Determination of a stochastic friction coefficient depending on a randomly rough surface

The goal of this contribution is to calculate the the friction coefficient for a scanned surface of a worn brake pad. The data shows that the asperities can be approximated by paraboloids which allows to calculate the contact force and area with the Hertz contact model if the deformation is elastic. The friction force is calculated with the Bowden-Tabor approach which suggests that the friction force is the force to shear apart contacting asperities. This is considered to be the dominant friction cause in dry contact. To generate many surfaces with similar peak statistics the spectral decomposition is used. The friction coefficient and it's stochastic properties is calculated for these surfaces. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation analysis of a stochastic non-smooth friction model

Non-smooth friction systems such as systems with dry friction show several bifurcation phenomena. The discontinuity of these so called slip-stick vibrations makes these systems interesting and there has been a lot of research in this field, see for example Hinrichs [1]. Due to the non-smooth friction force even the deterministic system shows a rich bifurcation behavior. Measurements indicate that the friction coefficient which plays a large role in the system behavior is not deterministic but can be described as a friction characteristic with added white noise. Therefore, the stochastic characteristic is introduced into the non-smooth system and the change of the bifurcation behavior is studied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation analysis of a stochastic non-smooth friction model

Non-smooth systems with stochastic parameters are important models e.g. for brake and cam follower systems. They show special bifurcation phenomena, such as grazing bifurcations. This contribution studies the influence of stochastic processes on bifurcations in non-smooth systems. As an example, the classical mass on a belt system is considered, where stick-slip vibrations occur. Measurements indicate that the friction coefficient which plays a large role in the system behavior is not deterministic but can be described as a friction characteristic with added white noise. Therefore, a stochastic process is introduced into the non-smooth model and its influence on the bifurcation behavior is studied. It is shown that the stochastic process may alter the bifurcation behavior of the deterministic system. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gauss色噪声激励下含黏弹力摩擦系统的随机响应分析

研究了Gauss色噪声激励下含黏弹力、弱非线性阻尼的摩擦振子的随机响应.将适用于光滑系统的随机平均法推广到了非光滑摩擦系统,进而得到系统振幅、位移及速度的稳态概率密度函数.同时结合材料的黏弹性,研究了摩擦力和Gauss色噪声对系统响应的影响.研究表明,摩擦力、黏弹力及噪声项的相关参数均可引起随机P-分岔,并且在一定范围内系统响应对摩擦力极为敏感.此外,理论结果与Monte Carlo 模拟结果吻合较好,验证了方法的有效性.     

Hysteresis Effects in Abrasive and Frictional Contacts

Grinding is a very complex and high dynamical material removal process with stochastically distributed grain engagements and strong varying local contact conditions. Over a long time only macroscopic effects are analyzed and predicted by empirical relations. To understand the dynamical behavior also local effects must be considered. Therefore, the local contact conditions and especially the time-dependent friction coefficient are analyzed. One detected effect is the dependency of the friction coefficient on the normal forces and on their time history, so a hysteresis loop occurs for increasing and decreasing values. With the force dependent friction coefficient local and dynamic effects are physically interpretable. In contrast, the global mean friction coefficient is constant over the entire force range which describes only quasi-stationary effects. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fixed-point iteration approach for multibody dynamics with contact and small friction

Acceleration–force setups for multi-rigid-body dynamics are known to be inconsistent for some configurations and sufficiently large friction coefficients (a Painleve paradox). This difficulty is circumvented by time-stepping methods using impulse-velocity approaches, which solve complementarity problems with possibly nonconvex solution sets. We show that very simple configurations involving two bodies may have a nonconvex solution set for any nonzero value of the friction coefficient. We construct two fixed-point iteration algorithms that solve convex subproblems and that are guaranteed, for sufficiently small friction coefficients, to retrieve, at a linear convergence rate, the unique velocity solution of the nonconvex linear complementarity problem whenever the frictionless configuration can be disassembled. In addition, we show that one step of one of the iterative algorithms provides an excellent approximation to the velocity solution of the original, possibly nonconvex, problem if for all contacts we have that either the friction coefficient is small or the slip velocity is small.Subject Index. 70E55, 75M10, 75M15, 90C33A partial version of this work has appeared in the proceedings of the NATO Advanced Studies Institute on Virtual Nonlinear Multibody Systems, Prague, 2002.     

Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.     

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.     

The numerical analysis of the influence of the blankholder force and the friction coefficient on the value of the drawing force

In this paper the numerical analysis of the drawing process with the blankholder is presented. The influence of the value of the blankholder force on the value of the drawing force and drawpiece height is examined. A different friction coefficient of the sheet metal for die and blankholder is considered. A defect at the lack of the blankholder is presented. The analysis in the system ANSYS/LS-DYNA is passed. The material model with the mixed hardening, isotropic and isothermal is used. Technological parameters of the sheet-metal to drawing, the die, the stamp and the blankholder from the literatures are chosen. Examples results of computer simulations are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Friction of polymeric materials. 4. effect of temperature and relative sliding velocity on the coefficient of friction of flastomers

The relationship between the coefficient of friction of elastomers and the temperature and sliding velocity has been calculated for various specific pressures on the assumption that the frictional force is low and that heating up of the surface of the bodies rubbing against each other can be neglected. Qualitative agreement with experimental results was obtained. The changes in the coefficient of friction observed are explained in terms of the temperature dependence of the volumetric properties of the elastomer, Young's modulus, and the shearing strength. A formula is proposed for calculating the rate of deformation of the material in the contact zone, which depends on the radius of curvature of the hump and the sliding velocity.     

Modeling and simulations of the clutch dynamics using approximations of the resulting friction forces

The work presents the effectiveness of a certain class of approximate models of the resulting friction force found during simulations of a simplified model of clutch dynamics. The friction models are based on integral expressions assuming fully developed sliding and Coulomb's classical law of friction on each element of the planar contact. Special approximations of the integral model of the friction force and the moment are proposed, which are based on Padé approximants and their generalizations. The system of clutch dynamics is simplified to a friction disk on a rotating master disk. Two different configurations are investigated, including the coaxial and non-coaxial arrangement of the disks. The model based on the generalization of Padé approximants is compared with the corresponding approximants based on the Taylor series expansion and with the model using the integral expressions for the resultant friction force and torque components.     

The interaction of surface topography and friction dynamics in brake systems described by a Cellular Automaton

The friction coefficient μ , which is the quotient of the friction force R and the normal force N is in principal not a stationary material parameter, but also dependent on for instance the relative velocity, the normal load, the temperature, the climate conditions, the location and the event itself. The dynamics in the boundary layer between a brake disc and a brake pad is closely linked with the surface topography dynamics. Growing and destroying processes of hard, thin patches, carrying the friction power, determine the time-dependence of the friction coefficient. This interaction between friction and wear has already been simulated with a set of differential equations [2-4], which give an idea about the equilibrium of flow in the contact zone and which are able to describe the fading effect, for example. Based on this assumption we discretised the boundary layer with a Cellular Automaton [5], which makes it possible to have a more detailed look at the processes in the contact area. This paper will show new conclusions concerning the interdependencies of the friction behaviour and the surface topography. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic flocking dynamics of the inertial spin model with state-dependent noises

We study stochastic flocking dynamics of the inertial spin (IS) model with state-dependent noises. The IS model was considered to describe the collective behaviors of starling flocks moving with constant speed. Unlike mechanical flocking models extensively studied in the literature, this model incorporates an internal dynamic observable, namely spin (internal angular momentum) in addition to mechanical observables (position and velocity), and it describes how spin interacts with mechanical observables. In previous works, emergent dynamics of the deterministic counterparts for the IS model and its mean-field limit have been investigated under some specific setting in which network topology is multiplicatively separable. In this work, we present sufficient frameworks for stochastic flocking dynamics of the IS model, which state-dependent noises vanish at the equilibria of the deterministic IS model. The proposed frameworks are in terms of coupling strength, friction, and inertial coefficients, and our asymptotic convergence results for sample paths are given in both an almost sure and an expectation sense. We have also conducted several numerical experiments to verify our analytical results and to explore what can be studied further in future work     

A nonlinear fracture differential kinetic model to depict chaotic atom motions at a fatigue crack tip based on the differentiable manifold methodology

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. As demonstrated in the paper, chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.     

Existence and estimate of positive solutions to a nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids

This paper presents a rigorous proof of existence and uniqueness of solutions to laminar boundary layer flow in power law non-Newtonian fluid. A theoretical estimate for skin friction coefficient is given, which is characterized by a power law exponent. The reliability and efficiency of the proposed estimate formula are verified by numerical results with a good agreement. The estimate formula can be successfully applied to give the value of the skin friction coefficient.     

A DAE formulation for geared rotor dynamics including frictional contact between the teeth

A DAE approach is presented for geared rotor dynamics simulations with rigid helical evolvent gears. It includes the normal contact force between the teeth as well as tangential components. Given the evolvent tooth flank geometry of gear 1 and gear 2 [1], the contact line and the velocity difference in the contact are found. The requirement of no penetration of the teeth yields a non-holonomic constraint and the contact normal force. The friction caused force and moment are obtained by applying Coulomb's friction model. This approach is used to investigate the dynamics of two ideal rotors with translational DoFs, which are connected by gears to one another. The driving rotor has a given angular speed, while the driven rotates unrestrainedly and is connected to a rotational damper. Because of the periodic friction terms, the solution is periodic. A direct time integration or a harmonic approach can be used for the numerical computation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic models for first-order kinetics of biochemical oxygen demand with random initial conditions, inputs, and coefficients

Biochemical oxygen demand (BOD) is a parameter of prime importance in surface water pollution studies and in the design and operation of waste-water treatment plants. A general, stochastic analytical model (denoted S1) is developed for the temporal expectation and (heteroscedastic) variance of first-order BOD kinetics. The model is obtained by integrating the moment equation, which is derived from the mathematical theory of stochastic differential equations. This model takes into account random initial conditions, random inputs, and random coefficients, which appear in the model formulation as initial condition (σ_O²), input (σ_l²), and coefficient (σ_c²) variance parameters, respectively. By constraining these three variance parameters to either vanish or to be nonnegative, model S1 is allowed (under appropriate combinations of the constraints) to split into six stochastic “submodels” (denoted S2 to S7), with each of these submodels being a particular case of the general model. Model S1 also degenerates to the deterministic model (denoted D) when each of the variance parameters vanish. The deterministic parameters (i.e., the rate coefficient and the ultimate BOD) and the stochastic variance parameters of the seven models are estimated on sets of replicated BOD data using the maximum likelihood principle. In this study, two (S5 and S7) of these seven stochastic models are found to be appropriate for BOD. The stochastic input (S5) model (i.e., null initial condition and coefficient variance parameters) shows the best prediction capabilities, while the next best is the stochastic initial condition (S7) model (i.e., null input and coefficient variance parameters).     

Generation of surface waves through friction: FEM investigation

Sliding friction between two bodies can generate elastic vibration. This study uses a finite-element model comprising an elastic body sliding against a flat rigid surface with constant coefficient of friction. For the elastic body a structured topography is taken into account. The model shows traveling surface waves, which depend on the asperities of the sliding surface. It can be shown that the surface structure and its inertia are the cause for elastic waves in the contact region. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-linear stochastic programming model based on expectation and variance and its application in transportation problem

The stochastic transportation problem involves in many areas such as production scheduling, facility location, resource allocation, logistics management. Constructing an operable solving method has important theoretical and practical value. In this paper, we first analyze the characteristic and deficiencies of the existing stochastic programming methods, such as higher computation complexity. We then give the concept of reliability coefficient and a quasi-linear processing pattern based on expectation and variance. We further analyze the relationship between reliability coefficient and reliability degree, also give the selecting strategy of reliability coefficient. Based on that, we establish a quasi-linear programming model for stochastic transportation problem, and we analyze its performance by a case-based example. The results indicate that this model has good interpretability and operability. It can effectively solve the transportation problem under complex stochastic environment or with incomplete information.