BIFURCATIONS OF PERIODIC MOVEMENTS WITH HITS TWO MASS DYNAMIC SYSTEM

The problems of the dynamics and stability of vibro-impact systems today constitute an independent section of the applied theory of oscillations. The interest in these problems is primarily due to the wide use in practice of machines and technologies that use systematic shock interactions as the basis of work processes. Vibrating hammers, vibro-impact tools, shock absorbers, disc brakes, machines for vibro-impact testing, devices for vibrotransport of piece and bulk cargo, vibroseparation, volumetric vibro-processing - this is not a complete list, which gives an idea of the diversity of technological uses of vibro-impact systems and range of issues requiring the application of the theory of these systems. Vibro-impact systems, as compared with conventional oscillatory systems, have additional parameters that characterize for one-dimensional systems, the gaps in shock pairs and the coefficients of restoring the speed upon impact. Previously, one of the authors found conditions for the existence and stability of periodic motions of a body moving horizontally using a belt mechanism due to the force of dry friction located inside the container, which performs straight-line harmonic oscillations. This model and its particular cases reflect the dynamics of both systems with shock interactions and systems with friction. We also note that some nonautonomous systems with one degree of freedom are inherent in some properties of multidimensional systems. In this paper, we study the evolution of periodic motions with impacts depending on one of the parameters (the other parameters are assumed to be fixed) and a general analysis of the period doubling bifurcation for periodic motions with two impacts is carried out.

1. Kobrinsky A. A., Kobrinsky A. E. Two-dimensional vibro-impact systems. Monograph. - M.: Science, 1981. 336 p.

2. Bespalova L.V. On the theory of a vibro-impact mechanism // Izv. An. USSR. 1957. Ser. OTN. № 5. P. 3‒14.

3. Di Bernardo M., Feigin M. I., Hogan S. J., and Homer M. E. Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems // Chaos, Solitons and Fractals, 1999. V. 10. P.1881−1908.

4. Ivanov A.P. Dynamics of systems with mechanical collisions. M: “International Education Program”, 1997. 336 p.

5. Ivanov A.P. Fundamentals of the theory of systems with friction. M. ‒ Izhevsk: SIC “Regular and chaotic dynamics”, Izhevsk Institute of Computer Science, 2011. 304 p

6. Andronov A. A., Witt A. A., Khaikin S. E. Oscillation Theory. M.: FizMatgiz, 1956. 915 p.

7. Teifel A., Steindl A., Troger H. Classification of nonsmooth bifurcations for an oscillator with friction // Problems of Analytical Mechanics and Theory of Stability. Sat scientific articles dedicated to the memory of Academician V.V. Rumyantsev. Moscow: NPU RAS, 2009. p. 161−175.

8. Figurina, T. Yu. Optimal control of the motion of a two-body system along a straight line // Izvestia RAN. Theory and control systems, 2007. № 2. S. 65- 71.

9. Chernousko, F. L. On the motion of a body containing a mobile internal mass // Dokl. Acad. Science, 2005. T. 405, № 1. P. 1-5.

10. Chernousko, F. L. Analysis and optimization of body movement, controlled by moving internal mass // Applied Mat. and Mechanics, 2006. T. 70, no. 6. pp. 915-941.

11. Lyubimtseva O.L. On the stability of periodic motions of a system with a vibrating limiter // Bulletin of the Nizhny Novgorod University. N. I. Lobachevsky. Series: Mathematical Modeling. Optimal control, 2012. №.2 (1). P. 184‒189.

12. Neimark Yu. I. The method of point mappings in the theory of nonlinear oscillations, parts I, II, III. // Proceedings of higher educational institutions, the series “Radio fizika”. 1958. V. 1. № 1, 2, 5‒6.

13. Aleksandrov, M.P., Lysyakov, A.G., Fedoseev, V.N., Novozhilov, M.V. Brake Devices: A Handbook. M.: Mashinostroenie, 1985. 312 p.