Vibration Analysis Based on the Method of Mathematical Prototyping of Energy Processes

Oscillatory motions can occur during the functioning of technical systems, living cells, and atmospheric phenomena. Therefore, one of the most important practical tasks encountered in modern engineering and technology is the task to analyze the occurrence of oscillatory motions, particularly auto-oscillations. The most precise method to investigate conditions of vibrations occurrence in a particular system is a mathematical modeling of the system in question. However, this method is the most time consuming. Therefore, this paper is devoted to development of analytical criteria for conditions of oscillations in the system. In the general case for analysis and mathematical modeling of the systems of different physical and chemical nature the authors have proposed a method of mathematical prototyping which incorporates laws of mechanics, theory of electric and magnetic circuits, electrodynamics, modern nonequilibrium thermodynamics. Since the method for mathematical prototyping of energy processes incorporates general physical laws, such a method gives adequate models of dynamics of physical and chemical processes (not contradicting the general physical laws). Therefore, the method for mathematical prototyping of energy processes proposed earlier by the authors is the basis for the method of analyzing the conditions of occurrence of oscillatory movements proposed in the present work.