  This paper deals with a version of the two-timing method which describes
various `slow' effects caused by externally imposed `fast' oscillations. Such
small oscillations are often called \emph{vibrations} and the research area can
be referred as \emph{vibrodynamics}. The governing equations represent a
generic system of first-order ODEs containing a prescribed oscillating velocity
u, given in a general form. Two basic small parameters stand in for the inverse
frequency and the ratio of two time-scales; they appear in equations as regular
perturbations. The proper connections between these parameters yield the
\emph{distinguished limits}, leading to the existence of closed systems of
asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to
demystify) the choices of a slow variable, and (ii) to give a coherent
exposition which is accessible for practical users in applied mathematics,
sciences and engineering. We focus our study on the usually hidden aspects of
the two-timing method such as the \emph{uniqueness or multiplicity of
distinguished limits} and \emph{universal structures of averaged equations}.
The main result is the demonstration that there are two (and only two)
different distinguished limits. The explicit instruction for practically
solving ODEs for different classes of u is presented. The key roles of drift
velocity and the qualitatively new appearance of the linearized equations are
discussed. To illustrate the broadness of our approach, two examples from
mathematical biology are shown.
