OFF-LINE INTERACTION OF THE NONLINEAR DYNAMIC SYSTEMS

S. E. Ivanov, G. I. Melnikov


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Abstract

Mathematical model of many mechanical systems is the system of dynamic equations with polynomial structure and periodic or constant parameters. Such mechanical systems are widely applied in the dynamics of a vibration guard of devices. Research of nonlinear systems with a finite number of freedom degrees represents a complicated actual problem in comparison with linear systems. Research of nonlinear systems does not come to definition of a finite number of private solutions because nonlinear systems do not possess superposition property of solutions. Mathematical model of nonlinear dynamic system with three degrees of freedom which contains polynomials to the fourth degree from phase variables is considered. Algorithmic formulas of an off-line interaction method of nonlinear dynamic systems are presented for the given model research. The nonlinear mathematical model of dynamic system is transformed to the autonomous form, and principal parameters of dynamic system are defined. The algorithm of an off-line interaction for research of nonlinear system protected from vibrations with three degrees of freedom is presented. In case of vibration guard solution of problems linear systems are widely used though linearity of functions does not approximate precisely enough the system performance, causing analysis errors. The problem of deriving and research of more exact nonlinear system model protected from vibrations is solved. The nonlinear system protected from vibrations with three degrees of freedom and nonlinear right members in the form of a polynomial of the third degree from phase variables with constant and periodic parameters is considered. The system consists of a vibration guard plant established on two platforms, one under another, the lower one is put on the vibrating foundation. Exterior harmonious perturbation exerts influence upon the foundation. Elastic system elements are supposed to be described by polynomials of the third degree and damping elements have nonlinear cubic performance. As a result of method application the nonlinear system will be transformed to the more simple autonomous form and the amount of parameters of nonlinear dynamic system is essentially reduced. The autonomous system contains essentially less parameters, than initial system, without making worse solution quality. Method application simplifies essentially the research of transitive and settled processes of nonlinear dynamic systems. The solved research problem for the system protected from vibrations in nonlinear statement is new and has theoretical and practical value.


Keywords: off-line interaction of nonlinear systems, research methods, nonlinear systems with three degrees of freedom

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