Piezoelectric Constitutive Equations Research Articles

Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers

Langevin transducers are employed in several applications, such as power ultrasound systems, naval hydrophones, and high-displacement actuators. Nonlinear effects can influence their performance, especially at high vibration amplitude levels. These nonlinear effects produce variations in the resonant frequency, harmonics of the excitation frequency, in addition to loss of symmetry in the frequency response and “frequency domain hysteresis”. In this context, this paper presents a simplified nonlinear dynamic model of power ultrasound transducers requiring only two parameters for simulating the most relevant nonlinear effects. One parameter reproduces the changes in the resonance frequency and the other introduces the dependence of the frequency response on the history of the system. The piezoelectric constitutive equations are extended by a linear dependence of the elastic constant on the mechanical displacement amplitude. For introducing the frequency hysteresis, the elastic constant is computed by combining the current value of the mechanical amplitude with the previous state amplitude. The model developed in this work is applied for predicting the dynamic responses of a 26 kHz ultrasonic transducer. The comparison of theoretical and experimental responses, obtained at several input voltages around the tuned frequency, shows a good agreement, indicating that the model can accurately describe the transducer nonlinear behavior.

Synergistic use of smart materials for vibration-based energy harvesting

Vibration-based energy harvesting is an approach where available mechanical vibration energy is converted into electrical energy that can be employed for different purposes. This paper deals with the synergistic use of smart materials for energy harvesting purposes. In essence, piezoelectric and shape memory alloys are combined to build an energy harvesting system. The combined effect of these materials can increase the system performance and reduce some limitations. The possibility to control the mechanical stiffness under vibration by a shape memory alloy (SMA) element can provide the ability to tune resonant frequencies in order to increase the output power. The analysis is developed considering a one-degree of freedom mechanical system where the restitution force is provided by an SMA element. The electro-mechanical coupling is provided by a piezoelectric element. Linear piezoelectric constitutive equation is employed together with the Brinson’s model for SMA element. Numerical simulations are carried out showing different responses of the system indicating that the inclusion of the SMA element can be used to extend the operational range of the system.

Numerical simulation of piezoelectric effect of bone under ultrasound irradiation

The piezoelectric effect of bone under ultrasound irradiation was numerically simulated using an elastic finite-difference time-domain method with piezoelectric constitutive equations (PE-FDTD method). First, to demonstrate the validity of the PE-FDTD method, the ultrasound propagation in piezoelectric ceramics was simulated and then compared with the experimental results. The simulated and experimental waveforms propagating through the ceramics were in good agreement. Next, the piezoelectric effect of human cortical bone on the ultrasound propagation was investigated by PE-FDTD simulation. The simulated result showed that the difference between the waveforms propagating through the bone without and with piezoelectricity was negligible. Finally, the spatial distributions of the electric fields in a human femur induced by ultrasound irradiation were simulated. The electric fields were changed by a bone fracture, which depended on piezoelectric anisotropy. In conclusion, the PE-FDTD method is considered to be useful for investigating the piezoelectric effect of bone.

Analysis of Energy Harvesting Performance for <inline-formula><tex-math>$d_{15}$</tex-math> </inline-formula> Mode Piezoelectric Bimorph in Series Connection Based on Timoshenko Beam Model

Shear strain is exactly described by combining the piezoelectric constitutive equations of d 15 mode with the Timoshenko beam model. We provide an electromechanical model to evaluate the energy harvesting performance of shear mode cantilevered piezoelectric bimorph in series connection. The output peak voltage-frequency and the output peak power-frequency curves are simulated by using the proposed model at different load resistances. The results are in good agreement with the finite element simulation results, indicating that the proposed model is valid for evaluating the energy harvesting performances of the cantilevered piezoelectric bimorph. The variations of the output peak voltage and power with load resistance change for excitation are also discussed. Moreover, the effect of proof mass on the energy harvesting performance is analyzed at certain load resistance. The proposed model can be used for the structure optimized design and performance improvement of shear mode piezoelectric energy harvesting.

Modeling and simulation of macro-fiber composite layered smart structures

Piezo fiber composite material, macro-fiber composite (MFC), is increasingly applied in engineering, due to its high flexibility and strong actuation forces. This paper develops a linear electro-mechanically coupled finite element (FE) model for composite laminated thin-walled smart structures bonded with orthotropic MFCs having arbitrary piezo fiber orientation. Two types of MFCs are considered, namely, MFC-d31 in which the d31 effect dominates the actuation forces, and MFC-d33 which mainly uses the d33 effect. The FE model is developed based on the Reissner–Mindlin hypothesis using linear piezoelectric constitutive equations. The present results are compared with ANSYS and experimental results reported in the literature (Bowen et al., 2011). Afterwards, isotropic or composite structures with cross-ply laminates, integrated with MFC-d31 or -d33 patches having different fiber orientation, are simulated under a certain electric voltage on the MFC patches.

Equivalent circuit method for resonant analysis of multilayer piezoelectric-magnetostrictive composite cantilever structures

In this paper, we discuss a theoretical model by the equivalent circuit method for magnetoelectric (ME) response at resonant frequencies corresponding to bending mode oscillations. A dynamic theoretical model of arbitrary multilayer cantilever with alternating piezoelectric (PE) and magnetostrictive (MS) layers on a substrate (Sub) was conducted. The equivalent circuit of the bending mode vibration has obtained by solving the piezoelectric, magnetostrictive and substrate constitutive equations under boundary conditions. The dependence of resonant frequencies and resonant ME coefficients on the material properties and structural dimensions is investigated. The ME coefficients were normalized by the total thickness of MS and PE layers and quality factor. As a demonstration, our theory for multilayer composite cantilever was then applied to specifically Metglas-AlN-Si systems. Our results demonstrate that the structural dimensions strongly influenced the neutral plane in the composite cantilever and then in turns strongly influenced the ME response. In addition, characteristics of multilayer ME cantilever in static (low frequency) station will be analyzed. The theory shows good agreement with experimental data. Our discussion may be helpful to the designing and fabrication of the giant ME cantilever with more realistic and optimal geometry.

X-ray diffraction measurement of residual stress in sol-gel grown lead zirconate titanate thick films on nickel-based super alloy substrate

Residual compressive stress of Pb(Zr0.52Ti0.48)O3 thick films was investigated using residual strains derived from X-ray diffraction patterns. Sin2ψ method was applied for the 5, 10 and 15 µm sol-gel derived thick films annealed at 700°C for 1 hr as high frequency structural health monitoring square-shape transducers of 10×10 mm, deposited onto the curved nickel-based super alloy substrates. A triaxial model was proposed based on piezoelectric constitutive equations, and Bragg’s law at a large diffraction angle (∼89o) was utilized considering the electromechanical coupling factor as well as elastic, dielectric and piezoelectric constants. Thickness variations led to a significant change in residual stress magnitudes delineated from more-accurate triaxial model compared to small angle plane-stress results not considering the piezoelectric coupling effects.

Sensitivity and Noise Evaluation of a Bonded Magneto(elasto) Electric Laminated Sensor Based on In-Plane Magnetocapacitance Effect for Quasi-Static Magnetic Field Sensing

The quasi-static magnetic field detection of a layer-bonded magneto(elasto) electric (ME) laminate has been investigated by measuring the in-plane electric capacitance via its interdigital electrodes close to the piezoelectric resonant frequency. The ME-layered composite is considered as a stress-induced dielectric effect because there is practically no direct response of the electric capacitance to an external magnetic field. The sensitivity is dominated by the magnetoelastic coupling in the magnetic layer and on the stress induced by the permittivity change in the piezoelectric layer. The low-frequency magnetocapacitance effect is sensitive to an external magnetic bias which can modulate the electric permittivity by producing a stress. The magnetoelastic coupling is another important parameter for this magnetic field detection mode. For a given magnetic field, the amplitude of the magnetostriction is directly related to this parameter as well. Therefore, an optimal magnetic bias can maximize the induced strain or stress which is coupled into the piezoelectric layer through the change of the electric permittivity in this layer. To evaluate the sensitivity and the noise performance by the magnetocapacitance effect, we have used the piezoelectric and magnetic constitutive equations to predict the permittivity dependence. Experimentally, this sensor achieved an equivalent magnetic noise spectral density, presently still limited, by the noise of the detection electronics, ~100 pT/√Hz at 1 Hz and offered a dc detection capability. With the model and experimental nonlinear factors, an equivalent sensor noise spectral density close to the pT/√Hz can be ultimately predicted considering the mechanical loss limitation of the sensor.

Experimental verification of a bridge-shaped, nonlinear vibration energy harvester

This paper reports a comprehensive modeling and experimental characterization of a bridge shaped nonlinear energy harvester. A doubly clamped beam at large deflection requires stretching strain in addition to the bending strain to be geometrically compatible, which stiffens the beam as the beam deflects and transforms the dynamics to a nonlinear regime. The Duffing mode non-linear resonance widens the frequency bandwidth significantly at higher frequencies than the linear resonant frequency. The modeling includes a nonlinear measure of strain coupled with piezoelectric constitutive equations which end up in nonlinear coupling terms in the equations of motion. The main result supports that the power generation is bounded by the mechanical damping for both linear and nonlinear harvesters. Modeling also shows the power generation is over a wider bandwidth in the nonlinear case. A prototype is manufactured and tested to measure the power generation at different load resistances and acceleration amplitudes. The prototype shows a nonlinear behavior with well-matched experimental data to the modeling.

Theoretical investigation of magnetoelectric effect in multilayer magnetoelectric composites

The aim of this paper is the modeling of an arbitrary magneto–elasto-electric multilayer constitute by magnetostrictive, piezoelectric and elastic substrate layers under free–free boundary conditions. A dynamic theory corresponding to the driving frequency of magnetoelectric (ME) response coupled flexural and extensional oscillations was constructed. In the proposed model, the influence of thickness dependence of stress, strain and magnetic, electric fields within a sample stack are taken into account. The results of ME voltage coefficient dependent on driving frequency are obtained based on magnetostrictive, piezoelectric and elastic constitutive equations. As a demonstration, the numerical results are given for the Terfenol-D/PZT/Si material system with magnetic field excitation parallel and electric polarization perpendicular to the multilayer, which constituted by bilayer Terfenol-D/PZT placed on a Si elastic substrate. The theoretical model for static ME response in such structures is also established. Based on the model, some calculations on ME response are conducted and discussed. Our results show that in both the static and resonant ME response, the dimensional parameters of the layer sequence, piezoelectric fraction and substrate thickness essentially determined the ME voltage coefficient. Furthermore, the influences of load impedance on the magnetoelectric coupling coefficients are also investigated.

Analyzing the installation angle error of a SAW torque sensor

When a torque is applied to a shaft, normal strain oriented at ±45° direction to the shaft axis is at its maximum, which requires two one-port SAW resonators to be bonded to the shaft at ±45° to the shaft axis. In order to make the SAW torque sensitivity high enough, the installation angle error of two SAW resonators must be confined within ±5° according to our design requirement. However, there are few studies devoted to the installation angle analysis of a SAW torque sensor presently and the angle error was usually obtained by a manual method. Hence, we propose an approximation method to analyze the angle error. First, according to the sensitive mechanism of the SAW device to torque, the SAW torque sensitivity is deduced based on the linear piezoelectric constitutive equation and the perturbation theory. Then, when a torque is applied to the tested shaft, the stress condition of two SAW resonators mounted with an angle deviating from ±45° to the shaft axis, is analyzed. The angle error is obtained by means of the torque sensitivities of two orthogonal SAW resonators. Finally, the torque measurement system is constructed and the loading and unloading experiments are performed twice. The torque sensitivities of two SAW resonators are obtained by applying average and least square method to the experimental results. Based on the derived angle error estimation function, the angle error is estimated about 3.447°, which is close to the actual angle error 2.915°. The difference between the estimated angle and the actual angle is discussed. The validity of the proposed angle error analysis method is testified to by the experimental results.

A uniform model for direct and converse magnetoelectric effect in laminated composite

Direct and converse magnetoelectric effects are two opposite processes, which were integrated in a model based on equivalent circuit. Following the piezoelectric and magnetostrictive constitutive equations, we obtained magnetic-mechanical-electric equations, and then built a symmetric equivalent circuit about the magnetoelectric coupling. We equivalently analyzed the direct and converse magnetoelectric effects with this equivalent circuit. The numerical calculations fit the reported results very well, including the magnetoelectric changes with the thickness ratio and frequency, the relative lower converse magnetoelectric resonance frequency, and the phase increase of π around the direct and converse magnetoelectric resonance frequencies. This model helps us understand the magnetoelectric properties deeply, especially for the converse magnetoelectric effect.

Thermo-electro-mechanical impedance based structural health monitoring of plates

This study presents a modeling of a plate structure in impedance-based structural health monitoring (ISHM) method using piezoelectric wafer active sensor (PWAS) considering temperature effects. For modeling of the plate, in plane and transverse vibrations of the plate has been considered. Rayleigh–Ritz method was used to discretize the system using three characteristic functions: one of them is plate characteristic function, the other one is Chebyshev pseudo spectral characteristic function and Discrete Singular Convolution (DSC) characteristic functions. After discretization, impedance of PWAS is determined using piezoelectric constitutive equations. In addition to the temperature effect, parameters uncertainly including Young’s modulus and density were considered in the modeling, because these uncertainties are affected on the impedance of PWAS. So, the Monte Carlo simulation is applied to consider parameters uncertainly on PWAS impedance.

Measurement of the internal stress and electric field in a resonating piezoelectric transformer for high-voltage applications using the electro-optic and photoelastic effects

The high output voltages from piezoelectric transformers are currently being used to accelerate charged particle beams for x-ray and neutron production. Traditional methods of characterizing piezoelectric transformers (PTs) using electrical probes can decrease the voltage transformation ratio of the device due to the introduction of load impedances on the order of hundreds of kiloohms to hundreds of megaohms. Consequently, an optical diagnostic was developed that used the photoelastic and electro-optic effects present in piezoelectric materials that are transparent to a given optical wavelength to determine the internal stress and electric field. The combined effects of the piezoelectric, photoelastic, and electro-optic effects result in a time-dependent change the refractive indices of the material and produce an artificially induced, time-dependent birefringence in the piezoelectric material. This induced time-dependent birefringence results in a change in the relative phase difference between the ordinary and extraordinary wave components of a helium-neon laser beam. The change in phase difference between the wave components was measured using a set of linear polarizers. The measured change in phase difference was used to calculate the stress and electric field based on the nonlinear optical properties, the piezoelectric constitutive equations, and the boundary conditions of the PT. Maximum stresses of approximately 10 MPa and electric fields of as high as 6 kV/cm were measured with the optical diagnostic. Measured results were compared to results from both a simple one-dimensional (1D) model of the piezoelectric transformer and a three-dimensional (3D) finite element model. Measured stresses and electric fields along the length of an operating length-extensional PT for two different electrical loads were within at least 50 % of 3D finite element simulated results. Additionally, the 3D finite element results were more accurate than the results from the 1D model for a wider range of electrical load impedances under test.

Disturbance rejection control for vibration suppression of piezoelectric laminated thin-walled structures

Abstract Thin-walled piezoelectric integrated smart structures are easily excited to vibrate by unknown disturbances. In order to design and simulate a control strategy, firstly, an electro-mechanically coupled dynamic finite element (FE) model of smart structures is developed based on first-order shear deformation (FOSD) hypothesis. Linear piezoelectric constitutive equations and the assumption of constant electric field through the thickness are considered. Based on the dynamic FE model, a disturbance rejection (DR) control with proportional-integral (PI) observer using step functions as the fictitious model of disturbances is developed for vibration suppression of smart structures. In order to achieve a better dynamic behavior of the fictitious model of disturbances, the PI observer is extended to generalized proportional-integral (GPI) observer, in which sine or polynomial functions can be used to represent disturbances resulting in better dynamics. Therefore the disturbances can be estimated either by PI or GPI observer, and then the estimated signals are fed back to the controller. The DR control is validated by various kinds of unknown disturbances, and compared with linear-quadratic regulator (LQR) control. The results illustrate that the vibrations are better suppressed by the proposed DR control.

A Three-Dimensional Model of the Effective Electromechanical Impedance for an Embedded PZT Transducer

A three-dimensional model of the effective electromechanical impedance for an embedded PZT transducer is proposed by considering the interaction between a PZT patch and a host structure. By introducing an effective mechanical impedance, the coupled electromechanical admittance formulations are derived using the piezoelectric constitutive equations. Then, a modified methodology for monitoring structure changes using an electromechanical impedance (EMI) technique is proposed. In the proposed method, the changes in the host structure are monitored by using the “active” part associated with the structural mechanical impedance, which is extracted from the measured raw admittance signatures. The strength gain of a concrete beam with embedded PZT transducers during the curing age was monitored with the proposed methodology. The experimental results demonstrate that the use of the “active” part is more sensitive as opposed to the raw admittance signatures for structural health monitoring (SHM).

Analyses for Thickness Vibration of Piezoelectric Ring

The thickness vibration of piezoceramic ring is investigated by theoretical analysis, numerical simulation, and experimental measurement. Based on the piezoelectric constitutive equations and the hypothesis that a ring has no strain in circumference when vibrating, the thickness vibration mode of ring is treated as a radial longitudinal extended vibration equivalently, the equations was simplified and the resonant frequency of thickness vibration is derived. The resonant frequency is also simulated with finite element software and measured by Impedance Analyzer. Theoretic calculation of resonant frequency is compared with the simulation and measurement. The errors between them are analyzed.

Vibration analysis for piezoceramic ring

Abstract The vibration property of piezoceramic ring is investigated by theoretical analysis, numerical simulation, and experimental measurement in this paper. Based on the piezoelectric constitutive and kinematical equation, the expression for resonant frequency of piezoceramic ring is derived by setting the boundary of vibrating ring as free. The expressions are calculated with numerical method especially for vibration modes in radial and wall-thickness direction. The relationship between resonant frequencies of two modes and the radius (or thickness) is obtained. The results calculated are compared with measurements by Impedance Analyzer, and the errors between them are analyzed in the paper.

Active structural vibration control: Robust to temperature variations

d-form augmented piezoelectric constitutive equations which take into account temperature dependence of piezoelectric strain coefficient (d31) and permittivity (∈33), are converted into e-form. Using e-form constitutive equations, a finite element model of a smart two dimensional plate instrumented with piezoelectric patches is derived. Equations of motion are derived using Hamilton's variational principle. Coupled equations of motion are uncoupled using modal analysis. Modal state vectors are estimated using the Kalman observer. The first mode of smart cantilevered plate is actively controlled using negative first modal velocity feedback at various temperatures. Total control effort required to do so is calculated using the electro-mechanical impedance method. The temperature dependence of sensor voltage, control voltage, control effort and Kalman observer equations is shown analytically. Simulation results are presented using MATLAB. Variations in (i) peak sensor voltage, (ii) actual and estimated first modal velocities, (iii) peak control voltage, (iv) total control effort and (v) settling time with respect to temperature are presented. Active vibration control performance is not maintained at temperature away from reference temperature when the temperature dependence of piezoelectric stress coefficient ‘e31’ and permittivity ‘∈33’ is not included in piezoelectric constitutive equations. Active control of vibrations becomes robust to temperature variations when the temperature dependence of ‘e31’ and ‘∈33’ is included in piezoelectric constitutive equations.

Analysis of vibration waveforms of electromechanical response to determine piezoelectric and electrostrictive coefficients

We developed a possible method to determine both coefficients of piezoelectricity (d) and electrostriction (M) at the same time by a waveform analysis of current and vibration velocity in the resonance state. The waveforms of the current and vibration velocity were theoretically described using the equations of motion and piezoelectric constitutive equations, considering the dissipation effect. The dissipation factor of the d coefficient and M coefficient is dielectric loss tangent tan δ. The waveforms measured in all of the ceramics, such as Pb(Zr,Ti)O(3) (PZT), Pb(Mg,Nb)O(3) (PMN), and 0.8Pb(Mg(1/3)Nb2/3)O(3)-0.2PbTiO(3) (PMN-PT), were well fitted with the calculated waveform. This fitting produced both the d and M coefficients, which agreed with those determined via the conventional methods. Moreover, the respective contributions of both piezoelectricity and electrostriction to the d value determined in the resonance-antiresonance method were clarified.