A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators

 An efficient finite element model is presented for the static and dynamic piezothermoelastic analysis and control of FGM plates under temperature gradient environments using integrated piezoelectric sensor/actuator layers. The properties of an FGM plate are functionally graded in the thickness direction according to a volume fraction power law distribution. A constant displacement-cum-velocity feedback control algorithm that couples the direct and inverse piezoelectric effects is applied to provide active feedback control of the integrated FGM plate in a closed loop system. Numerical results for the static and dynamic control are presented for the FGM plate, which consists of zirconia and aluminum. The effects of the constituent volume fractions and the influence of feedback control gain on the static and dynamic responses of the FGM plates are examined. Received: 13 March 2002 / Accepted: 5 March 2003 The work described in this paper was supported by a grant awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1024/01E).     

Active control of FGM plates subjected to a temperature gradient: Modelling via finite element method based on FSDT

An efficient finite element formulation based on a first‐order shear deformation theory (FSDT) is presented for the active control of functionally gradient material (FGM) plates with integrated piezoelectric sensor/actuator layers subjected to a thermal gradient; this is accomplished using both static and dynamic piezothermoelastic analyses. The formulation based on FSDT can be applied to a range of relatively thin‐to‐moderately thick plates. A constant displacement‐cum‐velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied to provide active feedback control of the integrated FGM plate in a self‐monitoring and self‐controlling system. Numerical results for the control of bending and torsional deflections and/or vibrations are presented for a FGM plate comprising zirconia and aluminium. The effects of constituent volume fraction and the influence of feedback control gain on the static and dynamic responses of the FGM plates are examined in detail. Copyright © 2001 John Wiley & Sons, Ltd.     

Modification of dynamic characteristics of FGM plates with integrated piezoelectric layers using first‐ and second‐order approximations

This paper proposes a method for calculating the required changes in the physical properties of plates made of functionally graded materials (FGM) in order to achieve the desired eigenfrequency shifts in the structure. A finite element formulation based on the classical laminated plate theory (CLPT) is presented for an FGM plate with integrated piezoelectric layers. Using this formulation, an efficient method based on the first‐order and second‐order approximations in Taylor expansion is expressed to calculate the corresponding changes in the plate modal properties due to changes in parameters which characterize the structure's dynamic behaviour. An initial sensitivity analysis is carried out to identify the regions within the structure where modifications are most effective on the structure's dynamic characteristics. The proposed algorithm is applied to a case study with two different boundary conditions and the results are validated against exact solutions. The influence of structural modifications on dynamic response of the plate is also studied using the Newmark‐β method. Copyright © 2006 John Wiley & Sons, Ltd.     

A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems

In general, the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem. Further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex. Several approaches have been proposed to overcome this singularity such as Nelson's method and the modal method. In the present work, a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems. With the proposed method, the exact eigenderivatives can be obtained by solving a first‐order linear algebraic system of equations. The method is applied on a 104 DOF ventilator–rotor system, which is used as an example of an asymmetric damped system with distinct eigenvalues. The diameter of the shaft has been chosen as the design parameter. The comparison of the computational time shows that the proposed method is more efficient than both Nelson's approach and the modal method. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal shape control of functionally graded smart plates using genetic algorithms

This paper deals with optimal shape control of functionally graded smart plate containing patches of piezoelectric sensors and actuators. The genetic algorithm (GA) is designed to search for optimal actuator voltage and displacement control gains for the shape control of the functionally graded material (FGM) plates. The work extends the earlier finite element formulations of the two leading authors, so that it can be readily treated using genetic algorithms. Numerical results have been obtained to study the effect of the shape control of the FGM plates under a temperature gradient by optimising (i) the voltage distribution for the open loop control, and (ii) the displacement control gain values for the closed loop feedback control. The effect of the constituent volume fractions of zirconia, through varying the volume fraction exponent n, on the optimal voltages and gain values has also been examined.     

The global modal parameterization for non‐linear model‐order reduction in flexible multibody dynamics

In flexible multibody dynamics, advanced modelling methods lead to high‐order non‐linear differential‐algebraic equations (DAEs). The development of model reduction techniques is motivated by control design problems, for which compact ordinary differential equations (ODEs) in closed‐form are desirable. In a linear framework, reduction techniques classically rely on a projection of the dynamics onto a linear subspace. In flexible multibody dynamics, we propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non‐linear holonomic constraints and to preserve the Lagrangian structure. The construction of this submanifold follows from the definition of a global modal parameterization (GMP): the motion of the assembled mechanism is described in terms of rigid and flexible modes, which are configuration‐dependent. The numerical reduction procedure is presented, and an approximation strategy is also implemented in order to build a closed‐form expression of the reduced model in the configuration space. Numerical and experimental results illustrate the relevance of this approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element modeling of active control of functionally graded shells in frequency domain via piezoelectric sensors and actuators

 A flat-shell element is presented for the active control of functionally graded material (FGM) shells through integrated piezoelectric sensor/actuator layers. The finite element formulation based on first-order shear deformation theory (FSDT) can be applied to shells ranging from relatively thin to moderately thick dimensions. A constant gain displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied to provide active control of the integrated FGM shell in a self-monitoring and self-controlling system. Frequency response characteristics of the FGM shell containing the piezoelectric sensors/actuators are analyzed in the frequency domain. The effects of constituent volume fraction and the influence of feedback control gain values on the dynamic responses of the FGM shell system are examined in detail. Received 13 November 2000     

压电智能结构的一种模态控制新方法

提出了一个用于压电智能结构振动控制的模态控制方法。就智能梁给出了压电模态传感器与压电模态致动器的新设计方法以及相应的模态控制方法，并对相应的观测溢出与控制溢出问题进行了分析，给出了抑制这些溢出的措施。     

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non‐unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal‐based reduced‐order model of the system. Copyright © 2012 John Wiley & Sons, Ltd.     

Eigenderivative analysis of asymmetric non‐conservative systems

In general, the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation. For this reason the eigenvalues and eigenvectors and consequently their derivatives become complex. Expressions for the first‐ and second‐order derivatives of the eigenvalues and eigenvectors of these linear, non‐conservative systems are given. Traditional restrictions of symmetry and positive definiteness have not been imposed on the mass, damping and stiffness matrices. The results are derived in terms of the eigenvalues and left and right eigenvectors of the second‐order system so that the undesirable use of the first‐order representation of the equations of motion can be avoided. The usefulness of the derived expressions is demonstrated by considering a non‐proportionally damped two degree‐of‐freedom symmetric system, and a damped rigid rotor on flexible supports. Copyright © 2001 John Wiley & Sons, Ltd.     

Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches

A generic static and dynamic finite element formulation is derived for the modelling and control of piezoelectric shell laminates under coupled displacement, temperature and electric potential fields. The base shell is of functionally graded material (FGM) type, which consists of combined ceramic–metal materials with different mixing ratios of the ceramic and metal constituents. A multi‐input–multi‐output (MIMO) system is applied to provide active feedback control of the laminated shell using self‐monitoring sensors and self‐controlling actuators through a close loop. Numerical studies clearly show the influence of the positional configurations of sensor/actuator pairs on the effectiveness of static and dynamic control for the shell laminates. The effects of the constituent volume fractions on the static and dynamic responses of the shell laminate are also elucidated. Copyright © 2002 John Wiley & Sons, Ltd.     

Fully coupled non‐linear analysis of piezoelectric solids involving domain switching

Domain switching is the cause of significant non‐linearity in the response of piezoelectric materials to mechanical and electrical effects. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The constitutive equations are non‐linear. Moreover, due to the domain switching phenomenon, the resulting governing equations become highly non‐linear. The corresponding non‐linear finite element equations are derived and solved by using an incremental technique. The developed formulation is first verified against a number of benchmark problems for which a closed‐form solution exists. Next, a cantilever beam made of PZT‐4 is studied to evaluate the effect of domain switching on the overall force–displacement response of the beam. A number of interesting observations are made with respect to the extent of non‐linearity and its progressive spread as the load on the beam increases. Copyright © 2002 John Wiley & Sons, Ltd.     

Inverse problems of linear feedback control systems: Stabilization,tracking and disturbance rejection

Abstract

Recent developments in the fractional representation approach of linear feedback control systems give some promise for a comprehensive unification of various design methodology. This paper presents some results of the fractional representation theory in terms of mapping of simple algebraic functions defined on appropriate domains of free design parameters. We emphasize that the symmetry of the closed‐loop feedback system could provide us some informations which are useful for the robustness analysis of the closed‐loop system.     

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。     

Hybrid‐stress six‐node prismatic elements

This paper presents three novel hybrid‐stress six‐node prismatic elements. Starting from the element displacement interpolation, the equilibrating non‐constant stress modes for the first element are identified and orthogonalized with respect to the constant stress modes for higher computational efficiency. For the second element, the non‐constant stress modes are non‐equilibrating and chosen for the sake of stabilizing the reduced‐integrated element. The first two elements are intended for three‐dimensional continuum analysis with both passing the patch test for three‐dimensional continuum elements. The third element is primarily intended for plate/shell analysis. Shear locking is alleviated by a new assumed strain scheme which preserves the element accuracy with respect to the twisting load. Furthermore, the Poisson's locking along the in‐plane and out‐of‐plane directions is overcome by using the hybrid‐stress modes of the first element. The third element passes the patch test for plate/shell elements. Unless the element assumes the right prismatic geometry, it fails the patch test for three‐dimensional continuum elements. It will be seen that all the proposed elements are markedly more accurate than the conventional fully integrated element. Copyright © 2004 John Wiley & Sons, Ltd.     

Feedback control of vibrating plates using piezoelectric patch sensors and actuators

An analysis of the solutions for various feedback control laws applied to vibrating simply supported plates is evaluated. The control is carried out via a piezoelectric patch sensor and patch actuator. By considering an integral equation formulation, which is equivalent to the differential equation formulation, the analytical results are investigated. The conversion is accomplished by introducing an explicit Green’s function. The feedback controls implemented include displacement, velocity, and a combination of these. A numerical comparison of eigenvalues is presented to illustrate the efficacy of the method and to contrast the effects of the controls. The results presented in the study can be used for benchmarking solutions based in numerical or approximation approaches.     

Active vibration control of composite plate with optimal placement of piezoelectric patches

The active vibration control of a composite plate using discrete piezoelectric patches has been investigated. Based on first order shear deformation theory, a finite element model with the contributions of piezoelectric sensor and actuator patches to the mass and stiffness of the plate was used to derive the state space equation. A global optimization based on LQR performance is developed to find the optimal location of the piezoelectric patches. Genetic algorithm is adopted and implemented to evaluate the optimal configuration. The piezoelectric actuator provides a damping effect on the composite plate by means of LQR control algorithm. A correlation between the patches number and the closed loop damping coefficient is established.     

Damping characteristics of magneto‐rheological fluid absorber based on closed‐loop proportion integration differentiation control

Aiming at investigating the working of magneto‐rheological fluid absorber, a test system used for detecting the damping characteristics of the magneto‐rheological fluid absorber was designed. The test system included sensors, data acquisition card, and so on. The vibration signals were detected by the sensors, put into LabVIEW data acquisition system, and then collected through the processing of closed‐loop Proportion Integration Differentiation control algorithm. Considering that the feedback current could be output to the magneto‐rheological fluid absorber by the data acquisition card, thus the damping force could be changed. The test results showed that the amplitude of the structure vibration response was obviously reduced through the closed‐loop Proportion Integration Differentiation control algorithm, and the control damping characteristics of the magneto‐rheological fluid absorber was improved.