Mesh‐free models for static analysis of smart laminated composite beams

This paper is concerned with the development of mesh‐free models for the static analysis of smart laminated composite beams. The overall smart composite beam is composed of a laminated substrate composite beam and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the distributed actuator layer of the smart beam. A layer‐wise displacement theory and an equivalent single‐layer theory have been used to derive the models. Several cross‐ply substrate beams are considered for presenting the numerical results. The responses of the smart composite beams computed by the present new mesh‐free model based on the layer‐wise displacement theory excellently match with those obtained by the exact solutions. The mesh‐free model based on the equivalent single‐layer theory cannot accurately compute the responses due to transverse actuation by the piezoelectric actuator. The models derived here suggest that the mesh‐free method can be efficiently used for the numerical analysis of smart structures. Copyright © 2016 John Wiley & Sons, Ltd.     

An adhesively laminated plate element for PZT smart plates

An adhesively laminated element taking into consideration peel stress is developed for a piezoelectric smart plate. In this novel finite element analysis formulation, a four node piezoelectric element is firstly derived, and an adhesive element of finite thickness with both shear and peel stiffness is sandwiched between two collocated four node plate elements to form an adhesively laminated element for a piezoelectric smart plate. In this framework of finite element analysis, because the displacement filed in this adhesively laminated element is continuous and a plate element is derived based on the Reissner–Mindlin plate theory, and thus it can be accurately applied to a thin or moderately thick host plate with bonded or debonded piezoelectric actuators and sensors. The formulation is performed for an isotropic host plate and a fiber reinforced laminate plate. Numerical results are presented to compare with those of the exact solutions for smart beams, and validate with the experimental results of the isotropic and composite host plates available in the literature. Using the present finite element analysis formulation, energy transfer stresses in the adhesive and equivalent forces induced in the host plate are investigated. The present formulation is demonstrated to allow debondings of piezoelectric patches and the debonding detection.The authors are grateful to the support of the Australian Research Council via a Discovery Projects grant (grant No: DP0346419).     

压电功能梯度层合梁的力-电-热耦合梁单元及最优振动控制

给出了一个压电功能梯度层合梁振动分析的两节点力-电-热耦合梁单元,并将其用于功能梯度层合梁的振动最优控制。在这个多场耦合梁单元中,功能梯度材料的等效力学性能用Voigt或Mori-Tanaka模型表征;梁的位移场用Shi改进的三阶剪切变形板理论描述;压电层的电势场用Layer-wise理论分层表征,且呈高阶非线性电势场的压电层可离散成数个子层。用Hamilton原理推导了压电功能梯度梁的力-电-热耦合单元列式,用拟协调元法给出了多场耦合梁单元的高计算效率的显式单元刚度矩阵,以及采用线性二次型(LQR)最优控制算法进行压电功能梯度层合梁的最优振动控制。使用所得力-电-热耦合梁单元进行了压电功能梯度层合梁的静力和动力分析。数值算例表明,所得力-电-热耦合梁单元可靠、准确和高效,LQR最优控制算法得到最优控制电压可有效抑制功能梯度梁的振动且实现控制系统能量的优化。     

Benchmark analysis of piezoelectric bimorph energy harvesters composed of laminated composite beam substrates

This paper is concerned with the derivation of exact solutions for the responses of piezoelectric bimorph energy harvesters composed of laminated composite beam substrates. An electro-elastic finite element model is also developed based on the layer wise first order shear deformation theory for computing the responses of the bimorphs under general boundary and loading conditions. Both series and parallel connections of the piezoelectric layers of the bimorphs are considered. The responses computed by the finite element model excellently match with that obtained by the exact solutions. The induced electric potential in case of the bimorph in which the piezoelectric layers are connected in series is significantly larger than that in case of the bimorph with piezoelectric layers connected in parallel. If the thickness of the piezoelectric layers and the substrate remain same, the piezoelectric bimorph composed of antisymmetric angle-ply substrate beam is capable of inducing more electric potential than the bimorphs with cross-ply substrate beams. Also, if the bimorph is cantilever, it induces significantly more electric potential than when it is simply supported. Optimum thickness of the piezoelectric layers of the bimorph and unimorph harvesters has been determined. Most importantly, it is found that the bimorph with its piezoelectric layers connected in series performs significantly better than the unimorph if the mass and volume of the piezoelectric layers and the substrates remain same. The results presented here may serve as the benchmark results for verifying experimental and numerical models.

     

Analysis of smart damping of laminated composite beams using mesh free method

This paper is concerned with the development of mesh free model for the performance analysis of active constrained layered damping (ACLD) treatments on smart laminated composite beams. The overall structure is composed of a substrate laminated composite beam integrated with a viscoelastic layer and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the active constraining layer of the smart beam and the viscoelastic layer acts as the constrained layer. A layer wise displacement theory has been used to derive the models. Both symmetric cross-ply and antisymmetric angle-ply laminated beams are considered for the numerical analysis. It is observed that ACLD treatment significantly improves the active damping properties of the substrate beam. The numerical results also reveal that the triangular ACLD treatment is more effective than the rectangular ACLD treatment of same thickness and volume for active damping of smart composite beams.     

Active constrained layer damping of geometrically nonlinear vibration of rotating composite beams using 1-3 piezoelectric composite

In this paper, an analysis for active constrained layer damping (ACLD) of rotating composite beams undergoing geometrically non linear vibrations has been carried out. Commercially available vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) material has been used as the material of the constraining layer of the ACLD treatment. A finite element (FE) model has been derived to carry out the analysis. The substrate beam is considered thin and hence, first order shear deformation theory (FSDT) and von-Karman type nonlinear strain–displacement relations are used to derive the coupled electromechanical nonlinear FE model. The rotary effect has been suitably modelled by incorporating extensional strain energy due to centrifugal force. The Golla–Hughes–McTavish method has been employed to model the constrained viscoelastic layer of the ACLD treatment in the time domain. The numerical responses revealed that the ACLD treatment with 1-3 PZC constraining layer efficiently performs the task of active damping of geometrically nonlinear vibrations of the rotating composite beams. The effects of the fibre orientation angles of the angle-ply substrate beams and the 1-3 PZC constraining layer on the ACLD of the geometrically nonlinear vibrations have been investigated. Also, the effect of the thickness variations of the 1-3 PZC layer and the viscoelastic constrained layer on the damping characteristics of the overall rotating composite beams has been studied.     

Reddy高阶组合梁的非线性有限元分析

该文基于Reddy高阶梁理论,提出了小变形双层组合梁的隐式运动学假定;应用拉格朗日乘子法,将该隐式关系引入到组合梁的最小势能原理,得到了考虑各子梁和粘结滑移层非线性材料特性的高阶组合梁非线性位移法有限单元,且该单元可以容易地转化为非线性Timoshenko和Euler-Bernoulli组合梁有限单元。随后,该研究分别应用提出的Reddy、Timoshenko和Euler-Bernoulli组合梁有限单元对双跨连续钢-混凝土组合梁进行了准静力分析,考察剪切效应对组合梁构件的挠度、粘结层滑移和截面应力的影响,且参数分析了组合梁的跨高比对剪切效应的影响。参数分析表明:短粗组合梁结构往往表现出显著的剪切效应,Newmark假定不再适用。     

Exact Solutions for the Analysis of Piezoelectric Fiber Reinforced Composites as Distributed Actuators for Smart Composite Plates

Performance of a layer of piezoelectric fiber reinforced composite (PFRC) material as the distributed actuator for smart composite plates has been investigated in this paper. The investigation is performed by finding the exact solutions for static analysis of simply supported symmetric and anti-symmetric cross-ply laminated plates integrated with a layer of PFRC material. The results suggest the potential use of PFRC materials for the distributed actuators of smart structures with both thick and thin substrate composite plates.     

Exact Solutions for the Analysis of Piezoelectric Fiber Reinforced Composites as distributed Actuators for Smart Composite Plates

Performance of a layer of piezoelectric fiber reinforced composite (PFRC) material as the distributed actuator for smart composite plates has been investigated in this paper. The investigation is performed by finding the exact solutions for static analysis of simply supported symmetric and anti-symmetric cross-ply laminated plates integrated with a layer of PFRC material. The results suggest the potential use of PFRC materials for the distributed actuators of smart structures with both thick and thin substrate composite plates.     

Shape control of smart composite plate with non-rectangular piezoelectric actuators

Quasi-static shape control of a smart structure may be achieved through optimizing the applied electric fields, loci, shapes and sizes of piezoelectric actuators attached to the structure. In this paper, a finite element analysis (FEA) software has been developed for analyzing static deformation of smart composite plate structures with non-rectangular shaped PZT patches as actuators. The mechanical deformation of the smart composite plate is modeled using a 3rd order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation is verified by comparing with experimentally measured deformation. Numerical results are obtained for the optimum values of the electric field in the PZT actuators to achieve the desired shape using the linear least square (LLS) method. The numerical results demonstrate the influence of the shapes of actuators.     

Smart damping of geometrically nonlinear vibrations of composite shells using fractional order derivative viscoelastic constitutive relations

In this article, a three-dimensional fractional order derivative model has been developed for the constrained viscoelastic layer of the active constrained layer damping (ACLD) treatment of laminated composite shells undergoing geometrically nonlinear vibrations. The constraining layer of the ACLD treatment is made of vertically/obliquely reinforced 1–3 piezoelectric composites and acts as the distributed actuator. A three-dimensional smart nonlinear finite element model has been developed. Several numerical results are presented to check the accuracy of the present three-dimensional fractional derivative model of the constrained viscoelastic layer for smart damping of geometrically nonlinear vibrations of laminated composite shells.     

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.     

考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁解析解

通过在Newmark 模型中引入(1)描述横向非均匀分布的纵向位移的翘曲形函数和(2)描述钢梁腹板剪切变形的Timoshenko 梁假定,建立了一个能考虑滑移、剪力滞后和剪切变形的钢-混凝土组合梁模型,并推导了均布荷载作用下的解析解。最后通过4 个算例验证了模型和解析解的正确性和适用性,并显示了考虑组合梁剪切变形的必要性。另外,算例还表明,在组合梁的三维有限元建模中采用Timoshenko 梁单元来考虑钢梁的剪切变形会导致显著的误差。     

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.     

变截面压电层合梁自由振动分析

考虑压电材料的质量效应和刚度效应,将表面粘贴或埋入式压电悬臂梁看作变截面梁,研究压电材料对智能结构固有特性的影响。基于一阶剪切变形理论导出压电层合梁的抗弯刚度和横向剪切刚度,计及梁的剪切变形和转动惯量,采用Timoshenko理论推导变截面压电层合梁的频率方程。给出了T300/970压电层合梁和硬铝压电层合梁的前3阶固有频率,并和有限元结果、等截面梁的计算结果进行比较。计算表明,压电材料对压电结构固有频率和固有振型的影响显著,在以振动控制为目标的压电结构动力学建模过程中,有必要考虑压电材料的质量和刚度。     

Adhesive element modelling and weighted static shape control of composite plates with piezoelectric actuator patches

A novel finite element model is presented for static and dynamic analysis of composite plates integrated with a laminated piezoelectric layer, a host laminated composite plate and an adhesive layer between them. A new adhesive element is developed which includes both peel and shear effects in the adhesive layer based on first‐order shear deformation plate theory. The thin adhesive layer between the piezoelectric layer and the host plate is modelled by assuming that it carries constant shear and peel strains throughout its thickness. In addition, a weighted static shape control scheme for finding the optimal voltage distribution for static shape control is given. By selecting different weighting matrices, a variety of items such as displacements, slopes, curvatures, strains and even generalized forces, can be included in finding the optimal actuating voltage for static shape control. The present model is validated by comparing with those results available in the literature. The numerical results show that the weighted linear least method can give a satisfactory voltage distribution to best match the desired shape. Copyright © 2004 John Wiley & Sons, Ltd.     

Super convergent shear deformable finite elements for stability analysis of composite beams

The super convergent finite beam elements are newly presented for the spatially coupled stability analysis of composite beams. For this, the theoretical model applicable to the thin-walled laminated composite I-beams subjected to the axial force is developed. The present element includes the transverse shear and the warping induced shear deformation by using the first-order shear deformation beam theory. The stability equations and force–displacement relationships are derived from the principle of minimum total potential energy. The explicit expressions for the seven displacement parameters are then presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the element stiffness matrix is determined using the force–displacement relationships. In order to demonstrate the accuracy and the superiority of the beam element developed by this study, the numerical solutions are presented and compared with the results obtained from other researchers, the isoparametric beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The effects of shear deformation, boundary condition, fiber angle change, and modulus ratios on buckling loads are investigated in the analysis.     

含压电片复合材料层合板的高阶计算模型

给出了一种分析含任意内埋压电片复合材料层合板的高阶耦合模型, 板的位移场采用三阶剪切理论, 并提出了压电片中电势场在厚度方向的三次分布模式, 可以更精确地描述力、电耦合作用下电场的非均匀分布。在平面应力的假设下给出了简化的压电材料本构方程, 推导了基于该模型的压电层合板有限元计算公式, 并对双压电片梁的弯曲和层合板的变形控制进行了计算, 压电梁的弯曲计算结果与解析结果吻合良好, 表明本文的模型和公式是精确有效的。     

Spectral element model for axially loaded bending–shear–torsion coupled composite Timoshenko beams

The use of frequency-dependent spectral element matrix (or exact dynamic stiffness matrix) in structural dynamics is known to provide extremely accurate solutions, while reducing the total number of degrees-of-freedom to resolve the computational and cost problems. Thus, in this paper, the spectral element model is developed for an axially loaded bending–shear–torsion coupled composite laminated beam which is represented by the Timoshenko beam model based on the first-order shear deformation theory. The high accuracy of the spectral element model is then numerically verified by comparing with exact theoretical solutions or the solutions obtained by conventional finite element method. For the numerical verification, the finite element model is also provided for the composite laminated beam.     

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。