Similarity and Analogousness in Dynamical Systems and Their Characteristic Features

Mathematical models describing technically oriented dynamical systems are generally rather complex. Very time-consuming interactive procedures have to be used when selecting the structure and parameters of the system. Direct enumeration of options using such procedures can be avoided by applying a number of means, in particular, dimension methods and similarity theory. The use of dimension and similarity theory along with the general qualitative analysis of the system can serve as an effective theoretical research method. At the same time, these theories are simple. Using dimension and similarity theory, it is possible to draw conclusions when considering phenomena that depend on a large number of parameters, but so that some of them become insignificant in certain cases.
The combined method of using the theory of similarity, analogousness and methods developed by the authors for testing the drive model provides insight into its dynamics, controllability and other properties. The proposed approach is based on systematization and optimization of the process of forming a dimensionless model and similarity criteria, its focus on solving the formulated problem, as well as on special methods of modeling and processing of simulation results. It improves the efficiency of using similarity properties in solving analysis and synthesis problems. The advantage of this approach manifests itself in the ultimate simplification of the dimensionless model compared to the original model. The reduced (dimensionless) model is characterized by a high versatility and efficiency of finding the optimal and final solution in the selection of parameters of the real device, as it contains a significantly smaller number of parameters, which makes it convenient in solving problems of analysis and, in particular, synthesis of the system.
Dimension methods and similarity theory are successfully applied in the study of dynamical systems of different classes. The problems that arise are mainly related to the selection of a rational combination of the main units of measurement of physical quantities, the transition to dimensionless models and the formation of basic similarity criteria. The structure and the form of the dimensionless model depend on the adopted units of measurement of the variables appearing in the equations of the model and on the expressions assigned to its coefficients. Specified problems are solved by researchers, as a rule, by appealing to their intuition and experience. Meanwhile, there exist well-known systematized approaches to solving similar problems based on the method of the theory of analogousness.