The solvability of the equations of motion of mechanical systems with sliding friction

The problem of the existence of local classical solutions of the equations of motion of holonomic systems with sliding friction is investigated. The conditions for the equations of motion to be solvable for the accelerations enable estimates to be obtained of the coefficients of friction, within the framework of which (from the point of view of the existence of solutions) Coulomb's laws can be used to describe the dynamics of mechanical systems with sliding friction.     

Attraction for autonomous mechanical systems with sliding friction

Issues on attraction in autonomous mechanical systems with ideal holonomic bilateral constraints acted upon by potential gyroscopic dissipative forces and forces of sliding friction are considered. In particular, the semi-invariance of ω-limit sets and the conditions for the dichotomy of such systems are established. The investigation is based on the invariance principle using several Lyapunov functions, combining the methods of [1] with the La Salle invariance principle [2, 3] applied to autonomous systems with a discontinuous right-hand side.     

Structural sliding equations for the tracking control of mechanical systems with active structure

The term of active structure is herein used (instead of variable structure) to emphasize that one deals with mechanical systems, some mechanical parameters of which (such as mass, inertia, rigidity) can be varied in time to account for environmental variations. One then has an internal structural control in addition to the usual external one in the form of forces or torques. By using a well-known classical result in physics, which characterizes a mechanical system by its Lagrangian only, one can obtain a new class of physically meaningful sliding equations for these systems. They are less severe regarding the mechanical constraints on the system, and they lower the amplitude of the chattering phenomenon. These systems can be controlled in a straightforward way by means of Barbashin-Utkin's variable structure technique.     

Quasiregular solutions for systems of hypoelliptic differential equations

L'vov State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 1, pp. 25–31, October, 1990.     

Periodic solutions for systems of second order differential equations

We consider vector differential equtions x″ + g(t, x, x′) = f(t), x?Rⁿ, and show the existence of periodic solutions under conditions which show that growth conditions on g required in earlier results are not needed.     

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.     

Pathwise uniqueness for solutions of systems of stochastic differential equations

A general theorem which obtains pathwise uniqueness for solutions of systems of Ito stochastic differential equations is given. It is shown that this theorem contains as special cases basic criteria which generalize Ito's result in which the coefficients satisfy Lipschitz conditions in the second variable. Also some new results which assume t-dependent modulus of continuity conditions on the coefficients are given as corollaries. The main result is established by means of Lyapunov type functions and comparison principle techniques.     

The stability of the set of equilibrium positions of autonomous mechanical systems with sliding friction

Problems of the stabiity of the set of equilibrium positions of autonomous differential equations, to which the equations of motion of mechanical systems with sliding friction can be reduced, are considered (see [1, 2]). The structure of the equations, due to the specific features of the system, is taken into account substantially and the general properties of motions, investigated previously in [3, 7], and the sets which arise when analysing the equations and which possess the properties of the absolute sector are used.     

Bounded solutions of second-order systems of differential equations

Sufficient conditions for the existence, uniqueness and stability of the bounded (in ) solution of the second-order systems of nonlinear differential equations are given.     

On the relationship between solutions of delay differential equations and infinite-dimensional systems of differential equations

We study the limit properties of solutions for a class of systems of ordinary differential equations as the number of equations and a certain parameter grow unboundedly. We show that the sequence of functions formed by the last components of solutions of such systems has a repeated limit. The limit function is a solution of a delay differential equation. Estimates of the convergence rate are obtained.     

Approximated solutions in rational form for systems of differential equations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

非线性混合阶微分方程组的正解

主要研究了由不同阶微分方程构成的非线性微分方程组的正解的存在性问题,通过对非线性项进行限制,运用锥理论中的不动点定理,得到了方程组正解存在性的多个结果.     

On the rapidly decreasing solutions for some systems of differential equations with unbounded coefficients

In this paper we continue the study of solvability of the non-homogeneous system of linear differential equations

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.     

On the estimation of solutions for some linear systems of differential equations

In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real lineR. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.     

Periodic solutions of singularly-perturbed systems of differential equations with impulse effect

Summary The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

---

Zusammenfassung Wir betrachten das Problem der Existenz, Eindeutigkeit und der asymptotischen Darstellung periodischer Lösungen singulär gestörter Systeme von Differentialgleichungen mit Impulsen. Die Studie wird mittels speziell dem Problem angepaßter Modifikation der Randwertfunktionsmethode ausgeführt.