. Conclusions have been drawn regarding the usefulness of the considered methods and the compatibility of results. The object of the analysis was a bevel gear characterised by an 18:43 gear ratio and arc tooth line, and manufactured according to the spiral generated modified-roll method.

The mathematical model of tooth contact analysis consists of both the mathematical model of tooth generating and the mathematical model of operating gear set. The first model is used to generate tooth flanks of the pinion and the ring gear in the form of grids of points. Then, such tooth surfaces are used for the tooth contact analysis performed with the other model. It corresponds to the no-load gear meshing condition. The finite element method model was built on the basis of the same tooth flanks obtained with the former model. The commercial finite element method software Abaqus was used to perform two instances of the contact analysis: a very light load, corresponding to the former no-load condition, and the operating load condition. The results obtained using the two models, in the form of the contact pattern for no-load condition, were compared. The effect of heavy load on contact pattern position, shape and size was shown and discussed.

The mathematical models correctly reproduce the shape, position and size of the contact pattern; thus, they can be reliably used to assess the quality of the bevel gear at the early stage of its design.

Determination of the correct geometry of the flank surfaces of the gear and pinion teeth through the observation of contact pattern is a fundamental step in designing of a new aircraft bevel gear.

A possibility of the independent use of the mathematical analysis of the contact pattern has been shown, which, thanks to the compatibility of the results, does not have to be verified experimentally.

To make a derivation of the load‐carrying capacity of elastohydrodynamic lubrication for special operating conditions, i.e. extremely heavy loads or extremely low rolling speeds based on the Newtonian fluid model by taking the Grubin‐type EHL inlet zone analysis, justify the load‐carrying capacity of elastohydrodynamic lubrication film in these operating conditions, and propose future trends of the research in EHL and mixed EHL based on the obtained results in the present paper.

A Grubin‐type EHL inlet zone analysis is carried out for the isothermal EHL of line contacts in special operating conditions, i.e. extremely heavy loads or extremely low rolling speeds based on the Newtonian fluid model. Comparison is made between the central EHL film thickness in line contacts, respectively, predicted by conventional EHL theories and accurately predicted from the present analysis for these operating conditions. An interpretation is made for the EHL film thickness in these operating conditions by taking the approach of the transportation and flow of the fluid through elastohydrodynamic contact when the EHL film is, respectively, thick and molecularly thin in the Hertzian zone. Conclusions are drawn on the load‐carrying capacity of EHL, EHL contact regimes and mixed EHL regimes in these operating conditions.

The present EHL inlet zone analysis shows that the EHL film thickness in the Hertzian zone is on the nanometer scale and the lubricant is non‐continuum across the film thickness in the Hertzian zone at relatively heavy loads in line contact EHL when the dimensionless rolling speed is lower than the dimensionless characteristic rolling speed U_ch=0.0372W^1.50/G. In this case, the central EHL film thickness in line contact EHL predicted by the conventional EHL theory may be several orders of magnitudes higher than that accurately predicted. This difference may be greater for heavier loads.The present results for line contact EHL based on the Newtonian fluid model show that in line contact EHL, for relatively heavy loads and the dimensionless rolling speed lower than the dimensionless characteristic rolling speed U_ch=0.0372W^1.50/G, the EHL analysis needs to further incorporate the lubricant non‐continuum effect across the film thickness in part of the lubricated area to investigate the EHL film thickness and the EHL film pressure in the contact in this very low film thickness condition; only the results based on such an analysis are believable for the EHL stage where the lubricant film thickness in the Hertzian zone approaches to zero and then vanishes; the results for EHL based on the Newtonian fluid model is unable to conclude that the EHL film thickness in the Hertzian zone is zero and dry contact occurs between the contact surfaces in EHL in any operating condition for ignoring the lubricant non‐continuum regime governing the EHL stage preceding the occurrence of the zero lubricant film thickness in EHL.

A very useful source of information for academic scientists, engineers and tribologists who are engaged in the study and application of the theory of elastohydrodynamic lubrication.

A derivation is first carried out for the isothermal EHL of line contacts in extremely heavy loads or extremely low rolling speeds by taking the Grubin‐type EHL inlet zone analysis by the present paper. Results and conclusions on the load‐carrying capacity of EHL in these operating conditions are first strict and thus convincing. These results are also original in clarifying the future trends of the researches in EHL and mixed EHL.

A new algorithm to solve contact body problems is presented. The contact boundaries are divided into finite elements, and contact element matrices and vectors are computed using a modified variational principle. The contact conditions are incorporated into the total functional using the Lagrange multiplier method. This algorithm considers surface contacts rather than point contacts, and computes contact tractions directly. The Coulomb friction law is used as the friction condition. A transformation matrix based on static equilibrium is developed to solve extensive sliding problems. This algorithm is extended to the elasto‐plastic analysis and several example problems are presented to demonstrate the algorithm.

The purpose of this paper is to study the influence of friction on static and dynamic characteristics, as well as the strength and lifetime of a flexible three-axes computer numerical control (CNC) machine tool.

The machine tool is first modeled by using finite element method to analyze static structure with frictionless surface-to-surface contact type. Because the machine tool structure is becoming more and more sophisticated over time, the significant influence of contact conditions between structural elements on the dynamic characteristics of the whole structure must be considered. To examine the dynamic effects caused by inertia forces and displacement of moving bodies on contact stress, the coefficient of friction between two contact bodies is added to perform dynamic simulation and compare the results with the static analysis results.

Distribution of stress and contact forces in solid-flexible contact is also studied by using the fundamental dynamic characteristics of a bushing joint.

Finally, the influence of dynamic structure, cutting conditions and material properties on the strength and lifetime of the CNC machine tool is discussed by using fatigue analysis. Consequently, this research can be used for efficient simulation of structural dynamics, lifetime assessment and interactions of the real CNC machine with the machine tool structure in a virtual environment.

The finite element method (FEM) has become the prevalent technique used for analyzing physical phenomena in the field of structural, solid and fluid mechanics. The output of scientific papers is fast growing and professionals are no longer able to be fully up‐to‐date with all the relevant information. The purpose of this paper is to provide a bibliographical review on the application of FEM in mechanical engineering, specifically for the analyses and simulations of gears and gear drives from the theoretical as well as practical points of view.

The following topics on gears and gear drives are handled from the computational points of view: gears in general, spur gears, helical gears, spiral bevel and hypoid gears, worm gears and other gear types and gear drives. The paper is organized into two parts. In the first one each topic is handled in a short text, relevant keywords are presented and current trends in applications of finite element techniques are briefly mentioned. The second part lists references of papers published for the period 1997‐2006.

This bibliography is intended to serve the needs of engineers and researchers as a comprehensive source of published papers on design, analysis and simulation of gears and gear drives.

The bibliography listed is by no means complete but it gives a comprehensive representation of different finite element applications on the subjects. It will save time for readers looking for information dealing with described subjects, not having an access to large databases or willingness to spend time with uncertain information retrieval.

This paper gives a review of the finite element techniques (FE) applied in the analysis and design of machine elements; bolts and screws, belts and chains, springs and dampers, brakes, gears, bearings, gaskets and seals are handled. The range of applications of finite elements on these subjects is extremely wide and cannot be presented in a single paper; therefore the aim of this paper is to give FE researchers/users only an encyclopaedic view of the different possibilities that exist today in the various fields mentioned above. An Appendix included at the end of the paper presents a bibliography on finite element applications in the analysis/design of machine elements for 1977‐1997.

Some frictional contact problems are characterized by significant variations in the location and size of the contact area occurring in the process of deformation. When this feature is combined with strongly non‐linear, path‐dependent material behaviour, difficulties with convergence of the typically used iterative processes can be encountered. Demonstrates this by analysis of press‐fit connection, a typical problem in which both of those characteristics can be present. Offers an explanation as to the possible source of those difficulties. Suggests in support of this explanation, two simple modifications of the usual iterative schemes. In spite of their simplicity, they are found to be more robust than those usual schemes which are normally used in numerical analysis of similar problems.

This paper aims to present the model and method involving multi-body system dynamic analysis, finite element quasi-statics contact analysis and numerical calculation of elastohydrodynamic lubrication (EHL), according to the cam wear prediction using Archard’s model. Cam–follower kinematic pairs always work under wear because of concentrated contacts. Given that a cam and follower contact often operates in the mixed or boundary lubrication regime, simulation of cam wear is a multidisciplinary problem including kinematic considerations, dynamic load and stress calculations and elastohydrodynamic film thickness evaluations.

Multi-body system dynamic analysis, finite element quasi-statics contact analysis and numerical calculation of EHL are applied to obtain the dynamic loads, the time histories of contact pressure and the oil film thicknesses in cam–follower conjunctions to predict cam wear quantitatively.

The wear depth of the cam in the valve train of a heavy-load diesel engine is calculated, which is in good agreement with the measured value in the practical test. The results show that the cam–tappet pair operates under a mixed lubrication or boundary lubrication, and the wear depths on both sides of the cam nose are extremely great. The wear of these points can be decreased significantly by modifying the local cam profile to enlarge the radii of curvature.

The main value of this work lies in the model and method involving multi-body system dynamic analysis, finite element quasi-statics contact analysis and numerical calculation of EHL, which can give good prediction for the wear of cam.

This paper aims to distinguish the relationship between the morphology characteristics of different scales and the contact performance of the mating surfaces. Also, an integrated method of the spectrum analysis and the wavelet transform is used to separate the morphology characteristics of the actual machined parts.

First, a three-dimensional (3D) surface profilometer is used to obtain the surface morphology data of the actual machined parts. Second, the morphology characteristics of different scales are realized by the wavelet analysis and the power spectral density. Third, the reverse modeling engineering is used to construct the 3D contact models for the macroscopic characteristics. Finally, the finite element method is used to analyze the contact stiffness and the contact area of the 3D contact model.

The contact area and the nominal contact pressure Pn have a nonlinear relationship in the whole compression process for the 3D contact model. The percentage of the total contact area of the macro-scale mating surface is about 70 per cent when the contact pressure Pn is in the range of 0-100 MPa, and the elastic contact area accounts for the vast majority. Meanwhile, when the contact pressure Pn is less than 10MPa, the influence factor (the relative error of contact stiffness) is larger than 50 per cent, so the surface macro-scale morphology has a weakening effect on the normal contact stiffness of the mating surfaces.

This paper provides an effective method for the multi-scale separation of the surface morphology and then lays a certain theoretical foundation for improving the surface quality of parts and the morphology design.