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International audience; Experimental and theoretical results on the nonlinear dynamics of a homogeneous thin beam equipped with piezoelectric patches, presenting internal resonances, are provided. Two configurations are considered: a unimorph configuration composed of a beam with a single piezoelectric patch and a bimorph configuration with two collocated piezoelectric patches symmetrically glued on the two faces of the beam. The natural frequencies and mode shapes are measured and compared with those obtained by theoretical developments. Ratios of frequencies highlight the realization of 1:2 and 1:3 internal resonances, for both configurations, depending on the position of the piezoelectric patches on the length of the beam. Focusing on the 1:3 internal resonance, the governing equations are solved via a numerical harmonic balance method to find the periodic solutions of the system under harmonic forcing. A homodyne detection method is used experimentally to extract the harmonics of the measured vibration signals, on both configurations, and exchanges of energy between the modes in the 1:3 internal resonance are observed. A qualitative agreement is obtained with the model.

The chapter presents modal responses and nonlinear interactions of a multi-physics nonlinear beam. It is composed of a piezoelectric material which is patched on a two-dimensional nonlinear Euler–Bernoulli beam. The spatio-temporal variables of governing equations of the composite beam are separated. Traced frequencies and mode shapes of the overall beam show modifications of its modal response due to the piezoelectric patch and its position on the beam. Studying the system in time domain via a multiple scale technique, leads to detection of its responses as function of frequency of excitation which present strongly nonlinear response due to nonlinearity of the beam and also presence of the piezoelectric patch.