This article presents the development of the “TB-ANALYSIS” software package, created in the MATLAB environmentfor mathematical modeling and analysis of the resonant characteristics of objects with moving boundaries.This package is designed to solve one-dimensional boundary value problems describing the dynamicsof objects of variable length. The main objectives of this work are: creating software for studying solutions toboundary value problems and modeling resonant phenomena; testing the effectiveness of the software package;and developing an algorithmic description of its functionality. The key capabilities of the software suiteenable numerical solutions of model equations, analysis of the influence of various parameters on resonanceproperties, and optimization of model parameters to prevent resonance phenomena. The software suite’s testingresults and examples of its practical application are presented. 
Introduction. Studying the dynamics of objects with moving boundaries, described by one-dimensionalboundary value problems, is of significant interest for modern engineering and scientific calculations. Of particularrelevance is the analysis of the resonance properties of objects capable of causing critical conditions andstructural failure. Therefore, the development of a specialized software tool for comprehensive mathematicalmodeling and analysis of resonance characteristics is a pressing issue.The TB-ANALYSIS software suite presented in this paper was developed to address these issues. Its key capabilities include: studying solutions to boundary value problems using asymptotic and approximate analytical methods; performing a factor analysis of resonance properties; and determining conditions for preventing resonance phenomena. The suite is implemented as a standalone application in the MATLAB environment and has undergone extensive testing.
Materials and Methods. The TB-ANALYSIS software package was developed in the MATLAB environment usingnumerical computation capabilities and a graphical interface. The package’s architecture includes four interconnected modules, accessed through the start window. The main modules are: studying solutions to boundaryvalue problems, analyzing resonance properties, and controlling resonance phenomena. Three methodsare implemented to solve boundary value problems: an analytical method for replacing variables in a systemof differential-difference equations; an asymptotic method for constructing solutions to homogeneous integro-differential equations; and an approximate analytical method for constructing solutions to integro-differentialequations. Intelligent method selection is performed automatically depending on the type of problembeing solved. To assess the accuracy of calculations, a special error estimation procedure is implemented basedon an analysis of the accuracy of integral calculations.
Results. The software package was tested on various model boundary value problems. The effectiveness of thenumerical methods and the correctness of the algorithms are demonstrated. To demonstrate the capabilities ofthe package, a study of the transverse vibrations of a viscoelastic rope of variable length lying on an elastic foundationwas conducted. Graphs of the oscillation amplitude versus time and boundary velocity were obtainedfor various model parameters. The influence of the medium resistance, viscoelasticity, and substrate stiffnesscoefficients on the resonance characteristics was analyzed. It was found that the oscillation amplitude at zerovalues of the corresponding coefficients is an upper bound for cases where the coefficients are nonzero.A calculation error estimation method was developed that takes into account the accuracy of integral calculationsand the number of terms in the solution. It was shown that the error in calculating the squared amplitudecan be effectively estimated and controlled during the calculation process.
Discussion and Conclusion. The developed TB-ANALYSIS software package is an effective tool for studyingthe dynamics of objects with moving boundaries. The main advantages of the package include: versatility forsolving various classes of boundary value problems; intelligent selection of solution methods; a user-friendlyinterface; and built-in tools for evaluating calculation accuracy.Numerical experiments confirmed the performance of the package and demonstrated its potential for analyzing complex dynamic processes. The obtained results allow for a deeper understanding of the nature of resonant phenomenain objects of variable length and the development of effective methods for controlling resonant modes.Prospects for further development of the complex are associated with expanding the class of problems solved,implemented implementation of additional numerical methods and optimization of computational algorithmsfor working with more complex models.

Для цитирования: Литвинова К. В. Разработка программного комплекса для анализа колебательных и резонансных явлений в механических системах с движущимися границами // Современные проблемы компьютерных и информационных наук : труды международного конгресса, Москва, 20-22 ноября 2025 г. – Москва : МГУ, 2026. – С. 39-46. doi: https://doi.org/10.25559/IT-EDU.OIT.978-5-521-24977-0_6