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In order to analyze and predict the performance of mechanical structure and system, the tangential contact of solid-liquid interface under normal and tangential loads is studied, and the tangential stiffness model of rough surfaces is established by analyzing various contact states of the asperity under normal elastic-plastic deformation and tangential stick-slip. Based on the contact model of closed oil pits based on the influence of rough surfaces, on the basis of fully considering the influence of gradual application of load on the number of closed oil pit, and then changing the load distribution of solid-liquid interface, a liquid tangential stiffness model is established. The tangential stiffness of solid-liquid interface is obtained by two parts in parallel. Through simulation and experimental analysis, the effects of normal load, deformation and lubricant viscosity on the tangential stiffness of solid-liquid interface are revealed, and the tangential stiffness with or without lubricating medium is compared and analyzed. The results show that the tangential stiffness of solid-liquid interface and solid-solid interface increases nonlinearly with the increase of normal load and deformation. Due to the contribution of liquid stiffness, the tangential stiffness of solid-liquid interface is always slightly higher than that of solid-solid interface. The greater the viscosity of the lubricating medium, the greater the tangential stiffness of the solid-liquid interface. At low load, liquid contact is dominant, and the tangential stiffness of solid-liquid interface is low. Under moderate and heavy loads, the solid contact is dominant, and the tangential stiffness of the solid-liquid interface increases rapidly. Therefore, the tangential contact stiffness of the solid-liquid interface in mixed lubrication can be effectively improved by increasing the normal load and increasing the viscosity of the lubricating medium.

Prestressed anchor cables are widely used in ocean engineering, the accurate and efficient prestress measurement is crucial for ensuring their safety and stability. In this study, a new theoretical model based on the Hertz contact theory and fractal theory was proposed to establish the relationship between tangential contact stiffness (TCS) at the interface of the anchorage and the anchor bedding plate in the external anchor head section. This model was validated through indoor static load tests, and its feasibility for practical application was explored by indoor dynamic load tests. The results demonstrate that both tangential load and tensile load are the primary factors determining the value of TCS. TCS decreases with increasing tangential load, but increases with growth in tensile force. The calculated theoretical values based on the proposed model align closely with test data from indoor static load tests, indicating the efficiency of the proposed model in prestress evaluation. The correlation trends between TCS and tensile force observed in dynamic load tests exhibit good consistency with those found in static load tests, which also align with calculated values based on the proposed model, demonstrating its feasibility in dynamic load tests. The proposed model is modified by considering the interactions between the micro-convex bodies, and other parameter such as fractal dimension D need to be considered when applying the proposed model. These findings can provide valuable insights for the application of prestressed anchor cables in ocean engineering. • A new contact model based on fractal theory was established for the contact interface description. • The correlation between the tangential contact stiffness and tensile force can be well described by the proposed model. • The present model exhibits greater feasibility compared to the previous methods. [ABSTRACT FROM AUTHOR]

An experimental method that can quantify relative slip was developed using digital image correlation (DIC) in order to evaluate the sliding portion. The bridge-type test setup was designed to establish the fretting contact condition. The relative displacements between the contact surfaces were determined by DIC methods. Based on the evolution and distribution of relative slip, the transitions from gross slip to partial slip on the contact surface were found throughout all tests. This result indicated that the fretting scar was closely correlated to relative slip. The variation of relative slip corresponding to the stick-slip state was consistent with the tangential force coefficient. Besides, the load amplitude was an important factor for fretting fatigue damage, which can affect the stick-slip state. [ABSTRACT FROM AUTHOR]

This paper proposes a modified tangential contact stiffness model considering friction's effect, which is the first key step to establish the dynamic model of the fixture-workpiece system, and this is the foundation of vibration suppression for the manufacturing process of aerospace blades. According to Love's elastic deformation, the model's derivation process starts with the potential function in each coordinate axis's direction respectively. The generalized Hertz contact theory is employed to calculate the contact forces in this model. The symmetrical characteristic of the contact area has simplified the derivation process to obtain the eventual tangential contact stiffness model. A validation experiment focusing on a tangential stiffness measuring is achieved by putting two spherical objects in contact together to get the tangential contact stiffness. Based on the data collected in this experiment, a comparison with a most similar existed model is carried out, and the result shows that the relative error of this modified model are all less than 10%, while the original model's (the most similar model) relative error exceeding 50% captures more than 3/4 of the 30 data sets randomly selected in each experiment group, and that means the modification of this paper brings great improvement to the contact stiffness model. [ABSTRACT FROM AUTHOR]

An experimental method that can quantify relative slip was developed using digital image correlation (DIC) in order to evaluate the sliding portion. The bridge-type test setup was designed to establish the fretting contact condition. The relative displacements between the contact surfaces were determined by DIC methods. Based on the evolution and distribution of relative slip, the transitions from gross slip to partial slip on the contact surface were found throughout all tests. This result indicated that the fretting scar was closely correlated to relative slip. The variation of relative slip corresponding to the stick-slip state was consistent with the tangential force coefficient. Besides, the load amplitude was an important factor for fretting fatigue damage, which can affect the stick-slip state.

This paper proposes a modified tangential contact stiffness model considering friction’s effect, which is the first key step to establish the dynamic model of the fixture-workpiece system, and this is the foundation of vibration suppression for the manufacturing process of aerospace blades. According to Love’s elastic deformation, the model’s derivation process starts with the potential function in each coordinate axis’s direction respectively. The generalized Hertz contact theory is employed to calculate the contact forces in this model. The symmetrical characteristic of the contact area has simplified the derivation process to obtain the eventual tangential contact stiffness model. A validation experiment focusing on a tangential stiffness measuring is achieved by putting two spherical objects in contact together to get the tangential contact stiffness. Based on the data collected in this experiment, a comparison with a most similar existed model is carried out, and the result shows that the relative error of this modified model are all less than 10%, while the original model’s (the most similar model) relative error exceeding 50% captures more than 3/4 of the 30 data sets randomly selected in each experiment group, and that means the modification of this paper brings great improvement to the contact stiffness model.

Turbomachinery and other jointed structures are carefully designed to optimise their dynamic response and prevent unwanted high-cycle fatigue failures due to vibration. Advanced numerical models are employed to predict the often nonlinear dynamic responses, but their reliability is partially limited by the lack of understanding of the friction mechanisms between the vibrating contact interfaces. Although several high-frequency friction rigs have been developed at different institutions to measure contact parameters such as friction coefficient and contact stiffness, a lack of direct comparisons prevents a throughout understanding. To address this issue, a comparison of these contact parameters has been performed by employing the high-frequency friction rigs of Imperial College London and Politecnico di Torino. A test plan was designed to cover a wide experimental space by testing the friction rigs to their limits and measuring hysteresis loops under a range of normal loads and displacement amplitudes at room temperature. Measurements from the two very different experimental setups are compared, showing a good level of agreement for the friction coefficient, but also highlighting some differences, especially for the contact stiffness. New insights are provided into the physics of these contact parameters and specific guidelines are given to improve contact models used for nonlinear dynamic analysis. [Display omitted] • Compared high-frequency hysteresis loops from friction rigs of two institutions. • Friction coefficient showed good agreement, but contact stiffness varied up to 70% • Contact stiffness strongly depends on the extension and distribution of wear spots. • Contact stiffness decreases by five times from stuck loops to loops in gross slip. • Guidelines proposed for enhanced contact parameter use in friction joint models. [ABSTRACT FROM AUTHOR]

On the basis of the M-B fractal model,and combining with distribution formula of two cylinders contact area,a calculation model of tangential contact stiffness for gear joint surface is set up by using anisotropic fractal geometric theory.The influence of the main parameters in the model on the gear fractal contact model is obtained through emulation of MATLAB.The results indicate that the joint surface tangential contact stiffness increases with increasing material property or gear teeth,but decreases with increasing fractal roughness or ratio of total tangential load to total normal load.When fractal dimension is smaller（D < 1.7）,the gear joint surface tangential contact stiffness increases as fractal dimension increases.However,when fractal dimension becomes larger（D≥1.7）,the gear joint surface tangential contact stiffness decreases as fractal dimension increases.

In this work, the role of the contact stiffness in the measurement of principal variables in fretting wear tests is assessed. Several fretting wear tribometers found in the literature, including one developed by the authors, are analysed and modelled using numerical methods. The results show the importance of the tribosystem stiffness and tangential contact stiffness in the displacement sensor calibration and in the correct numerical modelling of fretting wear tests, especially for flat-to-flat contact configuration. The study highlights that, in most cases, direct comparisons between fretting results with severe wear obtained with different tribometers cannot be performed if the contact stiffness is not properly considered during the development of the experiments.

Vibration of a blade couple damped by a dry friction contact in the shroud is investigated by means of hysteresis loops and response curves analysis. The studied system is excited by one harmonic external force in a frequency range near to the lowest eigenfrequency of real blades. Blades are connected by means of a damping element consisting of dry friction part linked in series with linear spring. This "stick-slip" damping element is supposed to be either weightless or of a very small mass which models the mass of elastically deformed parts of contacting bodies near the friction surface. Two approximate mathematical models of "stick-slip" dry friction elements are suggested and analysed. The response curves of blade couple connected by stick-slip damping element are presented for different values of slip friction forces and two values of mass of elastically deformed parts.

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This study demonstrates optical measurements of a real contact area and tangential contact stiffness in a rough contact interface between a soft adhesive elastomer and a glass plate. A friction tester developed in this study employs a rough contact interface between a rough rubber plate that is made of cross-linked poly-dimethyl siloxane (PDMS) and a smooth optical glass (BK7) hemisphere. This friction tester also equips a transmission optical system that is composed of a white light-emitting diodes (LED) light source, an objective lens system, and a charge coupled device (CCD) camera, for the in situ observation of the space distribution of the real contact area in the apparent contact region. Based on a simplified analysis, in accordance to the measurements of the transmitted light intensity, the amount of the real contact area was determined without any complicated calibrations. As a result, it was found that the amount of the real contact area linearly increased with normal load. Furthermore, using the digital image correlation (DIC) method, the time changes in the surface displacement field within the apparent contact region were visualized from the onset of the sliding motion to the onset of the steady sliding. These results provided a direct estimation of the tangential contact stiffness that resulted from the tangential deformation at the contacting asperity layer.

Vibration of a blade couple damped by a dry friction contact in the shroud is investigated by means of hysteresis loops and response curves analysis. The studied system is excited by one harmonic external force in a frequency range near to the lowest eigenfrequency of real blades. Blades are connected by means of a damping element consisting of dry friction part linked in series with linear spring. This ''stick-slip'' damping element is supposed to be either weightless or of a very small mass which models the mass of elastically deformed parts of contacting bodies near the friction surface. Two approximate mathematical models of ''stick-slip'' dry friction elements are suggested and analysed. The response curves of blade couple connected by stick-slip damping element are presented for different values of slip friction forces and two values of mass of elastically deformed parts. [ABSTRACT FROM AUTHOR]

The tangential contact stiffness of cylindrical asperities is investigated using macro- and micro-mechanisms in this study. A microanalysis model is developed and the tangential contact stiffness of elliptically parabolic asperities on the contact surface is determined. The shape influence coefficient of the cylindrical contact is defined, and its rationality is evaluated. The influence of asperity distribution on the rough surface is determined, and the tangential contact stiffness macroanalysis model is constructed based on fractal theory. The mathematical expression to determine the tangential contact stiffness of the macroscopically cylindrical contact is generated, and the effects of influential factors on tangential contact stiffness are explicitly evaluated. Numerical results show that the tan-gential contact stiffness of asperities is determined by several factors, such as material properties, applied loads, fractal dimension, and surface shape. [ABSTRACT FROM AUTHOR]

Digital image correlation (DIC) is a full field three dimensional measurement technique that can quantify displacements and strains of a surface. In this paper, digital image correlation is used as a slip measurement technique during coupon scale fretting fatigue experiments. Slip measured with the novel DIC technique is compared to conventional slip measurement techniques as clip gauges and modified clip gauge measurements proposed by Wittkowsky et al. Slip measurements with the DIC system show lower slip values and higher tangential contact stiffness's compared to (modified) clip gauge measurements. Slip measured with DIC is obtained closer to the contact compared to clip gauges, eliminating the influence of elastic deformations or fitting parameters. During the fretting fatigue experiments are two equal contacts simultaneously tested. However, the slip of both contacts is not identical with outliers of more than 10% difference in slip amplitude. [ABSTRACT FROM AUTHOR]

The tangential contact stiffness is an important parameter used in non-linear dynamic analyses of jointed structures since it can strongly affect the prediction of resonance frequencies. Many experimental techniques are available for contact stiffness estimations, but the reliability of such estimations remains unknown due to a lack of comparative studies. This paper proposes a comparative study of contact stiffness measurements obtained with two experimental techniques: hysteresis loop measurements and Frequency Based Substructuring (FBS). Hysteresis loops are traditionally measured with dedicated friction test rigs to provide, amongst others, contact stiffness estimations through local interface measurements. The assumption with hysteresis measurements is that the measured parameters are independent of the dynamics of the test rig and can therefore be used as input for analyses of other structures, as long as loading conditions and contact interfaces are comparable. An alternative approach to identify the contact stiffness is FBS, which uses information from the overall system dynamics. FBS has the advantage that it can be applied to any structure, without the need of building ad-hoc test rigs, consequently giving a structure-specific information. Despite this advantage over hysteresis measurements, it is as of yet not well understood how accurately FBS can extract contact stiffness values. This paper presents FBS measurements and hysteresis loop measurements performed simultaneously on the same contact interface of a traditional high-frequency friction rig during vibration, thus enabling a cross-validation of the results of both techniques. This novel comparison validates FBS approaches against local hysteresis measurements and shows the strengths and limitations of both experimental methods, making it possible to improve the current understanding of the contact stiffness of jointed structures. • Contact stiffness measurements and guidelines for use in nonlinear dynamic analysis. • Comparison of FBS and hysteresis loops for contact stiffness estimations. • FBS can provide structure-dependent contact stiffness estimations. • Joint resonances strongly influence contact stiffness over a wide frequency range. • The shape of stuck hysteresis loops can indicate (unknown) structural resonances. [ABSTRACT FROM AUTHOR]

An experimental method for estimating the tangential contact stiffness of contact interface with controlled contact asperities is discussed. The tangential contact stiffness is analyzed from the shift of certain eigenmode frequency of the system resulting from the changes in the density of contact asperities and the application of normal force. The results show that for the case of contact combination with 189 contact asperities, the corresponding eigenmode frequency shifted from 1.01 kHz at normal force of 19 N to 1.64 kHz at normal force of 148 N. This shift corresponds to the increase of tangential contact stiffness from 34.08 MN/m at normal force of 19 N to 67.44 MN/m at normal force of 148 N. It is found that the corresponding eigenmode frequency and its shift were affected by the density of contact asperities as well as the normal force. In the case of contact combination with 397 contact asperities, the corresponding eigenmode frequency shifted from 2.36 kHz at normal force of 49 N to 3.9 kHz at normal force of 153 N, indicating the increase of the tangential contact stiffness from 93.91 to 140.73 MN/m. A simple power relationship is proposed to explain the quantitative relationship between the eigenmode frequency shift and the tangential contact stiffness value. [ABSTRACT FROM PUBLISHER]

Abstract: Friction-induced stick–slip of brake friction materials was investigated by examining the correlation between contact stiffness (k n) and surface topography changed by material wear at the sliding interface. By changing the surface roughness and thickness of the friction material, friction instability during sliding at low speeds was analyzed using a scale brake dynamometer. The results showed that the intensity and frequency of stick–slip were determined by the relative magnitudes of tangential contact stiffness (k t), bulk stiffness (k b), sliding velocity, and the difference (Δμ) between the static (μ s) and kinetic coefficients (μ k) of friction and these results were supported by the contact theories proposed by Greenwood and Williamson. While Friction materials with rough surfaces showed small stick–slip amplitudes due to small tangential contact stiffness (k t), friction materials with high compressibility exhibited pronounced stick–slip because the amplitude inversely varied with the bulk stiffness (k b). The high stick–slip propensity observed from heat affected friction materials also supported the correlation of the surface condition and contact stiffness (k n). [Copyright &y& Elsevier]

A general contact stiffness model is proposed in this paper to study the contacts between rough surfaces of machined plane joints. The proposed model uses fractal geometry for surface topography description, elastic-plastic deformation of contacting asperities, and size-dependent contact stiffness of microcontacts, where the contact stiffness is derived from Hertz contact theory. Three cast iron specimens are produced from different machining methods (milling, grinding, and scraping), and their rough surface profiles are extracted. The structure function method was used to calculate each profile's fractal dimension and scale coefficient. Both theoretical analysis and experimental results of contact stiffness are obtained for these specimens under different contact loads. The comparison between the theoretical contact stiffness and the experimental results at the interface indicates that the present fractal model for the contact stiffness is appropriate and the theoretical contact stiffness is consistent with the experimental data. [ABSTRACT FROM AUTHOR]

• Ultrasound is used to monitor non-transparent friction contacts during vibration. • Strong ultrasonic signal modulation through the contact interface during vibration. • Ultrasonic signal modulation is linked to stick/slip contact state transitions • Ultrasonic transmission is shown to decrease with increasing sliding amplitude. • Above-identified features enable real-time contact state detection via ultrasound. Complex high-value jointed structures such as aero-engines are carefully designed and optimized to prevent failure and maximise their life. In the design process, physically-based numerical models are employed to predict the nonlinear dynamic response of the structure. However, the reliability of these models is limited due to the lack of accurate validation data from metallic contact interfaces subjected to high-frequency vibration cycles. In this study, ultrasonic shear waves are used to characterise metallic contact interfaces during vibration cycles, hence providing new validation data for an understanding of the state of the friction contact. Supported by numerical simulations of wave propagation within the material, a novel experimental method is developed to simultaneously acquire ultrasonic measurements and friction hysteresis loops within the same test on a high-frequency friction rig. Large variability in the ultrasound reflection/transmission is observed within each hysteresis loop and is associated with stick/slip transitions. The measurement results reveal that the ultrasound technique can be used to detect stick and slip states in contact interfaces subjected to high-frequency shear vibration. This is the first observation of this type and paves the way towards real-time monitoring of vibrating contact interfaces in jointed structures, leading to a new physical understanding of the contact states and new validation data needed for improved nonlinear dynamic analyses. [ABSTRACT FROM AUTHOR]

The dynamic analysis model of axial piston pump was established; both the kinematics and dynamics simulation analysis were conducted by virtual prototyping approach. The displacement, velocity, acceleration and stress curves of the piston under different working conditions were investigated. In addition, a ball-in-socket contact model was established, and the effects of hydraulic pressure, piston radius and radial clearance on normal displacement, contact radius, maximum contact pressure, normal contact stiffness and tangential contact stiffness were analyzed comprehensively. The results indicate that the normal displacement, maximum contact pressure, contact radius, normal contact stiffness and tangential contact stiffness can be improved by enlarging the piston radius and decreasing the radial clearance. [ABSTRACT FROM AUTHOR]

In this work, the role of the contact stiffness in the measurement of principal variables in fretting wear tests is assessed. Several fretting wear tribometers found in the literature, including one developed by the authors, are analysed and modelled using numerical methods. The results show the importance of the tribosystem stiffness and tangential contact stiffness in the displacement sensor calibration and in the correct numerical modelling of fretting wear tests, especially for flat-to-flat contact configuration. The study highlights that, in most cases, direct comparisons between fretting results with severe wear obtained with different tribometers cannot be performed if the contact stiffness is not properly considered during the development of the experiments. [ABSTRACT FROM AUTHOR]