具有时滞反馈控制的双稳态压电-电磁式俘能器的随机动力学

孙成佳，靳艳飞，张艳霞; Sun Chengjia; Jin Yanfei; Zhang Yanxia

引 en

具有时滞反馈控制的双稳态压电-电磁式俘能器的随机动力学^*

孙成佳¹
机构：
1. 北京理工大学宇航学院力学系, 北京 100081
×
，靳艳飞¹
机构：
1. 北京理工大学宇航学院力学系, 北京 100081
×
，张艳霞²
机构：
2. 西安科技大学理学院数学系, 西安 710054
×

1. 北京理工大学宇航学院力学系, 北京 100081；
2. 西安科技大学理学院数学系, 西安 710054；

Stochastic Dynamics of the Bistable Piezoelectric and Electromagnetic Hybrid Energy Harvester with Time-Delayed Feedback Control^*

Sun Chengjia¹
Affiliation：
1. Department of Mechanics, Beijing Institute of Technology, Beijing 100081 , China
×
，Jin Yanfei¹
Affiliation：
1. Department of Mechanics, Beijing Institute of Technology, Beijing 100081 , China
×
，Zhang Yanxia²
Affiliation：
2. School of Sciences, Xi’an University of Science and Technology, Xi’an 710054 , China
×

1. Department of Mechanics, Beijing Institute of Technology, Beijing 100081 , China；
2. School of Sciences, Xi’an University of Science and Technology, Xi’an 710054 , China；

通讯作者:

靳艳飞,E-mail:jinyf@bit.edu.cn

中图分类号:TM619;TP212.9

文献标识码:A

文章编号:1672-6553-2023-21(8)-075-007

DOI:10.6052/1672-6553-2023-057

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参考文献
出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 双稳态压电-电磁俘能器的时滞动力学方程
2 系统的随机响应分析
2.1 能量包线随机平均法
2.2 系统响应分析
3 结论
参考文献

摘要

本文研究了色噪声激励下具有时滞反馈控制的双稳态压电-电磁混合式俘能器的动力学特性.首先,建立了非线性双稳态压电-电磁俘能器的集中参数型机电耦合运动方程,并基于能量包线随机平均法推导得到系统稳态概率密度函数和平均输出功率表达式,通过蒙特卡洛数值方法验证了理论解的正确性.其次,分析了时滞和反馈增益系数对系统输出功率的影响.研究结果表明：在一定参数范围内,随着位移时滞与速度时滞的增加,输出功率出现周期性变化,并在特定的时滞与反馈增益组合下达到最大值；调节反馈增益系数可以改变输出功率的相位和大小.说明通过合理地设置时滞反馈控制可以提高俘能器的采集功率,为随机激励下非线性振动俘能器的设计和优化提供一定的理论依据.

Abstract

In this paper, the dynamical behaviors of a bistable piezoelectric and electromagnetic hybrid energy harvester under the colored noise are studied, which is considered the effects of time-delayed feedback control. Firstly, the lumped parameter electromechanical coupling motion equation of a bistable piezoelectric and electromagnetic energy harvester is established. The expressions of stationary probability density (SPD) and average output power of the system are derived by adopting the improved stochastics averaging method of energy envelope. The obtained theoretical solution is verified by numerical results through Monte Carlo numerical method. Secondly, the influences of time delay and feedback gains on the system output power is analyzed. The results show that within a certain parameter range, the output power changes periodically with the increase of time delays in the path of displacement and velocity. An optimal combination of time delay and feedback gains exist to make the maximum of output power. The phase and value of output power can be changed by adjusting the feedback gains. That is, the harvesting performance can be greatly improved by choosing the proper design of time-delayed feedback control. Therefore, the dynamical analysis in this paper provides a theoretical basis for the parameter optimalization and design of nonlinear vibration energy harvester under random excitation.

关键词

压电-电磁混合式俘能器；能量包线随机平均法；时滞反馈控制；输出功率

Keywords

piezoelectric and electromagnetic hybrid energy harvester ； stochastics averaging method of energy envelope ； time-delayed feedback control ； system output power

引言
随着物联网技术的发展，无线传感器得到广泛的应用.为了充分发挥无线传感器的潜力，需要一种可持续的电源在恶劣情况下为传感器供电.振动能量采集为这一问题提供了解决方案^[1-2].振动在自然环境中普遍存在，例如：海洋平台受到的随机波浪载荷、路面不平度对运动车辆的作用、强地震引起的地面运动等，且真实环境振动中随机性是普遍存在的，因此亟需针对随机振动下非线性俘能器的动力学开展研究.传统的线性谐振式振动俘能器的频带窄，无法满足实际应用需求^[3].为了充分采集环境中的随机振动能量，研究者考虑了结构的非线性因素，提出非线性俘能器扩大频带宽度，提高系统的俘能特性^[4-7]，其中非线性双稳态俘能器相比于单稳态俘能器可以产出更高的输出功率，极大地提高俘能效率^[8-11].振动俘能器通常有电磁式、压电式、静电式，为了提高能量采集器的性能，研究人员将不同类型能量采集技术进行融合.压电-电磁混合式俘能器由于具有功率密度高、转换效率高、适应性强等优点^[12]受到了许多研究人员的关注.Yang等^[13]开发了一种基于悬臂梁结构的压电-电磁混合式能量采集器，并通过改进微加工线圈的设计和制造，减少了采集器的附加尺寸和重量.Hafizh等^[14]设计了一种压电与电磁转换模块相结合的采集器，成功完成了水流动能到电能的转换，并研究了不同扰动流体形状对发电性能的影响.Javed等^[15]提出了一种压电-电磁能量采集装置，并提出了系统的耦合动力学模型，分析了压电振动采集器和电磁振动采集器的外接负载阻值对起振速度、耦合阻尼、频率以及输出功率的影响.Zhang等^[16]利用数值方法分析了Lévy噪声作用下三稳态压电-电磁能量采集系统的随机分岔和随机共振现象.因此，压电-电磁混合式振动能量采集系统可以有效发挥二者的优势，提高了环境振动能量转换效率和俘能效果，并具有更高的环境适应性.
双稳态混合式振动俘能器已被证明是改善能量采集性能的较好解决方案，但是如果非线性系统设计或控制不当，可能会诱发复杂的非线性动力学现象，如分岔现象等^[17-18].通过引进控制方法，可以进一步改善环境随机激励和压电悬臂非线性振动的相互作用^[19-20].从振动控制的角度，时滞反馈控制可以调整系统的非线性振动并优化系统性能，使系统稳定性增强^[21-23].Jin等^[24-25]在具有时滞状态反馈的Duffing振子的研究中证明，选择适当的时滞参数和反馈增益可以提高系统的动态性能.Belhaq等^[26]研究了一种延迟非线性微机电系统，发现存在一种适当的时滞参数组合可以使周期内能量采集效率最大化.Yang等^[27]发现时滞反馈控制可以增强随机共振现象，从而获得更显著的系统响应和更高的输出功率.Jin等^[28]利用多尺度方法研究了窄带随机激励下具有时滞状态反馈的Duffing型能量采集器，确定了适当的时滞和反馈增益组合使得系统的能量转化率和平均输出电压极大的提高.可见，时滞反馈控制已成为提高振动能量采集效率的理想解决方案，并受到广泛关注.然而，大多数研究集中于时滞反馈控制下的压电能量采集器，而揭示时滞反馈控制下混合式振动能量采集器的复杂动态行为具有重要意义.
本文考虑双稳态压电-电磁能量采集器的非线性特性，建立了集中参数型机电耦合时滞反馈控制方程，利用色噪声模拟环境激励，基于能量包线的随机平均法得到系统运动输出的解析解表达式，分析了时滞、反馈增益、色噪声等对系统稳态响应和俘能的影响，并通过蒙特卡洛方法验证了理论方法的有效性.
1 双稳态压电-电磁俘能器的时滞动力学方程
考虑如下具有时滞状态反馈控制的机电耦合的混合式双稳态振动俘能系统，其结构示意图如图1所示.
该混合式双稳俘能器包括一个耦合到电路的机械振荡器，可通过压电模块将振动的机械能转换为电能；一个固定线圈，振荡器一端的永磁体与线圈产生相对位移从而产生感应电动势；以及一个时滞状态反馈控制器，设计为线性位移和速度的时滞状态反馈控制.压电单元与电磁单元的等效电路如图2 所示，其中R_p为压电阻值，R_E为线圈阻值， $\dot{X}$ 为悬臂梁尖端的磁铁运动速度.根据牛顿第二定律和基尔霍夫定律，具有时滞反馈控制的无量纲耦合双稳态混合俘能系统可以表示为
图1 双稳态压电-电磁混合式俘能系统的结构示意图
Fig.1 The schematic diagram of a bistable piezoelectric and electromagnetic hybrid energy harvester
图2 混合式能量采集器中压电和电磁单元的等效电路
Fig.2 The equivalent model of the piezoelectric and electromagnetic elements for hybrid energy harvester

\begin{matrix} \ddot{X} + β \dot{X} + \frac{d U_{0} (x)}{d x} + κ_{1} V + κ_{2} I = \\ μ X_{τ 1} + ν {\dot{X}}_{τ 2} + ξ (t) \end{matrix}

(1)

\dot{V} + α_{1} V = \dot{X}

(2)

\dot{I} + α_{2} I = \dot{X}

(3)

其中 $X_{τ 1} = X (t - τ_{1}) ， {\dot{X}}_{τ 2} = \dot{X} (t - τ_{2}) ， τ_{1} ， τ_{2}$ 为时滞参数，X为质量M的位移，V为压电模块采集电压，μ表示位移的反馈增益，v表示速度的反馈增益，β为阻尼系数，κ₁，κ₂分别表示压电模块和电磁模块的机电耦合系数，α₁，α₂分别表示压电模块和电磁模块的时间常数比，ξ（t）分别表示模拟环境激励的色噪声，统计学特性如下
$\begin{matrix} < ξ (t) > = 0 \\ < ξ (t) ξ (s) > = \frac{D}{c} e x p [- \frac{| t - s |}{c}] \end{matrix}$
其中D为噪声强度，c为噪声相关时间.
方程（1）中势函数U₀（x）表达式为U₀（x）=-1/2δ₁x²+1/4δ₃x⁴，其中δ₁，δ₃分别为线性刚度系数和立方刚度系数.通过对势函数分析可知，系统存在两个稳定的平衡点和一个不稳定的鞍点 $X_{3}^{*}$ ，其具体形式为 $X_{1}^{*} = \sqrt{δ_{1} / δ_{3}} ， X_{2}^{*} = - \sqrt{δ_{1} / δ_{3}} ， X_{3}^{*} = 0 .$
在下面的分析中，除特殊说明外，选取如下的参数值：β=0.02，κ₁=0.3，κ₂=0.3，α₁=0.05，α₂=0.05，δ₁=3，δ₂=3，D=0.005，c=0.03.
2 系统的随机响应分析
2.1 能量包线随机平均法
根据能量包线的随机平均法^[29]，引入广义调和函数，系统的位移和速度可以表示为

\begin{matrix} X (t) = A [H (t)] c o s {ω [H (t)] t + φ (t)} + \\ X_{i}^{*} [H (t)] \end{matrix}

(4)

\begin{matrix} \dot{X} (t) = - A [H (t)] ω [H (t)] \times \\ s i n {ω [H (t)] t + φ (t)} \end{matrix}

(5)

根据总能量H的不同，系统存在三种不同的周期运动：右侧势阱内的振动（i=1）、左侧势阱内的振动（i=2）和两势阱间的大范围往复振动（i=3）.由式（2）和式（3），此时电压、电流可近似表示为^[30]

V (t) = \int_{0}^{t} e^{- {α 1}_{} s} \dot{X} (t - s) d s

(6)

I (t) = \int_{0}^{t} e^{- {α 2}_{} s} \dot{X} (t - s) d s

(7)

其中

\begin{matrix} \dot{X} (t - s) = \dot{X} (t) c o s {ω [H (t)] s} + {X (t) - \\ X_{i}^{*} [H (t)] \times ω [H (t)] s i n {ω [H (t)] s}\} \end{matrix}

(8)

将式（8）代入式（6）和式（7）可得电压与电流表达式

\begin{matrix} V (t) = \frac{ω^{2} [H (t)]}{α_{1}^{2} + ω^{2} [H (t)]} \{X (t) - X_{i}^{*} [H (t)]\} + \\ \frac{α_{2}}{α_{1}^{2} + ω^{2} [H (t)]} \dot{X} (t) \end{matrix}

(9)

\begin{matrix} I (t) = \frac{ω^{2} [H (t)]}{α_{2}^{2} + ω^{2} [H (t)]} \{X (t) - X_{i}^{*} [H (t)]\} + \\ \frac{α_{2}}{α_{2}^{2} + ω^{2} [H (t)]} \dot{X} (t) \end{matrix}

(10)

结合式（9）和式（10），可将系统（1）解耦后的方程写为

\begin{matrix} \ddot{X} + \tilde{β} \dot{X} - δ_{1} X + δ_{3} X^{3} + \tilde{δ} {X (t) - \\ X_{i}^{*} [H (t)]\} = ξ (t) \end{matrix}

(11)

这里
$\begin{matrix} \tilde{β} = β + \frac{κ_{1} α_{1}}{α_{1}^{2} + ω^{2} (H)} + \frac{κ_{2} α_{2}}{α_{2}^{2} + ω^{2} (H)} + \\ \frac{μ}{ω (H)} s i n [ω (H) τ_{1}] - ν c o s [ω (H) τ_{2}] \\ \tilde{δ} = \frac{κ_{1} ω^{2} (H)}{α_{1}^{2} + ω^{2} (H)} + \frac{κ_{2} ω^{2} (H)}{α_{2}^{2} + ω^{2} (H)} - \\ μ c o s [ω (H) τ_{1}] - ν ω (H) c o s [ω (H) τ_{2}] \end{matrix}$
此时势能和总能量函数为

U (X) = - \frac{1}{2} δ_{1} X^{2} + \frac{1}{4} δ_{3} X^{4} + \frac{1}{2} \tilde{δ} X^{2}

(12)

H = \frac{1}{2} {\dot{X}}^{2} + U (X)

(13)

方程（11）中依赖于能量的频率ω（H）的表达式为

ω (H) = \frac{2 π}{T (H)} = 2 π {[\int_{x a}^{x_{b}} \frac{d X}{\sqrt{2 H - 2 U (X)}}]}^{- 1}

(14)

其中x_a和x_b分别为不考虑时滞影响下系统运动位移的最小值和最大值.由于势能函数的表达式（12）中含有未知的频率ω（H），此时可以利用迭代法对方程（14）进行求解得到频率^[31]，如图3所示.
图3 依赖于能量的系统频率，其中τ=1.3，τ₂=0.5，μ=0.01，v=0.01
Fig.3 The energy-dependent frequency under the given system parameters for τ=1.3, τ₂=0.5, μ=0.01, v=0.01
由公式（13）可得如下运动方程

\dot{X} = \pm \sqrt{2 H - U (X)}

(15)

\dot{H} = - \dot{X} (\tilde{β} \dot{X} - \tilde{δ} X_{i}^{*}) + \dot{X} ξ (t)

(16)

系统的能量是慢变量，则能量过程是一个慢变过程，可近似为一维马尔可夫过程，利用能量包络线的随机平均法可得对应的伊藤方程为

d H = m (H) d t + σ (H) d B (t)

(17)

其中，B（t）为标准布朗运动，m（H）和σ（H）为平均后的漂移系数和扩散系数.
$\begin{matrix} m (H) = - {⟨\dot{X} (\tilde{β} \dot{X} - \tilde{δ} X_{i}^{*})⟩}_{t} + \\ \frac{D}{1 + c^{2} ω^{2} (H)} \\ σ^{2} (H) = \frac{2 D}{1 + c^{2} ω^{2} (H)} {⟨{\dot{X}}^{2}⟩}_{t} \end{matrix}$
这里，时间平均为
$⟨ \cdot ⟩_{t} = \frac{1}{T} \int_{0}^{T} (\cdot) d t = \frac{1}{T} ∮ \frac{(\cdot)}{\dot{X}} d t = \frac{ω (H)}{2 π} ∮ \frac{(\cdot)}{\dot{X}} d x$
由伊藤方程（16）可写出对应的FPK方程，对其求解可得系统的稳态概率密度函数为

p (H) = \frac{N_{0}}{σ^{2} (H)} e x p [\int \frac{2 m (H)}{σ^{2} (H)} d H]

(18)

其中N₀为归一化常数.
由变换 $p （ X ， \dot{X} ） = p （ H ） / T （ H ）$ 可得此时解耦后的等效系统（11）的X（t）和 $\dot{X}$ （t）的联合稳态概率密度函数表达式

\begin{matrix} p (X, \dot{X}) = \frac{N_{0} [1 + c^{2} ω^{2} (H)]}{D} \times \\ e x p \{- \frac{\tilde{β} [1 + c^{2} ω^{2} (H)]}{D} \times \\ [\frac{1}{2} {\dot{X}}^{2} - \frac{1}{2} (δ_{1} - \tilde{δ}) X^{2} + \frac{1}{4} δ_{3} X^{4}]\} \end{matrix}

(19)

根据式（9）和（19）可得均方电压表达式

\begin{matrix} E [V^{2}] = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \{\frac{ω^{2} (H)}{α_{1}^{2} + ω^{2} (H)} [X - X_{i}^{*} (H)] + \\ {\frac{α_{1}}{α_{1}^{2} + ω^{2} (H)} \dot{X}\}}^{2} \times p (X, \dot{X}) d X d \dot{X} \end{matrix}

(20)

同理可得均方电流和功率表达式

\begin{matrix} E (I^{2}) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \{\frac{ω^{2} (H)}{α_{2}^{2} + ω^{2} (H)} [X - X_{i}^{*} (H)] + \\ {\frac{α_{2}}{α_{2}^{2} + ω^{2} (H)} \dot{X}\}}^{2} \times p (X, \dot{X}) d X d \dot{X} \end{matrix}

(21)

E (P) = κ_{1} α_{1} E (V^{2}) + κ_{2} α_{2} E (I^{2})

(22)

2.2 系统响应分析
在本小节中，为了验证上述理论结果的正确性，首先采用蒙特卡洛方法对系统式（1）~式（3）进行数值计算，在X与 $\dot{X}$ 取值范围[-2，2]内划分80个大小为0.5×0.5的网格，取样本数为10000，时间步长为0.05.根据式（19）绘制的联合概率密度图像如图4（a）所示，图4（b）为通过蒙特卡洛数值计算得到的系统平稳联合概率密度，通过对比图4（a）和图4（b）可知，平稳联合概率密度均具有双峰结构，且峰的高度一致，因此数值结果与解析计算结果一致，说明解析表达式（19）~式（22）的正确性及能量包线的随机平均法的有效性.
图4 系统的平稳联合概率密度，其中参数 τ₁=1.3，τ₂=0.5，μ=0.01，υ=0.01
Fig.4 Joint stationary probability density (SPD) with τ₁=1.3, τ₂=0.5, μ=0.01, υ=0.01
下面主要根据理论结果（22）来讨论时滞大小与反馈增益系数对于系统平均输出功率的影响.图5（a）给出了系统平均输出功率E[P]随τ₁、τ₂的增大而变化的曲线.对于固定的反馈增益系数，当τ₁，τ₂在[0，4]范围内变化时，E[P]在点（τ₁，τ₂）=（1.4，2）处达到最大值.即，当反馈增益系数固定时，可以通过选取最优的τ₁，τ₂使得系统的平均输出功率达到最大，实现混合式俘能器的动态性能的提高.在图5（b）中，取τ₁=0，μ=0，分析速度时滞反馈控制对系统响应的影响.由图可见，随着τ₂的增加，E[P]呈现周期性的变化.若τ₂=1固定，此时负速度时滞反馈控制下系统的E[P]明显高于正速度时滞反馈的情形；若取τ₂=2.5，正速度时滞反馈控制下系统的E[P]明显高于负速度时滞反馈的情形；当τ₂=2.0时，正、负速度时滞反馈控制下的平均输出功率相等，即E[P]=10^-3.因此，速度时滞反馈控制的效果依赖于时滞的选取.在图5（c）中，固定τ₁=1.4，μ=0.01，讨论E[P]随τ₂和v的变化趋势.通过对曲线的观察发现，对于固定的τ₂=2.5，随着速度反馈增益系数的增大，E[P]增大.因此，可以通过选取τ₂和v使系统平均输出功率达到最大，例如：说明位移时滞与速度时滞反馈控制的共同作用效果优于单一速度时滞反馈控制的情形.
图5 不同时滞和反馈增益下系统的平均输出功率
Fig.5 The mean output power of system for different time delays and feedback gains
3 结论
随机振动环境激励下非线性俘能器的动力学研究已成为机械、力学、材料、电子等诸多学科领域关注的重要问题.由于非线性结构特性的引入，在随机激励下振动俘能系统会产生一些新的非线性动力学现象，此时振动俘能器的随机动力学分析显得尤为重要.本文针对色噪声激励下双稳态压电-电磁混合式俘能系统，提出了通过线性位移和速度时滞反馈控制来提高系统的响应和采集效率.首先，建立了压电-电磁混合式双稳态俘能器的动力学方程，基于能量包线的随机平均法推导出系统输出响应的解析表达式，包括：系统平稳联合概率密度、均方电压、均方电流、平均输出功率.然后，利用蒙特卡洛方法得到系统输出响应的数值解，通过对比发现平稳联合概率密度均具有双峰结构完全，理论解与数值解一致，说明理论结果的正确性.最后，分析了时滞与反馈增益系数对系统响应的影响，研究结果表明：存在最优的时滞与反馈增益组合，使得系统的平均输出功率最大.此外，通过比较发现位移时滞与速度时滞反馈的联合作用比单个速度时滞反馈更有利于提高系统的俘能效率.
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基本信息

中图分类号: TM619;TP212.9
文献标识码: A
DOI: 10.6052/1672-6553-2023-057
文章编号: 1672-6553-2023-21(8)-075-007

基金信息

国家自然科学基金资助项目(12072025)；
北京市自然基金项目(1222015)；
陕西省自然科学基础研究计划项目(2022JQ-044)；
陕西省教育厅专项科研计划项目(22JK0462)；
Natural Science Foundation of China(12072025)；
Natural Science Foundation of Beijing(1222015 )；
Natural Science Basic Re-search Program of Shaanxi Province (2022JQ-044)；
Natural Science Foundation of Shaanxi Provincial Department of Education (22JK0462)；

引用信息

稿件历史

收稿日期: 2023-05-17

参考文献

[1] CHALLA V,PRASAD M G,FISHER F T.A coupled piezoelectric electromagnetic energy harvesting technique for achieving increased power output through damping match [J].Smart Materials and Structures,2009,18(9):95029-95011.
[2] 曹东兴,吴鹏,张伟,等.一种新型压电俘能器的振动特性分析及性能研究 [J].动力学与控制学报,2015,13(4)：300-307.CAO D X,WU P,ZHANG W,et al.Nonlinear vibrationof a novel piezoelectric vibration energy harvester [J].Journal of Dynamics and Control,2015,13(4):300-307.(in Chinese)
[3] 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,2016,46(8):791-807.DAI Z X,ZHANG Z,LIU X Y,et al.Research progress of broadband vibration energy harvesting technology using nonlinear method [J].Science China Technological Sciences,2016,46(8):791-807.(in Chinese)
[4] GAMMAITONI L,NERI I,VOCCA H.Nonlinear oscillators for vibration energy harvesting [J].Applied Physics Letters,2009,94(16):164102.
[5] 杨涛,周生喜,曹庆杰,等.非线性振动能量俘获技术的若干进展 [J].力学学报,2021,53(11)：2894-2909.YANG T,ZHOU S X,CAO Q J,et al.Some advances in nonlinear vibration energy harvesting technology [J].Chinese Journal of Theoretical and Applied Mechanics,2021,53(11):2894-2909.(in Chinese)
[6] DAQAQ M F,MASANA R,ERTURK A.On the role of nonlinearities in vibratory energy harvesting:a critical review and discussion [J].Applied Mechanics Reviews,2014,66:040801-040824.
[7] 郭抗抗,曹树谦.压电发电悬臂梁的非线性动力学建模及响应分析[J].动力学与控制学报,2014,12(1):18-23.GUO K K,CAO S Q.Nonlinear modeling and analysis of piezoelectic cantilever energy harvester [J].Journal of Dynamics and Control,2014,12(1):18-23.(in Chinese)
[8] 蓝春波,秦卫阳,带碰撞双稳态压电俘能系统的俘能特性研究 [J].物理学报,2015,64(21)：210501.LAN C B,QIN W Y.Vibration energy harvesting from a piezoelectric bistable system with two symmetric stops [J].Acta Physica Sinica,2015,64(21):210501.(in Chinese)
[9] HARNE R L,WANG K W.A review of the recent research on vibration energy harvesting via bistable systems [J].Smart Materials and Structures,2013,22(2):023001.
[10] 陈丽华,景浩,陈坦,等.微型双稳态板压电俘能器的仿真研究 [J].动力学与控制学报,2019,17(6)：537-545.CHEN L H,JING H,CHEN T,et al.The simulation of bi-stable micro piezoelectric energy harvesting [J].Journal of Dynamics and Control,2019,17(6):537-545.(in Chinese)
[11] MANN B P,OWENS B A.Investigations of a nonlinear energy harvester with a bistable potential well [J].Journal of Sound and Vibration,2010,329(9):1215-1226.
[12] ZHOU J W,HE L P,LIU L,et al.Design and research of hybrid piezoelectric-electromagnetic energy harvester based on magnetic couple suction-repulsion motion and centrifugal action [J].Energy Conversion and Management,2022,258:115504.
[13] YANG B,LEE C,KEE W L,et al.Hybrid energy harvester based on piezoelectric and electromagnetic mechanisms [J].Journal of Micro-Nanolithography Mems and Moems,2010,9(2):41-42.
[14] MUTHALIF A G,HAFIZH M,RENNO J,et al.A hybrid piezoelectric-electromagnetic energy harvester from vortex-induced vibrations in fluid-flow;the influence of boundary condition in tuning the harvester [J].Energy Conversion and Management,2022,256:115371.
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[20] CASSIDY I L,SCRUGGS J T,BEHRENS S.Optimization of partial-state feedback for vibratory energy harvesters subjected to broadband stochastic disturbances [J].Smart Materials and Structures,2011,20(8):85019-85013.
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具有时滞反馈控制的双稳态压电-电磁式俘能器的随机动力学*

Stochastic Dynamics of the Bistable Piezoelectric and Electromagnetic Hybrid Energy Harvester with Time-Delayed Feedback Control*

摘要

Abstract

关键词

Keywords

1 双稳态压电-电磁俘能器的时滞动力学方程

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2 系统的随机响应分析

2.1 能量包线随机平均法

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2.2 系统响应分析

3 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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具有时滞反馈控制的双稳态压电-电磁式俘能器的随机动力学^*

Stochastic Dynamics of the Bistable Piezoelectric and Electromagnetic Hybrid Energy Harvester with Time-Delayed Feedback Control^*