System approach to assessing dynamic states of technical objects: elements of concept of dynamic invariants

The scientific and methodological base is being developed within the framework of systems theory and system analysis in application to the tasks of assessment, control and formation of dynamic states of technical objects, the design schemes of which are mechanical oscillatory systems with concentrated parameters. In the context of the fundamental problems of the theory of system studies of forced movements under deterministic disturbances there are considered the problems, whose solutions are aimed at developing a methodological approach to assessing the dynamic states of technical objects operating under coherent vibration loads. The purpose of the study is to develop a method for evaluating, controlling and forming dynamic states of the objects under harmonic force loads. It is assumed that mechanical oscillatory systems formed by a solid body performing small steady-state oscillations relative to the position of the static equilibrium can be used as calculation schemes of the technical objects. To construct mathematical models, Lagrange formalism, Laplace integral transformations taking into account zero conditions and structural methods are used, which make it possible to consider mechanical oscillatory systems as dynamically equivalent structural schemes of automatic control systems. The dynamic state of an object is assessed by using a transfer function, which in the physical sense represents the interpretation of a dynamic compliance. The amplitude-frequency characteristics used to assess the dynamic state of the object present a parametric family, for which the first parameter is the coefficient of connectivity of external force disturbances, and the second parameter is the coordinate of the point for which the dynamic accuracy is defined. An infinite family of amplitude-frequency characteristics is regularized due to the introduction of dynamic invariants, which can be considered as generalized dynamic states of mechanical oscillatory systems. To determine the dynamic invariants of a family of amplitude-frequency characteristics, taking into account two parameters, a map of dynamic invariants is constructed, which divides the plane of parameters into disjoint sets using boundaries. A theorem is proved to determine the explicit analytic expressions of these boundaries.