Discrete mathematical structures in the analysis and synthesis of passive and active vibration isolating devices

Abstract

The article deals with discrete mathematical structures used for the analysis and synthesis of passive and active vibration isolation devices. It is shown that approaches of discrete mathematics in the form of kinematic graphs and their further development in the form of modified kinematic graphs make it possible to effectively model the functional interaction between the elements of such devices, describing their interaction with the corresponding finite mathematical structures. Such an approach to solving the problems of synthesis of vibration isolating devices makes it possible to effectively carry out a complete enumeration of possible options and can be the basis for the transition to the design of their industrial samples. The paper presents an analysis of existing passive and active vibration isolating devices on the basis of which, as an example, a synthesis of a fundamentally new passive vibration isolating device with elements of active systems was carried out. The basis of the synthesis is a modified kinematic graph, which is the result of the intersection of the sets of elements of the modified kinematic graphs, respectively, of passive and active vibration isolation devices. It has been established that such a modified kinematic graph has two cycles with one pole and an arc that defines an elastic connection included in both cycles. It is shown that the resulting modified kinematic graph of the synthesized device is very quickly transformed into the corresponding block diagram and can be constructively implemented as a prototype of this device. It is substantiated that the device synthesized in the presented work, according to its structural features, occupies an intermediate place in the hierarchy of vibration isolation devices between passive and active ones.