# Piezoelectric Energy Harvesting from Low-Frequency Vibrations Based on Magnetic Plucking and Indirect Impacts

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Piezoelectric Energy Harvester

_{p}of the layer; moreover, in view of the presence of a unique electrode along the beam’s axis, the voltage does not depend on the x coordinate and is constant along that axis. The voltage $V\left(t\right)$ at the electrode is the time-dependent degree of freedom. By applying the Euler–Lagrangian approach it is possible to obtain the governing differential system of ordinary differential equations (ODEs). Furthermore, the dynamics of the moving mass is described by the Newton’s second law applied to a rigid body subjected to an external acceleration, a magnetic force, and to the gravity field if the mechanism is placed to work in the vertical direction.

_{m}is the linear damping coefficient, ${k}_{l}$ is the linear elastic stiffness, m

_{z}is the activated mass [29] and ${C}_{e}$ is the capacitance equivalent to piezoelectric layers. $\theta $ is the coefficient related to the multiphysics coupling due to the piezoelectricity. The external force is, in this case, the sum of inertial forces due to vibrational input, the magnetic interaction force, the impact force and the gravity load in case the device works in the vertical direction. $\ddot{Z}$ is the input acceleration on the device, ${f}_{g}$ and ${f}_{G}$ are the gravity loads of the discretized cantilever and moving mass, respectively. ${f}_{mag}$ is the plucking magnetic force and a is the resulting acceleration due to Newton’s law on the moving mass and M is the moving mass. In system (1), the time rate of electrical charge is obtained under the hypotheses of the connection with a resistive load R, as common in energy harvesting investigations [1,4].

## 3. Design and Fabrication

^{3}in which there are two main slots used to host, respectively, the piezoelectric cantilever beam and the moving mass. Between the two slots, a free zone is used to set a gap distance between the magnets of less than 1.0 mm. The top and lateral faces of the device have been covered with a plexiglass plate, as shown in Figure 5a,b. The moving mass is a cube of 13 mm side length and with a mass of 0.017 kg. It is made of non-magnetic steel AISI 316 with the aim of not affecting the magnetic field provided by the equipped magnet. The Neodymium magnets are cubic with a 3 mm side length and with a magnetization of 1.32 T. The piezoelectric cantilever beam is a commercial bimorph element RS 285-784, RS Components

^{®}with the active layers connected in series. Its features are summarized in Table 1: the titanium shim is enclosed in two PZT layers. Considering the clamped zone of the piezoelectric beam, the effective length is 10.5 mm. A copper foil on FR4 has been glued on the bottom face of the device orthogonal to the guide direction to weld the electrical connections.

## 4. Experimental Results

#### 4.1. Free Vibration of the Piezoelectric Beam

_{0}equal to 1 MΩ and a capacitance C

_{0}of 14 pF. Figure 8a shows the voltage across the resistor as a function of time in case of a single indirect impact in the absence of moving mass in the slot. In this case, the indirect impact is generated by an external mechanical impulse imparted to the package. In other tests, the impulse is induced by the magnetic plucking, in attractive and repulsive configurations: the mass is pushed and forced to move in the slot, whose bottom is removed. Figure 8b reports the FFTs for all the aforementioned cases. The first eigenfrequency of the cantilever is 659.2 Hz, 665.2 Hz, and 668.3 Hz for repulsive, no magnetic interaction and attractive cases, respectively. Through a MATLAB

^{®}program, in which the modelling of the piezoelectric cantilever has been implemented [35], an eigenfrequency of 672.5 Hz is obtained without interactions. This value is 1% larger than the experimental one, because of the discretization approach.

_{M}= 13.72 is obtained without the magnetic interaction.

#### 4.2. Experimental Tests in Open Circuit

#### 4.3. Experimental Tests with a Connected Circuit

_{1}= 1 μF, as reported in Figure 13.

_{0}= 20 pF in parallel with a resistor R

_{0}> 200 TΩ. R

_{p}is the internal resistance of the piezoelectric layer, generally neglected in the modelling [4]. The triaxial accelerometer and the device have been tied to the wrist of a person. Tests have been performed both in case of shaking and running activities. The shaking has been performed mainly in the direction of the guide (z axis in Figure 3) for activating the mechanism. For the running activity, instead, the motion guide has been oriented orthogonally to the axis radius of the arm as in Figure 7b. In Figure 14 and Figure 15, typical accelerograms of these events are represented with also the Fast Fourier Transforms (FFTs), for shaking and running, respectively. In the graphs, the z-component is parallel to the motion axis of the mass. The time histories of acceleration have been recorded for each different test and they are comparable in terms of the amplitude of acceleration and in frequency content. In general, the acceleration related to shaking is larger than the one for running; moreover, the dominant frequency for shaking is around 6 Hz, whereas in the case of running, the dominant frequency is around 3 Hz.

_{1}is the storage capacitance and v

_{c}(t) is the measured voltage. Figure 16a shows v

_{c}(t) as a function of time in case of shaking while Figure 16b shows the corresponding energy harvested computed through Equation (2). In this case, the magnetic interaction improves the scavenged energy (blue and black curves) in comparison to the harvester with the presence of indirect impacts only (red curve). The repulsive configuration is more promising in terms of scavenged electrical energy, at parity of experimental conditions, compared to the attractive and indirect impacts configurations, since 253.41 μJ, 70.32 μJ and 37.35 μJ have been harvested, respectively. For the case of running activity, represented in Figure 17, the behavior is reversed. Within 40 s of operation, the system recovers 0.61 μJ, 2.47 μJ, and 4.30 μJ, in the cases of repulsive, attractive, and with only indirect impacts, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Illustration of the magnetic plucking mechanism: (

**a**) the mass approaches the cantilever, (

**b**) high frequency vibrations of the piezoelectric cantilever after the snap in case of attractive interaction.

**Figure 3.**Qualitative total potential energy (TPE) function with respect to tip displacement W, for two versions of the system in Equation (1): (

**a**) attractive configuration; (

**b**) repulsive configuration with inter-well (1) and intra-well (2) mechanisms.

**Figure 4.**Illustration of the package of the piezoelectric energy harvester designed with Solidworks©.

**Figure 5.**Assembled prototype (

**a**) lateral and (

**b**) top view. (

**c**) Detail of the arrangement of the permanent magnet inside the moving mass.

**Figure 7.**(

**a**) Close-up of the rectifier circuit board. (

**b**) Accelerometer and prototype tied to the wrist.

**Figure 8.**(

**a**) Measured voltage across R = 100 kΩ due to free oscillations in case of a single indirect impact in absence of moving mass in the slot. (

**b**) FFT responses of the open circuit voltage with a close-up on [0.2,1] kHz.

**Figure 9.**Open-circuit voltage in case of (

**a**) repulsive and (

**b**) attractive plucking for a single snap.

**Figure 10.**Open-circuit voltage for repulsive interaction combined with impacts (

**a**) single snap (

**b**) shaking with multiple snaps with indication of motion of mass (M) and impact phases.

**Figure 11.**Voltage response in case of (

**a**) linear system (

**b**) repulsive and indirect impacts mechanisms with a load resistance of R = 100 kΩ.

**Figure 12.**Close-up on voltage responses in case of load resistance, R = 100 kΩ (

**a**) repulsive case (

**b**) attractive case. MP—magnetic plucking, II—indirect impact.

**Figure 16.**(

**a**) voltage and (

**b**) energy harvested in case of shaking with capacitive circuit (C

_{1}= 1 µF).

**Figure 17.**(

**a**) voltage and (

**b**) energy harvested in case of running with capacitive circuit (C

_{1}= 1 µF).

**Figure 18.**Working cycles in the case of low-level input acceleration (such as for the running activity): (

**a**) repulsive interaction; (

**b**) attractive interaction.

Material | ρ [kg/m^{3}] | E [GPa] | ν [-] | d_{31} [pC/N] | ε_{33}^{s} [-] | t [mm] | Width [mm] | Length [mm] |
---|---|---|---|---|---|---|---|---|

Titanium | 4500 | 115 | 0.3 | - | - | 0.065 | 1.5 | 15 |

PZT | 7500 | 60 | 0.3 | 212 | 2000 | 0.280 per layer | 1.5 | 15 |

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**MDPI and ACS Style**

Rosso, M.; Nastro, A.; Baù, M.; Ferrari, M.; Ferrari, V.; Corigliano, A.; Ardito, R. Piezoelectric Energy Harvesting from Low-Frequency Vibrations Based on Magnetic Plucking and Indirect Impacts. *Sensors* **2022**, *22*, 5911.
https://doi.org/10.3390/s22155911

**AMA Style**

Rosso M, Nastro A, Baù M, Ferrari M, Ferrari V, Corigliano A, Ardito R. Piezoelectric Energy Harvesting from Low-Frequency Vibrations Based on Magnetic Plucking and Indirect Impacts. *Sensors*. 2022; 22(15):5911.
https://doi.org/10.3390/s22155911

**Chicago/Turabian Style**

Rosso, Michele, Alessandro Nastro, Marco Baù, Marco Ferrari, Vittorio Ferrari, Alberto Corigliano, and Raffaele Ardito. 2022. "Piezoelectric Energy Harvesting from Low-Frequency Vibrations Based on Magnetic Plucking and Indirect Impacts" *Sensors* 22, no. 15: 5911.
https://doi.org/10.3390/s22155911