Asymptotic solution of the mathematical model of nonlinear oscillations of absolutely elastic inextensible weightless string

Abstract

The mathematical model of nonlinear oscillations of weightless string is analyzed. To find an asymptotic solution of the problem, uniformly valid in a long interval of time, an averaged system of integral differential equations has been constructed. A method for constructing special approximations of its solutions is proposed.

Uniformly Valid Asymptotics for Carrier’s Mathematical Model of String Oscillations

“…In our paper, for the definite equations (1.3), we have applied the general method of averaging along characteristics the weakly nonlinear hyperbolic systems. The constructed averaging system allows finding uniformly valid asymptotic approximations of a polynomial form by applying the methods of our earlier work [17]. On the other hand, the theoretical analysis of the obtained averaged system (see [18,19]) allows determining the conditions of appearing combinatoric resonances.…”

Weak resonances of periodical nonlinear oscillations

“…Prie tokio pavidalo uždavinio gali būti pertvarkyta netiesinės stygos svyravimų lygtis, taikant mažojo parametro metodą [2] ir paliekant lygtyje eilės O(ε) ir O(ε 2 ) narius [2]. Taikymuose, tai sudaro dviejų vienmačių netiesiškai sąveikaujančių laukų sistemą.…”