On the variational formulation of the dynamics of systems with friction

We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered. For systems with ideal constraints the problem under discussion was solved by Lagrange in his “Analytical Dynamics” (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4]. The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.     

Analysis and numerical solution of a piezoelectric frictional contact problem

We consider a mathematical model which describes the frictional contact between an electro-elastic–visco-plastic body and a conductive foundation. The contact is modelled with normal compliance and a version of Coulomb’s law of dry friction, in which the stiffness and the friction coefficients depend on the electric potential. We derive a variational formulation of the problem and we prove an existence and uniqueness result. The proof is based on a recent existence and uniqueness result on history-dependent quasivariational inequalities obtained in [15]. Then we introduce a fully discrete scheme for solving the problem and, under certain solution regularity assumptions, we derive an optimal order error estimate. Finally, we present some numerical results in the study of a two-dimensional test problem which describes the process of contact in a microelectromechanical switch.     

A dynamically consistent model of the contact stresses in the plane motion of a rigid body

The problem of determining dry friction forces in the case of the motion of a rigid body with a plane base over a rough surface is discussed. In view of the dependence of the friction forces on the normal load, the solution of this problem involves constructing a model of the contact stresses. The contact conditions impose three independent constraints on the kinematic characteristics, and the model must therefore include three free parameters, which are determined from these conditions at each instant. When the body is supported at three points, these parameters (for which the normal stresses can be taken) completely determine the model, while indeterminacy arises in the case of a larger number of contact points and, in order to remove this, certain physical hypotheses have to be accepted. It is shown that contact models consistent with the dynamics possess certain new qualitative properties compared with the traditional quasi-static models in which the type of motion of the body is not taken into account. In particular, a dependence of the principal vector and principal moment of the friction forces on the direction of sliding or pivoting of the body, as well as on the magnitude of the angular velocity, is possible.     

Viscoelastic body colliding against a rigid wall with and without dry friction effects

The impact of a body against a rigid wall is studied. The body consists of a rigid block and a viscoelastic rod described by the distributed-order fractional model and in particular its solid-like and fluid-like special cases. Translatory motion of a body is studied in two cases: without and with the influence of dry friction. When present, dry friction is modeled by the Coulomb friction law. The problem is treated analytically by the use of the Laplace transform method and solutions are obtained in a convolution form.     

The conditions for the unique solvability of the equations of the dynamics of systems with friction

The problem of determining the generalized accelerations and reactions of constraints in systems with dry friction is investigated. The necessary and sufficient conditions for the unique solvability of the problem are obtained, applicable for cases of sliding and static friction. A geometrical approach is used, based on the introduction of a certain auxiliary parameter space divided into non-overlapping regions in terms of the number of possible types of motion. In each of these regions there are explicit expressions for the accelerations and reactions, which enable us, using piecewise-smooth mapping, to express, from the equations of motion, the generalized forces in terms of the parameters. The solution of the problem is equivalent to inverting the given mapping. A number of examples are given.     

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)     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.     

The motion of a body supported at three points on a rough plane

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.     

Self-excited oscillations of a two-mass oscillator with dry “stick-slip” friction

A system of two masses, moving along a single straight line, is considered. The first is connected by a spring to a fixed point, while the second is connected by a spring to the first and is in contact with a belt with dry friction moving with constant velocity. A piecewise-constant model of dry friction with different coefficients of friction, sliding and at rest, is used. The limit “stick-slip” type cycles are investigated analytically. It is shown numerically that in the case of equal masses there are forward and reverse limit cycles. The period of the oscillations of the forward and reverse cycles increases as the ratio of the stick and slip coefficients of friction increases, and decreases when the velocity of the belt increases. The reverse cycle exists for all values of the parameters of the problem, while the forward cycle exists up to a certain critical value of the ratio of the stick and slip coefficients of friction, and this critical value increases when the velocity of the belt increases.     

An asymptotic analysis of the solution in the neighbourhood of the corner point of a crack along the kinked interface of two media

The asymptotic behaviour of an elastic field in the neighbourhood of the corner point of a crack at the interface of different materials is investigated within the framework of plane elasticity, taking into account the contact of its surfaces and the possibility of their mutual slippage with dry friction. The problem is solved by the method of complex Kolosov-Muskhelishvili potentials. The results obtained enable one to estimate the angular range of existence of contact zones and the singularity of the stresses close to the corner point of the crack. It is shown that the formation of contact zones, taking into account the friction forces accompanying slippage, depends essentially on the magnitude of the angle of the interface kinking the elasticity moduli of the materials and the friction coefficient. Numerical calculations are carried out and the stress and displacement distributions in the neighbourhood of the corner point are obtained.     

Tensile-bending coupling in the limitting 1D beam equations resulting from the frictional contact

The asymptotic dimension reduction for beam contact problems without friction leads to decoupled ODE systems in the longitudinal variable, see [4]. This decoupling is due to the fact that friction is not taken into account. In this work we extend [2] by an alternative scaling of the tensile force in the 3D formulation that permits us to reduce the dimension of a contact problem with friction in tensile direction, too. The tension and the bending component in the direction of the contact normal are coupled due to the friction. The coupling of the displacement components is illustrated by a numerical example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incipient sediment transport for non-cohesive landforms by the discrete element method (DEM)

We introduce a numerical method for incipient sediment transport past bedforms. The approach is based on the discrete element method (DEM) [1], simulating the micro-mechanics of the landform as an aggregate of rigid spheres interacting by contact and friction. A continuous finite element approximation [2] predicts the boundary shear stress field due to the fluid flow, resulting in drag and lift forces acting over the particles. Numerical experiments verify the method by reproducing results by Shields [3] and other authors for the initiation of motion of a single grain. A series of experiments for sediments with varying compacity and constituting piles yields enhanced relationships between threshold shear stress and friction Reynolds number, to define incipient sediment transport criterion for flows over small-scale bed morphologies.     

The Vibrational Behavior of Bladed Disks in Consideration of Friction Damping and Contact Elasticity

Rotating turbine blading is subjected to fluctuating gas forces during operation that cause blade vibrations. One of the main tasks in the design of turbomachinery blading is the reduction of the vibration amplitudes of the blades to avoid high resonance stresses that could damage the blading. The vibration amplitudes of the blades can be reduced significantly to a reasonable amount by means of friction damping devices such as underplatform dampers. In the case of blade vibrations, relative displacements between the friction damping devices and the neighboring blades occur and friction forces are generated that provide additional damping to the structure due to the dry friction energy dissipation. In real turbomachinery applications, spatial blade vibrations caused by a complex blade geometry and distributed excitation forces acting on the airfoil accur. Therefore, a three dimensional model including an appropriate spatial contact model to predict the generalized contact forces is necessary to describe the vibrational behavior of the blading with sufficient accuracy, see [1] and [2]. In this paper the contact model presented in [2] is extended to include also local deformations in the contacts between underplatform dampers and the contact surfaces of the adjacent blades. The additional elasticity in the contact influences the resonance frequency of the coupled bladed disk assembly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a general dynamic history-dependent variational–hemivariational inequality

This paper is devoted to the study of a general dynamic variational–hemivariational inequality with history-dependent operators. These operators appear in a convex potential and in a locally Lipschitz superpotential. The existence and uniqueness of a solution to the inequality problem is explored through a result on a class of nonlinear evolutionary abstract inclusions involving a nonmonotone multivalued term described by the Clarke generalized gradient. The result presented in this paper is new and general. It can be applied to study various dynamic contact problems. As an illustrative example, we apply the theory on a dynamic frictional viscoelastic contact problem in which the contact is modeled by a nonmonotone Clarke subdifferential boundary condition and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the total slip.     

Robust a Simulation for Shallow Flows with Friction on Rough Topography

In this paper, we propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough topography. The major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface tracking. An analytical integration method is presented for the friction force term to remove the stiffness. In the vicinity of wet/dry interface, the numerical stability can be attained by introducing an empirical parameter, the water depth tolerance, as extensively adopted in literatures. We propose a problem independent formulation for this parameter, which provides a stable scheme and preserves the overall truncation error of $\mathbb{O}$∆$x^3$. The method is applied to solve problems with complex rough topography, coupled with $h$-adaptive mesh techniques to demonstrate its robustness and efficiency.     

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.     

Conditions for the onset of sliding in a plane system with friction

The problem of the plane motion of a rigid body along a fixed surface in the presence of dry (Coulomb) friction is considered. The constraint is assumed to be non-restraining. It is shown that the validity of a certain system of two inequalities of the same type guarantees that the surfaces maintain contact and that the body will continue to roll without sliding. These conditions are analysed in a few specific cases of mechanical systems.     

A qualitative analysis of the dynamics of a disc on an inclined plane with friction

The problem of the motion of a disc on an inclined plane with dry friction is investigated. It is shown that, if the friction coefficient is greater than the slope of the plane, the disk will come to rest after a certain finite time, and its sliding and rotation will cease simultaneously. The limit position of the instantaneous centre of velocities is indicated. The limit motions of the disc in the case when the ratio of the friction coefficient to the slope of the plane is equal to or less than unity: uniform sliding (in the case of a general position) and equiaccelerated sliding (always) of the disc along the line of greatest slope of the plane, respectively, are obtained. The case when the friction coefficient is equal to the slope, while the initial sliding velocity is directed upwards along the line of greatest slope, is an exception. In this case, the disc comes to rest after a finite time, and the sliding velocity and the angular velocity of the disc vanish simultaneously.     

Existence for an energetic problem in contact mechanics with dry friction

A simplified contact problem with dry friction is considered for a non–linear elastic system with two degrees of freedom. The simplification consits in neglecting kinetic energies or equivalently inertial forces. By proving existence it is shown under which conditions such simplifications are justifiable. The main focus is on the influence of a curved obstacle surface on the question of existence. The friction is modelled according to the Coulomb law and the coefficient of friction may vary along the obstacle surface. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic solution of contact problemsfor a relatively thick elastic layer when there are friction forces in the contact area

The plane contact problem of the theory of elasticity of the interaction between a punch, having a base in the form of a paraboloid,and a layer, taking Coulomb friction in the contact region into account, is considered. It is assumed that either the lower boundary of the layer is fixed or there are no normal displacements and shear stresses on it, and that normal and shear forces are acting on the punch. Here, the punch-layer system is in a condition of limit equilibrium, and the punch does not turn during the deformation of the layer. The case of quasi-statistics, when the punch moves evenly over the layer surface, can be considered similarly in a moving system of coordinates. The problem is investigated by the large-λ method (see [1–3], etc.), which is further developed here, namely, simple recurrence relations are derived for constructing any number of terms of the series expansion of the solution of the corresponding integral equation in negative powers of the dimensionless parameter λ related to the thickness of the layer.