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We study some spring mass models for a structure having some unilateral springs of small rigidity ε. We obtain and justify mathematically 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: T ε ∼ 1/ε as usual; or, for a new critical case, we can only expect: T ε ∼ 1/ √ ε. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes"doi:10.1093/imamat/hxq045 fatcat:jaodij5evrh45k7mzwxrx54jgm