近些年，随着煤矿采掘智能化水平发展，少人或者无人化开采模式对煤矿采掘装备位姿感知、健康监测等提出了前所未有的要求。传统有线监测方式由于复杂的布线和防爆设计，常常致使设备许多部位无法实现监测，已难以满足煤矿发展需求。无线监测技术在煤矿作业中受到了极大关注，其应用瓶颈主要在于各传感节点供电问题。传统化学电池供电寿命有限，维护成本高且存在环境污染风险。构建振动能量收集系统有望实现无线传感器件自供电，解决传感器布线难题，提升监测预警水平。为此，论文以设计适用于煤矿采掘激励的低频带压电能量采集器为研究对象，综合考虑采掘激励特性以及不同类型压电俘能器适用工况条件，提出两种新型压电俘能器结构，即悬臂式和双端固支式拱形梁压电俘能器。重点对悬臂式拱形梁压电俘能器展开机电耦合建模，动力学和俘能特性分析以及实验研究；对双端固支式拱形梁压电俘能器俘能特性进行探索。以期研究为拱形梁式压电俘能器设计及其工程化应用提供理论指导。论文主要研究工作如下：

对直梁与拱形梁结构展开应力分布与刚度特性分析，为压电俘能器设计提供理论指导。针对悬臂式拱形梁双稳态压电俘能器，建立拱形梁非线性恢复力模型，采用改进的磁偶极子法建立磁力模型，最后建立系统机电耦合方程。在简谐激励条件下，利用数值计算，结合系统分岔图、最大Lyapunov指数图以及Poincaré映射图等对系统动力学特性展开分析。最后，对比分析磁力耦合拱形梁双稳态、无磁力拱形梁以及传统直梁结构压电俘能器能量收集性能，实验验证了拱形梁双稳态压电俘能器性能的优越性和理论分析的正确性。

为改善磁力耦合拱形梁双稳态压电俘能器势阱特性对系统性能负面影响，提出采用偏置磁距方法补偿势阱非对称性，并建立具有偏置磁距的磁力模型和相应约束条件以实现精确补偿。研究表明，不同激励水平下，补偿后系统较原系统性能均有不同程度提升。随着激励水平增加，系统具有足够的能量突破势垒限制实现大幅响应，势阱非对称对性能的影响作用削弱；相应地，系统性能提升水平逐渐减弱。最后，通过实验验证理论分析的正确性和补偿策略的有效性。

为探究拱形梁双稳态压电俘能器在采掘激励下响应特性，采用窄带随机信号模拟采掘激励，建立窄带随机激励下系统动力学模型。基于Monte Carlo方法计算系统位移-速度联合概率密度函数，证明势阱非对称性在随机条件下对系统响应具有负面影响。基于势阱补偿后拱形梁双稳态压电俘能器动力学模型，研究窄带随机激励中心频率、激励强度对响应特性的影响。最后，提出采用中心频率与带宽构建的频率有效响应区域作为系统性能评价指标，对比分析窄带随机激励下拱形梁单稳态、拱形梁双稳态(无补偿)和拱形梁双稳态(补偿)压电俘能器俘能性能。研究表明，弱激励水平下，磁力耦合拱形梁单稳态压电俘能器具有最优俘能性能；随着激励水平的增加，拱形梁双稳态(补偿)压电俘能器具有频率最大有效响应区域，最后通过实验结果证明了理论分析的正确性。

为研究双端固支式拱形梁压电俘能器俘能特性，建立系统机电耦合方程，探究激励水平、结构参数和负载对系统俘能特性影响。结果表明，随着激励水平增大，系统依次展现出拟线性、软特性和软-硬混合非线性特性。尤其，软-硬混合特性会随着频率增加形成两个跳跃点，显著拓宽系统有效响应带宽。保持激励水平一定，合适的结构参数可以使系统在低激励水平呈现软-硬混合非线性响应，拓宽响应带宽。最后，实验证明双端固支拱形梁相对于传统直梁压电俘能器在功率输出及有效带宽性能方面的优越性。

最后，对拱形梁双稳态压电俘能器进行系统测试，进行功率输出特性实验、自俘能电容充电实验、自供电温度无线监测系统开发与测试，验证系统可行性，为压电式自俘能无线监测系统工程应用进行探索。

In recent years, with the development of intelligent level of coal mining, the less manned or unmanned mining mode put forward unprecedented requirements for the position sensing and health monitoring of coal mining equipment. Due to complex wiring and explosion-proof design, many parts of the equipment can not be monitored by traditional wired monitoring methods. Wireless monitoring technology receives great attention in coal mine operations, and its application bottleneck mainly lies in the power supply of each sensing node. Traditional chemical batteries have limited power supply life, high maintenance cost and risk of environmental pollution. The construction of vibration energy harvesting system is expected to realize the self-power supply of wireless sensors, thus solving the problems of sensor wiring and improving the monitoring and warning level. To this end, this dissertation mainly designs piezoelectric energy harvesters for low-frequency band in coal mining excitation. Considering the characteristics of the mining excitation and the applicable condition of different types of piezoelectric energy harvesters, the dissertation proposes two new types of piezoelectric energy harvester structures, i.e., cantilever-type and doubly clamped arch-shaped piezoelectric energy harvester, and focuses on the electromechanical coupling modeling, dynamics and capture characteristics analysis, and experimental study of the cantilevered arch-shaped piezoelectric harvester. Meanwhile, the harvestering characteristics of the arch-shaped doubly clamped piezoelectric harvester are explored. This work could contribute to providing theoretical guidance for the design of high-efficiency piezoelectric energy harvesters and their engineering application. The main contents are shown as follows:

The stress distribution and stiffness characteristics for the straight beam and arch-shaped beam are analyzed to provide theoretical guidance for the design of piezoelectric energy harvesters. Aiming at the arch-shaped bistable piezoelectric energy harvester, the nonlinear restoring force model of the arch-shaped beam is established, the magnetic force model is derived by the improved magnetic dipole method, and the electromechanical coupling equations of the system are established. Under harmonic excitation conditions, the dynamic responses are obtained by solving the coupling equations, and the effect mechanism of the excitation frequency and amplitude on large-amplitude periodic response is discussed and analyzed via the bifurcation diagram, the maximum Lyapunov exponent diagram, and the Poincaré map. Finally, the energy harvesting performance of the magnetically coupled arch-shaped bistable, the non-magnetic arch-shaped monostable and the conventional straight beam harvester are analyzed and compared, respectively. The performance’s superiority of the arch-shaped bistable piezoelectric harvester and the correctness of the theoretical analysis are verified by the experiments.

To improve the negative impact of the potential well characteristics of the magnetic coupling arch-shaped bistable piezoelectric harvester on system performance, it is proposed to compensate the potential well asymmetry by using the biased magnetic distance method. The corresponding magnetic model and constraint conditions are established to realize the accurate compensation. Compared with the original system, the performance of the compensated system is improved under different excitation levels. However, with the increase of the excitation level, the system could obtain enough energy to break through the barrier to realize large-amplitude response, and the effect of the potential well asymmetry on the performance is weakened. Accordingly, the performance enhancement level of the system is gradually weakened. Finally, the correctness of the theoretical analysis and the effectiveness of the compensation strategy are verified by experiments.

To investigate the response characteristics of the arch-shaped bistable piezoelectric energy harvester under mining excitation, the narrow-band random signal is used to simulate the mining excitation, and the corresponding dynamics model is established. Based on the Monte Carlo method, the joint probability density function of the system is calculated. It is proved that the asymmetry of the potential well has a negative effect on the system response under the random condition. The effects of the center frequency and excitation intensity of the narrow-band random excitation on the response characteristics are investigated for the compensated bistable piezoelectric energy harvester. Under narrow-band random excitation, the effective frequency response region constructed by the center frequency and bandwidth is proposed, which is used as the performance evaluation index to compare and analyze the performance of the arch-shaped monostable, the arch-shaped bistable(uncompensated) and the bistable(compensated) piezoelectric energy harvester. It is shown that the magnetically coupled arch-shaped monostable piezoelectric energy harvester has the optimum energy harvestering performance under the weak excitation level. With the increase of the excitation level, the arch-shaped bistable(compensated) piezoelectric energy harvester has the maximum frequency response region. Finally, the corresponding theoretical results are qualitatively verified by the experiments.

In order to study the energy harvesting characteristics of the arch-shaped doubly clamped piezoelectric harvester, the electromechanical coupling equations of the system are established, and the effects of excitation level, structural parameter and load resistance on the energy harvestering characteristics are investigated, respectively. The results show that the system orderly exhibits quasi-linear, softening nonlinear and mixed softening-hardening nonlinearity behavior with the increase of excitation level. In particular, the softening-hardening mixed characteristics could form two jump points, which can significantly broaden the effective response bandwidth of the system. Keeping the excitation level constant, the appropriate structural parameters can make the system present softening-hardening mixed nonlinear response under low excitation level, thus broadening the response bandwidth. Finally, the superiority of the arch-shaped doubly clamped piezoelectric energy harvester over the conventional counterpart in terms of power output and effective bandwidth is experimentally verified.

Finally, the arch-shaped beam bistable piezoelectric harvester is systematically tested. The power output characteristics, capacitor charging experiment and self-powered wireless monitoring system application are tested to verify the system’s feasibility and explore the engineering application of piezoelectric energyharvester.

[1] 中国煤炭报社.煤炭清洁利用[EB/OL].中国煤炭网要闻. 2021-12-21.

[2] 方圆,张万益,曹佳文,等.我国能源资源现状与发展趋势[J].矿产保护与利用,2018(04): 34-42+47.

[3] 王国法,王虹,任怀伟,等.智慧煤矿2025情景目标和发展路径[J].煤炭学报, 2018, 43 (02): 295-305.

[4] 佟占生,王春明,李凯.矿井无线传感器网络设计及应用[J].工矿自动化,2017,43(08): 98-100.

[5] Muduli L, Mishra D P, Jana P K. Application of wireless sensor network for environmental monitoring in underground coal mines: A systematic review[J]. Journal of Network and Computer Applications, 2018, 106: 48-67.

[6] Wu Y, Qiu J, Zhou S, et al. A piezoelectric spring pendulum oscillator used for multi-directional and ultra-low frequency vibration energy harvesting[J]. Applied energy, 2018, 231: 600-614.

[7] Sharma S, Kiran R, Azad P, et al. A review of piezoelectric energy harvesting tiles: Available designs and future perspective[J]. Energy Conversion and Management, 2022, 254: 115272.

[8] Chen X, Zhang X, Wang L, et al. An arch-linear composed beam piezoelectric energy harvester with magnetic coupling: Design, modeling and dynamic analysis[J]. Journal of Sound and Vibration, 2021, 513: 116394.

[9] Zhang N, Tao C, Fan X, et al. Progress in triboelectric nanogenerators as self-powered smart sensors[J]. Journal of Materials Research, 2017, 32(9): 1628-1646.

[10] Wang S, Miao G, Zhou S, et al. A novel electromagnetic energy harvester based on the bending of the sole[J]. Applied Energy, 2022, 314: 119000.

[11] 芮小博,李一博,曾周末.压电悬臂梁振动能量收集器研究进展[J].振动与冲击,2020, 39(17): 112-123.

[12] Paul K, Amann A, Roy S. Tapered nonlinear vibration energy harvester for powering Internet of Things[J]. Applied Energy, 2021, 283: 116267.

[13] Miao G, Fang S, Wang S, et al. A low-frequency rotational electromagnetic energy harvester using a magnetic plucking mechanism[J]. Applied Energy, 2022, 305: 117838.

[14] Muscat A, Bhattacharya S, Zhu Y. Electromagnetic Vibrational Energy Harvesters: A Review[J]. Sensors, 2022, 22(15): 5555.

[15] 张广义.微小型低频宽带高性能振动俘能器研究[D].北京:北京理工大学,2017.

[16] 王翔.基于非线性技术的压电-静电复合式振动能量收集器的研究[D].厦门:厦门大学,2018.

[17] Maamer B, Boughamoura A, El-Bab A M R F, et al. A review on design improvements and techniques for mechanical energy harvesting using piezoelectric and electromagnetic schemes[J]. Energy Conversion and Management, 2019, 199: 111973.

[18] Wang G Q, Liao W H. A bistable piezoelectric oscillator with an elastic magnifier for energy harvesting enhancement[J]. Journal of Intelligent Material Systems and Structures, 2017, 28(3): 392-407.

[19] Muthalif A G A, Nordin N H D. Optimal piezoelectric beam shape for single and broadband vibration energy harvesting: Modeling, simulation and experimental results[J]. Mechanical Systems and Signal Processing, 2015, 54: 417-426.

[20] Karadag C V, Ertarla S, Topaloglu N, et al. Optimization of beam profiles for improved piezoelectric energy harvesting efficiency[J]. Structural and Multidisciplinary Optimization, 2021, 63: 631-643.

[21] Ibrahim D S, Beibei S, Fatai S, et al. Numerical and experimental study of a gauge-shaped beam for improved performance of piezoelectric energy harvester[J]. Microsystem Technologies, 2021, 27(12): 4253-4268.

[22] Cao D, Gao Y, Hu W. Modeling and power performance improvement of a piezoelectric energy harvester for low-frequency vibration environments[J]. Acta Mechanica Sinica, 2019, 35: 894-911.

[23] Wang B, Luo X, Liu Y, et al. Thickness-variable composite beams for vibration energy harvesting[J]. Composite Structures, 2020, 244: 112232.

[24] Izadgoshasb I, Lim Y Y, Vasquez Padilla R, et al. Performance enhancement of a multiresonant piezoelectric energy harvester for low frequency vibrations[J]. Energies, 2019, 12(14): 2770.

[25] Xie X, Wang Z, Liu D, et al. An experimental study on a novel cylinder harvester made of L-shaped piezoelectric coupled beams with a high efficiency[J]. Energy, 2020, 212: 118752.

[26] Xie X, Wang Q. A mathematical model for piezoelectric ring energy harvesting technology from vehicle tires[J]. International Journal of Engineering Science, 2015, 94: 113-127.

[27] Yang Z, Wang Y Q, Zuo L, et al. Introducing arc-shaped piezoelectric elements into energy harvesters[J]. Energy Conversion and Management, 2017, 148: 260-266.

[28] Piyarathna I E, Thabet A M, Ucgul M, et al. Linear Segmented Arc-Shaped Piezoelectric Branch Beam Energy Harvester for Ultra-Low Frequency Vibrations[J]. Sensors, 2023, 23(11): 5257.

[29] Shang M, Qin W, Li H. Harvesting vibration energy by novel piezoelectric structure with arc-shaped branches[J]. Mechanical Systems and Signal Processing, 2023, 200: 110577.

[30] Zhang X, Guo Y, Zhu F, et al. A Linear-Arc Composite Beam Piezoelectric Energy Harvester Modeling and Finite Element Analysis[J]. Micromachines, 2022, 13(6): 848.

[31] Ding J, Lu M, Deng A, et al. A piezoelectric energy harvester using an arc-shaped piezoelectric cantilever beam array[J]. Microsystem Technologies, 2022, 28(8): 1947-1958.

[32] Paul K, Amann A, Roy S. Tapered nonlinear vibration energy harvester for powering Internet of Things[J]. Applied Energy, 2021, 283: 116267.

[33] Jin L, Gao S, Zhang X, et al. Output of MEMS piezoelectric energy harvester of double-clamped beams with different width shapes[J]. Materials, 2020, 13(10): 2330.

[34] Li P, Gao S, Niu S, et al. An analysis of the coupling effect for a hybrid piezoelectric and electromagnetic energy harvester[J]. Smart Materials and Structures, 2014, 23(6): 065016.

[35] Qin Y, Wei T, Zhao Y, et al. Simulation and experiment on bridge-shaped nonlinear piezoelectric vibration energy harvester[J]. Smart Materials and Structures, 2019, 28(4): 045015.

[36] 张广义,金磊,高世桥,等.双端固支梯形梁压电俘能器机电耦合模型与试验分析[J]. 北京理工大学学报,2018, 38(06): 600-605.

[37] Jin L, Gao S, Zhang X, et al. Output of MEMS piezoelectric energy harvester of double-clamped beams with different width shapes[J]. Materials, 2020, 13(10): 2330.

[38] Leadenham S, Erturk A. Nonlinear M-shaped broadband piezoelectric energy harvester for very low base accelerations: primary and secondary resonances[J]. Smart Materials and Structures, 2015, 24(5): 055021.

[39] Leadenham S, Erturk A. M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation[J]. Journal of Sound and Vibration, 2014, 333(23): 6209-6223.

[40] Chen K, Gao F, Liu Z, et al. A nonlinear M-shaped tri-directional piezoelectric energy harvester[J]. Smart Materials and Structures, 2021, 30(4): 045017.

[41] Shan X, Tian H, Chen D, et al. A curved panel energy harvester for aeroelastic vibration[J]. Applied Energy, 2019, 249: 58-66.

[42] Chen K, Ding X, Tian L, et al. An M-shaped buckled beam for enhancing nonlinear energy harvesting[J]. Mechanical Systems and Signal Processing, 2023, 188: 110066.

[43] 陈仲生.压电式振动能量俘获理论与方法[M].北京:国防工业出版社,2017.

[44] Williams C B, Yates R B. Analysis of a micro-electric generator for microsystems[J]. sensors and actuators A: Physical, 1996, 52(1-3): 8-11.

[45] Dutoit N E, Wardle B L, Kim S G. Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters[J]. Integrated Ferroelectrics, 2005, 71(1): 121-160.

[46] Roundy S, Wright P K. A piezoelectric vibration based generator for wireless electronics[J]. Smart Materials and Structures, 2004, 13(5): 1131.

[47] Shu Y C, Lien I C. Analysis of power output for piezoelectric energy harvesting systems[J]. Smart Materials and Structures, 2006, 15(6): 1499.

[48] 孙舒,曹树谦.双稳态压电悬臂梁发电系统的动力学建模及分析[J].物理学报,2012, 61(21): 95-106.

[49] Leng Y G, Gao Y J, Tan D, et al. An elastic-support model for enhanced bistable piezoelectric energy harvesting from random vibrations[J]. Journal of Applied Physics, 2015, 117(6): 064901.

[50] Yang W, Towfighian S. Low frequency energy harvesting with a variable potential function under random vibration[J]. Smart Materials and Structures, 2018, 27(11): 114004.

[51] 贺学锋,印显方,杜志刚,等.悬臂梁压电振动能采集器的集总参数模型和实验验证[J]. 纳米技术与精密工程,2012, 10(02): 108-112.

[52] Wang G, Lu Y. An improved lumped parameter model for a piezoelectric energy harvester in transverse vibration[J]. Shock and Vibration, 2014, 2014.

[53] Erturk A, Inman D J. On mechanical modeling of cantilevered piezoelectric vibration energy harvesters[J]. Journal of Intelligent Material Systems and Structures, 2008, 19(11): 1311-1325.

[54] Zou A, Shen X. Lumped Parameter Evaluation Model Analysis and Load Coupling Modeling of Piezoelectric Composite Cantilever Beam[J]. IETE Journal of Research, 2021: 1-10.

[55] Erturk A, Inman D J. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters[J]. Journal of Vibration and Acoustics, 2008, 130(4).

[56] Erturk A. Assumed-modes modeling of piezoelectric energy harvesters: Euler-Bernoulli, Rayleigh, and Timoshenko models with axial deformations[J]. Computers & Structures, 2012, 106: 214-227.

[57] Machu Z, Rubes O, Sevecek O, et al. Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations[J]. Sensors, 2021, 21(20): 6759.

[58] Chen Y, Yang Z, Chen Z, et al. Design, modeling, and experiment of a multi-bifurcated cantilever piezoelectric energy harvester[J]. Journal of Intelligent Material Systems and Structures, 2021, 32(20): 2403-2419.

[59] Abdelkefi A, Barsallo N. Comparative modeling of low-frequency piezomagnetoelastic energy harvesters[J]. Journal of Intelligent Material Systems and Structures, 2014, 25(14): 1771-1785.

[60] Tan T, Yan Z, Lei H, et al. Geometric nonlinear distributed parameter model for cantilever-beam piezoelectric energy harvesters and structural dimension analysis for galloping mode[J]. Journal of Intelligent Material Systems and Structures, 2017, 28(20): 3066-3078.

[61] Wang B, Li Z, Yang Z. A distributed-parameter electromechanical coupling model for a piezoelectric energy harvester with variable curvature[J]. Smart Materials and Structures, 2020, 29(11): 115015.

[62] 季文美,方同,陈松淇.机械振动[M].北京:科学出版社,2016.

[63] Krishnasamy M, Qian F, Zuo L, et al. Distributed parameter modeling to prevent charge cancellation for discrete thickness piezoelectric energy harvester[J]. Solid-State Electronics, 2018, 141: 74-83.

[64] Zhou W, Wang B, Lim C W, et al. A distributed-parameter electromechanical coupling model for a segmented arc-shaped piezoelectric energy harvester[J]. Mechanical Systems and Signal Processing, 2021, 146: 107005.

[65] 周潇雅.随机低频环境激励下压电俘能器研究[D].北京:北京理工大学,2018.

[66] Yang Z, Zhou S, Zu J, et al. High-performance piezoelectric energy harvesters and their applications[J]. Joule, 2018, 2(4): 642-697.

[67] Wang G, Liao W H, Zhao Z, et al. Nonlinear magnetic force and dynamic characteristics of a tri-stable piezoelectric energy harvester[J]. Nonlinear Dynamics, 2019, 97(4): 2371- 2397.

[68] Tang H, Hua C, Huang H, et al. Low-frequency vibration energy harvesting: a comprehensive review of frequency up-conversion approaches[J]. Smart Materials and Structures, 2022, 31(10): 103001.

[69] Tran N, Ghayesh M H, Arjomandi M. Ambient vibration energy harvesters: A review on nonlinear techniques for performance enhancement[J]. International Journal of Engineering Science, 2018, 127: 162-185.

[70] Su W J, Shih H A. A U-shaped multi-modal bi-directional piezoelectric energy harvester[J]. Applied Physics Letters, 2018, 113(7): 071905.

[71] Siddique A R M, Mahmud S, Van Heyst B. A comprehensive review on vibration based micro power generators using electromagnetic and piezoelectric transducer mechanisms[J]. Energy Conversion and Management, 2015, 106: 728-747.

[72] 代显智,张章,刘小亚,等.非线性宽频振动能量采集技术的研究进展[J].中国科学:技术科学,2016 (8): 791-807.

[73] 陈孝玉,张旭辉,左萌,等.非线性压电俘能技术研究现状及趋势分析[J].仪表技术与传感器,2020(12): 42-52.

[74] Erturk A, Hoffmann J, Inman D J. A piezomagnetoelastic structure for broadband vibration energy harvesting[J]. Applied Physics Letters, 2009, 94(25).

[75] Moon F C, Holmes P J. A magnetoelastic strange attractor[J]. Journal of Sound and Vibration, 1979, 65(2): 275-296.

[76] Erturk A, Inman D J. Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling[J]. Journal of Sound and Vibration, 2011, 330(10): 2339-2353.

[77] 唐炜,王小璞,曹景军.非线性磁式压电振动能量采集系统建模与分析[J].物理学报,2014,63(24): 76-89.

[78] Rubes O, Brablc M, Hadas Z. Nonlinear vibration energy harvester: Design and oscillating stability analyses[J]. Mechanical Systems and Signal Processing, 2019, 125: 170-184.

[79] 赵泽翔,王光庆,谭江平,等.双稳态压电振动能量采集器的时-频域动力学特性及实验研究[J].传感技术学报,2019, 32(08): 1200-1208.

[80] Zhang X, Yang W, Zuo M, et al. An arc-shaped piezoelectric bistable vibration energy harvester: Modeling and experiments[J]. Sensors, 2018, 18(12): 4472.

[81] Tan T, Yan Z, Ma K, et al. Nonlinear characterization and performance optimization for broadband bistable energy harvester[J]. Acta Mechanica Sinica, 2020, 36: 578-591.

[82] Shan G, Wang D F, Song J, et al. A spring-assisted adaptive bistable energy harvester for high output in low-excitation[J]. Microsystem Technologies, 2018, 24(9): 3579-3588.

[83] Wang G, Liao W H, Yang B, et al. Dynamic and energetic characteristics of a bistable piezoelectric vibration energy harvester with an elastic magnifier[J]. Mechanical Systems and Signal Processing, 2018, 105: 427-446.

[84] Friswell M I, Ali S F, Bilgen O, et al. Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass[J]. Journal of Intelligent Material Systems and Structures, 2012, 23(13): 1505-1521.

[85] Xu C, Liang Z, Ren B, et al. Bi-stable energy harvesting based on a simply supported piezoelectric buckled beam[J]. Journal of Applied Physics, 2013, 114(11): 42.

[86] 王胜杰,韩翻珍,郭翔鹰.新型双稳态屈曲梁结构的低频跳跃稳定性分析[J].哈尔滨工程大学学报,2022, 43(09): 1336-1341.

[87] Qian F, Zhou S, Zuo L. Approximate solutions and their stability of a broadband piezoelectric energy harvester with a tunable potential function[J]. Communications in Nonlinear Science and Numerical Simulation, 2020, 80: 104984.

[88] Cai Y, Fu J, Wu N, et al. A high-efficiency curved panel energy harvester featured by reduced stress concentration[J]. Energy Conversion and Management, 2022, 271: 116334.

[89] Nan W, Yuncheng H, Jiyang F. Bistable energy harvester using easy snap-through performance to increase output power[J]. Energy, 2021, 226: 120414.

[90] Liu W, Formosa F, Badel A. Optimization study of a piezoelectric bistable generator with doubled voltage frequency using harmonic balance method[J]. Journal of Intelligent Material Systems and Structures, 2017, 28(5): 671-686.

[91] Zou D, Liu G, Rao Z, et al. Design of vibration energy harvesters with customized nonlinear forces[J]. Mechanical Systems and Signal Processing, 2021, 153: 107526.

[92] Kumar P, Narayanan S, Adhikari S, et al. Fokker-Planck equation analysis of randomly excited nonlinear energy harvester[J]. Journal of Sound and Vibration, 2014, 333(7): 2040-2053.

[93] Xu M, Li X. Stochastic averaging for bistable vibration energy harvesting system[J]. International Journal of Mechanical Sciences, 2018, 141: 206-212.

[94] He Q, Daqaq M F. Daqaq.Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise[J]. Journal of Sound and Vibration, 2014, 333(15): 3479-3489.

[95] Hai-Tao L, Hu D, Xing-Jian J, et al. Improving the performance of a tri-stable energy harvester with a staircase-shaped potential well[J]. Mechanical Systems and Signal Processing, 2021, 159: 107805.

[96] Huang X. Stochastic resonance in a piecewise bistable energy harvesting model driven by harmonic excitation and additive Gaussian white noise[J]. Applied Mathematical Modelling, 2021, 90: 505-526.

[97] 朱位秋.随机振动[M].北京:科学出版社,1998.

[98] Zhou S, Yu T. Performance comparisons of piezoelectric energy harvesters under different stochastic noises[J]. AIP Advances, 2020, 10(3): 035033.

[99] Masana R, Daqaq M F. Response of duffing-type harvesters to band-limited noise[J]. Journal of Sound and Vibration, 2013, 332(25): 6755-6767.

[100] 吴娟娟,冷永刚,乔海,等.窄带随机激励双稳压电悬臂梁响应机制与能量采集研究[J].物理学报,2018, 67(21): 79-95.

[101] Khovanova N, Khovanov I, Davletzhanova Z. Nonlinear energy harvesting from random narrow-band excitations[J]. International Journal of Structural Stability and Dynamics, 2014, 14(08): 1440026.

[102] Huang D, Zhou S, Yang Z. Resonance mechanism of nonlinear vibrational multistable energy harvesters under narrow-band stochastic parametric excitations[J]. Complexity, 2019, 2019: 1-20.

[103] Zhou X, Gao S, Liu H, et al. Nonlinear hybrid piezoelectric and electromagnetic energy harvesting driven by colored excitation[J]. Energies, 2018, 11(3): 498.

[104] Jin Y, Zhang Y. Dynamics of a delayed Duffing-type energy harvester under narrow-band random excitation[J]. Acta Mechanica, 2021, 232: 1045-1060.

[105] Zhang Y, Jin Y, Zhang Z. Dynamics of a tri-stable hybrid energy harvester under narrow-band random excitation[J]. International Journal of Non-Linear Mechanics, 2023, 148: 104294.

[106] Wang S, Wang C, Gao Z, et al. Design and performance of a cantilever piezoelectric power generation device for real-time road safety warnings[J]. Applied Energy, 2020, 276: 115512.

[107] Zhang Y, Jin Y. Stochastic dynamics of a piezoelectric energy harvester with correlated colored noises from rotational environment[J]. Nonlinear Dynamics, 2019, 98: 501-515.

[108] Dietl J M, Garcia E. Beam shape optimization for power harvesting[J]. Journal of Intelligent Material Systems and Structures, 2010, 21(6): 633-646.

[109] Xie X, Wang Z, Liu D, et al. An experimental study on a novel cylinder harvester made of L-shaped piezoelectric coupled beams with a high efficiency[J]. Energy, 2020, 212: 118752.

[110] 陈路阳.三稳态压电俘能器动力学建模与响应特性研究[D].西安:西安科技大学, 2022.

[111] 薛晓.井下WSN节点自供能与功耗优化关键技术研究[D].武汉:中国地质大学, 2015.

[112] Stewart M, Weaver P M, Cain M. Charge redistribution in piezoelectric energy harvesters[J]. Applied Physics Letters, 2012, 100(7).

[113] 单辉祖.材料力学[M].北京:高等教育出版社,2004.

[114] Liang Z, Xu C, Ren B, et al. Theoretical analysis of energy harvesting performance for clamped–clamped piezoelectric beam[J]. Microsystem Technologies, 2015, 21: 815-823.

[115] 安朝,谢长川,杨超.弹性欧拉梁大变形分析的椭圆积分统一形式[J].北京航空航天大学学报,2021, 47(07): 1379-1386.

[116] 韩琪.拉弯型柔顺多稳态机构的精确建模方法研究[D].西安:西安电子科技大学, 2018.

[117] Awtar S, Shimotsu K, Sen S. Elastic averaging in flexure mechanisms: A three-beam parallelogram flexure case study[J]. Journal of Mechanisms and Robotics, 2010, 2: 317- 337.

[118] Ma F, Chen G. Modeling large planar deflections of flexible beams in compliant mechanisms using chained beam-constraint-model[J]. Journal of Mechanisms and Robotics, 2016, 8(2): 021018.

[119] 马付雷.柔顺机构中几何非线性变形问题的精确建模方法研究[D].西安:西安电子科技大学,2016.

[120] Awtar S, Sen S. A generalized constraint model for two-dimensional beam flexures: nonlinear load-displacement formulation[J]. Journal of Mechanical Design, 2010, 132: 81008

[121] Chen G, Ma F, Hao G, et al. Modeling large deflections of initially curved beams in compliant mechanisms using chained beam constraint model[J]. Journal of Mechanisms and Robotics, 2019, 11(1): 011002.

[122] 王光庆,张伟,刘创,等.非线性压电振动能量采集器的振动特性与实验研究[J].传感技术学报,2015, 28(10): 1494-1502.

[123] 张雨阳,冷永刚,谭丹,等.基于磁化电流法的双稳压电悬臂梁磁力精确分析[J].物理学报,2017, 66(22): 177-189.

[124] Charpentier J F, Lemarquand G. Optimal design of cylindrical air-gap synchronous permanent magnet couplings[J]. IEEE Transactions on Magnetics, 1999, 35(2): 1037- 1046.

[125] Wang G, Liao W H, Zhao Z, et al. Nonlinear magnetic force and dynamic characteristics of a tri-stable piezoelectric energy harvester[J]. Nonlinear Dynamics, 2019, 97: 2371-2397.

[126] Ren Z, Zhao H, Liu C, et al. Study the influence of magnetic force on nonlinear energy harvesting performance[J]. AIP Advances, 2019, 9(10).

[127] Zhang X, Zuo M, Yang W, et al. A tri-stable piezoelectric vibration energy harvester for composite shape beam: Nonlinear modeling and analysis[J]. Sensors, 2020, 20(5): 1370.

[128] Yang Y, Xiang H. A simple and precise formula for magnetic forces in nonlinear piezoelectric energy harvesting[J]. Nonlinear Dynamics, 2023, 111(7): 6085-6110.

[129] Erturk A, Inman D J. Piezoelectric Energy Harvesting[M]. Hoboken: John Wiley & Sons Ltd, 2011.

[130] IEEE Standard on piezoelectricity[S], New York: Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, 1987.

[131] Wang W, Cao J, Bowen C R, et al. Nonlinear dynamics and performance enhancement of asymmetric potential bistable energy harvesters[J]. Nonlinear Dynamics, 2018, 94: 1183-1194.

[132] 聂欣,张婷婷,靳艳飞.窄带随机激励下三稳态压电俘能器的动力学特性与实验研究[J].动力学与控制学报,2023, 21(03): 53-62.

[133] 朱位秋.随机动力学引论[M].北京:科学出版社,2017.

[134] Masana R, Daqaq M F. Response of duffing-type harvesters to band-limited noise[J]. Journal of Sound and Vibration, 2013, 332(25): 6755-6767.

[135] 蓝春波,秦卫阳,李海涛.随机激励下双稳态压电俘能系统的相干共振及实验验证[J].物理学报, 2015, 64(08): 76-83.

[136] Chen X, Zhang X, Guo Y, et al. Modeling and performance analysis of a curve-shaped based doubly clamped piezoelectric energy harvester (CD-PEH)[J]. Journal of Intelligent Material Systems and Structures, 2024, 35(1): 3-16.

[137] Yang Z, Zu J, Xu Z. Reversible nonlinear energy harvester tuned by tilting and enhanced by nonlinear circuits[J]. IEEE/ASME Transactions on Mechatronics, 2016, 21(4): 2174-2184.

[138] Beeby S P, Torah R N, Tudor M J, et al. A micro electromagnetic generator for vibration energy harvesting[J]. Journal of Micromechanics and Microengineering, 2007, 17(7): 1257.

[139] Wu Z, Xu Q. Design of a structure-based bistable piezoelectric energy harvester for scavenging vibration energy in gravity direction[J]. Mechanical Systems and Signal Processing, 2022, 162: 108043.

[140] Mei J, Fan Q, Li L, et al. Analytical Modeling of a Doubly Clamped Flexible Piezoelectric Energy Harvester with Axial Excitation and Its Experimental Characterization[J]. Sensors, 2021, 21(11): 3861.

[141] Liang H, Hao G, Olszewski O Z, et al. Ultra-low wide bandwidth vibrational energy harvesting using a statically balanced compliant mechanism[J]. International Journal of Mechanical Sciences, 2022, 219: 107130.

[142] Sriramdas R, Chiplunkar S, Cuduvally R M, et al. Performance enhancement of piezoelectric energy harvesters using multilayer and multistep beam configurations[J]. IEEE Sensors Journal, 2015, 15(6): 3338-3348.

[143] Hu G, Liang J, Lan C, et al. A twist piezoelectric beam for multi-directional energy harvesting[J]. Smart Materials and Structures, 2020, 29(11): 11LT01.