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The cable bracket is one of the crucial components of high-speed Electric Multiple Units (EMUs), which is fixed to the axle box by bolts to support the cable. As an important parameter of the joint surface, contact stiffness significantly affects the dynamic response and service life of the cable bracket. However, it is difficult to directly measure the contact stiffness of the joint surface. As a result, an identification method is proposed in this investigation to identify the contact stiffness. A finite element model incorporating both normal and tangential stiffness parameters is developed. According to the finite element model and modal testing results, an improved particle swarm optimization algorithm is employed to identify the contact stiffness of the joint surface. The effect of the contact stiffness on the vibrations of the cable bracket is investigated. The findings indicate that pre-tightening torque significantly influences the contact stiffness of the cable bracket. The contact stiffness has a great effect on the vibrations of the cable bracket. Optimal contact stiffness notably reduces vibrations of the cable bracket, thereby extending its service life. [ABSTRACT FROM AUTHOR]

In mechanical systems, the machining surfaces of various components, especially those subjected to precision machining such as the roller and track surfaces in feed systems, often exhibit non-Gaussian roughness characteristics, which significantly differ from the Gaussian surface model widely adopted in traditional contact models. In order to thoroughly investigate the influence mechanism of characteristic parameters of non-Gaussian rough surfaces on the contact performance of mechanical interfaces, this study specifically constructs a contact model for non-Gaussian rough surfaces. This model elaborately elucidates the complex effects of skewness and kurtosis, two key parameters, on contact force, actual contact area, and contact stiffness. Thus, it deepens the understanding of contact characteristics of non-Gaussian rough surfaces and provides a theoretical basis for the analysis and optimization of the contact performance between rollers and tracks in feed systems. [ABSTRACT FROM AUTHOR]

The validity of different available contact models is tested against experimental data for contact normal loading of artificially shaped granite stones. Three elastic models and two plastic models were tested with different combinations of contact shape (spherical, conical and flat) and roughnesses. It was found that the deviation from the Hertz model increased with surface roughness with a generally better representation of the data achieved by using models which consider surface roughness and the fractality of the surface. The validity of all these models seems to be limited to before excessive plastic deformations start taking place, after which the contact seems to deform under constant stiffness. [ABSTRACT FROM AUTHOR]

Contact tests on machined and natural granite showed that extensive plastic deformation which extends to the core shape is happening before the cross-over from the behaviour of an elastic rough surface to the Hertzian behaviour of an elastic smooth contact when all asperities have yielded in the surface. The plastic deformation, which was found to take place when the estimated maximum stresses at the contact reaches about 0.6 of the material hardness, affects the behaviour during normal loading as the material will start to deform at constant stiffness after reaching these stresses. The plastic deformation during lateral loading also affects the applicability of lateral loading models. The data yielded a much lower lateral stiffness which is around one order of magnitude less than that predicted by the available contact models. [ABSTRACT FROM AUTHOR]

The stick–slip friction observed in an atomic force microscope (AFM) experiment has been widely studied using the Prandtl-Tomlinson (PT) model or molecular dynamics (MD) simulations. However, the mechanisms of friction energy dissipation in AFM are still not well understood. Our detailed MD simulations of a benchmark system, a Pt metal tip sliding on the Au (111) surface, provide a method of computing the contact stiffness and damping between the tip apex and the metal surface. We revealed that the contact stiffness is largely dependent on the very first contact layer atoms of the tip apex, but essentially independent of the temperature and the atomic mass of the AFM tip, and is also less dependent on the normal load if the contact geometry remains unchanged in elastic contact. Furthermore, by connecting the atomic relaxation rate to the damping coefficient, an important parameter gauging the friction dissipation in the PT model but the choice of which is usually empirical, we demonstrate that this damping coefficient is dependent on the atomic structure of the tip apex and the intrinsic relaxation rate of the individual atoms in the contact layer. We use such mechanisms to calculate the two parameters and carry out Langevin dynamics simulation within the framework of the PT model for two friction systems: a small Pt tip consisting of 3956 Pt atoms and a large polycrystalline Pt tip consisting of 18,365 Pt atoms. Our simulation results show that both tip apexes are underdamped in a stick–slip friction. We also demonstrate that the results from the Langevin dynamics simulation using these two critical parameters compared remarkably well with the straightforward MD simulation results in a range of sliding velocity (V = 0.01 – 1 m/s). [ABSTRACT FROM AUTHOR]

Elastohydrodynamic lubrication (EHL) point contact occurs between two rough surfaces at the mesoscopic level, while the interaction of rough surfaces involves contact between asperities at the microscale level. In most cases, the contact between asperities within an interface takes the form of lateral contact rather than peak contact. Regions devoid of contact asperities are filled with lubricating oil. However, conventional models often oversimplify lateral contact forms as interactions between asperities and a smooth, rigid plane. These simplifications fail to accurately represent the true contact conditions and can lead to inaccuracies in the analysis of EHL's contact performance. To address this issue, we have developed a novel EHL interface model comprising two rough surfaces. This model allows us to explore the influence of asperity height, contact angle, and contact azimuth angle on EHL interface performance. [ABSTRACT FROM AUTHOR]

Contact stiffness–contact stress mapping relationship is the key to ultrasonic stress measurement. The interaction of asperities at the rough interface is very complicated. The complexity of the asperity interaction and the irregularity of the lubricant film distribution make it difficult for the existing acoustic models to accurately construct the mapping relationship. Therefore, this paper conducts experimental research on the contact stiffness of the lubricating interface, and constructs the contact stiffness–contact stress mapping relationship of the lubricating interface considering the influence of plastic deformation. The critical contact pressure is determined and used to analyze the influence of plastic deformation and elastic deformation on contact stiffness. The change of equivalent oil film thickness and stiffness under load was also analyzed, and the optimized analysis formula is obtained. Based on the load sharing concept, the contact stiffness–contact stress mapping relationship of lubricating interface was constructed. Finally, the proposed acoustic model was experimentally verified. [ABSTRACT FROM AUTHOR]

Background: The contact force–deformation relation between two spheres is one fundamental property which determines the mechanical response of the colliding bodies. In general, collisions of solid spheres occur in a very short time, so a highly accurate and a non-contact measurement method is required. Objective: In this study, a concept for the experimental determination of contact force–deformation curves of two colliding spheres is presented. Thereby two identical steel spheres collide by using a pendular setup and the sphere's velocities are measured by Laser Doppler Vibrometers (LDVs). Methods: The displacements are obtained by integrating the velocity with time and the accelerations are obtained by differentiating the velocity with time. From these values, the deformation and the contact forces can be calculated. Then, the elastic results are compared with the Hertzian theory of impact and experiments are conducted with elasto-plastic spheres. Results: Although the deformations are in good agreement with these analytical values, the influence of numerical differentiation is observed in the contact force. However, it is shown that the elastic contact force–deformation curve of two colliding spheres can be obtained with reasonable accuracy by using LDVs. For elasto-plastic spheres the coefficient of restitution became smaller than for the purely elastic case, and the force–deformation curve with hysteresis are measured. Conclusions: The results of this study are considered to be reasonable as far as comparisons with those for elastic collisions. Therefore, more detailed verification by numerical analysis, such as finite element analysis, is desirable. [ABSTRACT FROM AUTHOR]

Rockfill is a common irregular granular material used in most dam construction projects. The purpose of this paper is to investigate the distribution and size-dependent properties of the mechanical parameters describing the elastic properties and crushing strength of rockfill particles. These statistics can be used as a reference to calibrate the input parameters of numerical models when studying the macroscopic behavior of rockfill with particle breakage using the discrete element method. A series of limestone particles ranging in diameter from 20 to 240 mm were measured in this study using a single particle compression test. The elastic modulus, elastic contact stiffness, tensile stress and fracture force were then determined by characterizing each experimental force–displacement curve. Classical statistical methods were used. It has been shown that Weibull, lognormal and logistic functions can all represent the distributional features of the elastic modulus, tensile stress and fracture force, with the lognormal function being the optimal type here. As the grain size increases, the elastic modulus and tensile stress decrease, while the fracture force rises. Empirical models of power functions effectively reproduced these size-dependent laws. Meanwhile, the relationship between these parameters was also established. Finally, the lognormal function was adopted to express the randomness of the maximum elastic contact stiffness. Some suggestions were made after discussing the positive association between the maximum elastic contact stiffness and grain size. Moreover, the evaluation of the loading strain rates of individual particles tested shows that the present conclusions are applicable to quasi-static case. [ABSTRACT FROM AUTHOR]

A method of using a quadratic function of rotary asperities to be equivalent to asperity is established based on a measurement experiment on rough surface topography. A micro-contact model of a rough surface that includes elastic, elastic-plastic, and pure plastic deformation states is re-derived using the theory of contact mechanics and the method of probability statistics. A rough surface contact stiffness model under mixed lubrication is proposed in accordance with the rough surface contact stiffness, which consists of solid surface and lubrication medium contact stiffness. The influence of the lubrication medium, solid material, and asperity size parameters on contact stiffness is analyzed. Results show that elastic modulus, acoustic impedance, and the size parameters of the asperities of the solid part are the main factors that affect the contact stiffness of the solid part. The acoustic impedance of the lubricating medium mainly affects the contact stiffness of the lubrication part. The contact stiffness of the rough surface under mixed lubrication is a superposition of the solid and lubrication parts, and it is greater than that without the lubrication medium. The model established in this study fully considers the deformation state in practical engineering, which includes asperities and the lubrication state of rough surfaces. It provides a reference for the performance prediction of mechanical surfaces. [ABSTRACT FROM AUTHOR]

Understanding and modeling the contact behavior of particles provides enhanced constitutive laws in discrete-based simulations as well as a strong linkage between micro- and macroscopic behavior of granular materials. In the present study, we studied the influence of a polymer-based additive composed of Methocel cellulose ether on the tribological response of rough particles with grain-scale experiments. We performed both monotonic and cyclic tests on pairs of particles subjected to repeated shearing so that to explore the additional influence of abrasion in relation to previous loading history. Each sample was tested at first in a dry state and consecutively was immersed into the polymer-based fluid. The test results indicated increased friction at the contacts of the particles when immersed in the polymer-based additive, suggesting promising applications of the Methocel cellulose ether in improving the stability of geo-systems. For the shearing tests in a dry state, abrasion influences were amplified at higher magnitudes of normal load, whereas the polymer additive acted as a fluid film (or thin coating) mitigating in this way the continuous abrasion due to repeated shearing. We incorporated the Mindlin and Deresiewicz (M–D) contact model in a modified form to analyze the tangential response of the particles. It was found that the power of the modified M–D model was directly correlated with the natural logarithm of the microslip displacement threshold, when this threshold was normalized with respect to the normal load. Based on the analysis of the closed loops from the cyclic tests, the energy dissipation was found to be reduced when the samples were immersed in the polymer fluid compared with the tests in a dry state. The test results from the present work offer some insights into the micromechanical-based analysis of particulate materials and the interactions of rough particles in the presence of polymeric fluids with potential applications in contact mechanics modeling using discrete-based tools. These results also contribute to unveiling the prevailing micromechanisms in order to understand (or interpret) bulk behavior, thus the database from the present work may provide helpful guidance in understanding granular material behavior and the interactions between particles with polymer-based additives. [ABSTRACT FROM AUTHOR]

In situ changes in the nanofriction and microstructures of ionic liquids (ILs) on uncharged and charged surfaces have been investigated using colloid probe atomic force microscopy (AFM) and molecular dynamic (MD) simulations. Two representative ILs, [BMIM][BF4] (BB) and [BMIM][PF6] (BP), containing a common cation, were selected for this study. The torsional resonance frequency was captured simultaneously when the nanoscale friction force was measured at a specified normal load; and it was regarded as a measure of the contact stiffness, reflecting in situ changes in the IL microstructures. A higher nanoscale friction force was observed on uncharged mica and highly oriented pyrolytic graphite (HOPG) surfaces when the normal load increased; additionally, a higher torsional resonance frequency was detected, revealing a higher contact stiffness and a more ordered IL layer. The nanofriction of ILs increased at charged HOPG surfaces as the bias voltage varied from 0 to 8 V or from 0 to —8 V. The simultaneously recorded torsional resonance frequency in the ILs increased with the positive or negative bias voltage, implying a stiffer IL layer and possibly more ordered ILs under these conditions. MD simulation reveals that the [BMIM]+ imidazolium ring lies parallel to the uncharged surfaces preferentially, resulting in a compact and ordered IL layer. This parallel "sleeping" structure is more pronounced with the surface charging of either sign, indicating more ordered ILs, thereby substantiating the AFM-detected stiffer IL layering on the charged surfaces. Our in situ observations of the changes in nanofriction and microstructures near the uncharged and charged surfaces may facilitate the development of IL-based applications, such as lubrication and electrochemical energy storage devices, including supercapacitors and batteries. [ABSTRACT FROM AUTHOR]

When an electric current flow across an interface between two contacting solids a change in the electric potential occur over the interface. One part of the electric contact resistance is due to the constrictions at asperity contact regions. Barber has shown that the constriction resistance can be related to the interfacial contact stiffness. I discuss the relation between the classical (approximate) Holm theory for the constriction resistance and the (exact) theory involving the contact stiffness. [ABSTRACT FROM AUTHOR]

Mecanum wheels (MWs) are valuable components to automated guided vehicles (AGVs) and autonomous mobile robots (AMRs) for their versatile motion capabilities. Such characteristics are associated with the series of rubber-covered rollers obliquely attached around the circumference of the wheel rim. The rollers take turns in contact with the ground while MW is rotating. Even though MWs were designed and appear perfectly circular, vehicles equipped with MWs often exhibit periodic bouncing motion. Such a phenomenon hinders the application of MW in factories, such as panel manufacturing. This article sought the cause of the non-uniform contact stiffness associated with MWs and attempted to improve its design to mitigate the vibration problem. A hard-kill topology optimization scheme was proposed where interfacial elements between the steel core and the rubber layer can switch the material property to steer the contact stiffness toward the desired value. Nonlinear finite element analyses on the detailed 3D geometry of the roller system involving neo-Hookean material and contact simulation were performed for accurate numerical estimation in the optimization process. Due to elastic deformation, dual contacts occur at the transition of adjacent rollers; as a result, the total contact stiffness increases significantly compared to the other regions. The proposed optimization scheme successfully reached a design of uniform contact stiffness for the entire circumference of MW, including transition zones between adjacent rollers. Prototypes of the optimized MW were fabricated. Experiments on the vibration of an AGV with the old and the new MWs were conducted. The new MWs effectively reduced vibration amplitudes by approximately 30%. [ABSTRACT FROM AUTHOR]

Machined surfaces usually need to be loaded several times during assembly. The surface morphology after the simulation loading may be quite different from the physical loading, which subsequently causes differences between the contact stiffness after secondary simulation loading and secondary physical loading results. The fractal theory was used to reconstruct the measured surface morphology, followed by contact stiffness simulation and test. The results show that contact stiffness of new morphology obtained by the first simulated loading using the elastic-plastic deformation theory are smaller than the first loading test results, with an average error of −9.71 %. Contact stiffness of used morphology obtained by the first simulated loading using the pure elastic deformation theory are in good agreements with the second loading test results, with an average error of 4.71 %. However, the average contact stiffness of new morphology obtained by the second simulated loading using the elastic-plastic deformation theory, which is traditionally used for contact analysis, are 2.15 times larger than the second loading test result. These research results demonstrate a feasible finite element method (FEM) for solving the contact problem of rough surfaces that need to be assembled multiple times. [ABSTRACT FROM AUTHOR]

Stiffness characterization of mechanical interfaces is quite crucial for the analysis of several tribological behaviors. In the research of industrial robots, joints exist universally. The stiffness of different machine tools varies greatly, particularly for computer numerical control machine. Therefore, this research aims at providing an assessment of influence factors for stiffness of joint interfaces theoretically. Based on fractal roughness parameters independent of scale and contact mechanics theory, the contact area of joint interface is studied, and a multi-scale normal contact stiffness model and a multi-scale tangential contact stiffness model are proposed. Meanwhile, the problem of the deformation of any contact asperity is considered as three separate regimes. The laws of area-displacement and force-displacement under elastic-plastic regime are established. The transition which is in the deformation mechanism of asperity from elastic to plastic is consistent with classical contact mechanics. The analysis of numerical calculation results indicates the approximate linear relation among dimensionless normal load and key parameters. Moreover, these key parameters have been divided into two main categories for the multiscale model of joint interfaces, namely, fractal parameters, such as fractal dimension D and fractal roughness parameter G, and interfacial parameters. In addition, tangential load and friction factor are two important factors of the tangential stiffness. [ABSTRACT FROM AUTHOR]

The contact stiffness of the joint surface directly affects the static and dynamic mechanical behavior, and accuracy of the machine tool. A new elastoplastic contact stiffness model is proposed by considering continuous and smooth contact characteristics and substrate deformation. First, the interpolation interval of cubic Hermite polynomials is improved to meet the continuous and smooth change of contact parameters during asperity deformation. Then, the micro-contact mechanism considering substrate deformation is explored by establishing an asperity-substrate system model. Furthermore, combined with statistical principles, a new contact stiffness model of the joint surface is established. Finally, the correctness of the built model is verified by comparing with experimental data and different contact models. The simulation results show that the model changes continuously and smoothly in the three deformation regions. The substrate deformation mainly affects the asperities in the elastic contact stage. The smoother is the surface, the more significant is the influence of substrate deformation. [ABSTRACT FROM AUTHOR]

An approach to producing interfaces with tailored and repeatable normal contact stiffness using micropatterned surfaces is developed. A finite element model is first used to design square wave interfaces having a range of stiffnesses, and these are fabricated in polycarbonate via a microfabrication process. Results demonstrate that the contact stiffnesses of the fabricated interfaces are both tailorable and repeatable. The approach can be broadened to other materials and is useful for applications requiring specified interface stiffness. Finally, even with these deterministic interfaces, we show that low levels of roughness on the surface features are sufficient to produce a load-dependent contact stiffness at lower loads. Therefore, tailorability is mostly applicable above this limit where total contact stiffness converges to a load-independent value. [ABSTRACT FROM AUTHOR]

When two anisotropic rough surfaces contact, the interface is shown as the contact anisotropy or contact isotropy, which closely relates to the asperity contact azimuthal angles on the microscopic scale. The interfacial contact anisotropy/isotropy will influence the normal contact stiffness and damping of mechanical joints. A normal contact stiffness and damping model between two anisotropic rough surfaces is proposed, which includes the effects of the asperity shoulder-to-shoulder contact and adjacent asperity interaction. And the influence laws of the contacting match degree and the contact anisotropy on the normal contact stiffness and contact damping are revealed by a series of simulation experiments. Thereby, the normal contact stiffness and damping can be controlled by adjusting the surface topographic features and the contacting orientation between mechanical parts. [ABSTRACT FROM AUTHOR]

A new model to accurately obtain the contact stiffness of the mechanical joint surface is established, which satisfies the continuous smooth contact characteristics of the asperity and also considers the influence of the asperity interaction. In order to satisfy the continuous and smooth contact characteristics of the asperity, the interpolation interval of the Hermite polynomial function is improved to make the asperity contact stiffness curve smoothly transition in the elastic, elastoplastic and completely plastic stages. Further, the contact situation of the joint surface considering the asperity interaction is explored through micro-contact analysis and mechanism research. Finally, using statistical principles, the microscopic contact model is extended to the entire joint surface, and the contact stiffness model of the joint surface is established. The simulation results show that the asperity stiffness model satisfies the continuous smooth contact characteristics. Surface roughness is the main factor that affects the contact stiffness, and the influence of the asperity interaction cannot be ignored. [ABSTRACT FROM AUTHOR]

Using scanning electron microscopy (EDS analysis), magnetic force microscopy, and nanoindentation, a metallographic study of the magnetic structure and nanomechanical properties of sintered rare-earth magnets Nd–Dy–Fe–B of the USC-20L brand has been carried out (Ural Strip Casting Technology). The microstructure of the sintered Nd–Dy–Fe–B magnet of the USC-20L grade includes the following components: grains of the Nd2Fe14B phase are separated by lamellas of the phases enriched in neodymium. Inclusions Nd–29.1% Fe–6.2% C–2.2% O–1.4% Dy are located at the triple junctions of Nd2Fe14B grains. Inclusions Nd–4.5% Fe–9.1% O–6.7% C–4.5% Fe–2% Dy are along grain boundaries containing Nd and Dy oxides. The chemical composition of grains is Fe–25% Nd–6.9% C–1.6% Dy–1.4% B. It is found that, due to irregular grain growth, the interlayers of the enriched Nd phase are connected to each other in the region of grain junctions, causing a concentration of internal stresses and the appearance of a crack. The crack propagates along the grain boundary from one wetted joint of the grains to another, due to mechanical stresses. The phenomenon of intergranular wetting by a phase of Nd-enriched Nd2Fe14B/Nd2Fe14B grain boundaries is observed. It has been established that phases enriched with Nd can pseudo-incompletely (or pseudo-partially) wet such grain boundaries, i.e., form a nonzero contact angle along the grain boundaries and in triple joints. Based on the results of Magnetic Force Microscopy (MFM), a conclusion is made about the presence of a one-dimensional domain structure; domains cross grain boundaries. The presence of pores and inclusions of Nd and Dy oxides localized along grain boundaries is noteworthy. The average transverse domain size of the banded structure is ~1 μm, the energy of the domain wall γ ~ 14 kJ/m2, and the width of the domain wall δ ~ 0.6 × 10–9 m. The nanoindentation method is used to measure the values of nanohardness (H, GPa), elastic modulus (E, GPa), contact stiffness (S, N/m), elastic work (We, nJ) and plastic strain (Wp, nJ) in submicrovolumes of Nd2Fe14B grains. According to the measurement results, the minimum value of Nd2Fe14B grain adhesion Kint = 0.539 MPa m0.5 is estimated. [ABSTRACT FROM AUTHOR]

Establishing an accurate and effective stiffness model of the elastic contact surface is the basis for further modeling and analysis of machine tool dynamics. In this paper, a new elastic contact stiffness model is proposed that considers the bulk substrate deformation and modifies the Greenwood and Williamson microcontact model (GW model) of rough surfaces. Based on the Hertz contact theory and fixed-point iteration method, a single asperity contact model is created to analyze the effects of bulk substrate deformation and coating materials. To make the contact stiffness model more accurate and effective, two aspects are improved: One is to introduce the triangular distribution function to make the asperity heights distribution more consistent with the actual machined surface, and the other is to correct the defects in the micro-contact process. Comparing the finite element simulation results with the modal test data, the correctness of the proposed contact stiffness model is verified. The simulation results reveal the influence of distribution function, surface roughness and coating material on the contact characteristics of the joint surface. [ABSTRACT FROM AUTHOR]

Friction joints are one of the fundamental means used for the assembly of structural components in engineering applications. The structural dynamics of these components becomes nonlinear, due to the nonlinear nature of the forces arising at the contact interface characterised by stick-slip phenomena and separation. Advanced numerical models have been proposed in the last decades which have shown some promising capabilities in capturing these local nonlinearities. However, despite the research efforts in producing more advanced models over the years, a lack of validation experiments made it difficult to have fully validated models. For this reason, experimental techniques which can provide insights into the local dynamics of joints can be of great interest for the refinement of such models and for the optimisation of the joint design and local wear predictions. In this paper, a preliminary study is presented where ultrasound waves are used to characterise the local dynamics of friction contacts by observing changes of the ultrasound reflection/transmission at the friction interface. The experimental technique is applied to a dynamic friction rig, where two steel specimens are rubbed against each other under a harmonic tangential excitation. Initial results show that, with a controlled experimental test procedure, this technique can identify microslip effects at the contact interface. [ABSTRACT FROM AUTHOR]

The purpose of this study is to establish a calibration approach for connecting the macro elastic properties and micro contact properties of the DEM model based on Voronoi tessellation method. Analytical solutions were formulated to predict macro elastic properties for the DEM model with regular contact sets. Based on the combination of the analytical solutions and numerical calculations, a simple calibration approach has been established to predict the elastic modulus and Poisson's ratio of the DEM model with Voronoi tessellation. The influences from contact stiffnesses, block size and the spatial distribution of contacts can be comprehensively considered. Variations of the proposed approach were studied, which is size effected, and the uncertainties related to the approach were shown graphically. [ABSTRACT FROM AUTHOR]

The present paper is devoted to a theoretical analysis of sliding friction under the influence of in-plane oscillations perpendicular to the sliding direction. Contrary to previous studies of this mode of active control of friction, we consider the influence of the stiffness of the tribological contact in detail and show that the contact stiffness plays a central role for small oscillation amplitudes. In the present paper we consider the case of a displacement-controlled system, where the contact stiffness is small compared to the stiffness of the measuring system. It is shown that in this case the macroscopic coefficient of friction is a function of two dimensionless parameters—a dimensionless sliding velocity and dimensionless oscillation amplitude. In the limit of very large oscillation amplitudes, known solutions previously reported in the literature are reproduced. The region of small amplitudes is described for the first time in this paper. [ABSTRACT FROM AUTHOR]

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete-steel joint based on the fractal theory. The topography of contact surface exist in concrete-steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete-steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete-steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete-steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete-steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation. [ABSTRACT FROM AUTHOR]

In this paper, in view of microscopic surface topography characteristic, the relationship of microscopic surface topography characteristic and the dynamic characteristic of macroscopic system is established, and the influence of fractal contact stiffness on the stability and nonlinearity of modal coupling system is studied based on microscopic surface topography. According to the fractal characteristic of metal surface machined, the normal and tangential contact stiffness fractal models of joint surfaces are established and verified. In this paper, a critical two-degree-of-freedom modal coupling model is listed, the fractal contact stiffness obtained is embedded into oscillatory differential equation to study the influence of the coupling between friction coefficient and stiffness ratio of joint surfaces and the coupling between natural frequency and stiffness ratio of joint surfaces on the system stability, and the influence of fractal contact stiffness on the limit cycle of system is further analyzed. The above theoretical analysis can provide a reference for the design of suitable surface topography in the engineering. [ABSTRACT FROM AUTHOR]

The use of characteristics of the contact stiffness in the scientific literature to describe a discrete contact between machine components under normal and tangential loading has been analyzed. The application of a model of the discrete contact of rough surfaces, as well as the influence of the parameters of the contact surface microgeometry on the contact stiffness under normal and tangential loadings and shear has been studied. [ABSTRACT FROM AUTHOR]

A critical analysis was made of the Oliver and Pharr method for determination of the hardness and elastic moduli of materials by instrumented indentations with the continuous recording of the P- hdiagrams ( P is the force acting on the indenter, h is the approach of the indenter and a sample). Mistakes and insufficient justification were revealed in the basic theoretical relations of this method. In particular, this refers to an incorrect definition of the depth of the elastic contact h, which is the base of these relalations. New refined basic relations and formulas for the determination of hardness and elastic modulus are given, in which the above defects are eliminated and which are based only on the assumption of elastic unloading of the indenter according to the classic theories of the elastic contact. In addition to the foregoing, using the data of the P- h diagram measured in the arbitrary laboratory coordinate system, an improved method of the stable determination of the contact stiffness S = d P/dh at the P-h diagram position have been proposed in the commonly accepted theoretical coordinate system in which its basic classic model relations are recorded. These refinements have been derived without additional assumptions to hypotheses to the Oliver and Pharr method and additional experimental measurements. [ABSTRACT FROM AUTHOR]

The hardware-in-the-loop (HIL) simulation is necessary to test the contact dynamics of spacecraft in space. The stiffness parameter is important for the state recording and compliant control. A contact stiffness estimation method is proposed, where the force measurement delay and structural dynamics of the upper platform are compensated to guarantee the estimation accuracy. The force measurement delay and structural deformation of the upper platform lead to the delayed force signal and the position error respectively, and then the contact stiffness identification error. A phase lead based delay compensation is used to reconstruct the true contact force from the measured force. The structural deformation of the upper platform is compensated by a structural dynamics model to obtain the true contact position. Based on the compensated force and position signals, the stiffness identification is performed using the real time recursive least squares (RLS) method. The effectiveness of the proposed method is verified by simulations and experiments. In the experiments, the stiffness identification error is about ± 5 %, which is satisfactory for monitoring and control applications. [ABSTRACT FROM AUTHOR]

Shrink-fit holders are used widely in high-speed machining because of their high concentric and rigid clamp performance. In this paper, a method to determine the contact stiffness and contact force of shrink-fit tool-holder joint is proposed. The relationship between the contact stiffness and the normal contact load is deduced based on Hertz contact theory and fractal geometry theory. The contact force at the interface is obtained through finite element contact analysis. The identified contact stiffness is incorporated into the finite element model of the tool-holder assembly, which is used to predict the tool point frequency response function of the assembly. The predicted frequency response function agrees with that of the experiment, which verifies the proposed identification method. The effect of radial interference, tool insertion length, and rotational speed on contact behaviors between the shrink-fit tool-holder connection is determined, which is informative for the application and design of the tool-holder assembly. [ABSTRACT FROM AUTHOR]

The present paper is devoted to a theoretical analysis of sliding friction under the influence of oscillations perpendicular to the sliding plane. In contrast to previous works we analyze the influence of the stiffness of the tribological contact in detail and also consider the case of large oscillation amplitudes at which the contact is lost during a part of the oscillation period, so that the sample starts to 'jump'. It is shown that the macroscopic coefficient of friction is a function of only two dimensionless parameters-a dimensionless sliding velocity and dimensionless oscillation amplitude. This function in turn depends on the shape of the contacting bodies. In the present paper, analysis is carried out for two shapes: a flat cylindrical punch and a parabolic shape. Here we consider 'stiff systems', where the contact stiffness is small compared with the stiffness of the system. The role of the system stiffness will be studied in more detail in a separate paper. [ABSTRACT FROM AUTHOR]

The prediction of thermo-mechanical behavior for machine tools is important during the designing period. The performance of high-speed machine tools depends not only on their stiffness but also on their thermal behaviors. Especially, the mechanical and thermal characteristics of joints affect the behaviors of machine tools notably. The purpose of this paper is to establish a thermo-mechanical model for machine tools, by using finite element analysis (FEA) method to study temperature expansion and thermal deformation. Unlike the methods developed by other researchers that mainly focused on specific kind of joints, the proposed model studied various joints on machine tools. To improve the accuracy of calculation, this model took both the thermal contact resistance (TCR) and contact stiffness of the joints into account, and the coupling between the TCR and contact stiffness was also studied here. Based on the Hertz theory, fractal geometry was used to analyze the TCR and contact stiffness on the joints. Experiments were set up to test the transient temperature change and the thermal deformation characteristics of a vertical machine center (VMC). The results showed that the proposed model can accurately simulate thermal characteristics of machine tools. [ABSTRACT FROM AUTHOR]

The effect of various factors, such as the roughness parameters, the ratio of the hardnesses of materials of the parts, the average pressure, and the curvature radius of asperities, on the approach in the elastoplastic contact of the flat rough surfaces of two materials with similar hardnesses has been considered. In this case, asperities on the rough surface simultaneously undergo plastic deformation and penetrate into the counterface. [ABSTRACT FROM AUTHOR]

The stiffness of toolholder-spindle joint at high speeds plays an important role in the cutting efficiency and the machining accuracy. A double-locking toolholder (BTF type) is designed to improve the stiffness of joint. This paper presents a macro-micro scale hybrid method to determine the stiffness of double-locking toolholder-spindle joint at high speeds. In this method the finite element method and the three-dimensional fractal method are combined. It is assumed flat in macro-scale for the contact surfaces of joint. The finite element method is introduced to obtain the pressure distribution with the influence of centrifugal force at high speeds. In micro-scale, the contact surfaces are fractal featured and the three-dimensional fractal method is used to compute the stiffness based on the pressure. Experiments with BTF40-type toolholder are conducted to verify the efficiency of the proposed model in zero-speed case. The relationship between the stiffness and the technological parameters of the system can be derived based on the presented model. The upper limit of speed, the optimized range of each technological parameter are determined for obtaining the higher stiffness of joint. The results can provide theoretical basis for improving the cutting efficiency and the machining accuracy of high-speed machine tool. [ABSTRACT FROM AUTHOR]

The effects of roughness and fractality on the normal contact stiffness of rough surfaces were investigated by considering samples of isotropically roughened aluminium. Surface features of samples were altered by polishing and by five surface mechanical treatments using different sized particles. Surface topology was characterised by interferometry-based profilometry and electron microscopy. Subsequently, the normal contact stiffness was evaluated through flat-tipped diamond nanoindentation tests employing the partial unloading method to isolate elastic deformation. Three indenter tips of various sizes were utilised in order to gain results across a wide range of stress levels. We focus on establishing relationships between interfacial stiffness and roughness descriptors, combined with the effects of the fractal dimension of surfaces over various length scales. The experimental results show that the observed contact stiffness is a power-law function of the normal force with the exponent of this relationship closely correlated to surfaces' values of fractal dimension, yielding corresponding correlation coefficients above 90 %. A relatively weak correlation coefficient of 60 % was found between the exponent and surfaces' RMS roughness values. The RMS roughness mainly contributes to the magnitude of the contact stiffness, when surfaces have similar fractal structures at a given loading, with a correlation coefficient of −95 %. These findings from this work can be served as the experimental basis for modelling contact stiffness on various rough surfaces. [ABSTRACT FROM AUTHOR]

The thermal-mechanical character, which is difficult to ensure because of the lack of a corresponding theory and tool, has a significant effect on the dynamics of bearings. It even leads to a sudden failure of bearings in a working situation. In this research, a thermal deformation model was established, based on the analysis of temperature effect on the basic size of angular contact ball bearing. And the transmission from rolling size to bearing axial stiffness was explicit. On the basis of the variation of Hertz contact stiffness and the change of initial contact angle of angular contact ball bearing caused by temperature rise, a 'Thermo- mechanical' model of bearing was proposed. According to this model, using the corresponding calculation procedure programmed by MATLAB, the effect of bearing temperature on the axial stiffness has been studied. And the correctness of this model was verified with experiments. Some design suggestions have been made for the decision of bearing preload: to prevent the bearing failure caused by overheating. [ABSTRACT FROM AUTHOR]

To investigate the tangential contact stiffness of joint surfaces, the asperity contact of the joint surfaces is considered as an elliptically shaped contact, with the major and minor semi-axes of the contact areas assumed to follow a two-dimensional Gamma distribution. Hertz theory for the pressure distribution on an elliptical contact area is applied to solve for the tangential contact stiffness in a single asperity. Numerical results reveal a change in the tangential contact stiffness, with the influential factors being normal load, tangential load, eccentricity of elliptical asperities, correlation coefficient of asperity distribution, measured asperity height, and standard deviation of asperity distribution on joint surfaces. All of the results should provide theoretical guidance for evaluating or refining the tangential contact stiffness of joint surfaces. [ABSTRACT FROM AUTHOR]

A new expression for contact deformation is given, and the normal contact stiffness between single asperities is derived according to Hooke's law. A contact model between two ellipsoidal asperities is simulated by the FE method, the result compared with the theoretical solution. It is found that the curves of the normal contact stiffness versus the included angle in the principal curvature direction show similar trends and evolve as a cosine feature. The effects of the parameters on normal contact stiffness are found to show that normal contact stiffness increases and reaches the upper limit gradually with an increase in these parameters. [ABSTRACT FROM AUTHOR]

When a rubber block is squeezed against a nominal flat but rough surface, the rubber bottom surface will penetrate into the substrate roughness profile. The relation between penetration depth $$w$$ (or the average interfacial separation $$\bar{u}$$ ) and the applied squeezing pressure $$p$$ determines the (perpendicular) contact stiffness $$K=\hbox {d}p/\hbox {d}w=-\hbox {d}p/\hbox {d}\bar{u}$$ , which is important for many applications. We have measured the relation between $$p$$ and $$\bar{u}$$ for a rubber block squeezed against 28 different concrete and asphalt road surfaces. We find a linear relation between $${\mathrm{log}}p$$ and $$\bar{u}$$ , in agreement with theory predictions. The measured stiffness values correlate rather well with the theory prediction. [ABSTRACT FROM AUTHOR]

The paper is devoted to an experimental and theoretical investigation of the static friction force between a rapidly oscillating sample and a steel plate. The static frictional force is studied experimentally as function of the oscillating amplitude, the normal force and the contact geometry. A simplest model of tangent contact with a constant friction coefficient is proposed and shows a good agreement with experiment. The static friction force is proved to be a universal function of the ratio of the oscillation amplitude, the indentation depth and to the friction coefficient. [ABSTRACT FROM AUTHOR]

It has recently been suggested that many contact mechanics problems between solids can be accurately studied by mapping the problem on an effective one-dimensional (1D) elastic foundation model. Using this 1D mapping, we calculate the contact area and the average interfacial separation between elastic solids with nominally flat but randomly rough surfaces. We show, by comparison to exact numerical results, that the 1D mapping method fails even qualitatively. We also calculate the normal interfacial stiffness K and compare it with the result of an analytic study. We attribute the failure of the elastic foundation model to the incorrect treatment of the long-range elastic coupling between the asperity contact regions. [ABSTRACT FROM AUTHOR]

The digital image correlation (DIC) and ultrasound techniques have both previously been employed to measure the contact stiffness of real engineering interfaces, but a comprehensive comparison of these techniques has not previously been carried out. Such a comparison is addressed in the present paper. The principal novelty in this work is that DIC and ultrasound are used to simultaneously measure contact stiffness in the same tests and on the same contact interface. The results show that ultrasound measures somewhat higher contact stiffness magnitudes than DIC: at an average normal contact pressure of 70 MPa, ultrasound was around three times stiffer. Given that the techniques are vastly different in their measurement approach (DIC measures micron-scale relative displacements from external side-on images of the interface, while ultrasound uses the reflection of an Ångstrom scale ultrasonic perturbation from the interior of the interface itself), this level of agreement is thought to be encouraging. The difference in results can partly be explained by consideration of inherent physical differences between the techniques which have previously received little attention. Ultrasound measurement will always give the local elastic 'unloading stiffness' (even at a plastically deforming contact); whereas, a load-deflection technique like DIC, will give the 'loading stiffness'. The reason for this difference is discussed in the paper and tests carried out under increasing tangential load in the pre-sliding regime illustrate this difference experimentally. Under normal loading, the increase in real contact area obscures the effect to some extent as both DIC and ultrasound stiffnesses increase with normal load. The results suggest that rough interfaces may be satisfactorily modelled as a variable stiffness spring whose stiffness increases with contact pressure as the smooth contact case is approached. [ABSTRACT FROM AUTHOR]

The vibro-impact mechanism of a rotating shrouded blade with asymmetric gaps is investigated. The Frobenius method is employed to determine the dynamic frequencies and corresponding mode functions of the shrouded blade under the action of dynamic stiffness. The mode functions are used to truncate the governing partial differential equation of the shrouded blade to ordinary differential equations by using the Galerkin method. Taking into account the influence of the rotating speed, the contact stiffness of adjacent blades is assumed to be an equivalent contact spring stiffness which is determined by the bending dynamic stiffness of the blade. Then the average method is employed to obtain the dynamic responses of the primary, the sub-harmonic and the super-harmonic resonances of the shrouded blade with wake flow excitation, respectively. Finally, the dynamical responses of a shrouded blade with a set of typical material constants and geometrical parameters are given to illustrate the influence of the blade shroud gaps on the vibration amplitude. [ABSTRACT FROM AUTHOR]

The paper considers elastic contact of rough surfaces and develops a simple analytical expression for the stiffness of the contact under tangential loading, which predicts that the contact stiffness is proportional to normal load and independent of Young's Modulus. The predictions of this model are compared to a full numerical analysis of a rough elastic contact of finite size. The two approaches are found to be in good agreement at low loads, when the asperity spacing is large, but the numerical approach predicts much lower stiffnesses at medium and high loads. It is shown that the overall stiffness cannot exceed that of the equivalent smooth contact, and a simple means of modifying the analytical approach is proposed and validated. [ABSTRACT FROM AUTHOR]

The paper is devoted to the experimental, theoretical analysis and computer simulation of influence of elastic properties of contact stiffness and wheel-plate stiffness on the forces of vehicle-track interaction. Three types of wheels are considered with different contact stiffness and wheel-plate design. Exemplary simulation of freight car interaction with track which posses one corrugated rail for each type of wheel is presented. [ABSTRACT FROM AUTHOR]

Dynamic behaviors of ball-type contact surfaces under unbalanced bending loads are studied using point-to-point analysis, threedimensional finite element simulation based on the Hertz Contact Theory, and a modal test. Results derived from these models are very similar but the Finite Element Model provides the best results since it allows for more elements of study, such as the steel ball, carriage, rail etc. In the study, results also show that frequencies vary slightly, but there is an obvious change in shapes. Therefore, the contact stiffness in simulations must be properly selected with the conclusion that different external loadings may affect the dynamic characteristics of such structures significantly. [ABSTRACT FROM AUTHOR]

An experimental apparatus designed specifically for fretting experiments on mechanical lap joints is presented. A piezoactuator is used to impose fretting motion, and a tri-axial load cell is used to measure tangential force as well as possible misalignment forces. A laser nanosensor is employed to measure the relative motion between the joint halves. No post-processing and filtering of the data is needed to obtain the fretting response using this apparatus. Instead, raw data obtained from experiments with monolithic and 1-bolt aluminum and steel joints under various loading conditions suggest that noise, misalignment, stiffness and damping associated with the apparatus are minimal, and thus the fretting behavior of the mechanical lap joints is accurately captured. Analyses of typical fretting loops obtained by the proposed apparatus suggest that normal preload and maximum tangential displacement influence the critical joint parameters of stiffness and damping. Aluminum joints show a more compliant behavior with more energy dissipation compared to steel joints. [ABSTRACT FROM AUTHOR]

The paper describes experiments to investigate the frictional properties of a Titanium alloy (Ti-6Al-4V) and a Nickel alloy (Udimet 720) under representative engineering conditions. Flat fretting pads with rounded corners were clamped against a flat specimen and a servo-hydraulic tensile testing machine was used to apply cyclic displacement to the specimen. Slip displacement between the specimen and pad was measured remotely using an LVDT and locally using digital image correlation. The latter approach allowed accurate determination of the tangential contact stiffness from frictional hysteresis loops. The results obtained show that the contacts are significantly less stiff than would be predicted by a smooth elastic contact analysis. A finite element model of the experimental contact geometry was constructed and it was shown that good agreement with the experimental measurements of contact stiffness can be obtained with a suitable choice of elastic modulus for a compliant surface layer. [ABSTRACT FROM AUTHOR]

The development of reliable discrete element models to simulate the mechanics of granular media requires knowledge of the grain-to-grain contact laws of the material in question. We have conducted a series of normal and sliding contact experiments on material used in laboratory triaxial experiments to obtain such contact laws for DEM simulations of the experiments. The contact experiments employed segments of 14.72 mm-diameter spherical grains from the triaxial specimens and flat specimens of the same material. The spherical grains had a uniform diameter with a smooth surface finish. Monotonic and cyclic loading paths were applied in both the normal and sliding modes, and both sphere-sphere and sphere-flat contact behavior were examined. Force-displacement behavior and frictional loss were measured in all cases. The behavior was generally Hertzian in the normal contact experiments, which involved forces up to approximately 100 N. The normal contact stiffness increased from ≈2 to 15MN m−1 over the range of normal force examined. The sliding experiments employed several normal forces up to approximately 25 N, and produced a value of the coefficient of static friction of 0.28. The shear stiffness of the sliding contact increased with normal force, and ranged from 0.8 to 1.2MN m−1 under normal loads ranging from ≈1 to 7.5 N, respectively, for virgin contacts. The shear stiffness observed for the sphere-flat contact decreased with wear. Surface roughness measurements were obtained on both tested and untested regions of the spheres and flat specimens. The average roughness (Ra) for untested regions of the sphere and flat specimens were 270 and 230 nm, respectively. Repeated testing in the sliding mode reduced these values by 29–45% for the flat surfaces, and by 20% for the spherical contact. Frictional losses were observed in both the normal and sliding modes. In the sliding mode, frictional loss decreased with increasing normal load. We observed stable sliding (associated with significant contact movement under an increasing shear force) at forces that were below the macroscopic frictional limit and resulted in permanent displacement of the contact. There was generally a distinct threshold in shear force for this permanent sliding. The extent of sliding increased significantly with wear for the sphere-flat contact and was accompanied by a substantial drop in shear stiffness. [ABSTRACT FROM AUTHOR]