Open AccessINVESTIGATION OF THE NORMAL VIBRATION MODES STABILITY IN SOME ESSENTIALLY NONLINEAR SYSTEMS
Author(s) -
Natalia Goloskubova,
Yuri V. Mikhlin
Publication year - 2021
Publication title -
vìsnik nacìonalʹnogo tehnìčnogo unìversitetu "harkìvsʹkij polìtehnìčnij ìnstitut". matematične modelûvannâ v tehnìcì ta tehnologìâh
Language(s) - English
Resource type - Journals
ISSN - 2222-0631
DOI - 10.20998/2222-0631.2021.02.05
Subject(s) - mathematics , instability , nonlinear system , stability (learning theory) , mathematical analysis , lyapunov function , linear stability , singularity , physics , computer science , mechanics , quantum mechanics , machine learning
In the paper stability of nonlinear normal modes is analyzed by two approaches. One of them is the method of Ince algebraization, when a new independent variable associated with the unperturbed solution is introduced in the problem. In this case equations in variations are transformed to equations with singular points. The problem of determination of solutions corresponding to boundaries of the stability/ instability regions is reduced here to the problem of determination of functions that have singularity at the mentioned points. Such solutions can be obtained in the form of power series, which coefficients are satisfying a system of homogeneous linear algebraic equations. The condition ensuring the existence non-trivial solutions for such systems determines the boundaries between the stability / instability regions in the system parameter space. An advantage of the Ince algebraization is that we do not use the time-presentation of the solution when studying its stability. Other approach to investigating steady state stability is associated with the classical Lyapunov definition of stability. The analytical-numerical test proposed in the paper can be applied to a stability problem when the problem has no analytical solution. It also allows to obtain boundaries between the stability / instability regions in the system parameter space. In the present paper the first approach is used to analyze stability of normal vibration modes in the system of connected oscillators on the essentially nonlinear elastic support, and the second one is used to analyze stability of a horizontal vibration mode in the so-called stochastic absorber.