**1. Introduction**

The term energy harvesting is the process of capturing or harvesting wasted or unused ambient energy, such as temperature gradients, mechanical vibrations, radio frequencies, etc., and converting it into useable electrical energy. This differs from powering a system using traditional energy sources such as batteries, fuel cells, etc. Energy harvesting has received much attention in the last two decades from various disciplines, mainly due to its potential impact as a key technology enabling autonomous ultra-low-power electronics operating in remote and harsh environments without the need for large energy storage elements. Energy harvesting thus replaces conventional batteries, enhances the environmental friendliness of the system, and lowers the maintenance costs. In other words, it is a promising technology to power various applications ranging from structural health monitoring to medical implants.

Energy harvesting from mechanical sources including industrial machines, human activity, vehicles, building structures, and other environmental sources can be achieved through different conversion methods. Such approaches include piezoelectric, electromagnetic, electrostatic, and

magnetostrictive e ffects. Among the most promising sources for recovering energy are periodic vibrations generated by rotating machinery or environmental sources such as wind or ocean waves.

The present work addresses vibration-based energy harvesting systems based on the piezoelectric effect. Up to now, the piezoelectric e ffect has been extensively investigated and integrated in di fferent applications, such as sensors and actuators. Vibration energy harvesting converts vibration energy into electrical energy. The large availability of vibrations in industrial environments and the simultaneous need for sensing applications in such locations motivated the research in recent time.

Most often, a mechanical resonator is used to amplify the low vibration levels into usable deflections. Such systems consist usually of a tip-loaded clamped-free cantilever [1–3], which we refer to as 'first generation harvester concepts'. The drawback of such harvesters is that they operate efficiently only if the harvester's resonance frequency coincides with the dominant ambient vibration frequency. Any di fference between these frequencies will lead to a decrease in the power output. As realistic ambient vibration spectra exhibit multiple frequencies and vary over time as the vibration source is aging or changing in temperature for instance, 'first generation' energy harvesting schemes fail. In order to overcome this limitation, many research groups are addressing new resonator designs with an optimized active bandwidth, which enables the harvester to collect power on a broader frequency range. This research can be categorized in two groups. On one hand, numerous multimodal resonator designs, which are able to operate resonantly at multiple frequencies, have been introduced in [4–7]. Wu et al. [8] investigated a compact piezoelectric energy harvester, comprising of one main cantilever beam and an inner secondary cantilever beam. The system harvests power at two distinct frequencies. A novel trident (three-pronged spear) shaped piezoelectric energy harvester has been proposed by Upadrashta and Yang in [9] to collect power from wideband, low frequency, and low amplitude ambient vibrations. Lamprecht et al. [10] investigated how multiple vibration harvesters can be combined to a macroscopic array configuration by aiming not on high resonance peak powers, but on close spectral overlaps in lower power regions with a bandwidth 500 Hz. In [11] a multiresonant structure comprising a clamped–clamped piezoelectric fiber composite generator has been proposed by Qi et al., with side mounted cantilevers, which are tuned by added masses to resonate at individual frequencies, resulting in a wider harvesting bandwidth. Other works [12–14] proposed multiple concepts of multiresonant piezoelectric energy-harvesting devices, capable of harvesting power on a wider frequency range using both translational and rotational degrees of freedom.

On the other hand, other groups proved that integrating nonlinearity in the harvesting device broadens the operative bandwidth compared to the standard linear harvesters. The authors of [15–21] demonstrated the usability of bistable resonators for harvesting over a wider operational frequency range, by integrating permanent magnets positioned with respect to another permanent magne<sup>t</sup> on the resonator. In [22] a compact nonlinear multistable energy harvester array has been presented by Lai et al., for harvesting energy at low frequencies. Ho ffmann et al. [23,24] developed a harvesting system capable to autonomously adapt the magnetic field strength, acting on the resonator, by adjusting the orientation of a diametrically polarized permanent magnet. Another bistable rotational energy harvester operating at low frequencies has been proposed by Fu and Yeatman in [25,26]. The harvesting is achieved by magnetic plucking of a piezoelectric cantilever using a driving magne<sup>t</sup> mounted on a rotating platform. In [27–29] an autonomous tuning mechanism has been developed, allowing a compensation of the hysteresis, as well as maintaining the optimal working point. It allows the use of both coupling modes (attractive and repulsive), which enables the harvester to adapt its operating frequency to the dominant vibration frequency of the environment. Nammari et al. [30] proposed an enhanced novel design of a nonlinear magnetic levitation-based energy harvester, where the tuning effect is achieved by the magnetic and the oblique springs. In [31–34] we studied a dual-frequency piezoelectric energy harvester incorporating permanent magnets for bidirectional frequency tuning and presenting an increased useable frequency range compared to standard harvesters. Other groups used di fferent bandwidth broadening strategies, for example, shunted piezoelectric control systems have been proposed in [35,36]. Zhang and Afzalul [37] analyzed a broadband energy harvester with an array of piezoelectric bimorphs mechanically connected through springs. The operative bandwidth broadening can be achieved by carefully selecting the masses and adjusting the spring sti ffness.

In the present paper, we propose a so-called 'folded beam' resonator design for energy harvesting. It consists of an outer tip-loaded beam, mechanically connected to a pair of inner beams, which extend towards the fixed end. In contrast of [8], the resonator is fabricated from a single steel sheet. It uses permanent magnets at the free ends of the cantilevers for higher frequency agility, by making use of the nonlinearity provided for the nonlinear magnetic forces. It resonates at two distinct frequencies, spaced 10 Hz apart, in the range (50,100) Hz. Bidirectional frequency tuning was achieved using a similar approach as presented in [34]. It has been demonstrated that both modes can be tuned independently, together with the dual frequency feature, can provide a superior frequency agility and increase the operational bandwidth of the system compared to existing approaches. Furthermore, an optimized version of the resonator is presented in this paper, where the two power peaks could be nearer to each other and provide the same power levels. This feature further enhances the performances of such a harvester and shows the benefit of such a design compared to an array of simple harvesters. Due to the close resonances enabling the resonator to amplify the deformation of the outer beam and leading to a higher strain distribution on the inner beam compared to the aforementioned simple design, where the second cantilever will be directly connected to the same excitation. A control scheme, which uses an energy e fficient maximum- amplitude tracking algorithm, is optimized compared to [34]. A reduced model of the harvester has been developed and enabled us to fully simulate the behavior of the harvester in question. The position-dependent magnetic force obtained from magnetostatic simulations presented in [33] have been implemented together with the reduced model in ANSYS twin builder. The system is able to autonomously choose and tune the closest mode to the dominant vibration frequency to maintain maximum possible oscillation amplitude. Furthermore, we dropped the self-adaptive step size tuning control scheme due to the potential of higher power requirements needed for the fine-tuning operation. Finally, we experimentally characterized the behavior of the first prototype of such a piezoelectric harvester design and the result matched with the simulation results in terms of frequency and expected voltage output.

#### **2. Dual Frequency Piezoelectric Energy Harvester**

This section reviews shortly the dual-frequency harvester proposed in [32–34] together with its tuning approach. Continuing our modelling e ffort from [34] we advance the finite element (FE) model into a system-level model. For this purpose, we derive a reduced order model, integrate nonlinear tuning and damping forces, a control algorithm, and electrical circuitry into a more complex model.
