何燕丽, 赵翔^,2)西南石油大学土木工程与建筑学院,成都 610500

CLOSED-FORM SOLUTIONS FOR FORCED VIBRATIONS OF CURVED PIEZOELECTRIC ENERGY HARVESTERS BY MEANS OF GREEN'S FUNCTIONS ¹⁾

He Yanli, Zhao Xiang^,2)School of Civil Engineering and Architecture, Southwest Petroleum University, Chengdu 610500, China

通讯作者: 2) 赵翔,副教授,主要研究方向：动力学与控制. E-mail:zhaoxiang_swpu@126.com

收稿日期:2019-01-6接受日期:2019-04-10网络出版日期:2019-07-18

基金资助:

1) 国家自然科学基金资助项目.11702230

Received:2019-01-6Accepted:2019-04-10Online:2019-07-18
作者简介 About authors


摘要
本文运用格林函数法求解了曲梁压电俘能器在强迫振动下的解析解.运用微分法分析了压电层合曲梁结构面内各内力,根据曲梁压电 俘能器的动力学方程组,基于压电本构关系,建立了包含径向阻尼但不考虑俘能器曲梁结构部分的轴向力以及轴向惯性项的Prescott力 电耦合模型. 采用Laplace变换法求得了耦合振动方程的格林函数解.根据叠加原理和格林函数的物理意义,对耦合的系统方程解耦进而 求得强迫振动下曲梁压电俘能器的输出电压. 数值计算中,通过与现有文献的解析解进行对比,验证了本文解析解的有效性,并研究了阻 尼、电阻等重要物理参数对压电函数和谐振频率的影响.通过与有关传统直梁压电俘能器研究成果的对比,体现了曲梁压 电俘能器Prescott模型的高效集能特性. 数值分析研究表明：(1)使得曲梁俘能器达到最大输出电压时连接的最优负载电 阻为1 M$\Omega$;(2)通过更换适当的基底材料,降低材料的弹性模量,可以改变曲梁俘能器的高基频现象,以使结构适应 更复杂的工作环境,但这会导致俘能器的工作效率降低.
关键词： 曲梁俘能器;压电材料;格林函数;Laplace变换;力电耦合

Abstract
This article investigates the forced vibrations of curved piezoelectric energy harvesters by means of Green's functions. The differential method is used to analyze the in-plane forces of the cantilevered piezoelectric energy harvester. According to the governing equations of motion, the electromechanical coupled Prescott models are derived based on the piezoelectric constitutive relations, which the circumferential forcing and the circumferential inertia term can be negligible, and a damping effect, radial damping, is taken into account. Utilizing the Laplace transform, the explicit expressions of the Green's functions of the coupled vibration equations can be acquired. On the basis of the superposition principle and the physical interpretation of Green's functions, the coupled system is decoupled and the expression of the output voltage can be obtained analytically. The present model for the curved beam can be readily reduced to straight beam. In the numerical sections, the present solutions are verified by the results in some published references. By comparing with the result of traditional straight piezoelectric energy harvesters model, the high energy harvesting efficiency of the curved piezoelectric energy harvesters model in the thesis is demonstrated. It is apparent that the present model has a wider range of application than the existing ones. The influence of radial damping, Young's modules of two materials and some other essential physical parameters on the evaluation functions for output voltage and resonant frequency are discussed. This research suggests that to make the electric power reach the maximum value, the optimal resistive load is 1 M$\Omega$; the elasticity modulus for both piezoelectric material and structure material have a profound effect on the resonant frequency. By replacing the base materials with lower modulus of elasticity, the phenomenon of high frequency resonance can be improved to make the curved piezoelectric energy harvesters adapt to more complex working environment. However, the energy harvesting efficiency of the structure will be decline.
Keywords：curved energy harvester;piezoelectric materials;Green's function;Laplace transform;electromechanical


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本文引用格式
何燕丽, 赵翔. 曲梁压电俘能器强迫振动的格林函数解 ¹⁾. 力学学报[J], 2019, 51(4): 1170-1179 DOI:10.6052/0459-1879-19-007
He Yanli, Zhao Xiang. CLOSED-FORM SOLUTIONS FOR FORCED VIBRATIONS OF CURVED PIEZOELECTRIC ENERGY HARVESTERS BY MEANS OF GREEN'S FUNCTIONS ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(4): 1170-1179 DOI:10.6052/0459-1879-19-007

引言

目前,能源的收集利用问题依旧是研究的热点. 许多研究致力于将自然环境中被忽视的各种形式的能量进行转化并收集.利用压电俘能器可将海波激励、感应声波、振动等形式的 能量转化为电能.在不同的发电源中,机械振动在环境中容易获得,各种机械设备、建筑、道路、轨道均是机械振动的来源^[1], 因此将机械振动能转化为电能是实现能量转化最主要的形式,可分别通过电磁式、静电式、压电式可实现^[2].在能量转换过程中,压电 材料发挥了重要的作用.功率密度高,不依赖于外部磁场和电源而广泛应用于俘能器^[3-4].功能梯度压电材料是一类新型的 压电材料,具有沿预定方向连续平稳变化的材料性 能^[5-6].这种性能可以消除传统压电层合结构中的内部边界,降低界面应 力集中,提高智能器件的使用寿命和可靠性^[7].因而功能梯度压电材料具有广泛的应用前景. 但由于经济原因,目前PZT仍是压电俘能器中最常用的压电材料^[8-9].

压电俘能器的主体通常采用梁的形式. 对于压电直梁,多为考虑剪切变形和转动惯量的Timosh-enko梁模型^[10]以及不考虑剪切变形和转动惯量的Euler-Bernoulli梁 模型^[11].而基于Prescott方程,不考虑曲梁环向力以及环向惯性项^[12],仅考虑曲梁径向振动的Prescott模型目前还未 在压电俘能器中探究过^[13].

对于压电俘能器,通过添加不同的边界条件也展开了大量研究. 曹东兴等^[14]基于压电效应设计了简支新型压电俘能器结构.但是由于悬臂式俘能器的结构更简单,谐振频率低,与环境中的振动 频率相接近,在谐振状态下,能输出较高的能量,因此悬臂梁广泛用于低频振动的能量采集的^[15-16]. 郭抗抗^[17]通 过悬臂梁的边界条件,探究了较宽低频范围内悬臂式俘能器的压电响应和发电性能;岳国强^[18]对悬臂式俘能器性能进行了仿真 和实验研究,分析了最优输出功率以及阻尼比对其性能的影响;赵翔等^[19]基于Timoshenko假设,利用格林函数法对悬臂式直梁压电俘能器进行了动力学分析, 得到了强迫振动的解析解,并探究了各类因素 对压电响应的影响.周勇等^[20]提出了一种压电悬臂曲梁式俘能器,仿真结果表明了曲率半径对性能的影响,但其依旧是用于较低 谐振频率下俘获机械振动能量,俘能效率有限,仅对小范围谐振频率下的机械振动产生的能量才能进行利用.

虽然压电俘能装置的研究已经取得了大量成果,有了一些实际工程应用,但这些成果中的压电俘能器建模多采用直梁的形式,对曲梁 形式的压电俘能器的研究较少.随着机械系统的智能化微型化,工作的环境复杂化,直梁形式的俘能器已经不能满足现实工程技术的要求. 因此,曲梁俘能器的研究很有必要.本文所设计的曲梁模型针对中高谐振频率的机械振动能量的收集及其效率的进行探究,创新性地采 用Prescott模型利用格林函数法求解曲梁俘能器响应.该模型通过设计改造,也可以用于对较低谐振频率下的机械振动能量收集.

本文将建立一个力电耦合的曲梁压电俘能器模型并得到其强迫振动的解析解.基于压电本构关系,设计力电耦合的曲梁俘能器Prescott模型. 在振动控制方程中采用了分离变量法和Laplace变换求解力学平衡方程的格林函数.根据线性系统的叠加,耦合的曲梁俘能器系统解耦得 到一个独立的代数方程.运用格林函数法求解代数方程,推导了压电俘能器强迫振动下电压解析解.将曲梁俘能器与直梁俘能器的集能效 率进行对比分析,并在算例中探究曲梁俘能器各物理参数对压电响应和谐振频率的影响,为俘能器在不同谐振频率下的能量收集提供参 考依据.

1 曲梁压电俘能器模型的建立

俘能器采用受外载荷$f(s, t)$的曲梁(外层为压电 材料层,内层为结构层)作为主体结构,连接负载电阻$R_{l}$形成闭合回路 进行模拟,如图1所示. 假定压电层和结构层紧密贴合,梁的弧长为$L$. 图2所示为俘能器的横截面示意图,$b_{p}$和$b_{s}$为压电 层和结构层 的宽度;$h_{p}$和$h_{s}$ 分别为压电层和结构层的厚度;$h_{ a}$表示质量中心线($N.A)$到下表面的距离;$h_{b}$是质量中 心线至压电层底部的距离;$h_{ c}$是质量中心线到压电层上表面的距离;$h_{pc}$为质量中心线到压电层中性轴的距离.

图1

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图1曲梁压电俘能器模型

Fig. 1The model of curved piezoelectric energy harvester

图2

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图2曲梁压电俘能器横截面示意图

Fig. 2The cross section of the curved piezoelectric energy harvester



曲梁强迫振动的控制振动方程^[21]为

(1)$$ \left. \dfrac{\partial N}{\partial s} + \dfrac{Q}{R} = \mu \ddot {v} \ \dfrac{\partial Q}{\partial s}-\dfrac{N}{R}-c_1 \dot {w} = \mu \ddot {w} + f(s,t) \ \dfrac{\partial M}{\partial s} + Q = \gamma \ddot {\psi } \right\} $$
式中,$\ddot {v}(s,t)$,$\ddot {w}(s,t)$和$\ddot {\psi }(s,t)$分别为梁的轴向位移、径向位移及转角对时 间的二阶导数;$N(s, t)$表示轴向力,$Q(s, t)$表示剪力;$M(s, t)$为弯矩; $\mu $和$\gamma $分别表示单位长度梁的质量和转动惯量;$c_{1}$ 表示沿$r$轴方向的阻尼系数^[22],顶标"$ \dot{ \ } $"表示对时间的一阶导数.

有压电本构关系^[23]

(2)$ \left.\begin{array}{l} \sigma _{ss}^{p} = E_{p} (\varepsilon _{ss}^{p}-d_{31} E_{r} ) \\ \tau _{sr}^{p} = G_{p} (\gamma _{sr}^{p} + d_{15} E_{s} ) \\ \sigma _{ss}^{s} = E_{s} \varepsilon _{ss}^{s} , \ \ \tau_{sr}^{s} = G_{p} \gamma _{sr}^{s} \end{array}\right\} $
式中, $\sigma_{ss}^{p}$ ($\varepsilon_{ss}^{p})$和$\tau _{sr}^{p}$ ($\gamma _{sr}^{p})$分别是压电层的正应力(正应变)和切应力(切应变); $\sigma _{ss}^{s}$ ($\varepsilon_{ss}^{ s})$ 和$\tau _{sr}^{s}$ ($\gamma_{sr}^{s})$分别是结构层的正应力(正应变)和切应变(切应变);$E_{ p}$和$G_{p}$为压电层的杨氏模量和剪 切模量;$E_{s}$和$G_{ s}$分别为结构层的杨氏模量和剪切模量;$d_{31}$和$d_{15}$表示压电常数;$E_{r}$和$E_{s}$分 别为$r$轴方向和$s$轴方向的电场,与$r$轴方向的电场相比, $s$轴方向的电场十分微弱,因此$d_{15}$ $E_{ s}$这一项可以忽略不计^[24-25].

曲梁的弯矩$M(s, t)$, 剪力$Q(s, t)$, 轴力$N(s, t)$通过微分法求解,基于本构关系,并联合应变与位移、转角之间的关系^[26]可得

(3)$ M(s,t) = (EI)_{eff} \dfrac{\partial }{\partial s}\left( {\dfrac{\partial w}{\partial s}-\dfrac{v}{R}} \right) + \\ \qquad \vartheta _1 v(t)[H(s-s_1 )-H(s-s_2 )] $
(4)$ Q(s,t) = \kappa (G)_{eff} A\left( {\dfrac{\partial w}{\partial s}-\dfrac{v}{R}-\psi } \right) $
(5)$ N(s,t) = (E)_{eff} A\left( {\dfrac{\partial v}{\partial s} + \dfrac{w}{R}} \right) + \qquad \vartheta _2 v(t)[H(s-s_1 )-H(s-s_2 )] $
其中,$(EI)_{eff}$为有效刚度, $\vartheta _{1}$和 $\vartheta _{2}$为耦合系数,且

(6)$ (EI)_{eff} = \dfrac{E_{s} b_{s} (h_{b} ^2-h_{a} ^2) + E_{p} b_{p} (h_{c} ^2-h_{b} ^2)}{2} $
(7)$ \vartheta _1 =-\dfrac{E_{p} b_{p} d_{31} (h_{c} ^2-h_{b} ^2)}{2h_{p} } , \ \ \vartheta _2 = \dfrac{E_{p} b_{p} d_{31} (h_{c}-h_{b} )}{h_{p} } $
将各内力表达式(3) ~式(5)代入曲梁强迫振动的控制方程(1)得

(8a)$ (E)_ {eff} A{v}"-\dfrac{\kappa (G)_ {eff} A}{R^2}v-\mu \ddot {v} + \dfrac{(E)_{eff} A + \kappa (G)_ {eff} A}{R}{w}' - \\ \qquad \frac{\kappa (G)_ {eff} A}{R}\psi + \vartheta_2 v(t)[\delta (s-s_1 )- \delta (s-s_2 )] = 0 $
(8b)$ \kappa (G)_ {eff} A{w}"-\dfrac{(E)_ {eff} A}{R^2}w-c_1 \dot {w}-\mu \ddot {w}-\dfrac{(E)_ {eff} A + \kappa (G)_ {eff} A}{R}{v}'-\\ \qquad \kappa (G)_ {eff} A{\psi }'-\dfrac{\vartheta _2 v(t)}{R}[H(s-s_1 )-H(s-s_2 )] -f(s,t) = 0 \ $
(8c)$ (EI)_ {eff} {w}"' + \kappa (G)_ {eff} A{w}'-\dfrac{(EI)_ {eff} }{R}{v}"-\dfrac{\kappa (G)_ {eff} A}{R}v- \\ \qquad \kappa (G)_ {eff} A\psi-\gamma \ddot {\psi } + \vartheta _1 v(t)[\delta (s-s_1 )-\delta (s-s_2 )] = 0 $
其中,"$\prime $" "$\prime \prime $"和"$\prime \prime \prime $"分别表示对坐标$s$的一、二、三阶导数.由式(8a)得转角用位移可表示为

(9)$$ \psi = \dfrac{(E)_ {eff} AR}{\kappa (G)_ {eff} A}{v}"-\dfrac{1}{R}v-\dfrac{\mu R}{\kappa (G)_ {eff} A}\ddot {v} +\\ \qquad \dfrac{\kappa (G)_ {eff} A + (E)_ {eff} A}{\kappa (G)_ {eff} A}{w}' + \\ \qquad \dfrac{R}{\kappa (G)_ {eff} A}\vartheta _2 v(t)[\delta (s-s_1 )-\delta (s-s_2 )] $$
仅考虑曲梁的径向振动,因此采用Prescott模型. 根据Prescott方程,在式(9)中,$v$是曲梁在环向的位移,$v$对时间的二阶导数是环向的惯性项,当压电俘能器控制振动 方程中的剪切系数$\kappa $趋向于无穷大时即消除了环向位移及惯性项,使得模型退化为Prescott模型^[12].当取剪切模量$\kappa $为无穷大时,有

(10)$$ \psi = {w}'-\dfrac{1}{R}v $$
结合式(8b)与式(10)得

(11)$$ {v}' = \dfrac{1}{R}w-\dfrac{\mu R}{(E)_{eff} A}\ddot {w}-\dfrac{c_1 R}{(E)_{eff} A}\dot {w}-\dfrac{R}{(E)_{eff} A}f(s,t)-\\ \qquad \dfrac{\vartheta _2 v(t)}{(E)_{eff} A}[H(s-s_1 )-H(s-s_2 )] $$
将式(11)代入式(10)得

(12)$$ {\psi }' = {w}" + \dfrac{1}{R^2}w + \dfrac{\mu }{(E)_{eff} A}\ddot {w} + \dfrac{c_1 }{(E)_{eff} A}\dot {w} + \\ \qquad \dfrac{1}{(E)_{eff} A}f(s,t) + \dfrac{\vartheta _2 v(t)}{(E)_{eff} AR}[H(s-s_1 )-H(s-s_2 )] $$
对式(8c)整体求一阶导数,再与式(10)和式(11)结合得Prescott方程(13)

(13)$$ {{w}"}" + \dfrac{1}{R^2}{w}" + \Bigg[\dfrac{\mu }{(E)_{eff} A}-\dfrac{\gamma }{(EI)_{eff} }\Bigg ]{\ddot {w}}" + \dfrac{c_1 }{(E)_{eff} A}{\dot {w}}"-\\ \qquad \dfrac{\gamma \mu }{(EI)_{eff} (E)_{eff} A}\dot {\dddot {w}}-\dfrac{c_1 \gamma }{(EI)_{ eff} (E)_{eff} A}\dddot {w}-\dfrac{\gamma }{(EI)_{eff} R^2}\ddot {w} + \\ \qquad \Bigg (\dfrac{\vartheta _1 v(t)}{(EI)_{eff} } + \dfrac{\vartheta _2 v(t)}{(E)_{eff} AR}\Bigg ) [{\delta }'(s-s_1 )-{\delta }'(s-s_2 )]-\\ \qquad \dfrac{\vartheta _2 \ddot {v}(t)\gamma }{(E)_{eff} A(EI)_{eff} R} [H(s-s_1 )-H(s-s_2 )] + \\ \qquad \dfrac{1}{(E)_{eff} A}{f}"(s,t)-\dfrac{\gamma }{(E)_{eff} A(EI)_{eff} }\ddot {f}(s,t) = 0 $$
式(13)为曲梁压电俘能器模型的振动方程. 未退化的曲梁模型为经典模型,其控制振动方程为关于径向位移$W(s)$的六阶微分方程,而Prescott模型的控制振 动方程则是关于径向位移的四阶微分方程.

由图1可知,压电俘能器模型采用悬臂梁的形式,则边界条件为

(14)$$ w(0,t) = 0 , \ {w}'(0,t) = 0 , \ {w}"(L,t) = 0 , \ {w}"'(L,t) = 0 $$

2 力电耦合的电路控制方程

力电耦合的电路控制方程^[26]为

(15)$$ C_{p} \dfrac{dv(t)}{dt} + \dfrac{v(t)}{R_{l} } =-\beta \int_{s_1 }^{s_2 } {\dfrac{\partial ^2\psi (s,t)}{\partial s\partial t}} ds $$
其中

(16)$$ \left. C_{p} = \dfrac{\varepsilon _{33}^{s} b_{p} (s_2-s_1 )}{h_{p} } \\ \beta = E_{p} d_{31} b_{p} h_{pc} =-\dfrac{2h_{p} h_{pc} }{h_{c} ^2-h_{b} ^2}\vartheta _1 \right\} $$
式中,$\varepsilon _{33}^{s}$为介电常数. 结合式(10)和式(15),可以确定曲梁压电俘能器力电耦合的Prescott模型.

3 稳态的力电耦合压电俘能器模型

假设外部载荷$f (s,t)=F(s) {e}^{{i}\varOmega t}$,对轴向位移,径向位移,转角和电压作同样的假设,即

(17)$$ \left.\begin{array} v(s,t) = V(s) {e}^{{i}\varOmega t} , \ \ w(s,t) = W(s) {e}^{{i}\varOmega t} \\ \psi (s,t) = \varPsi (s) {e}^{{i}\varOmega t} , \ \ v(t) = \bar {V}{e}^{{i}\varOmega t} \end{array}\!\!\right\} $$
式中,$V(s)$, $W(s)$, $\varPsi (s)$和$\bar {V}$分别表示稳态的轴向位移、径向位移、转角和电压.

将式(17)分别代入式(12) ~式(15)分别得式(18) ~式(21)

(18)${\varPsi }' = {W}" + \Big(\dfrac{1}{R^2} + \dfrac{{i}\varOmega c_1-\mu \varOmega ^2} {(E)_{eff} A}\Big )W + \dfrac{1}{(E)_{eff} A}F(s) +\\ \qquad \dfrac{\vartheta _2 }{(E)_{eff} AR}\bar {V}[H(s-s_1 )-H(s-s_2 )] $
(19)$ {{W}"}"(s) + a_1 {W}"(s) + a_2 W(s) = b_1 F(s)-b_2 F(s)-\\ \qquad b_3 \bar {V}[{\delta }'(s-s_1 )-{\delta }'(s-s_2 )] + c\bar {V}[H(s-s_1 )-H(s-s_2 )] \ $
(20)$ \dfrac{{i}\varOmega C_{p} R_l + 1}{R_l }\bar {V} =-{i}\varOmega \beta \int_{s_1 }^{s_2 } {{\Psi }'(s)ds} $
(21)$ W(0) = 0 , \ \ {W}'(0) = 0 , \ \ {W}"(L) = 0 , \ \ {W}"'(L) = 0 $
\n其中

(22)$$ \left.\begin{array}{l} a_1 = \dfrac{1}{R^2} + \dfrac{\gamma \varOmega ^2}{(EI)_{eff} } + \dfrac{{i}\varOmega c_1-\mu \varOmega ^2}{(E)_{eff} A} \\ a_2 = \dfrac{({i}\varOmega c_1-\mu \varOmega ^2)\gamma \varOmega ^2}{(EI)_{eff} (E)_{eff} A} + \dfrac{\gamma \varOmega ^2}{(EI)_{eff} R^2} \\ b_1 = \dfrac{1}{(E)_{eff} A} , \\ \ b_2 = \dfrac{\gamma \varOmega ^2}{(EI)_{eff} (E)_{eff} A} \\ b_3 = \dfrac{\vartheta _2 }{(E)_{eff} AR} + \dfrac{\vartheta _1 }{(EI)_{eff} } , \ \ c =-\dfrac{\gamma \varOmega ^2\vartheta _2 }{(EI)_{eff} (E)_{eff} AR} \end{array}\!\!\right\} $$

4 稳态下压电俘能器的解析解

由式(19)可知,俘能器的稳态位移$W(s)$是由外荷载$F(s)$ 和电耦合效应$\bar {V} [ \delta' ( s-s_{1})-\delta' ( s-s_{2}) ]$,$\bar {V} [ H(s-s_{1})-H(s-s_{2}) ]$引起的,根据线性系统的叠加原理,稳态位移$W(s)$可以分解为$W_{1}(s)$, $W_{2}(s)$和$W_{3}(s)$,三个部分,即$W= W_{1}+ W_{2}+ W_{3}$. 位移$W_{1}(s)$, $W_{2}(s)$和$W_{3}(s)$则分别是由$F(s)$, $\bar {V}[\delta' (s-s_{1})- \delta' (s-s_{2})]$和$\bar {V} [H(s-s_{1})-H(s-s_{2})]$引起的. $W_{1}$, $ W_{2 }$ 和$W_{3}$分别为式(23)的解

(23a)$ {{W}"}"_1 ( s) + a_1 {W}"_1 ( s) + a_2 W_1 (s) =\\ b_1 {F}"(s)-b_2 F(s) $
(23b)$ {{W}"}"_2 ( s) + a_1 {W}"_2 ( s) + a_2 W_2 (s) =\\ b_3 \bar {V}[{\delta }'(s-s_1 )-{\delta }'(s-s_2 )] $
(23c)$ {{W}"}"_3 ( s) + a_1 {W}"_3 ( s) + a_2 W_3 (s) =\\ c\bar {V}[H(s-s_1 )-H(s-s_2 )] $
根据格林函数的物理意义,式(23a)的格林函数$G_{1}(s; s_{0})$为式(24)的解.

(24)$$ {{W}"}"_1 ( s) + a_1 {W}"_1 ( s) + a_2 W_1 (s) =\\ \qquad b_1 {\delta }"(s-s_0 )-b_2 \delta (s-s_0 ) $$
其中, $s_{0}$表示单位力作用的位置. 利用Laplace变换求得格林$H$函数解$G_{1}(s; s_{0})$, 再结合边界条件得

(25)$$ G_1 (s;s_0 ) = H(s-s_0 )\varphi _{11} (s-s_0 ) + \varphi _4 (s){W}"_1 (0) +\\ \qquad \varphi _5 (s){W}"'_1 (0) $$
$W_{1} (0), W_{1}'(0), W_{1}"(0), W_{1}"' (0)$分别为$W_{1}(s)$在$s =0$位置处的值及其各阶导数,且有

(26)$ \left.\begin{array}{l} \varphi _{11} (s-s_0 ) = \sum_{i = 1}^4 {A_i } (s)(b_1 s_i ^2-b_2 ) \\ \varphi _2 (\varphi _2 (s) = \sum_{i = 1}^4 {A_i } (s)(s_i ^3 + a_1 s_i ) \\ \varphi _3 (s) = \sum_{i = 1}^4 {A_i } (s)(s_i ^2 + a_1 ) \\ \varphi _4 (s) = \sum_{i = 1}^4 {A_i } (s)s_i \\ \varphi _5 (s) = \sum_{i = 1}^4 {A_i } (s) \end{array}\right\} $
(27)$ \left.\begin{array}{l} A_1 (s) = \dfrac{e^{s_1 s}}{(s_1-s_2 )(s_1-s_3 )(s_1-s_4 )} \\ A_2 (s) = \dfrac{e^{s_2 s}}{(s_2-s_1 )(s_2-s_3 )(s_2-s_4 )} \\ A_3 (s) = \dfrac{e^{s_3 s}}{(s_3-s_1 )(s_3-s_2 )(s_3-s_4 )} \\ A_4 (s) = \dfrac{e^{s_4 s}}{(s_4-s_1 )(s_4-s_2 )(s_4-s_3 )} \end{array}\!\!\right\} $
式(26)和式(27)中,$s_{i }$ $(i =1, 2, 3, 4)$是代数方程(28)的根

(28)$$ s^4 + a_1 s^2 + a_2 = (s-s_1 )(s-s_2 )(s-s_3 )(s-s_4 ) = 0 $$
根据叠加原理,位移$W_{1}(s)$可以通过对格林函数积分求解,即

(29)$$ W_1 (s) = \int_0^L F(\xi )G_1 (s ; \xi )d\xi $$
同样运用Laplace变换,式(23b)、式(23c)的格林函数解分别为$G_{2}(s; s_{0})$, $G_{3}(s; s_{0})$,表达式分别如下

(30)$ G_2 (s;s_0 ) = \bar {V}H(s-s_0 )\varphi _{12} (s-s_0 ) +\\ \qquad \varphi _4 (s){W}"_2 (0) +\varphi _5 (s){W}"'_2 (0) $
(31)$ G_3 (s;s_0 ) = \bar {V}H(s-s_0 )\varphi _{13} (s-s_0 ) +\\ \qquad \varphi _4 (s){W}"_3 (0) +\varphi _5 (s){W}"'_3 (0) $
其中

(32)$$ \left.\begin{array} \varphi _{12} (s-s_0 ) = \sum_{i = 1}^4 {b_3 } A_i (s)(-s_i ) \\ \varphi _{13} (s-s_0 ) = \sum_{i = 1}^4 {\dfrac{c}{s_i }} A_i (s) \end{array}\right\} $$
$W_{j} (0)$ $(j=2,3)$及其各阶导数由梁的边界条件决定.根据叠加原理,位移$W_{2}(s)$, $W_{3}(s)$可以分别表示为

(33)$ W_2 (s) = \int_0^L G_2 (s; \xi )[\delta (\xi-s_1 )-\delta (\xi-s_2 )]d\xi $
(34)$ W_3 (s) = \int_0^L G_3 (s ; \xi )[H(\xi-s_1 )-H(\xi-s_2 )]d\xi $
将边界条件式代入格林函数中,确定未知常数$W_{j}" (0), W_{j}"'(0)$ ($j=1, 2, 3$),得

(35)$$ {W}"_j (0) = \dfrac{\chi _j (s_0 )}{\alpha _1-\alpha _2 } , \ \ {W}"'_j (0) = \dfrac{\eta _j (s_0 )}{\alpha _1-\alpha _2 } $$
其中

(36)$$ \left.\begin{array}{l} \alpha _1 = {\varphi }"_4 (L){\varphi }"'_5 (L) , \ \ \alpha _2 = {\varphi }"'_4 (L){\varphi }"_5 (L) \\ \chi _j = {\varphi }"'_{1j} (L-s_0 ){\varphi }"_5 (L) , \ \ \chi _j = {\varphi }"_{1j} (L-s_0 ){\varphi }"'_5 (L) \end{array} \!\!\right\} $$
由此,可确定格林函数的解析式

(37a)$G_1 (s;s_0 ) = H(s-s_0 )\varphi _{11} (s-s_0 ) +\\ \qquad \varphi _4 (s)\dfrac{\chi _1 (s_0 )}{\alpha _1-\alpha _2 } + \varphi _5 (s)\dfrac{\eta _1 (s_0 )}{\alpha _1-\alpha _2 } $
(37b)$ G_2 (s;s_0 ) = \bar {V}\Bigg [H(s-s_0 )\varphi _{12} (s-s_0 ) + \qquad \varphi _4 (s)\dfrac{\chi _2 (s_0 )}{\alpha _1-\alpha _2 } + \varphi _5 (s)\dfrac{\eta _2 (s_0 )}{\alpha _1-\alpha _2 }\Bigg] $
(37c)$ G_3 (s;s_0 ) = \bar {V}\Bigg [H(s-s_0 )\varphi _{13} (s-s_0 ) +\\ \qquad \varphi _4 (s)\dfrac{\chi _3 (s_0 )}{\alpha _1-\alpha _2 } +\varphi _5 (s)\dfrac{\eta _3 (s_0 )}{\alpha _1-\alpha _2 }\Bigg ] $

5 力电耦合系统的解耦

根据线性系统的叠加原理以及Green函数的可叠加性对系统进行解耦. $W(s)$可写作

(38)$$ W(s) = \int_0^L F(\xi )G_1 (s; \xi )d\xi + \\ \qquad \int_0^L G_2 (s; \xi )[\delta (\xi-s_1 )-\delta (\xi-s_2 )]d\xi + \\ \qquad \int_0^L G_3 (s; \xi )[H(\xi-s_1 )-H(\xi-s_2 )]d\xi $$
将Green函数的解析式(37)结合式(22)可得出该模型输出电压

(39)$$ \bar {V} = \Bigg \{ -\int_{s_1 }^{s_2 } \Bigg[\int_0^L F(\xi )G_1 (s, \xi )d\xi \Bigg ]"ds + HA_1 + HA_2 \Bigg \} \Bigg / \\ \qquad \Bigg \{\dfrac{ {i}\varOmega C_{p} R_l + 1}{{ i}\varOmega \beta R_l } + \int_{s_1 }^{s_2 } \Big[ \bar {G}"_2 (s;s_2 )- \bar {G}"_2 (s;s_1 )\Big] ds + \\ \qquad HA_3 + HA_4 + HA_5 + HA_6 \Bigg \} \hskip 3cm $$
其中

(40)$ \left.\!\!\!\begin{array}{l} HA_1 = \Bigg(\dfrac{\mu \varOmega ^2-{i}\varOmega c_1 }{(E)_{eff} A}-\dfrac{1}{R^2} \Bigg )\int_{s_1 }^{s_2 } {\int_0^L {F(\xi )G_1 } } (s;\xi )d\xi ds \\ HA_2 =-\dfrac{1}{(E)_{eff} A}\int_{s_1 }^{s_2 } F(s)ds \\ HA_3 = \dfrac{\vartheta _2 }{(E)_{eff} AR}(s_1-s_2 ) \\ HA_4 = \int_{s_1 }^{s_2 } \! \Bigg \{\!\int_0^L [H(\xi-s_1 )-H(\xi-s_2 )]G_3 (s;\xi )d\xi \Bigg \}"ds \\ HA_5 = \Bigg(\dfrac{1}{R^2} + \dfrac{{i}\varOmega c_1-\mu \varOmega ^2}{(E)_{eff} A} \Bigg)\int_{s_1 }^{s_2 }\Big[ {\bar {G}"_2 } (s;s_2 )-{\bar {G}}"_2 (s;s_1 )\Big]ds \\ HA_6 = \Bigg(\dfrac{\mu \varOmega ^2-{i}\varOmega c_1 }{(E)_{eff} A}-\dfrac{1}{R^2}\Bigg )\int_{s_1 }^{s_2 } {\int_0^L {\bar {G}_3 (s;\xi )} } d\xi ds \end{array}\!\!\right\} $
(41)$ \bar {G}_2 (s;s_0 ) = \dfrac{G_2 (s;s_0 )}{\bar {V}} , \ \ \bar {G}_3 (s;s_0 ) = \dfrac{G_3 (s;s_0 )}{\bar {V}} $

6 数值结果及讨论

6.1 简谐激励

将简谐激励视为外荷载作用在曲梁压电俘能器上.若基础位移不等于零,梁的绝对位移为$w(s, t)$为基础位移$w_{b}(s, t)$与绝 对位移$w_{rel}(s, t)$的叠加^[24],对于简谐激励有

(42)$$ w_{b} (s,t) = A_0 {e}^{{i}\varOmega t} $$
即稳态基础位移$W_{b}=A_{0}$,进一步可得基础加速度$A_{b} =-\varOmega ^{2}A_{0}$,外 力$f (s, t)=F(s) {e}^{{i}\varOmega t}$可以写作^[24]

(43)$$ F(s) = (\mu \varOmega ^2- {i}\varOmega c_1 )A_0 $$
将式(43)代入输出电压表达式(39),简谐激励下的稳态电压为

(44)$$ \bar {V} = \Bigg \{-(\mu \varOmega ^2-{i}\varOmega c_1 )A_0 \int_{s_1 }^{s_2 } \Bigg [\int_0^L G_1 (s, \xi ) d\xi \Bigg]" ds + \\ \qquad HA_1 + HA_2 \Bigg\} \Bigg / \Bigg \{\dfrac{{i}\varOmega C_p R_l + 1}{{i}\varOmega \beta R_l } + \int_{s_1 }^{s_2 } \Big [\bar {G}"_2 (s;s_2 )- \\ \qquad\bar {G}"_2 (s;s_1 )\Big] ds + HA_3 + HA_4 + HA_5 + HA_6 \Bigg \} $$
其中

(45)$$ \left.\!\!\!\begin{array}{l} HA_1 = ( {i}\varOmega c_1-\mu \varOmega ^2 )A_0 \Bigg(\dfrac{1}{R^2} + \dfrac{{i}\varOmega c_1-\mu \varOmega ^2}{(E)_{eff} A}\Bigg )\cdot \\ \qquad \int_{s_1 }^{s_2 } \int_0^L {G_1 } (s;\xi )d\xi ds \\ HA_2 = \dfrac{\left( {\mu \varOmega ^2-{i}\varOmega c_1 } \right)A_0 }{(E)_{eff} A}(s_2-s_1 ) \\ HA_3 = \dfrac{\vartheta _2 }{(E)_{eff} AR}(s_1-s_2 ) \\ HA_4 =-\int_{s_1 }^{s_2 } \Bigg[\int_0^L G_3 (s;\xi )d\xi \Bigg ]"ds \\ HA_5 = \Bigg(\dfrac{1}{R^2} + \dfrac{{i}\varOmega c_1-\mu \varOmega ^2}{(E)_{eff} A}\Bigg)\int_{s_1 }^{s_2 } \Big [ \bar {G}"_2 (s;s_2 ) -\\ \qquad {\bar {G}}"_2 (s;s_1 )\Big] ds \\ HA_6 = \Bigg(\dfrac{\mu \varOmega ^2-{i}\varOmega c_1 }{(E)_{eff} A}-\dfrac{1}{R^2}\Bigg)\int_{s_1 }^{s_2 } {\int_0^L {\bar {G}_3 (s;\xi )} } d\xi ds \end{array}\!\!\right\} $$
在后续的数值算例中分析了函数$\vert \bar {V}/A_{b}\vert $,它是用参考标度$A_{b}$衡量的电压.

本文探究了厚度$h$,弧长$L$,受简谐激励的压电俘能器.为了说明,引入阻尼效应的无量纲阻尼比$\zeta _{1}$^[22] }$

(46)$$ \zeta _1 = \dfrac{c_1 }{2\mu \varOmega _0 } $$
其中, $\varOmega_{0}=\pi^{2}$ $(EI/\rho A)^{0.5}/L^{2}$ 是曲梁的一阶固有频率.

曲梁几何参数、压电参数取值如表1所示.

Table 1
表1
表1压电曲梁的几何参数、压电参数取值
Table 1Geometrical and electromechanical parameters of the beam

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6.2 解的有效性验证

本部分内容将验证曲梁俘能器强迫振动解析解的有效性.与直梁俘能器相比,曲梁考虑了轴力效应,若在曲梁俘能器模型中消除轴力效应,即令式(20)和式(21)中有等式(47)成立

(47)$$ \dfrac{1}{(E)_{eff} A} = 0 , \ \ \dfrac{\gamma \varOmega ^2}{(E)_{eff} A} = 1 $$
此时由式(20)得

(48)$$ {\varPsi }' = {W}" + \dfrac{1}{R^2}W $$
消除轴力后的控制方程的形式与式(24)相同,此时有

(49)$$ \left.\begin{array}{l} a_1 = \dfrac{1}{R^2} + \dfrac{\gamma \varOmega ^2}{(EI)_{eff} }\\ a_2 = \dfrac{ {i}\varOmega c_1-\mu \varOmega ^2 }{(EI)_{eff} } + \dfrac{\gamma \varOmega ^2}{(EI)_{eff} R^2} \\ b_1 = 0 , \ \ b_2 = \dfrac{1}{(EI)_{eff} }\\ b_3 = \dfrac{\vartheta _1 }{(EI)_{eff} } , \ \ c =-\dfrac{\vartheta _2 }{(EI)_{eff} R} \end{array}\!\!\right\} $$
根据前文的Laplace变换及格林函数法,结合电路控制方程,可得此时输出电压的表达式为

(50)$ \left.\begin{array} \bar {V} = \Bigg\{ ( {i}\varOmega c_{1}-\mu \varOmega ^2)A_0 \!\int_{s_1 }^{s_2 } \!\Bigg [\!\int_0^L \! G_1 (s, \xi )d\xi \Bigg]"\!ds + HA_1 \Bigg\} \Bigg / \\ \qquad \Bigg \{ \int_{s_1 }^{s_2 }\Big [\bar {G}"_2 (s;s_2 )-\bar {G}"_2 (s;s_1 )\Big ] ds + \\ \qquad \dfrac{ {i}\varOmega C_{p} R_l + 1}{ {i}\varOmega \beta R_l } + HA_2 + HA_3 + HA_4 \Bigg \} \end{array}\!\!\right\} $
(51)$ \left.\begin{array}{l} HA_1 = \dfrac{\left( {{i}\varOmega c_1-\mu \varOmega ^2} \right)A_0 }{R^2}\int_{s_1 }^{s_2 } \int_0^L {G_1 } (s;\xi )d\xi ds \\ HA_2 = \int_{s_1 }^{s_2 } \Bigg [\int_0^L G_3 (s; \xi )d\xi \Bigg ]"ds \\ HA_3 = \dfrac{1}{R^2}\int_{s_1 }^{s_2 } \Big[\bar {G}_2 (s;s_2 )-\bar {G}_2 (ss_1 )\Big ] ds \\ HA_4 =-\dfrac{1}{R^2}\int_{s_1 }^{s_2 } \int_0^L \bar {G}_3 (s;\xi ) d\xi ds \end{array} \!\!\right\} $
曲梁消除轴力效应后,若令俘能器的半径趋于无穷大,可将曲梁退化为直梁.所取曲梁的几何参数,压电参数与文献^[19]所 取保持一致. 根据消除轴力效应且半径无穷大时俘能器的解作与参考文献解的对比图. 由图3可知,去除轴力效应后的压电俘能 器与文献中直梁压电俘能器所得的响应效果在第一、二阶基本一致,第三阶稍有偏差.解的正确性得以验证,该数值算例也可以 用以验证新的模型.

图3

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图3本文解与文献解的对比

Fig. 3The comparison of present solutions and the solutions from reference



后续算例中,如无特殊说明取参数值$b= 0.01$, $h_{p} = 5.0 \times 10^{-4}$, $h_{s}= 5.0 \times 10^{-4}$, $\zeta_{1}=0.01$, $d_{31} =-1.90 \times 10^{-10}$, $R_{l}=100$, $E_{p} =6.6 \times 10^{10}$, $E_{s}=1.0\times 10^{11}$.

6.3 各物理参数对频率-压电响应的影响

本部分分别探究了电阻、阻尼、压电常数、弹性模量对压电响应和谐振频率的影响,为曲梁压电俘能器更好的设计应用提供理论依据.

如图4所示,响应随着电阻的增大逐步增强,电阻$R_{l}=1.0\times 10^{6} \Omega$和$R_{l}=1.0\times 10^{8} \Omega$时响应函数曲线基本重叠无明显差别. 在控制变量电阻变化的整个过程中,产生压电响应的谐振频率不变. 此俘能器最优负载电阻为1 M$\Omega $.

图4

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图4电阻对响应的影响

Fig. 4The frequency response of voltage with different resistive loads



分别选取 $\zeta _{1}=0.2$, 0.4, 0.8作图5探究阻尼比对频率-压电响应的影响情况.当阻尼比为0.8时,共 振峰基本消失,响应曲线趋于平缓,谐振频率在8 000$\sim $9 000 Hz范围内保持不变.

图5

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图5阻尼比对响应的影响

Fig. 5The frequency of voltage with different damping effects



在材料制备领域,通过低温烧结的方法和混合各种化学元素,例如锰和铅,压电常数很容易发生改变^[27-28].因此, 基于电压峰值来选取最优压电常数. 上述数值例子中所用的材料实际上是一种特殊类型的软压电材料.材料为: PZT-5A/5H,材料型号为3195HD. 这种软压电材料在最近的压电俘能器研究中得到了广泛的应用^[29-30]. 与电阻的影响趋势一致,如图6 所示,压电常数达到$d_{31}=-1.0\times 10^{-10}$前,响应随之增长. 继续增大压电常数则开始降低.谐振频率不随压电常数变化.

图6

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图6压电常数对响应的影响

Fig. 6The frequency of voltage with different piezoelectric constants



从模型自身结构来说,材料的弹性模量是一个重要的影响因素,可更换基底材料来实现其取值的不同以得到最大的输出电压. 本文分析了结构层弹性模量对频率-电压响应的影响. 由图7可知,结构层弹性模量对谐振频率的影响较大,不同的弹性模量均对应 不同的谐振频率,且频率的跨度较大. 可根据工作环境的不同选取基底材料.压电材料弹性模量对压电函数的影响效果与结构材料相似.

图7

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图7结构层弹性模量的影响

Fig. 7The frequency of voltage with different elasticity modulus of structure materials

7 结论

本文建立了曲梁压电俘能器的力电耦合模型,利用格林函数法研究了悬臂式曲梁压电俘能器的强迫振动.对于其他的边界条件, 比如简支,固支也可运用此求解方法进行求解.数值计算中,通过与传统直梁压电俘能器模型的文献解进行对比,验证了解的有效性. 通过探究阻尼,负载电阻,材料的弹性模量对响应结果的影响,得出结论：(1)该曲梁俘能器的最优负载为1 M$\Omega $;(2)通过选择弹性模量较低的基底材料,可以调低曲梁俘能器的高基频现象,但会导致俘能器的工作效率降低;(3)曲梁俘能器的工 作效率远远高于直梁俘能器.

参考文献 View Option 原文顺序
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