This paper reviews recent developments related to oscillatory systems, their impact on and relationship to the cases of smooth, essentially anharmonic (non-linearizable) potentials, and vice versa. Special methods of treatment that allow the response regimes in dynamic vibro-impact systems to be computed have been discussed. Mathematical models that approximate purely elastic as well as inelastic impact, with the help of smooth functions, are presented and illustrated by specific examples. The use of ideas based on non-smooth time transforms to treat essentially non-linear systems with smooth potentials has also been discussed. Special attention has been paid to uncommon applications of vibro-impact models.
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