  The purpose of the present work is to show that an adequate basis for
understanding the essentially nonlinear phenomena must also be essentially
nonlinear however still simple enough to play the role of a basis. It is shown
that such types of 'elementary' nonlinear models can be revealed by tracking
the hidden links between analytical tools of analyses and subgroups of the
rigid-body motions or, in other terms, rigid Euclidean transformation. While
the subgroup of rotations is linked with linear and weakly nonlinear
vibrations, the translations with reflections can be viewed as a geometrical
core of the strongly nonlinear dynamics associated with the so-called
vibro-impact behaviors. It is shown that the corresponding analytical approach
develops through non-smooth temporal substitutions generated by the impact
models.
