Design of laminated piezocomposite shell transducers with arbitrary fiber orientation using topology optimization approach

Sensor and actuator based on laminated piezocomposite shells have shown increasing demand in the field of smart structures. The distribution of piezoelectric material within material layers affects the performance of these structures; therefore, its amount, shape, size, placement, and polarization should be simultaneously considered in an optimization problem. In addition, previous works suggest the concept of laminated piezocomposite structure that includes fiber‐reinforced composite layer can increase the performance of these piezoelectric transducers; however, the design optimization of these devices has not been fully explored yet. Thus, this work aims the development of a methodology using topology optimization techniques for static design of laminated piezocomposite shell structures by considering the optimization of piezoelectric material and polarization distributions together with the optimization of the fiber angle of the composite orthotropic layers, which is free to assume different values along the same composite layer. The finite element model is based on the laminated piezoelectric shell theory, using the degenerate three‐dimensional solid approach and first‐order shell theory kinematics that accounts for the transverse shear deformation and rotary inertia effects. The topology optimization formulation is implemented by combining the piezoelectric material with penalization and polarization model and the discrete material optimization, where the design variables describe the amount of piezoelectric material and polarization sign at each finite element, with the fiber angles, respectively. Three different objective functions are formulated for the design of actuators, sensors, and energy harvesters. Results of laminated piezocomposite shell transducers are presented to illustrate the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical and analytical method for the design of piezoelectric modal sensors/actuators for shell‐type structures

In this paper we treat the problem of designing distributed piezoelectric modal sensors/actuators for cylindrically curved panels. The design problem is tackled as an optimization problem where the design variable is a function (the polarization profile of the electrode) that takes on three values only: ?1 (negative polarization), 0 (zero polarization or no piezoelectric material), 1 (positive polarization), and the objective function is connected with the frequency response of the transducer. For the model described here, we analytically prove that the electrode patterns that make it possible to ideally isolate particular vibration modes must entirely cover the piezoelectric lamina with either positive or negative polarization. Further, we propose an accurate numerical method for systematically designing these polarization patterns and a novel algorithm for parameterizing and visualizing them in 3d. Copyright © 2009 John Wiley & Sons, Ltd.     

压电梁振动的多输入多输出主动控制

对表面上贴有多个用作驱动器和传感器的压电陶瓷片的“压电梁”结构，导出了从驱动器到传感器的频响函数公式，作为压电结构设计和振动控制的数学模型。提出了压电梁对缓变周期扰动振动环境的多输入多输出振动抑制方法。     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

空分复用压电自感知执行器执行与传感效果研究

结构振动主动控制可以采用压电自感知执行器。空分复用解耦方法是实现压电自感知执行器的一种新方法，实质是采用几何方法解耦，即将压电片的一个完整电极分割为执行区和传感区以实现自感知。本文以悬臂梁为对象，以涡流位移计作为标准传感器，对两种电极分割方式的压电片的传感和执行效果进行了实验研究。通过测量压电梁的频率特性，证明了空分复用的压电陶瓷片同时兼有传感和执行两种功能。实验结果也表明传感区的敏感输出受到执行区激励电压的静电耦合的影响，利用悬臂梁存在反谐振点的特性，提出了一种定量测量静电耦合的方法，并测定了不同电极宽度、不同极间隙下的静电耦合系数。本文的工作为采用空分复用的压电自感知执行器进行振动主动控制奠定了基础。     

振动鲁棒控制研究Ⅱ—仿真算例

以柔性悬臂梁为对象,采用压电作动器和传感器,基于 控制理论设计鲁棒控制器,对梁进行振动控制。详细给出了控制器设计过程,包括模型参数误差,模型不确定性加权函数的选择以及性能加权函数的设计。对闭环系统进行了全面的仿真分析,在时域和频域对比了开环和闭环的性能,考察了参数摄动时闭环的鲁棒稳定性。分析结果验证了 控制理论用于结构振动控制的有效性。     

基于模态应变能分布的压电致动器/传感器位置优化遗传算法

本文由线弹性压电结构有限元动力方程,推导了压电智能结构的振动控制方程。建立了准确模拟层合压电结构动力行为的有限元模型。基于主结构模态应变能分布提出了一种新的优化目标函数,将压电致动器/传感器位置编号作为优化变量,建立了离散变量表示的智能结构优化问题,并通过二进制编码的遗传算法（GA）求解了该最优问题。以四边固支复合层合压电智能板为数值算例,采用比例反馈控制, 研究了最优位置配置致动器/传感器智能结构目标模态的控制效果。数值结果表明基于模态应变能分布的遗传算法所得优化解具有较好的振动控制效果。     

Modeling and characterization of piezoelectrically actuated bistable composites

This paper develops and validates a finite-element model to predict both the cured shape and snap-through of asymmetric bistable laminates actuated by piezoelectric macro fiber composites attached to the laminate. To fully describe piezoelectric actuation, the three-dimensional compliance [s(ij)], piezoelectric [d(ij)], and relative permittivity [ε(ij)] matrices were formulated for the macro fiber actuator. The deflection of an actuated isotropic aluminum beam was then modeled and compared with experimental measurements to validate the data. The model was then extended to bistable laminates actuated using macro fiber composites. Model results were compared with experimental measurements of laminate profile (shape) and snap-through voltage. The modeling approach is an important intermediate step toward enabling design of shape-changing structures based on bistable laminates.     

Vibration control of a simply supported cylindrical shell using a laminated piezoelectric actuator

Summary This paper develops a novel laminated piezoelectric actuator (LPA) to control the vibration of a cylindrical shell structure, which is fabricated through bonding multiple piezoelectric layers of the same property together. The electromechanically coupled equations of the system are derived based on the classic shell theory. A parametric study is then conducted to evaluate the effects of geometric and physical properties of the actuator on actuating forces. The results show that as the number of layers increases, the actuating forces per voltage produced by LPA in the axial, circumferential and radial directions of the shell all increase noticeably. The active vibration control of a simply supported cylindrical shell using LPA of different layer numbers is simulated as well under a velocity feedback scheme. It is indicated that with the same control voltage the LPA can obtain a better control performance than the conventional single layer piezoelectric actuator as expected and the targeted radial modal vibration of the shell is attenuated significantly.     

基于有限元法的周期压电智能梁滤波特性分析

周期结构具有通频和禁频特性,使其在动态载荷的滤波器、具有主动控制功能的结构研究中得到了重要应用。基于Timoshenko梁理论,考虑基梁和压电片的转动惯量和剪切效应,采用有限元法和传递矩阵法推导了波在周期性地粘贴压电片的Timoshenko梁中的传播模型,分析了几何尺寸和材料特性对其频带性质的影响,并与Bernoulli-Euler梁理论得到的结果进行了对比。研究表明,当基梁与压电层厚度比达到40时,禁带带宽减小了54%,因此对于周期结构中的深梁,应舍弃Bernoulli-Euler梁理论而采用Timoshenko梁理论建立的模型;对于不同尺寸和材料特性的压电周期结构,频带性质会有很大不同,可以通过调整结构的参数来改变其频带性质,从而改变波动在结构中的传播特性。     

新型智能隔振复合结构的动态特性研究

针对精密机械的微位移隔振问题,设计了一种以PVDF压电薄膜为作动器和传感器的新型智能隔振复合结构。根据压电方程推导出了层叠式PVDF压电薄膜作动器厚度变形量表达式,建立了该智能复合结构的隔振理论模型,采用LMS自适应控制算法,以Matlab和有限元混合建模分析方式对本智能隔振复合结构的动态特性进行研究。有限元模型的分析结果与Matlab计算数据一致,验证了本新型智能隔振复合结构对微位移隔振的有效性,其结论将为精密仪器、微纳米设备的微位移智能主动隔振奠定理论基础。     

Optimal shapes of PZT actuators for laminated structures subjected to displacement or eigenfrequency constraints

In this paper, dynamics, electromechanical couplings, and control of piezoelectric laminated cylindrical shells and rectangular plates are investigated. It is assumed that the piezoelectric layers are distributed on the top and bottom surfaces of the structures. First of all the governing equations and boundary conditions including elastic and piezoelectric couplings are formulated and solutions are derived. Then control of the plate/shells deflections and natural frequencies using high control voltages are studied in order to optimize the structural response. The present formulation of optimal design introduces boundaries of piezoelectric patches as new class of design variables. In addition, classical design variables in the form of ply orientation angles of orthotropic layers are also taken into account. For the actuator/actuator configuration, it was shown that the piezoelectric actuators can significantly reduce deformations/eigenfrequencies of the composite plate. Those effects were dependent on the value of the applied voltage. It was demonstrated that the proper choice of the actuator area is more efficient in reducing deflections/eigenfrequencies. The accuracy of optimal design are verified both with the aid of the FE package ABAQUS and using the standard Rayleigh-Ritz method. The results concerning active vibration control for axisymmetric cylindrical shells are also discussed.     

Experimental investigation on the piezo-composite actuator with piezoelectric single-crystal layer

Studies on muscle mimicking actuators have increased in the last two decades due to the possibility of various applications for compact lightweight actuators including small unmanned aircrafts, missile, and biomimetic robots. Piezoelectric materials have been used in a variety of applications ranging from shape control of structure and active vibration control of structure to noise suppression due to compact size and good frequency response. Conventional polycrystal piezoelectric ceramic materials, however, have limited actuating strains and displacement, hindering their use in actuators for small aerospace vehicles. In this study, the design and fabrication method of an actuator with a piezoelectric single-crystal layer were investigated to increase the actuation strain and displacement. From a comparison of the performance of the LIPCA-C2 and LIPCA-S prototypes, it was found that the new LIPCA-S2, which has much higher coefficient of the unimorph actuator, can generate an actuating displacement more than twice that of LIPCA-C2.     

Vibration suppression of laminated composite beams with a piezo-electric damping layer

An analytical formulation is derived for modelling the behaviour of laminated composite beams with integrated piezoelectric sensor and actuator. The major difference in approach to the solution compared to previous studies is that the analytical solution for active vibration control and suppression of smart laminated composite beams is presented in this paper. The governing equation is based on the first-order shear deformation theory (Mindlin plate theory), which is applicable for both thin and moderately beams, and includes the coupling between mechanical and electrical deformations. The voltage generated by the sensor layer and response of the beam to the actuator voltage can be computed independently. In this study, the new assumption of harmonic vibration is introduced, which includes both of the sine and cosine terms. Another contribution of this paper is introducing the transformation method of complex numbers to reduce the order of the governing equation of smart laminated beams. Thus, the exact solution of the reduced governing equation can be obtained by using MATLAB and the entire numerical results are presented. The behaviour of the output voltage from the sensor layer and the input voltage acting on the actuator layer is also studied. Graphical results are presented to demonstrate the ability of closed-loop system to actively control the vibration of laminated beams and it shows a good control effect. The influence of stacking sequence on the controlled transient response of the laminated beam is examined. Finally, the experiential formulation of the amplitude of beam vibration varying with the negative velocity feedback control gain has also been evaluated. The present method has a general application in this field of study.     

基于等几何方法的压电功能梯度板动力学及主动振动控制分析

针对表面粘贴有压电层的功能梯度板的动力学及主动振动控制问题，建立了一种基于三阶剪切变形理论的等几何分析求解方法。其中，功能梯度板的材料属性为板厚方向的幂函数分布，并假设电势沿着压电层的厚度方向呈线性变化。利用线性压电本构方程以及哈密顿变分原理，推导了压电功能梯度板的相关等几何分析有限元方程。通过分析压电智能结构的静态弯曲行为验证了该方法的有效性与精确性。运用模态叠加技术与Newmark-β直接积分法分析了两种不同结构的压电功能梯度板的动力学响应与主动振动控制问题。在主动振动控制分析中，引入了物理中面的概念避免当传感器与驱动器分别粘贴于功能梯度的上、下表面时，由拉伸-耦合效应引起的控制不稳定的问题，并着重分析了振动控制过程中两种结构传感器层和驱动器层的电压响应。     

风洞模型支杆系统设计及振动主动控制

研究了悬臂梁支撑系统在强流场环境下的振动抑制问题,并且基于跨声速实验设备中的模型支杆实际振动情况,采用压电陶瓷作为执行器设计了一套抑振器系统。对模型支杆进行了设计,对抑振器的轴向及周向安装位置进行了优化,以获得最佳的抑振器输出性能。基于PD控制算法,搭建了一套实验室模拟系统,在模型自由端施加加速度振幅为0.5 g,1.3 g,2 g的激励,实验结果表明,剩余振幅比为12%,10%,26%,取得了很好的振动抑制效果。     

基于自校正PID控制的智能悬臂梁振动控制

摘　要：由于智能结构的工作环境变化多端,各种性能参数会随着环境变化而变化,先前建好的模型不再适应设计好的控制律,本文应用压电双晶片的驱动传感一体化的特性,实现了智能悬臂梁的自适应控制研究。基于极点配置理论,采用了自校正PID控制方法在线实时设计了控制参数,解决了模型参数无法实时更新进而导致的控制精度低的问题。通过MATLAB的SIMULINK的数值仿真,得出了自校正PID控制方法在实现智能结构自适应振动控制中是可行的结论,并且通过搭建实验平台进行实验验证;利用压电双晶片的驱动传感特性,使智能悬臂梁的自由振动得到了有效控制。因此,基于自校正PID控制方法,采用压电双晶片对智能结构吸振减振提供了理论与实验的研究基础。     

Active vibration control of an arbitrary thick smart cylindrical panel with optimally placed piezoelectric sensor/actuator pairs

Active vibration suppression of a simply supported, arbitrarily thick, transversely isotropic circular cylindrical host panel, integrated with spatially distributed piezoelectric actuator and sensor layers, is investigated based on the linear three dimensional exact piezo-elasticity theory. To assist control system design, system identification is conducted by applying a frequency domain subspace approximation method based on N4SID algorithm using the first few structural modes of the system. The state space model is constructed from system identification and used for state estimation and development of control algorithm. The optimal electrode configuration for the collocated piezoelectric actuator–sensor pair is found by applying a genetic optimization procedure based on maximization of a quantifiable objective function considering the controllability, observability and spillover prevention of the identified system. A linear quadratic Gaussian (LQG) optimal controller is subsequently designed and simulated based on the identified model of optimally configured smart structure in order to actively control the system response in both frequency and time domains. The dynamic performance and effectiveness of the optimized vibration control system is demonstrated for two different types of external mechanical excitations (i.e., impulsive load and white noise disturbance). The accuracy of dynamic analysis is established with the aid of a commercial finite element package and the data available in the literature.     

悬臂Kagome夹心板独立模态空间振动控制研究

针对悬臂板挠度大、低频振动突出问题,对悬臂Kagome夹心板的振动主动控制进行研究。建立结构及压电作动器有限元模型;将独立模态空间控制与模态观测器相结合,提出悬臂Kagome夹心板的主动控制策略;针对突风载荷作用下夹心板基于独立模态空间的振动控制进行仿真,重点研究观测器极点对控制效果影响。结果表明,所提控制方法能显著提高悬臂Kagome夹心板结构的阻尼特性,观测器衰减系数越大控制效果越好;该夹心板在振动控制方面较传统板结构优势明显。     

基于拓扑优化的压电分流阻尼抑振实验研究

将结构拓扑优化引入压电分流振动抑制中,以压电元件的分布面积为设计变量,压电元件产生的电荷最大化为优化目标,对压电元件的拓扑进行了优化以获得最佳抑振效果。针对悬臂梁结构,得到了对不同的结构模态进行抑制时的压电元件最优拓扑构型。建立了带有压电分流阻尼系统的悬臂梁振动控制实验模型,将压电元件拓扑优化后的压电分流阻尼系统应用于悬臂梁多阶弯曲模态的振动响应抑制实验,并对比分析了带最优拓扑和非优拓扑压电元件的悬臂梁压电分流阻尼抑振效果。结果表明,对压电元件进行拓扑优化可以明显提高压电分流阻尼系统的抑振效果。