Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting

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Abstract

The paper introduces a multifunctional structural design combining superior mechanical wave filtering properties and energy harvesting capabilities. The proposed concept is based on the ability of most periodic structures to forbid elastic waves from propagating within specific frequency ranges known as phononic bandgaps. The bandgap density and the resulting filtering effect are dramatically enhanced through the introduction of a microstructure consisting of stiff inclusions which resonate at specific frequencies and produce significant strain and energy localization. Energy harvesting is achieved as a result of the conversion of the localized kinetic energy into electrical energy through the piezoelectric effect featured by the material in the microstructure. The idea is illustrated through the application to hexagonal truss-core honeycombs featuring periodically distributed stiff cantilever beams provided with piezoelectric electrodes. The multifunctional capability results from the localized oscillatory phenomena exhibited by the cantilevers for excitations falling in the neighborhood of the bending fundamental frequencies of the beams. This application is of particular interest for advanced aerospace and mechanical engineering applications where distinct capabilities are simultaneously pursued and weight containment represents a critical design constraint. The scalability of the analysis suggests the possibility to miniaturize the design to the microscale for microelectromechanical systems (MEMS) applications such as self-powered microsystems and wireless sensors.

Introduction

The demand for structures and materials with energy harvesting capabilities has grown in recent years in response to the proliferation of portable electronic devices, wireless sensors and microelectromechanical systems (MEMS). Modern microsystems are in fact often designed to be portable or to be operated remotely, and hence self-powered capability is highly desirable. In this paper, a novel class of multifunctional power-generating structures, solids, and devices is proposed based on the localization and conversion of the mechanical energy associated with the vibrational motion of the structural elements to which they are connected. The concept is schematically illustrated in Fig. 1a. The device is intended to work as a multifunctional vibration absorber separating two environments A and B, where it is assumed that A is subjected to externally applied mechanical loads and B needs to be powered and insulated from potentially harmful excitations. A natural benchmark application is represented by the problem of the insulation of sensitive self-powered electronic microsystems from environmental and engine-generated structural vibrations on board of space systems (To et al., 2008).

The proposed multifunctionality is achieved by designing the core of the device as a periodic structure and exploiting the wave propagation characteristics and the internal localization phenomena that are typical of this class of systems. A peculiar feature of periodic structures is represented by the phenomenon of phononic bandgaps. Bandgaps are defined as frequency intervals where elastic waves are forbidden from propagating as a result of wave interference due to the impedance mismatch generated by periodic geometric or material discontinuities. The existence of bandgaps makes periodic structures and materials extremely appealing as mechanical filters (Yang and Chen, 2008). Recent work has focused on the introduction of resonating microstructures to enhance the filtering properties of periodic domains. For example, Martinsson and Movchan (2003) have found that a microstructure of concentrated masses connected to the primary structure of a lattice by soft links leads to an increase in the number of bandgaps at low frequencies. Similar work has been done on epoxy matrices with stiff metallic inclusions (Jensen, 2003) and on sonic composite materials with spherical inclusions (Liu et al., 2000; Hirsekorn et al., 2006). These works suggest the possibility to apply the concept to the design of advanced material systems with superior mechanical properties.

In this work, we propose a class of multifunctional periodic assemblies containing microstructures featuring layers of piezoelectric material. The existence of a phononic bandgap in the band structure implies the availability of flat regions in the propagation modes immediately below and above the gap itself, in which the wave group velocity goes to zero. In these regions, the vibrational kinetic energy localizes in the form of an oscillatory motion of the internal structural elements, rather than being transferred across the material as propagating waves. In other words, the substructures behave as wave dampers and dynamic energy absorbers. The idea is to exploit the piezoelectric effect featured by the material in the substructures to convert into electrical energy the kinetic energy that localizes in the resonators for frequencies of excitation falling in the neighborhood of the bandgaps. By virtue of the energy localization and the resulting high strain fields, the microstructural deformation is ideal for maximizing the energy conversion effect. This criterion offers an additional motivation, besides the improvement of the global filtering properties, for maximizing the bandgap density in the phonon spectrum.

Multiple designs can be proposed based on locally resonant lattice structures and materials with different internal configurations. Two possible implementations are schematically depicted in Fig. 1b. System I is a “truss-core” hexagonal honeycomb lattice structure with a microstructure consisting of periodically distributed stiff piezoelectric cantilevers. The label “truss-core” typically refers to sandwich beams and plates with cores featuring hexagonal topologies developed across the thickness, as opposed to conventional honeycomb sandwich layouts (Ruzzene, 2004). An alternative configuration achieving similar effects is represented by system II in Fig. 1b. The design consists of a fiber-reinforced composite featuring a matrix of soft material and long periodically distributed stiff fibers coated with layers of highly deformable energy generating material, such as piezoelectric polymer, with considerably lower stiffness than that of the core and the matrix. In the remaining of the paper, configuration I is selected and detailed to illustrate the proposed idea.

The analysis of a full design scenario for the proposed honeycomb would result in a formidable engineering problem including manufacturing considerations that require to be addressed in successive stages. The work in this paper is limited to the analysis of the dynamic behavior of the system, with emphasis on the deformation mechanisms that are responsible for the bandgap phenomena, and the estimated power that can be harvested. The paper is organized in six sections including this introduction. Section 2 illustrates the proposed honeycomb configuration and the resulting band structure through unit cell analysis. Section 3 investigates the deformation mechanisms that are responsible for the bandgap generation phenomena and provides design suggestions. In Section 4 a harmonic analysis of finite structures is carried out to assess the transmissibility of the system and investigate the stress distribution. Section 5 is devoted to explain the energy harvesting effect and quantify the output power. Section 6 finally summarizes the main results of the work and provides recommendations for future investigations.

Section snippets

Modified honeycomb geometries for improved filtering performance

Most periodic lattices feature phononic bandgaps between high-frequency propagation modes. However, mechanical excitations are generally characterized by relatively low frequencies and therefore the design of mechanical filters based on lattice topologies is optimized when additional bandgaps are obtained in the low-frequency regime. One way to achieve this result consists of introducing microstructures featuring vibrational modes corresponding to certain types of standing waves in the global

Mechanisms of deformation and selection of parameters

The objective of this section is to investigate the deformation mechanism induced in the lattice by the resonating beams and verify the assumption that a link exists between the existence of stop bands in the band structure and the localized bending deformation of the cantilevers. Let us consider the case δ=2 as a reference solution. This configuration is representative of cantilever beams whose top and bottom surfaces are instrumented with layers of piezoelectric material having thickness h/2.

Harmonic response and filtering properties of finite lattices

Let us investigate the behavior of a finite lattice made of aluminum (Al,E=7.1×1010N/m2, ν=0.3, ρ=2.7×103kg/m3) consisting of the assembly of (20×20) unit cells of the kind depicted in Fig. 2b, with L=1m, β=115, γ=12 and δ=2. The goal of this analysis is twofold: first to verify the predictions about the filtering behavior made through unit cell analysis, second to observe the bending deformation taking place in the microstructure at the local resonance frequencies. The bottom side of the

Analysis of the power generation capability

Energy harvesting can be achieved as a byproduct of the deformation in the lattice microstructure. The power generation in the cantilevers can be estimated with a simplified method, commonly applied to study the piezoelectric effect in beams and plates, which allows calculating the voltage and power outputs for the electric circuit associated with each piezoelectric electrode through simple postprocessing of the calculated sectional stresses (see schematic in Fig. 15a). The method provides a

Concluding remarks

A multifunctional structure has been proposed based on a regular hexagonal honeycomb with piezoelectric microstructures. A finite element dynamic analysis has been carried out to show that the changes in the design enhance the filtering properties of the structure as a result of the onset of localized resonance phenomena. It has been proposed to employ the localized deformation pattern achieved in the frequency neighborhood of the bandgaps in order to harvest a certain amount of the kinetic

Acknowledgment

The support of this research by the National Science Foundation (NSF) is gratefully acknowledged.

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