The method of strained coordinates for vibrations with weak unilateral springs

We study some spring mass models for a structure having a unilateral spring of small rigidity ε. We obtain and justify an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: Tε ∼ ε−1 as usual; or, for a new critical case, we can only expect: Tε ∼ ε−1/2. We check numerically these results and present a purely numerical algorithm to compute “Non linear Normal Modes ” (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when becomes larger