English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we analyse the basic ideas of Vibrodynamics and the two-timingmethod. To make our analysis most instructive, we have chosen the Newton'sequation with a general oscillating force. We deal with its asymptoticsolutions in the high frequency limit. Our treatment is simple but general. Thetargets of our study are \emph{the distinguished limits} and \emph{theuniversal vibrogenic force}. The aim of \emph{the distinguished limitprocedure} is to identify how the small parameter can appear in an equation.The proper appearance of a small parameter leads to \emph{valid successiveapproximations}, and, in particular, to closed systems of averaged equations.We show, that there are only two distinguished limits. This means that Newton'sequation, with high-frequency forcing, has two types of interesting asymptoticsolutions. The key item in the averaged equations for all distinguished limitsis \emph{the unique vibrogenic force}. The current state-of-the-art in thisarea is: a large number of particular examples are well-known, effective andadvanced general methods (like the Krylov-Bogolyubov approach) are welldeveloped. However, the presented general and simple analysis of distinguishedlimits and the vibrogenic force, formulated as a compact practical guide, isnovel. An advantage of our treatment is the possibility of its straightforwarduse for various ODEs and PDEs with oscillating coefficients.

Distinguished Limits and Vibrogenic Force revealed by Newton's Equation with Oscillating Force