Nonlinear study of large deflection of simply supported piezoelectric layered-plate under initial tension

The nonlinear problem of large deflection of a simply supported piezoelectric layered plate under initial tension is studied. The approach follows von Karman’s plate theory for large deflection for a symmetrically layered isotropic case including a piezoelectric layer. The nonlinear governing equations are solved using a finite difference method, by taking the associated linear analytical solution as an initial guess in the numerical iteration procedure. The results for a nearly monolithic plate under a very low applied voltage are found to correlate well with available solutions for a single-layered case under pure mechanical loading and thus the present approach is validated. For three-layered plates made of typical silicon based materials, various initial tension and lateral pressure are considered, and different applied voltages up to a moderate magnitude are implemented. No edge effect was observed, in contrast to the cases of clamped plates in literature. In additions, varying the layer moduli seems to have an insignificant effect upon the structural responses of the layered plate. On the other hand, the piezoelectric effect tends to be apparent only in a low pretension condition. For a relatively large pretension, the effect of initial tension becomes dominant, yielding nearly unique solutions for the structural responses, regardless of the magnitudes of the applied voltage and the lateral pressure.     

Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators

This paper deals with the nonlinear vibration and dynamic response of simply supported shear deformable cross-ply laminated plates with piezoelectric actuators subjected to mechanical, electrical and thermal loads. The material properties are assumed to be independent of the temperature and electric field. Theoretical formulations are based on the higher order shear deformation plate theory and general von Kármán-type equation, which includes thermo-piezoelectric effects. Due to the bending and stretching coupling effects, a nonlinear static problem is first solved to determine the pre-vibration deformation caused by temperature field and control voltage. By adding an incremental dynamic state to the pre-vibration state, the equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of hybrid laminated plates. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply laminated plates. The results presented show the effects of temperature rise, applied voltage and stacking sequence on the nonlinear vibration and dynamic response of the plates.     

Moving load identification on a simply supported orthotropic plate

Different aspects of dynamic identification of moving loads from a vehicle traveling on top of a beam-slab type bridge deck are studied in this paper. The bridge deck is modeled as an orthotropic plate and the loads are modeled as a group of wheel loads or a group of axle loads moving on top of the bridge deck at a fixed spacing. The dynamic response of the bridge deck is obtained from the Hamilton principle with modal superposition. The equation of motion is formulated in state space and the resulting damped least-square identification problem is solved using the dynamic programming method with regularization on the solution. The combination of dynamic strain and velocity measurements, the number of analytical modes included in the identification, the effect of load eccentricity and the choice of regularization parameter are studied and discussed. The use of separate regularization parameter for each of these loads has been implemented and discussed. These considerations are further studied with experimental results from the laboratory. Recommendations on the selection of sensor number and sensor locations for moving loads identification are given.     

Interlaminar stresses of a rectangular laminated plate with simply supported edges subject to free vibration

The interlaminar stresses in a laminated rectangular orthotropic plate with four sides simply supported edges during free vibration was determined by using the integration method involving the dynamic inertia terms and displacements. The approximate stresses solutions are obtained under the effect of frequencies of vibration for four-layer symmetric cross-ply laminates with the ply configurations [0°/90°]_s and [90°/0°]_s, angle-ply laminates with the ply configuration [45°/−45°]_s. Numerical results show that the natural frequency has significant effects on the dominant interlaminar stresses in the stacking sequences [0°/90°]_s, [90°/0°]_s and [45°/−45°]_s.     

Bending of a simply supported circular plate subjected to elastic constraint for inclination at its edge

The bending problems of circular plates which are simply supported and subjected to elastic constraint for the inclination at the edge of the plate are treated, in this study, by an iterative method. Several kinds of the distribution of the elastic constraint for the inclination are considered, among which the stepwise distribution is contained. By this iterative method, any distribution of the elastic constraint can be fairly easily treated. Also, this iterative method is capable of dealing with non-linear constraints, as demonstrated by examples.     

Lateral buckling of a simply supported uniform diagrid

A partial difference equation governing the lateral buckling of a simply supported uniform diagrid is obtained using slope-deflexion equations for two orthogonally intersecting beams. After prescribing the boundary conditions, a solution is obtained in the form of an equation giving the buckling loads.     

Nonlinear dynamic response of a simply supported rectangular functionally graded material plate under the time-dependent thermalmechanical loads

An analysis on nonlinear dynamic characteristics of a simply supported functionally graded materials (FGMs) rectangular plate subjected to the transversal and in-plane excitations is presented in the time dependent thermal environment. Here we look the FGM Plates as isotropic materials which is assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The geometrical nonlinearity using Von Karman’s assumption is introduced. The formulation also includes in-plane and rotary inertia effects. In the framework of Reddy’s third-order shear deformation plate theory, the governing equations of motion for the FGM plate are derived by the Hamilton’s principle. Then the equations of motion with two-degree-of-freedom under combined the time-dependent thermomechanical loads can be obtained by using Galerkin’s method. Using numerical method, the control equations are analyzed to obtain the response curves. Under certain conditions the periodic and chaotic motions of the FGM plate are found. It is found that because of the existence of the temperature which relate to the time the motions of the FGM plate show the great difference. A period motion can be changed into the chaotic motions which are affected by the time dependent temperature.     

Lower-bound shakedown analysis of a simply supported plate carrying a uniformly distributed load and subjected to cyclic thermal loading

A lower-bound analysis is used to obtain the shakedown boundary for a simply supported circular plate carrying a uniformly distributed load subjected to cyclic thermal loading. Situations where the yield stress is constant and a function of position in the plate are examined. For low levels of thermal loading the results are in agreement with the upper-bound calculations of Ponter.     

Concepts for a class of novel piezoelectric self-locking long-stroke actuators

The friction-type motor is the most common type in the field of piezoelectric motors. One limitation of friction-type motors is their inability to achieve high output push force or torque. Based on the theory of self-lock, a novel mechanism for the linear piezoelectric motor is proposed. On the basis of the proposed mechanism, three prototype models have been developed. The new motors transfer the force and displacement generated by a piezoelectric actuator to the output directly, whereas the friction-type motor transfers via the induced friction between the stator and the rotor. The achieved positioning precision is within 10 nm, while the push force is up to 1,176 N. The new motors can be applied in cases where both high positioning precision and heavy load are essential.     

Hybrid controller of a linear piezoelectric walking stage relying on stack/shear piezoelectric actuators

This paper proposes a hybrid control strategy of a novel linear piezoelectric walking stage based on two sorts of piezoelectric actuators, which takes the load variation into account. The proposed stage consists of two parallel 4-bar lever amplification mechanisms with flexure hinges actuated by piezoelectric stacks to heighten the vertical distance (that is more tolerable to the assembly discrepancy), two compression springs (that is able to maintain a fixed linear position without powering), and two shear piezoelectric actuators (that can achieve longer and equivalent to walking motion) in a small form factor. The proposed stage has two operating modes, namely a coarse positioning mode with a more extensive travel range and a fine positioning mode with a nanometer-level resolution, to possess excellent performance for the linear piezoelectric walking stage of load variations. One multimodal switching controller and one feedforward-feedback controller conduct the coarse mode and fine mode, respectively. The optimal frequency for a specific load is obtained through a backpropagation neural network in the multimodal switching control. In the feedforward-feedback control, the inverse mathematical model based on the Bouc-Wen hysteresis model is used to mitigate the hysteresis effect in the feedforward part while the proportional–integral–derivative controller in the feedback part handles the external system disturbances. Experimental results show the proposed hybrid coarse/fine mode control strategy's effectiveness to satisfy an efficient and accurate positioning task.     

The deflection coefficient of a rectangular plate reinforced in the middle

In this paper, we investigated the deflection coefficients of rectangular plates that are reinforced in the middle. For reinforced plates with three different aspect ratios and two types of boundary conditions (simply supported and clamped), we derived the deflection coefficients with respect to the elastic modulus ratio and the relative length of the inner plate using the least-squares method. We performed a finite element analysis of the models, and calculated the deflection coefficients of reinforced plates in terms of the deflection coefficients of simple (nonreinforced) plates. The results can be extended to various types of reinforced rectangular plates.     

Investigation of design parameters for droplet generators driven by piezoelectric actuators

This paper presents a numerical analysis of a drop-on-demand (DOD) piezoelectric (PZT) actuated droplet generator. A finite difference numerical model was established to analyze the design parameters of droplet ejection. First, we discussed the influence of the driving conditions on the droplet ejection characteristics, such as the driving time and the driving volume change in the pressure chamber. The volume factor, an important design parameter, was proposed from the analysis. The ejected droplets can maintain the same ejection velocity at different nozzle diameters, as long as the volume factor remains the same. Two empirical formulas, based on the analysis data, are suitable for the design of PZT actuated droplet generator. The first empirical formula is a linear relationship between the droplet velocity and the volume factor with a slope of 0.3422 for different nozzle diameters. The second empirical formula defines the driving volume of PZT and nozzle diameter to eject the desired droplets. The geometrical design parameters of droplet generator, such as the nozzle thickness, the pressure chamber width and depth, as well as the driving conditions of the PZT actuator, are all included in the analysis. The sensitivity of geometrical design parameters which affect the droplet volume, the droplet velocity, and the lowest driving condition is established. The quantitative criterion for ejection of droplet, liquid jet, and no droplet is presented. The proposed empirical formula and figures provide easy-to-use tool for design of DOD PZT actuated droplet generators.     

Ultra-precise tracking control of piezoelectric actuators via a fuzzy hysteresis model

In this paper, a novel Takagi-Sugeno (T-S) fuzzy system based model is proposed for hysteresis in piezoelectric actuators. The antecedent and consequent structures of the fuzzy hysteresis model (FHM) can be, respectively, identified on-line through uniform partition approach and recursive least squares (RLS) algorithm. With respect to controller design, the inverse of FHM is used to develop a feedforward controller to cancel out the hysteresis effect. Then a hybrid controller is designed for high-performance tracking. It combines the feedforward controller with a proportional integral differential (PID) controller favourable for stabilization and disturbance compensation. To achieve nanometer-scale tracking precision, the enhanced adaptive hybrid controller is further developed. It uses real-time input and output data to update FHM, thus changing the feedforward controller to suit the on-site hysteresis character of the piezoelectric actuator. Finally, as to 3 cases of 50 Hz sinusoidal, multiple frequency sinusoidal and 50 Hz triangular trajectories tracking, experimental results demonstrate the efficiency of the proposed controllers. Especially, being only 0.35% of the maximum desired displacement, the maximum error of 50 Hz sinusoidal tracking is greatly reduced to 5.8 nm, which clearly shows the ultra-precise nanometer-scale tracking performance of the developed adaptive hybrid controller.     

Creative approach for the deflection formula according to relative length of supporting edge in a partially supported circular plate

The purpose of this investigation is to derive the maximum deflection formula of the circular plate with nonaxisymmetric boundary condition, such as chalk valves for vessel and artificial heart valves, with respect to the length ratio of supporting edge using elastic beam theory. To evaluate the deflection characteristics of this plate, we assumed the circular plate to have a cross-section that varies in the longitudinal direction, and derived the maximum deflection formula of this circular plate with a radius of ‘r’ in four boundary conditions. Then to verify the deflection formula that is derived from elastic beam theory, five different relative lengths of the supporting edge were adopted as the design parameter, and finite element analysis was carried out for each model.     

Transient bending of a piezoelectric circular plate

Thin, piezoelectric circular plates are frequently used as active components in transducer and smart materials applications. This paper reports on the exact, explicit solution for the transient motion of a piezoelectric circular plate, built-in or simply supported on the edge and electrically grounded over the entire surface. Expressed by elementary Bessel functions and obtained via exact inverse Laplace transforms, the solution enables the efficient calculation of accurate system parameters.     

Real-time inverse hysteresis compensation of piezoelectric actuators with a modified Prandtl-Ishlinskii model

This paper presents a novel real-time inverse hysteresis compensation method for piezoelectric actuators exhibiting asymmetric hysteresis effect. The proposed method directly utilizes a modified Prandtl-Ishlinskii hysteresis model to characterize the inverse hysteresis effect of piezoelectric actuators. The hysteresis model is then cascaded in the feedforward path for hysteresis cancellation. It avoids the complex and difficult mathematical procedure for constructing an inversion of the hysteresis model. For the purpose of validation, an experimental platform is established. To identify the model parameters, an adaptive particle swarm optimization algorithm is adopted. Based on the identified model parameters, a real-time feedforward controller is implemented for fast hysteresis compensation. Finally, tests are conducted with various kinds of trajectories. The experimental results show that the tracking errors caused by the hysteresis effect are reduced by about 90%, which clearly demonstrates the effectiveness of the proposed inverse compensation method with the modified Prandtl-Ishlinskii model.     

Output feedback integral control of piezoelectric actuators considering hysteresis

In this paper, output feedback integral control of piezoelectric actuators is considered with respect to the hysteresis effect. The linear dynamics of the piezoelectric actuator is modeled as a linear state space system with an input nonlinearity that considers the hysteresis effect. A proof of the Lyapunov stability of the system with integral control is presented, and a method for deriving the upper bound for the regulating gain is shown. A simple example is used to illustrate the approach, and then the approach is applied for tracking a step signal with an experimental single-axis piezoelectric actuator to verify that the system is stable.     

压电陶瓷执行器迟滞非线性补偿与最优控制

为提高空间望远镜精密稳像系统中压电驱动快摆镜(FSM)的摆动精度,对压电陶瓷执行器迟滞非线性补偿和控制技术进行研究。针对压电迟滞的非对称性以及Duhem模型求逆过程复杂的问题,对Duhem模型中的微分方程进行变换,直接建立Duhem非对称逆迟滞模型作为迟滞前馈补偿器,并利用免疫差分进化算法辨识模型参数。在Duhem逆模型补偿压电静态迟滞非线性的基础上,引入基于优化参考跟踪的线性二次型高斯(LQG-ORT)控制方法进一步提高压电执行器的动态定位精度,采用动态迟滞率相关自回归各态历经模型(ARX)建立状态空间方程,用于卡尔曼滤波器预测状态变量和控制器计算状态变量的最优控制系数矩阵。实验结果表明：直接建立的Duhem非对称逆迟滞模型能有效描述压电执行器非对称逆迟滞曲线,拟合均方根误差为0.635 9 V(0.5 Hz),相对误差为0.79%(0.5 Hz);实时跟踪幅值为24μm,频率范围1～80 Hz的目标位移信号,LQG-ORT算法的跟踪误差为0.065 5μm,相对误差为0.27%。