Cylindrical bending of piezoelectric laminates with a higher order shear and normal deformation theory

An analytical solution, based on a higher order shear and normal deformation theory, is presented for the cylindrical flexure of piezoelectric plates. The primary displacement terms are expanded in thickness coordinate and an exact nature of electric potential is obtained in actuator and sensing layers. The electric potential function is evaluated by solving a second order ordinary differential equation satisfying electric boundary conditions along thickness direction of piezoelectric layer. A unidirectional composite plate attached with distributed actuator and sensor layers is analyzed under electrical and mechanical loading conditions and comparison of results with exact solution is presented. Results for non-piezoelectric plates are also compared with elasticity and other solutions of cylindrical bending.     

Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies

This paper presents a numerical study concerning the active vibration control of smart piezoelectric beams. A comparison between the classical control strategies, constant gain and amplitude velocity feedback, and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, is performed in order to investigate their effectiveness to suppress vibrations in beams with piezoelectric patches acting as sensors or actuators. A one-dimensional finite element of a three-layered smart beam with two piezoelectric surface layers and metallic core is utilized. A partial layerwise theory, with three discrete layers, and a fully coupled electro-mechanical theory is considered. The finite element model equations of motion and electric charge equilibrium are presented and recast into a state variable representation in terms of the physical modes of the beam. The analyzed case studies concern the vibration reduction of a cantilever aluminum beam with a collocated asymmetric piezoelectric sensor/actuator pair bonded on the surface. The transverse displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and closed-loop classical and optimal control systems. The case studies allow the comparison of their performances demonstrating some of their advantages and disadvantages.     

Structural response of composite plates equipped with piezoelectric actuators

We propose the modelling of piezoelectric elements perfectly bonded on an elastic structure. The study aims at predicting the static and dynamic (vibration) electromechanical responses of the structure. The model is mostly based on the kinematic assumption of the Love–Kirchhoff thin plate theory including shear function with a quadratic variation of the electric potential along the thickness direction of the piezoelectric parts. A variational formulation of piezoelectricity leads to the equations of motion for an elastic plate equipped with piezoelectric elements. An important feature of the present investigation is that the stiffness and inertial contributions of the piezoelectric patch is not neglected. Moreover, the numerical simulations demonstrate the influence of the actuator position on the global and local responses of the elastic plate for two situations (i) bilayer and (ii) sandwich configurations. A number of benchmark tests are considered in order to characterize the plate deformation when applying an electric potential to the piezoelectric patch faces. Plate vibration problem is also presented and the frequencies for the axial and flexural modes are obtained. The spectra of vibration for the plate with a time-dependent electric potential are computed.     

Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy...     

Exact solution for static analysis of an intelligent structure under cylindrical bending

An exact static analysis of a simply supported intelligent structure under cylindrical bending is presented. The performance of the actuator layer of the intelligent structure to cause induced axial and transverse stresses in the substrate layer is investigated. The capability of the sensor layer to sense the deformations of the structure is shown in terms of induced electric potential on its surface. The distribution of electric potential across the thickness of the piezoelectric sensor and actuator implies that the linear variation can be adopted for choosing the electric potential function to model the laminated intelligent structure of complicated geometry and boundary conditions with refined approximate technique. The results presented here will be helpful to obtain correlation between theoretical and experimental results and to numerical analysts to verify their results.     

Distributed structural identification and control of shells using distributed piezoelectrics: Theory and finite element analysis

Distributed dynamic identification and vibration control of high-performance flexible structures has drawn much attention in recent years. This article presents an analytical and finite-element study on a distributed piezoelectric sensor and distributed actuator coupled with flexible shells and plates. The integrated piezoelectric sensor/actuator can monitor the oscillation as well as actively control the structural vibration by the direct/converse piezoelectric effects, respectively. Based on Maxwell's equations and Love's assumptions, new theories on distributed sensing and active vibration control of a generic shell using the distributed piezoelectrics are derived. These theories can be easily simplified to account for plates, cylinders, beams, etc. A new piezoelectric finite element is also formulated using the variational principle and Hamilton's principle. A piezoelectric micropositioning device was first studied; analytical solutions are compared closely with experimental and finite-element results. Distributed vibration identification and control of a zero-curvature shell-a plate-are also investigated.     

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.     

Exact solutions for dynamic analysis of composite plates with distributed piezoelectric layers

Exact solutions are presented for analyzing dynamics of composite plates with piezoelectric layers bonded at the top and the bottom surfaces. The expressions for mechanical displacements, stresses, electric displacements and potential are derived from constitutive relations and field equations for the piezoelectric medium under applied surface traction and electric potential. The procedure is illustrated with a simply supported symmetric cross-ply (0°/90°/0°) graphite–epoxy composite plate covered with piezoelectric material polyvinylidene fluoride (PVDF). Results are in good agreement with those obtained from finite element model.     

Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate

The aim of this paper is to derive a reduced model for a piezoelectric plate and to study its actuator and sensor capabilities. In the first part, we focus on the asymptotic modeling for thin plates formed by stacking layers of different piezoelectric materials. In the asymptotic model, the mechanical and electric unknowns are shown to be partly decoupled. In the second part, we study the actuator and sensor capabilities of this model. We use two discrete non-differentiable multi-objective optimization problems, which are solved by genetic algorithms. Several numerical results are reported.     

主动约束层阻尼梁有限元建模与动态特性研究

基于弹性、粘弹性和压电材料的本构关系,利用Hamilton原理,推导了主动约束层阻尼梁的有限元动力学模型．结合压电材料的机电耦合特性,采用自感电压的位移反馈,研究了主动约束层阻尼梁的闭环控制特性．求解了主动约束层阻尼简支梁的动态特性如固有频率、模态损耗因子及频率响应特性等．对被动控制、主动控制和主被动混合控制的控制效果进行了分析比较．研究了粘弹性层与约束层厚度等参数对减振控制效果的影响．     

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5?Pa), and a low-frequency (471.79?Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111?μW and peak-to-peak voltage of 0.2952?V are predicted.     

三阶剪切变形板的振动特性研究

对于中厚板或层合板而言,横向剪切变形的影响是显著的,采用三阶剪切变形理论比采用经典薄板理论和一阶剪切变形理论能更好的满足精度的要求,而且能更好地描述板的剪切变形和剪应力沿厚度方向的分布情况.本文用解析的方法研究了简支、自由和固定三种边界条件的任意组合下三阶剪切变形板的自由振动问题.首先应用哈密顿原理建立自由振动方程,再通过引入中间变量使得原来耦合的自由振动方程得到解耦和简化,基于分离变量法,利用边界条件得到基函数的表达式,利用Rayleigh-Ritz法,求得三阶剪切变形板在任意边界条件下的固有频率和振型.本文得到的结果可以为厚板在工程中的应用提供理论依据,具有较高的工程实际应用价值.     

Free vibration analysis of mindlin plates with parabolically varying thickness

The free transverse vibration characteristics of rectangular plates, including the effect of transverse shear deformation and rotary inertia, are presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions are presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.     

挠性智能梁的振动控制

研究采用共位配置的压电敏感器和致动器的挠性是臂梁的振动控制问题,建立了智能梁的模型,设计了一种线性反馈控制律,并应用无空维空间的ＬａＳａｌｌｅ不变原理和线性半群理论证明了当敏感器和控制器的分布使得系统能镇条件成立时,所设计的控制抑制了梁的振动。     

A finite element model for the analysis of viscoelastic sandwich structures

In this work a finite element model is developed for vibration analysis of active–passive damped multilayer sandwich plates, with a viscoelastic core sandwiched between elastic layers, including piezoelectric layers. The elastic layers are modelled using the classic plate theory and the core is modelled using the Reissener–Mindlin theory. The finite element is obtained by assembly of N “elements” through the thickness, using specific assumptions on the displacement continuity at the interfaces between layers. The lack of finite element plate-shell models to analyse structures with passive and active damping, is the principal motivation for the present development, where the solution of some illustrative examples and the results are presented and discussed.     

压电谐振驱动三足机器人的平面运动

设计并实现了一类利用压电陶瓷片作动,由三条曲梁足支撑的振动驱动机器人.建立了在一条足共振驱动下机器人水平运动的动力学方程,数值计算解释了摩擦作用下的运动机理,寻找到异性摩擦对运动方向、速度的影响和压电激励频率与运动速度间的关系.通过建立圆弧曲梁控制方程求解圆弧型足面内振动的固有频率及振型,设计了三组不同频率的圆弧曲梁足参数,实验制作了机器人模型,利用压电控制三足间振动的共振切换,实现了预想的三个方向的运动以达到平面运动的效果,实验测量了机器人的运动速度与理论计算吻合得较好.     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory

In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.

     

Self-learning active vibration control of a flexible plate structure with piezoelectric actuator

In this paper, an active vibration control (AVC) incorporating active piezoelectric actuator and self-learning control for a flexible plate structure is presented. The flexible plate system is first modelled and simulated via a finite difference (FD) method. Then, the validity of the obtained model is investigated by comparing the plate natural frequencies predicted by the model with the reported values obtained from literature. After validating the model, a proportional or P-type iterative learning (IL) algorithm combined with a feedback controller is applied to the plate dynamics via the FD simulation platform. The algorithms were then coded in MATLAB to evaluate the performance of the control system. An optimized value of the learning parameter and an appropriate stopping criterion for the IL algorithm were also proposed. Different types of disturbances were employed to excite the plate system at different excitation points and the controller ability to attenuate the vibration of observation point was investigated. The simulation results clearly demonstrate an effective vibration suppression capability that can be achieved using piezoelectric actuator with the incorporated self-learning feedback controller.     

The effect of nonlinear elasticity on the large amplitude free vibration behavior of elastic plates at small scale

The effect of nonlinear elasticity on the free vibration behavior of elastic plates has been evaluated by employing continuum mechanics model. The second-order non-linear stress–strain relationship has been considered and the Kirchhoff’s hypothesis has been applied on the elastic plate. The large deformation during vibration has also been considered. By using the Hamilton principle, the governing equations of the free vibration of the plate under different boundary condition have been obtained. In order to get the explicit solutions of the governing equations, the Galerkin’s method and the harmonic balance method have been utilized. The relationship between the vibration frequency and the vibration amplitude has been discussed and the vibration frequencies of different shaped plate have been compared. It is perceived that the nonlinear elasticity has a distinct effect on the free vibration of the plate.