The stability of equilibrium in systems with friction

Mechanical systems with non-ideal geometrical constraints are considered. The possible lack of uniqueness of the solution of the problem of determining the generalized accelerations and reactions with respect to specified coordinates and velocities is taken into account in solving the problem of the stability of an equilibrium state. A number of necessary and sufficient conditions of stability are obtained. It is shown that the results are also applicable in the case of unilateral constraints subject to the condition that a specific hypothesis concerning the character of the impacts on the constraints is adopted. A problem on the stability of a rigid body on a rough plane in the two-dimensional case is solved as an example.     

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.     

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

The dynamics of a disc on an inclined plane with friction in the dynamically consistent model of friction

The problem of the motion of a uniform circular disc of non-zero thickness on a fixed rough inclined plane is discussed. It is proved that, if the ratio of the coefficient of sliding friction to the slope of the plane is greater than unity, the disc comes to rest after a finite time. If this ratio is equal to unity, the limit motion of the disc is uniform slip along the line of greatest slope, and if this ratio is less than unity, the slip velocity of the disc increases without limit with time.     

Deformation and sliding friction of polyurethane in the presence of a lubricant

The relation between coefficient of friction and elastic and high-elastic strains is investigated with reference to polyurethane subjected to friction in various liquid media. It is shown that a definite relationship between these parameters does exist. The effect of various liquids on the elastic and high-elastic deformation of polyurethane is studied.Kiev Institute of Civil Aviation Engineers. Translated from Mekhanika Polimerov, Vol. 5, No. 2. pp. 357–359, March–April, 1969.     

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO.Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.     

On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability

For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10. pp. 1411–1416, October, 1999.     

Analytical existence of solutions to a system of nonlinear equations with application

We study the existence of analytical solutions to a system of nonlinear equations under constraints linked to the analysis of a road safety measure without computing second derivatives. We formally demonstrate this existence of solutions by applying a matrix inversion principle through Schur complement to a subsystem of equations derived from the main system. The analytical results thus obtained are used to construct a simple iterative procedure to look for optimal solutions as well as an initial solution adapted to data of each case study. We illustrate our results with simulated accident data obtained from the setting-up of a road safety measure. The numerical solutions thus obtained are then compared to those given through a classic Newton-Raphson type approach directly applied to the main system.     

Mass-spectrometric investigation of solid lubricating coatings subjected to friction in a deep vacuum

The life of solid lubricating coatings of the VNII NP type, based on molybdenum disulfides and various binders, has been experimentally investigated under deep vacuum conditions (10^–8–5 · 10^–9 torr) together with the composition of the gas released in the friction process. It is shown that both under atmospheric conditions and in a deep vacuum the life of the coatings depends on the chemical nature of the film-former. The depth of the vacuum also has an important influence on the life of the coatings, both the mechanism and the end result of this effect depending to a large extent on the physicochemical properties of the bind. On the interval 10^–1–10^–2 torr there is a sudden change in the life of the coating.Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1070–1075, November–December, 1970.     

Number and stability of nonlinear parametric oscillations in a system of weakly coupled oscillators with pendulum-type nonlinearity

The range of parameters for which the nonlinear system under study has three stable cycles with certain symmetry properties is determined. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 369–376, March, 1999.     

Numerical studies of the stability of steady-state solutions to a contact problem in coupled thermoelasticity

A finite element approximation is used to study the stability of steady-state solutions and the erratic behavior that is present in a problem of heat conduction through an elastic rod that may come into contact with a rigid wall. The quasi-static fully coupled theory of linear thermoelasticity is assumed and a heat exchange coefficient that depends on the pressure and the gap size is imposed across the region of contact.     

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.Partially supported by the Italian Consiglio Nazionale delle Ricerche.     

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.     

The stability of a non-potentially loaded discrete-continuous gyroscopic system with internal and external friction

The problem of the stability of the axial rotations of a symmetric rotor by means of an elastic shaft in the form of a thin straight rod, when there is a follower force and linear dissipation, is considered. Linearized equations for the displacement of the rotation of the rotor and the transverse plane oscillations of the rod are presented in a complex form as well as the boundary conditions. Using the standard procedure of a Laplace transformation with respect to time, relations for the images are obtained which are then solved by means of quasipolynomials of the complex transform parameter, and an asymptotic analysis of them is carried out. The effect of a follower force is investigated and its critical values are determined. The results of mathematical modelling for specific numerical values of the parameters are presented.     

Micro‐ and nanoscale modelling of dry friction with application to the wheel/rail contact

Friction is a phenomenon involving elastic interactions, plastic deformation and failure processes at different length scales. A model of dry friction is established based on the method of Movable Cellular Automata (MCA). The influence of material and loading parameters has been investigated within a large number of numerical simulations. The new friction law is applied to the calculation of stresses, deformations and tractive forces in wheel/rail contact with rough surfaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of sliding velocity on area of real contact and on force of friction in high-elastic materials

The velocity dependence of the force of friction and the area of real contact of SKN-18, SKN-26, and SKN-40 rubbers has been simultaneously determined on an optical instrument. In the region of small sliding velocities the area of real contact is virtually independent of the sliding velocity, while the force of friction increases in proportion to the logarithm of the sliding velocity, in accordance with the molecular-kinetic theory of friction. At high sliding velocities a deviation in the velocity dependences of both quantities is observed. The constant of friction has been determined over the entire velocity range and the resulting velocity dependence is examined in molecular terms.Mekhanika Polimerov, Vol. 3, No. 2, pp. 309–311, 1967