Multimodal coupling of periodic lattices and application to rod vibration damping with a piezoelectric network

Abstract

Elastic lattice of point masses can be a suitable representation of a continuous rod for the study of longitudinal wave propagation. By extrapolating the classical tuned mass damping strategy, a multimodal tuned mass damper is introduced from the coupling of two lattices having the same modal properties. The aim of the study is then to implement this multimodal control on a rod coupled to an electrical network. The electromechanical analogy applied to a lattice gives the required network and the energy conversion is performed with piezoelectric patches. The coupled problem is modeled by a novel semicontinuous transfer matrix formulation, which is experimentally validated by a setup involving a rod equipped with 20 pairs of piezoelectric patches. The broadband efficiency of the multimodal control is also experimentally proved with vibration reductions up to 25 dB on the four first resonances of the rod. At last, the practical interest of the network is pointed out as it limits the required inductance. This is confirmed by the present purely passive setup that only involves standard low value inductors.

Analysis of Longitudinal Waves in Rod-Type Piezoelectric Phononic Crystals

“…139-142), [30], which considers the Poisson's effect, the displacement field u " ru, v, ws T and the electric potential ϕ in a piezoelectric rod are assumed as u " " upx, tq,´ν 12 Bu Bx y,´ν 13 Bu Bx z…”

“…where u, v and w are the displacements of particle at (x, y, z) in the rod along x, y and z axes, respectively; ν 12 and ν 13 are the Poisson's ratios with respect to y and z axes, respectively; and r¨s T denotes the transposition of a matrix (or vector). The strain vector ε " rε x , ε y , ε z , γ yz , γ zx , γ xy s T and the electric field vector E " rE x , E y , E z s T can then be derived from the generalized strain-displacement relations …”

“…Therefore, rod-type phononic crystals containing smart material components have been presented in order to control the longitudinal waves in an active mode. Their frequency bands can be adjusted by tuning the temperature [6,7], magnetic [8] or electric [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] field to suit the external excitations.…”

“…Among these smart periodic rods, the periodic piezoelectric rods [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] have particularly been paid more attention, because the piezoelectric materials are widely used and the electric field is comparatively easier to control. Thus far, two kinds of rod-type piezoelectric phononic crystals have been presented to actively control longitudinal-wave bands via electromechanical coupling.…”

“…One kind is formed by periodically bonding shunted piezoelectric patches on the host rod, and the other is shaped through periodically arranging the piezoelectric constituent rods [9]. The attenuation of longitudinal vibrations/waves in rods with periodic shunted piezoelectric patches have been studied by Thorp et al [10], Chen et al [11] and Lossouarn et al [12,13] using the transfer matrix method (TMM). Here, we focus on the longitudinal vibrations/waves in periodically arranged piezoelectric rods.…”