A high-performance dual-mode energy harvesting with nonlinear pendulum and speed-amplified mechanisms for low-frequency applications

doi:10.1016/j.energy.2024.132553

Energy 能源

Volume 306, 15 October 2024, 132553
第 306 卷，2024 年 10 月 15 日，132553

https://doi.org/10.1016/j.energy.2024.132553 Get rights and content 获取权利和内容

Highlights 突出

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A novel speed-amplified dual-mode energy harvesting is proposed.
提出了一种新颖的速度放大双模能量收集。
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The system incorporates nonlinear pendulum and speed-amplified mechanisms.
该系统结合了非线性摆锤和速度放大机构。
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It achieves 3.79 W and 244 mW in rotation and vibration modes, respectively.
它在旋转和振动模式下分别达到 3.79 W 和 244 mW。
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It supports daily appliances and wireless monitoring for railways and humans.
它支持日常电器以及铁路和人类的无线监控。

Abstract 抽象

Vibrations induced by human motions and vehicle operations feature low-frequency, broadband, and time-varying. However, enhancing the power density of energy harvesting for various low-frequency applications presents a significant challenge. This paper proposes a high-performance dual-mode energy harvester (DM-EH) incorporating coupled nonlinear pendulum and speed-amplified mechanisms with gears for the simultaneous scavenging vibration and rotation energy. The pendulum serves as the energy capture unit for detecting vibration energy while ensuring the effective operation of the generation unit in rotational environments. The gears amplify the speed of the excitation and enable frequency up-conversion, thereby enhancing the energy conversion. Experimental results substantiate the DM-EH's ability to extract power from vehicle operations at speeds ranging from 60 to 540 rpm, as well as from human motions at frequencies of 0.7–1.5 Hz with 30° amplitudes. In rotation mode, the prototype achieves a maximum average direct current power of 3.79 W, while in vibration mode, it attains 244 mW. The prototype successfully powers portable electronics and supports battery-free triaxial acceleration and temperature multi-sensors during human motions and railway simulation tests. This prototype showcases immense potential as a sustainable power source for portable electronics and self-powered monitoring applications.
由人体运动和车辆操作引起的振动具有低频、宽带和时变等特点。然而，提高各种低频应用的能量收集功率密度是一项重大挑战。本文提出了一种高性能双模能量采集器（DM-EH），该能量采集器结合了耦合的非线性摆和速度放大机构，并带有齿轮，用于同时清除振动和旋转能量。摆锤用作能量捕获装置，用于检测振动能量，同时确保发电装置在旋转环境中有效运行。齿轮放大激励的速度并启用频率上转换，从而增强能量转换。实验结果证实，DM-EH 能够从 60 至 540 rpm 的车辆运行中获取动力，以及以 0.7-1.5 Hz 的频率和 30° 振幅从人体运动中提取动力。在旋转模式下，原型的最大平均直流功率为 3.79 W，而在振动模式下，它达到 244 mW。该原型成功地为便携式电子设备供电，并在人体运动和铁路模拟测试期间支持无电池的三轴加速度和温度多传感器。该原型展示了作为便携式电子产品和自供电监控应用的可持续电源的巨大潜力。

Graphical abstract 图形摘要

Keywords 关键字

Dual-mode

Energy harvesting

Nonlinear pendulum

Speed-amplified

Low-frequency

双模式

能量收集

非线性摆

锤

速度放大

低频

1. Introduction 1. 引言

5G, artificial intelligence, and the Internet of Things have taken off at a brisk speed over the years, creating an intelligent network via tremendous sensors deployed in various scenarios [1,2]. However, the energy supply limits its wide application, especially in some difficult-to-reach locations, such as rotational machines, wilderness operations, and marine environments [3,4]. The batteries as power sources are not sustainable, pollute the environment, and are heavy for large-capacity. How to effectively harvest the redundant energy in the environment and convert it into electricity for electronic devices and sensors, avoid frequent battery replacement to achieve long-term device operation, has brought widespread attention from academe and industries [5,6].
多年来，5G、人工智能和物联网迅速发展，通过部署在各种场景中的大量传感器创建了一个智能网络 [1,2]。然而，能源供应限制了其广泛应用，特别是在一些难以到达的地方，如旋转机器、野外作业和海洋环境[3,4]。作为电源的电池不可持续，污染环境，并且对于大容量来说很重。如何有效地收集环境中多余的能量并将其转化为电能，用于电子设备和传感器，避免频繁更换电池以实现设备的长期运行，已引起学术界和工业界的广泛关注 [5,6]。

The environment is replete with mechanical energy, thereby instigating the development of numerous transduction devices designed to convert mechanical vibrations into electrical energy [7,8]. These apparatuses exploit a variety of mechanisms, including electromagnetic [[9], [10], [11]], piezoelectric [[12], [13], [14]], and triboelectric [[15], [16], [17]] principles. Notably, ambient vibrations feature low-frequency, broadband, and time-varying, making efficient energy harvesting from such environmental sources exceedingly formidable [18,19]. The effectiveness of most vibration energy harvesters is heightened at higher excitation frequencies, given that the generated power is directly proportional to the cube of the vibration frequency [20]. Furthermore, a lower resonant frequency typically signifies a higher inertial mass or lower stiffness, consequently leading to the bulkiness of the harvesters [21]. To mitigate this challenge, innovative approaches such as frequency up-conversion techniques, including impact mechanism [[22], [23], [24]] and gear modulation [[25], [26], [27]] approach, have been proposed.
环境中充满了机械能，因此促使开发了许多旨在将机械振动转化为电能的转导装置[7,8]。这些装置利用了多种机制，包括电磁 [[9]、[10]、[11]、压电 [[12]、[13]、[14]] 和摩擦电 [[15]、[16]、[17]] 原则。值得注意的是，环境振动具有低频、宽带和时变的特点，这使得从此类环境源中高效收集能量变得非常强大[18,19]。鉴于产生的功率与振动频率的立方成正比，因此大多数振动能量收集器的效率在较高的激励频率下会更高[20]。此外，较低的谐振频率通常意味着较高的惯性质量或较低的刚度，从而导致收割机体积庞大 [21]。为了缓解这一挑战，已经提出了创新方法，例如频率上变频技术，包括冲击机构 [[22]、[23]、[24]] 和齿轮调制 [[25]、[26]、[27]] 方法。

The impact mechanism serves to transmute low-frequency excitation into high-frequency vibrations within the harvester through mechanical impacts or plucking [28], thereby accommodating low-frequency operational conditions and augmenting electromechanical conversion efficiency [29]. For example, the utilization of a ball [30] or a sliding mass [31] facilitates the conversion of human motion into high-frequency vibrations in a piezoelectric beam, culminating in an output voltage of 818 Hz when subjected to 5.8 Hz excitation [32]. The modulation of magnetic forces, designed to harmonize with the low-frequency lateral vibrations of a freight wagon, amplifies the resonant frequency of the power-generating beam, ultimately leading to a remarkable 150 % increase in electromechanical conversion efficiency [33]. Using a magnetic plucking mechanism converts omnidirectional and low-frequency vibration into high-frequency vibration of generating beam [34]. Furthermore, the implementation of a rotation-enhanced impact mechanism can boost output power by an astounding 682.8 % [35], primarily due to centrifugal forces that augment the impact's intensity [36]. The centrifugal force coupled with the nonlinearity surface impact mechanism improves the output power [37]. Nevertheless, it is imperative to acknowledge that, although the impact mechanism offers an efficient means to transfigure low-frequency motions into harvester's resonances, challenges concerning energy losses and mechanical wear persist and necessitate further enhancement. Additionally, it's worth noting that energy transfer occurs only once per cycle, imposing limitations on the prospects for elevating power density.
冲击机制的作用是通过机械冲击或拨动[28]将收割机内部的低频激励转化为高频振动[28]，从而适应低频运行条件并提高机电转换效率[29]。例如，使用球[30]或滑动质量[31]有助于将人体运动转换为压电束中的高频振动，在5.8 Hz激励下达到818 Hz的输出电压[32]。磁力的调制旨在与货车的低频横向振动相协调，放大了发电梁的谐振频率，最终使机电转换效率显著提高了 150% [33]。使用磁力拔弦机构将全向和低频振动转化为产生光束的高频振动[34]。此外，旋转增强冲击机构的实施可以将输出功率提高惊人的 682.8% [35]，这主要是由于离心力增加了冲击强度 [36]。离心力与非线性表面冲击机制相结合，提高了输出功率[37]。然而，必须承认的是，尽管冲击机制提供了一种将低频运动转化为收割机共振的有效方法，但有关能量损失和机械磨损的挑战仍然存在，需要进一步增强。此外，值得注意的是，能量转移每个周期只发生一次，这限制了提高功率密度的前景。

In parallel with the impact mechanism, several gear modulation techniques, including rack-pinion [38,39], ball screw [40,41], and lead screw [42,43] configurations, have been explored for the conversion of low-frequency vibrations into accelerated rotation. The rack-pinion mechanism, for instance, transforms reciprocating vibrations experienced in suspension, railway settings, and human motions into the high-speed rotation of the generator [[44], [45], [46]]. Moreover, this mechanism enhances the relative motion between magnets and coils, thus improving the output performance of the harvester [[47], [48], [49]]. While the ball screw encounters rotational speed limitations due to internal ball wear within the nut, the lead screw may confront challenges associated with friction and substantial damping forces [50]. These mechanical structures are proficient at converting low-frequency and irregular reciprocating motions into unidirectional rotation. Nonetheless, their substantial physical dimension presents installation challenges and potential operational safety concerns for freight trains or humans.
与冲击机制并行的同时，已经探索了几种齿轮调制技术，包括齿条-小齿轮[38\u201239]、滚珠丝杠[40\u201241]和丝杠[42\u201243]配置，用于将低频振动转换为加速旋转。例如，齿轮齿条机构将悬架、铁路设置和人体运动中经历的往复振动转化为发电机的高速旋转 [[44]， [45]， [46]]。此外，这种机制增强了磁铁和线圈之间的相对运动，从而提高了收割机的输出性能[[47]，[48]，[49]]。虽然滚珠丝杠由于螺母内部的滚珠磨损而遇到转速限制，但丝杠可能面临与摩擦和巨大阻尼力相关的挑战[50]。这些机械结构擅长将低频和不规则的往复运动转换为单向旋转。尽管如此，它们巨大的物理尺寸给货运列车或人类带来了安装挑战和潜在的操作安全问题。

Although the aforementioned harvesters could scavenge low-frequency energy through mechanical modulation, the energy loss and bulky size constrain the power density. Furthermore, for wearables such as smartwatches and wristbands, it is impractical to embed bulky energy harvesters as power sources. Besides, to the best of the authors' knowledge, there is rarely a report on the electromagnetic energy harvester that can harvest both the vibration and rotation energy for adapting to complex low-frequency environments. Therefore, this paper introduces a high-performance speed-amplified dual-mode energy harvesting (DM-EH) with a nonlinear pendulum mechanism, which not only harvests the vibration energy from human motions but also possesses the capability of scavenging the rotational energy from vehicle operations. The contributions of the DM-EH are as follows: (1) The pendulum serves a pivotal role in enabling the device's dual-mode operation. Specifically, it functions as the energy capture unit for detecting vibration energy while ensuring that the all-in-one generation unit operates effectively in rotational environments. (2) The prototype generates an average output power of 3.79 W and 244 mW with an optimal resistance load of 160 Ω, resulting in a power density of 182.58 mW cm⁻³ and 11.74 mW cm⁻³ at 540 rpm and a frequency of 1.5 Hz with an angle of 30°, respectively. (3) As the applications, the DM-EH exhibits the capability to power a range of daily electronic devices and commercial wireless sensors, including LED lights, smartwatches, smartphones, mobile power banks, acceleration sensors, and temperature sensors.
尽管上述收集器可以通过机械调制来清除低频能量，但能量损失和笨重的尺寸限制了功率密度。此外，对于智能手表和腕带等可穿戴设备，将笨重的能量收集器嵌入为电源是不切实际的。此外，据作者所知，很少有关于电磁能量采集器的报道可以同时收集振动和旋转能量以适应复杂的低频环境。因此，本文介绍了一种具有非线性摆锤机构的高性能速度放大双模能量收集（DM-EH），它不仅可以收集人体运动的振动能量，还具有从车辆运行中收集旋转能量的能力。DM-EH 的贡献如下：（1）摆锤在实现设备的双模式操作中起着关键作用。具体来说，它用作能量捕获装置，用于检测振动能量，同时确保一体式发电机组在旋转环境中有效运行。（2）原型产生的平均输出功率为 3.79 W 和 244 mW，最佳电阻负载为 160 Ω，在 540 rpm 和频率为 1.5 Hz、角度为 30° 时，功率密度分别为 182.58 mW ^cm-3 和 11.74 mW ^cm-3。（3）作为应用，DM-EH 能够为一系列日常电子设备和商用无线传感器供电，包括 LED 灯、智能手表、智能手机、移动电源、加速度传感器和温度传感器。

The paper's structural framework is delineated as follows: Section 2 elaborates on the design and working principle of the DM-EH. Section 3 presents modeling, dynamic analysis, and finite-element simulation of the prototype. Section 4 is devoted to the incorporation of experimental tests and the application of the harvester. The main findings are summarized in Section 5.
本文的结构框架描述如下：第 2 节详细阐述了 DM-EH 的设计和工作原理。第 3 节介绍了原型的建模、动态分析和有限元仿真。第 4 节专门介绍了实验测试和收割机的应用。主要发现总结在第 5 节中。

2. Design and working principle
2. 设计及工作原理

Fig. 1 provides an overview of the self-powered monitoring harvester designed for low-frequency environments, encompassing human motions, vehicle operations, and ocean waves. In recognition of the diverse nature of low-frequency motions, such as human limb vibration and the rotational movement of freight trains, this paper has engineered a high-performance speed-amplified dual-mode energy harvester (DM-EH) by integrating a pendulum mechanism into the frequency up-conversion technology.
图 1 概述了专为低频环境设计的自供电监控采集器，包括人体运动、车辆操作和海浪。认识到低频运动的多样性，例如人的肢体振动和货运列车的旋转运动，本文通过将摆锤机构集成到频率上变频技术中，设计了一种高性能的速度放大双模能量收集器（DM-EH）。

In Fig. 2a, a comprehensive structural overview of the DM-EH is provided through an exploded assembly diagram. The DM-EH is composed of three key components: a transmission unit, a generation unit, and a circuit. The transmission unit encompasses a sun gear, a planetary gear, and a substrate. The sun gear and planetary gear are positioned at the outer periphery of the substrate, with the sun gear's rotation located at the substrate's edge, while the planet gear's rotation is positioned in the central portion of the substrate, thus facilitating meshing between the sun gear and the planet gear. The generation unit incorporates a Halbach array magnet and a set of coils featuring permeability materials to augment the magnetic flux passing through the coils. The circuit includes an energy management module, encompassing a rectifier, voltage regulator, storage, and wireless sensing module.
在图 2a 中，通过分解装配图提供了 DM-EH 的全面结构概述。DM-EH 由三个关键部件组成：传输单元、发电单元和电路。传动装置包括一个太阳轮、一个行星齿轮和一个基体。太阳轮和行星齿轮位于基板的外周，太阳轮的旋转位于基板的边缘，而行星轮的旋转位于基板的中心部分，从而促进太阳轮和行星轮之间的啮合。该发电单元包括一个 Halbach 阵列磁体和一组具有磁导率材料的线圈，以增加通过线圈的磁通量。该电路包括一个能量管理模块，包括整流器、稳压器、存储和无线传感模块。

Fig. 2b describes the working principle and energy flows of the DM-EH. When the structure is installed on the swing vibration part, the planetary gear revolves around the sun gear in response to excitations. Concurrently, the planetary gear impels the permanent magnet to generate high-speed rotation. The coil is connected to the permanent magnet via bearings, ensuring that the coil remains stationary for the planetary gear. This arrangement results in relative motion between the coil and the permanent magnet. On the other hand, when the device is installed within the rotational component, the sun gear rotates in synchronization with rotation excitation. Subject to the combined influences of gravity torque, magnetic torque, and friction torque, the planetary gear orbits around the sun gear until it attains a state of equilibrium. In this state, the sun gear drives the planetary gear to produce high-speed rotation along with the permanent magnets. Under the condition of swing and rotational vibrations, both the permanent magnet and the coil experience relative motion. Hence the generation unit serves three vital functions: (1) functions as a counterweight, efficiently converting low-frequency vibrations into high-speed rotation, thus enhancing electromechanical energy conversion performance; (2) the smooth operation of the transmission unit by transforming the rotation of the sun gear into high-speed rotation of the planetary gear under anchorless condition of planetary gear; (3) the primary function is to convert mechanical energy into the relative motion of magnets and coils. According to Faraday's law of electromagnetic induction, the coil generates induced current as shown in Fig. 2c. It is worth mentioning that a group of coils was selected as a representative to demonstrate the power generation progress due to the circular array configuration. Besides, to facilitate the comparison of the harvester states, the magnets remain stationary while the coils rotate during the simulation.
图 2b 描述了 DM-EH 的工作原理和能量流。当结构安装在摆动振动部分时，行星齿轮响应激励围绕太阳轮旋转。同时，行星齿轮推动永磁体产生高速旋转。线圈通过轴承连接到永磁体，确保行星齿轮的线圈保持静止。这种布置导致线圈和永磁体之间的相对运动。另一方面，当设备安装在旋转部件内时，太阳轮与旋转激励同步旋转。受重力扭矩、磁扭矩和摩擦扭矩的综合影响，行星齿轮围绕太阳轮旋转，直到达到平衡状态。在这种状态下，太阳轮驱动行星齿轮与永磁体一起产生高速旋转。在摆动和旋转振动的情况下，永磁体和线圈都会经历相对运动。因此，发电机组具有三个重要功能：（1）起到配重的作用，有效地将低频振动转化为高速旋转，从而提高机电能转换性能;（2）在行星齿轮无锚的情况下，通过将太阳轮的旋转转化为行星齿轮的高速旋转，使传动装置的平稳运行;（3）主要功能是将机械能转化为磁铁和线圈的相对运动。根据法拉第电磁感应定律，线圈产生感应电流，如图 2c 所示。值得一提的是，由于圆形阵列配置，选择了一组线圈作为代表来演示发电进度。此外，为了便于比较收集器状态，在仿真过程中，当线圈旋转时，磁体保持静止。

This structural design offers the following advantages: (1) The generation unit facilitates the conversion of low-frequency vibrations to high-speed rotations, accommodating the low-frequency environments while meeting the high-frequency input requirements of the generation unit. (2) The prototype is capable of concurrently capturing both vibration and rotation motions, effectively harnessing multiple sources of excitation to improve the system's adaptability. (3) The generation unit serves as the inertial mass block of the pendulum, obviating the need for additional counterweights and simplifying the structural design.
这种结构设计具有以下优点：（1）发电单元有助于将低频振动转换为高速旋转，在满足发电单元的高频输入要求的同时适应低频环境。（2）原型能够同时捕捉振动和旋转运动，有效地利用多种激励源来提高系统的适应性。（3）发电单元用作摆锤的惯性质量块，无需额外的配重并简化了结构设计。

3. Modeling and dynamic analysis of the DM-EH
3. DM-EH 的建模和动力学分析

To further reveal the working principle and advantages of the DM-EH, an equivalent mechanical system model under human motion and vehicle operation is established to analyze its dynamic characteristics.
为了进一步揭示 DM-EH 的工作原理和优势，建立了人体运动和车辆运行下的等效机械系统模型，分析其动力学特性。

3.1. Theoretical modeling of the DM-EH under human motion
3.1. 人体运动下 DM-EH 的理论建模

A pendulum is used to simulate the human arm motion. As shown in Fig. 3a, the DM-EH is fixed at the end of the pendulum. The detailed parameters of the DM-EH are given in Table 1. The coordinate O₀ signifies the inertial frame, depicting motion exclusively in the X-direction, while coordinate O₁ represents the reference frame fixed to the sun gear of the DM-EH, and O₂ designates the reference frame anchored to the base of the DM-EH. The coordinate O₁ in the fixed coordinate system can be described as(1)

\begin{array}{l} x_{1} = x_{0} + L \sin θ \\ y_{1} = - L \cos θ \end{array}

钟摆用于模拟人的手臂运动。如图 3a 所示，DM-EH 固定在摆锤的末端。表 1 给出了 DM-EH 的详细参数。坐标 O₀ 表示惯性系，仅描述 X 方向的运动，而坐标 O₁ 表示固定在 DM-EH 太阳轮上的参考系，O₂ 表示锚定在 DM-EH 底座上的参考系。固定坐标系中的坐标 O₁ 可以描述为 (1)

\begin{array}{l} x_{1} = x_{0} + L \sin θ \\ y_{1} = - L \cos θ \end{array}

Table 1. Material properties and structural parameters of the proposed DM-EH.
表 1.所提出的 DM-EH 的材料性能和结构参数。

Parameter 参数	Value 价值	Parameter 参数	Value 价值
Number of magnets 磁铁数量	24	Size of magnets (ring) 磁铁尺寸（环）	R15 × 14 × 10 mm R15 × 14 × 10 毫米
Material of magnets 磁铁材料	NdFeB-N35 钕铁硼-N35	Norm B 标准 B	1.5 T 1.5 吨
Diameter of wire 线径	0.1 mm 0.1 毫米	Number of coil turns 线圈匝数	500
Internal resistance 内阻	160 Ω	Coil area 线圈面积	5 × 11 mm² 5 × 11 毫米²
The mass of the pendulum m 摆锤的质量 m	50.1 g 50.1 克	Moment of the pendulum I 钟摆时刻 I	23.22 g cm² 23.22 克^{cm 2}
Length of the pendulum 摆的长度l	15 mm 15 毫米	Mechanical damping c 机械阻尼 c	2e-4 2E-4 型
The radius of the sun gear r 太阳轮半径 r	13 mm 13 毫米	Transmission ratio n 传动比 n	5
Distance between O₀ and O₁ L O₀ 和 O₁L 之间的距离	50 cm 50 厘米	Costs of the prototype 原型的成本	10.25 USD 10.25 美元
Weight of the prototype 原型的重量	78.4 g 78.4 克	Size of the prototype 原型的大小	20.79 cm³ 20.79 厘米³

The velocity of the coordinate O₁ is as follows(2)

\begin{array}{l} {\dot{x}}_{1} = v_{0} + L \dot{θ} \cos θ \\ {\dot{y}}_{1} = L \dot{θ} \sin θ \end{array}

坐标 O₁ 的速度如下 (2)

\begin{array}{l} {\dot{x}}_{1} = v_{0} + L \dot{θ} \cos θ \\ {\dot{y}}_{1} = L \dot{θ} \sin θ \end{array}

The acceleration velocity(3)

\begin{array}{l} {\ddot{x}}_{1} = a_{0} + L α \cos θ - L ω^{2} \sin θ \\ {\ddot{y}}_{1} = L α \sin θ + L ω^{2} \cos θ \end{array}

加速度 (3)

\begin{array}{l} {\ddot{x}}_{1} = a_{0} + L α \cos θ - L ω^{2} \sin θ \\ {\ddot{y}}_{1} = L α \sin θ + L ω^{2} \cos θ \end{array}

The coordinate O₂ can be written as(4)

\begin{array}{l} x = x_{0} + L \sin θ + l \sin ϕ \\ y = - L \cos θ - L \cos ϕ \end{array}

坐标 O₂ 可以写成 (4)

\begin{array}{l} x = x_{0} + L \sin θ + l \sin ϕ \\ y = - L \cos θ - L \cos ϕ \end{array}

The velocity of the coordinate O₂(5)

\begin{array}{l} \dot{x} = v_{0} + L ω \cos θ + l \dot{ϕ} \cos ϕ \\ \dot{y} = L ω \sin θ + l \dot{ϕ} \sin ϕ \end{array}

坐标 O₂ (5)

\begin{array}{l} \dot{x} = v_{0} + L ω \cos θ + l \dot{ϕ} \cos ϕ \\ \dot{y} = L ω \sin θ + l \dot{ϕ} \sin ϕ \end{array}

的速度

The acceleration velocity of the coordinate O₂(6)

\begin{array}{l} \ddot{x} = a_{0} + L α \cos θ - L ω^{2} \sin θ + l \ddot{ϕ} \cos ϕ - l {\dot{ϕ}}^{2} \sin ϕ \\ \ddot{y} = L α \sin θ + L ω^{2} \cos θ + l \ddot{ϕ} \sin ϕ + L {\dot{ϕ}}^{2} \cos ϕ \end{array}

坐标 O 的加速度 O₂ (6)

\begin{array}{l} \ddot{x} = a_{0} + L α \cos θ - L ω^{2} \sin θ + l \ddot{ϕ} \cos ϕ - l {\dot{ϕ}}^{2} \sin ϕ \\ \ddot{y} = L α \sin θ + L ω^{2} \cos θ + l \ddot{ϕ} \sin ϕ + L {\dot{ϕ}}^{2} \cos ϕ \end{array}

The dynamic equation of the pendulum(7)

(I + m l^{2}) \ddot{ϕ} + F_{f} r + m g l \sin ϕ + F_{m} R = F l

摆 (7)

(I + m l^{2}) \ddot{ϕ} + F_{f} r + m g l \sin ϕ + F_{m} R = F l

锤的动力学方程

The electromagnetic damping force(8)

F_{m} = N P B I (t) L_{c} = \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2} n \dot{ϕ}}{R_{c} + R_{e}}

I (t) = \frac{E (t)}{R_{c} + R_{e}} = N \frac{d Φ / d t}{R_{c} + R_{e}} = N \frac{d B S / d t}{R_{c} + R_{e}} = N P B \frac{R^{2} L_{c} n \dot{ϕ}}{R_{c} + R_{e}}

电磁阻尼力 (8)

F_{m} = N P B I (t) L_{c} = \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2} n \dot{ϕ}}{R_{c} + R_{e}}

I (t) = \frac{E (t)}{R_{c} + R_{e}} = N \frac{d Φ / d t}{R_{c} + R_{e}} = N \frac{d B S / d t}{R_{c} + R_{e}} = N P B \frac{R^{2} L_{c} n \dot{ϕ}}{R_{c} + R_{e}}

The mechanical damping(9)

F_{f} = m c \dot{ϕ}

机械阻尼 (9)

F_{f} = m c \dot{ϕ}

The inertia driving force(10)

\begin{array}{l} F = m a = m \sqrt{{\ddot{x}}^{2} + {\ddot{y}}^{2}} = m (l \ddot{ϕ} + a_{0} \cos ϕ + L \ddot{ϕ} \cos θ \cos ϕ - L ω^{2} \sin θ \cos ϕ \\ + L \ddot{ϕ} \sin θ \sin ϕ + L ω^{2} \cos θ \sin ϕ) \end{array}

惯性驱动力 (10)

\begin{array}{l} F = m a = m \sqrt{{\ddot{x}}^{2} + {\ddot{y}}^{2}} = m (l \ddot{ϕ} + a_{0} \cos ϕ + L \ddot{ϕ} \cos θ \cos ϕ - L ω^{2} \sin θ \cos ϕ \\ + L \ddot{ϕ} \sin θ \sin ϕ + L ω^{2} \cos θ \sin ϕ) \end{array}

The dynamic equation of the pendulum(11)

\frac{I}{m l} \ddot{ϕ} + (\frac{c r}{l} + \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2}}{R_{c} + R_{e}} n) \dot{ϕ} + g \sin ϕ = (a_{0} \cos ϕ + L α \cos θ \cos ϕ - L ω^{2} \sin θ \cos ϕ + L α \sin θ \sin ϕ + L ω^{2} \cos θ \sin ϕ)

摆 (11)

\frac{I}{m l} \ddot{ϕ} + (\frac{c r}{l} + \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2}}{R_{c} + R_{e}} n) \dot{ϕ} + g \sin ϕ = (a_{0} \cos ϕ + L α \cos θ \cos ϕ - L ω^{2} \sin θ \cos ϕ + L α \sin θ \sin ϕ + L ω^{2} \cos θ \sin ϕ)

锤的动力学方程

To analyze the dynamic characteristics of the DM-EH, constant frequency excitation simulations were conducted within the frequency range of 0.7–1.5 Hz. Three arm swing angles (20°, 25°, and 30°) were selected as excitation levels to investigate the system's dynamic behavior. It is assumed that the arm swings at a constant speed, implying an angular acceleration α of 0. Fig. 3b and c illustrate the variation in DM-EH swing angle under different excitation levels with ODE 45 in Matlab 2019. It is worth noting that the pendulum exhibits nonlinear characteristics for its swing angle. Each point on the graph represents the amplitude of the swing at a specific frequency for each excitation cycle. The results show that with the increase of frequency and swing Angle, the swing Angle of the structure gradually increases, which means that the output voltage also increases.
为了分析 DM-EH 的动态特性，在 0.7-1.5 Hz 的频率范围内进行了恒频激励模拟。选择三个臂摆动角度（20°、25° 和 30°）作为激励水平，以研究系统的动态行为。假设手臂以恒定速度摆动，这意味着角加速度α为 0。图 3b 和 c 说明了 Matlab 2019 中 ODE 45 在不同激发水平下 DM-EH 摆动角的变化。值得注意的是，摆锤的摆动角度表现出非线性特性。图中的每个点都表示每个激励周期在特定频率下的摆动幅度。结果表明，随着频率和摆幅角的增加，结构的摆幅角逐渐增大，这意味着输出电压也随之增加。

3.2. Theoretical modeling of the DM-EH under vehicle operation
3.2. 车辆运行下 DM-EH 的理论建模

The coordinates O₀ and O₁ coincide with each other under the rotation condition, as shown in Fig. 4a. The dynamic equation of a friction pendulum can be described as(12)

\frac{I}{m l} \ddot{ϕ} + g \sin ϕ = (\frac{c r}{l} + \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2}}{R_{c} + R_{e}}) n (Ω - \dot{ϕ})

在旋转条件下，坐标 O₀ 和 O₁ 相互重合，如图 4a 所示。摩擦摆的动力学方程可以描述为 (12)

\frac{I}{m l} \ddot{ϕ} + g \sin ϕ = (\frac{c r}{l} + \frac{N^{2} P^{2} B^{2} R^{2} L_{c}^{2}}{R_{c} + R_{e}}) n (Ω - \dot{ϕ})

The pendulum's motion unfolds in two distinct stages: adhesion and detachment, representing the alternating interplay between static friction torque and dynamic friction torque (Fig. 4b), and the pendulum exhibits a piecewise nonlinearity. During the adhesion stage, the static friction torque and magnetic torque drive the friction pendulum and shaft to rotate synchronously. This continues until the accumulated torque reaches the point where it equals the sum of the maximum static friction torque and magnetic torque. In the detachment stage, the static friction torque converts into dynamic friction torque. However, as the dynamic friction torque is lower than the maximum static friction torque, the friction pendulum reverses its rotation under the influence of the applied torque(13)

(m_{c} c + \frac{N^{2} P^{2} B^{2} R^{3} L_{c}^{2}}{R_{c} + R_{e}}) i (Ω - \dot{ϕ}) = m_{c} g l

摆锤的运动分为两个不同的阶段：粘附和分离，代表静摩擦扭矩和动摩擦扭矩之间的交替相互作用（图 4b），摆锤表现出分段非线性。在粘附阶段，静摩擦力矩和磁力矩驱动摩擦摆和轴同步旋转。这种情况一直持续到累积扭矩达到等于最大静摩擦扭矩和磁扭矩之和的点。在分离阶段，静摩擦扭矩转化为动摩擦扭矩。然而，由于动摩擦扭矩低于最大静摩擦扭矩，摩擦摆在施加 (13)

(m_{c} c + \frac{N^{2} P^{2} B^{2} R^{3} L_{c}^{2}}{R_{c} + R_{e}}) i (Ω - \dot{ϕ}) = m_{c} g l

扭矩的影响下会反转其旋转

Variations in the rotational speed of the rotor magnet induce distinct motion patterns in the friction pendulum. With increasing external speed, the motion amplitude of the friction pendulum gradually escalates until it attains π/2, marking the point referred to as the critical speed. The cumulative energy buildup eventually prompts the friction pendulum to rotate by π, resulting in the planetary gear to orbit around the solar gear. When the rotor magnet's speed exceeds the critical speed, the coil generates motion relative to the shaft end cover of the freights, leading to device malfunction. Consequently, this section undertakes a theoretical analysis of the critical speed. When the friction pendulum synchronously rotates to an angle of π/2 under the combined influence of static friction torque and magnetic torque, the gravitational moment reaches its peak(14)

m g l = (\frac{c r}{l} + \frac{{(N B L_{c} R)}^{2}}{m l R_{c}}) \dot{ϕ}

转子磁铁转速的变化会在摩擦摆中引起不同的运动模式。随着外部速度的增加，摩擦摆的运动幅度逐渐增加，直到达到 π/2，标记称为临界速度的点。累积的能量积累最终促使摩擦摆旋转 π，导致行星齿轮围绕太阳能齿轮旋转。当转子磁铁的速度超过临界速度时，线圈会相对于货物的轴端盖产生运动，从而导致设备故障。因此，本节对临界速度进行了理论分析。当摩擦摆在静摩擦力矩和磁力矩的共同影响下同步旋转到 π/2 的角度时，重力矩达到峰值 (14)

m g l = (\frac{c r}{l} + \frac{{(N B L_{c} R)}^{2}}{m l R_{c}}) \dot{ϕ}

Fig. 4c illustrates the three-dimensional correlation of the critical mass, external excitation frequency, and resistance. As the excitation frequency rises, the external weight also increases. Simultaneously, the critical mass experiences a gradual reduction with the increase in resistance. This decrease in resistance leads to a decrease in circuit current, resulting in a diminished magnetic moment, which in turn weakens the external force responsible for upward pendulum movement.
图 4c 说明了临界质量、外部激励频率和电阻的三维相关性。随着激励频率的增加，外部重量也增加。同时，临界质量随着电阻的增加而逐渐减少。电阻的降低导致电路电流的减小，从而导致磁矩减小，进而减弱了负责钟摆向上运动的外力。

3.3. Finite-element of the generation unit
3.3. 生成单元的有限元

To achieve a reasonable allocation of the magnets in DM-EH, magnetic flux density around the coils’ boundary with different magnetic pole pairs was simulated by COMSOL Multiphysics 6.0. Compared with conventional arrayed magnets, the Halbach arrayed magnets generate concentrated one-side magnetic flux, leading to the production of more power in coils (Fig. 5a).
为了实现磁体在 DM-EH 中的合理分配，COMSOL Multiphysics 6.0 模拟了具有不同磁极对的线圈边界周围的磁通密度。与传统的阵列磁体相比，Halbach 阵列磁体产生集中的单侧磁通量，从而在线圈中产生更多的功率（图 5a）。

The increased pole pairs improve the value of the magnetic flux density, uniform the magnetic flux density, and add the waveform of magnetic flux density in one cycle as shown in Fig. 5b. Nevertheless, the trend of growth slows down when the magnetic pole pairs beyond 6. Besides, the increased magnetic pole pairs aggravate the difficulty of processing and installation of magnets. Therefore, the Halbach array with 6 magnetic pole pairs prototypes the DM-EH.
增加的磁极对提高了磁通量密度的值，均匀了磁通量密度，并在一个周期内增加了磁通量密度的波形，如图 5b 所示。然而，当磁极对超过 6 时，增长趋势会减慢。此外，磁极对数的增加加剧了磁铁加工和安装的难度。因此，具有 6 个磁极对的 Halbach 阵列是 DM-EH 的原型。

4. Performance investigation
4. 性能调查

4.1. Fabrication of the DM-EH
4.1. DM-EH 的制造

The DM-EMH consists of a generation unit, a transmission unit, and a circuit. Generation unit: a base connected to the three-phase coil through a sleeve, featuring an inner diameter of 3 mm, an outer diameter of 9.7 mm, and a height of 5 mm. The rotor contains a ring magnet with dimensions of R15 × 14 × 10 mm, comprising 6 pairs of Halbach array magnets. These magnets are embedded in the rotor and attached to the planetary gear using a shaft and MR63ZZ bearings. It is worth noting that magnetism materials encapsulate the coils to enhance the magnetic flux passing through them. Transmission unit: a 0.5 mold 50-tooth sun gear, in conjunction with an MR63ZZ bearing, was affixed to the base through an R3 × 10 mm lug boss. The sun gear diameter, center hole, and tooth thickness are 26 mm, 6 mm, and 5 mm, respectively. The sun gear engaged with a 0.5 mold 10-tooth planetary gear. Circuit: a three-phase rectifier is constructed using six A7 diodes, while zener diodes ensure a stable DC voltage within the capacitances.
DM-EMH 由一个发电单元、一个传输单元和一个电路组成。发电单元：通过套筒与三相线圈相连的底座，内径为 3 mm，外径为 9.7 mm，高度为 5 mm。转子包含一个尺寸为 R15 × 14 × 10 mm 的环形磁体，由 6 对 Halbach 阵列磁体组成。这些磁铁嵌入转子中，并使用轴和 MR63ZZ 轴承连接到行星齿轮上。值得注意的是，磁性材料封装了线圈以增强通过它们的磁通量。传动装置：一个 0.5 模 50 齿太阳轮，与 MR63ZZ 轴承一起，通过 R3 × 10 毫米凸台固定在底座上。太阳轮直径、中心孔和齿厚分别为 26 mm、6 mm 和 5 mm。太阳轮与 0.5 模 10 齿行星齿轮啮合。电路：三相整流器由六个 A7 二极管构成，而齐纳二极管确保电容内稳定的直流电压。

4.2. Characterization and measurements
4.2. 表征和测量

Ensuring exceptional electrical output from the harvester is paramount, particularly for portable power and battery-free wireless sensor applications. Therefore, the energy harvesting performance of the DM-EH in rotation mode was investigated, as depicted in Fig. 6a–c. The prototyped DM-EH was driven by a speed-controlled servo motor (model: HZXT-008, Wuxi Houde Automation Meter CO., LTD.), and the corresponding output voltage at various speeds was recorded and captured using an oscilloscope (model: DSOX1204G, Keysight). Employing Origin 2018 processes experimental data and generates visual images. Fig. 6b demonstrates the alternating current (AC) generated by the DM-EH ranging from 60 to 540 rpm, effectively covering the typical speed of 80 km h⁻¹ for freight trains in China [51,52]. Notably, the peak voltage reached 3.12 V at 60 rpm, gradually increasing with the growth of rotation speed, and eventually arriving at a peak of 25.14 V at 540 rpm. The direct current (DC) performance was further examined for most wireless sensors requiring DC voltage (Fig. 6c and d). It's evident that with increasing speed, the corresponding RMS voltage improved, as the induced voltage is highly dependent on the relative speed between the magnets and coils. Notably, the experiments involved testing 10 groups of DC voltage measurements, each conducted for 60 s at a constant speed, to ensure the validity of the experimental data. The DC voltage is a directly measured parameter, and the root mean square (RMS) values of this voltage output over 10 min were calculated.
确保采集器的出色电力输出至关重要，特别是对于便携式电源和无电池无线传感器应用。因此，研究了 DM-EH 在旋转模式下的能量收集性能，如图 6a-c 所示。原型 DM-EH 由调速伺服电机（型号：HZXT-008，无锡厚德自动化仪表有限公司）驱动，并使用示波器（型号：DSOX1204G，Keysight）记录和捕获各种速度下的相应输出电压。使用 Origin 2018 处理实验数据并生成视觉图像。图6b显示了DM-EH产生的交流电（AC），范围为60至540 rpm，有效地覆盖了中国货运列车80 km ^h-1的典型速度[51,52]。值得注意的是，峰值电压在 60 rpm 时达到 3.12 V，随着转速的增长而逐渐增加，最终在 540 rpm 时达到 25.14 V 的峰值。对于大多数需要直流电压的无线传感器，进一步研究了直流（DC）性能（图 6c 和 d）。很明显，随着速度的增加，相应的 RMS 电压会提高，因为感应电压在很大程度上取决于磁体和线圈之间的相对速度。值得注意的是，实验涉及测试 10 组直流电压测量，每组以恒定速度进行 60 秒，以确保实验数据的有效性。直流电压是直接测量的参数，计算了 10 分钟内该电压输出的均方根（RMS）值。

To assess the output power capability, the prototype was connected to a resistance box, where the resistance ranged from 10 to 500 Ω with steps of 10 Ω in the range of 10–300 Ω and 100 Ω in the range of 300–500 Ω, respectively. The output voltage and power versus resistance at typical frequencies revealed that the prototype achieved maximum power when the external resistance closely matched the coils' internal resistance (160 Ω). The maximum average power reached 59 mW - 3.79 W with optimal resistance at 60–540 rpm, resulting in the highest power density reaching 182.58 mW cm⁻³.
为了评估输出功率能力，将原型连接到一个电阻箱，其中电阻范围为 10 至 500 Ω，步长为 10 Ω，范围为 10-300 Ω，步长为 100 Ω，范围分别为 300-500 Ω。典型频率下的输出电压和功率与电阻的关系表明，当外部电阻与线圈的内阻（160 Ω）紧密匹配时，原型实现了最大功率。最大平均功率达到 59 mW - 3.79 W，在 60-540 rpm 时具有最佳阻力，导致最高功率密度达到 182.58 mW ^cm-3。

To analyze the arm swing angle during various body movements, gait analysis was conducted using an optical 3D motion capture system (Model: NOKOV Mars 4H), as depicted in Fig. 7a. This system accurately captured real-time 3D coordinates of each marker node on the human body during motion, facilitating subsequent 3D reconstruction and model analysis. The arm angle data under DM-EH on and off conditions during slow walking, fast walking, jogging, and rushing scenarios are presented in Fig. 7b. The results indicate that the arm swing angle has little change with the DM-EH compared to the DM-EH off conditions. To determine the swing frequency of the human arm during daily movements, the swing characteristics of the human arm at speeds ranging from 2 to 10 km h⁻¹ were characterized. Fourier transform analysis revealed the arm swing frequency ranges from 0.7 Hz to 1.5 Hz, as demonstrated in Fig. 7c.
为了分析各种身体运动过程中的手臂摆动角度，使用光学 3D 动作捕捉系统（型号：NOKOV Mars 4H）进行了步态分析，如图 7a 所示。该系统在运动过程中准确捕获人体上每个标记节点的实时 3D 坐标，便于后续的 3D 重建和模型分析。图 7b 显示了慢走、快走、慢跑和冲刺场景下 DM-EH 开关条件下的手臂角度数据。结果表明，与 DM-EH 关闭条件相比，DM-EH 的手臂摆动角度变化不大。为了确定人体手臂在日常运动中的摆动频率，表征了人体手臂在 2 至 10 km ^h-1 速度范围内的摆动特性。傅里叶变换分析显示手臂摆动频率范围为 0.7 Hz 至 1.5 Hz，如图 7c 所示。

To investigate the performance of the DM-EH at swing conditions, an experiment platform simulates the human motion as shown in Fig. 8a. A speed-controlled servo motor drove the crank linkage, and the corresponding output voltage at different frequencies and signals were recorded and saved by an oscilloscope (model: DSOX1204G, Keysight). It is worth noting that the speed of the motor and the length of the rocker tune the swing frequency and angle of the rod. Since the amplitude and frequency at 1.6 km h⁻¹ walking speed are approximately 20° and 0.7 Hz, the 4 km h⁻¹ fast walking corresponding to 25° and 0.9 Hz, and the 10 km h⁻¹ running speed matching to 30° and 1.5 Hz, the rotor frequency range was set 0.7–1.5 Hz to simulate the vibration of arms. Fig. 8b shows the performance of the DM-EH at the frequency range from 0.7 to 1.5 Hz for amplitudes of 20, 25, and 30°. It can be found that the increased amplitude or frequency promotes the output voltage due to the excitation acceleration highly depending on the frequency and amplitude. The RMS DC voltage is 2.13–6.25 V at the frequency from 0.7 to 1.5 Hz with 30° amplitudes. The maximum RMS DC power output achieves 244 mW, which is better than that of most counterparts as shown in Fig. 8c and d.
为了研究 DM-EH 在摆动条件下的性能，一个实验平台模拟了人体运动，如图 8a 所示。调速伺服电机驱动曲柄连杆，不同频率和信号的相应输出电压由示波器（型号：DSOX1204G，Keysight）记录并保存。值得注意的是，电机的速度和摇杆的长度会调节杆的摆动频率和角度。由于 1.6 km ^h-1 步行速度的振幅和频率约为 20° 和 0.7 Hz，4 km ^h-1 快速步行对应于 25° 和 0.9 Hz，10 km ^h-1 运行速度对应于 30° 和 1.5 Hz，因此将转子频率范围设置为 0.7-1.5 Hz 以模拟手臂的振动。图 8b 显示了 DM-EH 在 0.7 至 1.5 Hz 频率范围内振幅为 20、25 和 30° 的性能。可以发现，由于激励加速度，增加的幅度或频率会提高输出电压，这在很大程度上取决于频率和幅度。RMS 直流电压为 2.13–6.25 V，频率为 0.7 至 1.5 Hz，振幅为 30°。最大 RMS 直流功率输出达到 244 mW，优于大多数同类产品，如图 8c 和 d 所示。

4.3. Performance comparison
4.3. 性能比较

A performance comparison between the proposed energy harvester and the recently reported low-frequency energy harvesters was conducted, validating the operational capabilities of the DM-EH, as depicted in Fig. 9a and b and summarized in Table 2. The average output power per unit volume/mass and the operational frequency serve as the key metrics for assessing the energy harvester's performance. Normalized power density, defined as the output power divided by the harvester's volume/mass and the square of the frequency, provides a reliable metric for evaluating energy harvesting performance.
对拟议的能量收集器和最近报道的低频能量收集器进行了性能比较，验证了 DM-EH 的运行能力，如图 9a 和 b 所示，并在表 2 中总结。每单位体积/质量的平均输出功率和运行频率是评估能量采集器性能的关键指标。归一化功率密度，定义为输出功率除以采集器的体积/质量与频率的平方，为评估能量收集性能提供了可靠的指标。

Table 2. Comparison of DM-EH with recently reported low-frequency harvesters.
表 2.DM-EH 与最近报道的低频收割机的比较。

Mode 模式	Reference 参考	Operating conditions (excitation and resistance) 工作条件（激励和电阻）	Power (mW) 功率（mW）	Power density (mW cm⁻³ Hz⁻²) 功率密度（mW ^cm-3 ^Hz-2）	Power density (mW g⁻¹ Hz⁻²) 功率密度（mW g⁻¹ Hz⁻²）
Human 人	2023 (7)	2.4Hz, 12 mm, 400Ω 2.4Hz，12 毫米，400Ω	4.3 (AC) 4.3 （交流电）	∼8.55e-3	∼9.33e-3
	2024 (26)	7 km h⁻¹, 288Ω 7 km h⁻1,288Ω	73.01(AC) 73.01（交流）	/	∼4.5e-2
	2023 (27)	7 km h⁻¹, 80Ω 7 km h⁻1,80Ω	2034 (DC) 2034 （DC）	∼11.72	/
	2023 (38)	10 km h⁻¹, 100MΩ/20Ω 10 km h⁻1,100MΩ/20Ω	2000 (AC) 2000 （AC）	/	/
	2024 (45)	4Hz, 20° 4 赫兹，20°	930 (DC) 930 （直流）	∼0.31	∼0.32
	2021 (48)	1Hz, 25°, 60Ω 1Hz，25°，60Ω	1.46 (AC) 1.46 （交流电）	∼0.45	∼0.07
	2023 (49)	5Hz, 1g, 14MΩ/13kΩ/400Ω 5Hz， 1g， 14MΩ/13kΩ/400Ω	124 (AC) 124 （交流电）	∼0.53	/
	This work 这项工作	1.5Hz, 30°, 160Ω 1.5Hz，30°，160Ω	244 (DC) 244 （直流）	4.78	1.38
Vehicle 车辆	2024 (4)	220 rpm, 50Ω 220 转/分，50Ω	210 (DC) 210 （直流）	∼1.25e-2	∼5.66e-3
	2022 (9)	2050 rpm, 100 kΩ 2050 转/分，100 kΩ	7910 (AC) 7910 （交流）	∼9.57e-2	∼7.21119e-2
	2024 (14)	19.9Hz, 0.5g, 470 kΩ 19.9Hz，0.5g，470 kΩ	78.87 (AC) 78.87 （交流电）	∼2.04e-2	/
	2022 (22)	180 rpm, 5 m s⁻², 24.6Ω 180 rpm，5 m s⁻2,24.6Ω	13.13 (AC) 13.13 （交流电）	∼0.13	∼0.11
	2024 (23)	14.1Hz, 0.6g, e3.6Ω 14.1Hz，0.6g，e3.6Ω	307.8 (AC) 307.8（交流）	∼4.73e-3	∼3.3e-2
	2023 (41)	80 rpm, 798Ω 80 转/分，798Ω	965 (AC) 965 （交流）	∼0.43	/
	2024 (46)	1260 rpm, 164Ω 1260 转/分，164Ω	712 (DC) 712 （直流）	∼7.65e-2	∼1.87e-2
	This work 这项工作	540 rpm, 160Ω 540 转/分，160Ω	3794 (DC) 3794 （直流）	0.85	0.25

The DM-EH demonstrates superior performance, particularly in terms of the power density under vehicle operations. The incorporation of the pendulum mechanism and frequency up-conversion technology not only enhances energy harvesting efficiency but also broadens the operational bandwidth at low frequencies. However, due to its relatively small volume (Fig. 9c and d), its normalized power density does not achieve the optimal value compared to that reported in Reference 27 for low-frequency human motions. Additionally, to enhance wearability, the volume and mass of DM-EH should be further reduced. Furthermore, the output stability and mechanical durability, which are essential prerequisites for commercial viability, were rigorously assessed by recording the output voltage before and after 30 days at 120 rpm, as presented in Fig. 9e. The prototype's output stability, mechanical robustness, and system safety align seamlessly with the design requirements for wireless monitoring.
DM-EH 表现出卓越的性能，尤其是在车辆运行下的功率密度方面。摆锤机构和频率上变频技术的结合不仅提高了能量收集效率，还拓宽了低频的工作带宽。然而，由于其体积相对较小（图 9c 和 d），与参考文献 27 中报告的低频人体运动相比，其归一化功率密度并未达到最佳值。此外，为了提高耐磨性，应进一步减小 DM-EH 的体积和质量。此外，通过记录 120 rpm 下 30 天前后的输出电压，严格评估了输出稳定性和机械耐久性，这是商业可行性的必要先决条件，如图 9e 所示。原型的输出稳定性、机械稳健性和系统安全性与无线监控的设计要求无缝一致。

4.4. Practical applications
4.4. 实际应用

Considering that the daily appliances and commercial wireless sensors require DC power, a three-phase rectifier was paralleled with a 560 μF capacitor (model: 1210) and zener diodes (model: MCC349B) to ensure that the 220 μF 10V capacitor stores a stable DC voltage, as shown in Fig. 11. Fig. 10a illustrates the charging curves of the DM-EH for commercial capacitors with capacitances of 100, 220, and 470 μF, as well as supercapacitors with 0.22 and 1.5 F at 120 rpm. Notably, the DM-EMH efficiently charges capacitors exceeding 5 V within 5 s. Remarkably, the charging speeds are nearly identical for capacitors with capacitances of 100, 220, and 470 μF, highlighting its outstanding charging capability. Furthermore, the DM-EH was capable of charging supercapacitors in a shorter time. As displayed in Fig. 10b and c and Video S1, the harvester effectively powered an LED light while simultaneously providing power to a cellphone (model: Huawei P10, Huawei Technologies Co., Ltd) and a smartwatch (Honor Band 4, Hornor) at 120 rpm.
考虑到日常电器和商用无线传感器需要直流电源，将三相整流器与 560 μF 电容器（型号：1210）和齐纳二极管（型号：MCC349B）并联，以确保 220 μF 10V 电容器存储稳定的直流电压，如图 11 所示。图 10a 显示了电容为 100、220 和 470 μF 的商用电容器以及 120 rpm 时电容为 0.22 和 1.5 F 的超级电容器的 DM-EH 的充电曲线。值得注意的是，DM-EMH 可在 5 秒内有效地为超过 5 V 的电容器充电。值得注意的是，电容为 100、220 和 470 μF 的电容器的充电速度几乎相同，凸显了其出色的充电能力。此外，DM-EH 能够在更短的时间内为超级电容器充电。如图 10b 和 c 以及视频 S1 所示，收割机有效地为 LED 灯供电，同时以 120 rpm 的速度为手机（型号：华为 P10，华为技术有限公司）和智能手表（Honor Band 4，Hornor）供电。

The DM-EH, when integrated with a wireless sensor, could effectively provide feedback on the vehicle's operating conditions. Acknowledging the significance of temperature and vibration information within vehicles, a commercial wireless temperature sensor and an acceleration sensor were integrated with the DM-EH. Fig. 11 and Video S2 demonstrated that the prototype successfully drove both the acceleration (model: ICM-42688-P, TDK) and temperature (model: ET80B, Chip source) sensors, enabling them to transmit signals while operating at 150 rpm.
DM-EH 与无线传感器集成时，可以有效地提供有关车辆运行状况的反馈。认识到车辆内温度和振动信息的重要性，商用无线温度传感器和加速度传感器与 DM-EH 集成在一起。图 11 和视频 S2 表明，原型成功驱动了加速度（型号：ICM-42688-P，TDK）和温度（型号：ET80B，芯片源）传感器，使它们能够在以 150 rpm 的速度运行时传输信号。

The viability of the DM-EH as a practical sustainable energy for converting mechanical energy for human motions has been evaluated for its capacity to maintain wearable devices and commercial wireless sensors described in Fig. 12. The prototype drove acceleration wireless sensor (model: ICM-42688-P, TDK) to offer motion signal under 7 km h⁻¹ corresponding vibration frequency of arms 1.2 Hz (Video S3).
DM-EH 作为一种将机械能转换为人体运动的实用可持续能源的可行性已经评估了其维护可穿戴设备和商用无线传感器的能力，如图 12 所示。原型驱动加速度无线传感器（型号：ICM-42688-P，TDK），提供低于 7 km ^h-1 的运动信号，相应的手臂振动频率为 1.2 Hz（视频 S3）。

It is worth noting that the DM-EH could charge electronic devices such as the smartwatch (model: Honor Band 4, Honor) and the cellphone (model: Huawei P10, Huawei Technologies Co., Ltd) for positioning and health monitoring, and charge the mobile power bank (model: L2, 5200mAh, Solove) help meet higher power needs for wilderness operations (Video S4). Smartwatch and cellphone have increased their battery power from 0 % to 60 % and 30 % within an hour under a speed of 4 km h⁻¹ and 8 km h⁻¹, respectively. Besides, the prototype could power LED lights to improve visibility and increase exercise safety during the night. The vertical vibrations contribute to increasing the swing amplitude and enhancing the relative motion between the coils and magnets of the generation unit, thereby improving the output performance of the DM-EH under vibration conditions.
值得注意的是，DM-EH 可以为智能手表（型号：Honor Band 4，Honor）和手机（型号：Huawei P10，华为技术有限公司）等电子设备充电，用于定位和健康监测，并为移动电源（型号：L2,5200mAh，Solove）充电，有助于满足野外作业更高的电力需求（Video S4）。在 4 km ^h-1 和 8 km ^h-1 的速度下，智能手表和手机在一小时内分别将其电池电量从 0 % 增加到 60 % 和 30 %。此外，该原型可以为 LED 灯供电，以提高能见度并提高夜间运动的安全性。垂直振动有助于增加摆动幅度并增强发电单元的线圈和磁铁之间的相对运动，从而提高 DM-EH 在振动条件下的输出性能。

The DM-EH effectively captures the rotational motion of the wheelset with a 1/2-scale freight wagon. A control box adjusts the speed of triple-phase asynchronous motors, which drive the roller rig via a gearbox. The roller rigs stimulate the rotation of wheelsets in a 1/2-scale freight wagon. The rotational speed of the 1/2-scale wheelset was controlled at approximately 110 rpm and 80 rpm during the railway simulation tests for battery-free temperature and acceleration sensors, respectively. The stable DC voltage of the DM-EH reached 3.13V and 4.55 V at 110 rpm and 80 rpm, respectively. Since there are no cables in the freight wagons, the DM-EH holds significant value in harnessing rotational energy for self-powered monitoring of wheelset bearing temperature (Video S5) and acceleration (Video S6) in railway environments as presented in Fig. 13. The whole volume and weight of the prototype are 20.79 cm³ and 78.4 g, respectively. When considering the volume and weight of freight vehicles, the impact of the prototype on the vehicles is negligible. In addition, considering that the counterweight may flip π, leading to synchronous rotation between the coils and magnets and a sharp drop in output voltage for emergency braking or abnormal vibrations, the DM-EH served as an early warning sensor to assess motion state and improve the reliability of vehicles.
DM-EH 有效地捕捉了 1/2 比例货车的轮对旋转运动。控制箱可调节三相异步电机的速度，三相异步电机通过变速箱驱动滚筒钻机。滚轮钻机刺激 1/2 比例货车中轮对的旋转。在无电池温度和加速度传感器的铁路模拟测试期间，1/2 比例轮对的转速分别控制在大约 110 rpm 和 80 rpm。DM-EH 的稳定直流电压在 3.13 rpm 和 4.55 rpm 时分别达到 110V 和 80 V。由于货车中没有电缆，因此 DM-EH 在利用旋转能量对铁路环境中的轮对轴承温度（视频 S5）和加速度（视频 S6）进行自供电监测方面具有重要价值，如图 13 所示。原型的整个体积和重量分别为 20.79 厘米³ 和 78.4 克。在考虑货运车辆的体积和重量时，原型对车辆的影响可以忽略不计。此外，考虑到配重可能会π翻转，导致线圈和磁铁之间同步旋转，以及紧急制动或异常振动时输出电压急剧下降，DM-EH 用作评估运动状态和提高车辆可靠性的早期预警传感器。

5. Conclusion 5. 总结

A high-performance dual-mode energy harvester (DM-EH) is proposed to scavenge low-frequency vibrations generated by human motions and vehicle operations. The generation unit, serving as a pendulum, efficiently converts irregular vibrations into rotation energy while ensuring effective operation for anchorless installation in rotational environments. The transmission unit plays a pivotal role in converting low-frequency vibrations into high-frequency ones, enabling it to accommodate low-frequency environments and satisfy the high-frequency input requirements of the generation unit. Dynamic experimental results substantiate the DM-EH's performance when subjected to low-frequency vibration sources.
提出了一种高性能双模式能量收集器（DM-EH）来清除人类运动和车辆操作产生的低频振动。发电装置用作摆锤，有效地将不规则振动转化为旋转能量，同时确保在旋转环境中无锚安装的有效运行。传动装置在将低频振动转换为高频振动方面发挥着关键作用，使其能够适应低频环境并满足发电机组的高频输入要求。动态实验结果证实了 DM-EH 在受到低频振动源时的性能。

The DM-EH demonstrated its operational capability over a broad range, encompassing vehicle rotations from 60 to 540 rpm and human motions ranging from 0.7 to 1.5 Hz with 30° amplitudes. It achieved a peak direct current RMS power output of 3.79 W and 244 mW with an optimal resistance of 160 Ω at 540 rpm for vehicle operations and 1.5 Hz with 30° amplitudes for human motions, resulting in power densities of 182.58 and 11.74 mW cm⁻³, respectively. These performance levels surpass those of traditional low-frequency harvesters. Regarding applications, the prototype successfully powered daily electrical appliances (such as LED, smartwatches, cellphones, and mobile power banks), rapidly recharged capacitance and supercapacitor, and drove temperature and acceleration sensors during vehicle operations and human motions. These results represent a substantial step toward the potential deployment of self-powered wireless condition monitoring systems for both human motions and vehicle operations.
DM-EH 展示了其在广泛范围内的操作能力，包括 60 至 540 rpm 的车辆旋转和 0.7 至 1.5 Hz 的人体运动和 30° 振幅。它实现了 3.79 W 和 244 mW 的峰值直流 RMS 功率输出，车辆运行时的最佳电阻为 160 Ω，在 540 rpm 时的最佳电阻为 1.5 Hz，人体运动时振幅为 30°，功率密度分别为 182.58 和 11.74 mW ^cm-3。这些性能水平超过了传统的低频采集器。在应用方面，该原型成功地为日常电器（如 LED、智能手表、手机和移动电源）供电，为电容和超级电容器快速充电，并在车辆运行和人体运动期间驱动温度和加速度传感器。这些结果代表着朝着为人体运动和车辆操作部署自供电无线状态监测系统迈出了实质性的一步。

CRediT authorship contribution statement
CRediT 作者贡献声明

Zhixia Wang: Writing – original draft, Visualization, Validation, Methodology, Funding acquisition, Data curation, Conceptualization. Siwei Kang: Validation, Investigation. Hongzhi Du: Validation, Conceptualization. Pengju Feng: Validation, Data curation. Wei Wang: Writing – review & editing, Supervision, Funding acquisition.
王志霞：写作 – 原稿、可视化、验证、方法、资金获取、数据管理、概念化。康思魏：验证、调查。杜宏志：验证、概念化。冯鹏钩：验证、数据管理。王伟：写作 - 审查和编辑，监督，资金获取。

Declaration of competing interest
利益争夺声明

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.
作者声明，他们没有已知的竞争性经济利益或个人关系，这些利益或个人关系似乎可能会影响本文报告的工作。

Acknowledgments 确认

The work was supported by the National Natural Science Foundation of China (Grant Nos. 12302022, 12172248, 12132010, 12021002), the Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 22JCQNJC00780), and Vehicle Measurement, Control and Safety Key Laboratory of Sichuan Province (Grant No. QCCK2024-0019).
这项工作得到了国家自然科学基金（批准号 12302022、12172248、12132010、12021002）、天津市应用基金与先进技术研究计划（批准号 22JCQNJC00780）和四川省车辆测量、控制与安全重点实验室（批准号QCCK2024-0019）。

Appendix A. Supplementary data
附录 A. 补充数据

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The following are the Supplementary data to this article.

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Data availability 数据可用性

Data will be made available on request.
数据将应要求提供。

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View in Scopus Google Scholar
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Nano Energy, 114 (2023), Article 108630
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Cited by (0) 被引用次数（0）

View Abstract

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View in Scopus Google Scholar

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View in Scopus Google Scholar

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Energies, 15 (2022), p. 6539
Crossref View in Scopus Google Scholar

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A variable damping vibration energy harvester based on half-wave flywheeling effect for freight railways
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View in Scopus Google Scholar

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L. Ren, Y. Zhou, X. Zhang, H. Zhang, Y. Tan
Active self-powered human motion assist system
Smart Mater Struct, 33 (2024), Article 055003
Crossref View in Scopus Google Scholar

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Z. Wang, H. Du, W. Wang, Q. Zhang, F. Gu, A. Ball, et al.
A high performance contra-rotating energy harvester and its wireless sensing application toward green and maintain free vehicle monitoring
Appl Energy, 356 (2024), Article 122370
View in Scopus Google Scholar

[47] [47]
D. Hao, L. Kong, Z. Zhang, W. Kong, A.M. Tairab, X. Luo, et al.
An electromagnetic energy harvester with a half-wave rectification mechanism for military personnel
Sustain Energy Techn, 57 (2023), Article 103184
View in Scopus Google Scholar

[48] [48]
M. Cai, W. Liao
High-power density inertial energy harvester without additional proof mass for wearables
IEEE Internet Things, 8 (2021), pp. 297-308
Crossref View in Scopus Google Scholar

[49] [49]
C. Wang, Y. Ji, S. Lai, Y. Liu, Y. Hao, G. Li, et al.
A speed-amplified tri-stable piezoelectric-electromagnetic-triboelectric hybrid energy harvester for low-frequency applications
Nano Energy, 114 (2023), Article 108630
View in Scopus Google Scholar

[50] [50]
Z. Liu, X. Wang, E. Al Shami, N.J. Baker, X. Ji
A study of a speed amplified linear generator for low-frequency wave energy conversion
Mech Syst Signal Pr, 149 (2021), Article 107226
View in Scopus Google Scholar

[51] [51]
China Railway Transport Department
Introduction to railway
China Railway Publishing House (2022)
Google Scholar

[52] [52]
M. Gao, J. Cong, J. Xiao, Q. He, S. Li, Y. Wang, et al.
Dynamic modeling and experimental investigation of self-powered sensor nodes for freight rail transport
Appl Energy, 257 (2020), Article 113969
View in Scopus Google Scholar

Outline 大纲

Figures (14) 手办（14）

Tables (2)

Extras (6)

Energy 能源

Highlights 突出

Abstract 抽象

Graphical abstract 图形摘要

Keywords 关键字

1. Introduction 1. 引言

2. Design and working principle
2. 设计及工作原理

3. Modeling and dynamic analysis of the DM-EH
3. DM-EH 的建模和动力学分析

3.1. Theoretical modeling of the DM-EH under human motion
3.1. 人体运动下 DM-EH 的理论建模

3.2. Theoretical modeling of the DM-EH under vehicle operation
3.2. 车辆运行下 DM-EH 的理论建模

3.3. Finite-element of the generation unit
3.3. 生成单元的有限元

4. Performance investigation
4. 性能调查

4.1. Fabrication of the DM-EH
4.1. DM-EH 的制造

4.2. Characterization and measurements
4.2. 表征和测量

4.3. Performance comparison
4.3. 性能比较

4.4. Practical applications
4.4. 实际应用

5. Conclusion 5. 总结

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of competing interest
利益争夺声明

Acknowledgments 确认

Appendix A. Supplementary data
附录 A. 补充数据

Data availability 数据可用性

References

Cited by (0) 被引用次数（0）

Energy 能源

A high-performance dual-mode energy harvesting with nonlinear pendulum and speed-amplified mechanisms for low-frequency applications高性能双模能量收集，具有非线性摆锤和速度放大机制，适用于低频应用

Highlights 突出

Abstract 抽象

Graphical abstract 图形摘要

Keywords 关键字

1. Introduction 1. 引言

2. Design and working principle2. 设计及工作原理

3. Modeling and dynamic analysis of the DM-EH3. DM-EH 的建模和动力学分析

3.1. Theoretical modeling of the DM-EH under human motion3.1. 人体运动下 DM-EH 的理论建模

3.2. Theoretical modeling of the DM-EH under vehicle operation3.2. 车辆运行下 DM-EH 的理论建模

3.3. Finite-element of the generation unit3.3. 生成单元的有限元

4. Performance investigation4. 性能调查

4.1. Fabrication of the DM-EH4.1. DM-EH 的制造

4.2. Characterization and measurements4.2. 表征和测量

4.3. Performance comparison4.3. 性能比较

4.4. Practical applications4.4. 实际应用

5. Conclusion 5. 总结

CRediT authorship contribution statementCRediT 作者贡献声明

Declaration of competing interest利益争夺声明

Acknowledgments 确认

Appendix A. Supplementary data附录 A. 补充数据

Data availability 数据可用性

References

A high-performance dual-mode energy harvesting with nonlinear pendulum and speed-amplified mechanisms for low-frequency applications
高性能双模能量收集，具有非线性摆锤和速度放大机制，适用于低频应用

2. Design and working principle
2. 设计及工作原理

3. Modeling and dynamic analysis of the DM-EH
3. DM-EH 的建模和动力学分析

3.1. Theoretical modeling of the DM-EH under human motion
3.1. 人体运动下 DM-EH 的理论建模

3.2. Theoretical modeling of the DM-EH under vehicle operation
3.2. 车辆运行下 DM-EH 的理论建模

3.3. Finite-element of the generation unit
3.3. 生成单元的有限元

4. Performance investigation
4. 性能调查

4.1. Fabrication of the DM-EH
4.1. DM-EH 的制造

4.2. Characterization and measurements
4.2. 表征和测量

4.3. Performance comparison
4.3. 性能比较

4.4. Practical applications
4.4. 实际应用

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of competing interest
利益争夺声明

Appendix A. Supplementary data
附录 A. 补充数据