# Mathematical modelling and control of the evolution of dynamic systems interacting with medium

## The article proposes a new method of mathematical modeling evolutional systems interacting with medium. A cutting process or tribo-space formed in the contact area of two conjugate mechanical subsystems is considered as medium. The features of the medium depend not only on state coordinates of systems but also on trajectories. | Vietnam Journal of Mechanics, VAST, Vol. 33, No. 3 (2011), pp. 148 – 161 MATHEMATICAL MODELLING AND CONTROL OF THE EVOLUTION OF DYNAMIC SYSTEMS INTERACTING WITH MEDIUM . Zakovorotny1, Nguyen Dong Anh2 , Pham Dinh Tung1 1 Don State Technical University, Rostov-Don, Russia 2 Institute of Mechanics, Hanoi, Vietnam Abstract. The article proposes a new method of mathematical modeling evolutional systems interacting with medium. A cutting process or tribo-space formed in the contact area of two conjugate mechanical subsystems is considered as medium. The features of the medium depend not only on state coordinates of systems but also on trajectories. It is obtained that the parameters of the medium are performed as integral operators of equations like Volterra equations of 2nd type. The problems of control of evolutional systems are being analyzed. Keywords: Evolutional system, interact, medium. 1. INTRODUCTION Studying evolution of technical systems is traditional and corresponds to the problem of ensuring reliability. In particular, studying evolutional transformations of processes on cutting machines affects the development of wearing of instruments changing cutting products quality indexes [see . 1-5]. All those indexes characterize the external display of evolutional changes. The evolutional changes of dynamic characteristics of cutting process, which influences the parameters of geometrical quality of goods and the state of cutting process, perform a more important meaning. They affect the current characteristics of the irreversible transformations, which are characterized by trajectories of changing the power of irreversible transformations of the function of work being performed. In technical systems with the friction areas the evolutional changing of matrices of dynamic rigidity and dissipation of tribo-space, being formed in the areas of conjugation of the contacted elements, takes place. The article is devoted to studying evolutional changes of the .

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