Friction force model and rolling dynamics for a ball on a rough surface with presliding displacement taken into account

A model of sliding and spinning friction forces for a ball in the form of finite relations obtained by integrating the tangential stresses over the contact area whose parameters are determined by Hertz’s theory for the “ball-rough horizontal surface” tribological conjunction pair is supplemented with a model of rolling friction torques. The combined model is peculiar in that the presliding displacement effect in rolling and spinning friction torques is taken into account. It is shown that the ball motions in the presliding displacement zone are of quasilinear character and, under shock perturbations, have the form of damping vibrations in the three orientation angles. The numerical parameters of the rolling and spinning friction model are experimentally determined for the presliding displacement zones, while the sliding friction parameters and partly the spinning friction parameters are calculated. Mathematical modeling permits one to discover new properties of the ball, namely, its deceleration in rolling, the onset of damping vibrations at the beginning and end of motion, and the transient process parameters.     

Contact stresses: a short survey of models and methods of computations

Contact stresses are identified as normal and tangential forces between contacting solids. The normal stresses are modeled using unilateral and complementary conditions, elastic response and normal compliance. Friction laws describe the tangential traction. Friction of materials depends on pressure, sliding velocity, surface temperature, time of contact, surface roughness and presence of wear debris. Phenomenological, micro-mechanical and atomic-scale models as well as non-classical models of anisotropic and heterogeneous friction are important steps in the development of friction modeling. Sophisticated friction models are desirable in vibrating systems, materials processing, rolling contacts, rubber and polymers, geomechanics, bioengineering and living systems. Main numerical methods in contact mechanics are: finite element method, boundary element method and discrete element method. To include specific contact constraints, the following computing techniques are applied: Lagrange multipliers, penalty function, perturbated and augmented Lagrangian methods, mathematical programming methods. The advances of adhesion and impact modeling are outlined in this paper.     

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.     

三方程线性弹性-阻尼DEM模型及碰撞参数确定

建立了一种考虑法向接触力、切向接触力(含静滑动摩擦力及动滑动摩擦力)和力矩(含由切向力产生的力矩及静滚动摩擦力矩和动滚动摩擦力矩)的三方程线性弹性-阻尼离散单元模型,并将该模型应用到颗粒物料的三维数值模拟中,讨论了模型中几个重要碰撞参数--刚性系数、阻尼系数及摩擦系数的选择及其对计算结果的影响,同时也探讨了时间步长等计算参数对模拟结果的影响.为了验证算法和参数选择的正确性,本文对几个有代表性的颗粒系统进行了数值试验研究,并对计算结果进行了细致的分析,验证了新模型和参数选择的正确性.     

Mechanics of advancing pin-loaded contacts with friction

This paper considers finite friction contact problems involving an elastic pin and an infinite elastic plate with a circular hole. Using a suitable class of Green's functions, the singular integral equations governing a very general class of conforming contact problems are formulated. In particular, remote plate stresses, pin loads, moments and distributed loading of the pin by conservative body forces are considered. Numerical solutions are presented for different partial slip load cases. In monotonic loading, the dependence of the tractions on the coefficient of friction is strongest when the contact is highly conforming. For less conforming contacts, the tractions are insensitive to an increase in the value of the friction coefficient above a certain threshold. The contact size and peak pressure in monotonic loading are only weakly dependent on the pin load distribution, with center loads leading to slightly higher peak pressure and lower peak shear than distributed loads. In contrast to half-plane cylinder fretting contacts, fretting behavior is quite different depending on whether or not the pin is allowed to rotate freely. If pin rotation is disallowed, the fretting tractions resemble half-plane fretting tractions in the weakly conforming regime but the contact resists sliding in the strongly conforming regime. If pin rotation is allowed, the shear traction behavior resembles planar rolling contacts in that one slip zone is dominant and the peak shear occurs at its edge. In this case, the effects of material dissimilarity in the strongly conforming regime are only secondary and the contact never goes into sliding. Fretting tractions in the forward and reversed load states show shape asymmetry, which persists with continued load cycling. Finally, the governing integro-differential equation for full sliding is derived; in the limiting case of no friction, the same equation governs contacts with center loading and uniform body force loading, resulting in identical pressures when their resultants are equal.     

摩擦与滚阻对被动行走器步态影响的研究

本文研究了圆弧足被动行走器支撑足与地面间的摩擦系数和滚阻系数对被动行走器步态的影响. 首先分别利用扩展的 赫兹接触力模型和LuGre摩擦模型描述了支撑足与地面接触点处的法向支撑力和切向摩擦力,并考虑了行走过程中支撑足 所受的滚动摩阻;其次利用第二类Lagrange方程推导出了该系统的动力学方程,并通过与已有成果的对比确定 了合适的LuGre摩擦模型参数;最后仿真分析了摩擦系数和滚阻系数对被动行走器步态的影响. 研究发现:摩擦系数的改变 虽然对被动行走器行走的平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较小,但摩擦系数的减小 会改变其行走步态类型,如发生倍周期分岔甚至混沌现象;然而,滚阻系数的改变会对行走器行走的 平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较大,尚未发现滚阻系数的改变会引起其行走步态的变化.      

Non-steady-state elastic–plastic behavior of metal forming (rolling) with a generalized friction law

An analytical and generalized friction law is formulated and a three-dimensional large-strain non-steady-state elastic–plastic finite element analysis has been performed for rolling process. The contact/friction problem at the interface between the workpiece and the rolls is treated rigorously by using this new friction law. The numerical results, including the evolutions of roll torque and roll forces, and the interfacial normal and shear stresses, are presented and discussed.     

Fast evaluation of friction forces in traction-drive contacts including thermal effects

Summary  In automotive traction drives, power is transmitted by friction forces. The friction forces result from the shear stresses developed in lubricated and highly loaded contacts between rolling bodies. Due to the kinematics of a traction drive, shear velocities occur in both the rolling direction and perpendicular to it. Due to these shear velocities and by normal pressure, the lubricant is forced to build up shear stresses. The increase of the shear stresses may be modelled by a nonlinear viscous element. The describing differential equations are coupled by the equivalent shear stress, which defines the nonlinear behaviour of the element. A fast method is described to evaluate the coupled differential equations. By using a known analytical approximation for the equivalent shear stress, the differential equations are decoupled and can be solved analytically. In an iterative procedure the equivalent shear stress is updated, and the complete solution is found. The iterative method is extended to account for thermal effects in the contact. Received 17 June 1999; accepted for publication 26 October 1999     

Double frictional receding contact problem for an orthotropic layer loaded by normal and tangential forces

With the increasing research in the field of contact mechanics, different types of contact models have been investigated by many researchers by employing various complex material models. To ascertain the orthotropy effect and modeling parameters on a receding contact model, the double frictional receding contact problem for an orthotropic bilayer loaded by a cylindrical punch is taken into account in this study. Assuming plane strain sliding conditions, the governing equations are found analytically using Fourier integral transformation technique. Then, the resulting singular integral equations are solved numerically using an iterative method. The weight function describing the asymptotic behavior of the stresses are investigated in detail and powers of the stress singularities are provided. To control the trustworthiness and correctness of the analytical formulation and to compare the resulting stress distributions and contact boundaries, a numerically efficient finite element method was employed using augmented Lagrange contact algorithm. The aim of this paper is to investigate the orthotropy effect, modeling parameters and coefficients of friction on the surface and interface stresses, surface and interface contact boundaries, powers of stress singularities, weight function and to provide highly parametric benchmark results for tribological community in designing wear resistant systems.

     

Contact strength of a two-layer covering under friction forces in the contact region

Friction and antifriction composite materials of multilayer structure [1] have recently become very popular in the engineering industry. Antifriction materials are widely used in sliding bearings, and friction materials are widely used in brakes. In the first case, the friction forces between the contacting surfaces are negligible, but in the second case, they are rather large. We use two examples of two plane problems from the theory of elasticity concerning the interaction between a die and a base formed by two elastic layers with different mechanical properties, which are rigidly connected with each other and with an undeformable support, to study how the geometric and mechanical parameters of the problem affect the stress-strain state of such a base, both on its surface and at its internal points, and to find the optimal parameters ensuring the required operation resources of the friction units thus modeled. We assume that the die foot is parabola-shaped or plane, the normal and tangential stresses in the contact region are related to each other by the Coulomb law, and the die is subjected to normal and tangential forces. In this case, the die-two-layer base is in the limit equilibtrium, and the die does not rotate in the process of deformation of the layer. In this setting, the problems were studied in [2] by solving the integral equations (IE) by the asymptotic method of large λ (see [3–7], etc.), which permits finding the effective solution only for relatively large thicknesses of layers compared with the dimensions of the contact region. But in real friction units mentioned above, the layers can have rather small relative thicknesses, and the large λ method cannot be used. We note that the other asymptotic methods (e.g., see [3]) efficient in the case of relatively small thicknesses of layers cannot yet be adapted to the case of friction forces in the contact region. In the present paper, we propose to use the collocation method following the scheme given in [8] to solve the corresponding integral equation of the first kind with logarithmic kernel. This method allows one to obtain sufficiently exact solutions practically for all values of the parameters of the problem with relatively small expenditure of the computer time for modern computers. The contact problem for a two-layer base was used in [9] for a close statement of the problem without friction forces in the contact region.     

Variational principles for constrained systems: Theory and experiment

In this paper we present two methods, the nonholonomic method and the vakonomic method, for deriving equations of motion for a mechanical system with constraints. The resulting equations are compared. Results are also presented from an experiment for a model system: a ball rolling without sliding on a rotating table. Both sets of equations of motion for the model system are compared with the experimental results. The effects of various forms of friction are considered in the nonholonomic equations. With appropriate friction terms, the nonholonomic equations of motion for the model system give reasonable agreement with the experimental observations.     

含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性.      

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

On indentation of a pair of rigid punches connected by an elastic beam into an elastic half-plane with regard to the friction and adhesion forces in the contact region

The contact problem of indentation of a pair of rigid punches with plane bases connected by an elastic beam into the boundary of an elastic half-plane is considered under the conditions of plane strain state. The external load is generated by lumped forces applied to the punches and a uniformly distributed normal load acting on the beam.It is assumed that the contact between the punch and the elastic half-plane can be described by L. A. Galin’s statement, i.e., it is assumed that the adhesion acts in the interior part of each of the contact regions and the tangential stresses obeying the Coulomb law act on their boundaries.With the symmetry taken into account, the problem is stated only for a single punch, and solving this problem is reduced to a system of four singular integral equations for the tangential and normal stresses in the adhesion region and the contact pressure in the sliding zones. The solution of the constitutive system together with three conditions of equilibrium of the system of punches connected by a beam is constructed by direct numerical integration by the method of mechanical quadratures.As a result of the numerical analysis, the contact stress distribution functions were constructed and the values of the sliding zones and the punch rotation angle were determined for various values of the geometric, elastic, and force characteristics.     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

Rolling contact of a rigid sphere/sliding of a spherical indenter upon a viscoelastic half-space containing an ellipsoidal inhomogeneity

In this paper, the frictionless rolling contact problem between a rigid sphere and a viscoelastic half-space containing one elastic inhomogeneity is solved. The problem is equivalent to the frictionless sliding of a spherical tip over a viscoelastic body. The inhomogeneity may be of spherical or ellipsoidal shape, the later being of any orientation relatively to the contact surface. The model presented here is three dimensional and based on semi-analytical methods. In order to take into account the viscoelastic aspect of the problem, contact equations are discretized in the spatial and temporal dimensions. The frictionless rolling of the sphere, assumed rigid here for the sake of simplicity, is taken into account by translating the subsurface viscoelastic fields related to the contact problem. Eshelby's formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on the contact pressure distribution, subsurface stresses, rolling friction and the resulting torque. A Conjugate Gradient Method and the Fast Fourier Transforms are used to reduce the computation cost. The model is validated by a finite element model of a rigid sphere rolling upon a homogeneous vciscoelastic half-space, as well as through comparison with reference solutions from the literature. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity is performed. Transient and steady-state solutions are obtained. Numerical results about the contact pressure distribution, the deformed surface geometry, the apparent friction coefficient as well as subsurface stresses are presented, with or without heterogeneous inclusion.     

Multicomponent models of friction

The problem of motion of a homogeneous ball on a horizontal plane is considered. It is assumed that the contact patch is of spherical shape, whereas the pressure center does not coincide with the center of the contact patch and is displaced in the sliding direction of the ball. The friction force has two components that are parallel and perpendicular to the sliding velocity; the friction force moment has a vertical component and two horizontal components being parallel and perpendicular to the sliding velocity.     

A finite element study of the deformations, forces, stress formations, and energy losses in sliding cylindrical contacts

This work presents the results of a finite element analysis (FEA) used to simulate two-dimensional (2D) sliding between two interfering elasto-plastic cylinders. The material for the cylinders is modeled as elastic-perfectly plastic and follows the von Mises yield criterion. The FEA provides trends in the deformations, reaction forces, stresses, and net energy losses as a function of the interference and sliding distance between the cylinders. Results are presented for both frictionless and frictional sliding and comparisons are drawn. The effects of plasticity and friction on energy loss during sliding are isolated. This work also presents empirical equations thatt relate the net energy loss due to sliding under an elasto-plastic deformation as a function of the sliding distance. Contour plots of the von Mises stresses are presented to show the formation and distribution of stresses with increasing plastic deformation as sliding progresses. This work shows that for the plastic loading cases the ratio of the horizontal force to the vertical reaction force is non-zero at the point where the cylinders are perfectly aligned about the vertical axis. In addition, a “load ratio” of the horizontal tugging force to the vertical reaction force is defined. Although this is analogous to the common definition of the coefficient of friction between sliding surfaces, it just contains the effect of energy loss in plasticity. The values of the contact half-width are obtained for different vertical interferences as sliding progresses.