  The purpose of this paper is to prove ergodicity and provide asymptotic
formulae for probabilities of threshold crossing related to smooth
approximations of three fundamental nonlinear mechanical models: (a) an
elasto-plastic oscillator, (b) an oscillator with dry friction, (c) an
oscillator constrained by an obstacle (one sided or two sided) and subject to
impacts, all three in presence of white or colored noise. Relying on a
groundbreaking result on density estimates for degenerate diffusions by Delarue
and Menozzi (2010), we identify Lyapunov functions that satisfy appropriate
conditions leading to ergodicity (invariant measure and Poisson equation) and a
functional central limit theorem. These conditions appear in the very
fundamental works of Down, Meyn and Tweedie (1995) and Glynn and Meyn (1996).
From an applied mathematics perspective, an important consequence is the access
to asymptotic formulae for quantities of interest in engineering and science.
