The Method of Strained Coordinates for Vibrations with Weak Unilateral Springs

Stéphane Junca 1 Bernard Rousselet 1
1 JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : We study some spring mass models for a structure having a unilateral spring of small rigidity $\epsilon$. 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: $T_{\epsilon} \sim {\epsilon}^{-1}$ as usual; or, for a new critical case, we can only expect: $T_{\epsilon} \sim {\epsilon}^{-1/2}$. We check numerically these results and present a purely numerical algorithm to compute ``Nonlinear Normal Modes'' (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when $\epsilon$ becomes larger.
Keywords : piecewise linear unilateral spring nonlinear vibrations approximate nonlinear normal mode method of strained coordinates
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Stéphane Junca, Bernard Rousselet. The Method of Strained Coordinates for Vibrations with Weak Unilateral Springs. IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2011, 76 (02), pp.251-276. ⟨10.1093/imamat/hxq045 ⟩. ⟨hal-00395351v2⟩

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