Distinguished Limits and Vibrogenic Forces for the Newton's Equation with Oscillating Forces}}

In this paper, we discuss the basic ideas of Vibrodynamics and the two-timing method with the use of Newton's equation containing a general time-periodic force. We deal with asymptotic solutions in the high frequency limit. Accordingly, our small parameter is the dimensionless inverse frequency or the amplitude of oscillations. Our treatment is simple but general. The central targets are to analyse \emph{the distinguished limits} and to demonstrate the appearance of \emph{the universal vibrogenic force}. The aim of the distinguished limit procedure is to identify how the small parameter can appear in a given equation. This consideration leads to a \emph{valid procedure of successive approximations}, and then to obtaining related systems of averaged equations. We show, that there are at most two distinguished limits. This means that Newton's equation with periodic forcing has not more than two types of asymptotic solutions. The key term in these solutions is the vibrogenic force, which appears in the averaged equation for both distinguished limits. The current state-of-the-art in this area is: a large number of particular examples are well-known, and complex general methods (like the Krylov-Bogolyubov approach) are well developed. However, the presented general and simple analysis, formulated as a compact practical guide, is novel. An advantage of our treatment is the possibility of its straightforward use for various PDEs. This paper appears as a result of many years of teaching PhD students. We have deliberately chosen the most known equation (Newton's), and the most known example of high frequency forcing (a pendulum with oscillating pivot) for making our presentation most instructive.