Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

Abstract

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.

Study of an electromechanical nonlinear vibration absorber: Design via analytical approach

“…It should be mentioned that in this study we did not take into account possible internal resonances (Nayfeh and Balachandran, 1989) or multimodal interactions of the system (Abdelkefi et al, 2012; Garg and Dwivedy, 2019; Guillot et al, 2019, 2020; Lee et al, 2008). Moreover, the effects of inherent nonlinearities of the piezoelectric patch (Guyomar et al, 1997; Mam et al, 2016; Parashar and Wagner, 2004; Stanton et al, 2010) which follow the large deflection of the beam are also ignored.…”

“…The homogeneous beam is supposed to present a cubic nonlinearity coming from its geometrical nonlinearity activated by large deflection (Pai and Nayfeh, 1990; Silva, 1988). The effects of the nonlinearities (Guillot et al, 2020) of the piezoelectric patch are ignored supposing that the patch follows linear behavior. The circuit possesses a nonlinear capacitance creating a cubic term, an inductance, a resistance and a negative capacitance (Bricault et al, 2019; Silva et al, 2018).…”

Smart structures and materials: Vibration and control

“…On the other hand, Kraus et al (2020) dealt with a planar flexible robot equipped with three degrees of freedom planar active absorbers. Finally, Guillot et al (2020) investigated theoretically and experimentally the 1:3 internal resonance in a cantilever elastically isotropic beam with piezoceramic patches.…”

“…Additionally, 176 full papers were included in the e-book proceedings. The four articles in this issue are representatives of the ‘GS03: Smart systems ’ (Kraus et al, 2020), ‘MS07: Modelling and design of smart composite structures ’ (Araújo and Madeira, 2020) and ‘MS10: Vibration mitigation through electromechanical couplings ’ (Guillot et al, 2020; Høgsberg, 2020). These topics are in line with those of JVC, particularly ‘ Adaptive and smart structures ’, ‘ Vibration and control of structures ’, ‘ Vibration and control of machinery ’, ‘ Vibration absorbers ’, ‘ Structural control ’, and ‘ Structural acoustics ’.…”

On the effects of linear damping on the nonlinear Ziegler’s column

“…The nonlinear dynamics of Ziegler's column, equipped with piezoelectric devices (see, e.g., [41][42][43][44][45][46][47][48]), is also investigated in [49][50][51]: it is shown that, when suitable control strategies are applied (see [52]), the threshold of linear stability can be enlarged, and the limitcycle amplitude can be controlled, also in the presence of nonlinear damping.…”