Development of system ideas in the approach of the dynamics of oscillatory structures: the frequency function and forms of connectivityof the elements’ motions

This article develops new approaches to the formation of a methodological basis when evaluating the properties of oscillatory structures by the examples of mechanical oscillatory systems with lumped parameters, which are used as computational schemes for technical objects of technological and transport purposes. It considers the features of the system approach in evaluating the dynamic interactions of elements of mechanical oscillatory systems. Methods for evaluating the properties of mechanical oscillatory systems are developed on the basis of the characteristics that depend on the coefficients of the forms of motion of partial systems in the free oscillation mode. The paper introduces the concept of a frequency function that reflects the features of the ratio of potential and kinetic energy of the system. When applied to mechanical oscillatory systems with two degrees of freedom, it proposes and develops an algebraic method for constructing a frequency function that depends on the connectivity coefficient and reflects the dynamic features of a mechanical oscillatory system. For the model example, it is shown that the constructed frequency function coincides with the frequency energy function obtained in the framework of methods of structural mathematical modeling. A method for constructing frequency functions is developed to evaluate the features of the dynamic properties of mechanical oscillatory systems that display the properties of connectivity of the forms of the elements’ oscillatory motions. A relationship between the characteristic of elastic elements and the distribution of coefficients of forms that determine the extreme values of the frequency function is established. A number of forms of frequency functions for various variants of mechanical oscillatory systems are considered, including the limit parameters of rigidity that determine the degree of connectivity of mass-inertia elements of the system. The results of the solution are presented using model examples.