Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators

The active vibration control of a rectangular plate either partially or fully submerged in a fluid was investigated. Piezoelectric sensors and actuators were bonded to the plate, and the assumed mode method was used to derive a dynamic model for the submerged plate. The properties of the piezoelectric actuators and sensors, as well as their coupling to the structure, were used to derive the corresponding equations of their behaviour. The fluid effect was modelled according to the added virtual mass obtained by solving the Laplace equation. The natural vibration characteristics of the plate both in air and in water were obtained theoretically and were found to be consistent with the experimental results, and the changes in the natural frequencies resulting from submersion in fluid can be accurately predicted. A multi-input, multi-output positive position feedback controller was designed by taking the natural vibration characteristics into account and was then implemented by using a digital controller. The experimental results show that piezoelectric sensors and actuators along with the control algorithm can effectively suppress the vibration of a rectangular plate both in air and submerged in a fluid.     

Disturbance rejection-based robust control for micropositioning of piezoelectric actuators

Precise control of piezoelectric actuators used in micropositioning applications is strongly under the effect of internal and external disturbances. Undesired external forces, unmodelled dynamics, parameter uncertainties, time variation of parameters and hysteresis are some sources of disturbances. These effects not only degrade the performance efficiency, but also may lead to closed-loop instability. Several works have investigated the positioning accuracy for constant and slow time-varying disturbances. The main concern is controlling performance and also the presence of time-varying perturbations. Considering unknown source and magnitude of disturbances, the estimation of the existing disturbances would be inevitable. In this paper, a compound disturbance observer-based robust control is developed to achieve precise positioning in the presence of time-varying disturbances. In addition, a modified disturbance observer is proposed to remedy the effect of switching behaviour in the case of slow time variations. A modified Prandtl–Ishlinskii (PI) operator and its inverse are utilized for both identification and real-time compensation of the hysteresis effect. Experimental results depict that the proposed approach achieves precise micropositioning in the presence of estimated disturbances.     

On prestress stiffness analysis of bolt-plate contact assemblies

Bolt connections are among the most important connections used in structures. The stiffnesses of the bolt and of the connected members are the primary qualities that control the lifetime of the connection. The stiffness of the bolt can be estimated rather easily, in contrast to the member stiffness, but with finite element (FE) and contact analysis, it is possible to find the stiffness of the member. In the case of many connections and for practical applications, it is not suitable to make a full FE analysis. The purpose of the present paper is to find simplified expressions for the stiffness of the member, including the case when the width of the member is limited. The calculation of the stiffness is based on the FE, including the solution to the contact problem, and we express the stiffness as a function of the elastic energy in the structure, whereby the definition of the displacements related to the stiffness is circumvented. The contact analysis is performed using a method where iterations are not necessary, and the results are compared to alternative available results. New practical formulas for the stiffnesses are suggested.     

Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method

This paper presents an efficient meshless method for analyzing cracked piezoelectric structures subjected to mechanical and electrical loading. In this method, an element free Galerkin (EFG) formulation, an enriched basic function and some special shape functions that contain discontinuous derivatives are employed. Based on the moving least squares (MLS) interpolation approach, the EFG method is one of the promising methods for dealing with problems involving progressive crack growth. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing procedure required in the conventional finite element method (FEM) is avoided. The numerical results show that the proposed method can yield an accurate near-tip stress field in an infinite piezoelectric plate containing an interior hole. In another example studying a ceramic multilayer actuator, the proposed model was found to be accurate in the simulation of stress and electric field concentrations arround the abrupt end of an internal electrode. The project supported by the National Natural Science Foundation of China (10025209, 10132010, and 90208002), and the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7203/03E). The English text was polished by Yunming Chen.     

Nonlinear vibration analysis of piezoelectric bending actuators: Theoretical and experimental studies

Piezoelectric bimorph actuators are used in a variety of applications, including micro positioning, vibration control, and micro robotics. The nature of the aforementioned applications calls for the dynamic characteristics identification of actuator at the embodiment design stage. For decades, many linear models have been presented to describe the dynamic behavior of this type of actuators; however, in many situations, such as resonant actuation, the piezoelectric actuators exhibit a softening nonlinear behavior; hence, an accurate dynamic model is demanded to properly predict the nonlinearity. In this study, first, the nonlinear stress–strain relationship of a piezoelectric material at high frequencies is modified. Then, based on the obtained constitutive equations and Euler–Bernoulli beam theory, a continuous nonlinear dynamic model for a piezoelectric bending actuator is presented. Next, the method of multiple scales is used to solve the discretized nonlinear differential equations. Finally, the results are compared with the ones obtained experimentally and nonlinear parameters are identified considering frequency response and phase response simultaneously. Also, in order to evaluate the accuracy of the proposed model, it is tested out of the identification range as well.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation

The effects of applied voltage on the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators were examined in a combined experimental and numerical investigation. Experiments were performed to measure the strain near internal and surface electrodes at various electrical loading conditions. The finite element method was also used to solve the coupled electro-elastic boundary value problem. The strain, stress and electric displacement concentrations were calculated and a non-linear behavior induced by localized polarization switching was discussed. A comparison of strain concentration was made between experiment and simulation.     

Study on electroelastic field concentration around the electrode tip in multilayer ferroelectric actuators of two designs and their optimizations

Distribution of electromechanical field near electrode tips is closely related to the reliability of ferroelectric multilayer actuators. In this paper, the deformation and stress concentrations around the electrode tip in two multilayer actuator designs, partially and fully cofired, are investigated by means of experimental measurement and numerical simulations. The digital speckle correlation method (DSCM) is used to measure the full displacement field near the electrode tip with the high spatial resolution. The paths of electric breakdown and cracks initiated from the edge of electrodes were observed. With the proposed Double Gibbs free energy criterion, a fully coupled nonlinear electromechanical finite element method based on domain-switching mechanisms is developed and the simulation results agree well with the experiments. It is found that the crack-like “defects” in the partially cofired layered actuators, i.e. the interlayer gaps filled with soft insulating wax, can significantly reduce the maximum tensile stress level compared with that in “perfect” fully cofired actuators, which implies that the partially cofired design is more reliable than the fully cofired one. Further optimization on geometrical dimension of actuators is also carried out.     

A study on adjusted contact force laws for accelerated large scale discrete element simulations

The aim of this work is to systematically investigate the effect of the normal force law and the applied stiffness on the behavior of single particles and particle systems. A detailed review of the literature regarding altered stiffnesses and their use in force laws is provided. The effect on macroscopic simulation properties such as contact numbers, velocity profiles, discharge rates and quality of dispersion for different fractions of maximum overlap is studied in the case of a stirrer vessel and a rectan...     

Fracture mechanics and crack contact analyses of the active repair of multi-layered piezoelectric patches bonded on cracked structures

In this paper, two-dimensional plane strain finite element analyses of the active repair for cracked structures by using multi-layered piezoelectric patches have been studied. The reductions of stress intensity factors and strain energy density factor at the crack tips are obtained. Also, the repair voltages for various conditions are obtained for estimating the repair performances. Lower repair voltage is a better choice because it is low-energy-consuming and safer for the operation. From the results of numerical fracture mechanics, it shows the crack contact conditions must be considered in the analysis. However, the friction on the crack has few effects on the repair performances for this mode-I dominated case. The better design choices for the piezoelectric patch are as follows: increasing the layer number, increasing the patch length, and reducing the patch thickness. In additions, it is not a good idea to use higher input voltage that is larger than the repair voltage because it will enlarge the crack open near the crack tip. Too long patch length has no advantage for the active repair.     

Stiffness relations for piezoelectric indentation of flat and non-flat punches of arbitrary planform: Applications to probing nanoelectromechanical properties of materials

Stiffness relations for voltage-dependent contact mechanics of piezoelectric material are derived for an indenter of arbitrary planform under normal force, centrally or non-centrally applied, and electric charge distribution at the base. Relations between indentation depth, indentation force, electric potential and electric charge are explicitly given in terms of indenter's geometry and piezoelectric material constants. The analysis covers indenters with non-flat base approximated by a second-order surface; elliptic paraboloid is considered as an example. In the case of the elliptic non-flat planform, the derived stiffness relations are exact; otherwise, they are approximate and are shown to have good accuracy. The stiffness relations are given in elementary functions and are obtained by utilizing the recently established principle of correspondence between the piezoelectric and purely elastic problems. Besides contributing to extension of Hertzian mechanics to piezoelectric materials, these results are essential for quantitative interpretation of the scanning probe microscopy and piezoelectric nanoindentation data on piezoelectric, ferroelectric, and multiferroic materials.     

Monoharmonic vibrations and vibrational heating of an electromechanically loaded circular plate with piezoelectric actuators subject to shear strain

The paper addresses the forced flexural vibrations and dissipative heating of a circular viscoelastic plate with piezoactive actuators under axisymmetric loading. A refined formulation of this coupled problem is considered. The viscoelastic behavior of materials is described using the concept of complex moduli dependent on the temperature of dissipative heating. The electromechanical behavior of the plate is modeled based on the Timoshenko hypotheses for the mechanical variables and analogous hypotheses for the electric-field variables in the piezoactive layers of the actuator. The temperature is assumed constant throughout the thickness. The nonlinear problem is solved by a time stepping method using, at each step, the discrete-orthogonalization and finite-difference methods to solve the elastic and heat-conduction equations, respectively. A numerical study is made of the effect of the shear strain, the temperature dependence of the material properties, fixation conditions, and geometrical parameters of the plate on the vibrational characteristics and the electric potential applied to the actuator electrodes to balance the mechanical load Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 104–114, September 2008.     

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.     

A study of contact non-linearities in pin-loaded lugs: Separation,clearance and frictional slipping effects

Pin-loaded lugs with bush fitting are widely encountered in industrial applications to connect parts and transmit loads and motions. Due to their mechanical function, frictional contact inevitably takes place on the pin–bush and lug–pin interfaces, and can lead to non-linear behaviours occurrence under monotonic or periodic loading, such as bush–lug contact separation, pin–bush conforming contact effects in the presence of initial clearance, or bush–lug frictional slipping mechanisms. The aim of this paper is not to present new results of finite element simulations for lugs involving contact with friction but to provide a comprehensive study of those contact non-linearities through a dedicated analytical contact model.     

A modeling and vibration analysis of a piezoelectric micro-pump diaphragm

The vibration analysis of a micro-pump diaphragm is presented. A piezoelectric micro-pump is studied. For this purpose, a dynamic model of the micro-pump is derived. The micro-pump diaphragm is modeled as circular double membranes, a piezoelectric one as actuator and a silicon one for representing the membrane for pumping action. The damping effect of the fluid is introduced into the equations. Vibration analysis is established by explicitly solving the dynamic model. The natural frequencies and mode shapes are calculated. The orthogonality conditions of the system are discussed. To verify the results, the finite-element micro-pump model is developed in ANSYS software package. The results show that the two methods are well comparable.     

A method for the selection of sensor and actuator locations

A new and efficient technique for determining optimal locations of sensors and actuators of intelligent structures is presented. The optimization of sensor and actuator locations is based on the 1st order singular value perturbations of observability and controllability. Using this method the optimal placements of sensors and actuators of the intelligent structurer can be selected. Two numerical examples are given to demonstrate the applications of the method. The impulse responses of structures due to different locations of actuators with the same control law are analyzed in detail. The project supported by the National Natural Science Foundation of China and the Mechanical Technique Development Foundation of China     

Application of plane wave expansion and stiffness matrix methods to study transmission properties and guided mode of phononic plates

Numerical procedure based on plane wave expansion and stiffness matrix method is developed to calculate the transmission factor of a micro two-dimensional phononic plate. Calculations of the dispersion curve have been achieved by introducing particular functions which transform motion equations into an eigenvalue problem. The state vector has been generalized to a phononic material, it leads to a comparatively convenient matrix formulation. The influence of the layer number on the transmission factor is studied. In addition, our interest is focused on the observed gap and how it behaves when phononic structure undergoes a slight change. The result shows that if the central phononic layer is replaced by one or two homogeneous layers, guided modes originate inside the frequency band gaps.     

Energy density and release rate study of an elliptic-arc interface crack in piezoelectric solid under anti-plane shear

The anti-plane problem of an elliptical inhomogeneity with an interfacial crack in piezoelectric materials is investigated. The system is subjected to arbitrary singularity loads (point charge and anti-plane concentrated force) and remote anti-plane mechanical and in-plane electrical loads. Using the complex variable method, the explicit series form solutions for the complex potentials in the matrix and the inclusion regions are derived. The electroelastic field intensity factors, the corresponding energy release rates and the generalized strain energy density at the cracks tips are then provided. The influence of the aspect ratio of the ellipse, the crack geometry and the electromechanical coupling coefficient on the energy release rate and the strain energy density is discussed and shown in graphs. The results indicate that the energy release rate increases with increment of the aspect ratio of the ellipse and the influence of electromechanical coupling coefficient on the energy release rate is significant. The strain energy density decreases with increment of the aspect radio of the ellipse and it is always positive for the cases discussed. The energy release rate, however, can be negative when both mechanical and fields are applied.     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.