双不确定参数作用下双稳态能量采集系统的随机响应分析

张莹，高艺玲，段霞霞，贾万涛，岳晓乐; Zhang Ying; Gao Yiling; Duan Xiaxia; Jia Wantao; Yue Xiaole

引 en

双不确定参数作用下双稳态能量采集系统的随机响应分析^*

张莹^1,2
机构：
1. 西北工业大学数学与统计学院,西安 710129
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129
×
，高艺玲^1,2
机构：
1. 西北工业大学数学与统计学院,西安 710129
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129
×
，段霞霞^1,2
机构：
1. 西北工业大学数学与统计学院,西安 710129
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129
×
，贾万涛^1,2
机构：
1. 西北工业大学数学与统计学院,西安 710129
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129
×
，岳晓乐^1,2
机构：
1. 西北工业大学数学与统计学院,西安 710129
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129
×

1. 西北工业大学数学与统计学院,西安 710129；
2. 西北工业大学空天领域复杂性科学教育部重点实验室,西安 710129；

Stochastic Response Analysis of Bistable Energy Harvesting System with Double Uncertain Parametters^*

Zhang Ying^1,2
Affiliation：
1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China
×
，Gao Yiling^1,2
Affiliation：
1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China
×
，Duan Xiaxia^1,2
Affiliation：
1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China
×
，Jia Wantao^1,2
Affiliation：
1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China
×
，Yue Xiaole^1,2
Affiliation：
1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China
×

1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129 , China；
2. MOE Key Laboratory for Complexity Science in Aerospace Northwestern Polytechnical University, Xi’an 710129 , China；

通讯作者:

岳晓乐,E-mail:xiaoleyue@nwpu.edu.cn

中图分类号:O322

文献标识码:A

文章编号:1672-6553-2023-21(10)-072-013

DOI:10.6052/1672-6553-2023-095

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目录contents

摘要 Abstract
关键词 Keywords
1 双稳态能量采集系统的模型和响应
1.1 双稳态能量采集系统模型
1.2 双稳态能量采集系统的确定性响应
2 双稳态能量采集系统的随机响应
2.1 正交多项式逼近
2.2 两不确定参数对系统响应的影响
2.3 不确定参数强度对系统响应的影响
2.4 能量俘获性能
3 结论
参考文献

摘要

非线性多稳态能量采集系统模型复杂、结构参数较多,由于测量、加工、装配过程中不可避免的存在误差,这些结构参数都存在一定的不确定性.而关键参数的微小变化都可能对系统输出电压产生显著影响,特别当多不确定参数同时存在时,可能会对系统能量采集性能产生更加复杂的影响.因此,研究多不确定参数作用下非线性能量采集系统的随机行为十分必要.本文针对具有双不确定参数的双稳态能量采集系统,首先,利用Routh-Hurwitz定理分析了确定性双稳态系统平衡点的稳定性及其静态分岔特性；继而,借助正交多项式逼近法,将具有两个相互独立的不确定机电耦合系数的随机双稳态系统转化为等价确定性扩阶系统,使系统的随机响应问题转换为等价系统的响应问题；之后,从全局和局部两个角度出发,通过等价系统与确定性系统吸引子、吸引域、相轨、均方电压以及能量转化率的对比,揭示了不确定参数对系统动力学行为和发电性能的影响.结果表明,在一些敏感的参数区间内,两机电耦合系数的不确定性均会导致系统运动状态发生变化,且不确定参数强度越大,系统会越早通过倍周期分岔级联进入混沌状态；此外,在两机电耦合系数的不确定性作用下,系统均方电压均会出现一定程度的降低；且与电方程中机电耦合系数相比,机械方程中机电耦合系数对系统的影响更显著,当两不确定参数共同作用时,系统的能量采集性能变化更显著.

Abstract

It is known that there are many structural parameters in the nonlinear multi-stable energy harvesting system model. Due to the inevitable errors in the process of measurement, processing and assembly, these structural parameters are uncertain. Even slight variation of key parameters may lead to a significant influence on the output voltages. Especially when multiple uncertain parameters exist at the same time, it may have a more complex impact on the energy harvesting performance of the system. Therefore, it is of great value to study the stochastic behavior of nonlinear energy harvesting systems with multiple uncertain parameters. In this paper, a bistable energy harvesting system with double uncertain parameters is investigated. Firstly, the equilibrium point stability and static bifurcation of deterministic bistable systems are analyzed by using Routh-Hurwitz theorem. Then, by using the orthogonal polynomial approximation method, the stochastic bistable system(TSBS) with two independent uncertain electromechanical coupling coefficients is reduced into an equivalent deterministic extended-order system, so that the stochastic response problem of TSBS is transformed to the response problem of an equivalent system. After that, from both global and local perspectives, the effects of uncertain parameters on system dynamics and power generation performance are revealed by comparing the attractor, attraction domain, phase orbit, mean square voltage, and energy conversion rate between the equivalent system and the deterministic system. The results show that in some sensitive parameter ranges, the uncertainty of the two electromechanical coupling coefficients will lead to the change of the dynamical motion state of the system, and the greater the intensity of the uncertain parameters are, the earlier the system will enter the chaotic state through the period doubling bifurcation cascade. In addition, under the uncertainty of the two electromechanical coupling coefficients, the mean square voltage will decrease to a certain extent. Compared with the electromechanical coupling coefficient in the electrical equation, the electromechanical coupling coefficient in the mechanical equation has a greater impact on the system. Furthermore, when the two uncertain parameters both exist, the energy harvesting performance of the system changes more significantly.

关键词

双稳态能量采集系统；不确定参数；正交多项式逼近法；机电耦合系数

Keywords

bistable energy harvester ； uncertain parameter ； orthogonal polynomial approximation method ； electromechanical coupling coefficient

引言
近年来，随着微电子技术和集成电路的发展，各种电子器件快速进入到集成化，微型化和低功耗的时代，新型的能量采集技术备受关注^[1，2].而能量采集就是将自然界中广泛存在的太阳能、风能、声能、水势能、热能和振动能等各种能源转化为电能的一种新兴技术^[3].在能量采集技术的研究中，由于振动能具有分布广泛、洁净以及不受气候和地域等环境因素的影响等优点，将振动能量收集并将其转化为电能的能量采集技术成为了研究热点^[4，5].目前，基于振动的能量采集方式主要有三种：电磁式，静电式和压电式^[6-8].其中压电式能量采集方式因其发电性能好，结构简单，不依赖外部磁场和初始直流电压，被广泛应用于生物医学、海洋工程和国防军事等领域^[9].
Erturk^[10，11]等Ferrari^[12]对线性压电式能量采集系统进行了系统研究，结果表明，当外部激励的频率接近固有频率时，系统才能获得较高的能量，这很难与环境振动的宽频、变频特性匹配.基于此研究者们提出了许多优化结构，包括共振调谐^[13，14]，单稳态^[15，16]，双稳态^[17-21]，三稳态^[22，23]，多稳态^[24，25]以及多自由度能量采集系统^[26，27]等.理论分析和实验结果表明，在一定条件下，非线性技术可以提高能量采集系统的性能.
上面关于压电式能量采集系统的研究都是基于确定的结构参数和确定的数学模型进行的.然而，在实际工作环境中，受材料性质、测量、加工及装配误差等多种因素的影响，能量采集系统的结构参数不可避免地存在不确定性，这些结构参数的不确定性往往会导致系统动力学响应的不确定性^[28].Ali等^[29]分析了固有频率和阻尼的不确定性对系统平均输出功率和耦合系数的影响；Nanda^[30]使用正交法结合最大熵原理研究了不确定参数对系统能量采集性能的影响；Franco等^[31]通过蒙特卡洛模拟方法研究了多不确定参数共同作用和分别作用时系统输出功率的变化；周生喜等^[28]使用改进的区间扩展方法分别分析了随机质量、非线性刚度、阻尼、电容和机电耦合系数对双稳态能量采集系统的工作性能的影响，并通过蒙特卡罗仿真验证了该方法的有效性.上述研究表明，参数的不确定性可能会导致系统的动力学响应发生明显变化.因此，研究不确定参数作用下非线性能量采集系统的动力学行为，尤其是系统的能量俘获特性具有重要意义.
在实际问题中，系统会同时存在多个不确定参数，在它们共同作用下系统会产生更加复杂的动力学行为.因此，借助随机动力系统理论研究含多个不确定参数的压电式能量采集系统的动力学问题，对压电式振动能量采集系统的实验设计和结构优化具有重要的指导意义.目前，关于不确定参数的研究方法中概率相关方法理论相对成熟，并且已广泛应用于科学和工程领域.主要包括蒙特卡洛方法^[32]、随机摄动法^[33]和正交多项式逼近法等.其中，正交多项式逼近法由于不受不确定参数变异系数大小的限制，并且具有显著的计算优势得到了越来越多的关注.该方法最早源于Wiener^[34]在1938年首次提出的齐次混沌理论，但其只适用于含有高斯分布随机变量的模型，为了处理更一般的随机问题，Xiu^[35]对原方法进行了推广，提出广义多项式混沌展开法，该方法可以根据不确定参数的概率分布确定相应的收敛最优多项式.Leng^[36，37]和马少娟^[38]等利用该方法成功研究了不确定参数作用下非线性系统的动力学和控制问题.到目前为止，关于含有不确定参数的能量采集系统的理论研究还不完善，尤其是含有多个不确定参数的情形.Zhou等^[28]研究发现机电耦合系数的不确定性对系统输出电压的影响强于其他结构参数.因此，本文拟针对双稳态能量采集系统，同时考虑机械方程和电方程中的两机电耦合系数的不确定性，运用正交多项式逼近法将其随机响应问题转化为与之等价的确定性系统的响应问题，通过全局分析和局部分析，揭示两不确定机电耦合系数的共同作用对系统运动状态和俘能性能的影响机理，从而深入对不确定参数作用下压电式能量采集系统的认知，为压电式能量采集系统结构参数的优化设计和能量俘获效率的提高提供可靠的理论保障.
1 双稳态能量采集系统的模型和响应
1.1 双稳态能量采集系统模型
经典的非线性压电悬臂梁能量采集系统的简化模型如图1所示，其机械部分由一个根部粘有压电片的悬臂梁、一块固定在悬臂梁自由端的永磁铁和两个对称的靠近悬臂梁的外部磁铁构成.悬臂梁自由端的永磁铁与外部磁铁的磁极相反，形成非线性磁斥力，在基础激励作用下，悬臂梁发生振动，其自由端附近的压电片产生形变，压电材料内部发生极化现象，它的两个相对表面上出现正负相反的电荷，从而产生输出电压，实现了机械能到电能的转化.在该模型中，磁力是产生双稳态现象的根源，通过调整磁铁之间的间距和外部磁体的角度可以改变非线性磁斥力的大小，使系统拥有两个稳定的平衡点，一个不稳定的平衡点，从而表现出双稳态特性.
图1 压电能量采集器的简化模型
Fig.1 Schematics of a general piezoelectric energy harvester
简谐激励下压电悬臂梁双稳态能量采集系统的动力学方程为

\{\begin{matrix} m \ddot{\bar{x}} (τ) + \bar{c} \dot{\bar{x}} (τ) + \frac{d U (\bar{x})}{d \bar{x}} - \\ \bar{θ} \bar{v} (τ) = F c o s (Ω τ) \\ C_{p} \dot{\bar{v}} (τ) + \frac{1}{R_{l}} \bar{v} (τ) + \bar{θ} \dot{\bar{x}} (τ) = 0 \end{matrix}

(1)

式中， $\bar{x}$ （t），m和 $\bar{c}$ 分别表示悬臂梁自由端永磁体的横向位移; 等效质量和等效阻尼系数， $\bar{θ}$ 是机电耦合系数，C_p是压电片的等效电容， $\bar{v}$ （t）是通过负载电阻R_l的输出电压，F为激励幅值，Ω为激励频率.
$U （ \bar{x} ）$ 是能量采集系统的势函数，其形式如下

U (\bar{x}) = \frac{1}{2} k_{1} {\bar{x}}^{2} + \frac{1}{4} k_{3} {\bar{x}}^{4}

(2)

式中，k₁和k₃分别是线性和非线性恢复力系数.
无量纲的变量和参数定义如下：

\begin{matrix} x = \frac{\bar{x}}{l}, v = \frac{C_{P} \bar{v}}{l}, t = \tilde{ω} τ, \tilde{ω} = \sqrt{\frac{k_{1}}{m}}, l = \sqrt{\frac{k_{1}}{k_{3}}} \\ c = \frac{\bar{c}}{m l}, a = - \frac{k_{1}}{m l}, μ = \frac{k_{3} l}{m}, θ = \frac{\bar{θ}}{m l C_{P}} \\ λ = \frac{1}{C_{P} R_{l}}, g = \bar{θ}, A = \frac{F}{k_{1} l}, ω = \frac{Ω}{\tilde{ω}} \end{matrix}

(3)

因此，双稳态压电能量采集器的无量纲机电模型为

\{\begin{matrix} \ddot{x} (t) + c \dot{x} (t) - a x (t) + μ x^{3} (t) - \\ θ v (t) = A c o s (ω t) \\ \dot{v} (t) + λ v (t) + g \dot{x} (t) = 0 \end{matrix}

(4)

式中，x（t）是悬臂梁自由端永磁体的无量纲位移，c是无量纲阻尼系数，a和μ分别是无量纲线性和三次非线性刚度系数，θ和g分别是机械方程和电方程中的无量纲机电耦合系数，A为激励幅值，ω为激励频率，λ是时间常数比.
下面对系统（4）对应的自治系统进行静态分岔分析和稳定性分析.首先将方程（4）对应的自治系统表示为一阶微分方程

[\begin{matrix} \dot{x} \\ \dot{y} \\ \dot{v} \end{matrix}] = [\begin{matrix} y \\ - c y + a x - μ x^{3} + θ v \\ - λ v - g y \end{matrix}]

(5)

当a≤0时，系统存在一个平衡点（0，0，0）；当a＞0时，系统存在三个平衡点（0，0，0）和±（ $\sqrt{a / μ}$ ，0，0），下面分别讨论各个平衡点的稳定性.
平衡点（0，0，0）所对应的Jocabi矩阵是

(6)

相应的特征方程det（J₁-kI）=0可展开为

k^{3} + (c + λ) k^{2} + (c λ + g θ - a) k - a λ = 0

(7)

对于特征方程（7），根据Routh-Hurwitz判据可构造的行列式为

\begin{matrix} Δ_{1} = c + λ \\ Δ_{2} = |\begin{matrix} c + λ & 1 \\ - a λ & c λ + g θ - a \end{matrix}| \\ = (c λ + g θ) (c + λ) - a c \\ Δ_{3} = |\begin{matrix} c + λ & 1 & 0 \\ - a λ & c λ + g θ - a & c + λ \\ 0 & 0 & - a λ \end{matrix}| \\ = - a λ [(c λ + g θ) (c + λ) - a c] \\ = - a λ Δ_{2} \end{matrix}

(8)

当a＜0时，由于参数c，μ，θ，λ，g都大于0，故Δ₁＞0，Δ₂＞0，Δ₃＞0，根据Routh-Hurwitz定理可知特征方程（7）的所有特征根均有负实部，所以（0，0，0）是渐进稳定的平衡点；当a=0时，平衡点（0，0，0）对应的特征方程（7）出现零特征根，此时（0，0，0）为分岔点；当a＞0时，-aλ＜0，根据式（8）可知Δ₂和Δ₃必然一正一负，由Routh-Hurwitz定理可知特征方程（7）必然存在正实部的特征根，因此平衡点（0，0，0）是不稳定的。同理可知平衡点±（ $\sqrt{a / μ}$ ，0，0）是渐进稳定的.
综上可知，当a＜0时，系统只有一个稳定的平衡点（0，0，0）；当a=0时，系统的平衡点（0，0，0）是分岔点；当a＞0时，系统有一个不稳定的平衡点（0，0，0）和两个稳定的平衡点±（ $\sqrt{a / μ}$ ，0，0），此时系统呈现出双稳态特性.
由势函数（2）的形式可知，a＞0时，势函数存在一个势垒和两个势阱，其中势垒的极大值点对应的是系统的不稳定平衡点（0，0，0），势阱的两个极小值点对应系统的两个稳定平衡点±（ $\sqrt{a / μ}$ ，0，0），相应的势阱深度为a²/（4μ），势阱间距为2 $\sqrt{a / μ}$ .由此可见，线性刚度系数a和三次非线性刚度系数μ不仅会决定系统平衡点的个数，还会影响系统的势阱深度以及势阱间距。如图2展示了a=0.5时，μ取不同值时系统的势函数图.可见，随着μ的增大，系统的势阱深度越来越小，在相同的外部激励强度条件下，系统会更容易越过势垒在两势阱间做大幅运动；但势阱深度的减小同时也会导致势阱间距的减小，进一步使系统运动的位移幅值减小，最终影响系统能量俘获的效率.
图2 三次非线性系数的势函数图
Fig.2 Potential functions for
1.2 双稳态能量采集系统的确定性响应
在平衡点静态分岔及其稳定性分析的基础上，取系统（4）的参数为c=0.2，a=0.5，θ=0.05，A=0.12，ω=0.8，λ=0.05，g=0.5，分析双稳态系统的大幅运动和小幅运动情况.图3呈现了系统（4）在多初值情况下随三次非线性刚度系数μ变化的分岔图，从图中可见，系统在μ∈（0，2]上有丰富的动力学现象。特别地，当μ∈（0，0.25）时，系统有两个共存的周期1吸引子；当μ≈0.25时，系统产生了两个新的周期1吸引子，并在μ∈（0.25，0.32）内保持多吸引子共存状态；并随着μ的增大，其中两个1周期吸引子相继消失；当μ继续增大到μ≈0.49时，系统通过倍周期分岔级联进入混沌状态，并在μ∈（0.50，0.51）内出现了共存的稀有吸引子，当μ≈0.61时系统发生了融合激变，由两个共存的混沌吸引子融合为一个更大的混沌吸引子；随着μ进一步增大到0.91附近时，系统再次发生融合激变随即进入周期窗口，并持续在混沌和周期窗口间进行切换.
图3 三次项系数μ为分岔参数时，系统（3）的多值分岔图
Fig.3 Multivalued bifurcation diagram with the variation of Cubic nonlinear coefficient μ
图4 μ取不同值时，系统（3）的相图和均方电压图（a），（b）初值（-0.5，0.5，1.0）;（c），（d）初值（-0.5，-0.5，1.0）
Fig.4 Phase portraits and Mean square voltage diagram with the variation of μ (a) , (b) the initial condition (-0.5, 0.5, 1.0) ; (c) , (d) the initial condition (-0.5, -0.5, 1.0)
在系统分岔行为分析的基础上，观察参数μ变化时，系统不同运动状态对输出电压的影响机理.图4（b）和图4（d）表示了从不同初值出发时系统的均方电压随μ的变化情况.观察可知，在μ值较小时，从初值（-0.5，0.5，1.0）出发系统围绕着稳定平衡点（- $\sqrt{a / μ}$ ，0，0）做单阱小幅运动，从初值（-0.5，-0.5，1.0）出发系统围绕着稳定平衡点（ $\sqrt{a / μ}$ ，0，0）做单阱小幅运动，且它们的均方电压值都较小; 当μ=0.63时，系统将摆脱势阱的束缚呈现出大幅阱间运动，这时均方电压保持较高水平; 当进一步增加μ的值时，系统将继续保持阱间振动，但运动幅度逐渐减小，相应的均方电压也呈下降趋势. 因此优化三次非线性刚度系数，使系统具有最优的势阱深度和势阱间距，进而保持阱间大幅运动，可以有效提高双稳态能量采集系统的均方电压.
2 双稳态能量采集系统的随机响应
2.1 正交多项式逼近
在实际工程结构中，涉及到的不确定参数通常是有界的.因此，本文选取服从拱形分布的随机变量进行研究.设两不确定机电耦合系数的形式如下

θ = \bar{θ} + σ_{1} ξ_{1}, g = \bar{g} + σ_{2} ξ_{2}

(9)

式中， $\bar{θ}$ ， $\bar{g}$ 分别为θ，g的均值; ξ₁，ξ₂是定义在[-1，1]上服从拱形分布的独立的有界随机变量; σ₁，σ₂分别是随机变量ξ₁，ξ₂的随机强度; ξ_i（i=1，2）的概率密度函数^[37]为

p_{Ξ_{i}} (ξ_{i}) = \{\begin{matrix} \frac{2}{π} \sqrt{1 - ξ_{i}^{2}} & |ξ_{i}| ⩽ 1, \\ 0 & |ξ_{i}| > 1 . \end{matrix}

(10)

将式（9）带入系统（4），具有双不确定参数的双稳态能量采集系统可表示为

\{\begin{matrix} \ddot{x} (t) + c \dot{x} (t) - a x (t) + μ x^{3} (t) - \\ (\bar{θ} + σ_{1} ξ_{1}) v (t) = A c o s (ω t) \\ \dot{v} (t) + λ v (t) + (\bar{g} + σ_{2} ξ_{2}) \dot{x} (t) = 0 \end{matrix}

(11)

对应于本文研究的具有拱形分布的随机变量，选取相应的第二类Chebyshev多项式^[39]，其正交性表示为

(12)

系统（11）的响应是时间t和随机变量ξ₁，ξ₂的函数，根据正交多项式逼近理论，它的响应可表示为如下形式

\{\begin{matrix} x (t, ξ_{1}, ξ_{2}) = \sum_{i = 0}^{N} \sum_{j = 0}^{N} x_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) \\ v (t, ξ_{1}, ξ_{2}) = \sum_{i = 0}^{N} \sum_{j = 0}^{N} v_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) \end{matrix}

(13)

式中，N表示Chebyshev多项式的最高阶数.当 $N \to + \infty$ 时， $\sum_{i = 0}^{N} \sum_{j = 0}^{N} x_{i j} （ t ） H_{i} (ξ_{1}) H_{j} (ξ_{2})$ 和 $\sum_{i = 0}^{N} \sum_{j = 0}^{N} v_{i j} （ t ） H_{i} (ξ_{1}) H_{j} (ξ_{2})$ 等价于原系统（11）的响应x（t，ξ₁，ξ₂）和x（t，ξ₁，ξ₂）.我们取N=2，并将其代入（10）式，可以得到

\{\begin{matrix} \sum_{i = 0}^{2} \sum_{j = 0}^{2} {\ddot{x}}_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) + c \sum_{i = 0}^{2} \sum_{j = 0}^{2} {\dot{x}}_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) - a \sum_{i = 0}^{2} \sum_{j = 0}^{2} x_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) + \\ μ {[\sum_{i = 0}^{2} \sum_{j = 0}^{2} x_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2})]}^{3} - (\bar{θ} + σ_{1} ξ_{1}) \sum_{i = 0}^{2} \sum_{j = 0}^{2} v_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) = A c o s (ω t) \\ \sum_{i = 0}^{2} \sum_{j = 0}^{2} {\dot{v}}_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) + λ \sum_{i = 0}^{2} \sum_{j = 0}^{2} v_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) + \bar{g} \sum_{i = 0}^{2} \sum_{j = 0}^{2} {\dot{x}}_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) + \\ σ_{2} ξ_{2} \sum_{i = 0}^{2} \sum_{i = 0}^{2} {\dot{x}}_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2}) = 0 \end{matrix}

(14)

利用Chebyshev多项式之间的关系和其正交性对其进行化简，并关于随机变量，取期望ξ₁，ξ₂可将式（14）转化为如下等价确定性扩阶系统

\{\begin{matrix} {\ddot{x}}_{00} (t) + c {\dot{x}}_{00} (t) - a x_{00} (t) + μ X_{00} (t) - \bar{θ} v_{00} (t) - [σ_{1} v_{10} (t)] / 2 = A c o s (ω t), \\ {\dot{v}}_{00} (t) + λ v_{00} (t) + \bar{g} {\dot{x}}_{00} (t) + [σ_{2} {\dot{x}}_{01} (t)] / 2 = 0, \\ {\ddot{x}}_{01} (t) + c {\dot{x}}_{01} (t) - a x_{01} (t) + μ X_{01} (t) - \bar{θ} v_{01} (t) - [σ_{1} v_{11} (t)] / 2 = 0, \\ {\dot{v}}_{01} (t) + λ v_{01} (t) + \bar{g} {\dot{x}}_{01} (t) + \{σ_{2} [{\dot{x}}_{00} (t) + {\dot{x}}_{02} (t)]\} / 2 = 0, \\ {\ddot{x}}_{02} (t) + c {\dot{x}}_{02} (t) - a x_{02} (t) + μ X_{02} (t) - \bar{θ} v_{02} (t) - [σ_{1} v_{12} (t)] / 2 = 0, \\ {\dot{v}}_{02} (t) + λ v_{02} (t) + \bar{g} {\dot{x}}_{02} (t) + [σ_{2} {\dot{x}}_{01} (t)] / 2 = 0, \\ {\ddot{x}}_{10} (t) + c {\dot{x}}_{10} (t) - a x_{10} (t) + μ X_{10} (t) - \bar{θ} v_{10} (t) - \{σ_{1} [v_{00} (t) + v_{20} (t)]\} / 2 = 0, \\ {\dot{v}}_{10} (t) + λ v_{10} (t) + \bar{g} {\dot{x}}_{10} (t) + [σ_{2} {\dot{x}}_{11} (t)] / 2 = 0, \\ {\ddot{x}}_{11} (t) + c {\dot{x}}_{11} (t) - a x_{11} (t) + μ X_{11} (t) - \bar{θ} v_{11} (t) - \{σ_{1} [v_{01} (t) + v_{21} (t)]\} / 2 = 0, \\ {\dot{v}}_{11} (t) + λ v_{11} (t) + \bar{g} {\dot{x}}_{11} (t) + \{σ_{2} [{\dot{x}}_{10} (t) + {\dot{x}}_{12} (t)]\} / 2 = 0, \\ {\ddot{x}}_{12} (t) + c {\dot{x}}_{12} (t) - a x_{12} (t) + μ X_{12} (t) - \bar{θ} v_{12} (t) - \{σ_{1} [v_{02} (t) + v_{22} (t)]\} / 2 = 0, \\ {\dot{v}}_{12} (t) + λ v_{12} (t) + \bar{g} {\dot{x}}_{12} (t) + [σ_{2} {\dot{x}}_{11} (t)] / 2 = 0, \\ {\ddot{x}}_{20} (t) + c {\dot{x}}_{20} (t) - a x_{20} (t) + μ X_{20} (t) - \bar{θ} v_{20} (t) - [σ_{1} v_{10} (t)] / 2 = 0, \\ {\dot{v}}_{20} (t) + λ v_{20} (t) + \bar{g} {\dot{x}}_{20} (t) + [σ_{2} {\dot{x}}_{21} (t)] / 2 = 0, \\ {\ddot{x}}_{21} (t) + c {\dot{x}}_{21} (t) - a x_{21} (t) + μ X_{21} (t) - \bar{θ} v_{21} (t) - [σ_{1} v_{11} (t)] / 2 = 0, \\ {\dot{v}}_{21} (t) + λ v_{21} (t) + \bar{g} {\dot{x}}_{21} (t) + \{σ_{2} [{\dot{x}}_{20} (t) + {\dot{x}}_{22} (t)]\} / 2 = 0, \\ {\ddot{x}}_{22} (t) + c {\dot{x}}_{22} (t) - a x_{22} (t) + μ X_{22} (t) - \bar{θ} v_{22} (t) - [σ_{1} v_{12} (t)] / 2 = 0, \\ {\dot{v}}_{22} (t) + λ v_{22} (t) + \bar{g} {\dot{x}}_{22} (t) + [σ_{2} {\dot{x}}_{21} (t)] / 2 = 0 . \end{matrix}

(15)

响应的均值可表示为

\{\begin{matrix} E [x (t, ξ_{1}, ξ_{2})] = \sum_{i = 0}^{2} \sum_{j = 0}^{2} E [x_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2})] = x_{00} (t), \\ E [v (t, ξ_{1}, ξ_{2})] = \sum_{i = 0}^{2} \sum_{j = 0}^{2} E [v_{i j} (t) H_{i} (ξ_{1}) H_{j} (ξ_{2})] = v_{00} (t) \end{matrix}

(16)

方差可表示为

\{\begin{matrix} V a r [x (t, ξ_{1}, ξ_{2})] = E [x {(t, ξ_{1}, ξ_{2})}^{2}] - {\{E [x (t, ξ_{1}, ξ_{2})]\}}^{2} = x_{01}^{2} (t) + x_{02}^{2} (t) + \\ x_{10}^{2} (t) + x_{11}^{2} (t) + x_{12}^{2} (t) + x_{20}^{2} (t) + x_{21}^{2} (t) + x_{22}^{2} (t) \\ V a r [v (t, ξ_{1}, ξ_{2})] = E [v {(t, ξ_{1}, ξ_{2})}^{2}] - {\{E [v (t, ξ_{1}, ξ_{2})]\}}^{2} = v_{01}^{2} (t) + v_{02}^{2} (t) + \\ v_{10}^{2} (t) + v_{11}^{2} (t) + v_{12}^{2} (t) + v_{20}^{2} (t) + v_{21}^{2} (t) + v_{22}^{2} (t) \end{matrix}

(17)

对原随机系统（11）采用蒙特卡洛（Monte Carlo，简记为MC）方法进行方差的计算，并将其数值结果与式（17）的理论计算结果进行对比.从图5不难发现，数值模拟与理论结果基本吻合，验证了正交多项式逼近法（Orthogonal Polynomial Approximation，简记为OPA）的有效性.从数值模拟结果来看，系统处于周期运动（μ=0.4）时响应的方差小于分岔点附近（μ=0.49）和混沌运动状态（μ=0.59），可见周期运动时，系统响应在随机参数影响下离散程度更小.但总体上看，对于不同参数值时方差值均较小.
图5 系统响应的方差对比
Fig.5 Variance comparison of system responses
下面将进一步通过系统（15）的集合平均响应（Ensemble Mean Responses，简记为EMR）来探究随机双稳态俘能系统的非线性动力学特性，并通过对比系统（4）的确定性响应（Deterministic Response，简记为DR）和系统（15）的EMR来揭示两不确定参数对系统的影响机理.
2.2 两不确定参数对系统响应的影响
取系统参数c=0.2，a=0.5， $\bar{θ}$ =0.05，A=0.12，ω=0.8，λ=0.05， $\bar{g}$ =0.5时，针对不同三次非线性刚度系数，讨论双不确定参数作用下双稳态能量采集系统的全局和局部动力学特性. 由于系统（4）的相空间是三维的，其吸引子和吸引域难以充分呈现，因此将借助二维截面来分析系统的全局特性.
图6展示了μ=0.49时，双不确定参数对系统运动状态及输出电压的影响. 由图6（a）可见，DR和σ₁=0，σ₂=0.03时的EMR绕右侧稳定平衡点做周期为1的单阱运动; 但在不确定参数θ的影响下，当σ₁=0.03，σ₂=0和σ₁=0.03，σ₂=0.03时EMR已发生阱间跳跃，在左侧势阱内保持周期为2的运动，如图6（b）所示. 电压的时间历程图也体现了上述跳跃现象，见图6（c）和6（d）.
图6 μ=0.49时的DR和EMR，DR的初值为（x， $\dot{x}$ ，v）=（0.33，-0.76，1.0），EMR的初值为（x₀₀， $\dot{x}$ ₀₀，v₀₀）=（0.33，-0.76，1.0），（x_ij， $\dot{x}$ _ij，v_ij）=（0.0，0.0，0.0），（i≠0，j≠0）.
Fig.6 DR and EMR for μ=0.49. The initial condition of DR is (x, $\dot{x}$ , v) = (0.33, -0.76, 1.0) and EMR are (x₀₀, $\dot{x}$ ₀₀, v₀₀) = (0.33, -0.76, 1.0) , (x_ij, $\dot{x}$ _ij, v_ij) = (0.0, 0.0, 0.0) , (i≠0, j≠0)
进一步分析双不确定参数作用下DR和EMR的全局特性.图7（a）和图7（c）中，DR和σ₁=0，σ₂=0.03时的EMR的吸引子保持一致，均有两个周期为1的共存吸引子; 但当σ₁=0.03，σ₂=0和σ₁=0.03，σ₂=0.03时，EMR的吸引子已演化为周期为2的吸引子，且吸引子A1的吸引域B1的占比增大，见图7（b）和图7（d）.
图7 μ=0.49时，DR和EMR的全局特性
Fig.7 Global characteristics of DR and EMR for μ=0.49
图8 μ=0.59时，DR和EMR的全局特性
Fig.8 Global characteristics of DR and EMR for μ=0.59
随着μ的增加，系统通过倍周期分岔进入混沌.如图8所示，当μ=0.59时，在四组不同不确定性参数强度作用下，DR和EMR都存在两个混沌吸引子A₁和A₂，且吸引子的形态基本保持一致，吸引域具有明显的分形特性; 图8（b）和8（c）分别描述了单个不确定参数θ或g的影响下EMR的全局特性，可以看出图8（c）中EMR的全局特性与DR基本保持一致，但是图8（b）中EMR的两个吸引域交织更加紧密，并且吸引域B₂的范围扩大，表明系统对不确定参数θ更敏感; 图8（d）展示了两个随机参数共同作用下EMR的全局特性，发现其吸引域相较其他三种情形有明显的变化.
当μ进一步增大时，取μ=1.22，DR和EMR的全局特性如图9所示. DR存在两个周期吸引子A₁和A₂，它们的吸引域呈现分形特性. 在随机参数的影响下，系统的全局特性发生明显的变化，EMR在原有的两个周期吸引子A₁和A₂不变的同时，又出现一个新的混沌吸引子A₃，三个吸引子的吸引域B₁，B₂和B₃相互缠绕，动力学行为更加复杂. 对比图9中的四种情形可以发现，当σ₁=0，σ₂=0.03时EMR的吸引域B₁和B₂与DR相似，如图9（c）所示，说明不确定参数g对系统吸引域的影响较小; 由图9（b）可以看出，当考虑不确定参数θ时，即σ₁=0.03，σ₂=0，EMR的吸引域形状已发生明显变化，混沌吸引子A₃的吸引域B₃已侵蚀原有的吸引域B₁和B₂; 当两个不确定参数共同影响时，从图9（d）中不难发现，混沌吸引子A₃的吸引域B₃所占比例进一步增大，几乎占据了整个空间. 表1给出了DR和EMR中各吸引域的占比情况，从中可以发现，相比不确定参数g，在b不确定参数θ影响下，吸引域B₁和B₂减小的更加显著，而在两个不确定参数同时作用下，吸引域B₁和B₂进一步减小，吸引域B₃占比达到92.61%.
图9 μ=1.22时，DR和EMR的全局特性
Fig.9 Global characteristics of DR and EMR for μ=1.22
在以上全局分析的基础上，图10给出了μ=1.22时，从初值（0.33，-0.76，1.0）出发的DR和EMR的相图和频谱图，由图10可知，当只考虑单个不确定参数g时（σ₁=0，σ₂=0.03），EMR仍维持阱间周期运动，其频谱图也与DR保持一致，但其已跳跃到与DR对称的周期运动.当单个随机参数θ（σ₁=0.03，σ₂=0）和两个不确定参数同时作用（σ₁=0.03，σ₂=0.03）的情形时，EMR由阱间周期运动演化为阱间混沌运动，离散的频谱图被连续的频谱图代替.
表1 μ=1.22时，DR和EMR吸引域的占比情况
Table1 The proportion of DR and EMR attraction domains for μ=1.22
图10 μ=1.22时的DR和EMR，DR的初值为（x， $\dot{x}$ ，v）=（0.33，-0.76，1.0），EMR的初值为（x₀₀， $\dot{x}$ ₀₀，v₀₀）=（0.33，-0.76，1.0），（x_ij， $\dot{x}$ _ij，v_ij）=（0.0，0.0，0.0），（i≠0，j≠0）.
Fig.10 DR and EMR for μ=1.22. The initial condition of DR is (x, $\dot{x}$ , v) = (0.33, -0.76, 1.0) and EMR are (x₀₀, $\dot{x}$ ₀₀, v₀₀) = (0.33, -0.76, 1.0) , (x_ij, $\dot{x}$ _ij, v_ij) = (0.0, 0.0, 0.0) , (i≠0, j≠0)
2.3 不确定参数强度对系统响应的影响
为进一步揭示不确定因素对系统动力学行为的影响，下面将探索不确定参数强度变化对系统动力学行为的影响机理. 首先借助最大李雅普诺夫指数（Top Lyapunov Exponent，简记为TLE）来表征不确定参数对系统周期运动和混沌运动的影响，如图11（a）所示，随着参数θ的不确定性强度σ₁的增大，系统的TLE为正数的参数区间持续扩大，特别是在μ∈[0.4，0.55]和μ∈[0.9，1.0]时，这意味着在不确定参数θ的影响下，系统会更易进入混沌运动状态; 而当不确定参数g的强度σ₂改变时，如图11（b），系统的TLE没有发生太大的变化.
图11 μ∈（0，2]时，DR和EMR的TLE图
Fig.11 The TLE with the variation of μ under different intensities
基于以上分析，μ∈[0.4，0.55]，借助DR和EMR的分岔图从总体上分析不确定参数的影响. 如图12，在不确定参数作用下EMR保持和DR相似的动力学行为，即随着μ的增大，它们都经历了由1周期运动→对称破裂分岔产生2周期运动→倍周期级联进入混沌运动的过程.从局部来看，当μ＜0.45时，参数θ不确定性强度σ₁取0.03，0.05和0.1的EMR仍保持和DR相同的1周期运动; 当μ增大到分岔点附近时，EMR的分岔点随着σ₁的增大不断向左移动，系统会更早的进入混沌状态; 而图11（b）中不确定参数g的强度σ₂增大时，EMR仍保持稳定的周期运动，只是在分岔点附近发生了微小的改变.这也再次说明，参数g的不确定性对双稳态系统的动力学影响较小.
图12 μ∈（0.4，0.55]时，DR和EMR的分岔图
Fig.12 Bifurcation diagrams of DR and EMR for μ∈ (0.4, 0.55]
2.4 能量俘获性能
关于系统的能量俘获性能，可通过均方电压和能量转换率这两个重要指标来描述.其中能量转化率可表示为

η = \frac{P_{e}}{P_{m}}

(18)

P_e和P_m分别表示电能和机械能的有效功率值，计算公式如下

\begin{matrix} P_{e} = \sqrt{\frac{1}{t} \int_{0}^{t} {(P_{e}^{i n s})}^{2} d t} \\ P_{m} = \sqrt{\frac{1}{t} \int_{0}^{t} {(P_{m}^{i n s})}^{2} d t} \end{matrix}

(19)

式中， $P_{e}^{i n s}$ 和 $P_{m}^{i n s}$ 代表瞬时电能的功率值和瞬时机械能的功率值.
双稳态系统（4）的无量纲瞬时电能功率和机械能功率为

P_{e}^{i n s} = \frac{θ λ}{g} v^{2}, P_{m}^{i n s} = \dot{x} A c o s (ω t)

(20)

图13 μ∈（0，2]时系统的均方电压和能量转化率
Fig.13 Mean square voltage diagram and Energy conversion diagram for μ∈ (0, 2]
为更直观的表现随机参数对系统俘能性能的影响，图13展示了μ∈（0，2]时，DR和EMR的均方电压和能量转化率. 从图13（a），图13（c）和图13（e）中可以看出，系统的均方电压总体呈现先增大后减小的趋势. 其中DR的均方电压在μ=0.9处出现了一个峰值，这时的均方电压比μ=0.1时的提高了192%; 但在不确定参数θ的影响下，EMR的均方电压整体上都有所减小，并且σ₁越大，均方电压越小，均方电压的峰值越向右移动，σ₁取0.03，0.05和0.1时EMR的峰值均方电压比μ=0.1时的电压分别提高了161%，136%和124%. 另一个机电耦合系数g的b不确定性也会导致系统在某些取值处产生较小的输出电压，但它对均方电压峰值的影响不显著，图13（c）中可以清楚地看到σ₂=0.05和σ₂=0.1的均方电压与σ₂=0.03的均方电压几乎完全吻合，且它们的峰值均方电压与DR相差很小. 能量转化率方面，从图13（b）和图13（d）可以看出，受不确定参数θ的影响，转化率会有明显的下降趋势，而在不确定参数g的影响下，转化率的变化不大. 这说明系统对参数g的不确定性强度变化不敏感. 从图13（e）和图13（f）可以看出当两不确定参数同时作用于系统时，仅在随机强度较小时，系统的均方电压和能量转化率都出现了较大幅度的减小，表明两个不确定参数共同存在时，对系统能量俘获性能的影响较为显著.
3 结论
本文研究了具有两个不确定机电耦合系数的双稳态能量采集系统的动力学行为及其发电性能.首先，通过Routh-Hurwitz定理分析了确定性双稳态系统平衡点的稳定性和静态分岔特性，呈现了该系统的大幅运动和小幅运动.在此确定性行为分析的基础上，考虑了双稳态能量采集系统中机电耦合系数的不确定性，借助正交多项式逼近理论，对两不确定参数作用下双稳态系统的响应进行了求解，并探究了两个机电耦合系数及其随机强度对系统分岔、混沌、激变、吸引子变迁、大幅阱间运动以及能量俘获效率的影响.
结果表明在双不确定机电耦合系数的影响下，双稳态系统吸引子的形状和吸引域的占比都会发生变化，且参数不确定性强度越大，变化越明显；在一些敏感参数区间内，这两个参数的不确定性可能使得系统发生阱间跳跃，也会导致系统的倍周期分岔点前移，使得系统更早进入混沌状态，致使系统的均方电压和能量俘获效率降低.此外，与电方程中机电耦合系数相比，该系统对机械方程中机电耦合系数的不确定性更敏感，且两个不确定参数共同作用会对系统的能量俘获效率产生更显著的影响.
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基本信息

中图分类号: O322
文献标识码: A
DOI: 10.6052/1672-6553-2023-095
文章编号: 1672-6553-2023-21(10)-072-013

基金信息

国家自然科学基金资助项目(12172286, 11872306, 12172284)；
航空科学基金资助项目(201941053004)；
National Natural Science Foundation of China (12172286, 11872306, 12172284)；
Aeronautical Science Foundation of China (201941053004)；

引用信息

稿件历史

收稿日期: 2023-08-31

参考文献

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双不确定参数作用下双稳态能量采集系统的随机响应分析*

Stochastic Response Analysis of Bistable Energy Harvesting System with Double Uncertain Parametters*

摘要

Abstract

关键词

Keywords

1 双稳态能量采集系统的模型和响应

1.1 双稳态能量采集系统模型

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

1.2 双稳态能量采集系统的确定性响应

2 双稳态能量采集系统的随机响应

2.1 正交多项式逼近

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

2.2 两不确定参数对系统响应的影响

2.3 不确定参数强度对系统响应的影响

2.4 能量俘获性能

(18)

(19)

(20)

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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双不确定参数作用下双稳态能量采集系统的随机响应分析^*

Stochastic Response Analysis of Bistable Energy Harvesting System with Double Uncertain Parametters^*