PIEZOELECTRIC VIBRATION ENERGY HARVESTERS AND DYNAMIC ANALYSIS BASED ON THE SPINNING BEAM
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摘要: 为研究轴向载荷及梁上外激励共同作用下自旋梁结构强迫振动的压电振动能量采集问题, 文章提出运用格林函数法求解自旋梁压电俘能器强迫振动下的电压解析解. 基于Euler-Bernoulli梁理论, 采用扩展Hamilton原理及PZT-5A压电本构, 建立了自旋梁压电俘能器强迫振动的力电耦合模型. 采用Laplace变换法求得耦合振动方程的格林函数解, 并根据线性叠加原理和格林函数的物理意义, 对耦合的系统方程进行解耦, 进而求得强迫振动下自旋梁压电俘能器的电压解析解. 数值计算中, 通过与现有文献中的解析解以及实验结果进行对比, 验证了本文解的有效性, 并分别分析了自旋梁压电俘能器的压电响应与电阻、转速等重要物理参数之间的关系. 数值分析研究表明: (1) 自旋梁俘能器的压电响应随电阻阻值的增大而增大, 直至阻值达到最优负载电阻; (2)通过调高转速, 可以提高压电俘能器的最大输出电压; (3)通过降低轴向载荷, 可在保持俘能器高效工作的情况下改善俘能器的高基频现象.Abstract: In order to study the piezoelectric vibration energy harvesting problem of the forced vibration of a spinning beam structure under the combined effect of axial forces and external excitation on the beam, this paper proposes to use the Green's function method to solve the analytical solution of the voltage under the forced vibration of the spinning piezoelectric energy harvester. The extended Hamilton's principle and PZT-5A piezoelectric constitutive relationship are used to develop a force-electric coupling model for the spinning piezoelectric energy harvester of forced vibration based on the Euler-Bernoulli beam theory. Utilizing the Laplace transform, the explicit expressions of the Green's function of the coupled vibration equations can be acquired. Based on the linear superposition principle and the physical significance of the Green's function, the coupled system equations are decoupled to find the analytical solution of the voltage of the spinning piezoelectric energy harvester under forced vibration. In the numerical calculation, the validity of the solution of this paper is verified by comparing the present solution with the result of the existing literature as well as experimental result. The relationship between the piezoelectric response and physical parameters such as resistance and spinning speed of the energy harvester is analyzed separately. This research suggests that piezoelectric response of the spinning energy harvester increases with increasing resistance until the resistance reaches the optimal load resistance; the maximum output voltage of the energy harvester can be increased by turning up the spinning speed; by reducing the axial force, the high fundamental frequency of the energy harvester can be improved while maintaining the efficient operation of the energy harvester.
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引 言
近年来, 能量采集技术得到了大力发展, 许多研究致力于将自然环境中容易忽视的能量如太阳能、热能、振荡能等转换为电能. 在诸多能量来源中, 由各种机械设备、道路、建筑等产生的机械振动是主要能量来源之一, 因此将机械振动能转化为电能成为实现能量转化的普遍方式之一[1]. 其中, 绕x轴旋转的梁结构是工程应用中极为广泛的一种基本机械元件[2-3], 其在现代生产生活中广泛存在, 如机械中的角速度传感器[4]、航空航天中的旋翼、钻井中的钻杆等, 其中的诸多旋转元件都可以用自旋梁来模拟其运动状态及振动特征[5-6]. 由于自旋梁在高速旋转时产生的振动比普通梁更为强烈[7-10], 因此有必要对自旋梁进行振动能量采集.
压电振动能量采集最早由美国Inman等[11-12]提出, 其在Euler-Bernoulli梁假设下, 研究了悬臂式压电俘能器在基座横向运动时所产生的横向振动的精确解析解. Zhao等[13]应用格林函数法对悬臂式Timoshenko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电压响应的影响. 在直梁的基础上, 何艳丽等[14]用格林函数法对Timoshenko曲梁压电俘能器进行了动力学分析, 此外, 赵翔等[15]提出采用逆方法对含裂纹的曲梁压电俘能器进行损伤检测. Niazi等[16]提出了一种带动态放大器的悬臂压电−磁致收缩双稳态混合能量采集器, 该系统包括两个振动自由度和两个电自由度, 其可在提高发电功率的同时获得更宽的激励频带. 为了实现风能驱动压电能量采集器振动发电, Priore等[17]提出了一种基于涡激振动和驰振相互作用的气动弹性压电能量采集器, 其旨在优化俘能器在低风速范围内的俘能性能; Zhao等[18]通过引入Lamb-Oseen涡旋模型, 对悬臂式压电俘能器的涡激振动进行了动力学分析.
虽然对于压电振动能量采集研究已经形成了较为完整的理论体系, 部分理论成果也已付诸于实际工程中, 但这些成果中的压电俘能器大部分采用非旋转梁的形式, 虽有少量研究采用自旋梁形式, 但其结构均为矩形截面, 且无轴向载荷以及外激励作用的自由振动[19-20]. 周兰伟等[21]基于模拟退火算法对受轴向载荷作用的自旋梁压电分流电路进行优化, 但其未考虑梁上受外激励作用; Wang等[22]结合翘曲效应和自旋效应的影响, 建立了具有分布式压电传感器的非对称截面自旋梁的控制方程; Yang等[23]研究了表面粘贴压电薄膜的弹性梁自旋时的弯曲振动, 得到了自旋梁在交流电压激励下强迫振动的解析解. 由此可见, 随着机械系统的复杂化和智能化, 自旋梁俘能器的研究很有必要.
针对上述问题, 本文采用扩展Hamilton原理建立了轴向载荷以及梁上外激励共同作用下自旋梁压电俘能器强迫振动的力电耦合模型, 并创新性地采用格林函数法求解自旋梁俘能器的压电响应. 数值计算中将自旋梁俘能器与非旋转梁俘能器的集能效率进行对比分析, 探究了自旋梁俘能器的转速、电阻等重要物理参数对输出电压和谐振频率的影响,以期为自旋梁压电俘能器在不同谐振频率下的能量采集以及振动研究提供理论参考.
1. 压电俘能器模型的建立
俘能器采用受外载荷p(x,t)和轴向载荷T0的自旋直梁(外层为压电层, 内层为结构层)作为主体结构, p(x,t)与y方向的夹角为${\bar{\theta}}_{\text{0}}$, 其在y方向与z方向的分力分别为p1, p2, 连接负载电阻Rl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, 梁长为L. 图2所示为俘能器的横截面示意图, 外直径为D, 结构层直径为d, 压电层厚度为hp, x-y方向和x-z方向上的弯曲刚度为EI, Euler-Bernoulli梁绕x轴以速度$\bar{{ {{\varOmega}} }}$旋转, μ表示单位长度梁的质量.
根据Euler-Bernoulli梁理论, 梁的剪切变形和转动惯量很小, 在分析中忽略不计. 梁上任一点的变形可以用向量r表示为[24]
$$ {\boldsymbol{r}} = v{\boldsymbol{i}} + w{\boldsymbol{j}} $$ (1) 式中, v和w分别是y和z方向上的横向位移, i和j分别是y和z方向上的单位向量, 梁上任一点的速度可以表示为
$$ {\boldsymbol{v}} = {\boldsymbol{\dot r}} + {\boldsymbol{\bar \varOmega }} \times {\boldsymbol{\dot r}} = (\dot v - {\bar \varOmega }w){\boldsymbol{i}} + (\dot w + {\bar \varOmega }v){\boldsymbol{j}} $$ (2) 式中, $\bar {{ {\boldsymbol{\varOmega}} }}\text{ = }\bar {{ {{\varOmega}} }}{{\boldsymbol{k}}}$, k为x方向上的单位向量; “˙”表示对时间的一阶导数, 则压电俘能器的动能表达式为
$$ \begin{split} & T = \frac{1}{2}\int_0^L {{{\left| {\boldsymbol{v}} \right|}^2}\mu {\rm{d}}x} = \\ &\qquad \frac{1}{2}\int_0^L {\mu \left[ {{{\dot v}^2} + {{\dot w}^2} + 2{\bar \varOmega }(v\dot w - \dot vw) + {{{\bar \varOmega }}^2}({v^2} + {w^2})} \right]{\rm{d}}x} \end{split} $$ (3) 压电俘能器的势能为
$$ U{\text{ = }}\frac{1}{2}\iiint_{{\rm{volume}}} {{\sigma _{xx}}{\varepsilon _{xx}}{\rm{d}}x{\rm{d}}y{\rm{d}}z} $$ (4) 式中, ${\text{σ}}_{{xx}}$(${{ \varepsilon }}_{{xx}}$)是法向应力(法向应变). 根据PZT-5A的本构关系可以得到
$$ \sigma _{xx}^{\text{p}} = {E_{\text{p}}}(\varepsilon _{xx}^{\text{p}} - {d_{31}}{E_{\text{h}}}),\quad \sigma _{xx}^{\text{s}} = {E_{\text{s}}}\varepsilon _{xx}^{\text{s}} $$ (5) 本文中, “s”表示结构层参数, “p”表示压电层参数; d31表示压电常数; ${{E}}_{\text{h}}{ = }{-}{v}{(}{t}{)}/{{h}}_{\text{p}}$, Eh表示厚度方向上的电场强度, v(t)表示电路中的电压. 基于Euler-Bernoulli梁应变与位移的几何关系, 压电俘能器的势能为
$$\begin{split} & U{\text{ = }}\frac{1}{2}\int_0^L {\left[ {\int_0^{\frac{d}{2}} {{E_{\text{s}}}} {{(v''\rho )}^2} \cdot \text{π} \rho {\rm{d}}\rho + \int_0^{\frac{d}{2}} {{E_{\text{s}}}} {{(w''\rho )}^2} \cdot \text{π} \rho {\rm{d}}\rho } \right.} +\\ &\qquad \int_{\frac{d}{2}}^{\frac{D}{2}} {{E_{\text{p}}}{{\left(v''\rho - {d_{31}}\frac{{v(t)}}{{{h_{\text{p}}}}}\right)}^2} \cdot \text{π} \rho {\rm{d}}\rho }+ \\ &\qquad \left. { \int_{\frac{d}{2}}^{\frac{D}{2}} {{E_{\text{p}}}} {{\left(w''\rho - {d_{31}}\frac{{v(t)}}{{{h_{\text{p}}}}}\right)}^2} \cdot \text{π} \rho {\rm{d}}\rho } \right]{\rm{d}}x- \\ &\qquad \frac{1}{2}\int_0^L {{T_0}\left[ {{{v'}^2} + {{\left(v' - {d_{31}}\frac{{v(t)}}{{{h_{\text{p}}}}}\right)}^2}} \right.}+ \\ &\qquad \left. {{{w'}^2} + {{\left(w' - {d_{31}}\frac{{v(t)}}{{{h_{\text{p}}}}}\right)}^2}} \right]{\rm{d}}x= \\ &\qquad \frac{1}{2}\int_0^L {\left\{ {{{(EI)}_{{\text{eff}}}}({{v''}^2} + {{w''}^2}) + 2{C_{{\text{p1}}}}{{v(t)}^2}} \right.}- \\ &\qquad 2{C_{{\text{p2}}}}v(t)(v'' + w'') - {T_0}\left[ {2{{v'}^2} + 2{{w'}^2}} \right.+ \\ &\qquad \left. { 2{C_{{\text{p3}}}}{{v(t)}^2}\left. { - 2{C_{{\text{p4}}}}v(t)(v' + w')} \right]} \right\}{\rm{d}}x\\[-12pt] \end{split} $$ (6) 式中, “”表示对空间坐标x的导数, Es和Ep分别是结构层和压电层的杨氏模量, (EI)eff表示等效弯曲刚度, Cp1, Cp2, Cp3, Cp4表示耦合系数, 分别表示为$”表示对空间坐标x的导数, Es和Ep分别是结构层和压电层的杨氏模量, (EI)eff表示等效弯曲刚度, Cp1, Cp2, Cp3, Cp4表示耦合系数, 分别表示为
$$ \left.\begin{split} & {(EI)_{{\text{eff}}}} = \frac{{{E_{\text{s}}}\text{π} {d^4} + {E_{\text{p}}}\text{π} ({D^4} - {d^4})}}{{64}} \\ & {C_{{\text{p1}}}} = \frac{{{E_{\text{p}}}d_{31}^2\text{π} ({D^2} - {d^2})}}{{8h_{\text{p}}^2}},{C_{{\text{p2}}}} = \frac{{{E_{\text{p}}}{d_{31}}\text{π} ({D^3} - {d^3})}}{{24{h_{\text{p}}}}} \\ & {C_{{\text{p3}}}} = \frac{{d_{31}^2}}{{h_{\text{p}}^2}},{C_{{\text{p4}}}} = \frac{{{d_{31}}}}{{{h_{\text{p}}}}} \end{split}\right\} $$ (7) 非保守力p所做的虚功为
$$ \delta W = \int_0^L { -( {p_1}\delta v} - {p_2}\delta w){\rm{d}}x $$ (8) 根据Hamilton原理
$$ \delta\int_{{t_1}}^{{t_2}} {(T - U){\rm{d}}t} = \int_{{t_1}}^{{t_2}} {\delta W{\rm{d}}t} $$ (9) 式中, δ表示变分运算, 将式(3)、式(6)、式(8)带入式(9)并化简可得压电俘能器模型动力学方程为
$$ \begin{split} & {(EI)_{{\text{eff}}}}v'''' + 2{T_0}v'' + \mu \ddot v - \mu {{\bar \varOmega }^2}v - 2\mu \bar \varOmega \dot w - \\ &\qquad {C_{{\text{p2}}}}v(t){[\delta (x) - \delta (x - L)]^2}+ \\ &\qquad 2{T_0}{C_{{\text{p4}}}}v(t)[\delta (x) - \delta (x - L)] = {p_1} \end{split} $$ (10) $$ \begin{split} & {(EI)_{{\text{eff}}}}w'''' + 2{T_0}w'' + \mu \ddot w - \mu {{\bar \varOmega }^2}w + 2\mu \bar \varOmega \dot v- \\ &\qquad {C_{{\text{p2}}}}v(t){[\delta (x) - \delta (x - L)]^2} + \\ &\qquad 2{T_0}{C_{{\text{p4}}}}v(t)[\delta (x) - \delta (x - L)] = {p_2} \end{split} $$ (11) 由图1可知, 压电俘能器模型采用简支梁形式, 则边界条件为
$$ \left.\begin{split} & v(0/L,t) = 0,\quad w(0/L,t) = 0 \\ & v''(0/L,t) = 0,\quad w''(0/L,t) = 0 \end{split}\right\} $$ (12) 2. 力电耦合的电路控制方程
本节基于参考文献[25]中的方法建立了电路方程, 电路为欧姆电路. 根据参考文献[26], 对于自旋Euler-Bernoulli梁压电俘能器, 考虑以下压电本构关系
$$ {D}_{y} = {d}_{31}{E}_{\text{p}}{\varepsilon }_{xx}^{{y}}-{\varepsilon }_{33}^{\text{s}}\frac{v(t)}{{h}_{\text{p}}}\text{, }{D}_{z} = {d}_{31}{E}_{\text{p}}{\varepsilon }_{xx}^{{z}}-{\varepsilon }_{33}^{\text{s}}\frac{v(t)}{{h}_{\text{p}}} $$ (13) 式中, $ {{ \varepsilon }}_{\text{33}}^{\text{s}} $为介电常数, ${{ \varepsilon }}_{{xx}}^{{y}}$和${{ \varepsilon }}_{{xx}}^{{z}}$分别是y方向和z方向上的平均弯曲应变, 表示为
$$ {\varepsilon }_{xx}^{y} = -{h}_{\text{pc}}\frac{{\partial }^{2}v(x,t)}{\partial {x}^{2}}\text{, }{\varepsilon }_{xx}^{z} = -{h}_{\text{pc}}\frac{{\partial }^{2}w(x,t)}{\partial {x}^{2}} $$ (14) 式中, hpc表示质量中心线到压电层中性轴的距离. 将式(14)代入式(13)得
$$ \left.\begin{split} & {D_y} = - {d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}\frac{{{\partial ^2}v(x,t)}}{{\partial {x^2}}} - \varepsilon _{33}^{\text{s}}\frac{{v(t)}}{{{h_{\text{p}}}}} \\ & {D_z} = - {d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}\frac{{{\partial ^2}w(x,t)}}{{\partial {x^2}}} - \varepsilon _{33}^{\text{s}}\frac{{v(t)}}{{{h_{\text{p}}}}}\end{split}\right\} $$ (15) 根据参考文献[25], 电荷q(t)可以通过对电极区域上的电位移进行积分来获得
$$ \begin{split} & q(t) = \int_A {\boldsymbol{D}} \cdot {\boldsymbol{n}}{\rm{d}}A= \\ &\qquad - \int_0^L {\left[ {{d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}d\frac{{{\partial ^2}v(x,t)}}{{\partial {x^2}}} + \varepsilon _{33}^{\text{s}}d\frac{{v(t)}}{{{h_{\text{p}}}}}} \right.}+ \\ &\qquad \left. { {d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}d\frac{{{\partial ^2}w(x,t)}}{{\partial {x^2}}} + \varepsilon _{33}^{\text{s}}d\frac{{v(t)}}{{{h_{\text{p}}}}}} \right]{\rm{d}}x \end{split} $$ (16) 式中, D是电位移矢量, n是单位外法向矢量, 由压电俘能器产生的电流i(t)表示为
$$ \begin{split} & i(t) = \frac{{{\rm{d}}q(t)}}{{{\rm{d}}t}} = - \int_0^L \left[ {d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}d\frac{{{\partial ^3}v(x,t)}}{{\partial {x^2}\partial t}} +\right.\\ &\qquad \left.{d_{31}}{E_{\text{p}}}{h_{{\text{pc}}}}d\frac{{{\partial ^3}w(x,t)}}{{\partial {x^2}\partial t}} \right]{\rm{d}}x - 2\frac{{\varepsilon _{33}^{\text{s}}dL}}{{{h_{\text{p}}}}}\frac{{{\rm{d}}v(t)}}{{{\rm{d}}t}} \end{split} $$ (17) 基于${v}{(}{t}{) = }{{R}}_{{l}}{i}{(}{t}{)}$, 电路方程可以推导为
$$ {C_{\text{p}}}\frac{{{\rm{d}}v(t)}}{{{\rm{d}}t}} + \frac{{v(t)}}{{{R_{\text{l}}}}} = - \int_0^L {{\beta _1}\left[ {\frac{{{\partial ^3}v(x,t)}}{{\partial {x^2}\partial t}}\left. { + \frac{{{\partial ^3}w(x,t)}}{{\partial {x^2}\partial t}}} \right]{\rm{d}}x} \right.} $$ (18) 式中, ${{C}}_{\text{p}}\text{ = }{2}{{ \varepsilon }}_{\text{33}}^{\text{s}}{Ld}/{{h}}_{\text{p}}$, $\beta_{\text{1}}\text{ = }{{d}}_{\text{31}}{{E}}_{\text{p}}{{h}}_{\text{pc}}{d}$.
3. 稳态的力电耦合压电俘能器模型
假设梁上作用的简谐载荷p(x,t) = P(x)eiΩt, 其中Ω为载荷的激励频率, 则相应的p在y, z方向上的分力以及位移, 电压可以分别假设为如下形式
$$\left. \begin{split} & {p_1}(x,t) = {P_1}(x){{\rm{e}}^{{\text{i}}\varOmega t}},{p_2}(x,t) = {P_2}(x){{\rm{e}}^{{\text{i}}\varOmega t}} \\ & v(x,t) = V(x){{\rm{e}}^{{\text{i}}\varOmega t}},w(x,t) = W(x){{\rm{e}}^{{\text{i}}\varOmega t}},v(t) = \bar V{{\rm{e}}^{{\text{i}}\varOmega t}} \end{split} \right\}$$ (19) 其中, V(x), W(x)和$\bar {{V}}{(}{x}{)}$分别是稳态的平面内位移、平面外位移和电压. 为了简化系统, 进行分离变量, 消除时间变量t, 将式(19)代入式(10), 式(11)中可得分离变量后的控制方程为
$$ \begin{split} & {(EI)_{{\text{eff}}}}V'''' + 2{T_0}V'' - ({{\varOmega }^2} + {{{\bar \varOmega }}^2})\mu V - 2\mu {\text{i}}{\varOmega \bar \varOmega }W- \\ &\qquad {C_{{\text{p2}}}}\bar V{[\delta (x) - \delta (x - L)]^2} + 2{T_0}{C_{{\text{p4}}}}\bar V[\delta (x) - \delta (x - L)] = {P_1} \end{split} $$ (20) $$\begin{split} & {(EI)_{{\text{eff}}}}W'''' + 2{T_0}W'' - ({{\varOmega }^2} + {{{\bar \varOmega }}^2})\mu W + 2\mu {\text{i}}{\varOmega \bar \varOmega }V - \\ &\qquad {C_{{\text{p2}}}}\bar V{[\delta (x) - \delta (x - L)]^2} + 2{T_0}{C_{{\text{p4}}}}\bar V[\delta (x) - \delta (x - L)] = {P_2} \end{split} $$ (21) 简化后的压电俘能器稳态控制方程为
$$ \begin{split} & V'''' + {a_1}V'' - {a_2}V - {a_3}W = {b_1}{P_1} +\\ &\qquad{b_2}\bar V{[\delta (x) - \delta (x - L)]^2} - {b_3}\bar V[\delta (x) - \delta (x - L)] \end{split} $$ (22) $$ \begin{split} & W'''' + {a_1}W'' - {a_2}W + {a_3}V = {b_1}{P_2} +\\ &\qquad {b_2}\bar V{[\delta (x) - \delta (x - L)]^2}- {b_3}\bar V[\delta (x) - \delta (x - L)] \end{split} $$ (23) 式中
$$ \left.\begin{split} & {a_1} = \frac{{2{T_0}}}{{{{(EI)}_{{\text{eff}}}}}},{a_2} = \frac{{\mu ({{\varOmega }^2} + {{{\bar \varOmega }}^2})}}{{{{(EI)}_{{\text{eff}}}}}},{a_3} = \frac{{2\mu {\text{i}}{\varOmega \bar \varOmega }}}{{{{(EI)}_{{\text{eff}}}}}} \\ & {b_1} = \frac{1}{{{{(EI)}_{{\text{eff}}}}}},{b_2} = \frac{{{C_{{\text{p2}}}}}}{{{{(EI)}_{{\text{eff}}}}}},{b_3} = \frac{{2{T_0}{C_{{\text{p4}}}}}}{{{{(EI)}_{{\text{eff}}}}}} \end{split}\right\} $$ (24) 稳态电路方程
$$ \frac{{{\rm{i}}\varOmega {C_{\text{p}}}{R_{\text{l}}} + 1}}{{{R_{\text{l}}}}}\bar V = - \int_0^L {{\text{i}}\varOmega {\beta _1}\left[ {V''(x)\left. { + W''(x)} \right]{\rm{d}}x} \right.} $$ (25) 4. 压电俘能器稳态强迫振动的格林函数
从式(22)和式(23)可知, 压电俘能器的稳态位移由外部载荷P和电耦合效应${\bar{V}}{[}\delta{(}{x}{)-}\delta{(}{x}{-}{L}{)]}$2, ${\bar{V}}[\delta({x})- \delta{(}{x}{-}{L}{)]}$引起. 根据线性系统的叠加原理, 稳态位移V和W可以分为V1(W1), V2(W2)和V3(W3)三部分, 即V(W) = V1(W1) + V2(W2) + V3(W3), 每部分分别由P, ${\bar{V}}{[}\delta{(}{x}{)-}\delta{(}{x}{-}{L}{)]}$2和${\bar{V}}{[}\delta{(}{x}{)-}\delta{(}{x}{-}{L}{)]}$引起, V1(W1), V2(W2)和V3(W3)分别为Case1, Case2和Case3的解, 每部分又需分为两种情况进行计算, 即
Case1-1
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = {b_1}{P_1} $$ (26) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = 0 $$ (27) Case1-2
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = 0 $$ (28) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = {b_1}{P_2} $$ (29) Case2-1
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = {b_2}\bar V{[\delta (x) - \delta (x - L)]^2} $$ (30) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = 0 $$ (31) Case2-2
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = 0 $$ (32) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = {b_2}\bar V{[\delta (x) - \delta (x - L)]^2} $$ (33) Case3-1
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = - {b_3}\bar V[\delta (x) - \delta (x - L)] $$ (34) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = 0 $$ (35) Case3-2
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = 0 $$ (36) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = - {b_3}\bar V[\delta (x) - \delta (x - L)] $$ (37) 4.1 Case1-1的格林函数
根据格林函数的物理意义, Case1-1的格林函数G11(x; x0)为式(38)和式(39)的解
$$ V'''' + {a_1}V'' - {a_2}V - {a_3}W = {b_1}\delta (x - {x_0}) $$ (38) $$ W'''' + {a_1}W'' - {a_2}W + {a_3}V = 0 $$ (39) 式中, δ(·) 是Dirac函数, x0表示单位力作用的位置, 对式(38), 式(39)进行关于x的Laplace变换
$$ \begin{split} & \hat V(s,{x_0})F= {b_1}({s^4} + {a_1}{s^2} - {a_2}){{\rm{e}}^{ - s{x_0}}} +\\ &\qquad ({s^4} + {a_1}{s^2} - {a_2})V'''(0)+ s({s^4} + {a_1}{s^2} - {a_2})V''(0) +\\ &\qquad ({s^2} + {a_1})({s^4} + {a_1}{s^2} - {a_2})V'(0)+ ({s^3} + {a_1}s)\cdot\\ &\qquad({s^4} + {a_1}{s^2} - {a_2})V(0) + {a_3}W'''(0) +\\ &\qquad {a_3}sW''(0)+ {a_3}({s^2} + {a_1})W'(0) + {a_3}({s^3} + {a_1}s)W(0) \end{split} $$ (40) $$ \begin{split} & \hat W(s,{x_0})F = - {b_1}{a_3}{{\rm{e}}^{ - s{x_0}}} + ({s^4} + {a_1}{s^2} - {a_2})W'''(0) +\\ &\qquad s({s^4} + {a_1}{s^2} - {a_2})W''(0) + ({s^2} + {a_1})({s^4} + {a_1}{s^2} - {a_2})\cdot\\ &\qquad W'(0) + ({s^3} + {a_1}s)({s^4} + {a_1}{s^2} - {a_2})W(0) - {a_3}V'''(0)- \\ &\qquad {a_3}sV''(0) - {a_3}({s^2} + {a_1})V'(0) - {a_3}({s^3} + {a_1}s)V(0) \end{split} $$ (41) 其中
$$ F = {s^8} + 2{a_1}{s^6} + (a_1^2 - 2{a_2}){s^4} - 2{a_1}{a_2}{s^2} + a_2^2 + a_3^2 $$ (42) 参数s是复变量, ${W}{(0)}$, ${{W}}{{'}}{(0)}$, ${{W}}{{'}{'}}{(0)}$, ${{W}}{{'}{'}{'}}{(0)}$, ${V}{(0)}$, ${{V}}{{'}}{(0)}$, ${{V}}{{'}{'}}{(0)}$和${{V}}{{'}{'}{'}}{(0)}$ 是可由梁的边界条件确定的待定系数, 再将式(40)和式(41)进行Laplace逆变换, 即可以得到Case1-1情况下的格林函数
$$ \begin{split} & {G_{{V_{11}}}}(x,{x_0}) = H(x - {x_0}){\phi _1}(x - {x_0}) + {\phi _2}(x)V(0)+ \\ &\qquad {\phi _3}(x)V'(0) + {\phi _4}(x)V''(0) + {\phi _5}(x)V'''(0) + \\ &\qquad {\phi _6}(x)W(0) + {\phi _7}(x)W'(0) + {\phi _8}(x)W''(0) + \\ &\qquad {\phi _9}(x)W'''(0) \end{split} $$ (43) $$ \begin{split} & {G_{{W_{11}}}}(x,{x_0}) = H(x - {x_0}){\phi _{10}}(x - {x_0}) + {\phi _2}(x)W(0)+ \\ &\qquad {\phi _3}(x)W'(0) + {\phi _4}(x)W''(0) + {\phi _5}(x)W'''(0) + \\ &\qquad {\phi _{11}}(x)V(0) + {\phi _{12}}(x)V'(0) + {\phi _{13}}(x)V''(0)+ \\ &\qquad {\phi _{14}}(x)V'''(0) \end{split} $$ (44) 式中, H(x−x0) 是Heaviside函数, $\phi _{\text{1}}$和$\phi _{\text{1}\text{0}}$表示自旋梁系统的受迫振动项, 其余的$\phi _{{i}}$为自旋梁系统的自由振动项, 即自由振动模态, 它们的表达式分别为
$$\left. \begin{split} & {\phi _1}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){b_1}({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _{10}}(x) = \sum\limits_{i = 1}^8 { - {A_i}(x){b_1}{a_3}} \\ & {\phi _2}(x) = \sum\limits_{i = 1}^8 {{A_i}(x)({s_i}^3 + {a_1}{s_i})({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _3}(x) = \sum\limits_{i = 1}^8 {{A_i}(x)({s_i}^2 + {a_1})({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _4}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){s_i}({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _5}(x) = \sum\limits_{i = 1}^8 {{A_i}(x)({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _6}(x) = - {\phi _{11}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}({s_i}^3 + {a_1}{s_i})} \\ & {\phi _7}(x) = - {\phi _{12}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}({s_i}^2 + {a_1})} \\ & {\phi _8}(x) = - {\phi _{13}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}{s_i}} \\ & {\phi _9}(x) = - {\phi _{14}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}} \end{split}\right\} $$ (45) 其中, si是代数方程F(s) = 0即式(42)等于0的根. Ai的表达式分别为
$$\left. \begin{split} &{A_1}(x) = \frac{{{{\rm{e}}^{{s_1}x}}}}{ ({s_1} - {s_2})({s_1} - {s_3})({s_1} - {s_4})({s_1} - {s_5}) ({s_1} - {s_6})({s_1} - {s_7})({s_1} - {s_8}) } \\ & {A_2}(x) = \frac{{{{\rm{e}}^{{s_2}x}}}}{ ({s_2} - {s_1})({s_2} - {s_3})({s_2} - {s_4})({s_2} - {s_5}) ({s_2} - {s_6})({s_2} - {s_7})({s_2} - {s_8}) } \\ & {A_3}(x) = \frac{{{{\rm{e}}^{{s_3}x}}}}{ ({s_3} - {s_1})({s_3} - {s_2})({s_3} - {s_4})({s_3} - {s_5}) ({s_3} - {s_6})({s_3} - {s_7})({s_3} - {s_8}) } \\ & {A_4}(x) = \frac{{{{\rm{e}}^{{s_4}x}}}}{ ({s_4} - {s_1})({s_4} - {s_2})({s_4} - {s_3})({s_4} - {s_5}) ({s_4} - {s_6})({s_4} - {s_7})({s_4} - {s_8}) } \\ & {A_5}(x) = \frac{{{{\rm{e}}^{{s_5}x}}}}{ ({s_5} - {s_1})({s_5} - {s_2})({s_5} - {s_3})({s_5} - {s_4}) ({s_5} - {s_6})({s_5} - {s_7})({s_5} - {s_8}) } \\ & {A_6}(x) = \frac{{{{\rm{e}}^{{s_6}x}}}}{ ({s_6} - {s_1})({s_6} - {s_2})({s_6} - {s_3})({s_6} - {s_4}) ({s_6} - {s_5})({s_6} - {s_7})({s_6} - {s_8}) } \\ & {A_7}(x) = \frac{{{{\rm{e}}^{{s_7}x}}}}{ ({s_7} - {s_1})({s_7} - {s_2})({s_7} - {s_3})({s_7} - {s_4}) ({s_7} - {s_5})({s_7} - {s_6})({s_7} - {s_8}) } \\ & {A_8}(x) = \frac{{{{\rm{e}}^{{s_8}x}}}}{ ({s_8} - {s_1})({s_8} - {s_2})({s_8} - {s_3})({s_8} - {s_4}) ({s_8} - {s_5})({s_8} - {s_6})({s_8} - {s_7}) } \end{split}\right\} $$ (46) 4.2 Case1-2 ~ Case3-2的格林函数
根据格林函数理论, Case1-2 ~ Case3-2的求解过程与Case1-1的过程类似. 其格林函数表示为
Case1-2
$$ \begin{split} & {G_{{V_{12}}}}(x,{x_0}) = H(x - {x_0}){\phi _{15}}(x - {x_0}) + {\phi _2}(x)V(0) + \\ &\qquad {\phi _3}(x)V'(0) + {\phi _4}(x)V''(0) + {\phi _5}(x)V'''(0) + \\ &\qquad {\phi _6}(x)W(0) + {\phi _7}(x)W'(0) + {\phi _8}(x)W''(0) + \\ &\qquad {\phi _9}(x)W'''(0) \end{split} $$ (47) $$ \begin{split} & {G_{{W_{12}}}}(x,{x_0}) = H(x - {x_0}){\phi _{16}}(x - {x_0}) + {\phi _2}(x)W(0) + \\ &\qquad {\phi _3}(x)W'(0) + {\phi _4}(x)W''(0) + {\phi _5}(x)W'''(0)+ \\ &\qquad {\phi _{11}}(x)V(0) + {\phi _{12}}(x)V'(0) + {\phi _{13}}(x)V''(0)+ \\ &\qquad {\phi _{14}}(x)V'''(0) \end{split} $$ (48) Case2-1 ~ Case3-2
$$\begin{split} & {G_{{V_{{\text{ij}}}}}}(x,{x_0}) = \bar VH(x - {x_0}){\phi _{\text{m}}}(x - {x_0}) + {\phi _2}(x)V(0)+ \\ &\qquad {\phi _3}(x)V'(0) + {\phi _4}(x)V''(0) + {\phi _5}(x)V'''(0)+ \\ &\qquad {\phi _6}(x)W(0) + {\phi _7}(x)W'(0) + {\phi _8}(x)W''(0)+ \\ &\qquad {\phi _9}(x)W'''(0) \end{split} $$ (49) $$\begin{split} & {G_{{W_{{\text{ij}}}}}}(x,{x_0}) = \bar VH(x - {x_0}){\phi _{\text{n}}}(x - {x_0}) + {\phi _2}(x)W(0) + \\ &\qquad {\phi _3}(x)W'(0) + {\phi _4}(x)W''(0) + {\phi _5}(x)W'''(0)+ \\ &\qquad {\phi _{11}}(x)V(0) + {\phi _{12}}(x)V'(0) + {\phi _{13}}(x)V''(0)+ \\ &\qquad {\phi _{14}}(x)V'''(0)\end{split} $$ (50) 其中, i = 2, 3; j = 1, 2; m = 17, 19, 21, 23; n = m + 1.
$$\left.\begin{split} &{\phi _{15}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){b_1}{a_3}} \\ & {\phi _{16}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){b_1}({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _{17}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){b_2}({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _{18}}(x) = \sum\limits_{i = 1}^8 { - {A_i}(x){a_3}{b_2}} \\ & {\phi _{19}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}{b_2}} \\ & {\phi _{20}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){b_2}({s_i}^4 + {a_1}{s_i}^2 - {a_2})} \\ & {\phi _{21}}(x) = \sum\limits_{i = 1}^8 { - {A_i}(x){b_3}} ({s_i}^4 + {a_1}{s_i}^2 - {a_2}) \\ & {\phi _{23}}(x) = \sum\limits_{i = 1}^8 { - {A_i}(x){a_3}{b_3}} \\ & {\phi _{24}}(x) = \sum\limits_{i = 1}^8 { - {A_i}(x){b_3}} ({s_i}^4 + {a_1}{s_i}^2 - {a_2}) \\ & {\phi _{22}}(x) = \sum\limits_{i = 1}^8 {{A_i}(x){a_3}{b_3}} \end{split} \right\}$$ (51) 5. 确定格林函数的待定系数
基于式(12)中的边界条件${V}{(0)}{ = }{0}$, ${{V}}{{'}{'}}{(0)}{ = }{0}$, ${W}\text{(0)}\text{ = }\text{0}$, ${{W}}{{'}{'}}\text{(0)}\text{ = }\text{0}$, Case1-1和Case1-2的格林函数可以简化为
$$\begin{split} & {G_{{W_{{{1j}}}}}}(x,{x_0}) = H(x - {x_0}){\phi _{{n}}}(x - {x_0}) + {\phi _3}(x)W'(0) + \\ &\qquad {\phi _5}(x)W'''(0) + {\phi _{12}}(x)V'(0) + {\phi _{14}}(x)V'''(0) \end{split}$$ (52) $$\begin{split} & {G_{{W_{{{1j}}}}}}(x,{x_0}) = H(x - {x_0}){\phi _{{n}}}(x - {x_0}) + {\phi _3}(x)W'(0) + \\ &\qquad {\phi _5}(x)W'''(0) + {\phi _{12}}(x)V'(0) + {\phi _{14}}(x)V'''(0) \end{split}$$ (53) 其中, 当j = 1时, m = 1, n = 10; j = 2时, m = 15, n = 16.
Case2-1 ~ Case3-2的格林函数可以简化为
$$ \begin{split} & {G_{{V_{{{ij}}}}}}(x,{x_0}) = \bar VH(x - {x_0}){\phi _{{m}}}(x - {x_0}) + {\phi _3}(x)V'(0) + \\ &\qquad {\phi _5}(x)V'''(0) + {\phi _7}(x)W'(0) + {\phi _9}(x)W'''(0) \end{split} $$ (54) $$\begin{split} & {G_{{W_{{{ij}}}}}}(x,{x_0}) = \bar VH(x - {x_0}){\phi _{{n}}}(x - {x_0}) + {\phi _3}(x)W'(0) + \\ &\qquad {\phi _5}(x)W'''(0) + {\phi _{12}}(x)V'(0) + {\phi _{14}}(x)V'''(0) \end{split} $$ (55) 其中, i = 2, 3; j = 1, 2; m = 17, 19, 21, 23; n = m + 1. 常数${{W}}{{'}}{(0)}$, ${{W}}{{'}{'}{'}}\text{(0)}$, ${{V}}{{'}}\text{(0)}$, ${{V}}{{'}{'}{'}}\text{(0)}$可以由以下矩阵方程确定, 该方程是从其余4个边界条件导出的
$$ \left[ {\begin{array}{*{20}{c}} {{\phi _3}(L)}&{{\phi _5}(L)}&{{\phi _7}(L)}&{{\phi _9}(L)} \\ {{{\phi ''}_3}(L)}&{{{\phi ''}_5}(L)}&{{{\phi ''}_7}(L)}&{{{\phi ''}_9}(L)} \\ {{\phi _{12}}(L)}&{{\phi _{14}}(L)}&{{\phi _3}(L)}&{{\phi _5}(L)} \\ {{{\phi ''}_{12}}(L)}&{{{\phi ''}_{14}}(L)}&{{{\phi ''}_3}(L)}&{{{\phi ''}_5}(L)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {V'(0)} \\ {V'''(0)} \\ {W'(0)} \\ {W'''(0)} \end{array}} \right] = {\boldsymbol{T}} $$ (56) 其中
$$\begin{split} & {\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} { - {\phi _{{m}}}(L - {x_0})}&{ - {{\phi ''}_{{m}}}(L - {x_0})} \end{array}} \right. \\ &\qquad {\left. {\begin{array}{*{20}{c}} { - {\phi _{{n}}}(L - {x_0})}&{ - {{\phi ''}_{{n}}}(L - {x_0})} \end{array}} \right]^{\text{T}}} \end{split} $$ (57) 当为Case1-1时, m = 1, n = 10; 当为Case1-2时, m = 15, n = 16; 当为Case2-1 ~ Case3-2, m = 17, 19, 21, 23, n = m + 1且${{\boldsymbol{T}}}{ = }{{\boldsymbol{T}}}\cdot {\bar{V}}$.
6. 力电耦合系统的解耦
根据第5节, 基于线性系统的叠加原理和格林函数的可叠加性对系统进行解耦, 梁的位移V(x)和W(x)可表示为
$$\begin{split} & V(x,\bar \theta ;{x_0},{{\bar \theta }_0}) = \int\nolimits_0^L {\int\nolimits_0^{2\text{π} } {\left( {{G_{{V_{11}}}}(x;{x_0})\cos {{\bar \theta }_0}{P_1}({x_0}) + } \right.} } \\ &\qquad \left. {{G_{{V_{12}}}}(x;{x_0})\sin {{\bar \theta }_0}{P_2}({x_0})} \right){\rm{d}}{x_0}{\rm{d}}{{\bar \theta }_0}+ \\ &\qquad \bar V\int_0^L {\left( {{{\bar G}_{{V_{21}}}}(x;{x_0}) + {{\bar G}_{{V_{22}}}}(x;{x_0})} \right.} +\\ &\qquad \left. { {{\bar G}_{{V_{31}}}}(x;{x_0}) + {{\bar G}_{{V_{32}}}}(x;{x_0})} \right) \cdot [\delta ({x_0} - L) - \delta ({x_0})]{\rm{d}}{x_0} \end{split} $$ (58) $$ \begin{split} & W(x,\bar \theta ;{x_0},{{\bar \theta }_0}) = \int\nolimits_0^L {\int\nolimits_0^{2\text{π} } {\left( {{G_{{W_{11}}}}(x;{x_0})\cos {{\bar \theta }_0}{P_1}({x_0})} \right.} }+ \\ &\qquad \left. { {G_{{W_{12}}}}(x;{x_0})\sin {{\bar \theta }_0}{P_2}({x_0})} \right){\rm{d}}{x_0}{\rm{d}}{{\bar \theta }_0} +\\ &\qquad \bar V\int_0^L {\left( {{{\bar G}_{{W_{21}}}}(x;{x_0}) + {{\bar G}_{{W_{22}}}}(x;{x_0})} \right.} +\\ &\qquad \left. { {{\bar G}_{{W_{31}}}}(x;{x_0}) + {{\bar G}_{{W_{32}}}}(x;{x_0})} \right) \cdot [\delta ({x_0} - L) - \delta ({x_0})]{\rm{d}}{x_0} \end{split} $$ (59) 式中, ${{G}}_{{{V}}_{{ij}}}{ = }{\bar{V}}\cdot{{\bar{G}}}_{{{V}}_{{ij}}}$, ${{G}}_{{{W}}_{{ij}}}{ = }{\bar{V}}\cdot{{\bar{G}}}_{{{W}}_{{ij}}}$, i = 2, 3; j = 1, 2. 将式(58)和式(59)代入式(25)中可得电压${\bar{V}}$的表达式
$$\begin{split} &\bar{V} = \left\{-{\rm{i}} \varOmega \beta_1 \int_0^L\left[\int _ { 0 } ^ { L } \int _ { 0 } ^ { 2 \text{π} } \left(G_{V_{11}}\left(x ; x_0\right) \cos \bar{\theta}_0 P_1\left(x_0\right)+ G_{V_{12}}\left(x ; x_0\right) \sin \bar{\theta}_0 P_2\left(x_0\right)\right) {\rm{d}} x_0 {\rm{d}} \bar{\theta}_0\right]^{\prime \prime} {\rm{d}} x-\right. \\ &\qquad \left.{{\rm{i}} \varOmega \beta_1 \displaystyle\int_0^L\left[\int _ { 0 } ^ { L } \int _ { 0 } ^ { 2 \text{π} } \left(G_{W_{11}}\left(x ; x_0\right) \cos \bar{\theta}_0 P_1\left(x_0\right)+G_{W_{12}}\left(x ; x_0\right) \sin \bar{\theta}_0 P_2\left(x_0\right)\right) {\rm{d}} x_0 {\rm{d}} \bar{\theta}_0\right]^{\prime \prime} {\rm{d}} x}\right\}\Biggr/\\ &\qquad \left\{{\dfrac{{\rm{i}} \varOmega C_p R_1 + 1}{R_1} + {\rm{i}} \varOmega \beta_1\left[\displaystyle\int_0^L \displaystyle\sum_{\substack{{i}=2,3 \\{j}=1,2}}\left(\bar{G}_{V_{{ij}}}^{\prime \prime}(x ; L) - \bar{G}_{V_{{ij}}}^{\prime \prime}(x ; 0)\right) {\rm{d}} x +\int_0^L \sum_{\substack{{i}=2,3 \\{j}=1,2}}\left(\bar{G}_{W_{{ij}}}^{\prime \prime}(x ; L)-\bar{G}_{W_{{ij}}}^{\prime \prime}(x ; 0)\right) {\rm{d}} x\right]}\right\}\\[-12pt] \end{split} $$ (60) 7. 数值结果与讨论
7.1 简谐激励
将简谐激励视为外载荷作用在自旋梁压电俘能器上. 若基础位移不等于零, 梁的绝对位移w(x,t)为基础位移wb(x,t)与绝对位移wrel(x,t)的叠加[27], 对于简谐激励有
$$ {w_{\text{b}}}(s,t) = {A_1}{{\rm{e}}^{{\rm{i}}\varOmega t}},{v_{\text{b}}}(s,t) = {A_2}{{\rm{e}}^{{\rm{i}}\varOmega t}} $$ (61) 即稳态的基础位移Wb = A1, Vb = A2, 进一步可得基础加速度${{A}}_{{\text{b}}_{\text{1}}}\text{ = }-{{ \varOmega }}^{\text{2}}{{A}}_{\text{1}}$, ${{A}}_{{\text{b}}_{\text{2}}}\text{ = }-{{ \varOmega }}^{\text{2}}{{A}}_{\text{2}}$; 外力${p}{(}{x}{,}{t}{)}{ = }{P}{(}{x}{)}{{{\rm{e}}}}^{{{\rm{i}} \varOmega t}}$可以写作
$$ {P_1}(s) = \mu {{\varOmega }^2}{A_1},{P_2}(s) = \mu {{\varOmega }^2}{A_2} $$ (62) 其中, A1, A2分别表示平面内位移和平面外位移的振幅. 将式(62)代入电压表达式(60)中, 可得简谐激励下的稳态电压为
$$ \begin{split} &\bar{V}=\left\{-{\rm{i}} \varOmega \beta_1 \mu \varOmega^2 A_1 \int_0^L\left[\int _ { 0 } ^ { L } \int _ { 0 } ^ { 2 \text{π} } \left(G_{V_{11}}\left(x ; x_0\right) \cos \bar{\theta}_0+ G_{W_{11}}\left(x ; x_0\right) \cos \bar{\theta}_0\right) {\rm{d}} x_0 {\rm{d}} \bar{\theta}_0\right]^{\prime \prime} {\rm{d}} x-\right. \\[-4pt] &\qquad \left.{{\rm{i}} \varOmega \beta_1 \mu \varOmega^2 A_2 \displaystyle\int_0^L\left[\int _ { 0 } ^ { L } \int _ { 0 } ^ { 2 \text{π} } \left(G_{V_{12}}\left(x ; x_0\right) \sin \bar{\theta}_0+G_{W_{12}}\left(x ; x_0\right) \sin \bar{\theta}_0\right) {\rm{d}} x_0 {\rm{d}} \bar{\theta}_0\right]^{\prime \prime} {\rm{d}} x}\right\}\Biggr/\\[-7pt] &\qquad \left\{{\dfrac{{\rm{i}} \varOmega C_{\mathrm{p}} R_1+1}{R_1}+{\rm{i}} \varOmega \beta_1\left[\displaystyle\int_0^L \displaystyle\sum_{\substack{i=2,3 \\ j=1,2}}\left(\bar{G}_{V_{i j}}^{\prime \prime}(x ; L)-\bar{G}_{V_{i j}}^{\prime \prime}(x ; 0)\right) {\rm{d}} x+ \int_0^L \sum_{\substack{i=2,3 \\ j=1,2}}\left(\bar{G}_{W_{i j}^{\prime \prime}}^{\prime \prime}(x ; L)-\bar{G}_{W_{i j}}^{\prime \prime}(x ; 0)\right) {\rm{d}} x\right]}\right\} \end{split} $$ (63) 在本节中, 为了简便起见, 引入无量纲化参数
$$ \bar \varOmega '{\text{ = }}\frac{{\bar \varOmega }}{{{\varOmega _0}}} $$ (64) 其中, ${{ \varOmega }}_{\text{0}}\text{ = }{\text{π}}^{\text{2}}\sqrt{{EI}\mu}/{{L}}^{\text{2}}$是Euler-Bernoulli梁的一阶固有频率, 轴向力T0通过最小Euler临界屈曲载荷${{T}}_{\text{cr}}\text{ = }{\text{π}}^{\text{2}}{EI}/{{L}}^{\text{2}}$进行归一化.
7.2 解的有效性验证
7.2.1 数值计算验证
本小节通过俘能器模型的退化解与已发表文献的数值计算结果进行对比, 验证本文解析解的有效性. 通过去除压电层, 并将轴力设为0, 设定与文献[28]中相同的材料参数
$$ EI\text{ = }582.996{\text{ N·m}}^{\text{2}}\text{, }L\text{ = }1.29\text{ m}\text{, }\mu \text{ = }2.87\text{ kg/m} $$ (65) 分别计算得出了不同转速下的前6阶固有频率, 将其与文献[28]中固有频率进行对照, 结果如图3所示. 从图中可以看出, 去除压电层和轴向载荷后的前六阶固有频率与文献中的数值计算结果基本吻合, 验证了本文解的有效性.
7.2.2 实验验证
本小节通过与文献[29]中的实验结果对比验证本文解的有效性. 文献[29]所采用的实验设备如图4所示, 通过设定与文献中相同的材料参数, 分别计算得出转速为300 r/min时前3阶固有频率, 与文献中实验的对照结果如表1所示. 从表中可以看出, 数值计算结果与实验结果基本吻合, 进一步验证了本文解的有效性.
表 1 $\bar{{ {\boldsymbol{\varOmega}} }}$ = 300 r/min时前3阶固有频率对比Table 1. Comparison of first three natural frequencies between the present and Ref. [29] ($\bar{{ {{\varOmega}} }}$ = 300 r/min)Mode Experiment/Hz Present/Hz Diff/% 1 31.577 29.490 6.61 2 97.745 98.592 0.87 3 206.386 209.207 1.37 在后续算例中, 分析了$\left|{{\bar{V}}}_{\text{0}}\right|\text{ = }\text{|}{\bar{V}}\text{/}[({{A}}_{{\text{b}}_{\text{1}}}\text{ + }{{A}}_{{\text{b}}_{\text{2}}})\text{/2] |}$, 它是用参考$\text{(}{{A}}_{{\text{b}}_{\text{1}}}\text{ + }{{A}}_{{\text{b}}_{\text{2}}}\text{)/2}$衡量的电压. 如无特殊说明, 自旋梁几何参数, 压电参数取值如表2所示.
表 2 压电自旋梁的几何参数、压电参数取值Table 2. Geometrical and electromechanical parameters of the spinning beamParameters Value Parameters Value substructure layer diameter, d/m 0.019 Outer diameter, D/m 0.029 Young’s modulus of substructure layer, Es/Pa 1.0 × 1011 Young’s modulus of piezoelectric layer, Ep/Pa 6.6 × 1010 electrode start distance from the base, x1/m 0 electrode end distance from the base, x2/m 0.1 length of the beam, L/m 0.1 resistance load, Rl/Ω 1.0 × 103 thickness of piezoelectric layer, hp/m 0.005 permittivity, ε33/(nF·m−1) 15.93 piezoelectric constant, d31/(m·V−1) −1.9 × 1010 spinning speed, $\bar{ { \varOmega } }{'}$ 2.5 axial load, T/Tcr 0.2 angel, ${\bar{\theta} }_{\text{0} }$/(°) π/3 7.3 各物理参数对频率−压电响应的影响
本部分分别探究了转速、电阻、压电常数和轴力对压电响应和谐振频率的影响, 为自旋梁压电俘能器更好的设计应用提供理论依据.
作为自旋结构, 转速是影响自旋梁结构振动的关键因素之一. 图5分别选取了$\bar{{ \varOmega }}{'}$ = 1.5, 2.0, 2.5, 3分析转速对频率−压电响应的影响情况. 从图5中可以看出转速对谐振频率的影响较大, 并且转速越高, 俘能器的基频越高, 最大输出电压也越高, 在高频激励时的频率跨度越小. 从力学的角度出发, 自旋梁在两个方向上的1阶频率随转速的提高是由离心硬化效应引起的, 即离心力使梁的刚度提高. 此外, 自旋梁俘能器的工作效率远高于非旋转梁俘能器, 这主要是由于高速旋转加剧了梁的振动.
图6分析了负载电阻对频率−压电响应的影响. 如图所示, 压电响应随着负载电阻的增大逐步增强, 电阻Rl = 1.0 × 103 Ω和Rl = 1.0 × 104 Ω时电压响应函数曲线除在低频激励下有少许差别外, 在中高频激励下响应曲线基本重叠无明显差别, 而当负载电阻高于1.0 × 104 Ω即Rl = 2.0 × 104 Ω时, 其产生的电压响应曲线与Rl = 1.0 × 104 Ω时的曲线几乎无差别, 同时无论是低频激励还是高频激励, 产生压电响应的谐振频率不随负载电阻的变化而变化. 因此该俘能器的最优负载电阻为10 kΩ.
在压电材料制备领域, 通过低温烧结的方法与多种化学元素混合, 如锰和铅, 材料的压电常数很容易发生改变. 因此, 基于压电响应来选取最优压电常数. 本文数值算例中所用的压电材料为一种特殊类型的软压电材料: PZT-5A/5H, 材料型号为3195D, 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[30]. 如图7所示, 与电阻趋势一致, 压电响应随着压电常数的增大而增大, 而谐振频率不随压电常数发生变化.
图8选取了T0/Tcr = 0.1, 0.2, 0.3, 0.4分别分析轴向压缩载荷和轴向拉伸载荷对频率−压电响应的影响情况. 从图中可以看出无论是轴向压缩载荷还是轴向拉伸载荷对电压响应的谐振频率都会产生影响, 但其相对于转速对谐振频率的影响较小. 在中低频激励时, 轴向载荷越大, 压电响应越大, 而在高频激励时压电响应随着轴向载荷的增大而逐渐减小. 两者不同的是, 中低频激励时谐振频率的跨度随着轴向压缩载荷的增大而逐渐减小, 而随着轴向拉伸载荷的增大逐渐增大.
8. 结 论
本文首先运用能量法建立了受轴向载荷和梁上外激励共同作用下的自旋梁压电俘能器力电耦合模型, 其次利用格林函数法推导出该模型的电压解析解, 最后在数值计算中, 通过与现有文献解以及实验结果进行对比, 验证了本文解的有效性, 并探究了自旋梁俘能器在负载电阻, 转速等物理参数影响下电压响应的变化情况, 以期为自旋梁俘能器在不同谐振频率下的能量采集以及动力学分析提供理论参考, 结论如下:
(1) 自旋梁俘能器的压电响应随电阻阻值的增大而增大, 直至阻值达到最优负载电阻;
(2) 自旋梁的转速越高, 压电响应越高, 但基频也越高, 应根据实际工程情况选择合适的转速;
(3) 自旋梁俘能器的工作效率远高于非旋转梁俘能器.
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表 1 $\bar{{ {\boldsymbol{\varOmega}} }}$ = 300 r/min时前3阶固有频率对比
Table 1 Comparison of first three natural frequencies between the present and Ref. [29] ($\bar{{ {{\varOmega}} }}$ = 300 r/min)
Mode Experiment/Hz Present/Hz Diff/% 1 31.577 29.490 6.61 2 97.745 98.592 0.87 3 206.386 209.207 1.37 表 2 压电自旋梁的几何参数、压电参数取值
Table 2 Geometrical and electromechanical parameters of the spinning beam
Parameters Value Parameters Value substructure layer diameter, d/m 0.019 Outer diameter, D/m 0.029 Young’s modulus of substructure layer, Es/Pa 1.0 × 1011 Young’s modulus of piezoelectric layer, Ep/Pa 6.6 × 1010 electrode start distance from the base, x1/m 0 electrode end distance from the base, x2/m 0.1 length of the beam, L/m 0.1 resistance load, Rl/Ω 1.0 × 103 thickness of piezoelectric layer, hp/m 0.005 permittivity, ε33/(nF·m−1) 15.93 piezoelectric constant, d31/(m·V−1) −1.9 × 1010 spinning speed, $\bar{ { \varOmega } }{'}$ 2.5 axial load, T/Tcr 0.2 angel, ${\bar{\theta} }_{\text{0} }$/(°) π/3 -
[1] Parinov IA, Cherpakov AV. Overview: state-of-the-art in the energy harvesting based on piezoelectric devices for last decade. Symmetry, 2022, 14: 765-814 doi: 10.3390/sym14040765
[2] Ebrahimi R, Ziaei-Rad S. Nonplanar vibration and flutter analysis of vertically spinning cantilevered piezoelectric pipes conveying fluid. Ocean Engineering, 2022, 261: 112180 doi: 10.1016/j.oceaneng.2022.112180
[3] 张云顺, 赵香帅, 王万树. 旋转轮胎中压电悬臂梁离心距优化. 压电与声光, 2022, 44(1): 89-100 (Zhang Yunshun, Zhao Xiangshuai, Wang Wanshu. Optimization of centrifugal distance of piezoelectric cantilever beam in rotating tires. Piezoelectricity and Sound and Light, 2022, 44(1): 89-100 (in Chinese) doi: 10.11977/j.issn.1004-2474.2022.01.017 Zhang Yunshun, Zhao Xiangshuai, Wang Wanshu. Optimization of centrifugal distance of piezoelectric cantilever beam in rotating tires. Piezoelectricity and Sound and Light, 2022, 44(1): 89-100 (in Chinese) doi: 10.11977/j.issn.1004-2474.2022.01.017
[4] Qu YL, Jin F, Yang JS. Vibrating flexoelectric micro-beams as angular rate sensors. Micromachines, 2022, 13: 1243 doi: 10.3390/mi13081243
[5] Yang S, Hu HJ, Mo GD, et al. Dynamic modeling and analysis of an axially moving and spinning Rayleigh beam based on a time-varying element. Applied Mathematical Modelling, 2021, 95: 409-434 doi: 10.1016/j.apm.2021.01.049
[6] Xu H, Wang YQ, Zhang YF. Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method. Archive of Applied Mechanics, 2021, 91: 4817-4834 doi: 10.1007/s00419-021-02036-7
[7] Lee H. Dynamic response of a rotating Timoshenko shaft subject to axial forces and moving loads. Journal of Sound and Vibration, 1995, 181: 169 doi: 10.1006/jsvi.1995.0132
[8] Mamandi A, Kargarnovin MH. Nonlinear dynamic analysis of an axially loaded rotating Timoshenko beam with extensional condition included subjected to general type of force moving along the beam length. Journal of Vibration and Control, 2013, 19: 2448-2458 doi: 10.1177/1077546312456723
[9] Ouyang HJ, Wang MJ. A dynamic model for a rotating beam subjected to axially moving forces. Journal of Sound and Vibration, 2007, 308: 674-682 doi: 10.1016/j.jsv.2007.03.082
[10] Zu JWZ, Han RP. Natural frequencies and normal modes of a spinning Timoshenko beam with general boundary conditions. Journal of Applied Mechanics, 1992, 59: 197-204 doi: 10.1115/1.2899488
[11] Erturk A, Inman DJ. Issues in mathematical modeling of piezoelectric energy harvesters. Smart Materials and Structures, 2008, 17: 065016 doi: 10.1088/0964-1726/17/6/065016
[12] Erturk A, Inman DJ. On mechanical modeling of cantilevered piezoelectric vibration energy harvesters. Journal of Intelligent Material Systems and Structures, 2008, 19: 1311-1325 doi: 10.1177/1045389X07085639
[13] Zhao X, Yang EC, Li YH, et al. Closed-form solutions for forced vibrations of piezoelectric energy harvesters by means of Green’s functions. Journal of Intelligent Material Systems and Structures, 2017, 28: 2372-2387 doi: 10.1177/1045389X17689927
[14] 何燕丽, 赵翔. 曲梁压电俘能器强迫振动的格林函数解. 力学学报, 2019, 51: 1170-1179 (He Yanli, Zhao Xiang. Closed-form solutions for forced vibrations of curved piezoelectric energy harvesters by means of green’s functions. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51: 1170-1179 (in Chinese) 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. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51: 1170-1179 (in Chinese) doi: 10.6052/0459-1879-19-007
[15] 赵翔, 李思谊, 李映辉. 基于压电振动能量俘获的弯曲结构损伤监测研究. 力学学报, 2021, 53: 3035-3044 (Zhao Xiang, Li Siyi, Li Yinghui. The research on damage detection of curved beam based on piezoelectric vibration energy harvester. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53: 3035-3044 (in Chinese) doi: 10.6052/0459-1879-21-452 Zhao Xiang, Li Siyi, Li Yinghui. The research on damage detection of curved beam based on piezoelectric vibration energy harvester. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53: 3035-3044 (in Chinese) doi: 10.6052/0459-1879-21-452
[16] Niazi K, Parsi MJK, Mohammadi M, et al. Nonlinear dynamic analysis of hybrid piezoelectric-magnetostrictive energy-harvesting systems. Journal of Sensors, 2022, 2022(S): 8921779
[17] Priore ED, Romano GP, Lampani L. Coupled electro-aeroelastic energy harvester model based on piezoelectric transducers, VIV-galloping interaction and nonlinear switching circuits. Smart Materials and Structures, 2023, 32: 075012 doi: 10.1088/1361-665X/acdb15
[18] Zhao X, Zhu DW, Li YH. Closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester under a fluid vortex. Journal of Vibration and Acoustics, 2022, 144: 021010 doi: 10.1115/1.4051773
[19] Li W, Yang XD, Zhang W, et al. Free vibrations and energy transfer analysis of the vibrating piezoelectric gyroscope based on the linear and nonlinear decoupling methods. Journal of Vibration and Acoustics, 2019, 141: 041015 doi: 10.1115/1.4043062
[20] Li W, Yang XD, Zhang W, et al. Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities. Acta Mechanica Sinica, 2019, 35: 879-893 doi: 10.1007/s10409-019-00851-4
[21] 周兰伟, 陈国平, 孙东阳等. 基于模拟退火算法的旋转梁压电分流电路优化. 振动. 测试与诊断, 2016, 36: 315-320 (Zhou Lanwei, Chen Guoping, Sun Dongyang, et al. Optimization of rotating beam piezoelectric shunt circuit based on simulated annealing algorithm. Journal of Vibration,Measurement &Diagnosis, 2016, 36: 315-320 (in Chinese) Zhou Lanwei, Chen Guoping, Sun Dongyang, et al. Optimization of rotating beam piezoelectric shunt circuit based on simulated annealing algorithm. Journal of Vibration, Measurement & Diagnosis, 2016, 36: 315-320 (in Chinese)
[22] Wang J, Li D, Jiang J. Coupled flexural-torsional vibration of spinning smart beams with asymmetric cross sections. Finite Elements in Analysis and Design, 2015, 105: 16-25 doi: 10.1016/j.finel.2015.06.008
[23] Yang JS, Fang HY. Analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2002, 49: 798-804 doi: 10.1109/TUFFC.2002.1009338
[24] Banerjee J, Su HJC. Development of a dynamic stiffness matrix for free vibration analysis of spinning beams. Computers and Structures, 2004, 82: 2189-2197 doi: 10.1016/j.compstruc.2004.03.058
[25] Erturk A, Inman DJ. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of Vibration and Acoustics, 2008, 130: 041002 doi: 10.1115/1.2890402
[26] Li X, Ding H, Chen W. Three-dimensional analytical solution for a transversely isotropic functionally graded piezoelectric circular plate subject to a uniform electric potential difference. Science in China, Series G: Physics, Mechanics and Astronomy, 2008, 51: 1116-1125 doi: 10.1007/s11433-008-0100-z
[27] Danesh-Yazdi AH, Elvin N, Andreopoulos Y. Green׳s function method for piezoelectric energy harvesting beams. Journal of Sound and Vibration, 2014, 333: 3092-3108 doi: 10.1016/j.jsv.2014.02.023
[28] Zhu K, Chung J. Nonlinear lateral vibrations of a deploying Euler–Bernoulli beam with a spinning motion. International Journal of Mechanical Sciences, 2015, 90: 200-212 doi: 10.1016/j.ijmecsci.2014.11.009
[29] Perng YL, Chin JH. Theoretical and experimental investigations on the spinning BTA deep-hole drill shafts containing fluids and subject to axial forces. International Journal of Mechanical Sciences, 1999, 41: 1301-1322 doi: 10.1016/S0020-7403(98)00091-5
[30] Leadenhan S, Erturk A. Unifie nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation. Nonlinear Dynamics, 2015, 79(3): 1727-1743 doi: 10.1007/s11071-014-1770-x
-
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13. 陈志强,刘战合,苗楠,冯伟. 基于增量学习的非定常气动力参数化降阶模型. 航空学报. 2021(07): 343-354 . 百度学术
14. 章飞,程芳. 某型飞机操纵面间隙非线性颤振时域分析. 航空工程进展. 2021(04): 99-104 . 百度学术
15. 黄锐,胡海岩. 飞行器非线性气动伺服弹性力学. 力学进展. 2021(03): 428-466 . 百度学术
16. 聂雪媛,郑冠男,杨国伟. 含间隙非线性机翼跨声速颤振时滞反馈控制. 北京航空航天大学学报. 2021(10): 1980-1988 . 百度学术
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18. 张家铭,杨执钧,黄锐. 基于非线性状态空间辨识的气动弹性模型降阶. 力学学报. 2020(01): 150-161 . 本站查看
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23. 唐强,张宁,李浩,雷志荣. 无人机自主控制系统简述. 测控技术. 2020(10): 114-123 . 百度学术
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28. 杨建忠,徐丹,杨士斌,李勇. 考虑非线性因素的阵风减缓系统建模与仿真. 飞行力学. 2018(04): 48-52 . 百度学术
29. 周强,李东风,陈刚,李跃明. 基于CFD和CSM耦合的通用静气弹分析方法. 航空动力学报. 2018(02): 355-363 . 百度学术
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