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Sensors and Actuators A 331 (2021) 112743
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Sensors and Actuators A: Physical
journal homepage: www.elsevier.com/locate/sna
Review
A review on vibration-based piezoelectric energy harvesting from the aspect of compliant mechanisms
Haitong Liang a,b, Guangbo Hao a,b,∗, Oskar Z. Olszewski a,∗
a Tyndall National Institute, University College Cork, Cork, Ireland b School of Engineering and Architecture-Electrical and Electronic Engineering, University College Cork, Cork, Ireland

article info
Article history: Received 23 September 2020 Received in revised form 10 March 2021 Accepted 2 April 2021 Available online 6 April 2021
Keywords: Piezoelectric energy harvesting Compliant mechanism Mechanical structures MEMS Normalized Power Density

a b s t r a c t
Piezoelectric energy harvesters (PEHs) promote the construction of a smarter world through powering electric devices with energy scavenged from environmental vibrations by means of piezoelectric effect. To enable the piezoelectric effect, piezoelectric materials are usually attached to mechanical structures (e.g. ﬂexible beams) that can deform due to mechanical vibration and produce strain in piezoelectric material. The mechanical structure for energy harvesting in nature is a Compliant Mechanism (CM). A large variety of structural solutions have been proposed aiming to expend the working frequency range and maximizing the energy output of PEHs. To advance PEHs, a comprehensive review on existing structural solutions and materials is necessary. According to structural characteristics of current PEHs from the aspect of CMs, designs in state of the art are analysed and categorized into ﬁve conﬁgurations, mono-stable, multi-stable, multi-degrees-of-freedom, frequency up-conversion and stress optimization. For each conﬁguration, working principles and compatibilities with miniaturization to MEMS scale are analysed and assessed. Additionally, several CMs are ﬁrst proposed for PEHs in different conﬁgurations as inspirations and references to prompt the development of PEHs. Piezoelectric materials are also important factors in enhancing the energy harvesting performance. Characters of several widely adopted piezomaterials are summarized and compared. The metric of Normalized Power Density (NPD) is introduced to compare and assess the energy generation capability of PEHs with several widely-used piezoelectric materials and in different scales. A NPD-Volume graph is ﬁrst presented based on the data collected in literature. It shows that PEHs with PZT have the highest NPD and stable energy generation performance in a wide volume range. Both the structural categorization and NPD-Volume graph provide guidance and reference for design and optimization of PEHs.
© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Mono-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Frequency tuning through modifying proof mass, m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2. Frequency tuning through adjusting stiffness, k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1. Stiffness adjustment with geometric parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2. Nonlinear stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Multi-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1. Bi-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1. Magnetically bi-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2. Mechanically bi-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
∗ Corresponding authors at: Tyndall National Institute, University College Cork, Cork, Ireland.
E-mail addresses: G.Hao@ucc.ie (G. Hao), zbigniew.olszewski@tyndall.ie (O.Z. Olszewski).
https://doi.org/10.1016/j.sna.2021.112743 0924-4247/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

H. Liang, G. Hao and O.Z. Olszewski

Sensors and Actuators A 331 (2021) 112743

2.2. Tri-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3. High-order-stable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Multi-degrees-of-freedom structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1. Multi-mode structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2. Multi-direction structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4. Frequency up-conversion structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1. Mono-stable ﬁrst-stage oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2. Bi-stable ﬁrst-stage oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3. First-stage oscillators in other forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5. Stress optimization structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.1. Stress distribution optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2. Stress ampliﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6. Piezoelectric materials and energy generation capability of PEHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7. Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Introduction
Powering sensors and circuit systems with energy scavenged from ambient environment is extremely promising, not only because it is eco-friendly, cost-efﬁcient, but also this technology breaks the limitations posed by traditional batteries. It enables the distribution of electronics into inaccessible environment, such as deep wells, nuclear factories, and even human bodies, where changing or recharging batteries are challenging, risky or impossible. Beneﬁts will also be seen in the ﬁeld of IoT (Internet of Things) in particular [1], where numerous sensors are powered in a network for data exchanging wirelessly. In addition, energy harvesters themselves can be used as sensors for extreme value estimates [2]. Thanks to the advances in large scale integration technology and the improvement in low-energy-cost electronics over the past decades, the development and application of energy harvesting have been highly promoted [3,4].
In general, environmental energy exists in various forms, such as light, heat, electromagnetic ﬁelds, and mechanical vibrations. Harvesting energy from mechanical vibrations is more attractive, partially because mechanical vibrations are ubiquitous and independent of the season, weather and locations. Vibrational harvesters have relatively high energy density (about 0.3 ␮W/mm3) [5,6] and the availability of low-power circuits and sensors (␮W – mW range) [4] makes vibrational energy harvesting a suitable energy source for various application scenarios.
In principle, the vibrational energy harvester is analogous to an electronic power source which forces electrons to ﬂow in electronic circuits. To realize the electromechanical coupling between mechanical and electric domains in vibrational energy harvesters, there are four main mechanisms available: piezoelectric [5,7,8], electromagnetic [9–12], electrostatic [13–15] and triboelectric [16–18]. Because of its numerous advantages, piezoelectric conversion principle has received most attention. This is because the transducers have relatively simple conﬁgurations and high conversion efﬁciency. Furthermore, advances in the piezoelectric MEMS and CMOS ﬁlm technologies offer high–volume, wafer scale fabrication and integration capabilities. According to these advantages merits, this review focuses on piezoelectric energy harvesters (PEHs).
Resonance-based PEHs have large energy outputs because resonance signiﬁcantly ampliﬁes vibrational amplitudes and the stress generation in the piezoelectric materials. However, resonance

inherently induces a high Q-factor or, in other words, a narrow bandwidth, which means a slight shift from the resonant frequency will lead to a dramatic drop of the energy output. Fig. 1(a) shows a sharp energy output line subject to frequency representing the resonance-based PEHs. For practical applications, harvesters are expected to have a wide frequency bandwidth, as illustrated in Fig. 1(b). According to the literature survey based on the Core Collection of Web of Science with topic keywords: (energy harvest*) AND (piezoelectric) AND (vibration) AND (wide bandwidth) OR (large bandwidth) OR (broad bandwidth), growing attentions have been drawn over the past 10 years as reﬂected from the number of articles published from 2010 to 2019 as shown in Fig. 2. Although various approaches have been proposed to broaden the working bandwidth of PEHs [3,4,19–21], this problem still exists with signiﬁcant worldwide research ongoing addressing many open issues.
Instead of utilizing traditional rigid joints, compliant mechanisms (CMs) transfer motion, load and energy through the deformation of ﬂexible components [22–24], which can generate large stress/strain in deformed areas. For PEHs, stress/strain is required by piezoelectric effect to transfer vibrational energy into electric energy. It is thus practical to combine CMs with PEHs for enhancing the energy scavenging capability. In addition, other advantages of CMs, such as highly-alleviated friction, less or no assembling procedure, reduced mass, low cost, good compatibility to MEMS techniques etc. [24,25], will also beneﬁt PEHs. In particular, the monolithic feature of CMs further enhances the scalability of PEHs from macro-scales into micro-scales. Therefore, it is necessary to carry out a comprehensive review on current PEHs from the aspect of CMs to explore potential solutions for the narrow-bandwidth problem. Such a review has not been reported.
This paper analysed and categorized current PEHs into ﬁve conﬁgurations based on their structural characteristics. The categorization is summarized in Fig. 3. Extended discussions on each conﬁguration are detailed from Section 1 to Section 5. The compatibility of each conﬁguration to MEMS process is also analysed and assessed. At the end of each section, several compliant structures suitable for PEHs are proposed as inspirations and references for researchers in this area. In Section 6, the energy generation capability of PEHs with several widely adopted piezoelectric materials in different scales are compared and assessed in a NPD (Normalized Power Density)-Volume graph. Conclusions and prospects are given in Section 7.

2

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Sensors and Actuators A 331 (2021) 112743

Fig. 1. Energy output- frequency graphs. (a) Resonance-based PEHs. (b) Ideal PEHs.

Fig. 2. The number of articles published on vibrational PEHs targeting a wide frequency bandwidth (year 2010 - 2019).

Fig. 4. Two main concepts of mono-stable PEHs for frequency tuning. (a) Modifying the mass m. (b) Adjusting the stiffness k.

1. Mono-stable structures
Mono-stable oscillators have single stable positions. Once the external stimulation stops, mono-stable oscillators eventually return to their equilibrium positions when the dynamic energy is totally damped off. PEHs in this conﬁguration maximize the energy generation by resonance, normally in their ﬁrst-order natural mode with the lowest resonant frequency and lowest stimulating energy requirement. Resonance signiﬁcantly ampliﬁes vibrational amplitudes and then the stress/strain generation for piezoelectric effect, leading to high energy outputs. Structural simplicity is another advantage of this conﬁguration. Cantilever beams and clamped-clamped beams are two most commonly used monostable structures for PEHs. The mono-stable PEHs can be described

through a mass-spring vibration systems and their resonant frequency, fr , can be calculated with the equation [8,26]:

fr

=

1 2

k m

(1)

Here, m represents the mass of this harvesting device, and k is the stiffness ratio.
Resonance-based devices work at the single resonant frequency [27] while environmental frequencies spread along a wide range (normally in the low frequency range below 150 Hz [4,28]). Therefore, a wide and tuneable working frequency range in a low spectrum is expected from mono-stable PEHs. According to Eq. (1), there are two ways to achieve this: (i) modifying the mass, m and (ii) adjusting the stiffness, k, as shown schematically in Fig. 4.

Fig. 3. Categorization of PEHs reported in literature bases on structural characteristics. 3

H. Liang, G. Hao and O.Z. Olszewski

Fig. 5. Cantilever beam with tip mass.

Based on these two concepts, tuneable mono-stable PEHs for a wide frequency range can be classiﬁed into two groups.

1.1. Frequency tuning through modifying proof mass, m

According to Eq. (1), a larger mass leads to a lower resonant frequency when the stiffness ratio is given. Based on this principle, researchers attached proof masses on mono-stable PEHs to tune the resonant frequencies. Roundy et al. reported an energy harvester composed of a bimorph piezoelectric cantilever beam with tip-mass [29]. A power output of 375 mW was obtained at 120 Hz. Leland et al. utilized a piezoelectric bimorph attached tip-mass to power a wireless sensor for environmental monitoring [30]. The average power generated was 23.9 ␮W at 27 Hz. Erturk et al. presented a closed-form solution to predict the energy generation of piezoelectric cantilever beams with tip masses [31]. In experiments a maximum electric power of 23.9 mW/g2 was obtained at 45.6 Hz. Liu et al. successfully fabricated this cantilever tip-mass structure with MEMS techniques obtaining very low operation frequency of 36 Hz [32]. Defosseux et al. also developed a cantilever tip-mass harvester in micro scale [33]. A maximum output power of 0.62 ␮W was observed when the excitation acceleration is 0.25 g at 214 Hz. Side effects of attaching proof mass for frequency tuning include increased total weight and decreased power density of the whole device. Targeting this issue, Berdy et al. showed a novel solution of replacing the proof mass with well-designed sensor node electronic in their mono-stable energy harvesting device [34]. A maximum power output of 198 ␮W was obtained at 35 Hz.
Taking the geometric position of the proof mass into consideration, as indicated in Fig. 5, the equation of the resonant frequency in Eq. (1) can be rewritten as [8,35]:

fr

=

1 2

Eωh3 12ml3

·

8r4

+

r2 + 6r + 2

14r3

+

21 2

r2

+

4r

+

2 3

(2)

E is the Yong’s modulus of the material used for the cantilever, w, h, and l are the width, thickness and length of the cantilever respectively. r is the ratio between the position of tip mass gravity centre, x, and the length of the cantilever, l, which is x/l. m is the proof mass, considering the cantilever mass is negligible. According to this equation, a close distributed proof mass results in a high resonant frequency, while a far distributed mass leads to a low resonant frequency. Mass distribution in the direction perpendicular to the cantilever has no effect on the natural frequencies. Wu et al. veriﬁed this frequency tuning method both theoretically and experimentally with their prototype which was composed of a cantilever beam and a screwed tip proof mass with changeable gravity centre [36]. Through adjusting the relative position of the mass centre, a wide working frequency range (from 130 Hz to 180 Hz) was achieved. Schaufuss et al. adjusted the gravity centre through an auxiliary mass in their energy harvesting device to match the resonant frequency with driving frequency [37]. An effective working frequency bandwidth of 42 Hz–55 Hz was reported. Somkuwar et al. deployed two cylindrical masses in the tip container of a piezo-

Sensors and Actuators A 331 (2021) 112743

electric cantilever [38]. The equivalent gravity centre of these two masses changed when they moved to different positions and an auto-tuned resonant frequency range of 22 Hz–35 Hz was observed in experiments. Shin et al. developed an ultra-wide bandwidth PEH composed of a doubly clamped oscillating beam and a movable proof mass [39]. The resonant frequency of the device changed when the proof mass slid along the beam during device operation. This approach provided a bandwidth of 36 Hz. Shi et al. mounted a miniature stepper motor on their cantilever PEH to adjust the positon of the proof mass dynamically and an effective frequency bandwidth of 6 Hz was achieved [40]. In another approach, Jackson et al. ﬁlled the mass cavity with a liquid and used this to change the mass centre of gravity [41]. Inﬂuences of liquid density and viscosity on the performance of the nonlinearly-distributed mass were investigated.
CMs have medium degree of ﬂexibility between rigid parts and liquids, and they provide alternative solutions for the concept of changing the centre of gravity for response broadening. During vibration of PEHs, proof masses made with compliant structures are able to change the equivalent position of gravity centre through the deformation of ﬂexible components under the action of centrifugal force. The compliant proof mass can be fabricated in one piece with technologies like 3D printing etc. Thus, the whole device keeps monolithic. The deformation of the compliant proof mass can be pre-designed, leading to a controllable working frequency range. These unique advantages are not comparable for both rigid and liquid proof masses. One possible solution of the compliant proof mass for PEH is proposed and illustrated in Fig. 6. This compliant tip mass has 3D structures made of ﬂexible materials, such as plastics or rubber. During oscillating, the equivalent position of gravity centre changes and this dynamically tunes the device resonance frequency and broadens its frequency response.

1.2. Frequency tuning through adjusting stiffness, k

In addition to the modiﬁcation of proof mass, adjusting stiffness is the second way for frequency tuning of mono-stable PEHs. Stiffness is determined by several factors, including geometric parameters, loading conditions etc., and the categorization in this section is based on these factors.

1.2.1. Stiffness adjustment with geometric parameters Cantilever beams with rectangular cross section are widely used
for mono-stable PEHs. The stiffness ratio, k, of such cantilever beams can be calculated with the equation [8]:

k

=

3EI l3

=

Ewh3 4l3

(3)

Where I is the cross-section moment of inertia. According to this equation, the stiffness of the cantilever beam can be modiﬁed through changing geometric parameters (w, h and l), when the structural materials (represented with Young’s modulus E in this equation) is chosen. Length, l, and thickness, h, are more inﬂuential due to their third-order power.
Jackson et al. reported a MEMS PEH with ultra-low resonant frequency (28.5 Hz) through reducing the cantilever thickness to 5 ␮m [42]. A peak power of 29.8 ␮W was observed in their experiments. One shortage of reducing the thickness is that the inevitably weakened strength of the beam and therefore also decreasing the device reliability. Increasing the length of the cantilever can lower the resonant frequency as well and to keep a compact footprint, expended cantilevers are arranged in meandering shapes. Scornec et al. achieved a tunable frequency range of 7 Hz by adjusting the effective length of the piezoelectric cantilever in their energy harvesting device [43]. Berdy et al. adopted a meandered cantilever in their PEH and a power output of 118 ␮W was obtained under

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Sensors and Actuators A 331 (2021) 112743

Fig. 6. Cantilever beam with deformable proof mass on tip for PEH. (a) Isometric view. (b) Side view.

Fig. 7. Meandered cantilever beams for PEH [44].
external acceleration of 0.2 g at 49.7 Hz in experiments [44]. To eliminate the torsional stress and stress concentration, Sharpes et al. introduced a symmetric structure which contained two meandered cantilever beams in their PEH [45]. Experiments showed an energy output of 81.3 ␮W at 68.125 Hz with the acceleration of 0.1 g. Karami et al. investigated the possibility of powering peacemaker with a zigzag piezoelectric cantilever [46]. 10 ␮W power was expected to be generated under the excitation of heart beat. Liu et al. experimentally veriﬁed the low-frequency character of a S-shaped cantilever through the direct comparison with a straight counterpart for PEHs in MEMS scale [47]. To further reduce the natural frequencies of meandered cantilever beams, Apo et al. replaced straight branches with arc-based shapes [48]. Numerical models demonstrated that the frequency reduction reached up to 40 %. Wen et al. designed a MEMS PEH with a lengthened PVDF cantilever beam in a spiral shape and the working frequency of this device was about 20 Hz [49]. A MEMS-based spiral cantilever was also utilized in the PEH developed by Song et al. [50] and with an excitation of 0.25 g at 68 Hz, 23.3 nW was obtained from a ﬁve-turn spiral MEMS energy harvester. Based on the structure developed by Berdy et al., one typical meandered cantilever for PEH is shown in Fig. 7.
1.2.2. Nonlinear stiffness When a mono-stable beam is compressed axially, its lateral
stiffness is reduced. Conversely, tensile loads harden the stiffness. Frequency tuning through external loading has been adopted in PEHs and veriﬁed experimentally. Eichhorn et al. altered the resonant frequency of a cantilever beam from 380 Hz to 292 Hz by applying mechanical stress [51]. Leland et al. compressed a doubly clamped mono-stable beam axially on both sides and the resonant frequency was reduced from 250 Hz to just over 200 Hz [27]. AlAshtari et al. introduced tensile stress in their bimorph PEH through the attractive force between a ﬁxed magnet and the tip magnet [52]. Experiments showed that the resonant frequency was increased by 70 %. It is noteworthy that if the mono-stable beam is compressed to buckle where the axial load exceeds the critical loading point, its original mono-stable positon collapses, and two new stable states emerge. This bi-stability of a buckled beam has caused great research interest in PEHs and will be introduced in section 2 where multi-stable devices are addressed.

During vibration, loading condition and stress distribution inside PEHs change dynamically due to the geometric deformation, which brings a nonlinear stiffness. This stiffness nonlinearity enables a broader working frequency range of mono-stable PEHs. To describe this vibrational nonlinearity, a cubic item is introduced in the differential equation of a mass-damping-stiffness oscillating system under a harmonic vibration, as presented in Eq. (4), which is also called Dufﬁng equation [8,53].

mx¨ + cx˙ + kx + ˛x3 = Asin (ωt)

(4)

x is the oscillating displacement. Asin(ωt) is the external sinusoidal excitation and ω indicates the excitation frequency. x˙ , x¨ are the ﬁrst and second differentiation of x with respect to time, t, which represent speed and acceleration respectively. x3 represents the stiffness nonlinearity of the vibrating system. Constant coefﬁcients, m, c, k, ˛ and A, are system mass, damping rate, linear stiffness ratio, nonlinear stiffness ratio and excitation amplitude, respectively. In general, a nonlinear stiffness ratio, ˛, determines the nonlinearity of this oscillating system. The frequency response of displacement, x, corresponding to different nonlinear stiffness ratio, ˛, is shown in Fig. 8.
When the nonlinear stiffness ratio, ˛, is zero, Dufﬁng Eq. (4) describes the oscillation of a linear mono-stable oscillator under sinusoidal excitation. In this case, displacement, x, reaches maximum at the resonance frequency with a narrow working bandwidth, as shown in Fig. 8(a). When ˛ is non-zero, frequency hysteresis emerges. If ˛ is negative, the frequency response curve bends towards lower frequency domain, which is called stiffness softening, as indicated in Fig. 8(b). Conversely, a positive ˛ leads to stiffness hardening and the frequency response curve bends towards the higher frequency region, as illustrated in Fig. 8(c). For both stiffness softening and hardening, bended solid curves indicate a wider frequency bandwidth, but this depends on frequency sweeping directions. The internal resonance is also a nonlinear vibration phenomenon [54]. In addition to dufﬁng nonlinear oscillations, internal-resonance-based structures have been introduced in the ﬁeld of energy harvesting to broaden the frequency bandwidth [55–59]. As shown in Fig. 8(d), in this approach the response characteristic bends towards higher and lower frequency which can broaden the bandwidth regardless of the direction of the frequency sweep. Dashed lines, indicating the unstable condition that cannot be reached and will be replaced by the jump-up or jump-down behaviours between solid lines [60].
The concept of using stiffness nonlinearity in PEHs for expanding the operation bandwidth has been reported in literature and practical implementation includes three approaches: (i) geometric restriction, (ii) preloading and (iii) stoppers. In the ﬁrst approach, a clamped-clamped beam is widely used whereby a beam bends and stretches during oscillation, as shown in Fig. 9. Tensional stress generated along the beam results in a hardening stiffness. Piezoelectric materials attached on deformed structures convert stress energy into electric energy by piezoelectric effect. Gafforelli et al. provided an in-depth analysis on the stiffness nonlinearity of clamped-

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Fig. 8. Frequency response of mechanical oscillators with (a) ␣ = 0 (Linear stiffness); (b) ␣ < 0 (Nonlinear stiffness representing spring softening effect); (c) ␣ > 0 (Nonlinear stiffness representing spring hardening effect); (d) Internal resonance.

Fig. 9. Clamped-clamped beam and deformation during vibration [61]. (a) Original position. (b) Deformed beam with bending and stretching.
clamped PEHs [61]. Leadenham et al. developed a PEH based on a M-shaped doubly-clamped beam where stiffness nonlinearity was obtained under very low vibration levels (below 0.1 g and at around 5 Hz) [62]. Marzencki et al. utilized the clamped-clamped monostable structure in MEMS harvester [63]. Experiments veriﬁed an over 36 % adjustable frequency spectrum under 2 g excitation. Similarly, Hajati et al. developed the PEH with the doubly-clamped beam in MEMS scale [64]. A working bandwidth exceeding 20 % of the highest frequency was observed.
Preloading is the second approach to create stiffness nonlinearity. This preloading can be applied by either mechanical forces or magnetic coupling. Schematic structures of mono-stable cantilevers with different preloading conditions are illustrated in Fig. 10. The dynamically changing preloading conditions during vibration lead to the stiffness nonlinearity. To lower the working frequencies of oscillators, compression is normally adopted. Masana et al. explained the contribution of the axial preloading on the stiffness nonlinearity with an analytical model [65]. A broad working frequency range was observed in their experiments due to the stiffness hardening of a preloaded clamped-clamped beam in the pre-buckling condition [66]. Chen et al. analytically veriﬁed the wide operation frequency bandwidth introduced by stiffness hardening through compressing a clamped-clamped beam to its pre-buckling status [67]. Lin et al. applied external preloading on their piezoelectric cantilever with the nonlinear magnetic interaction between the tip magnet mounted on the cantilever and the stationary magnet ﬁxed nearby [68]. A dynamic response over a

much wider frequency range than that of a standard cantilever was obtained. Stanton et al. applied external load on the cantilever beam by the inactive force between the tip magnet and external magnets in their PEH [69]. Both stiffness softening and hardening could be achieved with the same device through adjusting the relative position of the magnets distributed. Sebald et al. expanded the working frequency range of a piezoelectric cantilever to 450 % through introducing the interaction between the tip magnet and external magnets [70]. The maximum power output was decreased by 60 % in report. Instead of a ﬁxed external magnet, Tang et al., utilized an oscillating magnet to interact with the tip magnet of the cantilever to achieve stiffness-softening nonlinearity [71]. Veriﬁed by experiments, the working bandwidth was nearly doubled and power output was increased by 41 % compared with the counterpart with a ﬁxed external magnet. To enhance the dynamic performance of the structure as shown in Fig. 10(b) with stiffness hardening nonlinearity, Yu et al. introduced a slider which can move freely along the cantilever [72]. Based on the interaction between the cantilever and sliding mass, the system could automatically catch the high-energy orbit increasing the power output by more than 25 times when compared to the counterpart with a ﬁxed end mass.
For constructing of the PEHs based on the internal resonance phenomenon, a primary nonlinear oscillator is connected with a linear auxiliary oscillator as illustrated in Fig. 11(a). The nonlinear structures as shown in Fig. 10 are options for the primary oscillators. Chen et al. ﬁrst proposed an energy harvester with the two-to-one internal resonance and a bi-stable oscillator was chosen as the primary oscillator in their device [55]. The wide bandwidth under a sinusoidal excitation in both frequency sweep directions was theoretically demonstrated. Although the harvester is originally based on electromagnetic mechanism, the structure proposed is transferable for PEHs. Also, Jiang et al. investigated the PEH devices based on the internal resonance, however, in their structure a buckled (preloaded) beam was used as the primary oscillator [56]. The wide bandwidth in two frequency sweep directions was also demonstrated and theoretically analysed. Xiong et al. employed the nonlinear structure as shown in Fig. 10(b) as the primary oscilla-

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Fig. 10. Schematic structures of mono-stable piezoelectric cantilevers with (a) Mechanical pre-load [66] or (b) Magnetic coupling [68].

Fig. 11. Schematic structures of nonlinear oscillators based on (a) Internal resonance [55] and (b) NESs [80].

Fig. 12. A nonlinear PEH with a mechanical stopper [87] (contact aided CM [81]). (a) Schematic structure. (b) Piecewise-linear stiffness ratio.

Fig. 13. Schematic potential energy graph of multi-stable structures. (a) Mono-stable type. (b) Bi-stable type. (c) Tri-stable type. (d) High-order-stable type. 7

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Fig. 14. Schematic structures of magnetic bi-stable energy harvesters based on (a) Attractive forces (type I) [93] and (b) Repulsive forces (type II) [94] created by ﬁxed permanent magnet located in close proximity to the vibrating structure.

Fig. 15. Purely mechanical bi-stable structures for PEHs. (a) Post-buckling type [100]. (b) Pre-shaped type.

tor in their l-shaped PEH with the internal resonance [58,59]. A comprehensive study was carried out and the broadened frequency bandwidth was experimentally veriﬁed. Xie et al. reported a Tshaped PEH device that used a clamped-sliding nonlinear beam as the primary oscillator in connection with a linear cantilever [57]. The wide frequency bandwidth of this harvester was analytically and experimentally demonstrated. The nonlinear energy sinks (NES) responding to wide-range frequency excitation have also attracted the attentions of researchers to improve the performance of PEHs [73–79]. Different from the connecting sequence of linear and nonlinear oscillators in those internal-resonance-based structures [55–59], nonlinear oscillators (as NESs) are mounted on the linear (primary) oscillators in the NES-based structures as shown in Fig. 11(b). Under certain conditions, the vibrational energy is irreversibly pumped from the primary oscillator to the coupled nonlinear oscillators [80]. In the PEH developed by Zhang et al. [75], a NES, which consisted a prepressed clamped-clamped beam with nonlinear stiffness, was utilized to connect a primary oscillator. Experiments veriﬁed the harvester had a broadband response from the oscillation of the NES while the vibration of the primary oscillator was suppressed. In the NES-based PEH developed by Xiong et al., a piezoelectric cantilever with a magnetic tip interacting with adjacent magnets was used as the nonlinear oscillator [76]. The wide frequency bandwidth under external harmonic excitation was demonstrated. Zhang et al. proposed the application for the NES-based PEH into a special scenario in the spacecraft [77]. The PEHs based on the internal resonance and NESs can be classiﬁed into multi-degrees-of-freedom structures as summarized in section 3. They are introduced here to highlight the effect of stiffness nonlinearity in widening the frequency bandwidth.
Mono-stable PEHs coupled with stoppers also can exhibit stiffness nonlinearity. However, in this scenario the stiffness hardening happens sharply rather than gradually. A typical structure of this nonlinear PEH with a mechanical stopper and its characteristic curve of stiffness are given in Fig. 12. Once the cantilever’s vibrating amplitude exceeds the distance, d, to the ﬁxed stopper, their contact results in a shorter oscillating cantilever ﬁxed at the contact position. The stiffness of the system is hardened from k1 up to k2 because of the dimension effect. Therefore, the overall system owns a piecewise-linear stiffness, as shown in Fig. 12(b). From the aspect of CMs, nonlinear-stiffness structures coupling with stoppers are also called contact aided CMs [81]. Halvorsen et al. veriﬁed

the stiffness-hardening nonlinearity caused by a mechanical stopper both theoretically and experimentally [82]. A wide frequency range (110 Hz–132 Hz) was observed. Zeng et al. symmetrically arranged a mechanical stopper on each side of the piezoelectric cantilever beam for Energy harvesting [83]. A bandwidth of 7.3 Hz (51.2–58.5 Hz) was achieved. Olszewski et al. developed a PEH to scavenge energy from wires carrying AC current and the frequency bandwidth was broadened by 250 % through introducing a mechanical stopper [84]. Instead of placing stoppers on the base, Hu et al. arranged the stoppers onto a primary oscillating system in their PEH [85]. The overall device showed piecewise linear stiffness as well and a wider frequency bandwidth was obtained. To expand the effective frequency spectrum of a S-shaped piezoelectric beam in MEMS scale, Liu et al. introduced hardening stiffness through applying a mechanical stopper in their device [86]. The efﬁcient frequency bandwidth was increased from 3.4 Hz to 11.1 Hz. The inﬂuence of stop distances and input acceleration on the frequency broadening effect were also investigated in their research. Liu et al. developed a MEMS harvester with a low resonance frequency of 36 Hz [32]. The working frequency bandwidth was widened to 17 Hz through assembling the device on a carrier with limited spacer. In spite of the advantage of widened bandwidth brought by mechanical stoppers, the energy conversion efﬁciency declines because of the reduced vibrating amplitude at a continuous frequency sweeping range.
1.3. Discussion
In terms of miniaturization, various mono-stable PEHs (linear or nonlinear) have been proposed due to their structural simplicity, as this has been introduced in this section. With special design, CMs exhibit stiffness nonlinearity. Howell et al. carried out a systematic research on nonlinear compliant structures [23]. Hao et al. proposed a framework for designing CMs with nonlinear stiffness and several nonlinear compliant structures based on this concept were presented [25]. Because of the inherent beneﬁts of CMs, these nonlinear compliant structures provide references and inspirations for the design of nonlinear PEHs. In addition, mono-stable oscillators normally own the simplest structures. They can be seen as the basic elements and theoretical foundations for the construction and design of PEHs in more complex conﬁgurations in the following sections.

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Fig. 16. Schematic structure of the bi-stable pre-shaped composite plate [110]. (a) The ﬁrst/original stable position. (b) The second stable position with residential stress.

2. Multi-stable structures
Unlike devices with mono-stable conﬁguration, which has a single stable point along the vibrating range, multi-stable structures can stay in balance in multiple positions. Each stable position represents local potential-energy minimum, and it is therefore also called the potential energy well. Between every two adjacent potential energy wells, there exists one potential energy barrier, indicating the energy required for oscillators to jump from one stable position to another. Schematic graphs of potential energy patterns along the deformation range for mono-stable and multi-stable types are given in Fig. 13. According to the number of stable positions, multi-stable conﬁgurations are subdeﬁned as bi-stable, tri-stable, quad-stable etc. Theoretically, PEHs in multi-stable conﬁgurations have nonlinear stiffness because of the changing stiffness during oscillating. Multi-stable conﬁgurations are introduced in a separate section is to highlight their unique features for application into PEHs.
The presence of multiple stable positions brings PEHs into diverse vibrating behaviours. Depending on the excitation magnitude of environmental vibration, multi-stable oscillators vibrate in three manners, i.e. intra-well, chaotic and inter-well vibrations respectively [88]. When the environmental excitation is weak and kinetic energy provided is lower than that required to snap through energy barriers, the multi-stable oscillator can only oscillate locally around current stable positon. In this case, multi-stable oscillators show the same dynamic character as the mono-stable devices. When the environmental excitation is larger, the oscillator can jump from one stable position to another, and then an inter-well vibration is obtained. Chaotic oscillation will be triggered when the external excitation energy is between the two conditions above. In such situation, multi-stable oscillators behave in an unpredictable (chaotic) manner. To gain more energy output from PEHs, inter-well vibration is preferable because the large deformation of oscillators while snap-through behaviours lead to a higher stress generation. Therefore, multi-stable PEHs with shallower and wider energy barriers can generate more energy even under weak excitations [89,90]. Due to these advantages, the multi-stable conﬁguration (especially bi-stable type) has drawn more attention in recent years [20,88,91,92]. Sharing the same working principles, multi-stable type devices can be seen as derivations of bi-stable oscillators.
2.1. Bi-stable structures
With designing a bi-stable oscillator the essence is to create an energy barrier between two potential energy wells. This potential energy barrier can be understood as the work of a resistance force against the vibrating motion in a certain area. Based on the nature of this impeding force, bi-stable prototypes reported can be cate-

gorized into two groups: magnetically and mechanically bi-stable structures.
2.1.1. Magnetically bi-stable structures Magnetic forces used to create bi-stable PEHs can be either
attractive or repulsive that can be achieved by different magnet alignment. Typical structures of magnetic bi-stable PEHs are given schematically in Fig. 14. A piezoelectric cantilever with a tip magnet is repelled from its original mono-stable position by attractive forces (type I) or repulsive forces (type II) from closely located ﬁxed magnets. This results in two new stable energy positions, as described in Fig. 14(b).
Erturk et al. proposed the concept of high-energy orbits for bi-stable piezo-magneto-elastic energy harvesters (type I), as described in Fig. 14(a) [93]. It was veriﬁed experimentally that the bi-stable harvester outperformed its mono-stable counterpart in terms of generated power with the difference in the order of one magnet. To further enhance the performance of bi-stable PEHs in type-I, Zhou et al. optimized the orientation of two external magnets by introducing a small tilt [95]. To further improve the working efﬁciency of bi-stable magnetic PEHs, Lan et al. lowered the potential energy barrier between two potential energy wells through imposing an extra small magnet in between of two magnets in typeI structure [90]. Both experiments and theoretical analysis veriﬁed that bi-stable energy harvesters with lower energy barriers could be triggered in to an inter-well vibration more easily than those with higher barriers. In other words, a lower energy barrier makes bi-stable oscillators more sensitive to weak vibrations. A broad lowfrequency operation range of 4–22 Hz was obtained. Ferrari et al. designed a bi-stable energy harvester based on structural type II [96]. Comparison experiments with respect to mono-stable cantilever counterpart showed that the voltage output was increased by 88 %. To express the oscillating nonlinearity of bi-stable PEHs in structural type II, Stanton et al. developed an analytical model which could predict key oscillating characteristics accurately [94]. Under the give excitation amplitude, the dynamic performance of bi-stable PEHs highly depends on the potential energy patterns [97]. Using the Type-II bi-stable structure schematically shown in Fig. 14(b), Yang et al. proposed a hybrid resonator with internal resonance through replacing the ﬁxed tip magnet with a moveable magnet connected with a spring [98]. It was theoretically demonstrated that the frequency bandwidth of this hybrid device could be doubled compared with that of the bi-stable oscillator with two ﬁxed magnets.
In terms of miniaturization, the presence of magnets makes the fabrication process complex and expensive. A MEMS harvester in bi-stable style II has been fabricated by Baglio et al. on a die (7 mm × 7 mm) [99]. Enhancement both in energy output and effective frequency bandwidth was obtained in comparison experiments with a mono-stable device under Gaussian white noise excitation. Ferrari

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et al. reported a MEMS bi-stable piezo-magneto-elastic harvester in the same bi-stable type [96]. Triggered by Gaussian white noise with an upper bandwidth limit of 500 Hz, the maximum output voltage reached above 1 V in experiments.

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2.1.2. Mechanically bi-stable structures Some mechanical structures have bi-stable characters and
enables bi-stable PEHs to be fabricated purely mechanically. A ﬂexible beam buckles when it is over-compressed axially. The original single stable positon collapses and two new stable positions appear. This simple post-buckled structure provides the ﬁrst approach to develop bi-stable PEHs mechanically. Bistability can also be created through pre-shaped compliant structures without any loading, which is the second method for designing mechanical bi-stable PEHs. Fig. 15 indicates these two mechanical bi-stable structures schematically. Although, a post-buckling structure and a pre-shaped structure may have similar geometric proﬁles, they are different from each other in fabrication, working features etc. Another difference between them is that pre-shaped bi-stable structure is stress-free in its original stable position, while the post-buckling type stores stress energy in both stable positions.
Cottone et al. created a bi-stable energy harvester in the pastbuckling type [100]. Compared with the unbuckled counterpart, the enhancement in both oscillation amplitude and energy output was observed under the white noise excitation. Sneller et al. attached a proof mass in the central part of the post-buckled clampedclamped beam to enhance its dynamic performance [101]. It was proved theoretically and experimentally that the attached central mass can both broaden the frequency range and lower the energy barrier for snap-through oscillation. With the similar doubly clamped bi-stable beam, Zhu et al. further introduced magnetic forces through the central poof mass to prompt the energy generation [102]. This magnetic force triggered the second buckling mode and made the jumping behavior between two equivalent positions easier. To reduce the inﬂuence from a ﬁxed potential energy shape and enhance the energy generating performance of bi-stable conﬁguration, Hosseinloo proposed an adaptive bi-stable PEH [103]. The enhanced performance was analytically veriﬁed. Andò et al. introduced a cost-efﬁcient way to develop bi-stable energy harvesters in post-buckling style [104]. Both electrodes and piezoelectric layers were printed on a ﬂexible PET (Poly Ethylene Terephthalate) substrate through a commercial inkjet printer. Purely mechanical bi-stable structures are more friendly to miniaturization compared with those based on magnetic force because of the absence of magnets. Andò et al. theoretically veriﬁed the advantages of a purely mechanical bi-stable structure for PEHs in MEMS scale [105]. Xu et al. fabricated a doubly clamped bi-stable energy harvester successfully in MEMS scale [106]. They utilized residential stress, rather than pre-loading, to create two stable points out of the fabrication plane. Experiments have veriﬁed a wide operating frequency range (from 50 Hz to 150 Hz) at a low acceleration of 0.2 g. For the pre-shaped bi-stable structure, it has been widely researched in the area of CMs [107–109], but the application for PEHs is rare. This is mainly because of the large positive stiffness at the initial position, which corresponds to a high resonant frequency.
Bi-stability can also be achieved through planar compliant structures in a pre-curved shape and this compliant structure has been adopted for PEHs [110]. A pre-curved composite plate stays in balance in this original shape. Under sufﬁcient environmental excitation, it jumps to the second stable positon with an opposite curvature (as shown in Fig. 16) because of the ﬂexibility that carbon ﬁbre epoxy owns. This pre-curved composite plate can also be seen as a special case of the pre-shaped type (in Fig. 15(b)) in 3D. Piezoelectric patches bonded on planar surfaces convert stress energy

Fig. 17. Schematic structure of tri-stable piezoelectric energy harvesters utilizing repulsive force [117].
Fig. 18. Schematic structure of quad-stable piezoelectric energy harvester utilizing magnets [123].
into electric energy during the switching motion between two stable positions. Arrieta et al. and Betts et al. conducted in-depth research on the application of pre-shaped bi-stable composites for PEHs [111,112]. Both theoretical analysis and experiments veriﬁed that bi-stable composite laminates could broaden the frequency bandwidth and improve the energy output of PEHs. Arrieta et al. established a low order model to capture the dynamic response of the bi-stable composite plate [113]. The accuracy of this model was proved experimentally with a bi-stable composite plate. Instead of ﬁxing the geometric central point, Arrieta ﬁxed one edge of the pre-curved composite plate in a cantilevered manner in the PEH developed [114]. Bi-stability and high power conversion in a wide working bandwidth was observed in experiments.
2.2. Tri-stable structures
One more stable position is introduced along the vibrating trace of tri-stable oscillators than bi-stable oscillators. With the three equilibrium positions, more complex oscillating response to environmental vibrations is caused. Under sufﬁcient external excitation, tri-stable oscillators jump between not only two adjacent potential wells but also two outer stable positions of the potential energy pattern (as shown in Fig. 13(c)). This larger vibrating amplitude beneﬁts PEHs with more stress generation and higher energy output.
For PEHs with tri-stable structures, the height of the energy barriers (or the depth of the energy wells) and their distributed positions are two key factors [115]. Shallower potential energy wells increase the sensitivity of tri-stable harvesters to weak vibrations and inter-well oscillating can be triggered more easily [116,117]. A wider distance between two adjacent energy wells enlarges the deformation during inter-well vibration [118]. Due to the signiﬁcant inﬂuence of potential energy barriers and wells on the dynamic performance, the design of the potential energy pattern is critical to the development of tri-stable PEHs. Theoretically, tri-stability could be created magnetically or mechanically. However, current tri-stable PEHs reported in literature were all based on magnetic forces (either attractive or repulsive). Because of

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Fig. 19. Multi-stable compliant structures suitable for PEHs [128]. (a) Tri-stable compliant structure. (b) Quad-stable compliant structure.

the reduced number of external magnets utilized, repulsive force is more attractive in creating potential energy barriers for magnetic tri-stable PEHs. The schematic structure of such a magnetic tri-stable harvester is illustrated in Fig. 17.
Kim et al. [119] and Li et al. [117] investigated the inﬂuence of the geometric distribution of external magnets on the multi-stability of a cantilever-magnet device. With the same structure shown in Fig. 17, a continuous transition from mono-stable to bi-stable and then tri-stable could be achieved through adjusting the geometric parameters of d and a. The depth of potential energy wells and the relative distance between the stable positions can also be modiﬁed through changing geometric parameters, which provides a possible approach to design the potential energy patterns and dynamic behaviours of multi-stable PEHs. Zhou et al. carried out continuous and in-depth research on tri-stable piezoelectric energy harvesting [115,120]. The equivalent restoring force of the tri-stable PEHs was described as a high order polynomial for a more accurate expression of the dynamic behavior [120]. It was veriﬁed theoretically and experimentally that a wide working bandwidth at low frequency scale can be obtained under the weak vibrational excitation for tristale PEHs [121]. Mei et al. applied a tri-stable energy harvester to scavenge energy from rotational vibrations of a rolling vehicle wheel [122]. A wide effective rotational speed range (from 240 rpm to 440 rpm) was obtained. Utilizing the piezo-magnetic device shown in Fig. 17 with different magnetic distribution, Li et al. concluded that the tri-stable PEH showed a lower frequency threshold and higher voltage density than the bi-stable counterpart [117]. It should be noted that it is hard to compare tri-stable oscillators and bi-table oscillators without considering the depth and relative distance of the potential energy wells.
2.3. High-order-stable structures
To further enhance the advantages of multi-stable PEHs, more stable positions were created along the vibrating trace of highorder-stable conﬁgurations, such as quad-stable, Penta-stable structures. As shown in Fig. 13(d), there are n energy wells and n − 1 energy barriers distributed along the deformation range. More complex inter-well oscillating manners will be triggered under sufﬁcient external excitation.

Fig. 20. Potential energy-deformation pattern of inﬁnite-stable oscillators for PEH.
Like tri-stable embodiments introduced in section 2.2, energy barriers of high-order-stable oscillators are only created by magnetic forces (repulsive forces in particular) in report. Purely mechanical embodiments have not been seen in literature yet. The schematic structure of a piezo-magneto-elastic quad-stable harvester is illustrated in Fig. 18. Three external magnets interact with the tip magnet on the piezoelectric cantilever through repulsive forces. In this way, three potential energy barriers are created in between with every two energy wells of the four.
Zhou et al. carried out both theoretical analysis and experiments on the piezo-magnetic quad-stable energy harvesters [123,124]. Inﬂuences from external magnets distribution on the potential energy pattern of the quad-stable oscillator were investigated. In their experiments, dense snap-through vibrating behaviours among four stable positions were observed under weak excitation which brought large deformation and large energy output. Wang et al. adopted attractive magnetic force to create potential energy barriers in their magnetic quad-stable PEH [125]. The novel part was that a cantilever-surface contact was introduced to optimize the stress distribution and up-convert resonating frequencies. Experiments validated that 4.2 times more power was generated compared with its linear counterpart under a low-intensity vibration.

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Based on the same magnetic principle, higher-order multistable PEHs were also developed. Huang et al. theoretically analysed the dynamic response of a penta-stable energy harvester to environmental vibrations [88]. It was revealed that penta-stable PEHs owned more complex oscillating behaviours, including lowenergy intra-well vibrations, middle-energy inter-well vibrations and global inter-well vibration. Zhou et al. developed a penta-stable PEH in the cantilever-magnet conﬁguration which contained a piezoelectric cantilever with tip magnet and four external magnets [126]. With shallow and largely distanced potential energy wells, frequent inter-well oscillation was observed in experiments. Kim et al. proposed a method to create penta-stability with a bimorph cantilever beam attached with a soft magnetic tip and only two external magnets [127]. The mechanism of this penta-stability was thoroughly explained and the dynamic and energetic performance of this was analysed.
2.4. Discussion
In summary, the spirit of designing a multi-stable PEHs is creating multiple equilibrium positions isolated by potential energy barriers along the vibrating trace. According to the multi-stable PEHs summarized in this chapter, magnets were widely adopted to create the multi-stability. The multi-stability and dynamic response can be easily modiﬁed through adjusting the number and geometric distribution of external magnets. However, the presence of multiple magnets makes miniaturization complex and difﬁcult. Except bi-stable conﬁguration, high-order multi-stable PEHs have not been either fabricated purely mechanically or in MEMS scale.
Multi-stable CMs provide multi-stability purely mechanically through the deformation of ﬂexure segments. Their monolithic character enables the fabrication in different scales with accessible techniques, which makes multi-stable CMs a great choice for PEHs. According to authors’ knowledge, PEHs based on multistable CMs (except bi-stable type) have not been reported yet. Howell has conducted systematic research on multi-stable CMs [23,24]. Chen et al. introduced a design principle on multi-stable CMs through the synthesis of several single bi-stable compliant structures [128–130]. Based on their research, some examples of multi-stable CMs suitable for PEHs are presented in Fig. 19. Fig. 19(a) shows a tri-stable compliant structure and a quadstability CM is illustrated in Fig. 19(b). Integrating multiple bi-stable CMs into a sophisticated arrangement, Jin et al. [131] and Zareei et al. [132] developed multi-stable CMs with propagating motion, which could be potentially converted into PEHs in certain application scenarios.
If the number of stable positions keeps going up to extreme, such a structure will be able to stay balanced in a continuous range. Correspondingly the height of energy barriers will decline to zero and the potential energy pattern will be in a basin shape with a ﬂat base as shown in Fig. 20. As a special case of multistable conﬁguration, this can be regarded as the inﬁnite-stable type. Statically balanced CMs (SBCMs) make such concept come true. Statically balanced behavior can be achieved through combining a positive-stiffness structure with another structure with negative stiffness. Based on this construction principle, SBCMs in various forms have been developed [133,134]. In terms of the application of this inﬁnite-stability in PEHs, Sergio has analysed the performance theoretically [135], but further detailed research on the effective bandwidth and dynamic performance is still needed.
3. Multi-degrees-of-freedom structures
To widen the working frequency range and increase the energy output of PEHs, several mono-stable oscillators can be integrated

Sensors and Actuators A 331 (2021) 112743
in one single device. These mono-stable oscillating unites resonate simultaneously or individually and bring multiple degrees of freedom in the whole system level. Based on their main functions, multi-degrees-of-freedom (multi-DoF) PEHs can be sorted into two groups, multi-mode type for broadening the effective frequency bandwidth and multi-direction type aiming at scavenging vibrational energy from different directions.
3.1. Multi-mode structures
A continuous frequency range can be approximately represented by a group of isolated frequency points from this range. Based on this concept, a simple method to widen the effective frequency bandwidth of PEHs is to integrate several individual resonators with different natural frequencies to form a multi-DoF oscillator with multiple resonating modes (multi-mode). Monostable sub-resonators can be integrated in two methods: in parallel as an array and in series as a chain, which is shown schematically in Fig. 21. The serial type can also be seen as a single resonator with multiple resonant modes at different frequencies.
Xue et al. and Farokhi et al. numerically analysed the operating frequency bandwidth of multi-mode PEHs in array type [138,139]. It was concluded that the operating frequency bandwidth could be efﬁciently expanded through integrating more mono-stable resonating units. Shahruz proposed a typical multi-mode PEH with a set of piezoelectric cantilevers with different geometric parameters arranged in parallel [136]. The expanded working frequency bandwidth was veriﬁed theoretically. Feng et al. developed a micro multi-mode harvester containing four piezoelectric cantilevers in parallel [140]. In this prototype, different resonant frequencies were obtained through diversely distributed proof masses along four identical cantilever beams. A wide frequency range (from 300 Hz to 800 Hz) was achieved. Al-Ashtari et al. constructed a PEH with three piezoelectric cantilevers to broaden the efﬁcient frequency bandwidth [141]. Dhote et al. arranged several compliant orthoplanar beams tangentially in a monolithic plane [142]. Together with the stiffness-hardening effect during vibrating, an extended frequency bandwidth of 35 Hz and a maximum peak-to-peak voltage of 4 V were obtained in experiments. For multi-mode PEHs in parallel type, the energy generation efﬁciency is normally low because there is only one mono-stable unit working corresponding to the excitation frequency whiles other units keep static. To improve the energy output, Zhang et al. connected every two adjacent beams with a spring to induce coupling action when either of the two resonators worked [143]. In the bi-resonant PEH proposed by Li et al., two parallel piezoelectric cantilevers coupled with each other through mechanical impact [144]. A wide frequency range of 14 Hz was obtained at the acceleration of 1 g. Deng et al. introduced magnetic interaction among parallel piezoelectric cantilevers through tip magnets [145]. A synergetic poly-stable character was observed in overall scale due to the diverse relative positions of oscillating units. In experiments the power density was increased to 760 % compared with the piezoelectric cantilever array without magnetic interaction.
Connecting mono-stable resonators in series is the second approach to construct multi-mode PEHs. Roundy et al. gave an clear explanation on this concept [137]. He proposed that a multi-mode oscillator contained n spring-mass-damper systems connected in series. The i + 1 th sub-resonator had a resonant frequency one bandwidth away from the resonant frequency of the i th subresonator. A wide operating frequency range was obtained in the whole system scale. Based on this concept, Hu et al. developed a harvester containing three stages of cantilever units in a folded series way [146]. Three operating frequencies corresponding to the bending motion of three cantilever stages were obtained. Tang et al. reported a dual-mass PEH which connected two single degree of

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Fig. 21. Schematic structures of multi-mode PEHs. (a) Parallel style [136]. (b) Serial style [137].

freedom systems in series [147]. A better energy generation capability of this dual-mass harvester was proved with a numerical model. Tadesse et al. attached piezoelectric crystals along a slim cantilever beam to create multiple resonant modes [148]. Expanded frequency range was veriﬁed both numerically and experimentally. Gong et al. proposed a folded structure for the construction of a multi-mode harvester [149]. Energy output peaks were observed at two resonant frequencies (97 Hz and 120.9 Hz respectively) in experiments. Moon et al. proposed a method for broadening the working frequency bandwidth of a cantilever PEH through narrowing the gap between the ﬁrst two resonant frequencies [150]. This was achieved through tuning parameters of the proof mass.
3.2. Multi-direction structures
In real world, environmental vibrations not only spread along a wide frequency spectrum, but also come from various directions. In practical application, it is hard to align vibration directions of energy harvesters with those of external vibrations all the time. To solve this problem and further improve the energy scavenging efﬁciency, PEHs sensitive to multiple vibrational directions were also investigated. Similar with multi-mode PEHs, multi-direction PEHs also integrate sub-oscillators in two ways: in parallel and in serial, as illustrated schematically in Fig. 22.
Blad et al. proposed a metric to design two-direction PEHs integrating mono-stable oscillators in series [6]. Two dimensionless parameters: the relative strength of vibrations, Pv, and the relative dimension of the design space, Pl, were identiﬁed. Based on their principle, a two-direction harvester, which vibrated in and out of the fabrication plane, was proposed. Zhou et al. developed a multi-direction harvester which consisted several piezoelectric beams linked in series and in a zigzag shape [152]. With the ﬂexibility in two perpendicular directions, this meandering harvester was able to scavenge energy from vibrations with a wide direction range (from 0◦ to 90◦). To harvest the vibrational energy from multiple directions, Hu et al. proposed a twisted piezoelectric cantilever beam which could be regarded as the serial style in Fig. 22(b) [153]. A substantial power output was obtained in experiment when the excitation came from any directions in the plane perpendicular to the axis of the twisted beam. Multi-direction PEHs were also developed based on the isotropic characteristics of some oscillators for vibrations from different directions. Xu et al. attached a pendulum at the tip of a piezoelectric cantilever beam which was pulled to be deformed when the pendulum was stimulated by external vibrations from multiple directions [154]. Yang et al. utilized a doubly clamped elastic rod which was sensitive to vibrations from arbitrary directions in the vertical plane in the reported multi-direction PEH [155]. When the doubly clamped elastic rod was excited to vibrate, the central piezoelectric ceramics were compressed so that the vibrational energy was converted into the electric energy. For the parallel type, Andò et al. developed a two-direction harvester composed of two piezoelectric cantilever beams perpendicular to each other [156]. In experiments, a maximum energy output of 3.2 ␮W was observed from each cantilever beam. Su et al. introduced magnetic repulsive forces in their tri-directional

piezoelectric harvester [157]. Three mono-stable resonators were capable to scavenge energy from three mutually orthogonal directions separately. Because of the magnetic interaction, other two resonators would be triggered to resonant when one of the three oscillators was actuated by the environmental vibration. Chen et al. developed a dandelion-like PEH to capture energy from ambient vibrations in different directions, and this character was veriﬁed experimentally [151].
3.3. Discussion
Multi-mode and multi-direction conﬁgurations share the same constructing concept which is integrating mono-stable oscillators in parallel or in series. The main difference between them is that multi-mode conﬁguration is aimed at broadening the effective frequency bandwidth, while the multi-direction conﬁguration targets on scavenging energy from multiple directions. Due to the variety of mono-stable oscillators, distribution types, and the number of mono-table oscillators integrated, the possible form of multi-DoF PEHs shows a good diversity. Based on the sophisticated research on mono-stable PEHs (introduced in section 1), the design and fabrication of multi-mode PEHs are not difﬁcult. In terms of miniaturization, the presence of magnets and 3D structures bring some difﬁculties.
A wider effective frequency bandwidth and higher energy output from multiple vibrational directions can be obtained through simply integrating more mono-stable units in one device, which is a solution for current problems of PEHs. However, some disadvantages, such as increasing volume, large weight, high cost, low power density etc. come along and a better solution is expected to be proposed.
4. Frequency up-conversion structures
Resonance signiﬁcantly ampliﬁes vibrational amplitudes of oscillators, and the energy output of PEHs at resonance is much higher than that when they are off resonance. However, the energy generation efﬁciency can still be low when the resonance is triggered by low-frequency vibrations because of the low-frequency deformation cycle of the piezoelectric elements. To improve the energy generation efﬁciency of PEHs working in such conditions, frequency up-conversion is then proposed to stimulate higher-frequency vibrations of piezoelectric elements when the environmental frequencies are low.
For frequency up-conversion PEHs, two stages of oscillators are integrated. First-stage oscillators, or driving oscillators, have low natural frequencies and they resonant with environmental vibrations. Second-stage oscillators, or generating oscillators, have higher natural frequencies and they vibrate freely when they are triggered by ﬁrst-stage oscillators. Energy harvesting process is mainly completed by the second-stage piezoelectric oscillators. Interactive forces between the two stages of oscillators can be either mechanical (such as direct impact) or magnetic. Vibration character of these two stages of oscillators is shown in Fig. 23. From this aspect, frequency up-conversion devices can also be seen as fre-

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Fig. 22. Schematic structures of multi-direction PEHs. (a) Parallel style [151]. (b) Serial style [6].

Fig. 23. Vibrational amplitudes of both stages of oscillators subject to time.
quency ampliﬁers. In literature, ﬁrst-stage oscillators are reported in various forms, but second-stage oscillators are mainly monostable resonators with high resonance frequencies. In this chapter, frequency up-conversion PEHs are sorted and introduced according to their ﬁrst-stage oscillators utilized.
4.1. Mono-stable ﬁrst-stage oscillators
Cantilever beams are adopted as ﬁrst-stage oscillators in frequency-up-conversion PEHs due to their structural simplicity. Fig. 24 schematically shows two types of interaction manners between the ﬁrst-/second-stage cantilevers in such PEHs. They are mechanical impact (Fig. 24(a)) and magnetic interaction (Fig. 24(b)) respectively. Gu took a piezoelectric cantilever beam as the driving oscillator in between of two more rigid piezoelectric cantilevers as second-stage oscillators in his frequency-up-conversion harvester [158]. Through the mechanical impact, the second-stage oscillators were triggered to vibrate at their natural frequency of 280 Hz when the environmental frequency was 20.1 Hz. Compared with the ﬁrst-stage piezoelectric cantilever only, the power density was increased from 13.6 ␮W/cm3 to 93.2 ␮W/cm3 in experiments. Ferrari et al. developed a similar frequency-up-conversion harvester based on the mechanical impact between two piezoelectric cantilever beams [159]. Experiments showed that the output voltage was roughly doubled and the efﬁcient bandwidth was widened compared with the condition without interaction between cantilevers. Liu et al. utilized a low-frequency cantilever in the normal shape and a meandered cantilever as the ﬁrst-stage oscillators separately in two MEMS frequency-up-conversion PEHs developed [47,160]. High-frequency cantilevers were chosen as second-stage oscillators. A wide operation frequency bandwidth of 14 Hz (from 12 Hz to 26 Hz) was observed in experiments. The power density reached 159.4 ␮W/cm3 under the acceleration of 0.8 g at 25 Hz. In addition to mechanical impact Zhang et al. introduced extra rope-driven interaction between the ﬁrst-/second-stage oscillators [161]. Both simulation and experiments veriﬁed a 4 times wider working frequency bandwidth from this impact-rope hybrid device than the conventional device based on impact only. Tang et al. uti-

lized magnetic interaction between tip magnets on the two stages of oscillators for frequency up-conversion [162,163]. The whole device had a compact footprint with all cantilevers arranged in the same plane. A peak power of 6.89 ␮W was scavenged from a running bus. Chen et al. adopted magnetic interaction as well for the frequency up-conversion in their harvester [164]. In Experiments, the second-stage oscillator vibrated at 85 Hz while the external excitation frequency was 40 Hz and an energy output of 0.3 mW was obtained. Magnetic interaction was also adopted by Wickenheiser et al. in the frequency-up-conversion PEH [165]. In this device the only piezoelectric cantilever beam acted as both ﬁrstand second-stage oscillators. When this cantilever vibrated with a large amplitude, closely distributed external magnets interacted with the tip magnet and caused local small-amplitude oscillation along the cantilever at a higher frequency. Galchev et al. also utilized magnetic interaction between the ﬁrst-stages oscillator (a clamped-clamped PZT beam in a spiral shape) and the secondstage oscillator (a copper spring with suspended metal mass) in the frequency up-conversion harvester [166]. An average power of 3.25 ␮W was generated under an acceleration of 1 g at 20 Hz, and an effective frequency bandwidth of 24 Hz was created. Panthongsy et al. developed a frequency-up-conversion PEH as a ﬂoor tile to harvest the energy from human footsteps [167]. The input frequency of the upper cover was up-converted to the resonant frequency the piezoelectric cantilever through magnetic interaction.
4.2. Bi-stable ﬁrst-stage oscillators
Bi-stable oscillators show better dynamic performance than mono-stable oscillators as introduced in section 2.1. Therefore, they have been also utilized as ﬁrst-stage oscillators in frequencyup-conversion PEHs. Fig. 25 schematically shows two distributing manners of bi-stable structures as ﬁrst-stage oscillators. Secondstage oscillators can be amounted on the base (in Fig. 25 (a)) or on the proof mass of the ﬁrst-stage oscillator (in Fig. 25(b)). Andò et al. adopted a clamped-clamped bi-stable compliant beam as the ﬁrst-stage oscillator and two piezoelectric cantilever beams as the second-stage oscillators in their frequency-up-conversion harvester [168,169]. In a low frequency range (0.5−7 Hz), a maximum power output of 416 ␮W was obtained with the acceleration of 13.35 m/s2. Jung et al. attached second-stage piezoelectric cantilevers on the central proof mass of the ﬁrst-stage bi-stable beam instead of on the frame to achieve frequency up-conversion in their PEH [170]. Second-stage oscillators were actuated to oscillate by impulsive acceleration during the snap-through buckling of the ﬁrst-stage bi-stale oscillator. A maximum power output of 131 ␮W was obtained under the base excitation at 30 Hz. To lower the energy threshold for snap-through buckling, Han et al. ﬁxed the ﬁrst-stage bi-stable compliant beam onto two ﬂexible sidewalls on each side [171]. Due to the outward moving tendency of the ﬂexible sidewalls, the PEH device was sensitive to weak excitations with an acceleration of 0.5 g. Inspired by the auditory hair bundle structure,

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Fig. 24. Structures of frequency-up-conversion harvesters utilizing cantilvers as ﬁrst-stage oscillators. (a) Coupling through mechanical impact [160]. (b) Coupling through magnetic interactions [163].
ing a mild activity. Pillatsch et al. utilized a rotating pendulum as the ﬁrst-stage oscillator [176]. The second-stage piezoelectric cantilever was triggered by the magnetic interaction between the magnets attached on both stages of oscillators. The energy density of 23.2 ␮W/cm3 was obtained with the external acceleration of 20 m/s2 at 2 Hz. Fan et al. adopted rolling elements (cylinder or ball) as ﬁrst-stage oscillators [177,178]. The second-stage piezoelectric cantilevers were triggered by magnetic coupling forces between the tip magnets and rolling elements under external excitations. Minh et al. used a micro metal ball guided by a cylindrical hole as the ﬁrst-stage oscillator in their frequency-up-conversion harvester in MEMS scale [179]. Under environmental excitation, the metal ball was stimulated to impact the piezoelectric cantilever beam arranged above. In experiments, a maximum power output of 34.6 nW was measured with the external acceleration of 39 m/s2 in a broad working frequency range (from 20 Hz to 150 Hz).

4.4. Discussion

Fig. 25. Structures of frequency-up-conversion harvesters utilizing bi-stable resonators as ﬁrst-stage oscillators. (a) Two stages of oscillators are distributed separately [169]. (b) Second-stage oscillators are mounted on the ﬁrst-stage oscillator [170].
Kim et al. developed a bi-stable compliant mechanism as the ﬁrststage oscillator for the frequency up-conversion harvester [172]. With this ﬂexure structure, the free vibration of the second-stage piezoelectric oscillator at 80 Hz could be triggered by infra-low environmental vibrations at 1–10 Hz, and A maximum power output of 1.8 ␮W at 7 Hz was obtained in experiments. To power the peacemaker with energy from heart beats, Jackson et al. developed a shock-induced MEMS harvester which could be stimulated by the low-frequent high-acceleration heart impulses (60–240 beats per minute) and worked at its high resonant frequency of 376 Hz [173]. In this application scenario, the heart can be seen as the bi-stable ﬁrst-stage oscillator while the harvester applied in heart acts as the second oscillator.
4.3. First-stage oscillators in other forms
In addition to mono-stable or bi-stable oscillators, ﬁrst-stage oscillators can be in other forms. Tang et al. created a nonlinear ﬁrst-stage oscillator through constraining a sliding magnet in a lubricated tube with magnetic ends [174]. Second-stage piezoelectric cantilevers were triggered by the magnetic interaction between the sliding magnet and tip magnets. An average energy output of 10 ␮W was obtained in the frequency range of 10 Hz–22 Hz with the acceleration of 1 g. Yeo et al. utilized an eccentric rotor (as the ﬁrst-stage oscillator) to stimulate six circularly arranged PZT beams (as the second-stage oscillators) to scavenge the energy from low-frequency vibration sources [175]. A power output of 40–50 ␮W was produced when the device was worn on the wrist dur-

Frequency-up-conversion PEHs improve the energy generation efﬁciency through amplifying the low excitation frequencies up to higher resonant frequencies of energy-generating oscillators. Drawbacks of current frequency-up-conversion harvesters in literature include complex assembling process, large volume, difﬁculties in miniaturization etc. Some CMs with frequency-upconversion character can be solutions for these problems. Tolou et al. proposed several CMs with such frequency-amplifying functions [180,181]. One of the reported structures and its working concept are illustrated in Fig. 26. Based on the elastic deformation and buckling behaviours of ﬂexible thin beams, the monolithic mechanism can double the input frequency and export the ampliﬁed frequency at output end. Higher frequency ampliﬁcation ratios are achievable through concatenating several frequency-doubling CMs in series. Beneﬁts of such compliant frequency ampliﬁers include no assembling, friendliness to miniaturization, compact footprints etc. CMs also provide options for the ﬁrst-stage oscillators in frequency-up-conversion PEHs. Pellegrini concluded that SBCMs were suitable for the ﬁrst-stage oscillators because of their ultra-low resonant frequencies [135]. With numerical analysis, the energy generation performance was conﬁrmed to be much better than other frequency-up-conversion PEHs in state of the art. However, only theoretical analysis was carried out and prototypes based on this concept have not been reported yet.
5. Stress optimization structures
Mechanical stress is critical for PEHs because it is basically required by the piezoelectric effect. Therefore, operations on mechanical stress, such as optimizing stress distribution and maximizing stress generation, provide new approaches to enhance the energy generation performance of PEHs. In this section, PEHs based on this concept are focused and summarized.

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Fig. 26. Frequency doubling compliant structure proposed by Tolou et al. [181]. (a) Mechanical structure. (b) The displacement of input and output points with respect to time.

Fig. 27. Piezoelectric cantilever in trapezoidal shape for optimizing the stress distribution [182].
5.1. Stress distribution optimization
For piezoelectric cantilevers in rectangular shape, one serious problem during oscillation is the stress concentration on beams near the clamped end, while there is nearly zero stress at the free end. The unevenly distributed stress leads to a low energy conversion efﬁciency and short life-durance of the device. Targeting this problem, piezoelectric cantilevers with optimized shapes, such as triangular, trapezoidal etc. were proposed. The trapezoidal piezoelectric cantilever is shown schematically in Fig. 27. Roundy et al. reported that a cantilever PEH in the trapezoidal shape could double the energy output compared with that in a rectangular shape [137]. In the comparison experiments carried out by baker et al., the energy density of trapezoidal piezoelectric cantilevers was 30 % higher than that of rectangular counterparts [182]. Goldschmidtboeing et al. reported that triangular-shaped cantilever beams had larger tolerable excitation amplitude and higher power output than rectangular cantilever beams for PEHs [183]. Montazer et al. compared the energy harvesting performance of piezoelectric cantilever beams in three different geometries, i.e. near edge width quadratic, half quadratic and Trapezoidal [184]. It was veriﬁed with simulation and experiments that the near edge width quadratic shape had much higher power extraction than other two types. In MEMS scale, Jackson et al. compared the performance of PEHs with cantilever beams in trapezoidal, wide rectangular and narrow rectangular shapes through both ﬁnite-element simulation and experiments [185]. It was concluded that trapezoidal beams showed more balanced character in power density and frequency bandwidth. Minh et al. utilized a triangular piezoelectric cantilever in their MEMS frequency-up-conversion device as the secondstage oscillator to optimize the stress distribution and increase the energy output [179]. Different from shape optimization, CMs showed another solution in tackling the problem of unevenly distributed stress in the cantilever PEHs. In the prototype developed by Ma et al., a polyvinylidene diﬂouoride (PVDF) unimorph was ﬁxed on the base at one end, and the other free end was attached with a complaint structure containing two ﬂexure hinges and a

Fig. 28. Schematic structure of a compliant stress ampliﬁer for PEH.
proof mass for a more uniform stress distribution [186]. Compared with traditional cantilever beams in experiments, 50 % more power was produced but with a lower maximum stress under the acceleration of 0.31 g. With the same device based on the CM, Yeo et al. enhanced the energy generation performance through utilizing PZT ﬁlms instead of PVDF ﬁlms [187]. A power output of 149 ␮W was achieved under the excitation of 6 Hz with 0.1 g acceleration.
5.2. Stress ampliﬁcation
In the scenario where energy is scavenged from mechanical impacts, piezoelectric stacks are used. To improve the energy generation efﬁciency, external impulsive force applied on piezoelectric stacks is expected to be multiplied. CMs have been adopted as such stress ampliﬁers in PEHs. A compliant structure with stressamplifying function is schematically shown in Fig. 28. Based on the deformation of compliant joints and lever principle, the input load will be multiplied with a certain ratio (determined by geometric parameters) and then applied on piezoelectric stacks. Zhang et al. developed a compliant load ampliﬁer in a pedal PEH to maximize voltage outputs [188]. The monolithic compliant structure was designed based on pseudo-rigid-body and topology theories. In experiments, a maximum power of 427 ␮W was generated under the load of 15 N at 1 Hz. Wen et al. designed a force-ampliﬁcation CM for harvesting energy from human walking [189]. Integrating four stages of force ampliﬁers in a monolithic form, the device ampliﬁed the input force 17.9 times and a maximum power of 50.8 mW was observed with the input force of 65.73 N. Feenstra et al. replaced the strap buckle of a backpack with a piezoelectric stress-ampliﬁer to scavenge energy from wearers’ motions [190]. This integrated compliant structure converted and multiplied the tensile force to a larger compressive force on the piezoelectric stack. A mean energy output of about 0.4 mW was obtained in testing. For scavenging the vibration energy of pressure ﬂuctuations in pipeline systems, Cao et al. introduced a compliant force ampliﬁer to increase the force applied on the piezoelectric stack [191]. Such

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Table 1 Comparison of piezo-materials widely used in PEHs.

Piezo-materials PZT (Pb[ZrxTi1-x]O3)

Advantages
1. Excellent piezoelectric properties [192]. 2. High Curie temperatures [4].

KNN (K0.5Na0.5NbO3) ZnO AlN
PVDF (Polyvinylidene ﬂuoride)

1. Lead-free perovskites ceramics [196,197]. 2. High piezoelectric constants, good Young’s modulus, moderate dielectric properties and Curie temperatures [197,198,199].
Good semiconducting properties; multiple forms are available [200].
1. Compatible to MEMS process. Various operations can be done on it [3,202]. 2. Excellent physical properties: high thermal conductivity, high breakdown voltage, high resistivity and corrosion resistance [202,203]. 3. Biocompatible [204,205]. 1. Outstanding ﬂexibility, low density, good stability, tough [207]. 3. Available in the market for easy applications.

Sensors and Actuators A 331 (2021) 112743
Disadvantages 1. Lead contained in PZT is toxic [193]. 2. Polarization process is required for piezoelectric effect [194,195]. High leakage current density caused by element loss because of the thermalization problem [3].
1. Moderate piezoelectric properties [201]. 2. ZnO is unstable and reacts with other IC materials; 3. It is hard to reproduce its properties [202]. 1. Moderate piezoelectric properties [3,201].
2. Residual stress in AlN during fabrication has an inﬂuence on the piezoelectric performance [206].
1. Poling process is required [208,209,210].
2. Moderate piezoelectric properties [3,207].

Table 2 Energy generation capability of PEHs from literature.

Number Power (␮W)

Acceleration Volume

(g)

(mm3 )

Frequency NPD (␮W/ Piezoelectric

(Hz)

mm3/HZ/g2) materials

1

118

0.2

2

29.3

0.05

3

23,900

1

4

375

0.25

5

0.0855

1

6

257

1

7

2.13

2

8

0.32

0.75

9

280

0.08

10

0.0233

0.25

11

240

0.4

12

13.98

1

13

1.11

0.39

14

43

2

15

1530

0.4

16

2.16

1

17

2.765

2.5

18

1.288

2

19

1

1

20

13

1

21

15,300

0.69

22

1.4

2

23

5.3

0.5

24

22

4

25

13.9

0.2

26

0.731

1

27

3.62

1

28

1.1

1

29

1.25

1

30

0.00046

0.5

31

3.5

0.2

32

60

2

33

0.62

0.275

34

0.18

1

35

6.9

0.2

36

0.8

2

37

128

1

38

34.78

2

39

0.669

1

40

18.56

1.75

41

40.9

1

42

1.0688

1

43

100.833

0.431

44

112.8

0.5

588 3106 3520 1000 10.1 56.4 0.652 0.769 50,000 0.11 14,025 18.6 0.99 1850 16,416e 0.78 0.425 0.612 11.9 0.153 1140 0.48 4.05 0.021 464 0.306 2.01 11.22 11.5 105.125 30.7 31 2.8 1.63 15 0.504 14.16 12.76 2.125 280 41.19 267 48.7 132.6

49.7 27 45.6 120 47 97.6 461.15 183.8 26 68 67 235 528 2 20.1 608 255.9 214 892 2297 29 870 126 1300 76 1509 132 1036 1300.1 489.9 149 572 214 853 599 1495 58 572 315 30 164 102.9 100 34.4

1.01E-01 1.40E-01 1.49E-01 5.00E-02 1.80E-04 4.67E-02 1.77E-03 4.02E-03 3.37E-02 4.98E-02 1.60E-03 3.20E-03 1.40E-02 2.91E-03 2.90E-02 4.55E-03 4.07E-03 2.46E-03 9.42E-05 3.70E-02 9.72E-01 8.38E-04 4.15E-02 5.04E-02 9.85E-03 1.58E-03 1.36E-02 9.46E-05 8.36E-05 3.57E-08 1.91E-02 8.46E-04 1.37E-02 1.29E-04 1.92E-02 2.65E-04 1.56E-01 1.19E-03 9.99E-04 7.21E-04 6.05E-05 3.89E-05 1.11E-01 9.89E-02

PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT PZT KNN KNN KNN ZnO ZnO AlN AlN AlN AlN AlN AlN AlN AlN AlN PVDF PVDF PVDF PVDF PVDF

Mechanical structures (corresponding sections in this review)

Source

Mono-stable (section 1.2) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Nonlinear Mono-stable (section 1.2.2) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Nonlinear mono-stable (section 1.2.2) Mono-stable (section 1.2.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Frequency up-conversion (section 4) Frequency up-conversion (section 4) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Nonlinear Mono-stable (section 1.2.2) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Multi-mode (Section 3.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1) Mono-stable (section 1.1)

Berdy et al. 2012 [44] Leland et al. 2004 [30] Erturk et al. 2009 [31] Roundy et al. 2004 [29] Liu et al. 2011 [32] Shen 2009 [218] Shen 2009 [218] Shen 2009 [218] Challa et al. 2008 [219] Song et al. 2017 [50] Zhu et al. 2011 [220] Lei et al. 2011 [221] Park et al. 2010 [222] Pillatsch et al. 2014 [176] Gu 2010 [158] Fang et al. 2006 [223] Lee et al. 2009 [224] Lee et al. 2009 [224] Kanno et al. 2012 [198] Isarakorn et al. 2011 [225] Cho et al. 2014 [212] Muralta et al. 2009 [226] Morimoto et al. 2010 [227] Hajati et al. 2011 [64] Durou et al. 2010 [228] Minh et al. 2013 [229] Won et al. 2016 [230] Kanno et al. 2012 [198] Wang et al. 2015 [231] Tao et al. 2019 [232] Jackson et al. 2014 [185] Elfrink et al. 2009 [233] Defosseux et al. 2012 [33] Hirasawa et al. 2010 [234] Elfrink et al. 2010 [235] Marzencki et al. 2008 [236] Andosca et al. 2012 [237] Dow et al. 2014 [238] Jackson et al. 2013 [239] Kim et al. 2018 [240] Montazer et al. 2018 [184] Shen 2009 [218] Cao et al. 2011 [241] Song et al. 2017 [242]

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Fig. 29. NPD-Volume graph of PEHs.

stress-amplifying compliant PEHs in MEMS scale have not been reported in literature yet according to authors’ knowledge.
6. Piezoelectric materials and energy generation capability of PEHs
In addition to mechanical structures, piezoelectric materials also play critical roles in PEH. Popular piezoelectric materials for PEHs include PZT (Pb[ZrxTi1-x]O3), KNN (K0.5Na0.5NbO3), ZnO, AlN and PVDF (Polyvinylidene ﬂuoride). For a clear comparison, main characteristics of these piezo-materials are summarized in Table 1.
Due to their speciﬁc physical characters, different piezomaterials were utilized in PEHs with different structures, sizes and application scenarios. The energy generation capability varies from a device to another. To compare their energy generation performance, a normalized criterion is required. We adopt the metric of Normalized Power Density (NPD), Power/(volume·frequency·acceleration2), considering the device volume, excitation frequencies and stimulating accelerations. The volume of the PEH device is an important parameter because it qualitatively reﬂects structural conﬁgurations (which have been introduced above), fabrication techniques used, application conditions etc. We present the NPD of different PEHs and the device volume in one graph to illustrate their relationship clearly. A large amount of data on energy generating performance of PEHs has been collected and summarized [3,44,211], however, it has not been organized in such a way. Initial data of PEHs from literature are listed in Table 2. Estimated Figures based on original publications are marked with ‘e’. Volume here is the active volume of devices. Based on the data in Table 2, the NPD-Volume graph of PEHs is ﬁrst shown in Fig. 29. It should be noted that inaccuracy exists in this NPD-Volume graph due to the three main reasons: 1) device volumes may be calculated in different ways by authors because no standard is established for this yet [212]. 2) The same piezomaterial can be fabricated in various forms which shows different

performances. For example, PZT can be in ceramics, ﬁbres, and even ﬂexible ﬁlms which are newly developed [213,214]. 3) The number of samples is limited since not all publications on PEHs provide sufﬁcient ﬁgures wanted. Therefore, this NPD-Volume graph of PEHs shows the NPD-Volume tendency and relationship qualitatively.
Based on this NPD-Volume graph, several conclusions can be drawn. 1) PZT owns relatively the most stable power generating capability across a wide device volume range. It also represents the largest NPD among all these piezo-materials. Because of its outstanding performance, PZT has the highest popularity. 2) AlN shows higher power density in a larger device size. 3) The NPD of KNN and ZnO has a tendency to decline when the device volume increases. 4) Around the device volume of 100 mm3, the NPD of PVDF varies among different PEHs. In addition, according to the column of mechanical structures in Table 2, it is noteworthy that mono-stable cantilever beams are still the most popular structural type for PEHs realized. Although the NPD statistic on PEHs in this paper is not comprehensive, it can still provide reference and suggestions for material selection, structural design and optimization of PEHs.
7. Conclusion and prospects
A wide variety of solutions for PEHs has been reported in literature. However, the majority of them are still based on resonance. Although resonance signiﬁcantly ampliﬁes vibrational amplitudes and then increases power outputs of PEHs, it inherently brings a high Q-factor or, in other words, a narrow bandwidth. This means a slight shift from the resonant frequency will lead to a dramatic drop of the energy output. In addition, resonance restricts the miniaturization of PEHs because of the size effect, i.e. smaller sizes lead to higher resonating frequencies [215]. Fundamental frequencies of MEMS structures can be in the kHz level which is rare in most application environment. Ideally, PEHs are supposed to be sensitive to weak vibrations in a broad frequency range. Such devices

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with satisﬁed performance have not been realized and solutions are expected to be explored.
Stress is basically required for piezoelectric effect. Therefore, the essence for creating ideal PEHs is to generate sufﬁcient stress in a wide frequency spectrum. CMs transfer load, motion and energy through deformations of ﬂexible components, and a great deal of stress is generated naturally during the working process, which fulﬁls the stress requirements of piezoelectric effects for energy scavenging. What’s more, monolithic CMs show more beneﬁcial aspects for PEHs, such as highly-alleviated friction, less or no assembling procedure, reduced mass, low cost, good compatibility to MEMS etc. CMs, such as SBCMs, thus show large practical value in tackling current problems in PEHs. This has been veriﬁed by the good performance of PEHs utilizing CMs.
For a better understanding of state-of-the-art solutions, this review analysed and categorized PEHs into ﬁve groups, mono-stable, multi-stable, multi degree of freedom, frequency up-conversion and stress optimization, according to their structural characteristics. In each section, the compatibility of structural conﬁgurations to MEMS process was also discussed. Some structural conceptions based on CMs principles were ﬁrst proposed for different conﬁgurations as both references and inspirations for researchers in this area. Apart from mechanical structures, piezoelectric materials directly determine the energy generating capability of PEHs. The metric of NPD was introduced to compare and assess the energy generation capability of PEHs with different piezo-materials and in different scales. It is noteworthy that power limit [216,217] is another ﬁgure of merit to evaluate the energy generation efﬁciency of PEHs with various structures and piezoelectric materials. Based on the data collected in literature, the NPD-Volume graph on PEHs was ﬁrst drawn. PEHs with PZT owned the highest NPD among all piezo-materials discussed here and stable energy generation in a large volume range. The energy generating performance of PVDF is moderate, but its mechanical ﬂexibility is very outstanding. Both of the categorization and the NPD-Volume graph of PEHs provide guidelines and reference for the piezo-material selection, structural design and optimization of PEHs for researchers in this area. In particular, the huge application value of CMs in PEH is illustrated and highlighted in this review. With the increasing attention on the application of CMs in PEHs and the development of ﬂexible piezoelectric materials, such as ﬂexible PZT ﬁlms [213], more PEHs bases on ﬂexible structures with better energy generating performance are expected to be developed.
Declaration of Competing Interest
The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper.
Acknowledgements
This research is ﬁnancially supported by the project of EnABLES. EnABLES (http://www.enables-project.eu/) has received funding from the EU Horizon 2020 research & innovation programme, under Grant Agreement No. 730957.
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Biographies
Haitong Liang is currently a PhD student in Tyndall National Institute/School of Engineering and Architecture, University College Cork, under the supervision of Dr. Guangbo Hao and Dr. Oskar Z. Olszewski. His research topic is piezoelectric energy harvesting based on compliant mechanisms. He received his B.S. degree in Mechanical Engineering from Taiyuan University of Technology (2012) and the M.S. degree in Mechanical Engineering from Beihang University (2015). Before his doctoral study, he had 3-year experience in mechanical design as an engineer.

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Dr. Guangbo Hao is a Senior Lecturer with University College Cork, Ireland. His research areas lie in design of compliant mechanisms and robotics and their applications in precision engineering, energy harvesting and biomedical devices. He obtained his PhD degree in Mechanical Engineering from Heriot-Watt University in 2011. He is a member of ASME and an elected member of the ASME Mechanisms and Robotics Committee. He is serving as the Editor-in-Chief of the IFToMM afﬁliated journal: Mechanical Sciences and the Associate Editor of ASME Journal of Mechanisms and Robotics. He has won some accolades including the 2017 and 2018 ASME Compliant Mechanisms Awards in a row. He has published over 140 peer-reviewed journal papers. More can be found from the website: https://sites.google.com/site/doctorghao/
Oskar Z. Olszewski received the B.Sc. degree in Electronics from the Gdynia Maritime Academy, Gdynia, Poland, in 2002, the M.Eng. degree in Electronic Engineering from the Cork Institute of Technology, Cork, Ireland, in 2004, and the PhD degree in Microelectronics from the University College Cork, Cork, Ireland, in 2010. He is a senior researcher at Tyndall National Institute, Cork, Ireland, where he is involved in research in the ﬁeld of MEMS technology. His interests include device design and modelling, process development, and device characterization. In particular, he currently works on piezoelectric (AlN and AlScN) resonators and actuators for various MEMS applications.

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