Features of the system concepts of the dynamics of mechanical vibrational structures under asymmetric connections

The paper is devoted to a new approach to the formation of the methodological basis of system analysis in application to problems of the dynamics of mechanical oscillatory systems taking into account the forces of viscous friction. A mechanical oscillatory system with two mass-inertia elements connected to the support surfaces and to each other by elastic-dissipative elements is considered. A method for estimating the free movements of a mechanical oscillatory system is developed based on the characteristics expressed in terms of the parameters of the lever connections between the partial blocks in the form of a gear ratio. Mathematical models are built on the basis of Lagrange formalism, algebraic methods, and the theory of functions of a complex variable. A matrix method is proposed for constructing frequency functions and damping functions for mechanical oscillatory systems with two degrees of freedom based on the ratio of the potential and kinetic energy of the system, taking into account the forces of viscous friction. For a mechanical system in which the connection of elements with a single reference surface is reset, a graphic analytic analysis of the extreme properties of the corresponding frequency functions and damping functions is performed. A topological approach is proposed.

SYSTEM REPRESENTATIONS OF DYNAMIC PROCESSES IN MECHANICAL OSCILLATORY SYSTEMS WITH SPECIAL FEATURES

Рассматриваются подходы в формировании методологического базиса системного анализа колебательных структур на примере упруго-диссипативных механических систем с двумя степенями свободы. Целью исследования является разработка метода оценки свойств механических колебательных систем с учетом сил вязкого трения на основе частотных функций и функции демпфирования в зависимости от коэффициентов форм связности. Для построения математических моделей используется формализм Лагранжа, матричные методы, элементы теории функций комплексной переменной. Представлены понятия частотной функции и функции демпфирования, отражающие особенности соотношения потенциальной, кинетической энергии системы с учетом сил вязкого трения, представленных диссипативной функцией. Разработан метод построения частотной функции и функции демпфирования, отражающих динамические особенности свободных движений с учетом сил трения в зависимости от коэффициента форм связности. На основе полученных общих аналитических выражений частотных функций и функций демпфирования проведен анализ особенных вариантов механических колебательных систем, представляющих интерес на начальном этапе исследования. Разработанный метод построения частотной функции и функции демпфирования может быть использован для отображения динамических форм связности движений механических колебательных систем. Предложенный метод построения частотной функции и функции демпфирования может быть обобщен на механические колебательные системы, рассматриваемые в различных системах координат. Approaches to the formation of a methodological basis for the system analysis of oscillatory structures on the example of elastic-dissipative mechanical systems with two degrees of freedom are considered. The aim of the study is to develop a method for evaluating the properties of mechanical oscillatory systems with account for viscous friction forces based on frequency functions and damping functions depending on the coefficients of connectivity forms. To build mathematical models, we use Lagrange formalism, matrix methods, and elements of the theory of functions of a complex variable. The concepts of the frequency function and the damping function are presented, which reflect the features of the ratio of the potential and kinetic energy of the system, taking into account the viscous friction forces represented by the dissipative function. A method is developed for constructing the frequency function and damping function that reflect the dynamic features of free movements, taking into account the friction forces depending on the coefficient of connectivity forms. Based on the obtained General analytical expressions of frequency functions and damping functions, special variants of mechanical oscillatory systems that are of interest at the initial stage of research are analyzed. The developed method for constructing the frequency function and the damping function can be used to display dynamic forms of connectivity of movements of mechanical oscillatory systems. The proposed method for constructing the frequency function and the damping function can be extended to mechanical oscillatory systems considered in different coordinate systems.