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Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting Stefano Gonella a, Albert C. To b,c, Wing Kam Liu a, a b c

Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

a r t i c l e in fo

abstract

Article history: Received 15 July 2008 Received in revised form 29 October 2008 Accepted 9 November 2008

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. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Energy harvesting Microstructure Phononic bandgaps Vibration Piezoelectric effect

1. 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.

Corresponding author. Tel.: +18474917094; fax: +1 847 491 3915.

E-mail address: [email protected] (W.K. Liu). 0022-5096/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2008.11.002

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Fig. 1. Multifunctional design: (a) application as a vibration absorber and (b) internal conﬁgurations.

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

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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. 2. 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 lattice (Martinsson and Movchan, 2003). The idea is based on the principle of dynamic damping: in a dynamic damper, the suppression of vibrations in a structure (primary structure) is achieved through the transfer of kinetic energy to an auxiliary structure, which is connected to the primary and oscillates at its natural frequency. Based on these considerations, a variation of standard honeycomb geometries is proposed by introducing the conﬁguration of Fig. 2a, consisting of the standard unit cell, or representative volume element (RVE), of a regular hexagonal honeycomb (internal angle y ¼ 30 ) with three additional cantilever beams directed along the directions deﬁned by the following angles: 90 , 210 and 330 . Let L0 be the length of each cantilever, where L0 ¼ gL

(1)

and L, h denote the length of the other beams in the unit cell and the beam thickness, respectively. For the remaining of the paper, it will be assumed that g ¼ 12, unless otherwise speciﬁed. The wave propagation characteristics of a periodic structure can be obtained by means of a simple unit cell analysis through the application of periodic propagation conditions, according to Bloch theorem. A ﬁnite element formulation of the wave propagation problem and a discussion of the dispersion relations for classical honeycomb geometries are given in Gonella and Ruzzene (2008a), where the standard terminology used in honeycomb mechanics is also recalled. Applying the technique to the structure of Fig. 2a yields the band structure depicted in Fig. 3a, where the frequency axis is normalized with respect to the ﬂexural resonance of a simply supported beam of length L sﬃﬃﬃﬃﬃﬃﬃ p2 EI w1 ¼ 2 (2) rA L where E; r are the properties of the selected material and A; I the wall cross sectional area and inertia of the lattice beams. A comparison with the band structure of a regular hexagonal honeycomb immediately reveals that the addition of the cantilevers is responsible for profound changes in the dynamic behavior of the structure. In the range O 2 ½5 10, for instance, the density of stop bands is dramatically increased, as different mechanisms of deformation give rise to almost ﬂat propagation modes. From the standpoint of the ﬁltering behavior, this alone represents a considerable improvement in the design. A more subtle change in the band structure is observed at lower frequencies, where a narrow bandgap appears between the third and fourth mode, as shown in the magniﬁed view of Fig. 3b. The latter phenomenon can be linked to the local oscillatory bending deformation of the cantilevers that represents the foundation for the use of the honeycomb as an energy harvesting device. Being shorter than the beams in the primary structure, the cantilevers have higher stiffness and act as stiff inclusions for the periodic lattice. Their bending natural frequencies can be evaluated analytically as functions of the geometric and material properties as sﬃﬃﬃﬃﬃﬃ b 4E 2 ðkn LÞ (3) OnC ðdÞ ¼ 3r w1 L

Fig. 2. Unit cell for honeycomb lattice with enhanced properties: (a) unit cell with cantilevers and (b) unit cell with stiffened cantilevers.

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Fig. 4. Schematic of a cantilever beam with layers of piezoelectric material.

where kn L is a set of non-dimensional modal parameters for clamp-free beams (Hodges and Pierce, 2002) and b is the beam slenderness ratio. Eq. (3) yields O1C ¼ 1:425, which indeed falls inside the ﬁrst bandgap. Let us consider now the unit cell of Fig. 2b. In this conﬁguration, the stiffness of the cantilevers is further increased by 0 bringing the thickness to a new value h related to the original thickness as 0

h ¼ dh

(4)

such that d is an indirect measure of the stiffening effect. The motivation for this change in the unit cell geometry is twofold. From an energy harvesting standpoint, the additional thickness models the layers of piezoelectric material that need to be introduced in the design to achieve energy conversion, according to the schematic of Fig. 4. From a vibration ﬁltering viewpoint, as it has been observed that the beams behave as a resonant microstructure, it seems reasonable to enhance this effect by increasing the stiffness and the inertia of the resonators. It is important to establish a precise relation between d and the entity of the induced bandgap phenomena, in order to determine whether a value of d exists such that the ﬁltering performance of the structure is maximized. Fig. 5 shows the evolution of the lattice band structure for d ¼ 1:4 and 1.8 in the frequency interval O 2 ½0 5. It can be noticed that the width of the lowest bandgap, which is related to the oscillatory phenomena in the cantilevers, grows with d and its location shifts towards higher frequencies. The dependency of the position of the gap upon the stiffening factor follows from the expression of the fundamental natural frequency of the cantilevers, which can be shown to grow linearly with d according to the expression sﬃﬃﬃﬃﬃﬃ db 4E 2 ðk1 LÞ (5) O1C ðdÞ ¼ 3r w1 L The impact of the increased stiffening goes beyond the location of the lowest bandgap, as the whole band structure appears to be deeply modiﬁed. In particular, it can be noticed that the slope of the non-dispersive part of the acoustic modes decreases with d. It is possible to provide an explanation for this phenomenon based on the knowledge of the homogenized properties of the honeycomb. Regular hexagonal lattices are in fact known to feature directional behavior at higher frequencies while being isotropic in the low-frequency long-wavelength limit (Srikantha Phani et al., 2006). In this frequency neighborhood, their behavior can be described by homogenized partial differential equations which assume the form of the classical elastodynamic equations for a two-dimensional homogeneous isotropic solid under plane stress conditions (Gonella and Ruzzene, 2008b). Two wave velocities can be estimated in terms of the lattice homogenized

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Fig. 5. Band structure for hexagonal honeycomb with stiffened cantilevers: (a) d ¼ 1:4 and (b) d ¼ 1:8.

parameters as sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K þ G cP ¼ cS ¼

sﬃﬃﬃﬃﬃﬃ G

r

r

(6)

where K , G and r are the effective bulk modulus, shear modulus and density of the lattice. These quantities correspond to the slopes of the acoustic modes in the band structure. Because the addition of the cantilevers preserves the angular symmetry of the unit cell, the isotropy of the lattice is also conserved and the expressions of Eq. (6) still hold. K , G are equivalent static elastic properties of the lattice and are not affected by the auxiliary structure as the cantilever beams do not alter the elastic connection between neighboring cells. On the contrary, the effective density r ¼ rr¯ grows with d, as the relative density r¯ depends on the additional mass introduced by the cantilevers for the same volume occupied by the RVE, and, as shown by Gonella and Ruzzene (2008b), can be expressed as pﬃﬃﬃ 1 3 1þ d b r¯ ¼ 2 (7) 2 3 The expression of r¯ for a generic d and the reference one for standard hexagonal honeycombs, denoted by ðÞHEX and obtained by setting d ¼ 0 in Eq. (7), are ﬁnally used to construct the quantity sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cP cS 1 (8) w¼ ¼ ¼ cPHEX cSHEX 1 þ 12 d which provides a precise metric of the reduction in slope of the acoustic modes as a function of d. 3. 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 d ¼ 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. For metallic beams made of aluminum (Al; E ¼ 7:1 1010 N=m2 , n ¼ 0:3, r ¼ 2:7 103 kg=m3 ), it is fair to assume that the mechanical properties of the piezoelectric layers are not too far from those of the core. Under this 0 assumption, the whole layered cantilever can be modelled as a beam of thickness h ¼ dh ¼ 2h with the same cross sectional material properties of its core. In accordance to the quantity O1C calculated in Eq. (5), the band structure shown in Fig. 6a reveals the presence of a signiﬁcant bandgap in the O 2 ½2:4 2:9 interval. The deformation mechanism occurring in the lattice in the frequency neighborhood of the ﬁrst bandgap can be inferred by looking at the propagation mode shapes relative to the dispersion branches located immediately below and above the bandgap (see dots in Fig. 6a). The deformed shapes are shown in Fig. 7

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Fig. 6. Comparison of bandgaps in the low-frequency region for (a) d ¼ 2 and (b) d ¼ 4.

Fig. 7. Mode shapes for the 6th and 7th mode for d ¼ 2: (a) VI mode—Point O; (b) VI mode—Point A; (c) VI mode—Point B; (d) VII mode—Point O; (e) VII mode—Point A and (f) VII mode—Point B.

and reveal considerable deformations of the cantilever beams. Moreover, the cantilevers feature ﬁrst-mode-like deformed shapes, in agreement with the fact that the selected frequencies fall in the neighborhood of the ﬁrst fundamental frequency of a clamp-free beam with equal geometric and material properties. From an energy harvesting perspective, this deformation pattern is ideal, as the performance of beam-like piezoelectric devices is maximized when the deformed shape is such that the highest stress value is observed at the root of the beam. The dependency of bandgap location and size upon d presents a tradeoff. While higher values of d cause an undesired shift towards higher frequencies, they are responsible for wider gaps, as a consequence of the higher inertia of the 0 cantilevers. In order to maximize the ﬁltering effect, one could be tempted to privilege a design with h bh, as suggested by the inspection of Fig. 6b, obtained for d ¼ 4, which features a large bandgap between the 6th and 7th mode. However, a choice based on this criterion alone would compromise the power generation capability of the honeycomb. Analysis of the mode shapes for frequencies falling in the neighborhood of the bandgap (Figs. 8a–f corresponding to the squares in Fig. 6b) shows in fact that, for d ¼ 4, a different deformation mechanism takes place in the structure. Most of the energy is conﬁned in the primary structure, while negligible deformation is observed in the cantilevers, as a consequence of their increased stiffness and inertia. When d becomes very large, each cluster of stiff cantilevers behaves as a rigid structure which either 0 undergoes rigid motion or does not displace at all. In the limit, for h bh, the stiffened nodes become internal clamps and the original structure can be seen as an assembly of clamped frames of the kind shown in Fig. 9a. Accordingly, the band structure asymptotically tends to the limit case of Fig. 9b, which features a single region of forbidden propagation, interrupted by ﬂat modes. The modes are characterized by frequencies coinciding with the natural frequencies of the frame of Fig. 9a listed in Table 1 and corresponding to the deformed shapes of Fig. 10. Since the ﬂat modes are associated with

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Fig. 8. Mode shapes for the 6th, 7th and 9th mode for d ¼ 4: (a) VI mode—Point O; (b) VI mode—Point A; (c) VI mode—Point B; (d) VII mode—Point O; (e) VII mode—Point A; (f) VII mode—Point B; (g) IX mode—Point O; (h) IX mode—Point A and (i) IX mode—Point B.

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Table 1 First four natural frequencies of a clamped reverse-Y-shaped frame. nMODE

n¼1

n¼2

n¼3

n¼4

On FRAME

1.5347

2.1638

4.8350

5.4260

standing waves conﬁned between sets of internal clamps, they have constant frequency along the contour of the Brillouin zone and therefore do not feature any dependency upon the direction of propagation. In this scenario, the lattice features an outstanding ﬁltering performance, as the rigid joints act as internal barriers preventing waves from propagating through

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Fig. 10. First four mode shapes of a clamped reversed-Y-shaped frame: (a) I mode; (b) II mode; (c) III mode and (d) IV mode.

the structure. However, this comes at the expense of the compliance of the piezoelectric cantilevers and therefore of the energy harvesting capability of the design. The choice of the optimal value for d needs to be done in light of a clear tradeoff between the efﬁciency of the honeycomb as a damper on one side and energy harvesting and weight considerations on the other. As a side note, it is worth noticing that, for d ¼ 4, resonance of the cantilevers is still achieved when the frequency of excitation matches the fundamental frequency of the stiffened beams. This, however, happens far from the frequency range of interest, producing the narrow bandgap located between the 9th and 10th mode, as conﬁrmed by the deformed shapes relative to the 9th propagation mode plotted in Figs. 8g–i, corresponding to the dots in Fig. 6b, which again feature signiﬁcant bending motion of the cantilevers. 4. Harmonic response and ﬁltering properties of ﬁnite lattices Let us investigate the behavior of a ﬁnite lattice made of aluminum (Al; E ¼ 7:1 1010 N=m2 , n ¼ 0:3, r ¼ 2:7 103 kg=m3 ) consisting of the assembly of (20 20) unit cells of the kind depicted in Fig. 2b, with L ¼ 1 m, 1 b ¼ 15 , g ¼ 12 and d ¼ 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 primary structure is constrained to simulate a clamp, while the top side is subject to a pressure load modelled as an array of vertical forces applied to the nodes of the free surface. Let us consider a harmonic excitation of the form F ¼ F 0 eiot

(9)

3

where F 0i ¼ 10 N on each loaded node, and a frequency sweep O 2 ½0 5. The ability of the device to suppress vibrations can be quantiﬁed by evaluating the force transmissibility f deﬁned as

f¼

Ft F0

(10)

where F 0 and F t are the mean magnitudes of the nodal forces applied on the top surface and those transmitted to the constraints, respectively. The transmissibility is plotted versus the normalized excitation frequency in Fig. 11a and compared to the corresponding band structure (Fig. 11b) to highlight the agreement between unit cell and global analysis. It can be seen that the transmissibility plot conﬁrms the distribution of bandgaps obtained with a unit cell analysis. Let us focus our attention on the distribution of deformation and stress in the lattice under a harmonic load with frequency O ¼ 2:4, corresponding to the onset of the ﬁrst bandgap for d ¼ 2. The global deformation of the lattice is shown in Fig. 12a. According to the notion of bandgap, it can be observed that the primary structure does not undergo large deformation, as most of the nodal displacement is conﬁned to the region neighboring the nodes where the load is applied. Moreover, the transmissibility drops to a mere f ¼ 103 , showing that mechanical damping is effectively achieved. On the other hand, a careful inspection of Fig. 12a reveals the presence of regions in the lattice where the stiff cantilevers undergo signiﬁcant deformation. Figs. 13a and b show a detail of the deformed lattice corresponding to the area inscribed in the rectangle highlighted in Fig. 12a, where the magnitude of the localized deformation is particularly signiﬁcant. An ampliﬁcation factor 108 is applied to allow proper visualization of the deformation pattern occurring in the microstructure. It can be noticed that the cantilevers undergo huge bending motion, while the hexagons of the primary structure are marginally deformed. Fig. 13b shows the stress distribution in the cantilevers undergoing large bending motion. Similar calculations are performed for a second frequency of interest (O ¼ 2:98), located immediately above the upper boundary of the ﬁrst bandgap. The results are shown in Figs. 12b, 14a and b. Considerable ﬁltering effect is achieved (f ¼ 1:5 102 ) and the deformation pattern in the cantilevers feature again high ﬁrst bending mode deﬂections (Fig. 14a) leading to high stress values at the root of the beams (Fig. 14b). 5. 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

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Fig. 12. Nodal displacement (color intensity is proportional to displacement): (a) O ¼ 2:4 and (b) O ¼ 2:98.

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 ﬁrst estimate of the power that can be generated by devices at different length scales and under different loading conditions. Fig. 15b shows the equivalent electric circuit for a piezoelectric generator, where subscripts ð ÞS and ð ÞL , respectively, refer to the source, which is modelled as a series of the resistance RS and the capacitance C S , and the load, which is characterized by the impedance Z L . For a beam whose free surface is coated with a layer of piezoelectric material, the voltage on the poling surfaces of the electrode is given by V ¼ g 31 t p s11max

(11)

where g 31 is the piezoelectric voltage constant for the selected material, t p is the thickness of the electrode and s11max is the axial stress calculated on the free surface and averaged along the entire length of the beam. The label s11max is chosen as a remainder of the fact that the cross sectional stress for a beam in bending varies linearly with the distance from the neutral axis, as shown in Fig. 15a, and is thus largest at the free surface. It has been shown (Rizzoni, 2001) that the power generated

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Fig. 13. Magniﬁed view of region with extreme localized deformation (O ¼ 2:4): (a) distribution of displacement (m) and (b) distribution of axial stress (N=m2 ).

Fig. 14. Magniﬁed view of region with extreme localized deformation (O ¼ 2:98): (a) distribution of displacement (m) and (b) distribution of axial stress (N=m2 ).

Fig. 15. Voltage output estimation of a piezoelectric beam: (a) cross sectional stress distribution for a beam in bending and (b) equivalent electric circuit for a piezoelectric generator.

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by a vibrating beam can be expressed, for a purely resistive circuit, as P¼

2 V˜ RL

(12)

ðRS þ RL Þ2

where V˜ is the root mean square (RMS) source voltage value in phasor notation, while RL and RS are the load and source resistances, respectively. Clearly, while the voltage only depends upon the material properties of the electrode and the stress state in the deformed beam, the power is also a function of the parameters of the equivalent electric circuit. Inspection of Eq. (12) reveals that maximum power is achieved when RL ¼ RS , yielding 2

Pmax ¼

V˜ 4RS

(13)

The efﬁciency of the proposed power generation device relies on the possibility to store a certain amount of energy from each piezoelectric beam in the lattice. This calls for the need to design a smart way to wire the beams and construct the appropriate electric circuit. The resulting challenges at the manufacturing level can have a huge impact on the ﬁnal design of the device and will constitute the bulk of future research investigations. It is hereby assumed that each electrode can be wired independently so that the total power Ptot generated by the device is simply calculated as the sum of the maximum power values P max produced by each piezoelectric cantilever. Let us consider the hexagonal honeycomb of Fig. 12 with characteristic beam length L ¼ 250 mm and stiffening factor d ¼ 2, under the same boundary conditions previously used for the stress analysis. The selected parameters correspond to a rectangular structure of size 7:5 mm 8:7 mm and resulting occupied area A ¼ 64:95 mm2 , which are typical values for a large MEMS device. Let us consider a harmonic load applied on the top nodes characterized by F 0 ¼ 25 N and nondimensional frequency O ¼ 2:37, corresponding to a point in the band structure immediately below the onset of the ﬁrst bandgap. The piezoelectric constant g 31 ¼ 9:5 103 V m=N and the resistances RS ¼ RL ¼ 330 000 O complete the set of parameters. The effects of mechanical damping on the efﬁciency of the energy harvesting process can be severe, as damping is expected to signiﬁcantly reduce the amplitude of the resonance oscillations in the microstructure, and need to be accounted for in the simulation. Proportional damping can be easily introduced in the model by assuming a complex Young’s modulus E ¼ Eð1 þ iZÞ for all beam elements in the structure, where E is the nominal modulus and Z is the loss factor. Normal values for Z suggested for slender and compliant structural elements like aluminum beams are 104 . However, in the case of frame structures, higher factors (typically Z 102 ) are commonly considered to account for the internal joints. For the presented honeycomb conﬁguration, Z 103 is likely to represent sufﬁcient damping, since the beams that are responsible for the power generation are cantilevers and therefore less affected by the damping associated with the internal clamps. Tables 2 and 3 show the power generated Ptot and the power density, deﬁned as the power generated per unit area, P A ¼ P tot =A, under increasing damping, for O ¼ 2:37 and 2:98. The two frequencies correspond to a point in the band structure immediately below and immediately above the ﬁrst bandgap, respectively. The power predictions are in line with the power outputs of existing power-generating devices for microsystems based on the piezoelectric effect (Wang and Song, 2006). It is also important to check whether the stress states in the deformed beam elements preserve the structural integrity of the lattice. It is found that, in the limit scenario of undamped systems, the maximum Von Mises stresses for O ¼ 2:37 and 2:98 are sVMmax ¼ 3322 and 15 381 psi, respectively, both lower than the value s ¼ 20 000 psi typically indicated as the upper limit for beams with rectangular cross section subjected to shear tip loads. Table 2 Estimated power output—O ¼ 2:37.

Z

Power output (mW)

Power density (mW=mm2 ) 3:5 101

Undamped

22:8

104

16:0

2:5 101

103

2:1

3:2 102

102

0:4

5:7 103

Table 3 Estimated power output—O ¼ 2:98.

Z

Power output (mW)

Power density (mW=mm2 )

Undamped

769:9

1:2 101

104

587:6

9:0 100

103

115:0

1:7 100

102

10:8

1:7 101

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S. Gonella et al. / J. Mech. Phys. Solids 57 (2009) 621–633

1010

1010

10

0

10−5 Ω=2.37

10

1 μm

Ω=2.98

1 μm

10 F0 [N]

10 N 100 N 1000 N F0

F0

10−5 10−10

1 mm 1m

2

1000 N Ω=2.98

100

100 N

1 mm 1m

−10

10−15 105

Ω=2.37

105

L PA [μW / mm2]

PA [μW / mm2]

105

10 N

L

103

10−15 10−6

10−4

10−2

100

L [m]

Fig. 16. Parametric analysis of power density: (a) P A versus F 0 and (b) PA versus L.

Finally, a parametric analysis is conducted to investigate the dependency of the power upon the magnitude F 0 of the applied excitation and the characteristic dimension L of the honeycomb unit cell. It is observed that the power density grows quadratically with F 0 , i.e. P A / F 20 , as a result of the fact that the maximum stress s in the cantilevers varies linearly with F 0 . Interestingly, the stress distribution observed in the resonant cantilevers is the same showcased by a single clampfree beam under harmonic tip shear force with magnitude F 0 . The effect of L is also interesting. It is observed that, if the number of unit cells in the lattice is ﬁxed, the total power P tot does not change with L as a result of the fact that the strain in the beams is conserved. However, because larger values of L produce structures with a larger surface area, the power density PA decreases with L as the number of stiff cantilevers per unit area decreases. The last consideration suggests that this design is especially efﬁcient for small scale applications such as MEMS. These results are summarized in Figs. 16a and b for the two frequencies O ¼ 2:37 and 2:98, where F 0 2 ½10 103 N and L 2 ½106 1 m in logarithmic scales. 6. 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 energy available in the oscillating members. It has been shown that the deﬂection of the piezoelectric cantilevers can lead to the generation of an amount of power that is sufﬁcient to power micromechanical systems. The elements of novelty and the strengths of the proposed device that have been highlighted in the analysis can be summarized in three main points: (1) While most existing energy harvesting devices exploit the interaction of piezoelectric elements (microcantilevers, nanowires) with external agents, such as ﬂuid ﬂows or mechanical probes, the proposed device relies on internal mechanisms that are triggered by the natural exposure of the system to external vibrations. (2) The device minimizes the risk of damaged components by shocks and friction, as the active microstructure is embedded and shielded from the external environment. The increased robustness has important consequences on the lifespan of the system. (3) The system operates in a fail-safe mode. If the cantilevers fail, the structural integrity is not compromised as the structure still performs as a lightweight truss-core honeycomb with satisfactory load bearing capabilities. Future research efforts will be directed towards the implementation of a fully coupled electromechanical analysis of the device. As a result of the electromechanical coupling arising in the system, the actual mechanical behavior of the lattice could be considerably different from the one obtained through simple mechanical analysis. The inverse piezoelectric effect can in fact be responsible for non-negligible deformations induced in the structure by the electric ﬁeld established in the piezoelectric material (Li et al., 2007; Wang et al., 2008). In this sense, one of the goals is to explore the possibility to control the bandgap distribution of the structure by electrically driving the piezoelectric microstructure, in order to achieve a tunable semi-active ﬁlter. A complete coupled problem can only be considered once the cantilevers are explicitly modelled as layered piezoelectric beams and enough knowledge becomes available of the load and the circuitry. This detailed analysis, in which all the beam layers are modelled independently, will also remove the approximation resulting

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from the assumption, implicitly made throughout this work, that the piezoelectric electrodes can be treated as layers of elastic material. It is instead known that a piezoelectric material in a closed electrical circuit shows inelastic behavior. This simpliﬁcation might be responsible for signiﬁcant changes in the predicted properties of the structure. All these issues will be addressed when a more precise conﬁguration of the device (including the associated electric circuit), a clear application and the manufacturing constraints are clearly identiﬁed. On the experimental side, the main challenge will consist of ﬁnding a rational and efﬁcient way to provide each cantilever in the lattice with its own electric circuit without increasing the weight of the device or jeopardizing the robustness of the honeycomb. This issue is expected to become particularly challenging when the honeycomb is miniaturized for applications as a multifunctional micromechanical ﬁlter for MEMS. In this scenario substantial design modiﬁcations might become necessary to face potential difﬁculties at the manufacturing stage.

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