Fractal Contact Model Research Articles

Friction Mechanism and Characteristics of Reticle Based on Contact Theory

Abstract With the development of integrated circuits, the structure of chips becomes more and more complex, and the processing cost increases accordingly. To improve the productivity of lithography, the acceleration of the reticle stage should be increased to reduce the positioning time. However, the increase of acceleration will cause the relative slip between the reticle and the vacuum chuck, which seriously affects the accuracy and the product yield of lithography. To suppress the slippage, the friction mechanism and the characteristics between the reticle and the chuck are studied in this article. First, based on the Kogut–Etsion contact model and the Majumdar–Bhushan (MB) fractal contact model, the maximum static friction coefficient model between nano-scale surfaces was established. Then, the surface morphology parameters of reticle and chuck adsorption surface were obtained by atomic force microscopy (AFM) scanning. Finally, the maximum static friction force experiments show that the MB model is more suitable for the study of friction mechanism between reticle and vacuum chuck, and the model is more instructive for the suppression of reticle slip.

A Stiffness Model for Bolted Joints Considering Asperity Interactions of Rough Surface Contact

Abstract A revised fractal contact model considering asperity interactions is proposed. The displacement of the mean of asperity heights is used to represent the effects of the asperity interactions. Then, the critical contact area will be dependent on the contact load, and the contact stiffness will be an integral whose integrand is an implicit expression. The fractal dimension and the fractal roughness are obtained by the measurement of the surface profile to calculate the theoretical contact stiffness. The measurement of deformation is conducted to obtain the actual contact stiffness for verification, and the results show that the proposed model is closer to the experimental results than other models without considering asperity interactions. Once the contact stiffness is determined, a new equivalent normal stiffness model for bolted joints considering the contact of two rough surfaces is also proposed. Since the contact stiffness is dependent on the clamped force, the equivalent normal stiffness for bolted joints is calculated iteratively at given initial preload and external separating force. Different from the classical model, the equivalent normal stiffness for bolted joint decreases with the external separating force increases, and this stiffness loss will become larger with initial preload decreases. In this sense, the proposed equivalent normal stiffness model is a way to determine the suitable initial preload for different sizes of bolts when the stiffness loss is restricted to a certain range.

Load-Bearing Capacity Investigation of Friction Pairs in Hydro-Viscous Drive Based on Fractal Contact Theory

Abstract Hydro-viscous drive has gained extensive application in fans, pumps, belt conveyors, and scraper conveyors for step-less speed regulating, soft starting, and overload protection. In the mixed friction stage, the state of contact and load-bearing capacity are always changing with the pressure of control oil. In order to predict the load-bearing capacity of friction pairs, a description of the frictional and hydrodynamic lubrication states was first made. Oil film’s pressure distribution and load-bearing capacity were analyzed with the average flow model. The fractal contact model of the asperity surface was proposed for simulating mutual contact state and contact load between the asperities. The influences of fractal parameters on the contact performance of friction pairs were conducted. Eventually, test rig of load-bearing capacity was established. As indicated by the experiment, asperity contact area and load-bearing capacity of the asperity contact increase slowly at first and then rapidly with reduced film thickness ratio. Correspondingly, oil film load-bearing capacity progressively reduces. Due to the reduction of film thickness ratio, friction pairs’ load-bearing capacity increases gradually. The load-bearing capacity of friction pairs is affected by the fractal dimension and scale factor. Results of the experiment well conform to the theoretical values, which illustrates that friction pairs’ load-bearing capacity can be accurately predicted and computed using the average flow model and M-B fractal contact model. The calculation offers a theoretical reference to the survey about friction and torque properties of the hydro-viscous drive.

An improved model for meshing characteristics analysis of spur gears considering fractal surface contact and friction

Considering the influence of roughness on the normal contact stiffness of gears, a fractal contact model suited for gear pair contact has been established. On the basis of modified fractal contact model, the asperity contact stiffness is calculated, thus the fractal contact compliance can be obtained. By substituting the nonlinear Hertz contact compliance with the fractal contact compliance, the effects of roughness on the time-varying mesh characteristics (including mesh stiffness (TVMS), loaded static transmission error (STE), load sharing ratio (LSR) and contact stress (CS)) of gears then can be discussed. The numerical simulation results show that the contact stiffness increases with the increment of the fractal dimension and the reduction of the fractal scale characteristic coefficient. In addition, the effect of friction on the mesh characteristics has also been investigated. As the coefficient of friction increases, the variation of the mesh characteristics in the single-tooth engagement region will increase correspondingly. Besides, the influence of multi-parameters on the mesh characteristics of gear pairs has been conducted. It can be found from the parametric studies that the amplitudes of gear mesh stiffness and contact stress are nonlinear with the changes of fractal parameter and friction coefficient.

Research on the loading–unloading fractal contact model between two three-dimensional spherical rough surfaces with regard to friction

A loading–unloading fractal contact model between two three-dimensional spherical rough surfaces with regard to friction is developed. The contact model of single asperity for the load–unload cycle with regard to friction is obtained. The expressions between contact load and real contact area in the loading and unloading processes in different deformation stages of three-dimensional spherical contact surfaces are deduced. The impacts of some parameters in the model are discussed by numerical analysis. The numerical results show that the critical contact load and the critical real contact area in different deformation stages of spherical contact surfaces with friction are positively related to the fractal dimension and the curvature radius, negatively correlated with the fractal roughness, influenced by the contact type. In the elastoplastic stage of spherical contact surfaces with friction, the real contact area in the unloading process is larger than that in the loading process with the same contact load; for a given contact load, the real contact area of the loading–unloading process, as well as the differences of the real contact area in the loading–unloading processes, increases with the increase in the fractal dimension and the curvature radius, decrease in the fractal roughness, or the substitution of internal contact for external contact.

Thermal contact conductance between balls and bearing rings

The thermal contact conductance (TCC) between balls and bearing rings is predicted by proposing the TCC model between the curved and spherical surfaces. The Weierstrass-Mandelbrot function is applied to characterize the microscopic topography of surfaces. The fractal contact model is constructed by considering the elastic, elastic-plastic, and plastic deformations of microprotrusions, and then an effective contact coefficient is built to modify the fractal contact parameters. Finally, the TCC modeling method is proposed by considering the elastic and plastic TCCs of different contacting microprotrusions with different sizes. The experiments of TCCs were conducted under different conditions to verify the validity of the proposed model. The results show that the predicted TCCs fit well with measured values and that the predictive accuracy can be improved greatly by considering the effective contact coefficient. Then the effect of critical factors on TCCs is discussed, and the TCC increases with the external load, and increases with the fractal dimension D and decreases with the scaling constant G. The TCC of the dynamic lubricated ball bearing is much greater than that of the static lubricated ball bearing and the TCC of the static lubricated ball bearing is much greater than that of the static non-lubricated ball bearing.

Effect of joint interfacial contact stiffness on structural dynamics of ultra-precision machine tool

The contact stiffness of the joint interface significantly affects the dynamic characteristics of a machine tool structure. Herein, an improved theoretical model is proposed to calculate the joint interfacial contact stiffness. This model is established based on fractal theory. To mitigate the plastic and elastic contact problems, the Stronge contact model is adopted to express the contact force in the plastic regime innovatively, whereas the elastic contact force is interpreted using Hertz theory; the domain extension factor is compensated to revise the microcontact size distribution. Furthermore, the rotational contact stiffness is first derived, which renders the contact model more comprehensive. A representative test structure is designed, and the contact model is validated by comparing the simulated contact stiffness with experimental results. The transfer matrix method for multibody systems is incorporated with fractal contact model to establish a dynamic model of an ultra-precision machine tool; this ensures that the dynamic characteristics of the machine tool can be predicted more precisely. Experiments involving a vibration response test and a modal test demonstrate the feasibility of the fractal contact and dynamic models. Based on these two models, the effect of the interfacial contact stiffness on the structural dynamics is investigated. As the preloads at two joints loosen by 15%, the maximum vibration displacements increase by 1.74% and 2.90%, respectively, indicating that the joint interface between the column and lathe bed is more sensitive. The procedure presented herein will facilitate the design and optimization of the complex structures.

An improved shell-fastener model for modelling C/SiC composite bolted joints with rough surface characteristics

An improved shell-fastener model is proposed to study the mechanical behavior of C/SiC composite bolted joints with the rough surface (millimetric magnitude). Based on the axial compressive experiment, the normal contact stiffness is obtained by a modified fractal normal contact model, which is also validated by using a solid FE model and a hybrid FE model (shell-fastener and solid nuts). The equivalent axial stiffness and shear stiffness used in the shell-fastener model for the bolted joints are determined combining with normal contact stiffness. In comparison of the experimental results, the improve shell-fastener model with equivalent stiffness under pre-tightening load can efficiently and accurately describe the in-plane tension of C/SiC composite bolted joints. The proposed model has the potential to study the large C/SiC composite structures with a large number of bolted joints.

Analysis of Contact Strength for Nutation Transmission with Conical Movable Teeth by Fractal Theory

Background: Nutation transmission with conical movable teeth is a patent about a new type of spacial movable tooth drive that was developed based on the bevel planetary gear drive with a difference in the number of small teeth. Objective: The aim of this study was to establish the fractal contact model and calculate the contact strength of the center gear in the nutation transmission with conical movable teeth based on the M-B fractal contact theory and also to demonstrate the validity of the present solution by comparing it with the FEM results and the Hertz contact results. Methods: Based on the fractal theory, the relationship between the loads and the area for the tapered roller teeth in contact with the center disk was obtained, and the four basic parameters were simulated to analyze the effect of the fractal contact model in MATLAB. Results: The accuracy of the fractal contact model was verified using numerical simulation and analysis of the relationship diagram for the real contact area and load and the contact area ratio chart with different parameters. Conclusion: Based on the FEM theory, the Hertz elastic contact theory, and the fractal theory, the tooth surface contact strength of the center disk was compared and analyzed, which enabled us to demonstrate the validity of the present solution, and measures were taken to optimize and improve the followup design of the nutation drive.

Research on dynamic characteristics of oil-bearing joint surface in slide guides

The application of oil-bearing joint that is widely used in various mechanical equipment can effectively improve the dynamic characteristics of integrated machineries. A fractal contact model of oil-bearing joint considering the effect of asperity interaction is proposed in this article. The equivalent stiffness and damping of oil-bearing joint are divided into a solid contact portion and a liquid contact portion. On the basis of elastic mechanics and fractal theory, the deformation of a single asperity of solid contact part is deduced. The stiffness and damping model considering the asperity interaction are obtained from the relationship between normal load and deformation. The stiffness and damping expression of the liquid contact part are also deduced based on the generalized Reynolds equation. It is concluded that the equivalent stiffness increases with the increase of the fractal dimension and the decrease of the fractal roughness parameter. At the same time, increasing the hydrodynamic viscosity can demonstrably improve the contact stiffness. Experiment shows that the proposed model is more in line with the actual load-stiffness relationship. The model considers the factor of asperity interaction and provides a theoretical basis for predicting the dynamic characteristics of oil-bearing joint. Highlights An accurate theoretical model of the dynamic characteristics for the oil-bearing joint surface was established. Identification of the dynamic properties of the oil-bearing joint surface using frequency response functions. The factors influencing the stiffness of solid portion and liquid portion were studied, respectively.

Wear Prediction Model Based on the Fractal Contact Characteristics

A new fractal wear model is proposed. The new model is based on the multi-scale fractal contact model. Studying the elastoplastic and plastic state of contact asperities, the new wear model is established. The relationship between the amount of contact surface wear and the fractal dimension D, the characteristic coefficient G and the material property constant C is studied. Besides, wear coefficient of the rough contact surface is predicted. It can be seen from the results that the influence of the scale effect on contact surface must be considered, because of the asperities with the same deformation ?? will be in different contact state. The relationship between amount of wear in the new model and amount of wear in existing model has the same trend. When D = 1.45, the amount of wear is the lowest. With the increasing of G reflecting the roughness of contact surface, the wear amount is increasing. The material constant also has a certain influence on the wear amount, and the material with high hardness has low wear. The contributions of the amount of asperity wear (elastoplastic and plastic) have the same effect to the total wear amount. Besides, the result of new fractal wear coefficient and the result of classical have the same trend, however, the new wear model is more precise than the classical wear model and closer to the actual state.

Modeling of nonlinear interaction and its effects on the dynamics of a vibrator-ground system

The nonlinear interaction between a vibrator and the ground is one of the main sources of signal distortion in land exploration. So, it is crucial to describe this nonlinear interaction and study how it affects the outgoing wave radiated by the vibrator. For this purpose, an approach is developed to obtain the equivalent ground stiffness and damping using a fractal contact model with the half-space method. Both the impacts of surface topography and the dynamics of ground are accounted for innovatively in this approach. The results show that both the ground stiffness and damping increase with the total deformation load but decrease with the excitation frequency and roughness. In addition, a modified two-degree-of-freedom vibration model of the vibrator–ground system is proposed to calculate the output force of the vibrator for different rough surface topographies. A field test is used to verify the theoretical model, and it is concluded that compared with the theoretical excitation signal, the output force varies with the excitation frequency. The amplitude of the output force in the high-frequency phase decreases for a rougher contact surface, as does the resonance frequency of the system. Moreover, reducing the mass of the baseplate is an efficient way to improve the high-frequency output.

Thermal contact conductance modeling of baring outer ring/bearing housing interface

Currently, no model could be used to readily predict the TCR of baring outer ring/bearing housing interface, then a thermal contact conductance (TCC) model between cylindrical surfaces with transition fits is constructed. The topography is described by the fractal geometry to characterize the multiscale and self-affinity behaviors and two stochastic variables are introduced into the fractal function to guarantee the random and disordered features. Then the contact states of contact points are analyzed, and a new fractal inner contact model is established. An effective contact factor is introduced to modify the contact parameters. Eventually, the predictive model of TCCs is constructed by summing up the TCCs of all the contacting protrusions with different scales. To demonstrate the validity of the TCC model, the measuring setup of TCCs was developed, and then the tests were conducted. The results show that the average deviations between the measured data and predicted TCCs are 10.25% and 8.73% for SS304 and carbon steel, respectively. When the effective contact factor is not considered, the average deviations between the predicted TCCs and measured data are 25.47% and 20.61% for SS304 and carbon steel, respectively. Then the necessity to introduce the effective contact factor is verified. For SS304, the prediction accuracy of the present model is 15.22%, 56.36%, and 31.55% higher than that of the present model without effective contact factor, M-T model, and Zou model, respectively. For carbon steel, the prediction accuracy of the present model is 11.88%, 63.76%, and 34.87% higher than that of the present model without effective contact factor, M-T model, and Zou model, respectively. Finally, the effects of the surface roughness, the protrusion top radius, the effective contact factor, and the transition fit on TCC are discussed. The results show that TCC decreases with surface roughness. The TCC with different protrusion top radii is greater than that with identical top radius. The TCC increases with the effective contact factor slowly and then the growth rate increases rapidly. Besides, the TCC increases with transition fit sharply at the early stage, and then increases slowly for both SS304 and carbon steel.

Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces

In this paper, a stiffness model of contact surfaces based on a modified three-dimensional fractal contact model is built, which is in accordance with the experiment results. Additionally, the static, dynamic, and stable behaviors of the bolt joint between the spindle box and the machine bed are analyzed. The mathematical relationship between fractal parameters of the surface topography and the stiffness of the system was established to accurately study its static behaviors. Asymmetric curves are observed from the load–deflection results and the nonlinear stiffness characteristic is also presented. It is shown that both the stress and the stiffness increase with the increase of the displacement near the static equilibrium position. Meanwhile, a simplified model without the consideration of roughness is compared with joint interfaces composed from milling, scraping, and grinding surfaces. Numerical calculation was employed to investigate effects of design parameters on the system under harmonic excitation when the processing method, excitation force, bolt pre-tightening force, topography parameters, and other structural parameters, i.e., nominal contact area, joint thickness and bolt number, are eventually regarded as the control parameters. The aim of the article is to analysis the influence of these parameters, including surface morphology, on nonlinear characteristics of the bolt interface with fractal contact surfaces andto provide some references to improve the characteristics.

The contact characteristics analysis for rod fastening rotors using ultrasonic guided waves

Rod fastening rotors are widely used in aeroengine and contact characteristics between rotor disks are critical to the stable operation of rotors. The value and distribution of preload loading on rotor disks have great influence on the contact characteristics. Thus, in this paper, the contact characteristics analysis includes the preload measurement and uneven-distributed preload detection which can mainly determine the contact state of rotor disks. Directly measuring preload and detecting uneven-distributed preload of rotor disks are difficult tasks because of the complicated structure of rod fastening rotors. This paper proposes novel methods to measure the preload and detect uneven-distributed preload using ultrasonic guided waves. Ultrasonic guided wave is a kind of stress wave propagating in the solid structure. It is very sensitive to the contact between rotor disks surface. The relationship between the guided wave energy and preload is investigated in both simulation and experiment. And the relation curve is drawn which can be used to measure preload. Through the loading and unloading experiments, the relationship between the guided wave and uneven-distributed preload is acquired and some laws are found which can instruct the detection of uneven-distributed preload. A detection method based on the axisymmetric ring sensor array is subsequently proposed to detect the uneven distribution of preload. The results of simulation and experiments demonstrate that these methods using ultrasonic guided waves are credible to measure preload and detect the uneven-distributed preload, and are suitable to both pair-rotor disks and multi-rotor disks. To simulate the real condition of contact surface, the fractal contact model based on Weierstrass-Mandelbrot (W-M) function is adopted to model the rough surface of rotor disks in simulation.

Thermal contact conductance between rollers and bearing rings

To predict the TCC between rollers and bearing rings, an integrated thermal contact conductance (TCC) model between cylindrical surfaces was proposed. The surface topography was characterized by the Cantor set theory, and the structure function of the surface topography was used to identify the equivalent fractal dimension. Then the fractal contact model was established, and an effective contact factor was constructed to modify the contact parameters by considering the contact between two cylindrical surfaces. Finally, the discrete and continuous models were established to predict the TCC between rollers and bearing rings. To verify the validity of the TCC model, the experiments of TCCs were conducted and then the comparison between the measured and predicted TCCs was performed. The results show that the maximum and average errors between the predicted TCC and the measured data are 10.21% and 8.14%, respectively. Besides, the effects of the key factors on the TCC between rollers and rings were discussed. The TCC increases with the contact pressure because the critical generation nc increases with the contact load. Moreover, the smaller the fractal dimension, the rougher is the surfaces, and then the smaller is the TCC. The TCC between rollers and bearing rings increases with the dimensionless contact area. Besides, the TCC increases with the rotational speed for the fully run-in roller bearing, and the TCC of the dynamic lubricated roller bearing is much higher than that of the static lubricated roller bearing and that of the static non-lubricated roller bearing.

A detachable design method of large-sized structure for heavy duty machine tool based on joint surface dynamics characteristic analysis

Heavy duty and ultra-heavy duty machine tools are used to manufacture large parts or super large parts in aerospace, ship, transportation, and energy industries. The weight of a structural part of ultra-heavy duty machine tools will reach more than 100 tons, because of which both manufacturing and transportation become very difficult. In this paper, a detachable design method for large-sized structures of heavy duty machine tools is presented. The proposed method aims to make large-sized structure into several detachable sections that can be bolted together in order to be casted and transported and remanufactured more easily. To analyze the influence of the joint surface, a three-dimensional fractal contact model based on the influence of domain expansion factor is used to identify stiffness and damping of the joint surface. On the basis of the analysis of the dependence of contact stiffness and damping of joint surface on dynamic characteristics of structural part, a detachable optimization model of a super span beam is established, and the particle swarm optimization algorithm is adopted to carry out the optimization. After that, the dynamic characteristics of the optimized design is analyzed and verified by finite element analysis. Based on the simulated verification, a detachable beam of heavy duty gantry machine was designed by the proposed method and verified by the field test. The illustration example shows that the large span beam can be detached into three sections that can be bolted together by high strength bolts, and it also has satisfied performance.

New Fractal Contact Model Considered Multi-scale Levels

New Fractal Contact Model Considered Multi-scale Levels

Normal Stiffness model of bolted joint based on the modified three-dimensional elastic-plastic contact fractal model

Bolted joints have a significant effect on the dynamical behaviour of assembled structures. Accurate contact stiffness model of bolted joint is crucial in predicting the dynamic performance of bolted structures. In this research, a modified three-dimensional fractal model of normal contact stiffness is presented to more accurately predict the dynamic characteristic of a bolted assembly. In the present model, while the contact deformation exceed the critical value the elastic-plastic contact is used to take the place of Hertz elastic contact in the traditional M-B model, and revise the drawback that elastic-plastic contact occurs before elastic contact in existing elastic-plastic fractal models. As the increase of contact force, the plastic deformation of single asperity is considered again, which is omitted in the two type fractal models. An experimental set-up with two T-shaped specimen is designed conducted to verify the efficiency of the proposed model. Comparing with the ZX elastic-plastic fractal model, the present model can predict contact stiffness and dynamic performance more accurately, particularly, while for higher contact loads. The results show that the modified elastic-plastic fractal contact model can be used to more accurately predict the stiffness and dynamic characteristic of bolted structures in the machine tools.

A Contact Mechanics Model for Rough Surfaces Based on a New Fractal Characterization Method

There are mainly two kinds of contact mechanics models for rough surfaces. One is based on the statistical characteristic parameters and depends on the measurement scale of rough surface topography. The other is based on the fractal parameters, which is independent of the measurement scale. However, most of the contact models for rough surfaces based on fractal theory use the size that is corresponding to the contact area of an asperity or the sample length as the base diameter of an asperity to describe the initial profile of asperities. As a result, the obtained deformation mechanism of asperities is not correct. To solve this problem, a new fractal characterization method for rough surfaces based on the fractal dimension [Formula: see text], fractal roughness [Formula: see text] and the highest asperity height is proposed, and then a fractal contact model independent of the measurement scale is established. The contact mechanism of asperities and variation trends of the real contact area and contact stiffness are discussed. When the contact pressure of the rough surface is less than its yield strength, its normal contact stiffness can be expressed as the first derivative of the contact pressure versus the normal compression, regardless of the deformation forms of asperities.