A method to enhance the nonlinear magnetic plucking for vibration energy harvesters

Vibration-based energy harvesting (VEH) consists in collecting kinetic energy from mechanical vibrations of the surroundings and converting it into electrical energy. In the last two decades, VEH has drawn attention because it is a valuable option to extend the durability of batteries for portable electronics and, possibly, to introduce self-powering devices. The proposed energy conversion mechanisms are based on piezoelectric materials [1], electromagnetic generators [2], and electrostatic interaction [3]. In particular, piezoelectric harvesters have received the most attention because they allow for a direct conversion from kinetic to electrical energy, leading to larger power densities and a more straightforward integration in micro-power sources [4, 5]. In recent years, researchers have been exploring the performance of biocompatible, non-toxic piezoelectric materials in order to develop environmentally friendly applications [6].

Typical vibration sources cover a frequency bandwidth that goes from few Hz (human walking or running) to hundreds of Hz (domestic and industrial machinery) [7]. Even though several groups have proposed to work at resonance with environmental vibrations [8], this is not the most viable option from the perspective of electromechanical conversion efficiency. Indeed, the energy harvested at low frequencies (1–100 Hz) is not enough to power microsystems [9]; moreover, the frequency of ambient vibrations can vary significantly over time.

A strategy to overcome these limitations is the frequency up-conversion mechanism [9‐11]: low-frequency environmental vibrations are converted into high-frequency vibrations of the harvester, which can work at their resonance frequency to optimize the electromechanical conversion efficiency. Several frequency up-converting devices are based on the plucking mechanism, which consists in the deflection and the release of the harvester, e.g. a piezoelectric cantilever, to excite a high-frequency vibration. The plucking mechanism is obtained either by mechanical contact or by magnetic interaction. In the first case, a piezoelectric beam is put in free oscillations when plucked by a plectrum [12], by notches on a moving structure [13] or by a sliding mass [14]. In the other case, a permanent magnet (PM) deflects and releases a magnetized harvester through contactless magnetic interaction; both repulsive and attractive configurations have been studied [15, 16], as well as both rotational [17‐19] and traslational mechanisms [20, 21]. The advantage of magnetic plucking is the absence of contact on the brittle piezoelectric material; indeed, damages due to wear worsen the conversion efficiency of the piezoelectric beam [22]. However, long-range magnetic interaction could hinder a proper release phase, which ideally should be as sharp as possible to excite high-frequency vibrations of the harvester; for this reason, very high plucking velocities are needed to maximize the efficiency [22]. Therefore, this aspect can represent a significant operational limitation for harvesters in the context of low-speed applications (less than 1 m/s), such as those related to human motion. Although, as previously described, the application of magnetic plucking has been widely exploited, the possibility of enhancing the frequency up-conversion process through the engineering of the magnetic force to make the interaction more impulsive has not been explored.

The aim of this paper is to propose a novel technique for manipulating magnetic plucking, and thus to improve its impulsiveness. Our idea is to shield one or both permanent magnets with other magnets of opposite polarity, in order to alternate attractive and repulsive force regimes, and sharpening the release step. The detailed description of our idea is presented in the Sect. 2 of this article. Subsequently, in Sect. 3, we provide an analytical model, numerical simulations to highlight the advantage of the proposed technique, and experimental validation using a custom setup. In Sect. 4, we apply the technique to energy harvesting, simulating a meso-scale case study. We show that, by properly adjusting the thickness of the shielding, a significant amplification in the plucking dynamics (i.e., displacement-velocity portrait of the harvester) can be achieved. Closing remarks are provided in Sect. 5.

The basic idea for magnetic field manipulation is to exploit a device called magnetic flux concentrator (MFC) appropriately placed in the source region of the magnetic field. The MFC is essentially a block of suitably shaped soft ferromagnetic material. The principle of operation is based on the high magnetic permeability of such a material, which forces the flux lines in the surrounding space to follow the path indicated by its own geometry. The field lines are then free to expand into spatial regions where the permeability remains low, giving rise to an engineered pattern of the magnetic field. An illustration of this principle is shown in Fig. 1a with reference to an inductor for electromagnetic heat treatments. Researchers also used this principle recently, to improve the gap-to-gap heating of wind power gear [23], to control heating cycles at high temperatures [24] or in hot stamping steel sheets [25]. Gao et al. [26] in 2016 focused on studying the effect of geometric size on concentrator performance. In the MEMS field generally, flux concentrators are instead used to increase sensitivity and to redirect the field in Hall effect sensors. A study of general application on the material and shape was made in 2013 by Sun et al. [27]. Maspero et al. [28] recently made a MEMS magnetometer with flux concentrators. As an alternative to the flux concentrator, it is possible to use the patterned deposition of magnetic material [29] via the high-rate deposition triode sputtering technique.

In this work, a similar concept to the MFC is applied, with a specific application to frequency up-conversion via magnetic plucking in piezoelectric vibration energy harvesters. The idea is similar to manipulate the flux lines of the magnetic field generated by the permanent magnets in the harvesting system using additional material around them, as explained in detail below. The motivation, as previously mentioned, lies in the fact that it is not always possible to have an impulsive phenomenon on the harvester in the operational context of low-frequencies and velocities (e.g. human motion). The MFCs are applied to an energy harvester consisting of two dynamical systems: a bimorph piezoelectric cantilever beam and a moving mass along a predefined path transversal to the beam’s longitudinal axis, as will be shown in Sect. 4. Both the beam and the mass are equipped with cuboidal permanent magnets, enabling reciprocal interaction through magnetic force. This interaction occurs as the moving mass gets close to the cantilever tip through a low-frequency input vibration in a real frequency up-conversion system. If the velocity of the interaction between the PMs is very low, the plucking of the harvester does not occur due to the long-range nature of the magnetic force [30], and only a quasi-static phenomenon is observed without significant energy conversion. The main goal of this study is to increase the impulsiveness of the magnetic interaction by sharpening the magnetic force between the equipped magnets with respect to their relative distance. To achieve it, the four sides of the so-called main PMs parallel to the magnetization direction (4 over 6 faces in a parallelepiped case) are covered with additional PMs having an inverted polarization, as depicted in Fig. 1b. The main difference between MFCs and PMs is that MFCs are soft materials that become magnetized with respect to the magnetic field to which they are subjected. PMs, on the other hand, have a fixed magnetization and for high coercive fields, are not affected by an external field. The choice of using PMs to carry out the shielding derives from preliminary analyses in which greater efficiency was noted in the shaping of the plucking compared to the case of the soft material. Such a fact is due to the high capability of the PM to mantain constant the magnetization. It is also shown that by covering both or just one of the main PMs the technique still maintains its validity. The case with only one shielded magnet is studied both for completeness of investigation and because its effectiveness can be highly useful in scenarios where the same harvester is to be used, without altering the original tip mass, in different contexts of velocities and acceleration. In such a scenario, the magnetization arrangement on the moving magnet can be changed in the device without directly intervening on the harvester, thus avoiding the risk of damaging. In the Fig. 1c, d, the contour plot of the magnetic field and the magnetic field lines are shown for the unshielded and shielded cases, respectively, based on our finite element analyses. As it can be seen, for the shielded magnet (Fig. 1d) the magnetic field is more localized around its geometry, in comparison with the unshielded magnet (Fig. 1c). All the PMs in the systems are made of Neodymium-Boron-Iron alloy (NdFeB). These magnets have a higher magnetic energy density compared to ferrite magnets, allowing for the use of very small magnets, which are useful for creating highly compact systems. Additionally, their magnetic field intensity ensures sufficiently high interaction forces. The advantages have favored their selection since the early studies in this field [31].

The concept of magnetic shielding is now applied to the nonlinear dynamics of a piezoelectric vibration energy harvester. The scheme to be considered is represented in Fig. 10. A piezoelectric bimorph with a tip magnet is triggered through magnetic interaction by a moving mass also equipped with a magnet. The mass is externally driven, at constant velocity \({\bar{v}}\). The nonlinear constitutive model with consideration of elastic hysteresis and nonlinear coupling proposed in [36] is assumed for the piezoelectric material. Ferroelectric hysteresis, on the other hand, is not considered since typically in energy harvesting, the electric field in the piezoelectric layers is well below the coercive value. The focus of the following simulations is on the effect that the technique can provide on the nonlinear dynamic response of the oscillator. The configuration is assumed repulsive between the main (central) magnets since it provides a bistable system, more interesting for energy harvesting [33].

A lumped parameter model is adopted, as suggested in [33, 35]. The transducer is modeled with linearized kinematics, being typically very stiff, and the Euler-Bernoulli beam theory is adopted. Only two degrees of freedom (dofs) are assumed: U for the displacement of the cantilever tip and V for the voltage across the electrodes. A resistive load R is supposed to be connected to the harvester, and the piezoelectric layers are connected in series. The nonlinear second-order differential system governing the problem is the following:The coefficients of the system (2) are computed as integral over the volume of the structure. m is the modal mass of the first mode, \(b_1\), \(k_1\) \(\theta _1\) are the linear damping, stiffness, and coupling coefficients, respectively. The coefficients \(b_2\), \(k_2\), and \(\theta _2\) are the nonlinear ones. C is the effective capacitance of the transducer. Extensive details on the modeling calculations can be found in [36] and [37]. \(F_{mag}\) is the magnetic force whose variation is caused by the relative motion between the moving mass and the cantilever tip. However, since the mass is supposed to be driven externally at a constant velocity, its dynamics does not appear in the system (2). This system has been implemented in a MATLAB^® program and is integrated with the ODE23s solver [38]. To perform numerical simulations, the same data used in [37] are assumed for the materials. All physical and geometrical characteristics are summarized in Table 1.

In the following simulations, a gap distance between the magnets equal to 0.5 mm is assumed. Two magnetic force patterns are considered:With this layout, it is possible to compare fairly the performance, using the same amount of magnetic material and having identical frequency of vibration. Three cases of electric circuits have been considered: the arbitrary value of R=100 k\(\Omega\) just to have a comparison also in terms of power, short circuit (R=0), and open circuit (\(R\longrightarrow \infty\)). In the short circuit, the effect of the resistor is zero and it can be seen that the technique works on a purely mechanical oscillator. In the same way, in the open circuit, it can be seen that the technique works as well even when the stiffening effect provided by the resistor is maximum [39]. Figures 11, 12, 13 show the phase portrait of the nonlinear response for the three aforementioned circuit conditions, in case of a single plucking with \({\bar{v}}=0.8\) m/s (reasonable for human motion), without (a) and with (b) the shielding. The benefit of the technique in terms of vibration is clear: the shielded cases describe much larger orbits, in terms of displacements and velocities, which in the shielded case is even higher than the unshielded case. With a resistance of R=100 k\(\Omega\) and a shielded magnet (Fig. 11b), the orbits are clearly discernible owing to the further damping effect of the electrical circuit. In cases of short circuit (Fig. 12) and open circuit (Fig. 12), the system experiences less damping, resulting in a significantly denser appearance of the orbits, which remain larger compared to the unshielded scenario. Figure 14 shows the different dynamics over time. Figure 15 shows the trend of the simulated instantaneous power for \(R=100\,\hbox {k}\Omega\), in case of absent (a) or present (b) shielding at the same input velocity. With the proposed solution, the results show a gain in terms of peak power of about 17 times. The technique has also been simulated for velocities lower than 0.8 m/s, within the range of 0.2–0.7 m/s, which are pertinent to human motion applications. Figure 16 illustrates the comparison of phase portraits between the unshielded harvester (a) and the shielded one (b). The observations made at 0.8 m/s (Figs. 11, 12, 13, 14, 15) are further corroborated at lower velocities. Specifically, at 0.2 m/s, both cases exhibit a quasi-static regime, characterized by the absence of the structural mode in the dynamic response. However, a higher voltage is attained in the shielded case, attributed to the sharper force-distance curve, although not reported here for conciseness. At 0.3 m/s, the unshielded harvester remains quasi-static, while the shielded case displays high-frequency vibrations, compared to the unshielded counterpart. At 0.4 m/s, even the unshielded harvester initiates oscillations, albeit with minimal displacements, velocities, and voltages when compared with the shielded case. This trend persists across velocities of 0.5 m/s, 0.6 m/s, and 0.7 m/s. It is important to emphasize that the space-dependent force, with assigned fixed velocity, illustrates the concept of the operational principle, through a realistic example of piezoelectric harvester. Since the moving magnet is externally driven, further motion laws, such as harmonic or band-limited colored noise, would not change the conceptual outcome of this research. The parameter influencing the dynamic response is the interaction velocity [37]. For completeness, Table 2 summarizes the simulated performance results with system (2) across the entire velocity range considered (0.2–0.8 m/s). The table allows for a comparison of the results for the cases with and without shielding in terms of RMS voltage (\(V_{RMS}\)), peak power (\(P_p\)), and average power (\(P_m\)). The latter is calculated using Joule’s law across the resistor, with the RMS voltage computed from a single plucking of the beam (\(P_m=V_{RMS}^2/R\)). Peak power is particularly useful for illustrating how the impulsiveness of the phenomenon increases with shielding as velocity increases, from approximately 5.5 times for the slowest case of 0.2 m/s up to about 17 times for the fastest case of 0.8 m/s. Across the entire velocity range considered, the more impulsive dynamics of the shielding case consistently ensure better performance compared to the unshielded case.

In this work, a new technique in the field of magnetic frequency up-conversion of energy harvesters (or oscillators in general) has been presented. Experiments and simulations have shown that the magnetic force between permanent magnets can be manipulated in the space domain. This manipulation occurs by shielding one or more main magnets with additional magnetic material polarized in the opposite direction to the main magnet. With the proposed technique, it is therefore possible to realize dynamic forcing with an intentionally designed impulsiveness. The technique is highly useful where there is a need for an efficient frequency up-conversion in a low-velocity context (e.g. < 1 m/s), as in the case of human motion. A computational study carried out on a bimorph piezoelectric harvester, has shown that for a relative velocity between the magnets of 0.8 m/s with a 100 k\(\Omega\) resistor, there is a 17x gain in terms of peak power for the shielded case. In the considered velocity range, the shielding technique consistently yields superior performance in terms of power compared to the case without shielding. Moreover, the flexibility of the technique extends its applicability also in sensing and actuation context.