Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

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)     

Sensitivity study of the contact situation between the brake pad and the disc

The contact situation between the brake pad and the disc during the braking process is of particular importance concerning the squeal behavior of brake systems. After the braking process, the surface topography of brake pads can be measured using a confocal microscope. An algorithm to calculate the contact between two surfaces has been developed at the Institute of Dynamics and Vibrations. The algorithm calculates the deflection of asperities under a normal load regarding an elastic material behavior. A normal load is applied to a measured surface topography of the brake pad; the counter body (brake disc) is represented by a flat surface. The potential contact area, the locally distributed forces, deflections and normal stiffness of the pad are computed. Since there is an uncertainty in the relative position between the pad and the disc and hence the real contact situation during the braking process is not known, different contact situations must be considered during the simulations. Concerning various tilt angles of the pad that can arise during the vibrations of the brake system, a sensitivity study has been carried out. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instability of thermoelastic interaction between a half-space and a rigid base through a thin liquid layer

We investigate the instability of thermoelastic interaction between elastic and rigid half-spaces through a liquid interlayer under the conditions of heat transfer across the interfaces. Due to the small thickness of the liquid layer, its influence on the temperature field is taken into account by the thermal resistance of the contact between the bodies, which depends on the normal displacement of the boundary of the elastic body. The pressure inside the liquid is equal to the external pressure applied to the bodies. We determined the critical value of the external heat flow for which the instability becomes possible in such a system and studied the dependence of this value on the parameters of the elastic half-space, the thickness of the liquid layer, and its thermal conduction. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 76–82, April–June, 1998.     

Vibrational asperity interaction in slowly sliding contacts

When two elastic bodies slide against each other, contacting asperities carry tangential load. In the case of slow sliding, each asperity can show a stick-slip-like movement that can be modelled as an integrate-and-fire (IF) oscillator. Asperities influence each other by an elastic coupling which allows a globally synchronized state instead of local events. For two neighboured asperities, the dynamics of coupled IF oscillators is investigated in view of synchronization modes, energy dissipation and their significance for the tribological contact. Additionally, the influence of a relaxating coefficient of stiction, i.e. a time-dependent firing threshold, is discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional contact problems with friction for a composite elastic wedge

Contact problems for a composite elastic wedge in the form of two joined wedge-shaped layers with different aperture angles joined by a sliding clamp, where the layer under the punch is incompressible, are studied in a three-dimensional formulation. Conditions for a sliding or rigid clamp or the absence of stresses are set up on one face of the composite wedge. The integral equations of the problems are derived taking account of the friction forces perpendicular to the edge of the wedge. The method of non-linear boundary integral equations of the Hammerstein type is used when the contact area is unknown. A regular asymptotic solution is constructed for an elliptic contact area. By virtue of the incompressibility of the material of the layer in contact with the punch, this solution retains the well known root singularity in the boundary of the contact area when account is taken of friction.     

Friction induced flutter instability - on modeling and simulation of brake-squeal -

The issue of this contribution are self–excited vibrations due to sliding friction between moving elastic bodies. Starting from Hamiltons principle, the normal contact is implemented by two different approaches. Assuming discretizations in a spatial coordinate frame, the structure of the resulting equations of motions is outlined. For a simple model of a disc–brake, stability charts of the steady–state are presented, which show a considerable influence of the motion of the disc on the stability behavior. Finally, the influence of penalty approaches to the normal contact is addressed shortly. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Two Dimensional Model for the Dynamical Contact of Elastic Rough Surfaces

To model the contact behavior including dynamical effects, a two dimensional mechanical model of elastic rough contact is developed. This model can simulate the contact behaviour between two rough surfaces depending on normal pressure, sliding speed and roughness profiles. The contact between two rough surfaces is reduced to a rough rigid and a rough elastic layer. The elastic layer is modeled by point masses connected by spring-damper elements. The total system is described by coupled ODEs. The number of ODEs and thus the degree of freedom of the model depends on the varying contact conditions. The contact conditions are monitored during the simulation and the simulation interrupts, in case the contact conditions change. The equations of motion are then adapted with respect to the contact constraints. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem)

The plane problem of the sliding contact of a punch with an elastic foundation when there is friction and wear is considered. Assuming the existence of a steady solution in a moving system of coordinates, relations are derived between the sliding velocity, the wear, the contact stresses and the displacements for an arbitrary dependence of the wear rate on the contact pressure. Taking into account the presence of a deformation component of the friction force, an equation is written for the balance of the mechanical energy for the punch - elastic base system considered. It is shown that the equality of the work of the external force in displacing the punch to the losses due to friction and the change in the shape of the foundation due to wear is satisfied when the work done by the contact stresses on the increments of the boundary displacements is equal to zero, and the frictional losses must be determined taking into account the non-uniformity of the distributions of the shear contact stresses and the sliding velocity in the contact area. Two special cases of the foundation in the form of a wide and narrow strip are considered, for which the total coefficient of friction is calculated, taking into account the deformation component of the friction force.     

Regularization of 2D frictional contacts for rigid body dynamics

Classic rigid body mechanics does not provide frictional forces acting in a 2D contact interface between two bodies during sticking. This is due to the statical undeterminacy related with this problem. Many technical systems, e.g. disk clutches, have such surface-to-surface contacts and it is sometimes desirable to treat them as rigid body systems despite the 2D contact. Alternatively it is possible to model the systems using elastic instead of rigid bodies, but this might lead to certain drawbacks. Here a new regularization model of such 2D contacts between rigid bodies is proposed. It is derived from a material model for elasto-plasticity in continuum mechanics. Only dry friction is taken into account. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

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)     

The Cyclic Thermomechanical Coupled Problem of Thermal Gradients in Friction Railway Disc Brakes

The emergency braking distance of a TGV train at a speed of 320 km/h is almost 3000 m. Dry running brakes are reliable due to their predictable response to external stress and are thus used in such applications. The kinetic energy is dissipated proportionately into the brake disc and brake pad. This induced dissipation of energy and the high frequency of brake application cause high temperatures. These immense temperature changes could cause macroscopic cracks leading to failure of discs and accidents [1]. Generally, hot spotting describes the development of thermal localizations and can lead to early damage, early wear, pad performance loss, and squeal noise [2]. The aim of the present study is to improve a disc-pad transient numerical model by use of a coupled thermomechanical method. It is based on full 3-d thermomechanical calculations taking disc rotation into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spatial contact problem for an elastic layer and wavy punch when there is friction and wear

The spatial (three-dimensional) problem of the wear of a wavy punch sliding over an elastic layer bonded to a rigid base, assuming there is complete contact between the punch and the layer, is considered. It is assumed that there is Coulomb friction and wear of the punch. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the harmonic functions in which are represented in the form of double Fourier integrals, after which the problem reduces to a linear system of differential equations. It is established that the harmonics constituting the shape of the punch and the contact pressure are shifted with respect to one another in time along the sliding line of the punch. The velocity of this shift depends on the longitudinal and transverse frequencies of the harmonic, that is, dispersion of the waves is observed.     

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)     

Spatial contact problems for rough elastic bodies under elastoplastic deformations of the unevenness

On the basis of /1, 2/, a model is constructed for the contact between a rigid stamp and a rough body taking elastoplastic deformations of the unevenness into account. The contact model for rough bodies with elastic deformations of the unevenness is a special case. A classical approach utilizing boundary integral equations is applied in the mathematical formulation of the contact problem. Under quite general assumptions (for instance, the multiconnectedness of the contact domain desired), the uniqueness and existence of the solution are investigated. A method is developed to determine the contact pressure, the closure of the bodies, and also the contact area which consists of two parts in the general case, a zone of elastoplastic deformation of the unevenness and a zone of their elastic deformation. The efficiency of the method is shown in examples of new contact problems. The solution is represented in a convenient form for analysing the influence of the roughness. This is of considerable value for material testing by a contact method. A fairly complete survey of research on contact problems for rough bodies can be found in /1–4/.     

Asymptotic analysis of a thin‐layer device with Tresca's contact law in elasticity

In this paper, we consider a thin elastic layer between a rigid body and an elastic one. A Tresca law is assumed between the two elastic bodies. The Lamé coefficients of the thin layer are assumed to vary with respect to its height ϵ. This dependence is shown to be of primary importance in the asymptotic behaviour of the device, a critical case leading to a non‐classical contact law when deleting the bond. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamics of a two-degrees-of-freedom system with friction and heat generation

A novel thermomechanical model of frictional self-excited stick-slip vibrations is proposed. A mechanical system consisting of two masses which are coupled by an elastic spring and moving vertically between two walls is considered. It is assumed that between masses and walls a Coulomb friction occurs, and stick-slip motion of the system is studied. The applied friction force depends on a relative velocity of the sliding bodies. Stability of stationary solutions is considered. A computation of contact parameters during heating of the bodies is performed. The possibility of existence of frictional auto-vibrations is illustrated and discussed.     

On the formulation and investigation of a spatial contact problem for elastic bodies under mixed friction conditions

A spatial contact problem is formulated and investgated for rough elastic bodies which touch each other under mixed friction conditions: the elastic bodies are separated in one part of the contact domain by a layer of viscous incompressible liquid (lubricant), while in the other they are in direct contact (such conditions are characteristic for roller bearings, gear transmissions, etc.). The problem is reduced to a system of nonlinear integro-differential and integral equations and inequalities in the contact domain, part of the external boundary, and a number of inner boundaries that are unknown in advance, but separate the lubricated and unlubricated zones. Special cases are problems of dry and completely lubricated contact. A formulation is given for the problem for the case when the materials of the bodies are identical. The problem of mixed friction is considered in strongly drawn out contact. Sections of the contact domain in which the interaction between the bodies is direct or by means of the lubrication layer are investigated using asymptotic methods.