DYNAMICS OF AN ELASTIC SYSTEM WITH TIME-VARYING PARAMETERS AND A NONLINEAR DAMPING FUNCTION IN THE PRESENCE OF A LOCALIZED EXTERNAL DISTURBANCE

The development of analytical and numerical methods for studying the dynamic characteristics of nonlinear elastic systems is an urgent task in mathematical physics. They are used in various fields, such as the aviation industry, shipbuilding, building construction, mechanical engineering, rocketry, as well as in the production of engines and industrial equipment.The Dirac function serves as a powerful mathematical tool without which many physical disciplines are unimaginable. It is a fundamental concept widely used in the field of physics. The importance of the Dirac δ-function is particularly emphasised in mathematical physics. This applies to such fields as electrodynamics, gas dynamics, optics, acoustics, quantum mechanics, aerodynamics, and hydrodynamics. Therefore, there is a need to create effective hybrid approximate analytical and numerical research algorithms based on advanced asymptotic methods. We are talking about those systems whose parameters undergo changes in time and space under localised external influence, which is mathematically described by the δ-Dirac function. It is important to emphasise that only very special cases allow us to obtain exact analytical solutions for such problems. The latter are usually represented by singular value differential equations with variable coefficients or their systems.Complications also arise due to the presence of parameters near the highest derivative of the equation and the nonlinear component. The analytical- numerical algorithm for solving the problem of nonlinear dynamics of an elastic system with time-varying parameters and a nonlinear damping function taking into account the influence of an external localized disturbance is discussed.The results of calculations using the proposed analytical algorithm with the use of computer algebra are presented. The key points of the analysis are compared with the data of direct numerical integration of the basic differential equation of the problem under study. Using the computer algebra program “Mathematica”, graphs of the calculation results and graphs of comparison with direct numerical integration are constructed.