Periodic solutions of dry friction problems

Dry friction problems lead to discontinuous differential equations, e.g. to

Dynamic multi-layer models of dry friction

Determining the friction law between solids as a function of material and loading parameters remains a topical and still unsolved problem of tribology. In the rail-wheel-contact the law of sliding friction determines not only the maximum tractive forces but also defines the conditions for the occurrence of instabilities. Recent simulations of irreversible processes in micro-contacts with the method of movable cellular automata (MCA) have shown that dynamic aspects do play an essential role in these processes. Inelastic processes of plastic deformation, detaching of particles and their reintegration into the surfaces as well as processes of mechanical mixing occur only in a thin surface layer called quasi-fluid layer. Studying the properties of the quasi-fluid layer is thus a key for understanding friction. To better understand the physical nature of the quasi-fluid layer and especially the physical nature of the viscosity of the quasi-fluid state, multi-layer models of an elasto-plastic solid have been studied. In the layer model it becomes particularly clear that the physical reason for occurrence of the quasi-fluid layer is the bi-stability of the friction law between layers. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical modeling of friction stir welding considering dry and viscous friction

In this paper, the investigation of a pre-immersion period of Friction Stir Welding process (FSW) from the mechanical point of view using analytical considerations is presented. This initial period is a part of the real technological process, when a FSW instrument reaches the touchdown point with the workpiece, preheats the contact zone, but does not penetrate the material. This approach enables an insight into FSW process on a level, where the system parameters are known and an experimental evaluation of theoretical results is possible. Beside these considerations an analytical estimation of the power input of dry and viscous friction during instrument rotation and translation in parallel is shown. These phenomena are investigated including friction between a FSW instrument and a workpiece. The influences of both dry and viscous friction are studied. Experiments have been done to validate the calculated results.     

Damping by dry friction of dynamic loads in a fiber-reinforced composite

Conclusions The accurate analytical solutions of a number of nonlinear problems of impacts on semiinfinite and finite fibers, interacting with the matrix in accordance with the dry friction law, were obtained. We examined the cases of both unidirectional motions and reversed motions caused by unloading, and also oscillatory motions. The results can be used to calculate the energy dissipated in the separating sections of the composite in shock effects. It was shown that the amount of energy dissipated in the dynamic effect is considerably greater than in the quasistatic effect. Thus, dissipation in the system with friction depends strongly on the nature of load application with time.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 28–37, January–February, 1986.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Removing "silver" type microcracks in polymethyl methacrylate glass by dry friction

"Silver" type microcracks in polymethyl methacrylate are self-removed on aging. Dry friction can assist in their accelerated removal; by intensifying relaxation processes and causing flow in the surface layer of the polymer, it is able to restore both the polymer structure in the zones of "silver" type microcracks and also to age obvious microcracks. The restored polymer quality is stable. Dry friction raises the resistance of polymethyl methacrylate to splitting in failure.Lenin Civil Aviation Engineering Institute. Translated from Mekhanika Polimerov, No. 3, pp. 465–470, May–June, 1970.     

Evolution of the resonance vibrations of an oscillator with dry friction

We study the problem of free and forced vibrations of a mechanical system modeling the functioning of seismic isolation systems of “kinematic foundation” type in the case when the sliding friction damper is not centrally located. We determine the damping characteristics in the case of free vibrations and the critical values of the damping parameter for the case of forced vibrations. For the case of resonance vibrations with one degree of freedom we establish the relation of the increase in the amplitude of the vibrations to time at subcritical values of the friction. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Forced vibrations of an oscillator in the presence of dry friction

We study the vibrations of an oscillator with two degrees of freedom in the presence of dry friction. We compare the nature of the damping of the free oscillations in a straight line with the general case. For the forced vibrations we determine the way in which the critical values of friction at which there exist periodic motions depend on the parameters of external action in resonance mode.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 49–54.     

Penetration rate prediction for percussive drilling via dry friction model

A dynamic model of percussive drilling assuming a dry friction mechanism to explain the experimentally observed drop in penetration rate is presented. The inherent nonlinearity of the discontinuous impact process is modelled as a frictional pair, and this can generate the pattern of the impact forces close to reality. Despite quite radical simplifying assumptions, the model is able do describe the fall of material removal rate for a higher static loading with a good agreement to experimental investigations.     

Second order singular periodic problems in the presence of dry friction

We prove, via an approach by ordinary differential equations, the existence of oscillations for second order inclusions with restoring terms with singularities both of repulsive and attractive type and with a dry friction term.

     

Chaotic solutions in differential inclusions: chaos in dry friction problems

The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.

     

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)     

Straight chains of interconnected mass points under the action of non-symmetric dry friction

The motion of a chain of interconnected equal material points placed on a rough straight line and connected by equal kinematical constraints is considered. It is supposed that the system experiences a small non-symmetric Coulomb dry frictional force acting from the line side upon each mass point and opposite to the direction of motion. The magnitude of this force depends on the direction of motion. To the equation of motion the procedure of averaging is applied. The expression for the stationary “on the average” velocity of the motion of the system as a single whole is found. The obtained theoretical results can be used for the design of worm-like vibration robots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The equilibrium of a rigid body on a plane with anisotropic dry friction

The problem of the conditions for the static equilibrium of a body, resting on a rough plane at one, two or three points, is considered. It is assumed that an arbitrary system of active forces is applied to the body, while the friction on the rough supporting plane is anisotropic. This model generalizes the well-known isotropic model of Coulomb dry friction. Explicit analytic formulae, which express the necessary and sufficient conditions for static equilibrium, are obtained. The investigation procedure uses the idea of an anisotropic force of static friction, which enables analytical results for the equilibrium conditions to be obtained more easily.     

The model of dry friction in the problem of the rolling of rigid bodies

The Contensou model of combined dry friction [1] is considered. The problem of integrating the shear stresses over the contact area is solved in terms of elementary functions, unlike the solution in [1], reduced to elliptic quadratures. The problem of the rolling of a homogeneous sphere over a plane with dry friction is investigated.     

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.