Advanced asymptotic approaches and perturbation theory methods in the study of the mathematical model of single-frequency oscillations of a nonlinear elastic body

Abstract

A combination of asymptotic methods in nonlinear mechanics with basic techniques of perturbation theory to study a mathematical model of the nonlinear oscillation system is proposed in the paper. The system under consideration describes the torsional vibrations of an elastic body, where its elastic properties are under the nonlinear law. The relationships presented as the ordinary differential equations are obtained due to the proposed procedure. Therefore, the main parameters of the single-frequency oscillations and the resonance conditions can be determined. There are proposed applications of the obtained results to the optimization problem concerning the processing equipment.

Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam

“…where θ 1 is the frequency of the perturbing force that acts on the shaft. Equations ( 14), (15) are similar in structure to equations ( 9), (10). The only difference is to change some coefficients but the very nature of the dynamic process is similar to the longitudinal oscillations of the beam.…”

“…Therefore, all the results for the problem of longitudinal oscillations of prismatic beams can be extended to the problems of torsional oscillations of shafts of the circular cross-section by simply replacing the designations. Therefore, the solution to equation ( 13) is defined similar to ( 9), (10) as follows:…”

“…With the help of the theory of nonlinear oscillations and the asymptotic method used in the current work, it is possible to establish the optimal characteristics of nonlinearly-elastic systems. Employing mathematical models ( 9), (10), and ( 14), (15), it is possible to expand the operating conditions of machines. Our results make it possible to utilize the equipment more efficiently in the event of fluctuations that almost always occur during operation.…”

“…moreover, the parameters a and j for a resonant case are determined by a system of differential equations (10). The law of changing the frequency of longitudinal oscillations of the beam (in accordance with equation ( 9)) is found from the following system:…”

“…Mathematical models of such oscillations are described by differential equations, which differ from linear equations with constant coefficients only by the presence of sufficiently small terms [9]. Therefore, effective in the study of models of such systems are asymptotic methods of nonlinear mechanics [10].…”