Hertzian Contact Research Articles

Particulate behaviour of soft granular materials: a case study on lentils

In this article, lentils are used as a case study to characterise the particulate behaviour of soft granular materials. Experiments were carried out on a single lentil particle or a pair of lentils. The single particle or a pair of particles in contact were compressed vertically to crushing or to a fixed vertical load. Then, in inter-particle tests, pairs of particles were slid over each other at a constant vertical load, and the tangential stiffness and coefficient of friction were estimated. The pairs of lentil particles in contact were also subjected to repeated normal and tangential loading. The presence or absence of the cover of lentil particles (shell) was found to affect their behaviour significantly under these loading conditions. The lentil particles have a very compliant shell and stiff core in normal loading; the stiffness of the shell is constant, and the core follows Hertz contact law. The lentil particles show less variability in their crushing strength, with high Weibull modulus (~ 10), in comparison to other natural granular materials like sand. With repeated cycles of vertical loading, the contact between a lentil pairs of particles becomes more stiff and less damp. In tangential loading, the coefficient of friction between lentil particles decreases with normal force while the contact stiffness increases. Further, in cyclic tangential loading, the coefficient of friction decreases and the contact stiffness increases with cycles. A simple contact model is also proposed to use in discrete element simulations.Graphical abstract

Piezoresistive Cantilever Microprobe with Integrated Actuator for Contact Resonance Imaging

A novel piezoresistive cantilever microprobe (PCM) with an integrated electrothermal or piezoelectric actuator has been designed to replace current commercial PCMs, which require external actuators to perform contact-resonance imaging (CRI) of workpieces and avoid unwanted “forest of peaks” observed at large travel speed in the millimeter-per-second range. Initially, a PCM with integrated resistors for electrothermal actuation (ETA) was designed, built, and tested. Here, the ETA can be performed with a piezoresistive Wheatstone bridge, which converts mechanical strain into electrical signals by boron diffusion in order to simplify the production process. Moreover, a new substrate contact has been added in the new design for an AC voltage supply for the Wheatstone bridge to reduce parasitic signal influence via the EAM (Electromechanical Amplitude Modulation) in our homemade CRI system. Measurements on a bulk Al sample show the expected force dependence of the CR frequency. Meanwhile, fitting of the measured contact-resonance spectra was applied based on a Fano-type line shape to reveal the material-specific signature of a single harmonic resonator. However, noise is greatly increased with the bending mode and contact force increasing on viscoelastic samples. Then, to avoid unspecific peaks remaining in the spectra of soft samples, cantilevers with integrated piezoelectric actuators (PEAs) were designed. The numbers and positions of the actuators were optimized for specific CR vibration modes using analytical modeling of the cantilever bending based on the transfer-matrix method and Hertzian contact mechanics. To confirm the design of the PCM with a PEA, finite element analysis (FEA) of CR probing of a sample with a Young’s modulus of 10 GPa was performed. Close agreement was achieved by Fano-type line shape fitting of amplitude and phase of the first four vertical bending modes of the cantilever. As an important structure of the PCM with a PEA, the piezoresistive Wheatstone bridge had to have suitable doping parameters adapted to the boundary conditions of the manufacturing process of the newly designed PCM.

Contact force calculation and evolution analysis of granular systems based on micro-CT experiment

The calculation of inter-granule contact force in three-dimensional (3D) granular systems is a key and challenging aspect of granular mechanics research. Two elastic rubber balls are used as research objects for in-situ flat pressing Micro-CT experiments. Based on the Hertzian contact theory and Tatara large deformation contact theory, the contact model of elastic balls is verified, and the theoretical formula of the contact force of elastic balls based on the experiment is obtained. Taking the 3D granular systems as research object, in-situ probe loading experiment of micro-CT is carried out to obtain the 2D image sequence of the granules, after a series of digital transformations, the digital body images emerge, the contact force networks of the 3D granular systems under different loading conditions are obtained by constructing pore network models. The contact force distribution and evolution law of the granular systems are analyzed. The relation among the number of strong contacts, the distribution evolution, and the stability of the granular system is explored. The results show that the two elastic ball contact model conforms to the Hertzian contact theory and Tatara large deformation contact theory, and the contact force fitting formula based on experiment can characterize the contact force between two granules reasonably and effectively. The contact force of granules under probe loading is distributed in a net-like pattern starting from the contact point of the indenter and gradually transmitted to the lower and the surrounding area. The trend of average contact force is consistent with the trend of the contact times, showing a significant phase transition. With the increase of contact times, the frequency of particle compression increases, resulting in a greater contact force between granules, ultimately stabilizing at about 10.5 N. The number of strong contacts accounts for 45% to 50% of the total number of contacts, distributed throughout the whole granular system and supporting the network structure of the granular system. The larger values are concentrated below the indenter and exhibit a branching distribution. In the loading process, an equilibrium point is established at <i>z</i> = 14 mm, where the number of strong contacts reaches the peak. The network structure of strong contact force is spread throughout the entire 3D granular system, establishing the main skeleton that can withstand external loads. As the loading continues, the total value of strong contact forces increases, and their distribution within the granular system becomes more uniform.

The effect of relative position on the motion and interactions of particle sedimentation under gravity in a finite-width vertical channel

Understanding the settling behaviour of particles and the interactions between fluid and particles in channelled fluids is crucial for various natural and industrial processes. The research presented in this study focuses on elucidating the sedimentation modes of dual particles within finite-width channels, employing the Arbitrary Lagrangian-Eulerian (ALE) method in conjunction with Hertz contact theory. We aim to understand how the initial relative position affects the dynamic characteristics of particle settlement. The threshold values of α = 35° and gap ratio (the distance between particles relative to their diameter) L/D = 3.5 are deemed critical. The phenomenon of particles ‘kissing’ does not manifest when the initial angle α ≤ 35°. As the α increases, the DKT (Drafting-Kissing-Tumbling) process recurs at earlier stage. In all cases where dual-particle are arranged in a vertically aligned pattern (D P1 = D P2 or DP1> DP2 ), the settling behaviour of particles will eventually achieve a steady state characterized by the DKT process when L/D ≤ 4.0. As the gap ratio (L/D) increases, the time taken for the kissing phenomenon is prolonged. Additionally, the two particles exchange positions and the larger particle taking the lead in the settling process. In scenarios where dual particles are arranged side by side, the kissing phenomenon does not occur. Instead, the trajectory of the particles deviates from the centreline and shifts towards the channel wall due to the Magnus effect when L/D ≤ 2.0. When L/D ≥ 3.0, particles settle as though they were a single particle. In cases involving multiple particles, the hindering effect of the wall becomes significant and can not be ignored, leading to the observation of Rayleigh-Taylor instability development. As a result, the overall settlement pattern tends to be ‘concave’.

Dynamic modeling and analysis of wind turbine gear-bearing coupling system considering tooth root crack and slicing coupling effect

Tooth root cracks would change the meshing stiffness excitation of the gears, increasing the vibration and noise of wind turbine gearbox. Currently, there is relatively little literature that focuses on the slicing coupling effect of cracked gears, leading to incomplete accuracy in solving the time-varying mesh stiffness (TVMS) of gears under cracked conditions, making it difficult to accurately reflect the impact of root crack excitations on the dynamic behavior of wind turbine gear-bearing coupled systems. In this work, The improved analytical calculation method is proposed for the TVMS and system vibration coupling solution of the cracked gear pair, considering the slicing coupling effect. The dynamic model of the wind turbine high-speed stage gear set is developed using the lumped parameter approach, and the rolling bearing is modeled using the Hertz contact theory. Then, the excitations of the gear-bearing coupling system are developed using an improved slicing method. Lastly, the dynamic model of the wind turbine gear-bearing coupling system that considers tooth root cracks and the slicing coupling effect is established and validated. The results indicate that The TVMS modification algorithm considering the slicing coupling effect can better depict the dynamic change of the gear meshing stiffness under the crack failure. Cracks could reduce the TVMS of the gear pair, neglecting the coupling effect between the sliced gear tooth gains the potential risks of overestimating the influence of the gear crack depth on the TVMS. In addition, Gear cracks excitation may alter the dynamic contact load amplitude of the bearing rolling elements, resulting in a tendency for the load-bearing area to decrease. Moreover, When the cracked teeth are involved in meshing, The decrease in mesh stiffness will result in a larger amplitude periodic shock response to the system vibration displacement and a sideband phenomenon near the mesh frequency, neglecting the coupling effect between the sliced gear tooth or simplifying the bearing to a linear support stiffness matrix gain the potential risks of underestimating the influence of the gear crack depth on the vibration displacement of the gear-bearing coupling system.

Geometric generation and meshing characteristics analysis of a new S-shaped gear drive

A new type of S-shaped gear drive with a tooth profile combining involute and circular-arc curves is proposed in this paper. Tooth profile design includes convex involute, convex circular-arc, concave involute, and concave circular-arc curves is carried out. Geometric generation and mathematical model of tooth surfaces of the S-shaped gear are established based on imaginary rack method. Utilizing Hertz contact theory, the calculation formula of contact stress of tooth surfaces is provided. Furthermore, simulation analysis of contact stress based on finite element method under various working conditions is studied using ANSYS/Workbench software. And the comparisons of stress results with general involute gear are also provided. Dynamic characteristics analysis of the generated tooth surfaces is put forward using the established virtual prototype. Normal contact force, circumferential force, axial force and radial force of tooth surfaces both S-shaped gear and involute gear are analyzed. Research results show that the developed S-shaped gear has better transmission performance than involute gear. Further study on dynamic analysis and manufacturing method will be carried out.

Theoretical modeling of the interfacial shear and contact stresses of 2D braided composite cylinders

AbstractThis research aims to develop an analytical model to predict the interfacial shear stress distribution and contact stress of 2D braided composite cylinders. For this purpose, a tubular structure including a helix braid on a cylindrical mandrel is considered. The interfacial shear stress and axial tensile stress in the composite structure was estimated by applying shear lag assumptions. The contact normal stress and adhesion energy were estimated by applying the assumptions of Johnson‐Kendall‐Robert (JKR) and the Hertzian contact stress theory. The results show that the geometry of the braid and different braid yarn paths have a significant influence on the interfacial shear stress, normal stress in the contact zone, and adhesion energy of composite cylinders. As a novel approach, the proposed model considers the effect of surface roughness in the composite. The results of a parametric study indicated that the composite cylinders with larger braiding angles have lower debonding initiation force. The results also showed that the yarn diameter affected the pullout force in composite cylinders by a large extent.Highlights Normal contact stress and adhesion energy of composite cylinders are affected by the braid yarn path. The 30° composite presented the largest adhesion energy and normal contact stress. Normal tensile stress and interfacial shear stress are influenced by the reinforcement path in the composite. Normal tensile stress and interfacial shear stress in composite cylinders decreased with increased braid angle.

Fragility Analysis of Power Transmission Tower Subjected to Wind–Sand Loads

With the increasing construction of power transmission towers in desert regions for the transportation of wind or solar energy, structural safety under wind and sand loads has become critical. Current design codes primarily account for wind loads on these towers, overlooking the effects of sand impact. This study presents a new model to simulate sand–steel interactions and evaluates the fragility of transmission towers under both wind-only and combined wind–sand loads. The impact model is grounded in Hertz contact theory, with equations of motion derived for the interaction between wind-driven sand particles and structural members, solved via the central difference method. A parametric study investigates the effects of wind speed and sand particle mass: (1) impact forces and maximum deformations increase with wind speed, with impact duration initially decreasing up to 20 m/s and then gradually increasing; (2) an increase in sand particle mass leads to greater impact deformation, force, and duration. Fragility analysis, using incremental dynamic analysis, reveals that sand particles significantly amplify the tower’s response at high wind speeds and increase failure probability across all wind attack angles. These findings highlight the importance of incorporating sand-impact effects in the design and assessment of power transmission towers in desert environments to ensure structural safety and reliable operation of critical energy infrastructure.

Utilizing Hertzian contact model for a vibro-impact unimorph bistable energy harvester to increase working frequency bandwidth and harvested power

The typical bistable energy harvester (TBEH) systems usually have a limited working frequency bandwidth and low performance in energy conversion. This paper proposed a unilateral vibro-impact piezoelectric cantilever beam energy harvester. The novel vibro-impact bistable energy harvester (VIBEH) improves the drawbacks of a TBEH system. Despite previous research, in this paper, the collision process is modeled by nonlinear Hertzian contact theory. Nonlinear contact law adds strong nonlinearity to the harvester. By using the extended Hamilton principle and Euler-Lagrange equation, the governing equations of the energy harvester are derived and solved numerically. The existing nonlinearities cause frequency leaks and affect the type of motion (inter-well or intra-well motion) by changing the equilibrium points of the system. By applying nonlinear tools like Poincare diagrams, bifurcation plots, and phase portraits, the effects of periodic or chaotic oscillations will be studied on the performance of the VIBEH system. Finally, the effects of barrier stiffness, initial gap, barrier location, and electrical load are determined on the performance of the VIBEH system. The obtained results showed that using the VIBEH system could broaden the working frequency bandwidth by about 71.9%. Furthermore, the optimized values of impacting properties and electrical load are examined.

Nonlinear Thermomechanical Low-Velocity Impact Behaviors of Geometrically Imperfect GRC Beams.

This paper studies the thermomechanical low-velocity impact behaviors of geometrically imperfect nanoplatelet-reinforced composite (GRC) beams considering the von Kármán nonlinear geometric relationship. The graphene nanoplatelets (GPLs) are assumed to have a functionally graded (FG) distribution in the matrix beam along its thickness, following the X-pattern. The Halpin-Tsai model and the rule of mixture are employed to predict the effective Young modulus and other material properties. Dividing the impact process into two stages, the corresponding impact forces are calculated using the modified nonlinear Hertz contact law. The nonlinear governing equations are obtained by introducing the von Kármán nonlinear displacement-strain relationship into the first-order shear deformation theory and dispersed via the differential quadrature (DQ) method. Combining the governing equation of the impactor's motion, they are further parametrically solved by the Newmark-β method associated with the Newton-Raphson iterative process. The influence of different types of geometrical imperfections on the nonlinear thermomechanical low-velocity impact behaviors of GRC beams with varying weight fractions of GPLs, subjected to different initial impact velocities, are studied in detail.

Research of tangential cyclic loading on mechanical properties between cosine contact surfaces

Abstract The investigation of the mechanical properties of contact surfaces under tangential cyclic loading is very important for understanding the stiffness and damping of combined structures. In this paper, using Hertzian contact theory and Coulomb's friction law, the contact mechanical mechanism of a common cosine cylindrical contact model subjected to tangential loading, unloading, and reloading is investigated, and the load-displacement curves (hysteresis curves) and the energy equations of a complete displacement cycle are deduced. The results show that under the action of tangential cyclic load, the shear stress increases immediately, and the micro slip zone outside the contact area gradually expands inward until it becomes a slip zone. At this point, the hysteresis curve changes from a "shuttle shaped" to a "quadrilateral" shape. In addition, nonlinear finite elements are used for static simulation to verify the accuracy of the hysteresis curve, tangential stiffness, and energy dissipation analysis of the cosine contact model under different axial crossing angles. The energy equation's fluctuation with tangential displacement under different conditions, as well as the impact of friction coefficient, axial intersection angle of the contact surface, external load, and cosine surface characteristics on the mechanical properties of the contact surface. With the increase of surface roughness or normal load, energy increases nonlinearly with the increase of displacement. Finally, when the Von Mises stress reaches the yield limit, reciprocating displacement loading will lead to plastic accumulation, resulting in wear and tear.

Application of the non-circular sprocket in the timing chain drive system

As a fundamental component of hybrid vehicles, the timing chain drive system significantly improves fuel economy while minimizing transmission error. A non-circular sprocket is used in a timing chain drive system to improve the dynamics characteristics of the system. In the scheme to reduce the crankshaft speed fluctuation, the relationship between the transmission ratio and the camshaft rotation angle is determined by analyzing simulation and experimental results. The mathematical relationship between the transmission ratio and the pitch curve of the non-circular sprocket is deduced. The pitch curve and phase angle of the non-circular sprocket are calculated. Combined with the self-constructed dynamics model of the hydraulic tensioner, a complete three-dimensional timing chain drive system simulation model is established based on the recursive algorithm and Hertz contact theory. The non-circular sprocket is installed into the system at the initial mounting angle. Dynamics simulations are carried out taking into account actual operating conditions. The simulation results prove that the use of a non-circular crankshaft sprocket can effectively reduce speed fluctuation of the camshaft sprocket, transmission error, and maximum chain tension when the crankshaft speed fluctuates regularly. This research proposes and validates a method to improve the dynamics characteristics of the system.

Modeling and Dynamic Analysis of Double-Row Angular Contact Ball Bearing–Rotor–Disk System

This article presents a general numerical method to establish a mathematical model of a bearing–rotor–disk system. This mathematical model consists of two double-row angular contact ball bearings (DRACBBs), a rotor and a rigid disk. The mathematical model of the DRACBB is built on the basis of elastic Hertz contact by adopting the Newton Raphson iteration method, and three different structure forms are taken into account. The rotor is modeled by employing a finite element method in conjunction with Timoshenko beam theory, and the rigid disk is modeled by applying the lumped parameter method. The mathematical model of the bearing–rotor–disk system is constructed by the coupling of the bearing, rotor and disk, and the dynamic response of the bearing–rotor–disk system can be solved by employing the Newmark-β method. The validation of the above mathematical model is demonstrated by comparing the proposed results with the results from the existing literature and finite element software. The dynamic characteristics of the DRACBBs and the dynamic response of the bearing–rotor–disk system are investigated by parametric study. A dynamic characteristic analysis of the DRACBB is conducted to ensure the optimal structure form of the DRACBB under complex external loads, and it can provide a reference for the selection of the structural forms of DRACBBs.

Electrical contact with dielectric breakdown of interfacial gap

Abstract Electrical contact is fundamental to almost every aspect of modern industry including the fast-growing electric vehicle industry. In metallic contacts in atmospheric conditions, most of the electrical current passes via the micro-junctions formed between two electrodes. The classic electrical contact theory predicts an infinite current density at the circular contact periphery. In the present work, we explore the influence of the dielectric breakdown of air outside the contact area on the electrical contact interface. Incorporating the discharging boundary condition governed by the modified Paschen law, we develop the numerical model as well as two sets of closed-form solutions for low applied voltage cases where two electrodes are in solid-solid contact and complete separation, respectively. For Hertzian contact, the present work theoretically proves that the ignorance of discharge can lead to a singular current density at the contact periphery and an overestimation of the electrical contact resistance. The current density monotonically increases along the radial direction to a finite value at the contact area periphery and followed by a monotonic drop within the discharge zone. The present study serves as a foundation for the modeling of discharging rough surface electrical contact and sheds light on the machine element surface damages caused by the electrical discharge machining.

On elastic deformations of cylindrical bodies under the influence of the gravitational field

Background Elastic deformations of gravitating cylindrical bodies are relevant for state-of-the-art photonic experiments, as they affect the physical properties of materials under consideration, impacting wave propagation. This is of key importance for a recently planned experiment to explore the influence of the gravitational field on entangled photons propagating in waveguides. The purpose of this work is to determine these elastic deformations as functions of temperature, pressure, and of the gravitational field. We thus determine the deformations of the body due to changes of the gravitational field, and obtain stringent bounds on the control of temperature and pressure so that the effects of the associated elastic deformations on the photons propagating in a waveguide are smaller than the phase shifts associated with the change of the gravitational field. Methods We use the methods of linear elasticity, including thermoelasticity, to determine the stresses and strains of the medium. For this, the symmetry of the cylinder allows us to solve the problem by using Mitchell’s solutions of the equations satisfied by the Airy functions. The boundary conditions are implemented by an approximation of the Hertz contact method. Results We calculate the displacements, the stresses and strains for several classes of boundary conditions, and give explicit solutions for a number of physically motivated configurations. The influence of the resulting deformations on the planned GRAVITES experiment is determined. Conclusions The results are relevant for fiber interferometry experiments sensitive to the effects of the gravitational field on photon propagation. Our calculations give stringent bounds on the environmental variables, which need to be controlled in such experiments.

Study on thermal elastohydrodynamic lubrication of face gear pairs with low sliding ratio

The relative sliding interaction between tooth surfaces has a significant effect on the lubrication efficiency. In this study, the non-Newtonian characteristic is taken into consideration when examining the point contact thermal elastohydrodynamic lubrication (TEHL) behavior of a low sliding ratio (LSR) face gear pair. By using the instantaneous rotation axes method, tooth contact analysis (TCA) was carried out. Meanwhile, a combination of the Hertz contact model and finite element analysis (FEA) was adopted in order to conduct load tooth contact analysis (LTCA). Subsequently, a TEHL model for the face gear pair was established, incorporating the non-Newtonian behavior of the lubricant. Multi-grid (M-G) and Gauss-Seidel (G-S) methods were employed to improve the computational efficiency as well as convergence accuracy for relatively dense grids. The lubrication behaviors of LSR face gears and traditional face gears under full film lubrication conditions were compared. Additionally, the lubrication characteristics of face gears with the machined three-dimensional morphology were also clarified. The model and computations were validated through comparison with literature data. The findings reveal that improving contact performance can be achieved by designing gear pairs with low sliding ratios.

Quasi-conformal versus Hertz superposition: a comparison for two-dimensional contact

Hertz contact theory results in a relation between the normal contact force and the rigid indentation of the contact surfaces, the force is proportional to the indentation to the power of 3/2, and a semi-ellipsoidal traction distribution. It is a valid theory for non-conformal contact between smooth surfaces (curves in 2D) of elastic bodies. This force model can be derived by considering the Taylor polynomials of degree 2 at the contact point as approximations for the surfaces/curves. In the 2D case, it depends only on the curvature radii of the curves and is undefined when they are equal, the equality of radii violates the non-conformality.The Hertzian force is usually applied to the contact of solid particles as a model compensating for the small local deformations that occur in the contact region. We consider the 2D contact of a rigid convex particle with a rigid concavity enforcing the contact constraint with a barrier method, an approach that prevents overlapping of the bodies, resulting in a force that acts on the contact point. The geometries and the motion are specially designed to show the coalescence of two contact points into one. It illustrates two possible issues with convex-concave contact, the non-uniqueness of contact points and the conformality. We compare the results with a deformable finite element model for equivalent bodies and motion showing that there is a transition between a Hertzian and a non-Hertzian behavior. We conclude that a criterion for the conformality of curves must include the ever-present gap when a barrier method is used to enforce the constraint and the Hertzian force is not suitable for the coalescence of contact points.

Meshing model and temperature field analysis of small module gear with curve configuration

This paper provides a new approach for the design of small module gear with curve configuration, and studies the meshing temperature rise of the generated tooth surface under dry running condition. To predict the meshing temperature of small module gear with curve configuration, the normal load and motion velocity of two contact points are analyzed based on vector mapping method, and the contact stress and contact deformation are obtained based on Hertzian contact theory. The frictional heat flux at different speed and torque is calculated as the heat source, and the convective heat transfer coefficient of tooth surfaces are calculated as the heat dissipation condition. Finite element model of temperature field of small module gear with curve configuration is established, and the meshing temperature field is obtained. Variation trend of the maximum steady-state temperature is in good agreement with the experimental test results. Simulation results of flash temperature also agree well with Blok theory. The results show the rotational speed and the torque are high, the meshing temperature rise of the gear pair will be high. The maximum temperature is distributed in the middle area of single tooth on the first contact trace of the gear.

Analysis of the load distribution law and influencing factors of large-load three-stage synchronous ball screw

Abstract Based on the Hertz contact theory, the contact relationship between adjacent balls and lead screw, and the nut raceways is established for a three-stage synchronous ball screw. A model of axial stiffness for the three-stage synchronous ball screw under large load is established. Considering both the axial stiffness of the screw and the deformation of the screw and nut, the distribution of working contact angles and loads of all balls in the three-stage ball screw is established. Taking a three-hinge-point erection mathematical model as an example, a transmission model of the three-stage synchronous ball screw is established, plan the erection trajectory, and analyze and calculate the deformation and force of the balls in the three-stage synchronous ball screw under different load conditions and the force of a single ball in the erection process. The calculation results show that, during the erection process, the load distribution, the radial clearance, the normal deformation between balls and screw and between balls and nut of different balls in each stage of the ball screw are different.

Energy Loss in Frictional Hertzian Contact Subjected to Two-Dimensional Cyclic Loadings

We investigate the effect of three different harmonically varying loads as a function of the friction coefficient on energy loss in a three-dimensional discrete uncoupled frictional contact problem. Three loading cases include (1) a normal force is constant and a tangential force varies, (2) normal and tangential forces both vary, but the loading and unloading curves are identical, and (3) normal and tangential forces both vary, but the loading and unloading curves are different. For a higher coefficient of friction, three loading cases show different characteristics. If a normal force is constant and a tangential force varies, there is always some slip, but dissipation tends asymptotically to zero at large coefficient of friction. If normal and tangential forces both vary, but the loading and unloading curves are identical, there is no slip and no dissipation above a critical coefficient of friction. If the loading and unloading curves are different, dissipation occurs for all values of the coefficient of friction, and we expect that the dissipation is asymptotic to the relaxation damping value as the coefficient of friction approaches infinity. For lowering coefficient of friction, the three loading cases show similar behavior. Dissipation increases and reaches a maximum just before a state where gross slip is possible.