RESEARCH ON ACTION BEHAVIOR OF MACRO FIBER COMPOSITES APPLIED IN CANTILEVER STRUCTURES
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摘要: 压电纤维复合材料(macro fiber composite, MFC)是由美国NASA研发的综合性能优秀的新型压电材料, 掌握MFC的力学行为有利于将其投入结构变形控制、减振降噪和健康监测等领域. 目前对MFC宏观力学行为的研究中缺乏对作动力与驱动电压直接关系的研究, 现有的以MFC为作动器的应用中所使用力电关系的精度有限, 不利于将MFC投入更精密的使用场景. 针对此问题, 文章采用经典板理论, 考虑了MFC与受控结构的相对尺寸, 推导了MFC对悬臂结构的作动力方程. 为兼顾计算的准确性和便利性, 建立了考虑MFC叉指电极真实电场的精细有限元模型开展压电静力仿真, 给出将MFC叉指电极的弯曲电场简化为匀强电场的修正系数. 搭建MFC-悬臂梁结构的实验装置, 对有限元模型和作动力公式加以验证. 精细模型与简化模型的仿真结果与实验进行对比, 总体误差较小, 验证了模型和修正系数的可靠性. 理论计算与仿真结果对比, 误差在1%以内, 表明所得作动力预测公式在较大的宽度比范围内均具有较好精度. 建立的MFC作动力模型对MFC应用于悬臂结构的变形控制和振动抑制有一定指导意义.Abstract: Macro fiber composite (MFC) is a new kind of piezoelectric material with excellent performances developed by NASA Langley Research Center. An accurate understanding of the mechanical behavior of MFC helps to apply it to structural deformation control, vibration control, noise reduction, health monitoring and other fields. However, there are few researches on the macroscopic mechanical behavior of MFC in the literature, especially about the relationship between the actuating force and the voltage. The limited accuracy of the relationship between the actuating force and the voltage is not conducive to apply MFC to more precise application scenarios. In the present paper, the actuation equation of MFC acting on a cantilever structure is derived by considering the relative dimensions of MFC and the controlled structure to overcome this problem based on classic plate theory. For sake of the balance of the accuracy and convenience for the simulation, a detailed finite element model considering the bending electric field of the interdigitated electrode of MFC is established to carry out piezoelectric-static simulation. A correction coefficient is introduced for simplifying the electric field of the interdigitated electrode. A piezoelectric-static experiment is carried out to validate the finite element model and the actuation formula for a MFC-cantilever beam. The simulation results of the detailed model and the simplified model are compared with the experiment. The relative error between the simulations and the experiment is little, which verifies the reliability of the model and the correction coefficient of the curved electric field. The correctness of the theoretical analysis is verified by the simulation results, which indicates the actuation prediction formula has a good precision in a wide range of width ratio between MFC and the controlled structure. The actuating model of MFC established in this paper has a certain guiding significance for the application of MFC to the deformation control and vibration suppression for cantilever structures.
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引 言
压电材料是一种能将机械能和电能相互转换的智能材料, 可利用其正、逆压电效应制作传感器和作动器. 传统的压电材料中, 压电陶瓷材料脆性大, 难以应用至曲面结构[1]; 压电高分子聚合物材料压电性能弱, 需要大范围铺设才能起到明显效果[2], 这些缺点限制了压电材料在工程中的应用. 压电纤维复合材料[3](macro fiber composite, MFC)是美国NASA兰利研究中心[4]研发公布的新型压电材料, 具有体积小、质量轻、柔韧性好和压电性能强等优点. 其优秀的综合性能吸引了大量研究人员去尝试使用MFC替代传统的压电材料开展研究工作, 目前MFC已在悬臂结构的变形控制[5-8]、振动控制[9-12]和结构监测[13-15]等领域广泛应用.
为充分开发MFC的工作潜力, 首先需要准确把握其性能参数. 许多学者开展了相关研究, Brett[16]研究了MFC各组分对压电性能的影响并给出了MFC微观结构的详细参数, Deraemaeker等[17-18]基于均匀场理论推导了MFC均质化的混合规则并建立了MFC压电相的代表性体积元(representative volume element, RVE)模型, 提供了计算MFC均质化后的等效性能参数的方法, 在此基础上, Prasath等[19] 考虑了MFC的黏结层和电极层的结构建立了精细度更高的RVE模型, 文献[20-23] 则考虑了不同应用的应用场景, 对RVE的电边界条件进行修正, 提供了完备且精确的MFC等效性能参数.
其次, 需要构建MFC智能结构模型, 准确预测其静力或动力学行为. 黄丹丹等[24]开展了MFC-悬臂梁结构的压电有限元分析, 报道了MFC微观叉指电极的边界条件对仿真结果的影响. 李贇等[25]和刘宽等[26]分别使用温度比拟法和载荷比拟法作为替代来模拟MFC的力-电耦合行为, 与实验进行对比, 得到了不错的有限元结果, 为MFC有限元仿真研究提供了更多的思路. Pandey等[27]考虑了温度对MFC工作性能的影响, 开展了有限元和实验研究. Xu等[28]和刘宽等[29]针对MFC在结构静态变形领域的应用对其蠕变行为进行研究. 目前关于MFC与受控结构的宏观力学行为研究并未涉及或仅采用了相对简单的力-电关系, 相关应用的控制精度有限. 有必要深入研究MFC作动力与加载电压的直接关系, 以获取精确的作动力方程, 来提高MFC作动器的控制精度, 进而将其投入更精密的应用场景. 刘宽等[26]和Zhang等[30]分别推导了MFC作用于梁和平板结构的作动力方程, 并评估了一系列材料参数对MFC出力大小的影响, 但其并未深入考虑结构的边界条件, 从而忽略了结构相对尺寸因素的影响.
本文考虑MFC与结构的相对尺寸, 推导MFC作用于悬臂结构的作动力方程; 建立考虑MFC叉指电极边界条件的精细有限元模型开展仿真计算, 并在此基础上对传统的匀强电场模型进行修正; 开展了MFC-悬臂梁结构的静力实验对模型和公式进行验证, 以期为MFC在结构变形和振动抑制领域的应用提供参考.
1. MFC作用于悬臂结构的作动方程
1.1 MFC压电本构方程
压电材料的本构方程为
$$ {\boldsymbol{\varepsilon }} = {\boldsymbol{S\sigma }} + {\boldsymbol{d}}{{\boldsymbol{E}}_{ef}} $$ (1) 式中, ε和σ为应变和应力向量, S为柔度矩阵, d为应变压电系数矩阵, Eef为电场强度向量.
本文的研究对象为Smart Material公司生产的纤维方向与其长度方向一致的MFC, 包括P1型(d33模式)和P2型(d31模式) 2种, 通常约定3方向为MFC极化方向, 2方向为MFC平面的横向, MFC的微观结构及相应的坐标系如图1所示. 其中, P1型MFC额定的工作电压范围为−500 ~ 1500 V, 易产生较大变形, 常作为作动器使用, 因此后续推导以P1型MFC为对象.
MFC为正交各向异性材料, 通电极化后, 应变压电系数除d31, d32, d33, d24, d15 5个分量外均等于0, 且有d31 = d32, d24 = d15, 则式(1)展开成矩阵形式为
$$ \begin{split} & \left\{ {\begin{array}{*{20}{l}} {{\varepsilon _1}} \\ {{\varepsilon _2}} \\ {{\varepsilon _3}} \\ {{\varepsilon _4}} \\ {{\varepsilon _5}} \\ {{\varepsilon _6}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{l}} {{s_{11}}}&{{s_{12}}}&{{s_{13}}}&0&0&0 \\ {{s_{21}}}&{{s_{22}}}&{{s_{23}}}&0&0&0 \\ {{s_{31}}}&{{s_{32}}}&{{s_{33}}}&0&0&0 \\ 0&0&0&{{s_{44}}}&0&0 \\ 0&0&0&0&{{s_{55}}}&0 \\ 0&0&0&0&0&{{s_{66}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{l}} {{\sigma _1}} \\ {{\sigma _2}} \\ {{\sigma _3}} \\ {{\sigma _4}} \\ {{\sigma _5}} \\ {{\sigma _6}} \end{array}} \right\} + \\ &\qquad \left[ {\begin{array}{*{20}{l}} 0&0&{{d_{31}}} \\ 0&0&{{d_{32}}} \\ 0&0&{{d_{33}}} \\ 0&{{d_{24}}}&0 \\ {{d_{15}}}&0&0 \\ 0&0&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{E_{ef1}}} \\ {{E_{ef2}}} \\ {{E_{ef3}}} \end{array}} \right\}\\[-1pt]\end{split}$$ (2) 式中, sij (i, j = 1, 2, ···, 6)为柔度矩阵分量, dki (i = 1, 2, ···, 6; k = 1, 2, 3)为应变压电系数分量, Eefk (k = 1, 2, 3)为电场强度分量. εi和σj为应变及应力分量, 部分简写下标对应关系为{ε4 ε5 ε6}T = {γ23 γ31 γ12} T/2, {σ4 σ5 σ6}T = {τ23 τ31 τ12} T.
对于P1型MFC, 其垂直于平面方向(1方向)的电极具有相同的电势, 正负极沿长度方向(3方向)交错排列, 称为叉指电极, 产生的弯曲电场见图2(a). 由于压电纤维极化后仅受到3方向的电场会产生明显作用, 为了便于计算, 研究中通常将弯曲电场简化修正为沿3方向的匀强电场[31-32], 则电场Eef各分量与电压U的关系可表示为
$$ {E_{ef1}} = {E_{ef2}} = 0,{E_{ef3}} = \frac{{\eta U}}{t} $$ (3) 式中, t为一对相邻正负电极对之间的距离, 对于P1型MFC, t = 0.5 mm; η为修正系数, 绝大多数研究中η = 1, 此时弯曲电场简化为图2(b)所示平行极板产生的匀强电场.
由于MFC厚度远小于平面尺寸, 采用平面应力假设, 并忽略剪切变形, 式(3)代入式(2)得MFC应力σ为
$$ \left\{ {\begin{array}{*{20}{c}} {{\sigma _2}} \\ {{\sigma _3}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {{c_{22}}}&{{c_{23}}} \\ {{c_{32}}}&{{c_{33}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _2}} \\ {{\varepsilon _3}} \end{array}} \right\} - \left[ {\begin{array}{*{20}{c}} {{e_{32}}} \\ {{e_{33}}} \end{array}} \right]\left\{ {\frac{{\eta U}}{t}} \right\} $$ (4) 式中cij (i, j = 2, 3)为MFC刚度矩阵分量
$$ \begin{split} & {c_{22}} = \frac{{{E_2}}}{{1 - {\nu _{32}}{\nu _{23}}}},{\text{ }}{c_{23}} = \frac{{{\nu _{32}}{E_2}}}{{1 - {\nu _{32}}{\nu _{23}}}} \\ & {c_{32}} = \frac{{{\nu _{23}}{E_3}}}{{1 - {\nu _{32}}{\nu _{23}}}},{\text{ }}{c_{33}} = \frac{{{E_3}}}{{1 - {\nu _{32}}{\nu _{23}}}} \end{split} $$ eij (i, j = 2, 3)为MFC应力压电系数分量
$$ \left[ {\begin{array}{*{20}{c}} {{e_{32}}} \\ {{e_{33}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{c_{22}}}&{{c_{23}}} \\ {{c_{32}}}&{{c_{33}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{d_{32}}} \\ {{d_{33}}} \end{array}} \right] $$ 1.2 MFC作用于悬臂结构的作动方程
MFC-平板结构示意图如图3所示. 其中MFC长、宽和高分别为a, b和h, 定义板的长、宽和高与MFC一致, 记为A, B和H (h $\ll $ H). 取平板垂直于3方向的一侧表面作为悬臂边界的固定端, 假定受控板为各向同性材料, 则板的应力σbe为
$$ \left\{ {\begin{array}{*{20}{c}} {{\sigma _{be2}}} \\ {{\sigma _{be3}}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {c'_{22}}&{c'_{23}} \\ {c'_{32}}&{c'_{33}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _2}} \\ {{\varepsilon _3}} \end{array}} \right\} $$ (5) 式中cʹij (i, j = 2, 3)为板的刚度矩阵分量
$$ c'_{22} = c'_{33} = \frac{{{E_{be}}}}{{1 - {\nu _{be}}^2}},{\text{ }}c'_{23} = c'_{32} = \frac{{{\nu _{be}}{E_{be}}}}{{1 - {\nu _{be}}^2}} $$ 假定结构整体变形符合平截面假设, 近似认为结构的中面位于平板H/2处, 进而MFC与平板的应变ε沿厚度方向可表示为
$$ {\varepsilon _2}(x) = {\varepsilon _{02}} + {k_2}x,{\varepsilon _3}(x) = {\varepsilon _{03}} + {k_3}x $$ (6) 式中, ε0i (i = 2, 3)为结构中面的应变, ki为斜率.
将式(6)代入式(4)和式(5), 则结构各部分应力可表示为
$$ \begin{split} &\left\{ {\begin{array}{*{20}{c}} {{\sigma _2}} \\ {{\sigma _3}} \\ {{\sigma _{be{\text{2}}}}} \\ {{\sigma _{be3}}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {{c_{22}}}&{x{c_{22}}}&{{c_{23}}}&{x{c_{23}}} \\ {{c_{32}}}&{x{c_{32}}}&{{c_{33}}}&{x{c_{33}}} \\ {c'_{22}}&{xc'_{22}}&{c'_{23}}&{xc'_{23}} \\ {c'_{32}}&{xc'_{32}}&{c'_{33}}&{xc'_{33}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _{02}}} \\ {{k_2}} \\ {{\varepsilon _{03}}} \\ {{k_3}} \end{array}} \right\} -\\ &\qquad \left[ {\begin{array}{*{20}{c}} {{e_{32}}} \\ {{e_{33}}} \\ 0 \\ 0 \end{array}} \right]\left\{ {\frac{{\eta U}}{t}} \right\}\end{split} $$ (7) 对于悬臂结构, 截面的内力和弯矩平衡满足以下方程
$$ \iint_{{S_{bh}}} {{\sigma _3}{\text{d}}x{\text{d}}y} + \iint_{{S_{BH}}} {{\sigma _{be3}}{\text{d}}x{\text{d}}y} = 0 $$ (8) $$ \iint_{{S_{ah}}} {{\sigma _2}{\text{d}}x{\text{d}}z} + \iint_{{S_{AH}}} {{\sigma _{be2}}{\text{d}}x{\text{d}}z} = 0 $$ (9) $$ \iint_{{S_{bh}}} {{\sigma _3}x{\text{d}}x{\text{d}}y} + \iint_{{S_{BH}}} {{\sigma _{be3}}x{\text{d}}x{\text{d}}y} = 0 $$ (10) $$ \iint_{{S_{ah}}} {{\sigma _2}x{\text{d}}x{\text{d}}z} + \iint_{{S_{AH}}} {{\sigma _{be2}}x{\text{d}}x{\text{d}}z} = 0 $$ (11) 式中, Sah和Sbh为MFC截面, SAH和SBH为板截面.
本文假定, 受控结构的截面在MFC作用区域外也存在应力, 由于研究对象为悬臂梁或悬臂板, 认为截面同一高度位置应力相等, 记平板与MFC的长、宽及刚度的比值为: Ra = A/a, Rb = B/b, Re = Ebe/E3. 式(8) ~ 式(11)沿截面宽度积分后化简为
$$ \int_{\frac{H}{2}}^{\frac{H}{2} + h} {{\sigma _3}{\text{d}}x} + {R_b}\int_{ - \frac{H}{2}}^{\frac{H}{2}} {{\sigma _{be3}}{\text{d}}x} = 0 $$ (12) $$ \int_{\frac{H}{2}}^{\frac{H}{2} + h} {{\sigma _2}{\text{d}}x} + {R_a}\int_{ - \frac{H}{2}}^{\frac{H}{2}} {{\sigma _{be2}}{\text{d}}x} = 0 $$ (13) $$ \int_{\frac{H}{2}}^{\frac{H}{2} + h} {{\sigma _3}x{\text{d}}x} + {R_b}\int_{ - \frac{H}{2}}^{\frac{H}{2}} {{\sigma _{be3}}x{\text{d}}x} = 0 $$ (14) $$ \int_{\frac{H}{2}}^{\frac{H}{2} + h} {{\sigma _2}x{\text{d}}x} + {R_a}\int_{ - \frac{H}{2}}^{\frac{H}{2}} {{\sigma _{be2}}x{\text{d}}x} = 0 $$ (15) 将式(7)代入平衡方程式(12) ~ 式(15), 求解ε0i和ki (i = 2, 3)并代回至式(7), 则MFC沿平面长度方向的正应力σ3可由加载电压U和MFC与受控板的材料参数及尺寸参数表示. 除电压U为控制量外, 其余参数均已知, 进而, σ3沿MFC截面上积分即得到MFC沿平面长度方向的作动力F(U)
$$ F = - b\int_{\frac{H}{2}}^{\frac{H}{2} + h} {{\sigma _3}{\text{d}}x} $$ (16) 2. MFC叉指电极的简化修正
如前文所述, 包括本文在内的许多研究将叉指电极的弯曲电场近似为平行极板产生的匀强电场, 然而该近似方法精度有限, 黄丹丹等[24]的研究考虑MFC叉指电极结构开展有限元仿真, 并对MFC压电系数加以修正. 本文认为对压电系数进行修正是不恰当的, 叉指电极结构产生的弯曲电场, 其沿3方向(长度方向)的分量应当小于同电压下平行极板产生的匀强电场, 即修正系数η < 1. 因此本文建立了反映叉指电极微观构造的有限元模型开展压电静力仿真进行深入计算, 获取精确的修正系数η, 同时开展实验进行验证.
2.1 考虑MFC叉指电极边界的压电有限元分析
研究中使用的MFC型号为M-8514-P1型, 厚度为0.3 mm, 封装尺寸为101 mm × 20 mm, 作动尺寸(压电层)为85 mm × 14 mm. 为了后续实验MFC能产生足以观测的变形, 受控结构选用细长悬臂梁, 尺寸为450 mm × 20 mm × 2 mm. 本文采用MFC作动尺寸建立均质化模型, MFC黏贴于梁的中部, 二者长度方向一致, 示意图见图4.
梁的材质为铝, 测得其密度ρbe = 2462.2 kg/m3, 弹性模量Ebe = 59.3 GPa, 泊松比νbe = 0.3, MFC均质化后的等效参数如表1所示.
Definition Symbol Value engineering parameters ρ/(kg·m−3) 5440 E3/GPa 30.336 E2/GPa 15.857 ν32 0.31 G32/GPa 5.515 electric parameters d33/(pC·N−1) 460 d32/(pC·N−1) −210 permittivity coefficient εr33/(F·m−1) 1.45 × 10−8 微观尺度下MFC叉指电极长度为0.1 mm, 相邻叉指电极距离(中心间距)为0.5 mm[24]. 本文在有限元模型中的处理如下: 在MFC均匀化模型中的上下表面(垂直于1方向), 沿3方向(长度方向)在首尾两端设置长度为0.05 mm的电势表面, 内部设置空隙为0.4 mm, 长度为0.1 mm的电势表面. 正对的表面电势相同, 沿3方向交替设置正负电极的电势. 由于叉指电极的电场是反周期的, 为了正确模拟MFC的力学行为, 本文参考文献[34]的做法, 将相应区域MFC的压电系数设置为相反数, 上述电边界条件及材料设置完成后效果见图5(a).
通电后弯曲电场沿3方向的分量见图5(b). 图中可以看出该电场分量沿2方向的分布是均匀的, 因此本文通过加密MFC沿1和3方向划分的网格数开展网格敏感性研究. 表2列举了500 V电压负载下不同网格密度对有限元仿真结果的收敛性. MFC胞元(电场的一个反周期区域)沿1和3划分的网格数分别记为M1和M3; 梁上距自由端5 mm处的静挠度记为x, 修正系数η等于胞元内的平均电场$\bar E_{ef3} $与等电压下平行极板电场Eef3的比值.
表 2 500 V电压负载下不同网格密度的有限元仿真结果Table 2. Finite element simulation results of different mesh densities while U = 500 VModel M1 × M3 x/mm η 1 5 × 10 0.870 0.741 2 10 × 10 0.857 0.723 3 10 × 20 0.845 0.719 4 10 × 40 0.841 0.716 5 10 × 80 0.840 0.716 6 5 × 40 0.859 0.732 7 15 × 40 0.835 0.712 8 20 × 40 0.833 0.710 9 25 × 40 0.831 0.709 10 30 × 40 0.830 0.708 11 35 × 40 0.830 0.708 由表2可见, 1和3方向网格密度分别为30与40时, 仿真结果收敛, 因此后续数值仿真中均采用Model 10之情形. 需要说明的是, 本文所计算的修正系数是建立在结构发生小变形的基础上, 如果受控结构比较软, 通电后产生的变形大, 叉指电极间距离会发生显著变化, 会导致不同电压下的修正系数不是一个常数. 本文假定MFC通电后受控结构变形满足小变形假设, 将仿真所得平均电场与极板电场比值作为修正系数, 将叉指电极的弯曲电场简化修正为匀强电场.
2.2 实验验证
实验装置见图6, 由3个部分组成, 第1部分为MFC控制系统, 包括上位机、数据采集卡a和电压放大器, 上位机编程控制采集卡AO模块输出信号电压, 再由电压放大器产生控制电压, 用于驱动MFC; 第2部分为待测MFC-悬臂梁结构, 悬臂梁全长490 mm, 夹持长度为40 mm, 其余参数见上节, 测点选为距悬臂梁自由端5 mm处的中点; 第3部分为测量系统, 包括上位机、采集卡b和激光位移传感器, 激光位移传感器采集测点的挠度变化并转换成电信号, 借由采集卡AI模块储存至上位机.
施加400 ~ 1400 V的阶梯型电压, 阶梯步长为100 V. 每个阶梯包括加载和卸载两个阶段, 各持续30 s, 图7为测点的时间位移历程. 可以看出, 电压改变后测点的位置变化包括三个部分: (1) MFC受到不同电压负载产生弯矩引起的静挠度变化; (2) 电压突变引起的自由阻尼振动; (3) MFC的蠕变[28]. 其中, 自由阻尼振动不会导致静平衡位置的改变, 在此仅讨论弯矩引起的静挠度变化和蠕变. 一些研究中[24, 26]假定MFC的使用场景为变形控制等时间尺度较长的场景, 将蠕变稳定后的平衡位置作为通电后的平衡位置. 然而, 当MFC应用于振动控制等场景时, 电压会发生连续瞬时的变化. 此时蠕变等于各个电压值瞬时影响对时间的积分. 因此在测量弯矩引起静挠度时, 应当将蠕变排除. 本文将每次改变电压负载后, 测点在新的平衡位置发生自由阻尼振动时, 经过的第一个平衡点作为静平衡位置(图中已标出). 加载后的平衡位置(圆形标记), 蠕变作用的时间较短, 认为仅包含弯矩引起的静挠度改变; 电压卸载后由弯矩引起的静挠度消失, 而之前加载30 s累计产生的蠕变不会立即消失, 因此卸载后的平衡位置(三角形标记)偏离0点, 偏离量为相应的加载阶段产生的蠕变.
2.3 静挠度结果对比
表3对比了能够反映叉指电极结构的精细有限元仿真、用本文计算所得系数η施加相应大小的匀强电场的简化仿真和实验测量3种方法得到的测点的静挠度结果. 精细仿真与简化仿真结果分别记为FEM1和FEM2, 表中静挠度单位为mm. 可以看出, 2种仿真结果与实验测量的结果均吻合较好, 证明了有限元模型的可靠性, 该修正系数η提高了将叉指电极的弯曲电场近似为匀强电场时的精度.
表 3 不同电压下测点静挠度实验与仿真结果对比Table 3. Comparison of experimental and FE deflection results of the measuring point under different voltagesVoltage/V Experiment FEM1/FEM2 Error/% 400 0.663 0.664 0.15 0.667 0.61 500 0.848 0.830 −2.10 0.834 −1.51 600 1.015 0.996 −1.84 1.001 −1.38 700 1.194 1.162 −2.68 1.167 −2.26 800 1.376 1.328 −3.49 1.334 −2.88 900 1.555 1.494 −3.92 1.501 −3.47 1000 1.730 1.660 −4.05 1.668 −3.58 1100 1.904 1.826 −4.10 1.834 −3.68 1200 2.073 1.992 −3.91 2.001 −3.47 1300 2.236 2.158 3.49 2.168 −3.04 1400 2.396 2.324 −3.01 2.335 −2.55 3. 作动力公式的验证分析
图8为加载100 V电压时本文公式作动力F与Rb和Re的关系, 保持MFC的尺寸参数和材料参数不变, 通过调节梁的宽度和弹性模量得到Rb和Re, 其余参数沿用前文. 从图中可以看出, 同一MFC作用于宽度比Rb不同的结构时的作动力不同, 且对于刚度越小的材料变化越大.
为了验证该作动力公式, 改变有限元简化模型中梁的宽度与弹性模量开展仿真, 提取MFC截面内力进行计算, 与本文公式结果(仿真与公式均采用前文系数η对电压进行修正)进行对比, 汇总至表4, 作动力单位为N . 当Rb = 1时本文公式退化成文献[30]所述模型, 此时文献模型能得到精度较好的结果, Rb增大后文献公式开始出现误差, 本文误差则始终保持在1%以内, 表明本文所得MFC作用于悬臂结构的作动力预测公式在较大的宽度比范围内均具有较好精度.
表 4 不同Rb和Re下MFC的作动力理论计算及仿真结果对比(U = 100 V)Table 4. The comparison of the actuation of MFC between the theoretical and finite element results for different value of Re and Rb (U = 100 V)Re Rb Theoretical FEM Error/% 2 1 5.889 5.906 −0.29 2 6.813 6.821 −0.12 3 7.190 7.217 −0.38 4 7.394 7.423 −0.40 4 1 6.825 6.795 0.44 2 7.407 7.408 −0.02 3 7.624 7.645 −0.28 4 7.737 7.762 −0.32 6 1 7.207 7.178 0.36 2 7.629 7.633 −0.07 3 7.780 7.802 −0.28 4 7.858 7.883 −0.31 8 1 7.415 7.391 0.31 2 7.745 7.751 −0.12 3 7.861 7.884 −0.29 4 7.821 7.946 −0.32 4. 结论
本文推导了MFC作用于悬臂结构的考虑了MFC与结构宽度比的作动方程, 建立考虑MFC叉指电极边界条件的有限元模型进行仿真计算, 开展MFC-悬臂梁结构的静力实验, 通过分析得到以下结论.
(1) 对MFC叉指电极的弯曲电场进行研究, 发现MFC黏贴于结构时, 通电后其变形满足小变形假设, 忽略不同电压下叉指电极的间距变化, 因此可以将叉指电极弯曲电场修正为电场强度较小的匀强电场, 修正系数η = 0.708, 本文认为可以将该系数作为建议值推广至相关研究.
(2) 考虑MFC与受控结构的相对尺寸, 推导了MFC作用于悬臂结构的作动力方程, 与有限元仿真结果进行对比, 误差在1%以内, 本文所得MFC作用于悬臂结构的作动力预测公式在较大的宽度比范围内均具有较好精度.
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Definition Symbol Value engineering parameters ρ/(kg·m−3) 5440 E3/GPa 30.336 E2/GPa 15.857 ν32 0.31 G32/GPa 5.515 electric parameters d33/(pC·N−1) 460 d32/(pC·N−1) −210 permittivity coefficient εr33/(F·m−1) 1.45 × 10−8 
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表 2 500 V电压负载下不同网格密度的有限元仿真结果
Table 2 Finite element simulation results of different mesh densities while U = 500 V
Model M1 × M3 x/mm η 1 5 × 10 0.870 0.741 2 10 × 10 0.857 0.723 3 10 × 20 0.845 0.719 4 10 × 40 0.841 0.716 5 10 × 80 0.840 0.716 6 5 × 40 0.859 0.732 7 15 × 40 0.835 0.712 8 20 × 40 0.833 0.710 9 25 × 40 0.831 0.709 10 30 × 40 0.830 0.708 11 35 × 40 0.830 0.708
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表 3 不同电压下测点静挠度实验与仿真结果对比
Table 3 Comparison of experimental and FE deflection results of the measuring point under different voltages
Voltage/V Experiment FEM1/FEM2 Error/% 400 0.663 0.664 0.15 0.667 0.61 500 0.848 0.830 −2.10 0.834 −1.51 600 1.015 0.996 −1.84 1.001 −1.38 700 1.194 1.162 −2.68 1.167 −2.26 800 1.376 1.328 −3.49 1.334 −2.88 900 1.555 1.494 −3.92 1.501 −3.47 1000 1.730 1.660 −4.05 1.668 −3.58 1100 1.904 1.826 −4.10 1.834 −3.68 1200 2.073 1.992 −3.91 2.001 −3.47 1300 2.236 2.158 3.49 2.168 −3.04 1400 2.396 2.324 −3.01 2.335 −2.55
下载: 导出CSV
表 4 不同Rb和Re下MFC的作动力理论计算及仿真结果对比(U = 100 V)
Table 4 The comparison of the actuation of MFC between the theoretical and finite element results for different value of Re and Rb (U = 100 V)
Re Rb Theoretical FEM Error/% 2 1 5.889 5.906 −0.29 2 6.813 6.821 −0.12 3 7.190 7.217 −0.38 4 7.394 7.423 −0.40 4 1 6.825 6.795 0.44 2 7.407 7.408 −0.02 3 7.624 7.645 −0.28 4 7.737 7.762 −0.32 6 1 7.207 7.178 0.36 2 7.629 7.633 −0.07 3 7.780 7.802 −0.28 4 7.858 7.883 −0.31 8 1 7.415 7.391 0.31 2 7.745 7.751 −0.12 3 7.861 7.884 −0.29 4 7.821 7.946 −0.32
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