On the theory of the slipping state in dynamical systems with collisions: PMM vol. 36, n≗5, 1972, pp. 840–850

Abstract

For a wide class of systems with collisions we propose an approximate method for computing the slipping states, characterized by the same degree of completeness and labor-consumption as the known methods for computing motions, of simple types. For the case when the relative acceleration of the colliding bodies varies by a linear law on the final segment of the slipping state, we have obtained an analytic expression for the state's duration factor as a function of the velocity recovery factor under impact. Examples of the calculation of concrete models are considered. A comparison with results obtained by exact methods shows that the error does not exceed a few percents even for the first approximation. By a slipping state in a system with collisions we mean a motion accompanied on a finite time interval by an infinite sequence of instantaneous shock interactions between two fixed elements of the system. For a wide class of systems being considered the problem has been solved in [1–3] of determining in phase space the exact boundaries of the slipping state regions and of delineating the existence regions of periodic motions with a slipping state segment in parameter space. However, it is not advisable to recommend the use of the iterative procedure used in those papers as a practical calculation method because of the considerable consumption of labor. The approximate method of analyzing slipping states was suggested in [4], But, as was noted in [5, 6], when it was applied to con crete models, significant segments of the functions being computed could not be obtained successfully. In the present paper we propose an approximate method for calculating the slipping state regions in phase space and the existence regions of periodic motions with a slipping state segment in parameter space. The method is based on an analysis of point transformations of the shock interaction hyperplane into itself and on the following two idealizations; a) a certain new characteristic, which we introduce into consideration, namely, the duration factor of the slipping state, which depends only on the physical parameters of the system; b) the slipping state starting at some instant can also be treated as the motion of colliding masses with a superimposed kinematic constraint after their absolutely inelastic interaction [1]. In such an approach the approximateness of the method is connected with the approximateness of the idealizations adopted. However, the possibility remains of an unbounded refinement of the dynamic model being analyzed at the expense of choice of the instant taken as the start of the slipping state.