Greenwood-Williamson Model Research Articles

Multi-optimization of rod fastened rotor on the distribution of bending stiffness and unbalance

A rod fastened rotor (RFR), which contains serial disks, rods (or a rod), and a torque tube, is the main form of heavy duty gas turbines and light duty aero engines. The roughness of the interface is the main parameter affecting the bending stiffness of RFR, and the jumpiness of the disks is the intuitive representation of the unbalance of the rotor. In this paper, a multi-optimization algorithm, NSGA-Ⅱ, was used to improve the vibration behavior of the RFR-bearing system. First, the quantitative relation between the bending stiffness of the rotor and the roughness of the interface was set up by combining the Persson contact theory and the Greenwood–Williamson model, and the nondimensional bending stiffness (NS) was proposed as the optimization objective of roughness. Then, the optimization objective of the jumpiness of the disk, (A + B), was rendered based on the minimum unbalance force. Finally, the multi-optimization on the roughness and jumpiness of the RFR was carried out, and the vibration behavior was improved greatly. In addition, this multi-optimization method was applied to a practical RFR, and the experimental results showed that the vibration amplitude was reduced by about 82%. The work in this paper can provide the guidance for vibration reduction of the rotor system.

Analysis of the discrete contact characteristics based on the Greenwood-Williamson model and the localization principle

The contact of a rigid body with nominally flat rough surface and an elastic half-space is considered. To solve the contact problem, the Greenwood-Williamson statistical model and the localization principle are used. The developed contact model allows us to investigate the surface approach and the real contact area with taking into account the asperities interaction. It is shown that the mutual influence of asperities changes not only contact characteristics at the macroscale, but also the contact pressure distribution at the microscale. As follows from the results, the inclusion in the contact model of the effect of the mutual influence of asperities is especially significant for studying the real contact area, as well as the contact characteristics at high applied loads. The results calculated according to the proposed approach are in a good agreement with the experimentally observed effects, i.e., the real contact area saturation and the additional compliance exhaustion.

Acceleration mechanism of abrasive particle in ultrasonic polishing under synergistic physical vibration and cavitation: Numerical study

Ultrasonic technology is widely applied in the engineering ceramic polishing processes without the limitation of material properties and ideally integrated into computer numerical control system. Ultrasonic-induced cavitation and mechanical vibration effect could accelerate the motion of solid abrasives. The individual behaviors of microjet/shockwave of ultrasonic cavitation in gases and liquids, and micro-abrasives with simple harmonic vibrations in solids and liquids has been extensively studied. To conduct a systematic and integrated study of abrasives behavior in the polishing contact region involving abrasive, surround-workpiece wall, ultrasonic physical vibration, and ultrasonic cavitation impact, a novel model integrating the free abrasive motion velocity and fixed abrasive indentation depth under multi-scale contact was proposed according to Hertzian contact theory, Greenwood-Williamson model, indentation deformation theory, the basic equations of cavitation bubble dynamics, cavitation impact control equations, and Newton's law of motion equation. The effects of ultrasonic amplitude, ultrasonic frequency, preloading force and particle size on the proposed model were investigated by theoretical analysis and numerical simulations. Ultrasonic physical vibration mainly influences the dynamic gap and further influence the number of different abrasives. Furthermore, the indentation depth of fixed abrasive depends mainly on the abrasive geometry. As the contact gap and abrasive size decrease, the indentation depth gradually decreases. Under the synergistic effect of cavitation-induced shock wave and microjet, the velocity of free abrasive in this paper is generally 0–150 m/s, and the kinetic energy of free abrasive increases roughly linearly with increasing frequency and approximately as a quadratic function with increasing particle size. Increasing the preloading force leads to a reduction in the abrasive kinetic energy. Besides, the kinetic energy induced by the shock wave has a cliff-like increment at an amplitude of 0.7–0.8 μm. It is revealed that the abrasive kinetic energy is suppressed by the cavitation bubble expansion and collapse at smaller ultrasonic pressure amplitude and surround-wall distance. This research provides a theoretical reference for the modeling of potential defects and material removal on the workpiece surface caused by abrasive motion during polishing, and reduces the trial cost for parameter optimization in actual polishing processing.

Leakage Analysis of PEMFC Sealing System Considering Temperature Cycling

Addressing the leakage rate prediction problem of the PEMFC sealing system is of great importance for its stable operation. The leakage calculation method of the PEMFC sealing system is proposed based on thermal coupling. The parallel plate leakage model is improved to make it applicable to the leakage rate calculation with a variable channel cross-section height. The interface macroscopic contact pressure is solved using a finite element simulation for solid mechanics analysis. The Greenwood–Williamson model is used for contact mechanics analysis to solve for the average height of the leakage channel. The effects of the ambient temperature, compression ratio, bipolar plate misalignment, and gasket size on the cell sealing effect were studied numerically. The data indicated that as the ambient temperature, compression ratio, and gasket size became larger, the volume leakage rate began to be reduced. The leakage rate as a parameter to characterize the sealing performance, considering the influence of gasket size, operating conditions, and other factors on the cell sealing effect, is more meaningful for the durability study and failure analysis of PEMFC.

Planarization profile modeling using pad bulk deformation and real contact area analysis of macro-scale device pattern

In general, the shape of the polished pattern is not flat but has a rounded curved profile. Unlike micro-patterns that have similar scales to pad asperities, macro-patterns have a very large scale compared to asperities, so bulk deformation must also be considered. To derive the pad bulk deformation, real contact area (RCA) measurements were performed in this study. Based on the preceding contact model, a semi-empirical model for RCA and bulk deformation was derived. After including the constructed pad bulk deformation function in the existing Greenwood–Williamson model, a new model that can derive the material removal rate profile according to the position in the pattern was presented. Unlike the general upper and lower removal rate behavior, this model shows a unique behavior. At the beginning of polishing, the edge shows a higher removal rate than the center, but after a specific step height, the center has a higher removal rate due to the curved shape. Analysis was performed for comparison between the proposed model equation and the experimental value. When the polishing profile in the 10 mm pattern was compared with the previous model and the proposed model, respectively, the existing model did not predict the removal rate distribution according to the position in the pattern. However, this model has the advantage of predicting both over-polishing at the edge of the pattern and high removal rate at the bottom. In addition, it was confirmed through simulation that the 4 mm and 2 mm patterns had excellent matching properties.

Research and Analysis on Contact Resistance of Wheel and Insulated Rail Joint in High-Speed Railway Stations

Insulated rail joint (IRJ) as one of the components of the track circuit will be burned once the electric arc is generated when the train wheel passes through the insulated rail joint and the track circuit occurs the red-light band fault, which has a great impact on the safety of the train. Reducing the voltage at both ends of the insulated rail joint can effectively prevent the occurrence of electric arc. Clarifying the wheel-IRJ contact resistance is the basis for analyzing the voltage of wheel-IRJ. The effective apparent contact area of the wheel-IRJ contact was derived through the mathematical relation. Considering the surface roughness parameters, based on the electrical contact theory and the Greenwood-Williamson (GW) model, the mathematical models of the wheel-IRJ contact resistance and current density were established, and their influencing factors were analyzed. The results show that the contact resistance increased first and then decreased. The influences of load, roughness of the contact interface, and electric arc heat on the contact resistance were greater. Considering the influencing factors of contact resistance and current density, corresponding protective measures were proposed to reduce the potential difference at both ends of the insulated rail joint.

Friction of diene rubbers on rough floors considering viscoelastic properties in the high strain range

When rubber contacts a rough surface, it undergoes cyclic deformation due to friction against asperities on the surface. The generated friction force manifests as a hysteresis term. Hysteresis is determined by energy dissipation, which is related to the viscoelastic properties of rubber. However, these properties depend on the strain and frequency of the deformation. Here I conducted a friction test of rubber. The friction coefficient was found to depend on the friction conditions (normal force, sliding velocity, and surface roughness) and the material conditions (styrene content and filler content). The strain and frequency of rubber deformation were theoretically modeled using the Greenwood–Williamson model. The modeled friction coefficient increased with the energy loss ratio calculated from the measured viscoelastic properties.

Equivalent characteristic model of a rough contact interface based on virtual material method

The characteristic of rough contact interface has a major influence on overall performance of machines. To facilitate the setting of contact interface characteristics during the performance analysis of the whole machine, an equivalent characteristic model of a rough contact interface based on virtual material method is established. The normal and tangential properties of the rough contact interface per unit area are first experimentally conducted. Furthermore, according to the Greenwood-Williamson (GW) model and stress wave propagation conditions of discrete elements, the relationship between normal and tangential contact characteristics is explored by assuming the rough contact interface per unit area as a thin virtual layer. Based on virtual material method, the equivalent characteristic model of a rough contact interface is established, including equivalent virtual elastic modulus, Poisson’s ratio, thickness, and density. Finally, taking two step-shaped steel plates connected by four bolts for example, a numerical simulation of structural characteristics under different preloads is carried out. Numerical analysis results are then compared with tested results to verify the validity of the equivalent virtual material model of rough contact interface, and a good agreement between them is obtained.

An improved equivalent capacitance model of the triboelectric nanogenerator incorporating its surface roughness

Triboelectric nanogenerator (TENG) is a kind of device that generates electric energy in the external circuit through contact electrification and electrostatic induction. Recently, although the development of TENG applications is accelerating, the development of TENG fundamental theoretical model is relatively slow. At present, the latest TENG model has considered the distance-dependent and load-dependent and can better predict the open-circuit voltage. However, the TENG model considering the effect of surface roughness on capacitance has not been introduced yet, which may underestimate the equivalent capacitance of TENG, resulting in the underestimate of outputs such as short-circuit current. Here, A TENG model considering the effect of surface roughness on capacitance is established for the first time. Based on the established load-dependent model, the effect of surface roughness on TENG capacitance is analyzed by the Greenwood-Williamson model to determine a more accurate TENG equivalent capacitance. Compared with the load-dependent model without considering effect of surface roughness on capacitance, our model can better predict the outputs such as short-circuit current and transferred charge. The experimental results show that after considering the effect of surface roughness on capacitance, the average relative error between the calculated and measured results of the equivalent capacitance decreases from 47.47% to 11.41%, which is about 1/4 of the original error. Whether it is distance-dependent or load-dependent, the model can better predict the performance of TENG. The model in this paper provides a more comprehensive understanding of the working principle of TENG and more accurate output trend prediction, which can help to design a more efficient TENG device.

The Contact Mechanics for Indentation of Single Asperity and Rough Surfaces

Abstract A finite element analysis of a rigid sphere contact with a deformable elastic–plastic plat called indentation model is studied. The numerical results are applied on the rough surfaces contact of the Greenwood-Williamson (G-W) model. A series of the relationships of the rough surfaces contact parameters are obtained. The contact parameters of the indentation model and the flattening model are compared in detail, and the reasons for their differences are analyzed. In the case of single asperity contact, for ω/ωc > 1, the indentation model reaches the initial plastic yield while the flattening model is ω/ωc=1. In ω/ωc=10, the plastic yield reaches the contact surface for the first time, and the corresponding point of the flattening model is relatively earlier in ω/ωc=6. The contact parameters of the rough surface in different plasticity indexes are compared again. On the point of ψ = 0.5, the contact parameters of the flattening model and the indentation model coincide perfectly. For 0.5 < ψ < 4, the difference between the parameters curves become larger and larger. To the point of ψ = 4, when the distance difference reaches the maximum, it begins to decrease until the two curves are close to coincide again. The dimensionless elastic–plastic contact hardness is introduced. The relation between the real contact area and the contact pressure of the indentation model can be acquired quickly. The results show that the geometric shape of deformable contact parts has an important effect on the contact parameters, especially for the extension of the plastic deformation region within a specific range of plasticity index.

An improved contact model considered the effect of boundary lubrication regime on piston ring-liner contact for the two-stroke marine engines from the perspective of the Stribeck curve

The prediction of lubrication performance is required to be the basement of friction optimization for marine engines. This paper simulates the lubrication performance of marine engines based on statistical models which have the advantages of fast, efficient, and macroscopic fault location. Boundary lubrication exists in the piston ring-cylinder liner (PRCL) of two-stroke marine engines because of the harsher load, lower speed, and larger structure. It has been proposed that there would be tribofilm under boundary lubrication which has a significant influence on the contact. To understand the boundary lubrication, it is necessary to study the lubrication regime transition. In this paper, firstly, the coefficient of friction curve combined with the thickness ratio embodies the lubrication regime transition process of two-stroke engines under work conditions. However, the phenomenon that the coefficients under boundary lubrication are smaller than that of other regimes shows the non-objectivity of this curve. Therefore, the Stribeck curve is introduced for objectively evaluating the transition. Then, the calculation of asperities contact pressure under boundary lubrication, which Wen proposed, is introduced into the classic Greenwood-Williamson model, the problem that the original model cannot reflect the boundary lubrication regime in the form of the Stribeck curve is improved. Finally, the results are compared before and after modifying the model to verify this study’s practicability. It provides more precise asperities contact pressure for the tribofilm growth calculation from the perspective of the Stribeck curve under the PRCL statistical model in future work.

Contact acoustic nonlinearity effect on the vibro-acoustic modulation of delaminated composite structures

In recent years, Vibro-Acoustic Modulation (VAM) techniques for structural health monitoringhave received increasing attention. For such techniques, the sidebands and higher-order harmonics generated by double/single sinusoidal excitations are utilized to identify a series of damages. Currently, most VAM investigations are experimental, mainly involving signal processing, while few studies have paid attention to the mechanics of VAM generation. This paper presents a comprehensive investigation which studies the effects of Contact Acoustic Nonlinearity (CAN) on VAM for delaminated composite structures. The paper includes theoretical analysis, simulations, and experiments. Considering both a nonlinear contact constitutive model and the clapping/rubbing discontinuity, an approximate solution for nonlinear motional equation was established by using Fourier series expansion. A modified Greenwood-Williamson (GW) model for physical contact was implemented into the commercial finite element software ABAQUS by a UINTER subroutine, which described the contact behaviors between rough surfaces. The calculated signal responses from the delaminated composite plates were compared to experimental results. A good agreement was qualitatively and quantitatively achieved with acceptable error. Particularly, some specific features of higher-order sidebands existing in the experiment were identified. Results showed that the combined effect of the nonlinear contact constitutive model and the clapping/rubbing mechanism caused odd–even order differences. The asymmetry between the sidebands indicates the existence of amplitude and frequency modulations, which can be used to extract nonlinear damage indexes. These indexes are capable of characterizing the degree and range of damage.

Vibration-induced dynamic characteristics modeling of electrical contact resistance for connectors

The electrical contact resistance (ECR) of connectors fluctuates periodically under vibration stress. However, a theoretical model used to explain the dynamic characteristics of ECR has not been fully proposed. The dynamic characteristics of ECR are presented after an analysis of the measured results. Moreover, according to the Greenwood-Williamson model for elastic contact and the equivalent spring model for random rough surfaces, the change in the contact pressure of electrical connectors during vibration is studied. On this basis, combining with the Holm electrical contact theory, a dynamic model of ECR under vibration conditions is established. The model can theoretically explain the phenomena of ECR found in the experiments. A good correlation between experimental and predicted ECR fluctuation is achieved.

Crack nucleation in the adhesive wear of an elastic-plastic half-space

The detachment of material in an adhesive wear process is driven by a fracture mechanism which is controlled by a critical length-scale. Previous efforts in multi-asperity wear modeling have applied this microscopic process to rough elastic contact. However, experimental data shows that the assumption of purely elastic deformation at rough contact interfaces is unrealistic, and that asperities in contact must deform plastically to accommodate the large contact stresses. We therefore investigate the consequences of plastic deformation on the macro-scale wear response using novel elastoplastic contact simulations. The crack nucleation process at a rough contact interface is analyzed in a comparative study with a classical J2 plasticity approach and a saturation plasticity model. We show that plastic residual deformations in the J2 model heighten the surface tensile stresses, leading to a higher crack nucleation likelihood for contacts. This effect is shown to be stronger when the material is more ductile. We also show that elastic interactions between contacts can increase the likelihood of individual contacts nucleating cracks, irrespective of the contact constitutive model. This is supported by a statistical approach we develop based on a Greenwood–Williamson model modified to take into account the elastic interactions between contacts and the shear strength of the contact junction.

Dynamic characteristics of elastic ring squeeze film damper

Purpose The purpose of this paper is to present a numerical model to investigate the dynamic behavior and force coefficients of a compact squeeze film damper with dual film clearances adjusted by an elastic ring, known as elastic ring squeeze film damper (ERSFD). Design/methodology/approach The governing equations of ERSFD as well as the boundary conditions are obtained based on Reynolds equation. A simplified Greenwood–Williamson model is implemented to investigate the contact behavior between the elastic ring and the journal. The interactions between the films and the elastic ring are achieved by block iterative method. Findings The radial deformation as well as velocity of the elastic ring are captured to illustrate the pressure profiles of the inner and outer films. High-order frequency components related to the number of the boss N are observed on the frequency spectrum of the film force. The force coefficients of the ERSFD are constant for a wider range of non-dimensional whirling radius ε compared with conventional squeeze film damper. Originality/value The force coefficients of the ERSFD are obtained by assuming that the journal center moves in a circular centered orbit. High-order frequency components related to the number of bosses N are observed. These findings may provide helpful materials for the application of the ERSFD.

Metrology and numerical characterization of random rough surfaces—Data reduction via an effective filtering solution

Random rough surfaces appear in measurements as noisy signals varying spatially. Mathematically, there is no theoretical difference between such and time-varying signals. Hence, the extensive array of methods and analysis tools that have been developed for signal processing are available also for rough surfaces characterization. In both, the objective is to reduce the vast amount of data to just a few meaningful parameters that allow the application of other physical concepts. Particularly in contact mechanics, it is well known that the Greenwood–Williamson model requires three parameters for the calculation of the elastic deformation of rough surface asperities. The parameters are the roughness standard deviation, the equivalent asperity radius, and the asperity density. These parameters are byproducts of the spectral moments. The spectral moments have been employed for decades in many fields of engineering and science. For rough surfaces, for example, the work by McCool outlines a mathematical blueprint procedure on how to straightforwardly reduce the entire roughness data into the said three spectral moments. It is commonly claimed, however, that the said procedure inherently suffers from resolution problems, that is, a given surface shall have much different spectral moments depending on the sampling rate (or spacing). To study these issues, synthetic surfaces are generated herein using a harmonic waveform precisely as McCool had done. However, here the signals are contaminated by a white noise process with various magnitudes. A signal-to-noise ratio is defined and used to assess the quality of the signal, and the spectral moments are evaluated for various magnitudes of the noise. Since closed-from solutions are available for the spectral moments of the uncontaminated signal, the contaminated signals are evaluated vis-à-vis the exact anticipated values, and the errors are calculated. It is shown that using the common techniques (such as those outlined by McCool) can lead to enormous and unacceptable errors. Resolution is studied as well; it is shown to have an effect only in the presence of noise, but by itself it has no independent influence on the spectral moments. The venerable Savitzky–Golay smoothing filter is used on the noisy signals, showing some improvements, but the resulting spectral moments predicted still contain objectionable errors. A generalized exponential smoothing filter, G-EXP, is constructed, and it is shown to markedly moderate the errors and reduce them to acceptable levels, while effectively restoring the underlying surface physical characteristics. Moreover, the filtered signals do not suffer from resolution problems, where results, in fact, improve with higher (i.e., finer) resolutions. Fractal-generated signals are likewise discussed.

Discrete Greenwood–Williamson Modeling of Rough Surface Contact Accounting for Three-Dimensional Sinusoidal Asperities and Asperity Interaction

Abstract The Greenwood–Williamson (GW) model has been one of the commonly used contact models to study rough surface contact problems during the past decades. While this has been a successful model, it still has a number of restrictions: (i) surface asperities are spheres; (ii) the overall deformation must be assumed to be small enough, such that there is no interaction between asperities, i.e., they are independent of each other; and (iii) asperity deformation remains elastic. This renders the GW model unrealistic in many situations. In the present work, we resolve above restrictions in a discrete version of the GW model: instead of spherical asperities, we assumed that the surface consists of three-dimensional sinusoidal asperities which appear more similar to asperities on a rough surface. For single asperity mechanical response, we propose a Hertz-like analytical solution for purely elastic deformation and a semi-analytical solution based on finite element method (FEM) for elastic–plastic deformation. The asperity interaction is accounted for by discretely utilizing a modified Boussinesq solution without consideration of asperity merger. It is seen that the asperity interaction effect is more than just the delay of contact as shown in the statistical model, it also contributes to the loss of linearity between the contact force and the contact area. Our model also shows that: for elastic contact, using spherical asperities results in a larger average contact pressure than using sinusoids; when plasticity is taken into account, using a sphere to represent asperities results in a softer response as compared with using sinusoids. It is also confirmed that sinusoidal asperities are a much better description than spheres, by comparison with fully resolved FEM simulation results for computer-generated rough surfaces.

Analysis of contact and bending stiffness for Curvic couplings considering contact angle and surface roughness

This paper develops a method for calculating the contact and bending stiffness of a Curvic coupling, and investigates stiffness changes according to the coupling shape and surface roughness characteristics. The surface of the on-site Curvic coupling is chosen as reference for a most accurate simulation. The three parameters representing the surface roughness characteristics—the standard deviation of the asperity height distribution, the average radius of asperities, and the density of asperity on the nominal contact area—are calculated with a profile of the coupling surface through a random process: the contact problem between rough surfaces is tackled using the Greenwood-Williamson model, the Curvic coupling is modeled assuming that it has as many teeth as possible within the machining limits depending on the contact angle, and the tangential stiffness resulting from the contact angle is calculated by dividing into the stick and slip regions, and is taken into account in terms of total stiffness. With this, results showed that using Curvic couplings reduces stiffness than using flat disc couplings because of the contact angle, and that the standard deviation of rough surface height is the most crucial surface parameter affecting stiffness.

Theoretical and Experimental Study on Electrical Contact Resistance of Metal Bolt Joints

As a kind of fixed connection, bolt joints are widely used in engineering. In this paper, the analytic equations of the contact resistance under different situations are deduced by approximating the Gaussian distribution and the asperity deformation process based on the Greenwood–Williamson model. The equations explicitly describe the theoretical dependences of contact resistance on mechanical and electrical parameters of materials, surface topographic features, and the property of the surface oxidation film. The characteristic curves of contact resistance versus contact pressure for brass, copper, and their silver-coated counterparts are measured by using a specially designed bolted-joints structure. The experimental results are consistent with the theoretical model, indicating that, in the log–log plot, the contact resistance depends linearly on contact pressure with a slope ranging from −0.5 to −1. The analytic equations and experimental observations of contact resistance revealed in this paper can be used as a guidance for electrical contact engineering.

Numerical and experimental study on the effect of considering plastic and elastoplastic deformation of each asperity in dry contact of rough surfaces

Contact is one of the main means to transmit force between bodies, and a detailed study of contact mechanics provides the required knowledge toward a realistic prediction of the behavior of contacting bodies. In this study, an approach based on the Greenwood-Williamson model is developed in which all of the contact points on the contacting surfaces are observed momentarily. The surface roughness is experimentally measured. The surface asperities are treated as spline functions, and their properties such as radius of curvature are obtained by differentiating these functions. The response of each individual contact point is evaluated by solving contact equations related to a specific asperity in elastic, elastoplastic, and plastic regimes. The method is therefore called spatially resolved Greenwood-Williamson (SRGW) model. To validate the model, a new technique is used which is based on indentation. A series of indentation experiments are conducted on two contacting surfaces, and the elastic and plastic deformations of the asperities under different loads are measured. The percentage error of the proposed approach is lower compared to other methods such as GW and ZMC, which shows the higher accuracy of this new method. The results show that the presented model can predict experimental results with reasonable accuracy under different contact conditions.