Distinguished limit and two-timing method, shown by Newton's Equation

In this paper, we use the Newton's equation with a time-oscillating force for the demonstration of basic ideas of Vibrodynamics and two-timing method. Our treatment is simple but general. The central items are the use of the \emph{distinguished limit} and the universal appearance of a \emph{vibrogenic force}. The aim of the distinguished limit procedure is to established a closed system of averaged equations. The number of distinguished limits and related averaged equations is infinite, however the difference between them can be described in general terms. Their common part is the vibrogenic force, which (for different distinguished limits) appears in the averaged equations of different orders. The current state-of-the-art in this area is: a number of particular examples are well-known, but the presented general and simple/clarifying treatment is novel.