Nonsmooth modal analysis: from the discrete to the continuous settings

International audienceThis chapter addresses the prediction of vibratory resonances in nonsmooth structural systems via Nonsmooth Modal Analysis. Nonsmoothness in the trajectories is induced by unilateral contact conditions in the governing (in)equations. Semi-analytical and numerical state-of-the-art solution methods are detailed. The significance of nonsmooth modal analysis is illustrated in simplified one-dimensional space semi-discrete and continuous frameworks whose theoretical and numerical discrepancies are explained. This contribution establishes clear evidence of correlation between periodically forced and autonomous unilaterally constrained oscillators. It is also shown that strategies using semi-discretization in space are not suitable for nonsmooth modal analysis. The spectrum of vibration exhibits an intricate network of backbone curves with no parallel in nonlinear smooth systems

On the Cohomology of Invariant Variational Bicomplexes

Let Pi = E rarr M be a fiber bundle and let Gamma be an infinitesimal Lie transformation group acting onE. We announce various new results concerning the cohomology of the Gamma invariant variational bicomplex (OHgr Gamma *,* (Jinfin(E)), dH, dV) and the associated Gamma invariant Euler-Lagrange complex. As one application of our general theory, we completely solve the local invariant inverse problem of the calculus of variations for finite-dimensional infinitesimal Lie transformation groups