Abstract
The dynamics of mechanical systems, the operation of electromagnetic and electronic devices and devices, the principle of operation of a number of machines and mechanisms, engineering structures from various fields are often described by differential equations and their systems. Differential equations are often mathematical models of the movement and operation of various engineering objects. As a rule, such equations are solved by numerical methods for specific parameter values. These methods of solving differential equations are widely used in practice. However, these methods also have significant disadvantages. For example, the solution of differential equations is obtained for a specific object with the specified parameters. In this case, a solution is obtained for a single point in the parameter space of a set of similar objects, points in this space of the considered family of objects. A natural question arises: Is it possible to extend the results of the solution for a single point in space (a specific object) and the identified properties and regularities to other points (other objects) of the considered family? The purpose of this article is to identify conditions under which it is possible to generalize the results of solving differential equations with specific parameter values describing a single construction to the entire family of similar constructions, the entire space of parameters under consideration. The implementation of the identified conditions is illustrated by the example of solving the problem of “analyzing the dynamic properties of a mathematical model of a car with adaptive (adjustable) suspension of a new principle of action (developed by the authors), moving at a variable speed along an indirect profile of the road surface and developing recommendations for their radical improvement”.
Highlights
When constructing the laws of motion of dynamical systems [1], i.e., when solving the Cauchy problem (“initial” problem) [2,3] or when constructing a periodic solution [3] of differential equations of motion of dynamical systems, as a rule, the authors use a “numerical” solution
A good coincidence of the results obtained in this case indirectly confirms the validity of the approach we propose to implement the possibility of generalizing the results of studies of dynamical systems i.e., to implement the possibility of “non-local use of the results of local analysis”
The article considers the question of the validity of the non-local use of the results of local analysis in the numerical solution of systems of differential equations describing the motion of dynamical systems
Summary
When constructing the laws of motion of dynamical systems [1], i.e., when solving the Cauchy problem (“initial” problem) [2,3] or when constructing a periodic solution [3] of differential equations of motion of dynamical systems, as a rule, the authors use a “numerical” solution. At the end of the study, without justifying this in any way, the authors try to generalize their conclusions to the entire space of parameters of a dynamical system Such a transition, the transition to “non-local use of local analysis results”, is possible only if the dynamic system under study has some special properties [4]. The transition to “non-local use of local analysis results”, is possible only if the dynamic system under study has some special properties [4] This problem is well-known [4] and prompts every time the “numerical” construction of the laws of motion of dynamical systems, which indicates its undoubted relevance. We will consider its solution in more detail, using the example of solving the problem of “analyzing the dynamic properties of a car moving at a variable speed along an indirect profile of the road surface and developing recommendations for their radical improvement”
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