Optimal Vibration Fields in Problems of Modeling Dynamic States of Technical Objects
Authors: Eliseev A. V. , Kuznetsov N. K. 
Published: March 2024

Introduction. Vibration interaction control is timely in production processes related to liquid and bulk media, systems of solids experiencing kinematic or force disturbances. At the same time, there is no single methodological basis for the formation of vibrational interactions. The issues of constructing optimal vibration fields of technical objects have not been addressed. The objective of the study is to develop a structural approach to the development of mathematical models in the problems of formation, evaluation, and correction of vibration fields of technical objects under conditions of intense force and kinematic loads. The task is to build vibration fields that are optimal in terms of the set of requirements, with the possibility of selecting the criterion of optimality of the vibration field of a technical object.

Materials and Methods. A structural approach was used as the basic methodology. It was based on a comparison of mechanical vibratory systems used as computational schemes of technical objects, and structural schemes of automatic control systems, which are equivalent in dynamic terms. Lagrange formalism, elements of operational calculus based on Laplace integral transformations, sections of vibration theories, algebraic methods, and the theory of spline functions were used for structural mathematical modeling.

Results. An approach to the selection of criteria for the optimality of vibration fields based on minimizing the residual of vibration fields for various required conditions was proposed. The problem was considered within the framework of a mechanical vibratory system formed by solids. It was shown that the optimal vibration field was determined by an external disturbance and was to satisfy condition Ay̅ = b. There, A — matrix mapping the operator of conditions to the shape of the vibration field at control points; b — vector of values of vibration field characteristics; “–” above y meant the vibration amplitude of the steady-state component of the coordinate. To evaluate the field with account for noisy or unreliable requirements for dynamic characteristics, the smoothing parameter was used, indicating the priority of the criterion of optimality of the vibration field shape. The construction of a field for a mechanical vibratory system showed that the value of the vibration amplitudes of generalized coordinates remained constant when the frequency of external kinematic disturbances changed. Two approaches to the correction of the field optimality criteria were considered: equalization of the vibration amplitudes of the coordinates of a technical object and the selection of an energy operator.

Discussion and Conclusion. The development of the applied theory of optimal vibration fields involved, firstly, the correlation of the energy operator and the operator of the requirements for the shape of the vibration field in the theory of abstract splines. The second pair of comparable elements was the criterion of optimality of the vibration field and a system of requirements for the characteristics of the field at control points. The structural theory of optimal vibration fields improved in this way will find application in various industries. Accurate calculations in the formation, assessment, and correction of the states of systems under vibration loading are required in the tasks of increasing the durability of structures, improving measurements in complex vibratory systems, and developing new technologies and materials.