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.     

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).     

Bending of piezoelectric plates with a circular hole

Departing from the refined theory of transversely isotropic piezoelectric plates, an analytical solution for the bending of an infinite piezoelectric plate with a circular hole is obtained. Expressions of moment, stress and electric displacement concentration factors are presented in closed form. When the piezoelectric coupling is absent, the results reduce to the corresponding solutions for the transversely isotropic elastic plates. Some numerical results for PZT-6B piezoelectric ceramics are obtained and illustrated by figures. These results show that the effect of piezoelectric coupling on the concentration factors is not negligible.     

Active aeroelastic flutter analysis and vibration control of supersonic composite laminated plate

The active aeroelastic flutter analysis and vibration control at the flutter bounds of the supersonic composite laminated plates with the piezoelectric patches are studied. The piezoelectric patches are bonded on the top and bottom surfaces of the composite laminated plate to act as the sensor and actuator so that the active aeroelastic flutter suppression and vibration control for the supersonic laminated plate can be conducted. The unsteady aerodynamic pressure in supersonic flow is computed by using the supersonic piston theory. Hamilton’s principle with the assumed mode method is used to develop the governing equation of the structural system. The controller is designed by the velocity feedback and proportional feedback control algorithm, and the active damping and stiffness are obtained. The solutions for the complex eigenvalue problem are obtained by using the generalized eigenvalue methodology. The natural frequencies and damping ratios are also gotten. The aeroelastic flutter bounds of the supersonic composite laminated plate are calculated to investigate the characteristics of the aeroelastic flutter. The impulse responses of the structural system are calculated by using the Houbolt numerical algorithm to study the active aeroelastic vibration control. The influences of ply angle of the laminated plate and the control method on the characteristic of flutter and active vibration control are analyzed. From the numerical results it is observed that the aeroelastic flutter characteristics of the supersonic composite laminated plate can be improved and that the aeroelastic vibration response amplitudes can be reduced, especially at the flutter points, by the proportional feedback or the velocity feedback control algorithm using the piezoelectric actuator/sensor pairs. The effectiveness of the flutter control by the two control algorithms is also compared. The results of this study are of great significance to the flutter analysis and aeroelastic design of the aircraft.     

Active aeroelastic flutter suppression of a supersonic plate with piezoelectric material

The active aeroelastic flutter properties of supersonic plates are investigated by using the piezoelectric material. The piezoelectric material has been extensively used for the active vibration control of engineering structures. In this paper, the piezoelectric material is further used to improve the flutter characteristics of the supersonic plates. The equation of motion of the plate and piezoelectric material system is obtained by Hamilton’s principle with the assumed mode method. The supersonic piston theory is used to evaluate the aerodynamic load. By applying an appropriate external control voltage to activate the piezoelectric material, a displacement and acceleration feedback control strategy is used to obtain the active stiffness and active mass. Solving the eigenvalue problem of the equation of motion, the natural frequencies and damping ratios of the structural system are obtained. Furthermore, the aeroelastic flutter bounds are calculated, and the effects of feedback control gains on the active aeroelastic flutter characteristics of the structure are analyzed in detail. From the numerical results it is seen that the active stiffness and active mass have prominent effects on the flutter characteristics of the supersonic plates. The aeroelastic flutter properties can be greatly improved by introducing the active stiffness and active mass into the supersonic plate with the piezoelectric patch. With the increase of the feedback control gains, the active aeroelastic flutter properties for the lower order modes of the supersonic plate are gradually improved.     

On the three-dimensional stress field around a circular hole in a plate of arbitrary thickness

An analytical solution for the three-dimensional stress field in a plate of an arbitrary thickness, 2h, and weakened by a cylindrical hole of radius a is presented. Far away from the hole, the plate is subjected to a uniform tensile load, ₀, in a direction parallel to the plane of the plate. The solution is shown to be derivable from a general 3D solution, which the first author constructed in a previous paper. The analysis shows the stress concentration factor to vary across the thickness and to be sensitive to the value of the radius to thickness ratio, a/h. Furthermore, it is shown that for ratios of (a/h)4, the results predicted by plane stress theory are more than adequate for engineering applications. Finally, the transition between plane stress and plane strain appears to occur at a/h=0.5.Comment refers to an in-plane tensile load     

Stress intensity factors for cracks initiated from a center-holed plate with unsymmetrical lengths under tension

The values of stress intensity factors of unsymmetrical cracks initiated from hole edges under tension were calculated in a center-holed plate, that is a plate having a hole in the center of a front face. The correction factor for stress intensity in the case of the holed plate with cracks was compared with that of a center-cracked plate whose crack length was equal to the total of the hole diameter and the lengths of the cracks in the case of the holed plate. In the present study, to understand the variation of the stress intensity factor, counterplots of the stress intensity factor K_I and the correction factor F_I are employed. Also, the variation of ratio R_F of correction factors between the holed plate with crack and the center-cracked plate were investigated. Where non-dimensional hole diameter a/W is smaller than 0.2, the ratio R_F is less than 1.1 after crack growth length c/W reaches 0.1. Where, 2a, 2c and 2W are hole diameter, the growth length of the crack from hole edge, and plate width, respectively. Therefore, the stress intensity factor for cracks initiated from a holed plate can be approximated within 10% error to that of a center-cracked plate where a/W is smaller than 0.2, though the unsymmetrical cracks are initiated from the edges. Where a/W is larger than 0.3, R_F becomes larger than 1.1 in the wide range of c/W.     

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.     

Interfacial fracture of piezoelectric multilayer actuators under mechanical and electrical loading

The elastic stress and strain fields in a plate of finite thickness containing an elliptical hole are systematically investigated using the 3D finite element method. It is found that the stress and strain concentrations are different in the plate of finite thickness even if the plate is in an elastic state. The relation between the stress and the strain concentration factors depends on Poisson’s ratio, the hole’s geometric configuration and the plate thickness. The stress concentration factor is equal to the strain concentration factor only at the notch root of the plate surface. The stress (or strain) concentration factor at the notch root of the plate surface decreases rapidly with increasing thickness and becomes lower than the stress and strain concentration factors corresponding to the plane stress state or at the notch root of the mid plane. It is too low to reflect the overall stress concentration as the thickness increases or as the b/a ratio decreases. The maximum stress concentration factor occurs on the mid plane only when the plate is thinner than the transition thickness of the stress concentration factor. When the plate is thicker than the transition thickness of the stress concentration factor, the distance between the location of the maximum stress concentration factor and plate surface tends to be constant with increasing thickness for the plate with a given  b/a ratio. The differences between the maximum value and the surface value of the stress and strain concentration factors increase rapidly and tend to their respective constant values with increasing plate thickness. The smaller the b/a ratio, the larger these differences. The difference of the stress concentration factor is larger than that of the strain concentration factor in the same plate.     

Thickness vibrations of a piezoelectric plate with dissipation

The three-dimensional (3-D) equations of linear piezoelectricity with quasi-electrostatic approximation are extended to include losses attributed to the acoustic viscosity and electrical conductivity. These equations are used to investigate effects of dissipation on the propagation of plane waves in an infinite solid and forced thickness vibrations in an infinite piezoelectric plate with general symmetry. For a harmonic plane wave propagating in an arbitrary direction in an unbounded solid, the complex eigenvalue problem is solved from which the effective elastic stiffness, viscosity, and conductivity are computed. For the forced thickness vibrations of an infinite plate, the complex coupling factor K*, input admittance Y are derived and an explicit, approximate expression for K* is obtained in terms of material properties. Effects of the viscosity and conductivity on the resonance frequency, modes, admittance, attenuation coefficient, dynamic time constant, coupling factor, and quality factor are calculated and examined for quartz and ceramic barium titanate plates.     

压电元件驱动的功能梯度弹性薄板的屈曲

考虑功能梯度薄板,其上下表面嵌有压电执行元件.根据逆压电效应将电压转换成作用于板上的等效电载荷.假设梯度材料的弹性参数为板厚度方向坐标的幂函数,基于经典板理论,导出了功能梯度弹性薄板小挠度屈曲平衡微分方程.利用双三角级数展开法,得到了四边简支具有压电元件的功能梯度矩形板的临界屈曲载荷,在此基础上通过数值例子讨论了弹性板的几何尺寸、材料梯度指数的变化对临界电压(载荷)的影响.研究结果表明,材料的梯度指数对临界电压有重要影响,并且通过调整作用于执行元件上的电压的大小和方向,可实现对结构稳定性的有效控制.     

Hybrid damping of smart, functionally graded plates using piezoelectric, fiber-reinforced composites

This paper deals with the investigation of active, constrained layer damping (ACLD) of smart, functionally graded (FG) plates. The constraining layer of the ACLD treatment is considered to be made of a piezoelectric, fiber-reinforced composite (PFRC) material with enhanced effective piezoelectric coefficient that quantifies the in-plane actuating force due to the electric field applied across the thickness of the layer. The Young's modulus and the mass density of the FG plates are assumed to vary exponentially along the thickness of the plate, and the Poisson's ratio is assumed to be constant over the domain of the plate. A finite-element model has been developed to model the open-loop and closed-loop dynamics of the FG plates integrated with two patches of ACLD treatment. The frequency response of the plates revealed that the active patches of ACLD treatment significantly improve the damping characteristics of the FG plates over the passive damping. Emphasis has been placed on investigating the effect of variation of piezoelectric fiber angle in the constraining layer of the ACLD treatment on the attenuating capability of the patches. The analysis also revealed that the activated patches of the ACLD treatment are more effective in controlling the vibrations of FG plates when the patches are attached to the surface of the FG plates with minimum stiffness than when they are attached to the surface of the same with maximum stiffness.     

复合材料补片加固含椭圆孔钢板刚度分析

针对复合材料补片加固损伤钢板的受力特点, 建立三维弹性力学模型, 对复合材料补片双面加固轴向受拉平板进行了应力和变形分析。研究了粘贴界面的剪应力分布规律, 钢板和补片内轴向拉力的分布规律, 分析了加固后平板相对加固前的相对刚度。研究了含椭圆孔钢板经复合材料补片加固后相对加固前以及相对完好钢板的刚度。结果表明, 复合材料胶接修补含孔钢板的刚度随着补片厚度的增加而增加, 且增强的趋势逐渐趋于缓和。当补片层数大于5层后, 单纯依靠增加补片的厚度并不能有效提高修复效果。解析法与有限元法的计算结果及试验结果吻合得比较好。      

Stress concentrations in a plate with two unequal circular holes

This paper contains an exact solution for stresses which are produced in an infinite plate with two unequal circular holes by a uniform tension, an internal pressure or uniform shearing forces along a hole. Bipolar coordinates are used in the solution. Expressions of the hoop stress at the edges of holes are explicitly derived and the stress concentration factors are calculated. The stresses in an infinite plate with two contacted unequal circular holes are also discussed.     

2-2压电复合材料面板的压力放大作用

２－２压电复合材料应用普遍。材料中聚合物对压电陶瓷的应力作用，使性能得到施工改善。在复合材料两电极面上复盖薄面板，由于板的应力传递作用会使使接收国和辐射力产生放大效应，从而提高了接收灵敏度和发射响应。     

Catalogue of photoelastic fringes for a biaxially loaded infinite plate with a hole

This paper presents a typical catalogue of "virtual" photoelastic isochromatic fringes around the hole of a biaxially loaded infinite plate for two problems. Problem 1 relates to an infinite plate in which the circular hole is drilled first and then the loads are applied. Problem 2 is the residual stress problem in which the hole is drilled after the biaxial load is applied to the infinite plate. Use of this catalogue allows visualisation of the stress distribution around the hole, and a first approximation to the solution of the inverse problem using photoelasticity.     

内嵌倾斜压电柱复合材料板的压电振动特性分析

针对1-3型内嵌倾斜压电柱复合板结构,建立压电复合板结构的有限元模型,并推导出变形关系和形函数;应用拉格朗日方程,建立了单元结构的运动方程。针对不同倾斜角度压电柱复合板进行有限元仿真分析,研究了复合压电板的正压电特性,并对复合压电板的能量损耗因子进行分析。研究结果表明：在相同压力下,压电柱的倾斜角度对复合板的弯曲模态频率影响较大。随着压电柱体倾斜角度的增大,模态频率降低,弯曲变形增大;同时,倾斜柱体棱长变长,产生电压增大。悬臂板在压力作用下,沿宽度方向产生反对称电势。在压电陶瓷柱的倾角达到57时,损耗因子达到最大。采用内嵌式倾斜压电柱复合板结构,降低了压电板的脆性,保证大尺寸压电板的结构均匀性和应用。     

Reduction of the stress concentration factor in a homogeneous panel with hole by using a functionally graded layer

This work aims at understanding the effect of a radially heterogeneous layer around the hole in a homogeneous plate on the stress concentration factor. The problem concerns a single hole in a plate under different far-field in-plane loading conditions. By assuming a radial power law variation of Young’s modulus and constant value for Poisson’s ratio, the governing differential equations for plane stress conditions, and general in-plane loading conditions are studied. The elastic solutions are obtained in closed form and, in order to describe localized interface damage between the ring and the plate, two different interface conditions (perfectly bonded and frictionless contact) are studied. The formulae for the stress concentration factors are explicitly given for uniaxial, biaxial and shear in-plane loading conditions and comparisons with interface hoop stress values are performed. The solutions are investigated to understand the role played by the geometric and graded constitutive parameters. The results are validated with numerical finite element simulations in which some simplified hypotheses assumed in the analytical model, are relaxed to explore the range of validity of the elastic solution presented. In this way the results obtained are useful in tailoring the parameters for specific applications.     

Stress concentrations arising from a slot in a plate under biaxial stress

The stress distributions around a straight blunt slot under biaxial stress are found by adapting the classical solution to the stress concentration around an elliptical hole in an infinite plate under uniaxial stress. Firstly, the slot was converted into an equivalent thin ellipse of large major:minor axis ratio with its major equal to the length of the slot. The principle of superposition was then employed to find the stress state on an axis of symmetry at near and far locations from the notch tip. Stress states were found for any ratio of remotely applied stresses in directions perpendicular and parallel to the notch axis. The solutions were verified experimentally for specific tensile stress ratios by bonding strain gauges to an aluminium plate under hydraulic pressure.     

Energy trapping in power transmission through an elastic plate by finite piezoelectric transducers

We study transmission of electric energy through an elastic plate by acoustic wave propagation and piezoelectric transducers. Our mechanics model consists of an elastic plate with finite piezoelectric patches on both sides of the plate. A theoretical analysis using the equations of elasticity and piezoelectricity is performed. Energy trapping that describes the confinement and localization of the vibration energy is examined.