Abstract

Research and analysis of dynamic processes in oscillatory systems are closely connected to the establishment of exact or approximate analytical solutions to the problems of mathematical physics, which model such systems. The mathematical models of wave propagation in oscillatory systems under certain initial conditions at a fixed time are well known in the literature. However, wave processes in lengthy structures subject to an external force only and at the assigned states of the process at two points in time have been insufficiently studied. Such processes are modeled by a two-point time problem for the inhomogeneous wave equation in an unbounded domain t>0, x∈ℝ s . The model takes into consideration the assignment of a linear combination with unknown amplitude of oscillations and the rate of its change at two points in time. A two-point problem, generally speaking, is the ill-posed boundary value problem, since the respective homogeneous problem has non-trivial solutions. A class of quasi-polynomials has been established as the class of the existence of a single solution to the problem. This class does not contain the non-trivial elements from the problem's kernel, which ensures the uniqueness of solution to the problem. We have proposed a precise method to build the solution in the specified class. The essence of the method is that the problem's solution is represented as the action of a differential expression, whose symbol is the right-hand side of the equation, on some function of parameters. The function is constructed in a special way using the equation and two-point conditions, and has special features associated with zeroes of the denominator – the characteristic determinant of the problem. The method is illustrated by the description of oscillatory processes within an infinite string and a membrane. The main practical application of the constructed method is the possibility to adequately mathematically model the oscillatory systems, which takes into consideration a possibility to control the system's parameters. Such a control over parameters makes it possible to perform optimal synthesis and design of parameters for the relevant technical systems in order to analyze and account for special features in the dynamic modes of oscillations

Highlights

  • Головним практичним застосуванням розробленого методу є можливiсть адекватного математичного моделювання коливальних систем, яке враховує можливiсть керування параметрами системи

  • Adequate mathematical modeling of wave processes in oscillatory systems plays an important role in modern approaches to solving a series of scientific and engineering tasks, which arise in the problems on analysis, synthesis, and optimization of parameters for machine-building structures

  • Mathematical and numerical modelling are important in understanding the essence of phenomena and processes that are studied in modern mechanics and physics

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Summary

Intr oduction

Adequate mathematical modeling of wave processes in oscillatory systems plays an important role in modern approaches to solving a series of scientific and engineering tasks, which arise in the problems on analysis, synthesis, and optimization of parameters for machine-building structures. A relevant problem of the current state of research related to applied mechanics is the development of new, as well as the extension of existing, asymptotic approaches for studying the mathematical models of oscillatory systems. If magnitude U (t, x) and its derivative ∂t U (t, x) were determined not at the same time (t = 0), but at two moments (for example, when t = 0 and t = τ, where τ > 0), possibly very close moments, we would obtain the problem on finding a solution to equation (1), which, in domain t > 0, x ∈Rs, satisfies such two-point conditions for time. The task on enlarging a circle of oscillatory systems, whose mathematical models allow the possibility to apply precise analytical approaches to study, is relevant at the present stage of the development of mechanical theory of oscillations. Construction of a differential-symbol method to study the wave process, carried out in this work, is a pressing issue in the applied mechanics

Literature review and problem statement
The aim and objectives of the study
Forced oscillations of an infinite string with its two assigned profiles
Forced oscillations of an infinite membrane with two assigned profiles
Conclusions

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