The frequency response of dynamic friction: Enhanced rate-and-state models

The prediction and control of friction-induced vibration requires a sufficiently accurate constitutive law for dynamic friction at the sliding interface: for linearised stability analysis, this requirement takes the form of a frictional frequency response function. Systematic measurements of this frictional frequency response function are presented for small samples of nylon and polycarbonate sliding against a glass disc. Previous efforts to explain such measurements from a theoretical model have failed, but an enhanced rate-and-state model is presented which is shown to match the measurements remarkably well. The tested parameter space covers a range of normal forces (10–50 N), of sliding speeds (1–10 mm/s) and frequencies (100–2000 Hz). The key new ingredient in the model is the inclusion of contact stiffness to take into account elastic deformations near the interface. A systematic methodology is presented to discriminate among possible variants of the model, and then to identify the model parameter values.     

The frequency response of dynamic friction: A new view of sliding interfaces

Any attempt to predict friction-induced vibration, such as brake squeal in vehicle systems, requires as input some information about the dynamic friction force at the sliding interface. If the concern is only with the threshold of instability based on linearised theory, the quantity required is a type of frictional frequency response function over the range of possible unstable frequencies. This quantity has not previously been measured, and this paper reports the development and testing of a novel tribometer to measure it. Based on the concept of a pin-on-disc machine, controlled fluctuations in sliding speed are applied using an actuator and the resulting fluctuations in friction force are monitored in amplitude and phase. Sample results are shown, revealing parameter dependence, which is in some cases strikingly different from that of the mean friction force. The device has potential for providing necessary data on friction materials and also for discriminating between rival proposed theoretical models for dynamic friction.     

Instability of frictional contact states in infinite layers

The research reported in this paper is focused on the instability of equilibrium and steady sliding states of elastic orthotropic layers in the presence of unilateral obstacles with Coulomb friction with emphasis on a divergence type instability called directional instability of frictional contact states that cannot occur in isotropic layers. Analytic expressions and numerical solutions are provided for the instability mode and for the coefficient of friction at the onset of instability. A parametric study is done to investigate how this coefficient of friction and the instability mode vary with changes of the system parameters. For certain combinations of material data, significantly low coefficients of friction were required for the onset of instability. A finite element model that approximates the continuum and a lumped model that captures some of the features of the continuum are presented.     

Steady and transient sliding under rate-and-state friction

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μ_s and μ_k, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature.We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μ_s should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μ_s over a persistence length before dropping to the value μ_k. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μ_s in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μ_s is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.     

Stress field at a sliding frictional contact: Experiments and calculations

A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons’ friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient μ_d, but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressure-independence of μ_d is briefly discussed.     

Rolling friction as a viscoelastic dissipative process

The viscous additions to the stress tensor on the half–space surface over which a ball moves are calculated with the use of a viscosity (dissipative) tensor. The rolling friction force which corresponds to the Coulomb law and which is proportional to the velocity and which is simultaneously the lower estimate for the sliding friction force is found. Expressions for the radial and vertical displacements on the surface of an elastic half–space are given.     

基于LuGre摩擦模型的接触约束法旋转柔性梁斜碰撞研究

基于接触约束法和LuGre摩擦模型对在重力场作用下作大范围旋转运动的柔性梁系统和斜坡发生含摩擦斜碰撞的动力学问题进行研究.首先运用刚柔耦合的多体系统动力学理论对大范围运动的柔性梁进行离散化和动力学建模,在碰撞时采用冲量动量法求出跳跃速度,其次在法向上引入接触约束求解出碰撞力,在切向上采用LuGre摩擦模型分两种方式求解...     

Transition from stick to slip in Hertzian contact with “Griffith” friction: The Cattaneo–Mindlin problem revisited

Classically, the transition from stick to slip is modelled with Amonton–Coulomb law, leading to the Cattaneo–Mindlin problem, which is amenable to quite general solutions using the idea of superposing normal contact pressure distributions – in particular superposing the full sliding component of shear with a corrective distribution in the stick region. However, faults model in geophysics and recent high-speed measurements of the real contact area and the strain fields in dry (nominally flat) rough interfaces at macroscopic but laboratory scale, all suggest that the transition from ‘static’ to ‘dynamic’ friction can be described, rather than by Coulomb law, by classical fracture mechanics singular solutions of shear cracks. Here, we introduce an ‘adhesive’ model for friction in a Hertzian spherical contact, maintaining the Hertzian solution for the normal pressures, but where the inception of slip is given by a Griffith condition. In the slip region, the standard Coulomb law continues to hold. This leads to a very simple solution for the Cattaneo–Mindlin problem, in which the “corrective” solution in the stick area is in fact similar to the mode II equivalent of a JKR singular solution for adhesive contact. The model departs from the standard Cattaneo–Mindlin solution, showing an increased size of the stick zone relative to the contact area, and a sudden transition to slip when the stick region reaches a critical size (the equivalent of the pull-off contact size of the JKR solution). The apparent static friction coefficient before sliding can be much higher than the sliding friction coefficient and, for a given friction fracture “energy”, the process results in size and normal load dependence of the apparent static friction coefficient. Some qualitative agreement with Fineberg's group experiments for friction exists, namely the stick–slip boundary quasi-static prediction may correspond to the arrest of their slip “precursors”, and the rapid collapse to global sliding when the precursors arrest front has reached about half the interface may correspond to the reach of the “critical” size for the stick zone.     

DYNAMICS OF AN IN-LINE TUBE ARRAY SUBJECTED TO STEAM–WATER CROSS-FLOW. PART II: UNSTEADY FLUID FORCES

In this paper, experimental results of unsteady fluid-force measurements are reported. Important deviations of the measured fluid forces from their single-phase flow counterparts were uncovered. Most importantly, the resulting force coefficients are not simple functions of the reduced flow velocity U/fD, as is the case for single-phase flow. Test results at 0·5 MPa challenge the basic assumption of the existence of a time-invariant linear transfer function between tube displacement and the resulting fluid forces. Time–frequency analysis using Wigner–Ville transforms shows that the phase difference between tube displacement and the fluid force (an indicator of stabilizing or destabilizing fluid effects) undergoes significant variation under what may be considered steady flow conditions. This variation may explain the previously reported phenomenon of intermittent fluidelastic instability in two-phase flows.     

Brachistochrone with limited reaction of constraint in an arbitrary force field

The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate?x of the particle, i.e., u=d ² y/dx ². Applying Pontryagin??s maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.     

Multiple-zone sliding contact with friction on an anisotropic thermoelastic half-space

The paper studies a class of multiple-zone sliding contact problems. This class is general enough to include frictional and thermal effects, and anisotropic response of the indented material. In particular, a rigid die (indenter) slides with Coulomb friction and at constant speed over the surface of a deformable and conducting body in the form of a 2D half-space. The body is assumed to behave as a thermoelastic transversely isotropic material. Thermoelasticity of the Green–Lindsay type is assumed to govern. The solution method is based on integral transforms and singular integral equations. First, an exact transform solution for the auxiliary problem of multiple-zone (integer n > 1) surface tractions is obtained. Then, an asymptotic form for this auxiliary problem is extracted. This form can be inverted analytically, and the result applied to sliding contacts with multiple zones. For illustration, detailed calculations are provided for the case of two (n = 2) contact zones. The solution yields the contact zone width and location in terms of sliding speed, friction, die profile, and also the force exerted. Calculations for the hexagonal material zinc illustrate effects of speed, friction and line of action of the die force on relative contact zone size, location of maximal values for the temperature and the compressive stress, and the maximum temperature for a given maximum stress. Finally, from our general results, a single contact zone solution follows as a simple limit.     

On the sliding instabilities at rough surfaces

In this paper, the onset of sliding between two elastic half-spaces in contact, subjected to a tangential force, is studied within the framework of critical phenomena. First, it is shown that the contact domain between two rough surfaces is a lacunar set and that the distribution of contact stresses is multifractal. By applying an increasing tangential force, under constant normal load, the so-called regime of partial-slip comes into play. However, the continuous and smooth transition to full sliding, predicted by the classical Cattaneo-Mindlin theory, is not confirmed by the experiments, which show marked frictional instabilities. A numerical multi-scale procedure is proposed, taking into account the redistribution of stress, consequent to partial-slip, among the contact areas at all scales. It is shown that the lacunarity of the contact domain delays the onset of instability, when compared to compact Euclidean domains. Independently of the assumptions made for the frictional behaviour at the scale of the asperities (Coulomb friction for meso-scale asperities, adhesion for micro-scales), renormalization permits the critical value of the tangential force which provides the instability to be found. Moreover, the multifractal analysis of the domains where the shear resistance is activated captures the size-scale effects on the friction coefficient, currently evidenced by the experiments.     

Empirical dry-friction modeling in a forced oscillator using chaos

The coefficient of friction is measured during relative oscillation between sliding surfaces. Measurements are made during regular oscillations in which the excitation has a modulated amplitude, and during chaotic oscillations in which the excitation amplitude is fixed. The friction force is measured for paper on paper, and titanium on titanium. A friction law is derived based on observations from the measurements. This friction law is applied to a simulation model of an experimental forced oscillator. The simulated and experimental oscillators have similar qualitative dynamical features in the phase space.     

Sliding modes of two interacting frictional interfaces

In the context of rate-and-state friction, we report an extensive analysis of stability of the quasi-static frictional sliding of two parallel interfaces dividing a linear elastic solid sheared at a constant rate. One possibility for the frictional sliding is that the interfaces slip at equal rates, a steady state described as symmetric. However a steady-state friction law that is non-monotonic allows the competing possibility of an asymmetric steady state in which the interfaces slide at different rates. A rate-and-state law that delivers such behaviour and agrees with the experimental results of Heslot et al. [1994. Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Phys. Rev. E 49, 4973-4988] is proposed. Analytical results combined with numerical investigations performed with the continuation package Auto and direct time integration are used to compile the complete picture of the many bifurcations that exist between the diverse steady and oscillatory sliding modes. In addition to the control parameters corresponding to the driving velocity and the stiffness of the medium, we find that the geometrical details of the steady-state friction law determine the occurrence and nature of bifurcations. Pitchfork bifurcations from the symmetric to asymmetric steady states coincide with the extrema of the friction law; Hopf bifurcations occur in the velocity weakening regime of the friction law. Torus and period-doubling bifurcations of periodic orbits also occur, and lead to complicated dynamics. We also present results of numerical computations that illustrate the complex and versatile dynamics of the two-interface system. We anticipate that the dynamics found in our model should be verifiable by experiments.     

Self-sustained slip pulses of finite size between dissimilar materials

The problem of a steady-state slip pulse of finite size between dissimilar materials is studied. It is shown that for a Coulomb friction law, there is a continuous set of possible solutions for any slip propagation velocity and any slip length. These solutions are, however, nonphysical because they show a singular behaviour of the slip velocity at one extremity of the pulse, which implies a crack-like behaviour. In order to regularize these solutions, we introduce a modified friction law due to Prakash and Clifton (Experimental Techniques in the Dynamics of Deformable Solids, Vol. AMD-165, pp. 33-48; J. Tribol. 120 (1998) 97), which consists in introducing in the Coulomb friction law a relaxation time for the response of the shear stress to a sudden variation of the normal stress. Then, we show that even for a slip velocity-dependent characteristic time, the degeneracy of the solutions is not suppressed and a physical pulse is not selected. This result shows the absence of steady state self-healing pulses within the modified friction law and is consistent with recent finite-difference calculations (J. Geophys. Res. 107 (2002) 10).     

基于高斯原理的非理想系统动力学建模

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.      

Sliding friction in the presence of ultrasonic oscillations: superposition of longitudinal oscillations

Summary  It was often observed that friction forces can be reduced significantly if ultrasonic oscillations are superposed to the macroscopic sliding velocity. This phenomenon can be used to improve machining processes by addition of ultrasonic vibration to tools or workpieces, and forms the basis for many processes of ultrasonic machining. On the other hand, ultrasonic vibrations can be used to generate motion. The thrusting force of ultrasonic motors is provided to the rotor through friction. In the present paper, a simple theoretical model for friction in the presence of ultrasonic oscillations is derived theoretically and validated experimentally. The model is capable of predicting the reduction of the macroscopic friction force as a function of the ultrasonic vibration frequency and amplitude and the macroscopic sliding velocity. Received 22 November 2000; accepted for publication 6 February 2001     

Some equilibrium turbulent boundary layers

Using the Coles additive law of the wall and law of the wake for the mean velocity profile of a two-dimensional turbulent boundary layer, a differential equation for the friction and wake parameters is derived from the momentum integral equation with a view to finding out the conditions under which the boundary layer can exhibit equilibrium. It is predicted that equilibrium is possible for boundary layers in favorable pressure gradient over smooth as well as k-type rough walls. When the roughness height is allowed to increase linearly with the streamwise distance, equilibrium exists also in zero pressure gradient. For a d-type rough wall, equilibrium is possible for a certain range of pressure gradients, from favorable to adverse. Most of the predictions are verified by evaluating the friction and wake parameters from the available experimental data on mean velocity measurements.     

Wave-induced dynamics of a particle on a thin circular plate

In this paper, the dynamics of a particle placed on a thin circular plate carrying circumferential harmonic travelling wave is studied. Coulomb friction is used to model the particle–surface interaction. Distinct regions on the plate surface are identified where either of the three phases of particle motion, namely jumping, sliding and sticking, occurs. Also, the effect of wave frequency and the plate geometry on these regions is studied. Interestingly, there exists an optimum plate thickness for which the region of sliding is maximum. At certain wave frequencies, from the numerical simulations within sticking and sliding regions, it is observed that the average particle motion spirals inwards towards the plate centre. Such an average motion is observed whenever the particle is placed initially with a zero velocity relative to the plate surface. The Gedanken experiments discussed herein provide cogent explanations to all the observed average (slow) dynamics and are also found to be useful in predicting the slow dynamics of the particle a priori, that is, before the actual numerical simulations. The particle’s velocity couples the radial and tangential sliding friction components and is found to be the key physical feature that explains the observed behaviour. Also, it is observed that the plate surface excited by circumferential travelling waves can provide acoustic lubrication to a particle by reducing the limiting force required to move it relative to the surface. The methods discussed in this paper can be extended to study the dynamics of a group of particles (granular materials) and extended rigid bodies, interacting with such surface waves.