Pendulum Energy Harvesters: A Review

Abstract

In recent years, energy harvesters using pendulum systems have often been applied in ultra-low-frequency environments, such as ocean waves, human motion, and structural vibration. To illustrate the research progress in pendulum-type energy harvesting, a comprehensive review is provided in the present study. Specifically, single- and double-pendulum energy harvesters based on different energy-conversion mechanisms are separately grouped. In addition, different improvement techniques and design schemes used in studies on pendulum energy harvesters are summarized. Theoretical studies have explored the dynamic characteristics of single and double pendulums. Various key aspects, including the fundamental mechanisms, optimization methods, core structures, and applications, to improve the performance of single- and double-pendulum energy harvesters are discussed. Finally, several potential research directions and applications are proposed.

1. Introduction

Fossil fuels have been the major source of energy for decades [1]. However, with the excessive consumption of fossil energy, the problems of environmental pollution and the energy crisis have become a great challenge for humans at present [2]. Currently, batteries with limited energy storage require constant replacement, which is not conducive for the continued use of mobile electronics. Therefore, reliable and continuous energy resources that can substitute batteries must be developed [3]. In this context, the improvement of large-scale renewable energy sources, such as solar [1], tidal [4], and wind energy [5], is significant. These resources have abundant reserves worldwide and can generate megawatts of power. Energy-harvesting technologies capture energy from the environment and convert it into electrical energy [6]. With such energy harvesting techniques, devices that can store energy from the environment and power electrical equipment can be designed and manufactured [7]. These devices can not only contribute to the long-term use of electronics but also provide a solution to the energy crisis. The transduction mechanisms of energy-harvesting technologies can be divided into electromagnetic [8], piezoelectric [9], electrostatic [10], and triboelectric mechanisms [11], and these mechanisms have been extensively applied in energy harvesting research.

Energy harvesters adapted to different forms of energy resources have immense potential for application in various fields [12]. Wireless sensor networks (WSNs) are a classic example of energy harvesters [13]. In practical applications, wireless sensors for environmental monitoring must operate under diverse conditions [14]. However, the periodic replacement of the battery limits the lifetime of sensors and increases labor costs [15]. An energy harvester can directly convert energy in the environment [16], which is suitable as an energy resource for wireless sensors [17]. In addition, energy harvesting has applications in wearable electronics [18]. Furthermore, human movements, such as swinging of arms and bending of legs, contains a large amount of energy [19]. Energy-harvesting technologies can be used to efficiently harvest human motion energy, which can extend the life of self-powered wearable electronics and reduce necessity of battery replacement [20]. In the field of medicine, energy harvesters can derive energy of human organs to power implantable medical devices [21].

In previous studies, innovative structures and harvesting mechanisms have been presented to enhance the conversion efficiency and performance of energy harvesting [22]. These include magnetic levitation [23], multistable [24], self-tuning [25], and multi-directional [26,27,28] and curved structures [29,30]. Magnetic levitation energy harvesters comprise a hollow cylindrical container, coils, and magnets. This structure generates electrical power through the motion of the levitated magnet, which experiences a repulsive force from the fixed magnets. Current research on magnetic levitation energy harvesters concentrates on mono-stable to multi-stable structures [31]. A multistable energy harvester based on a magnetic cantilever beam and magnet is primarily used for piezoelectric energy harvesting [32]. The magnetic cantilever beam has multiple equilibrium positions, according to the magnet distribution [33]. The snap-through action of cantilever from one equilibrium position to another dramatically improves the mass velocity and performance of the energy harvester [34]. Owing to the nonlinear features of the multistable structure, the energy harvester can operate under broadband conditions [35]. Compared with the multi-stable structure, the self-tuning mechanism improves the response frequency range of energy harvesting using another approach. The self-tuning energy harvester can adjust its resonance frequency to adapt to environmental vibrations, significantly improving the working bandwidth. Some studies have focused on self-tuning energy harvesters, which comprise a fixed fixed-beam and a mobile proof mass [36,37]. When the resource frequency of the energy harvester does not match that of the environment, the mobile proof mass slides along the beam to adjust the resource frequency of the structure. A multidirectional energy harvester can be excited by vibrations from different directions [38]. According to the effectiveness, multi-directional energy harvesters can be classified as bi- and tridimensional structures [26]. In the actual environment, the direction of vibrations is variable and random, which affects the performance of unidirectional energy harvesters. Therefore, a multidirectional structure shows greater energy conversion efficiency and better environmental adaptability [39].

In addition to the above structures, pendulum systems can be applied for energy harvesting (Figure 1). Compared with the other structures, the pendulum system has richer dynamics and better ability to sustain motion. Although some reviews have summarized the various structures of energy harvesters, pendulum systems have not been analyzed as yet. To overcome this deficiency, the present study reviews research on energy harvesters based on pendulum systems, conversion mechanisms, basic configurations, and applications. In particular, Section 2 describes several common conversion mechanisms and summarizes research on single-pendulum energy harvesters corresponding to each mechanism. Section 3 presents the theoretical analyses and applications of double-pendulum systems in energy harvesting. Section 4 compares single- and double-pendulum energy harvesters. Finally, Section 5 presents conclusion regarding research on pendulum energy harvesters and their potential.

2. Energy Harvesters with a Single Pendulum

2.1. Characteristics of the Single Pendulum

Single pendulum is a classic nonlinear dynamic model. Studies on single pendulums have a long history. Compared with other structures, such as cantilever beams, a single pendulum exhibits different characteristics. Parametric resonance is a form of vibration that can be applied to energy harvesting. Unlike direct resonance, parametric resonance does not saturate due to linear damping [40]. However, damping may affect the initiation threshold amplitude. To solve this problem, Jia et al. [41] applied a pendulum structure for electromagnetic energy harvesting. The application of a pendulum structure can reduce the resonance amplitude threshold. In addition, some studies have shown that pendulum structures produced satisfactory effects on low-frequency and broadband energy-harvesting performance [42]. Dai et al. [43] equipped a vibration-energy harvester with a single pendulum. The authors established a theoretical model of the structure and calculated the ratio between the half-peak bandwidth and center frequency. Their analysis revealed that the bandwidth of the pendulum energy harvester was broadened. From the above analysis, the pendulum structure is suitable for increasing the conversion efficiency of the energy harvester under low-frequency conditions and for broadening the bandwidth.

2.2. Electromagnetic Energy Harvesting

Electromagnetic energy harvesters are based on the law of electromagnetic induction established by Faraday in 1831. As shown in Figure 2a, Faraday’s law of electromagnetic induction states that a change in the magnetic flux through a coil circuit can lead to the generation of an induced electromotive force [44]. Accordingly, electromagnetic energy harvesting creates a time-varying magnetic field, which generates an induced electromotive force across the circuit coil.

A classic approach to generate a time-varying magnetic field is to induce relative motion between the coil and magnet. Relative movement can be induced with two approaches: (1) a moving magnet and fixed coil and (2) a moving coil and fixed magnet [26]. As shown in Figure 2b, the electromagnetic energy harvester comprises a fixed magnet and coil. Mechanical energy in the environment, such as vibration and wind energy, can induce relative movement between the fixed magnet and coil. This induced electromotive force (EMF) can be expressed as follows:

$$E=n\frac{\Delta \varphi }{\Delta t}$$

(1)

The induced EMF is represented by E. The coil turns are denoted as n. Magnetic flux in each coil is represented by ϕ. Time is represented by t. According to Equation (1), the magnitude of the induced EMF is related to coil turns and change rate of magnetic flux. Compared with other energy conversion mechanisms, electromagnetic energy harvesters offer significant advantages. For instance, electromagnetic energy harvesters exhibit a better effect in large structures. In addition, they do not require an external voltage source.

Multiple studies have explored electromagnetic pendulum energy harvesters from the perspectives of application and structural optimization [45]. The major application of electromagnetic pendulum energy harvesters is in ocean energy harvesting. Ocean energy can be divided into tidal, wave, and ocean thermal energy [46]. Wave energy has abundant reserves worldwide, which is of great significance in alleviating the energy crisis [47]. Ding et al. [48] developed a low-frequency ocean energy harvester that could be used on underwater mooring platforms. As shown in Figure 3a, the primary structure of energy harvesters includes a pendulum rod that can rotate freely in the horizontal plane. The spindle of the pendulum rod is connected to a DC motor. When the ocean current flows in a certain direction, the platform can generate flow-induced vibrations. This movement can cause horizontal rotation of the pendulum rod. Experimental results showed that the maximum power reached 0.3 W. In addition, Daniil Yurchenko et al. [49] proposed a novel wave energy harvester, which is shown in Figure 3b. This device comprises a half-submerged ramp fixed to the seabed, a platform with a rotary pendulum system, and a buoy fixed to the platform. The buoy drives the platform to slide along the ramp due to the wave motion. This structure reduces the effect of gravity on the pendulum system and enhances the rotational potential of the pendulum system. The facilitation of the rotating motion of the pendulum significantly enhances the output power of the energy harvester. The above two systems primarily converted wave energy through the rotating movement of the pendulum. In addition, Cai et al. [50] optimized the conventional pendulum wave energy harvester from the perspective of adjustable frequency. Figure 3c shows the schematic of a double-mass pendulum oscillator. The major components of the energy harvester are two masses installed on the vertical screw bar. The positions of the two masses are adjusted. An electromagnetic energy converter is connected to the shaft of the vertical screw bar. The natural frequency of the structure can be changed by controlling the positions of the masses to match the wave frequency. According to experimental results, the output power could reach 100 mW at the wave height 0.1 m and wave period of 0.7 s.

In addition to wave energy, electromagnetic energy harvesters can be applied to convert human motion energy. The movement of the human body generates a large amount of energy that can be exploited, such as oscillation energy of the legs and arms [51]. Related studies have showed that heel falling to the ground of a human weighing 68 kg can theoretically generate approximately 67 W of power [52]. Therefore, the energy of human motion can satisfy the energy requirement of wearable electronics. Xie et al. [53] proposed a simple pendulum electromagnetic energy harvester that can be worn on the wrist to harvest kinetic energy. The primary structure is shown in Figure 4a. With the swing of a human arm, the magnetic pendulum can swing and produce a time-varying magnetic field [54] (Figure 4b). This harvester also has an urgent mechanism that can be driven by turning the handle and power outlet to charge mobile electronics. Simulation results showed that this harvester could generate 30 mW power with a 40 mm rotor weighing 50 g. Similarly, in another study [55], the performance of a magnetic pendulum energy harvester was analyzed. The authors fixed three pairs of fixed magnets on a pendulum and arranged coils on both sides of the rotor. Experimental results showed that the magnetic pendulum energy harvester could generate 11.3 mW power on average and 85.9 mW maximum power through human walking. Zhou et al. [56] presented a pendulum electromagnetic energy harvester based on the Halbach array. The primary structure is shown in Figure 5. As opposed to the systems described earlier, this structure uses only one large coil or four small coils, which decreases its size and weight. Four fixed magnets are arranged on the pendulum in the Halbach array to enhance the strength of the magnetic field. Experimental results for human motion on a treadmill showed that the harvester based on the Halbach array was able to generate the maximum power of 0.38 mW at a running speed of 8 km·h⁻¹.

In addition to energy harvesting, vibration reduction is also a function of the pendulum [57,58]. Therefore, pendulum structures have been utilized for both energy harvesting and vibration reduction. Kecik et al. [59] designed a coupled oscillator–pendulum system that can simultaneously realize energy conversion and vibration reduction. Figure 6a shows the primary structure of the system, comprising a tuned-mass damper and two independent energy-harvesting devices. The effects of magnetic levitation and rotating harvesters on vibration absorption have been analyzed. Simulation results revealed that rotating energy harvesters show superior energy conversion efficiency, while significantly affecting vibration absorption. In contrast, magnetic levitation harvesters have a lower output power but do not affect vibration reduction. Experimental results showed that magnetic levitation and rotation harvesters could generate the maximum output power of 36 and 64 mW, respectively. In addition to the performance of pendulum systems, their applications in vibration mitigation and energy conversion have been explored. For instance, structural vibration is a significant problem occurring during the operation of offshore wind turbines (OWTs) [60]. Therefore, OWTs have been equipped with various structures to absorb vibration. Jahangiri et al. [61] utilized an electromagnetic generator to replace a linear damper for energy harvesting. They developed a three-dimensional pendulum structure that could absorb vibrations and harvest energy in both x and y directions. The schematic is shown in Figure 6b. When the wind turbine vibrates, the swing of the pendulum induces the relative motion of the coils and magnets. Simulation results indicated that the displacement of the nacelle was reduced by approximately 70% and 77% in the axial and radial directions, respectively. Shen et al. [62] proposed a self-powered vibration reduction and monitoring system based on a pendulum structure. The primary structure is shown in Figure 6c. The analyses of vibration absorption and energy conversion efficiency under different excitation accelerations showed that the structural vibration was reduced by 49.6–65.8% using the pendulum system. The average electronic power of the electromagnetic generator could reach 312.4 mW, corresponding to the excitation acceleration of 0.05 g. In addition, some studies [61] and [62] analyzed the practical application of a pendulum-type energy harvester coupled with vibration reduction. However, these studies lacked experimental prototypes. Therefore, experiments on the performance of actual structures are warranted.

In addition to different application fields, some studies have focused on structural optimization. A mechanical motion rectifier (MMR), which is a type of electromagnetic energy harvester, can convert bidirectional movement to unidirectional rotation [63]. Further, some studies have optimized pendulum-type electromagnetic energy harvesters by equipping them with MMRs. For instance, Liang et al. [64] presented a pendulum energy harvester based on the bevel gear mechanism. Figure 7a shows the design of this electromagnetic energy harvester. A coaxially connected DC generator and frame constitute the pendulum, which can swing freely around the central shaft with external excitation. The conversion function from the swing of the pendulum to the unidirectional rotation of the DC generator is realized using three bevel gears. Experimental results showed that the bandwidth of the system with MMR was higher than that of the system without it. Moreover, Graves et al. [65] developed a motion rectifier using gear mechanism. As shown in Figure 7b, the oscillation energy of the pendulum is converted to unidirectional rotation of the DC motor through offset gearing. Experimental results indicated that this harvester was able to generate electrical power of 0.997 W at the excitation of 0.1 g and 0.75 Hz. The conventional gear mechanism performs well in a mechanism motion rectifier. However, it works under a strong excitation force and is difficult to apply in small harvesters. Therefore, energy harvesters with novel motion rectifiers have been designed. Figure 7c shows a pendulum energy harvester with a string-driven rectifier. This design uses a high-tensile strength rope and a one-way clutch as the motion rectifier, which simplifies the structure. Empirical evaluation indicated that the maximum normalized average power could reach 4.39 W·g⁻² [66]. In another study, Fan et al. [67] designed an ultra-low-frequency energy harvester based on a pendulum-plucked rotor, as shown in Figure 7d. In this harvester, a plectrum is fixed on the fan-shaped pendulum, which is composed of two parts: a longer flexible layer and a shorter rigid layer. The double-layered plectrum can convert the bidirectional swing motion of the pendulum into unidirectional rotation. At 40° swing amplitude and 4 Hz frequency, the harvester could generate an output power of 7.6 mW. The proposed structure realizes efficient energy harvesting in a low-frequency and low-amplitude environment and has a simplified configuration.

2.3. Piezoelectric Energy Harvesting

The piezoelectric effect is the basis of piezoelectric energy conversion. A piezoelectric material can generate a polarization phenomenon and produce voltage on the surface upon experiencing an external force, which is called the direct piezoelectric effect [68]. In contrast, the phenomenon in which a piezoelectric material generates mechanical deformation in an electric field is called the converse piezoelectric effect [69]. The primary working principle of piezoelectric energy conversion is shown in Figure 8. According to the external force and polarization directions, the working mode of the piezoelectric material can be divided into d₃₁ and d₃₃ modes [21].

The electromechanical coupling equation of the piezoelectric effect can be expressed as follows:

$$\{\begin{array}{l}\delta =s^E+dE\\ D=d\sigma +{\epsilon }^{T}E\end{array}$$

(2)

The strain and stress components are denoted by δ and σ, respectively. The electric field intensity and displacement are represented by E and D, respectively. Elastic compliance in a constant electric field is denoted as s^E, and dielectric constant at constant stress is denoted as ε^T. The piezoelectric coefficient is denoted as d.

Piezoelectric energy harvesting is widely used for vibration energy conversion [70,71]. Compared with other technologies, piezoelectric energy harvesting has higher energy density, higher voltage, and lower mechanical damping [9]. In addition, piezoelectric energy harvesters are easy to manufacture and have a simple structure [72]. Therefore, they can be applied to harvest wave [73] and human energy [74], smart grids [75], and convert structural vibrations [76], such as on bridges and roadways [77]. Many studies have been explored pendulum energy harvesting using piezoelectric materials [78]. For instance, Daniel et al. [79] proposed a low-consumption piezoelectric energy-harvesting system that can be placed at the sea bottom. The piezoelectric beam is driven by a plectrum fixed to the pendulum. This energy harvester was able to reach the maximum power density of 350 μW·cm⁻³. In another study, Zhang et al. [80] designed a novel multidirectional pendulum energy harvester based on homopolar repulsion. As shown in Figure 9a, the proposed system was applied to convert the inertial kinetic energy of driverless buses. Experimental results showed that the harvester could generate the maximum output power of 1.233 mW, and it could harvest multiple types of kinetic energy to self-powered sensors of driverless buses. This was confirmed in another similar study [81]. Additionally, Shukla et al. [82] developed a pendulum energy harvester (Figure 9b) that can be equipped on the human body to capture energy from waistline movement. The pendulum comprises multiple strikers that can bend compliant piezoelectric units. The maximum output power was able to reach 290 μW with PVDF units when the strikers were separated by 23 mm. Bao et al. [83] designed a handheld human motion energy harvester, as shown in Figure 9c. According to the inertial pendulum principle, the magnetic rotor can rotate and capture kinetic energy when the human body moves. Treadmill experiments revealed that the proposed structure was able o generate average power of 0.18 μW 6 km·h⁻¹. Wu et al. [42] presented a piezoelectric spring pendulum oscillator based on a binder clip structure. A spring pendulum is directly integrated with the piezoelectric element, which can improve energy harvesting performance. This harvester can simultaneously realize multidirectional and ultra-low frequency energy harvesting. Experimental results showed that the output power was 13.29 mW at 2.03 Hz.

In addition to studies in different application environments, the specific functions of pendulum piezoelectric energy harvesters, such as conversion of ultralow frequency and harvesting of multidirectional energy, have been explored. Compared with cantilever beams, the pendulum structure has a lower resonance frequency and is suitable for energy conversion from low-frequency vibrations.

Wu et al. [84] presented a frequency-up-converting harvester based on a pendulum component, as shown in Figure 10a. Specifically, they designed a 1:2:6 internal resonance mechanism to efficiently capture the ambient mechanical energy with a low vibration frequency. The energy-harvesting power of the piezoelectric energy harvester was approximately 2 mW at an excitation frequency of 2 Hz and 0.37 g. The proposed harvester exhibited excellent performance and significantly increased energy conversion efficiency in the low-frequency range. In another study, Fan et al. [85] developed a pendulum ball impact-excited piezoelectric energy converter to harvest low-frequency vibrations. Figure 10b shows the primary structure. Further, the performance of the single- and double-pendulum ball structures was compared. Experimental results showed that the output power of the double-pendulum ball structure is 43 μW at the vibration frequency of 2 Hz. Moreover, the output power of the harvester was effectively improved in a low-frequency environment, exhibiting immense application potential in low-frequency energy harvesting. Further, Pan et al. [86] developed a piezoelectric energy harvester comprising an inverted piezoelectric beam and pendulum. The mode coupling in the structure was beneficial for producing a snap-through motion. The maximum power of the harvester was 51.6 μW at stochastic excitations. Li et al. [87] investigated a harvester comprising a horizontally placed cantilever beam and pendulum. In this structure, the pendulum was able to swing freely under excitation and induce the dynamic buckling of the beam. The response of the harvester was analyzed under harmonic and random excitations. Furthermore, to increase conversion efficiency, multi-directional energy harvesting has become another optimization method for the pendulum structure. As such, Bao et al. [88] proposed a magnetic pendulum piezoelectric energy harvester for multidirectional energy harvesting, as shown in Figure 10c. In that study, internal resonance was induced through nonlinear coupling between the magnetic pendulum and cantilever beam. This harvester could capture vibrational energy in multiple directions through internal resonance. Experimental results indicated that the structure could generate the maximum output power of 0.64 mW at 7.5 Hz. Xu et al. [89] investigated a multidirectional energy harvester based on a pendulum ball that can freely swing in three-dimensional space (Figure 10d). Nonlinear coupling between the cantilever and pendulum ball enabled the conversion of vibrational energy in multiple directions, and this system showed a simpler structure and higher efficiency than conventional designs with multiple cantilever beams. Mo et al. [90] designed a piezoelectric energy harvester based on a U-shaped beam and pendulum. This harvester was able to capture multidirectional vibrational energy and operate at a low frequency.

2.4. Triboelectric Energy Harvesting

The triboelectric energy harvester is based on triboelectrification, as proposed by Wang et al. [91] in 2012. The key components of the triboelectric generator include two materials with different triboelectrification sequences. The two materials come into contact under the action of external mechanical energy. Subsequently, triboelectric charges are induced on the surfaces of the two materials. When the two materials are separated, the external circuit generates a current due to the impact of electric potential difference [47]. According to the working mode, energy harvesters can be divided into four categories [92], as shown in Figure 11.

Compared with other energy harvesters, triboelectric energy harvesters offer more options of materials. Simultaneously, triboelectric generators offer the merits of being lightweight, low-cost, and highly efficient at low frequencies [93]. Thus, triboelectric generators have been widely applied to capture the energy of water waves, human motion, and mechanical vibration, among others [94]. Additionally, a pendulum structure has been used to drive triboelectric generators. As shown in Figure 12a, Lee et al. [95] designed a triboelectric energy harvester driven by a pendulum. To enhance conversion efficiency, the authors used PDMS with a micro–nano-structured surface. Comparative experiments revealed that the output power of the harvester with a micro/nanostructured surface was four times higher than that of the harvester with an ordinary surface. He et al. [96] developed a radial-grating pendulum-structured triboelectric nanogenerator, as shown in Figure 12b. This generator comprises two coaxial sectorial structures: a rotor and stator. The swing of the rotor facilitates a rapid response to external mechanical forces from the generator. Thus, triboelectric harvesters can be used to capture energy from human motion and moving vehicles.

Ocean wave energy harvesting is a major application of triboelectric harvesters. For instance, Lin et al. [97] designed a spherical pendulum triboelectric generator, as shown in Figure 13a. The generator was able to capture water wave and wind energy in multiple directions. Rui et al. [98] developed a cylindrical pendulum-shaped triboelectric nanogenerator with a stator and rotor (Figure 13b). They equipped the rotor of triboelectric nanogenerators with FEP films. The FEP films can increase the contact area and decrease the friction force while facilitating energy conversion in two directions. The triboelectric energy generator achieved the average power of 0.87 mW under the action of water waves. Zhang et al. [99] designed an active resonance triboelectric energy harvester, as shown in Figure 13c. The primary structure of the system includes a single pendulum, tumbler structure, and triboelectric generator with a flexible ring multilayered structure. Under the interference of ocean waves, the single pendulum keeps swinging and drives the triboelectric energy harvester to contact and separate. Thus, current is generated in the circuit. In addition, the system can produce a resonance effect without coupling to ocean waves, which enhances conversion efficiency. Zhang et al. [100] developed an inverted pendulum-type multilayer triboelectric nanogenerator that could be applied in self-powered smart agriculture. The primary structure of the nanogenerator is shown in Figure 13d. The basic triboelectric energy harvester unit can generate separation and contact motions under the influence of water waves. Experimental results indicated that the mechanical energy capture efficiency, the ratio of available energy to convert mechanical energy, and the conversion efficiency of the available energy to electrical energy reached 0.42%, 12.98%, and 14.5%, respectively.

2.5. Hybrid Energy Harvesting

Hybrid energy harvesters have attracted much research attention in recent years [26]. Kinetic energy harvesters with a single conversion mechanism have limitations owing to the diversity of energy sources in the environment. Thus, although the pendulum structure exhibits excellent performance in energy harvesting, the combination of multiple conversion mechanisms can improve output power. In addition, each energy conversion mechanism has advantages and disadvantages; therefore, a hybrid energy harvester can integrate the merits of different structures [101].

For instance, Hou et al. [102] designed an electromagnetic-triboelectric hybrid generator based on a rotational pendulum. The primary structure is shown in Figure 14a. The rotation of the magnetic rotor induces current generation in the coil. Simultaneously, the pendulum rotor and blade produce a contact-separation-mode triboelectric generator. The maximum power density of the triboelectric and electromagnetic energy harvesters reached 3.25 and 79.9 W·m⁻², respectively, at 2 Hz frequency and 14 cm amplitude. Ren et al. [103] investigated a hybrid water-wave energy harvester that can be applied to long-distance wireless transmission. Figure 14b illustrates the structure of the wave energy harvester. The triboelectric module, which comprises two multilayered triboelectric energy harvesters, was combined with an electromagnetic module using a single pendulum. The harvester was applied to an offshore avoidance warning system. Xie et al. [104] proposed a non-resonant hybridized wave energy harvester. As shown in Figure 14c, the harvester has a magnet supported by a flexible pendulum. Four triboelectric energy harvesters are fixed on the hollow cylindrical shell, and a coil is fixed on the top cover of the shell. Under the swing of the pendulum, the magnet produces a changing magnetic field and drives triboelectric energy harvesting. Experimental results indicated that the maximum power of the triboelectric and electromagnetic energy harvesters reached 470 μW and 523 mW, respectively. A similar harvester is shown in Figure 14d, which was proposed by Zhang et al. [105]. This structure integrates triboelectric, electromagnetic, and piezoelectric energy harvesters. Owing to the bifilar pendulum structure, the energy harvester can capture both kinetic and gravitational potential energies of water waves. Experimental results showed that this system could achieve a power density of 358.5 W·m⁻³. In addition, a hybrid self-power vibration-monitoring sensor was designed by Fang et al. [106], as shown in Figure 14e. This system comprised a magnetic levitation structure as an electromagnetic energy harvester and a pendulum-like triboelectric energy harvester. Experiments were performed on a conveyor belt, which revealed that the harvester was able to charge a 6.8 mF capacitor from 0 to 4.35 V within 60 s.

3. Energy Harvesters with Double Pendulum

3.1. Theoretical Analysis of Double Pendulums

A double pendulum adds a degree of freedom to the basis of a single pendulum. However, a double pendulum exhibits complicated motion and rich dynamics [107]. Thus, double pendulums have attracted much research attention. A double pendulum comprises two rigid pendulae that can swing freely in the vertical plane. The primary structure of the double pendulum is shown in Figure 15.

The Lagrangian equations in Figure 15 can be defined as follows:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{\theta }}_i}\right)-\frac{\partial L}{\partial {\theta }_i}=0 \left(i=1,2\right)$$

(3)

The Lagrange term is denoted as L = K − II. The kinetic energy of the system is represented by K. The potential energy of the system is represented by equation #. The deflection angles of the two rods from the vertical line are represented by θ₁ and θ₂.

The kinetic and potential energies of the double pendulum can be defined as follows:

$$K=\frac{1}{2}{m}_1{l}_1^2{\dot{\theta }}_1^2+\frac{1}{2}{m}_2\left[l_1^2{\dot{\theta }}_1^2+l_2^2{\dot{\theta }}_2^2+2l_1{l}_2{\dot{\theta }}_1{\dot{\theta }}_2\cos \left({\theta }_1-{\theta }_2\right)\right]$$

(4)

$$\Pi =-\left(m_1+m_2\right)g{l}_1\cos {\theta }_1-m_{2}g{l}_2\cos {\theta }_2$$

(5)

The lengths of the two rods are represented by l₁ and l₂ and their masses by m₁ and m₂. The first study on a double-pendulum model was published in 1970. Lee et al. [108] analyzed the normal vibrations of a double pendulum. Specifically, they analyzed the solutions for special cases in which the two masses of the system were the same or the lengths of the two strings were the same. The normal modes of oscillation for the double pendulum are provided. Recent studies on the double pendulum focused on the periodic and chaotic motions of the system. For instance, Tomasz et al. [109] analyzed the chaotic features of a double pendulum using the Poincaré section, bifurcation diagrams, and Lyapunov characteristic exponents. They confirmed the transition of the double-pendulum system from a low-energy state to chaos with increasing energy. In another study, Nikolai et al. [110] investigated the dynamics and stability of a double pendulum for three cases: weightlessness, vertical, and horizontal. The Hamiltonian equations and Kapitsa method were applied for theoretical analysis, and good concordance was noted between the experimental and numerical results. Oiwa et al. [111] studied the Jacobian stability of a nonlinear double pendulum at equal initial angles of the double pendulum. Moreover, the association between the Jacobi stability and chaotic motion of the double pendulum was discussed using the Lyapunov exponent and Poincaré section. Wojna et al. [112] analyzed the chaotic and regular behaviors of a double-pendulum system with two repulsive permanent magnets. Experimental and numerical results showed good agreement. Bifurcation diagrams were used to confirm the transition from regular to chaotic motion. Rafael et al. [113] investigated the regular and chaotic motions of a double pendulum under different excitation conditions. The 0–1 test and Lyapunov exponents were applied to analyze the chaos of the pendulum in subharmonic and superharmonic excitations. Bifurcation diagrams indicated the chaos and stability of the double-pendulum system based on variable amplitudes.

3.2. Application of Double Pendulums in Energy Harvesting

Research on double-pendulum energy harvesting has started in recent years. The double pendulum exhibits a better energy conversion effect than the single pendulum. Basheer et al. [114] compared the power generation capacity of single, double, and Rott’s pendulums. Three prototypes were fixed to a rotating frame and connected to an electromagnetic generator, as shown in Figure 16a. Experimental results showed that the double pendulum achieved the highest output voltage. Furthermore, Zaouali et al. [115] designed a harvester for energy conversion with a rotating system, such as a wheel (Figure 16b). The authors fabricated an experimental setup to simulate a car wheel’s rotation and investigated the different structures of the single, double, and Rott’s pendulums. The output voltage of the prototypes was measured under two different conditions: constant and variable velocity. The double pendulum exhibited the maximum output power of 9.5 mW at 89 rpm.

Considering their effect on energy conversion, double pendulums have been applied in different environments, such as oceans and human motion. For instance, Kumar et al. [116] applied a double pendulum to electromagnetic energy conversion under ambient vibrations (Figure 17a). In addition, the effects of parametric factors, such as the mass ratio, length ratio, and magnetic mass, were evaluated. The experimental results were in good agreement with the numerical results. Carandell et al. [117] developed an offshore kinetic energy harvester based on a double pendulum system and flywheel. Experimental results indicated that the harvester was able to generate 0.18 mW power on average at a 1.43 m wave height and 0.29 Hz frequency. Furthermore, Chen et al. [118] designed a triboelectric-electromagnetic hybrid energy harvester based on a chaotic pendulum. As shown in Figure 17b, the double-pendulum system integrated the freestanding triboelectric and electromagnetic modes. The results of wave excitation experiments indicated that the maximum output power in the triboelectric and electromagnetic modes was 15.21 μW and 1.23 mW, respectively.

Chen et al. [119] proposed a rotational wind energy harvester based on a magnetic double-pendulum system. As shown in Figure 18a, the proposed structure generates electric power by coupling the double pendulum with nonlinear magnetic interaction. Owing to the dynamic characteristics of the double pendulum, this system can reduce the magnetic resistance torque at low rotational speeds. Experimental results indicated that the system was able to generate the maximum output power of 1.25 mW at a rotational speed of 557.31 rpm. Izadgoshasb et al. [120] applied a double pendulum system to human motion energy harvesting, as shown in Figure 18b. The piezoelectric cantilever beam is driven by a double pendulum through magnets. The authors compared three configurations using the mechanical shaker and human motion tests: cantilever beam, single pendulum, and double pendulum. The double pendulum exhibited the best effect. Sun et al. [121] expanded the bandwidth of a piezoelectric energy harvester by utilizing the internal resonance between the double pendulum and cantilever beam (Figure 18c). The experimental results showed that the double-pendulum harvester showed excellent broadband energy harvesting ability. Selyutskiy et al. [122] designed a flow-induced vibration energy harvester based on a double aerodynamic pendulum. In this system, the piezoelectric element is connected to the first pendulum and undergoes deformation under the rotation of the pendulum (Figure 18d). This harvester may be suitable for wind energy harvesting.

4. Discussion

Table 1. Comparison of various energy harvesters based on pendulum systems.

Investigators	Mechanism	Application	Structure	Working Condition	Average Output Power	Power Density	NPD
Ding et al. [48]	Electromagnetic	Ocean kinetic energy harvesting	Single pendulum	0.2 Hz and 0.24 m/s2	300 mW	-	3453.8 kg/m3
Cai et al. [50]	Electromagnetic	Ultra-low frequency wave energy harvesting	Single pendulum	0.12 m (wave height) and 0.7 s (wave period)	111 mW	-	-
Fan et al. [67]	Electromagnetic	Ultra-low frequency mechanical energy harvesting	Single pendulum	4 Hz and 40° (amplitude of swing)	7.6 mW	-	-
Zhou et al. [56]	Electromagnetic	Human motion energy harvesting	Single pendulum	8 km/h (running speed)	0.38 mW	23 μW/g	-
Zhang et al. [80]	Piezoelectric	Kinetic energy harvesting in driverless buses	Single pendulum	-	1.233 mW	-	-
Bao et al. [88]	Piezoelectric	Multi-directional energy harvesting	Single pendulum	7.5 Hz and 0.3 g	0.64 mW	-	-
Wu et al. [84]	Piezoelectric	Ultra-low frequency mechanical energy harvesting	Single pendulum	2 Hz and 0.37 g	2 mW	-	-
Rui et al. [98]	Triboelectric	Ocean wave energy harvesting	Single pendulum	0.8 Hz	0.87 mW	1.1 W/m3	-
Zhang et al. [99]	Triboelectric	Ocean wave energy harvesting	Single pendulum	-	12.3 mW	-	-
Xie et al. [104]	Triboelectric and electromagnetic	Ultra-low frequency water wave energy harvesting	Single pendulum	-	0.47 mW (Triboelectric) and 523 mW (electromagnetic)	-	-
Chen et al. [119]	Piezoelectric	Wind energy harvesting	Double pendulum	557.31 rpm	1.25 mW	-	-
Izadgoshasb et al. [120]	Piezoelectric	Human motion energy harvesting	Double pendulum	2 Hz	86.12 μW	-	-
Carandell et al. [117]	Electromagnetic	Ocean wave energy harvesting	Double pendulum	1.43 m (wave height) and 0.7 s (wave frequency)	179 μW	-	-
Chen et al. [118]	Triboelectric and electromagnetic	Ocean wave energy harvesting	Double pendulum	2.5 Hz	15.21 μW (Triboelectric) and 1.23 mW (Electromagnetic)	-	-
Zaouali et al. [115]	Electromagnetic	Energy harvesting for rotating system	Double pendulum	89 rpm	9.5 mW	-	-

summarizes studies on pendulum energy harvesters, primarily including the conversion mechanism, application, working conditions, and output power. In general, studies on the single pendulum are dominant, although a few studies have explored double pendulums. Overall, the double-pendulum systems showed better performance in energy harvesting than the single-pendulum system [114,115]. Thus, research on energy harvesters based on double-pendulum systems warrants further advancement. In particular, pendulum system energy harvesters are primarily applied for the conversion of ultra-low-frequency energy, such as oceanic kinetic and human motion energy. The output power of pendulum-like energy harvesters is mainly concentrated at the milliwatt and microwatt levels, which is suitable for low-power devices.

In the future, pendulum-type energy harvesters have the potential for optimization and improvement. As energy-harvesting technologies evolve, pendulum-type energy-harvesting systems can be integrated with several nonlinear structures to enhance performance. In recent studies, nonlinear magnetic interactions have been applied to single- [88] and double-pendulum systems [119], and noteworthy results have been obtained. In addition to magnetic interaction, other nonlinear systems, such as multistable structures [123], springs [124], multi-layer structures [125], and materials with nonlinear characteristics must be combined. A nonlinear system can increase the bandwidth and energy-conversion efficiency of energy harvesters. Furthermore, an intelligent algorithm is another strategy for optimizing energy harvesters. Specifically, intelligent algorithms, such as neural networks, have a strong nonlinear mapping ability [126]. Therefore, an intelligent algorithm can be applied to enhance energy harvesting performance [119,127].

From the perspective of engineering applications, current studies on pendulum-type energy harvesters focus on ocean and human motion energy. Therefore, the applications of pendulum system energy harvesters must be expanded to other fields. For instance, roadways and bridges are common engineering structures in life, which have the problem of structural vibrations [77]. In addition, hybrid energy harvesting is a promising approach. Complex internal coupling of hybrid systems consumes part of the electrical energy, thereby affecting the performance of energy harvesters [22]; however, this remains an effective method to enhance energy conversion efficiency.

5. Conclusions

The present article reviews research on the theoretical analyses of single and double pendulums and their applications in energy harvesting. The different energy-conversion mechanisms, design schemes, and optimal structures of pendulum-like energy harvesters have been summarized in the literature. Based on the existing research, single-pendulum energy harvesters are predominantly applied in ultra-low-frequency environments and for multi-directional energy harvesting. The working conditions of pendulum energy harvesters are <10 Hz, which mainly include ocean wave and human motion energy. Furthermore, single-pendulum systems can be combined with other functions, such as mechanical motion rectifiers and structural damping. In addition, energy harvesting based on double pendulums has attracted much research attention. Based on the above summary, we predict several future research directions and potential applications of pendulum-type energy harvesters:

(1)

From comparative experimental results, double-pendulum systems perform better than single-pendulum systems in terms of energy harvesting. However, studies on double-pendulum energy harvesters are relatively few, and research on energy harvesters based on double pendulums may be further advanced.

(2)

The combination of a nonlinear structure with a pendulum system energy harvester has been considerably optimized. The introduction of a nonlinear structure can improve the adaptive ability of energy harvesters. From the current studies, the energy harvesting performance can be effectively improved with the reasonable application of nonlinear systems.

(3)

Multi-physics coupling energy harvesting is another research topic. The coupling of different types of energy conversion mechanisms can broaden the operation bandwidth and improve output power.