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Edited by: Ettore Barbieri, Queen Mary University of London, UK

Reviewed by: Francesco Dal Corso, University of Trento, Italy; James Bowden, University of Nottingham, UK

Specialty section: This article was submitted to Mechanics of Materials, a section of the journal Frontiers in Materials

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The purpose of this work is to present recent advances in modeling and design of piezoelectric energy harvesters, in the framework of micro-electro-mechanical systems (MEMS). More specifically, the case of inertial energy harvesting is considered, in the sense that the kinetic energy due to environmental vibration is transformed into electrical energy by means of piezoelectric transduction. The execution of numerical analyses is greatly important in order to predict the actual behavior of MEMS devices and to carry out the optimization process. In the common practice, the results are obtained by means of burdensome 3D finite element analyses (FEA). The case of beams could be treated by applying 1D models, which can enormously reduce the computational burden with obvious benefits in the case of repeated analyses. Unfortunately, the presence of piezoelectric coupling may entail some serious issues in view of its intrinsically three-dimensional behavior. In this paper, a refined, yet simple, model is proposed with the objective of retaining the Euler–Bernoulli beam model, with the inclusion of effects connected to the actual three-dimensional shape of the device. The proposed model is adopted to evaluate the performances of realistic harvesters, both in the case of harmonic excitation and for impulsive loads.

The application of piezoelectric materials is continuously increasing, with different possible uses of both direct (conversion of mechanical into electric energy) and inverse effects. The latter is applied in actuators, e.g., in the case of micropumps (Ardito et al.,

In a previous paper (Gafforelli et al.,

The theoretical model is founded on an enrichment of the Euler–Bernoulli kinematic field, with additional strain contributions which aims at introducing some three-dimensional effects: the obtained model is denoted as modified transverse deformation (MTD). The modification is based on the width-to-length ratio and is driven by some considerations on the 3D behavior of piezoelectric films. Such an enriched model is unprecedented, to the author’s knowledge, and allows the user to obtain reliable results in a negligible time.

In this paper, the model is applied to some specific examples referred to inertial energy harvesters, which are usually represented by cantilever beams (see Figure

The paper is organized in this way: the proposed model is thoroughly described in Section

Linear piezoelectric constitutive equations are a combination of classical Hooke’s law, employed in continuum mechanics, and standard linear constitutive relation between electric and strain fields. The herein adopted notation is customary in the theory of piezoelectricity (IEEE Standard on Piezoelectricity,

The peculiarity of piezoelectricity is the third order coupling tensor _{mij}

33-mode: when applying an electric field along the polarization axis, the piezoelectric element stretches in this same direction (and vice versa);

31-mode: when applying an electric field along the polarization axis, the piezoelectric element shrinks in the orthogonal plane (and vice versa);

Shear mode: when applying an electric field orthogonal to the polarization axis, a shear occurs in the element plane (and vice versa).

According to the physics of piezoelectric coupling, only few coupling constants are non-zero. Indeed, _{333} is related to mode (a), _{311} and _{322} are related to mode (b), while _{113} and _{223} are related to mode (c). Voigt’s notation is usually employed to represent the constitutive equations. It is, therefore, possible to use vectors for representing second-order symmetric tensors and matrices for representing third and fourth order tensors. In this way, the piezoelectric coupling matrix

In the isotropic case, the complete expression of piezoelectric constitutive law reads

In the previous equations, the following symbols are used: _{ij}_{ij}_{j}_{j}_{hk}

It is worth noting that the MEMS energy harvesters are based on piezoelectric

The cantilever harvester is a piezolaminated beam clamped at one end section and free to oscillate on the other side. The dynamic response of the laminated beam can be easily modeled by employing standard beam theories for laminated composites. Herein, classical lamination theory (CLT) (Ballhause et al., _{1} axis lies along the beam’s length, so that it ranges between 0 and _{2} axis is along the beam’s width and ranges between −_{3} axis is across the beam’s thickness _{1}) only. In this paper, we consider a suitable modification to obtain the modified transverse deformation (MTD) model. The final model, in terms of the displacement components _{j}

The additional functions

The chosen model should fulfill the standard requirements for a beam theory: the in-the-thickness stress must be null, _{33} = 0 and the in-plane stress must be _{22} = 0 at _{2} = ±_{22} = 0 must be verified (limit situation of null transverse deformation, NTD); on the other hand, when Λ → ∞, the beam is extremely narrow and _{22} = 0 has to be guaranteed (limit situation of null transverse stress, NTS). These features can be obtained if the strain component _{22} is modified through the introduction of a “transversal shape function” _{Λ}:
_{Λ} and _{Λ}:

The modified transverse strain is obtained starting from the expression of _{22} in the NTS case; such an expression is then multiplied by the function _{Λ}. In that way, if Λ → ∞, then _{Λ} → 1 and the NTS situation is recovered; if Λ → 0, then _{Λ} → 0 and _{22} → 0, that correspond to the NTD case. The modified transverse strain, for isotropic piezoelectric material, reads

After some algebraic manipulations (see Gafforelli et al. (

Strains and displacements are linked via compatibility equations, and the kinematics of the beam as proposed in equation (_{13} is rigorously null on the whole beam; _{23} is proportional to the thickness-to-length ratio, which is very small, so its value can be neglected; _{12} is not equal to zero, but it has null mean on the cross-section since the “transversal shape function” _{Λ} is an even function with respect to _{2}. It is possible to conclude that the proposed model involves no shear contributions in the stiffness matrix, which governs the beam’s deformation.

The governing equations for the piezoelectric problem can be obtained by using the dissipative form of Euler–Lagrange equations:

The Lagrangian coordinates _{i}_{Λ} (which depends on _{2}, along the width).

An approximate solution is sought in the framework of the Rayleigh–Ritz procedure. First, the displacement field is expressed on the basis of a single time-variant parameter _{w}

Second, the electric potential is assumed to be linear across the thickness _{p}

The Lagrangian coordinates to be used in equation (

The governing equations are finally obtained

The coefficients are obtained by means of integration through the beam’s volume: _{M}_{E}

The final system of equations reads

The external force _{ext}_{ext}

The numerical validation of the proposed model has been obtained by means of critical comparisons with a 3D finite element (FE) model, which has been developed with the commercial code ABAQUS. In a preliminary stage, some simple static analyses have been used for the calibration of the parameters _{Λ} and _{Λ}; afterward, the validity of MTD hypotheses has been checked with reference to quasi-static and dynamic analyses. A simple cantilever has been adopted, with 2 layers only (2-μm PZT on 6-μm polysilicon substrate), and no tip mass is introduced. The beam’s length is

The complete validation campaign is described in Gafforelli et al. (_{r}_{M}_{M}

The analyses have been repeated by changing the resistance _{M}_{M}_{M}

The FE model has been used also in order to explore the validity of linear kinematics. In fact, the MTD model is based on the basic hypothesis of small strain and displacement. On the other hand, the FE analyses have been repeated after switching on the geometric non-linearity (finite strain and displacement). In that way, it has been possible to set the deformation limit until which the linear analysis can be adopted with a reasonable degree of approximation. This aspect will be treated in the next section.

Once validated, the model of the cantilever beam can be employed for the characterization and evaluation of the performances of cantilever harvesters. Parametric analyses with different geometrical features have been performed in order to analyze the influence of the beam’s length and of the piezoelectric layer’s thickness on the harvester response.

The laminate beam is composed of several layers, as summarized in Table _{M}

_{0} = 8.854 × 10^{–12} F/m

^{3}) |
_{31} (N/Vm) |
_{33} (N/Vm) |
||||||
---|---|---|---|---|---|---|---|---|

SiO_{2} |
Passivation | 0.30 | 2.33 | 70 | 0.27 | – | – | – |

Ru | Electrode | 0.10 | 4.50 | 447 | 0.30 | – | – | – |

PZT | Active | 0.5–2 | 7.70 | 100 | 0.30 | −12 | 20 | 2000 |

Pt | Electrode | 0.12 | 21.45 | 180 | 0.30 | – | – | – |

SiO_{2} |
Passivation | 0.62 | 2.33 | 70 | 0.27 | – | – | – |

Poly-Si | Structural | 5.00 | 2.33 | 148 | 0.33 | – | – | – |

SiO_{2} |
Passivation | 0.50 | 2.33 | 70 | 0.27 | – | – | – |

The amplitude is ^{2}), and the frequency is equal to the natural frequency of the device. The parametric analyses are intended to show the influence of the beam’s geometry (length and thickness) on the response of the harvester in terms of maximum displacement and voltage in open circuit conditions; maximum power generation is at the optimal resistance.

Figure _{P}_{max}_{max}_{P}_{P}

_{M}

_{M}

_{M}

These results are qualitatively and quantitatively in agreement with literature results on cantilever harvesters with dimensions similar to the ones considered herein (Muralt et al.,

It is worth reminding that the previous analyses have been done fixing the mechanical quality factor to 500. From a general point of view, the quality factor of an energy harvester should be as high as possible in order to improve the performances of the system, because high _{M}_{M}

_{M}

Cantilever piezoelectric beams are not only used as resonant energy harvesters but are also part of non-linear harvesters that consider jump phenomena or implement frequency-up-conversion. From a general point of view, such devices implement techniques that impulsively stimulate the piezoelectric beams, which then execute the energy conversion.

At this stage, the focus is more on the evaluation of the performances of the piezoelectric beam rather than the design of frequency conversion techniques. Similar to what has been done with resonant cantilever beams, parametric analyses have been performed in order to highlight the influence of the beam’s geometry on the conversion of energy under impulsive solicitations. The parametric study has been performed using the same cantilever beam used for resonant harvesters, but in this case, no seismic mass is provided at the free edge of the beam. The mechanical quality factor is again _{M}_{max}_{opt}_{E}ω_{r}^{−1}. It is worth reminding that the maximum peak power would be obtained at a value, which is roughly 1.7 times _{opt}

The peak displacement (Figure

The duration of the oscillation (_{osc}

Figure _{P}_{P}

The present paper is focused on parametric analyses of realistic MEMS PEHs, with the main purpose of pointing out the most important features to be considered in the design phase. The analyses have been carried out in the time domain (step-by-step analyses), and the computational burden has been substantially reduced by using the MTD model. In that way, a simple one-d.o.f. can be adopted without any loss in terms of accuracy. The proposed model can be improved by the introduction of non-linear kinematics, in order to simulate also the cases of large oscillations. Moreover, the design procedure of real PEH deserves a more detailed study of mechanical damping and of other possible external circuits (e.g., RLC circuits, non-linear electric behavior, etc.).

Two possible operation modes of PEHs have been considered. Power generation in resonant harvesters is directly proportional to the beam’s width and length, but a non-monotonic dependence on the PZT thickness has been highlighted. The peak power can reach reasonable levels, but that happens only in resonance conditions, with non-negligible displacements and high excitation frequency. For instance, resonant PEHs with _{P}_{max}_{r}

On the other hand, the cantilever piezoelectric beam shows good performances when impulsively solicited, possibly in conjunction with a FupC device. Remarkable peak power generation can be obtained (more than 25 μW in the example of Figure

RA – supervisor of reduced order modeling. AC – supervisor of the whole research activity. GG – developed the 1D model and FE simulations. CV – supervisor of the industrial applications. FP – developed the model for industrial applications. RZ – supervisor of the funding project.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors wish to thank the ENIAC Joint Undertaking, Key Enabling Technology, project Lab4MEMS, Grant no. 325622, for partial funding of this research.

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_{3}thin films by pulsed laser deposition