Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle

In the presented paper the equations of motion of a rotating composite Timoshenko beam are derived by utilising the Hamilton principle. The non-classical effects like material anisotropy, transverse shear and both primary and secondary cross-section warpings are taken into account in the analysis. As an extension of the other papers known to the authors a nonconstant rotating speed and an arbitrary beam’s preset (pitch) angle are considered. It is shown that the resulting general equations of motion are coupled together and form a nonlinear system of PDEs. Two cases of an open and closed box-beam cross-section made of symmetric laminate are analysed in details. It is shown that considering different pitch angles there is a strong effect in coupling of flapwise bending with chordwise bending motions due to a centrifugal force. Moreover, a consequence of terms related to nonconstant rotating speed is presented. Therefore it is shown that both the variable rotating speed and nonzero pitch angle have significant impact on systems dynamics and need to be considered in modelling of rotating beams.     

相似电路耦合模型及其在压电—梁结构分析中的应用

根据线性系统微分方程的相拟性确定机械系统的相似电路,由压电材料的本构和动力方法确定压电材料作为电耦合的变压器,从而用一具耦合的电路系统模拟智能结构的电学和力学行为,通过一个压电－梁智能结构的分析,证明了模型的可用性。     

旋转功能梯度智能结构的刚-柔耦合动力学分析

基于一次耦合模型理论建立了中心刚体-压电层-功能梯度材料智能梁系统的刚柔耦合动力学模型．研究了开环状态下将压电材料作为传感器的压电效应和质量刚度效应对系统动力学特性的影响．通过仿真算例与另两种不同建模理论（传统零次近似耦合模型、一次近似耦合模型）作了对比．随着中心刚体外驱动力矩的增大,零次近似耦合模型和一次近似耦合模型计算结果逐渐发散,而本文的一次耦合模型的计算结果始终保持收敛,较其他近似耦合模型具有一定优势．对三种不同的结构的计算结果表明,压电材料的压电效应对系统的动力学特性影响显著,压电材料的质量刚度效应也会影响智能梁的动力学行为,前者比后者的影响大得多．此外,功能梯度材料功能梯度指数对系统动力学特性的影响也较大．     

Active vibration control of thin constrained composite damping plates with double piezoelectric layers

Vibrations and the damping behaviour of thin constrained composite plates with double piezoelectric layers are analytically explored by using Fourier transformation and classical laminated plate theory. Electric potential equations in the double piezoelectric layers are solved with respect to closed and open circuit boundary conditions, an exterior dielectric slab and active control. The natural frequencies and loss factors of the constrained smart composite plates with passive control methods are not notably changed in comparison with those of the constrained composite plates without piezoelectric effects since vibrational energy does not efficiently convert to electrical energy. The loss factors of the composite plates with active constrained damping increase and the natural frequencies have significant variations as the proportional derivative gains increase. Transverse displacement power spectra of the piezoelectric composite plates with active control are compared with those of the piezoelectric composite plates with passive control showing that active control has the best suppression performance of vibrations for the constrained laminated plates with double piezoelectric layers. Radial power spectral density, phase angles and cylindrical-wave power spectral density are calculated. Interesting patterns of wave propagation are explained when plane wave expansion is used to obtain Bessel cylindrical waves.     

Nonlinear vibrations of blade with varying rotating speed

This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton??s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin??s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.     

Effect of electric boundary conditions on crack propagation in ferroelectric ceramics

In this paper, the effect of electric boundary conditions on Mode I crack propagation in ferroelectric ceramics is studied by using both linear and nonlinear piezoelectric fracture mechanics. In linear analysis, impermeable cracks under open circuit and short circuit are analyzed using the Stroh formalism and a rescaling method. It is shown that the energy release rate in short circuit is larger than that in open circuit. In nonlinear analysis, permeable crack conditions are used and the nonlinear effect of domain switching near a crack tip is considered using an energy-based switching criterion proposed by Hwang et al.（Acta Metal. Mater.,1995）. In open circuit, a large depolarization field induced by domain switching makes switching much more diffcult than that in short circuit. Analysis shows that the energy release rate in short circuit is still larger than that in open circuit, and is also larger than the linear result. Consequently,whether using linear or nonlinear fracture analysis, a crack is found easier to propagate in short circuit than in open circuit, which is consistent with the experimental observations of Kounga Njiwa et al.（Eng. Fract. Mech., 2006）.     

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Topology optimization of piezoelectric nanostructures

We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.     

Nonlinear modeling of tuned liquid dampers (TLDs) in rotating wind turbine blades for damping edgewise vibrations

Tuned liquid dampers (TLDs) utilize the sloshing motion of the fluid to suppress structural vibrations and become a natural candidate for damping vibrations in rotating wind turbine blades. The centrifugal acceleration at the tip of a wind turbine blade can reach a magnitude of 7–8g. This facilitates the use of a TLD with a relatively small fluid mass and with feasible geometric dimensions to mitigate the lightly-damped edgewise vibrations effectively. In the present paper, modal expansions are carried out directly on the velocity field and the free surface of the sloshing liquid in the rotating coordinate system. A formulation has been proposed leading to coupled nonlinear ordinary differential equations, which have been obtained through the Galerkin variational approach together with the modal expansion technique. Two models, with one sloshing mode and three sloshing modes, have been studied in the numerical simulation. It is shown that the one-mode model is able to predict the sloshing force and the damped structural response accurately, since the primary damping effect on the structure is achieved by the first sloshing mode of the fluid. Although it is unable to predict the fluid free-surface elevation equally well, the one-mode model can still be utilized for the design of TLD. Parametric optimization of the TLD is carried out based on the one-mode model, and the optimized damper effectively improves the dynamic response of wind turbine blades.     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

带集中质量的旋转柔性曲梁动力学特性分析

本文对带集中质量的平面内旋转柔性曲梁动力学特性进行了研究.基于绝对节点坐标法推导出曲梁单元,其中该曲梁单元采用Green-Lagrangian应变,并根据曲梁变形前后的曲率变化和曲率的精确表达式计算了曲梁单元弹性力所作的虚功.通过虚功原理,利用$\delta$函数和中心刚体与悬臂曲梁之间的固支边界条件,建立了带集中质量的旋转柔性曲梁非线性动力学模型.基于该模型,本文仿真计算了悬臂曲梁的纯弯曲问题和带有刚柔耦合效应的旋转柔性曲梁动力学响应问题,以此分别讨论了所提出曲梁单元的收敛性和动力学模型的正确性.进一步应用D'Alembert原理,将旋转曲梁等效为带离心力的无旋转曲梁,通过线性摄动处理得到系统的特征方程,以此分别研究了旋转角速度、初始曲率和集中质量对曲梁动力学特性的影响.最后重点分析了旋转曲梁的频率转向和振型切换问题,并阐述了两者之间的相互关系.研究结果表明:随着旋转角速度的增大,曲梁的频率特性与直梁的频率特性相近,以及曲梁拉伸变形占主导的模态振型会提前.      

An improved model of thin cylindrical piezoelectric layers between isotropic elastic media

Radially polarized piezoelectric fibers are now considered to be used in smart composites. The piezoelectric layers in these fibers are generally anisotropic in the transverse direction and therefore difficult to analyze when dynamic loads are involved. The present paper presents a theoretical study of the dynamic behavior of radially polarized cylindrical piezoelectric layers between isotropic elastic media. A new interphase model is developed to provide simple yet accurate evaluation of the dynamic response of such anisotropic elastic layers. Unlike the traditional interface-spring model, the current interphase model satisfies the equations of motion of the layers and can provide reliable prediction of the stress and displacement. A comparison between the developed model, the interface-spring model and the finite element analysis is conducted. The results clearly show the advantages of the current model over the traditional interface-spring model in simulating anisotropic layers. Numerical examples based on this interphase model for different interphase thicknesses, loading frequencies and material combinations are presented to evaluate the dynamic behavior of multilayered elastic media.     

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.     

The interaction between crack and electric dipole of piezoelectricity

Discrete dipoles located near the crack tip play an important role in nonlinear electric field induced fracture of piezoelectric ceramics. A physico-mathematical model of dipole is constructed of two generalized concentrated piezoelectric forces with equal density and opposite sign. The interaction between crack and electric dipole in piezoelectricity is analyzed. The closed form solutions, including those for stress and electric displacement, crack opening displacement and electric potential, are obtained. The function of piezoelectric anisotropic direction,p _a (θ)=cosθ+p _a sinθ, can be used to express the influence of a dipole's direction. In the case that a dipole locates near crack tip, the piezoelectric stress intensity factor is a power function with −3/2 index of the distance between dipole and crack tip. Supported by National Natural Science Foundation of China(No. 10072033)     

Vibration analysis of FG annular sector in moderately thick plates with two piezoelectric layers

Free vibration of functionally graded(FG) annular sector plates embedded with two piezoelectric layers is studied with a generalized differential quadrature(GDQ)method. Based on the first-order shear deformation(FSD) plate theory and Hamilton's principle with parameters satisfying Maxwell's electrostatics equation in the piezoelectric layers, governing equations of motion are developed. Both open and closed circuit(shortly connected) boundary conditions on the piezoelectric surfaces, which are respective conditions for sensors and actuators, are accounted for. It is observed that the open circuit condition gives higher natural frequencies than a shortly connected condition. For the simulation of the potential electric function in piezoelectric layers, a sinusoidal function in the transverse direction is considered. It is assumed that properties of the FG material(FGM) change continuously through the thickness according to a power distribution law.The fast rate convergence and accuracy of the GDQ method with a small number of grid points are demonstrated through some numerical examples. With various combinations of free, clamped, and simply supported boundary conditions, the effects of the thicknesses of piezoelectric layers and host plate, power law index of FGMs, and plate geometrical parameters(e.g., angle and radii of annular sector) on the in-plane and out-of-plane natural frequencies for different FG and piezoelectric materials are also studied. Results can be used to predict the behaviors of FG and piezoelectric materials in mechanical systems.     

Effect of rotation on frequency characteristics of a truncated circular conical shell

In the current dynamic model of rotating truncated conical shells, the expressions of centrifugal and coriolis accelerations and initial hoop tension were incomplete and some terms were missing. This might cause the frequency characteristics of rotating conical shells to be overestimated (or underestimated). Therefore, the effects of rotation upon frequency characteristics of rotating truncated conical shell are studied in the paper. Accurate expressions of centrifugal and coriolis accelerations and initial hoop tension are derived, and then a modified dynamic model for the rotating truncated conical shell is presented. The generalized differential quadrature method is utilized to obtain the natural frequencies. The influences of various boundary conditions and rotating speed on the free vibration of the conical shell are discussed in detail. Through comparison analysis, the errors in current model are also pointed out.     

非线性压电效应下压电弯曲执行器的动力分析

研究压电弯曲执行器在强电场作用下的非线性动力行为．考虑电致伸缩和电致弹性的非线性压电效应，导出了压电悬臂执行器变刚度的弯曲振动控制方程．利用非定常振动的渐近理论，讨论了弯曲压电执行器的动力特征．根据目前的非线性模型可以计算压电悬臂执行器的固有共振频率与电场的变化关系．结果表明压电执行器端头挠度谐振幅度随作用电场振幅的增大而增大，以及力学品质因数随电场振幅的增大而减少，并且与实验结果非常吻合．通过数值比较得到在电场频率随时间变化非常缓慢的情况下非定常振动问题可以近似地用定常振动来处理．