胡骏, 亢战^,2)大连理工大学工业装备结构分析国家重点实验室,大连 116024

TOPOLOGY OPTIMIZATION OF PIEZOELECTRIC ACTUATOR CONSIDERING CONTROLLABILITY ¹⁾

Hu Jun, Kang Zhan^,2)State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

通讯作者: 2) 亢战, 教授, 主要研究方向: 结构拓扑优化. E-mail:kangzhan@dlut.edu.cn

收稿日期:2019-01-7接受日期:2019-02-25网络出版日期:2019-07-18

基金资助:

1) 国家****科学基金项目.11425207 国家自然科学基金辽宁联合基金重点项目.U1508209

Received:2019-01-7Accepted:2019-02-25Online:2019-07-18
作者简介 About authors


摘要
压电作动器可以把电能转换成机械能,在结构主动振动控制中具有应用背景. 由于压电作动器的布局对振动控制效果影响很大,因此作动器布局优化一直是结构控制研究的关键之一. 为了提高压电结构控制能量的利用效率,本文提出了以提高结构可控性为目标的压电作动器的拓扑优化方法. 基于经典层合板理论对压电结构进行了有限元建模,并采用模态叠加法将动力控制方程映射到模态空间,推导了基于控制矩阵奇异值的可控性指标. 优化模型中,选取可控性指标指数形式为目标函数,将设计变量定义为作动器单元的相对密度,并基于人工密度惩罚模型构造了压电系数惩罚模型,给出了基于控制矩阵奇异值的可控性指标关于设计变量的灵敏度分析方法. 优化问题采用基于梯度的数学规划法求解. 数值算例验证了灵敏度分析方法和优化模型的有效性,并讨论了主要因素对优化结果的影响.
关键词： 拓扑优化;主动控制;压电作动器;可控性;奇异值

Abstract
Piezoelectric actuators can convert electrical energy into mechanical energy, and has application potential in active vibration control of structures. Since the layout of the piezoelectric actuators has a great influence on the vibration control effect, the optimization of the actuators has always been one of the key factors to structural control. In order to improve the efficiency of control energy in the piezoelectric structure, this paper proposes a topology optimization method for the layout design of piezoelectric actuators with the goal of improving structural controllability. The finite element modeling of the piezoelectric structure is carried out based on the classical laminate theory. The modal superposition method is used to map the dynamic governing equation to the modal space. The controllability index based on the singular value of the control matrix is derived. In the optimization model, the exponential form of the controllability index is chosen as the objective function, and the design variables are the relative densities of the actuator elements. Based on the Solid Isotropic Material Penalization method, an artificial piezoelectric coefficient penalty model is constructed. Sensitivity analysis for the controllability index is proposed based on the singular value of the control matrix. The optimization problem is solved by a gradient-based mathematical programming method. Numerical examples verify the effectiveness of the sensitivity analysis method and the optimization model and show the significance of the layout design of piezoelectric actuators. The influence of some key factors on the optimization results are discussed. It shows that the more piezoelectric materials, the better the controllability; the modes of interest in the objective function has a great influence on the layout of the piezoelectric actuators.
Keywords：topology optimization;active control;piezoelectric actuator;controllability;singular value


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本文引用格式
胡骏, 亢战. 考虑可控性的压电作动器拓扑优化设计 ¹⁾. 力学学报[J], 2019, 51(4): 1073-1081 DOI:10.6052/0459-1879-19-012
Hu Jun, Kang Zhan. TOPOLOGY OPTIMIZATION OF PIEZOELECTRIC ACTUATOR CONSIDERING CONTROLLABILITY ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(4): 1073-1081 DOI:10.6052/0459-1879-19-012

引言

压电材料因具有能量转换率高、响应快、性能稳定等优点,被广泛应用在结构的主动控制中. 由于压电传感器/作动器位置的 分布会直接影响振动控制效果,因此压电传感器/作动器布局优化的研究一直备受****们的关注^[1-4]. 然而,以往关于压电材料布局优化的研究大多基于启发式或智能算法,如遗传算法^[5-6]、粒子群算法^[7]等. 这些方法可以搜索全设计域,但计算量大,通常只能处理较少数量的设计变量,限制了这些方法在复杂工程设计中的应用.

拓扑优化是一种重要的结构优化设计方法,在过去的三十年,发展了均匀化法^[8]、变密度法^[9]、水平 集法^[10-11]、进化结构优化法^[12]和独立连续映射(ICM)法^[13-14]等. 特别是,不少国内外****研究了动力学拓扑优化问题^[15-17]. Du和Olhoff^[18]考虑了结构的频率优化问题. 彭细荣等^[13]推导了频响分析下位移幅值灵敏度的伴随法. Rong等^[19]采用序列二次规划优化了随机激励下结构的拓扑构型. 近几年,Vicente等^[20]优化了宏微观两种尺度下材料的布局以降低结构的响应. 朱继宏等^[21]考虑了谐波基加速度激励下的动力拓扑优化问题.

拓扑优化方法在压电智能结构中的应用也广受关注^[22-24]. Kang等^[25]同时考虑了压电层合板结构拓扑和控制 电压的优化. Wang等^[26]研究了具有嵌入式可移动作动器的柔性智能结构的拓扑优化问题. Yang等^[27]提出了以电压大小,压电作动器位置和基体板材料布局为设计变量结合精确变形控制的协同优化方法. Yoon等^[28]采用拓扑优化方法设计了声能聚焦器并提出一种改进的形态密度过滤函数使优化结果易于制备. 然而,考虑压电作动器主动控制的拓扑优化研究尚不多见.

Wang等^[29]基于控制矩阵的奇异值提出了可控性指标(controllability index, CI),并计算了作动器位置对指标的影响. 该可控性指标可以衡量在一定电能输入下,压电作动器输出能量的大小^[30]. Bruant 等^[31]考虑控制溢出对CI进行了修改,并以此为目标函数设计了压电作动器和传感器的布局. Dhuri等^[32]采用遗传算法研究了最大化CI和最小化结构固有频率变动的多目标优化问题.

可控性指标的大小和控制算法以及外激励形式没有关系^[1],而以往的压电结构拓扑优化大多基于预设的控制算法和外 激励载荷^[33-34],考虑可控性的拓扑优化研究可为不确定外激励形式或控制算法的压电作动器优化布局提供 一种解决方案. 但是,基于可控矩阵奇异值可控性的压电作动器的拓扑优化研究尚属空白.

本文提出了考虑可控性的压电作动器优化设计方法. 建立了基于经典层合板理论的压电智能结构的有限元数值分析模型,推导了 基于控制矩阵奇异值的可控性指标. 优化模型中,以作动器单元的相对密度为设计变量并构建了中间密度单元压电系数的惩罚模型;给出了可控性指标关于设计变量 的灵敏度分析方法,并采用基于梯度的数学计划方法实现对压电作动器的优化设计.

1 压电智能结构有限元动力模型

如图1所示为压电层合板结构. 在外激励和压电控制力作用下,压电结构的振动方程为^[33]

(1)$$ {\pmb M}\ddot {\pmb x}(t) + {\pmb C}\dot{\pmb x} (t) + {\pmb K}{\pmb x} (t) = {\pmb f}_{a} (t) + {\pmb f} (t) $$
式中,${\pmb M} \in {\pmb R}^{n\times n}$,${\pmb C} \in {\pmb R}^{n\times n}$ 和 ${\pmb K} \in {\pmb R}^{n\times n}$分 别为结构的质量矩阵、阻尼矩阵和刚度矩阵;$\ddot {\pmb x} (t) \in {\pmb R}^{n\times 1}$,$\dot{\pmb x} (t) \in {\pmb R}^{n\times 1}$和${\pmb x} (t) \in {\pmb R}^{n\times 1}$分别为加速度、速度和位移向量;${\pmb f} (t) \in {\pmb R}^{n\times 1}$和${\pmb f}_{a} (t) \in {\pmb R}^{n\times 1}$分别为外激励向量和主动控制力向量;$n$是结构总自由度数目.

图1

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图1压电层合板示意图

Fig. 1Schematic diagram for piezoelectric laminated plate



由于压电层合板结构包含基体层和压电作动器层,因此结构整体的刚度阵和质量阵可进一步表达为${\pmb M} = {\pmb M}_{h} + {\pmb M}_{a} $和${\pmb K} = {\pmb K}_{h} + {\pmb K}_{a} $,其中下标h和a分别表示基体层和作动器层.

直接求解方程(1)的计算量很大,而对于一般的工程结构,振动能量主要集中在低阶振动模态中,并且由于主动控制主要 应对低频振动,故引入常用的模态叠加法在保证精度的情况下降低计算量^[35]. 结构的位移可近似为

(2)$$ {\pmb x}(t) \approx {\pmb \varPsi} {\pmb \eta} (t) $$
式中,${\pmb \varPsi} = \left\{ { {\pmb\psi}_1 ,\;{\pmb\psi}_2 ,\; \cdots ,\;{\pmb\psi}_m } \right\}$和${\pmb\eta} = \left\{ {\eta _1 ,\;\eta _2 ,\; \cdots ,\;\eta _m } \right\}^{ T}$分别为对应的无阻尼系统的前$m$阶归一化模态矩阵和模态位移向量. 模态矩阵${\pmb \varPsi} $可通过求解如下的特征值问题获得

(3)$$ \left( { {\pmb K}-\omega _i^2 {\pmb M}} \right){\pmb\psi}_i = {\bf 0} $$
式中,$i = 1, 2, \cdots , m$;$\omega _i $为特征向量${\pmb\psi}_i $对应的圆频率.

对于低频振动为主的问题,也可采用模态加速法^[36-38]在一定程度上提高响应分析的精度.

假设结构的整体阻尼采用比例阻尼模型描述,则有

(4)$$ \tilde{\pmb C \ } = {\pmb \varPsi }^{T}{\pmb C}{\pmb \varPsi }= 2{\pmb Z}_{d} {\pmb \varOmega}_{ f} $$
式中,$\tilde{\pmb C \ } \in {\pmb R}^{m\times m}$为解耦的阻尼矩阵;${\pmb Z}_{d} \in {\pmb R}^{m\times m}$ 为阻尼系数组成的对角阵;${\pmb \varOmega}_{f} $为圆频率组成的对角矩阵,即${\pmb \varOmega} _{f} ={diag}\left( {\omega _1 , \omega _2 , \cdots , \omega _m } \right)$.

将方程(2)代入方程(1)并左乘${\pmb \varPsi}^{T}$,可得

(5)$$ \ddot {\pmb \eta }(t) + 2{\pmb Z}_{d} {\pmb \varOmega}_{f} \dot {\pmb \eta }(t) + {\pmb \varOmega}_{f}^2 {\pmb \eta} (t) = \\ \qquad {\pmb \varPsi}^{T}{\pmb f}_{a} (t) + {\pmb \varPsi}^{T}{\pmb f}(t) $$
压电控制力由施加在压电片上的电压产生,并由逆压电效应决定,满足如下关系式

(6)$$ {\pmb f}_{a} (t) =-{\pmb K}_{{u}\varphi } {\pmb \varPhi}_{a} \left( t \right) $$
式中,${\pmb \varPhi}_{a} \left( t \right)$为施加在压电作动器上的主动控制电压向量;${\pmb K}_{{u}\varphi }$是压电作动器的力电耦合矩阵;${\pmb K}_{{u}\varphi } $可表达为${\pmb K}_{{u}\varphi }= \int_{\varOmega _{a} } {\pmb B}_{u}^{T} e^{T}{\pmb B}_\varphi d\varOmega $,其中${\pmb B}_{u} $,${\pmb e}$,${\pmb B}_\varphi $和$\varOmega _{a} $分别是应变位移矩阵、压电系数矩阵、电场电压矩阵和作动器的体积域.

因此方程(5)可改写成如下振动控制方程

(7)$$ \ddot {\pmb \eta }(t) + 2{\pmb Z}_{d} {\pmb \varOmega}_{f} \dot {\pmb \eta }(t) + {\pmb \varOmega}_{f}^2 {\pmb \eta} (t) = {\pmb B}{\pmb \varPhi}_{a} (t) + {\pmb \varPsi}^{T}{\pmb f}(t) $$
式中,${\pmb B}$为控制矩阵并有${\pmb B} =-{\pmb \varPsi}^{T}{\pmb K}_{{u}\varphi } $.

2 可控性指标和拓扑优化模型

2.1 基于控制矩阵奇异值的可控性指标

压电控制力的内积可写为

(8)$$ {\pmb f}_{a} ^{T}{\pmb f}_{a} = {\pmb \varPhi}_{a} ^{T}{\pmb B}^{T}{\pmb B}{\pmb \varPhi }_{a} $$
对控制矩阵${\pmb B}$进行奇异值分解有

(9)$$ {\pmb B} ={\pmb M}{\pmb S}{\pmb N}^{T} $$
式中,${\pmb M}$和${\pmb N}$分别是控制矩阵的左右奇异向量;有${\pmb M} \in {\pmb R}^{m\times m}$,${\pmb M}^{T}{\pmb M} ={\pmb I}_m $,${\pmb N} \in {\pmb R}^{k\times k}$,${\pmb N}^{T}{\pmb N} = {\pmb I}_k $和

(10)$$ {\pmb S} = \left[ \!\! \begin{array}{cccc} {\sigma _1 } & { 0} & { 0} & {\bf 0} \\ { 0} & \ddots & { 0} & {\bf 0} \\ { 0} & { 0} & {\sigma _m } & {\bf 0} \end{array} \!\! \right] \in {\pmb R}^{m\times k} $$
式中,$\sigma _i \;\left( {i = 1,\;2,\; \cdots ,\;m} \right)$为矩阵${\pmb S}$的奇异值;$k$为作动器的数量. 这里假设作动器的数量比减缩的模态数量多,即$k > m$.

方程(8)可改写为

(11)$$ {\pmb f}_{a} ^{T}{\pmb f}_{a} ={\pmb \varPhi} _{a} ^{T}{\pmb N} {\pmb S}^2{\pmb N}^{ T}{\pmb \varPhi} _{a} $$
引入${\pmb \varPhi} = {\pmb N}^{T}{\pmb \varPhi}_{a} $可得(观察可知${\pmb \varPhi}^2 = {\pmb \varPhi}_{a}^2$)

(12)$$ {\pmb f}_{a}^{T}{\pmb f}_{a} = {\pmb \varPhi}^{T}{\pmb S}^2{\pmb \varPhi} $$
从能量的角度看,${\pmb \varPhi}$和${\pmb \varPhi}_{a} $是等价的,因此有

(13)$$ {\pmb f}_{a}^{T}{\pmb f}_{a}= \sum_{i = 1}^m \sigma _i^2 \phi _i^2 $$
式中,$\sigma _i $为矩阵${\pmb S}$的第$i$个奇异值;$\phi _i $为向量${\pmb \varPhi} $的第$i$个分量.

观察方程(13)可知,可通过提高矩阵${\pmb S}$的奇异值来提高系统的控制能量. 因此选取可控性指标(CI)为^[29]

(14)$$ W_{CI} = \prod\limits_{i = 1}^m {\sigma _i } $$
需要说明的是,以上的公式推导考虑了所有前$m$阶模态的可控性;若只考虑其中几阶模态的可控性,公式将有细微的改变,此处省略.

2.2 拓扑优化列式

本文以提高压电智能结构的可控性为目标设计压电作动器的布局,选取压电作动器层为设计区域,并假设基体层不变. 以压电材料的总用量为约束,作动器层单元的相对密度为设计变量. 由于迭代过程中可控性指标(CI)的数值将发生很大变化,故采用其对数形式为目标并最小化其负数. 优化列式为

(15)$$ \left.\!\!\begin{array}{ll} {\mathop {\min }\limits_\rho \;} & {J\left( \rho \right) =-\lg \prod\limits_{i = 1}^m {\sigma _i } } \\ {s.t.}\;\;\; & {\left( {K-\omega _i^2 M} \right)\psi _i = 0} , \ \ i = 1, 2, \cdots , m \\ & {\sum_{e = 1}^{N_{e} } {\rho _e V_e-f_{v} \sum_{e = 1}^{N_{e} } {V_e } } \leqslant 0} \\ & 0 < \rho _{\min } \leqslant \rho _e \leqslant 1 , \ \ e = 1, 2, \cdots , N_{e} \end{array}\!\!\right\} $$
式中,$\rho _e $为第$e$个压电单元的相对密度;$V_e $为第$e$个压电单元的体积;$f_{v} $为体积分数;$N_{e} $为单元的总数;$\rho _{\min } = 10^{-6}$为设计变量的下限.

采用SIMP模型,第$e$个压电单元的质量阵和刚度阵可表示为

(16)$$ \left.\begin{array} {\pmb M}_{a}^e = \rho _e \tilde {\pmb M \ }_{a}^e \\ {\pmb K}_{a}^e = (\rho _e )^{p_1 }\tilde{\pmb K \ }_{a}^e \end{array}\right\} $$
式中,$\tilde {\pmb M \ }_{a}^e $和$\tilde {\pmb K \ }_{a}^e $为相对密度为1时压电单元的质量阵和刚度阵;$p_{1} = 3$为惩罚系数. 由于基层板在优化过程中不变,低密度压电单元处将不会导致局部模态现象^[15].

类似于SIMP模型,引入惩罚模型对中间密度值的压电单元的压电系数进行惩罚,则有^[25]

(17)$$ {\pmb e} = \rho _e^{p_2 } \tilde {\pmb e} $$
式中,$\tilde {\pmb e}$为相对密度为1时压电材料的压电系数矩阵;$p_{2}$为压电系数的惩罚系数,本研究中取$p_{2} =1$.

3 灵敏度分析

由于优化模型的求解基于梯度优化算法,因此需要对目标函数关于设计变量做灵敏度分析. 为了提高计算效率,采用伴随变量法^{[34, 38-39]}进行求解.

对方程(15)中的目标函数$J$关于设计变量$\rho _e $求导,可得

(18)$$ \begin{array} \dfrac{ dJ}{ d\rho _e } =-\dfrac{ d\lg \Big (\prod\limits_{i = 1}^m {\sigma _i }\Big) }{ d\rho _e } = \\ \qquad-\dfrac{1}{\prod\limits_{i = 1}^m {\sigma _i } }\left( {\sum_{i = 1}^m {\dfrac{\partial \sigma _i }{ d\rho _e }} \prod\limits_{j = 1}^m {\sigma _j } } \right) , \ \ j \ne i \end{array} $$
上式包含奇异值对设计变量的导数. 通过求解${\pmb W} = {\pmb B}{\pmb B}^{T}$矩阵的特征值$\lambda _i = \sigma _i^2 $的导数可得到奇异值$\sigma_i $对设计变量的导数. ${\pmb W}$矩阵的特征方程为

(19)$$ \left( {{\pmb W}-\lambda _i {\pmb I} } \right) {\pmb \varphi} _i = {\bf 0} $$
对方程(19)关于$\rho _e $求导,并左乘${\pmb \varphi}_i^{T} $可得

(20)$$ {\pmb \varphi}_i^{T} \left( {\dfrac{\partial {\pmb W}}{\partial \rho _e }-\dfrac{\partial \lambda _i }{\partial \rho _e }{\pmb I}} \right) {\pmb \varphi}_i + {\pmb \varphi}_i^{T} \left( {{\pmb W }- \lambda _i {\pmb I}} \right)\dfrac{\partial {\pmb \varphi}_i }{\partial \rho _e } = 0 $$
由于$\left( {{\pmb W}-\lambda _i {\pmb I}} \right)$是对称矩阵,因此方程(20)的第2项为0,可得

(21)$$ \dfrac{\partial \lambda _i }{\partial \rho _e } = {\pmb \varphi}_i^{T} \dfrac{\partial {\pmb W}}{\partial \rho _e }{\pmb \varphi}_i $$
对关系式$\sigma _i^ = \sqrt {\lambda _i } $求导,并引入方程(21),则有

(22)$$ \begin{array}\dfrac{\partial \sigma _i }{\partial \rho _e } = \dfrac{1}{2\sqrt {\lambda _i } }\dfrac{\partial \lambda _i }{\partial \rho _e } = \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \dfrac{\partial {\pmb W}}{\partial \rho _e }{\pmb \varphi} _i =\\ \qquad \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \dfrac{\partial \left( {{\pmb B}{\pmb B}^{T}} \right)}{\partial \rho _e }{\pmb \varphi}_i = \\ \qquad \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \left( {\dfrac{\partial B}{\partial \rho _e }{\pmb B}^{T} + {\pmb B}\dfrac{\partial {\pmb B}^{T}}{\partial \rho_e }} \right) {\pmb \varphi}_i \end{array} $$
其中

(23)$$ \begin{array} \dfrac{\partial {\pmb B}}{\partial \rho _e } = -{\pmb \varPsi} ^{T}\dfrac{\partial {\pmb K}_{{ u}\phi } }{\partial \rho _e } = \\ \qquad -{\pmb \varPsi}^{T}{\pmb G}^{T}p_2 \rho _e^{(p_2-1)} \int_{\varOmega _{a}^e } {{\pmb B}_{u}^{T} \tilde {e}^{T}{\pmb B}_\varphi } d\varOmega \cdot {\pmb G} \end{array} $$
式中,${\pmb G}$为自由度转换矩阵.

将方程(22)和(23)代入方程(18)就可得到目标函数对设计变量的敏度.

4 数值算例和讨论

4.1 灵敏度分析验证

算例1 如图2,左端固支悬臂板的长宽分别为$a = 1.6$ m,$b = 1.2 $ m. 下层基 体板厚$t_{h} = 2\times 10^{-2}{m}$,密度$\rho _{h} = 2 700$ kg/m$^{3}$,杨氏模量$E_{h} = 6.9\times 10^{10}$ N/m$^2$,泊松比$\nu _{h} = 0.3$. 上层压电材料板厚$t_{a} = 5\times 10^{-4}$ m,其材料属性见表1. 假设阻尼系数矩阵为${\pmb Z}_{d} = \zeta _{d} {\pmb I}$,其中$\zeta _{d} = 1.0\times 10^{ -4}$. 优化模型中,目标函数考虑第1到第5阶模态的可控性,并取压电材料的体积分数为$f_{v} = 0.5$.

图2

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图2左端固支悬臂板示意图

Fig. 2A cantilever plate with left end fixed



Table 1
表1
表1压电材料属性
Table 1Material properties of piezoelectric material

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悬臂板划分成$N_{e} = 432(24\times 18)$个四节点Mindlin壳单元. 初始设计中,所有作动器单元的相对密度设为0.5. 采用本文所提灵敏度分析方法得到的目标函数关于设计变量的灵敏度结果如图3所示. 作为对比,分别采用有限差分法(finite difference method, FDM)计算了扰动为0.02%,0.1%和0.5%的灵敏度,得到的结果基本相同,并取扰动为0.1%的结果显示在图3中.

图3

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图3本文所提灵敏度计算方法和有限差分法计算的 灵敏度结果对比

Fig. 3Comparison of sensitives obtained by the proposed adjoint method and finite difference method



由图3可知,两种计算方法结果基本一致,这说明了本文灵敏度计算方法的有效性. 另外,本文灵敏度分析方法计算耗时2.147 s,有限差分法耗时342.148 s,可见本文方法更高效.

4.2 矩形悬臂板优化算例

算例2 考虑如图2所示左端固支矩形板作动器优化问题,板的长宽分别为 $a = 1.8$ m,$b = 1.2$ m. 材料属性、厚度、阻尼系数和体积分数与算例1相同. 在目标函数中考虑第1到第5阶模态的可控性. 将矩形板划分为$N_{ e} = 2 400$ $(60\times 40)$个四节点Mindlin壳单元,并设作动器单元初始的相对密度为0.5(初始设计). 采用移动渐近线算法(MMA)求解优化问题^[40].

优化经过49次迭代收敛,得到如图4所示的优化结果,结果显示灰色单元基本没有出现,因此本研究采用的压电材料人工模型 表现出了一定的自惩罚性. 图5为相应的目标函数和体积分数的优化迭代历史曲线;目标函数从36.539降低到33.986,即优化设计的可控性相对于初始设 计的可控性提高了12.8倍. 表2为初始设计和优化设计的前五阶固有频率,可以发现最大的频率变化只有3.26%.

图4

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图4考虑1$\sim $5阶模态可控性的拓扑优化结果

Fig. 4Topology optimization result for the controllability considering first five modes

图5

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图5可控性指标和体积分数的迭代历史曲线

Fig. 5Iteration history for controllability index and volume fraction



Table 2
表2
表2初始设计和优化设计的前5阶频率
Table 21st$\sim $5th order frequencies of the initial design and the optimized design

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为了进一步验证优化结果的有效性,这部分计算了优化设计与两个参考设计(如图6)在频率为80 Hz(悬臂板结构第五阶固有频率 附近)的简谐外激励作用下等增益速度反馈(constant gain velocity feedback, CGVF)^[41]和线性二次型最优控制(linear quadratic regulator, LQR)^[33]的主动控制效果,取减缩阶数$m$为40 (由于结构的第40阶频率在820 Hz左右,因此认为 对于80 Hz载荷的响应计算精度可以接受). 外激励作用在悬臂板的右端中部,方向垂直于板面向下,振幅为1$\times $10$^{3 }$ N;CGVF的反馈系数为$G_{ a}G_{c} = 1 \times 10^{6}$ V/A;LQR的权重矩阵${\pmb R}$取$m$阶单位阵,${\pmb Q}$取$\left[ \!\! \begin{array}{cc} {\bf 0} & {\bf 0} \\ {\bf 0} & {1\times 10^8 {\pmb I}^{m\times m}} \end{array} \!\! \right]$. CGVF和LQR的具体实施方法见文献^{[33, 41]}. 选取结构的动柔度和控制电压向量幅值的内积^[33]作为衡量控制效果的指标,结果见表3. 由表3可知,同样条件 下优化设计的动柔度(dynamic compliance)和能量消耗(energy consumption)都更小,可见优化设计在该结构CGVF和LQR控制中对能量的使用效率更高.

图6

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图6两个参考设计

Fig. 6Two reference designs



Table 3
表3
表3优化设计,竖条纹参考设计与横条纹参考设计在CGVF和LQR控制下控制效果的对比
Table 3Comparison of the control performance under CGVF and LQR for the optimized design, reference designs with vertical stripes and horizontal stripes

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为了研究体积分数对优化结果的影响,现考虑体积分数分别为0.4和0.6两种情况下的优化问题,优化结果如图7. 优化结果对应的可控 性指标分别为34.332和33.724,相对于体积分数为0.5的优化结果的可控性分别降低了29.25%和增加了29.95%,可见增加材 料的用量,可以提高所研究压电结构的可控性.

图7

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图7体积分数为0.4和0.6下的拓扑优化结果

Fig. 7Optimization results with volume fraction of 0.4 and 0.6

图7

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图7体积分数为0.4和0.6下的拓扑优化结果(续)

Fig. 7Optimization results with volume fraction of 0.4 and 0.6 (continued)



算例3 采用和算例2相同的模型,分别在目标函数中考虑第1, 2, 3, 4和5阶模态的可控性,体积分数都取为0.5,优化结果如图8. 由图可知,优化结果的拓扑构型发生了明显改变,即目标函 数中所考虑的模态对优化结果影响很大.

图8

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图8考虑不同模态可控性的拓扑优化结果

Fig. 8Optimization results for the controllability considering difference mode

5 结论

本文研究了考虑可控性指标的压电作动器拓扑优化设计问题. 首先推导了基于控制矩阵奇异值的可控性指标,构建了以此指标为目标函数的压电层合板结构的优化模型. 优化模型引入了对中间密度单元压电系数的惩罚. 进而,提出了可控性关于设计变量的灵敏度分析方法. 和差分法的对比验证了灵敏度分析方法的有效性. 数值算例得到了清晰的拓扑构型,优化设计相对于参考设计控制效果更好. 算例还讨论了体积分数和所选控制模态对优化结果的影响.

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