Piezoelectric Nonlinearity Research Articles

Dynamic Control of Piezoelectric Sandwich Beams with Multi-Frequency Excitations

In this work, we present a non-linear analysis of the dynamics of a sandwich beam with piezoelectric patches under multi-frequency excitations. The model takes into account geometrical non-linearities, piezoelectric non-linearities, and inertia. Hamilton’s principle is used to determine the electromechanical equation of motion, and Garlekin’s technique is used to determine the equivalent model of the piezoelectric generator. A mathematical methodology based on the multi-scale method for vibration control and stability is reviewed in this work to determine a system of amplitude and phase equations. The control of primary and secondary mode resonances is developed, and feedback effects are analyzed for small and large amplitudes of vibrations of sandwich beams. Frequency response curves are presented and discussed for various gain parameters.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

Vibration control of membrane structures by piezoelectric actuators considering piezoelectric nonlinearity under strong electric fields

Membrane structures, which are widely used in aerospace and civil engineering, are susceptible to large amplitude vibrations due to their low structural damping and high flexibility, leading to the degradation of their working performance. Strong electric fields are commonly needed for piezoelectric actuators to control larger amplitude vibration of membrane structures. However, the effect of piezoelectric nonlinearity due to strong electric fields on the control performance of membrane structures is still underexplored. Given the importance of piezoelectric nonlinearity due to strong electric fields, the present study aims to investigate the vibration control performance of piezoelectric actuators for membrane structures. First, based on the piezoelectric nonlinear constitutive equation, the actuation equation of the piezoelectric actuator is generated. Then, following D′Alembert's principle, a nonlinear dynamic control model of membrane structure containing actuation force is derived. Finally, active vibration control based on velocity feedback algorithms is realized. The results show that the predicted strain without considering the piezoelectric nonlinearity is 80 % smaller than the finite element structure at 500 V. Meanwhile, the errors of Root Mean Squared (RMS) control displacement without considering the piezoelectric nonlinearity are more than 24 % when vibration amplitudes achieve 10 % of the membrane thickness. The developed model provides theoretical implications for active vibration control of flexible structures with piezoelectric actuators in strong electric field environments.

A nonlinear analysis of electrically forced vibrations of piezoelectric plates with viscous damping near the thickness-shear mode

Based on the theory of nonlinear piezoelectricity, an approximate solution for electrically forced nonlinear thickness-shear vibrations of piezoelectric plates is introduced. The model considers an infinite piezoelectric plate subjected to an alternative voltage, incorporating 3rd-order and 4th-order elastic constants and viscous damping under finite deformation. The system's differential equations for steady-state forced vibrations are derived from the nonlinear pure thickness-shear vibration problem, and transformed so that the new electrical boundary conditions are free. After the application of Galerkin's method, the transformed differential equations lead to a cubic equation, capturing the nonlinear current-frequency curves of AT-cut quartz plates without the need of the quality factor Q, due to the introduced viscous damping. When the damping effect is ignored, the frequency response curves of the AT-cut quartz plate under a specific voltage are confirmed with the existing literature. The study emphasizes the presence of the critical voltage and the importance of the fourth-order elastic constant (c6666) in the stability, revealing lower c6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability.

Fractional-Order Modeling of Piezoelectric Actuators with Coupled Hysteresis and Creep Effects

A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. Offering a simpler structure and superior tracking performance compared to conventional models, the proposed fractional-order model parameters are identified using a method that integrates actuator dynamics and employs the particle swarm optimization algorithm. Experimental validation on a piezoelectric actuation platform confirms the model’s superior accuracy and simplified structure, contributing to a deeper understanding of piezoelectric actuation mechanisms and providing an efficient modeling tool for enhanced system performance.

Electromechanical coupling dynamics for a novel non-resonant harmonic piezoelectric motor

Most of the traditional piezoelectric motors are driven by the resonance of piezoelectric ceramics and can achieve a large output torque. However, these motors also have the defects of high speed and difficulty to adjusting the speed when working. To overcome these issues, a novel non-resonant harmonic rotary piezoelectric motor operating at low speed is presented. Electromechanically coupled vibration has a great influence on the dynamic performance of the piezoelectric motor, which may lead to instability of the output performance of the motor. Therefore, the electromechanical coupling dynamic characteristics of the proposed non-resonant piezoelectric motor are studied in this paper. The working principle of the motor is illustrated, and the electromechanical dynamic model and nonlinear dynamic equations are established. Utilizing the dynamic equations, the coupled natural frequencies, amplitude-frequency characteristics, and dynamic responses of the motor are developed. Meanwhile, the influence of parameters on natural frequency and nonlinear response is analyzed. What's more, the natural frequencies of the system are tested by building an experimental platform. Results show the minimum coupled natural frequency of the motor is 11231 rad/s. The length of the piezoelectric stack, the length, and mass block of the harmonic rod have an important influence on the natural frequency of the proposed non-resonant harmonic piezoelectric motor. The nonlinear piezoelectric effect has a larger effect on the forced response amplitude when the excitation frequency is far away from resonance, but a smaller effect on the forced response amplitude when the excitation frequency is near resonance. What's more, the maximum error between the theoretical natural frequencies and the test frequencies is 3%, which verifies the correctness of the theoretical model.

Electromechanical model and simple numerical analysis for a piezoelectric vibration energy harvester considering nonlinear piezoelectricity, nonlinear damping, and self-powered synchronized switch circuit

Establishment of an analytical technique that considers nonlinear piezoelectricity, nonlinear damping, and the connections to the self-powered synchronized switch circuit represents a challenging but practical issue in the development of piezoelectric vibration energy harvesters. The two-way coupled analysis method, which combines numerical software with a circuit simulator, can simulate the performance of the harvester, but it imposes a high computational load. We develop a numerical analysis technique that is more readily implemented to allow us to produce accurate predictions of the output power, displacement, and frequency response characteristics of this harvester with the switch circuit. First, we derive the electromechanical equation required for the miniature harvester, and we then determine the parameters in this equation using the harmonic balance method. Finally, the governing equation coupling the piezoelectric vibration energy harvester with the self-powered synchronized switch circuit is simplified, taking the form of a quartic equation with respect to the displacement. The simulation replicated a reduction in the displacement at higher resistive loads, wider variations in the resonance frequency, and the nonlinear stiffness softening characteristics that were observed experimentally. This paper provides a practical method for predicting the performance of the nonlinear piezoelectric vibration energy harvester with the synchronized switch circuit.

A Study on the Effect of Piezoelectric Nonlinearity on the Bending Behaviour of Smart Laminated Composite Beam

This paper presents a finite element analysis to model and analyze composite laminated beams with distributed piezoelectric actuators attached to the top and bottom surfaces considering nonlinear constitutive equations under a high electric field. The static response is presented for piezoelectric composite laminated beam using higher order electric field nonlinearity to assess the effect of electrostriction and elastostriction coefficient at a high electric field. A finite element approach based on higher-order shear deformation theory is applied for static analysis of composite laminated beams, varying the thickness and orientation of laminates, to verify the nonlinear effect under a high electric field. A good comparison of results is shown with the available results in the literature. The finding of the results highlights the importance of considering elastostriction term along with the electrostriction term in determining the deflection and stresses of the composite laminated beam.

Influence of doping and thickness on domain avalanches in lead zirconate titanate thin films

In undoped lead zirconate titanate films of 1–2 μm thick, domain walls move in clusters with a correlation length of approximately 0.5–2 μm. Band excitation piezoresponse force microscopy mapping of the piezoelectric nonlinearity revealed that niobium (Nb) doping increases the average concentration or mobility of domain walls without changing the cluster area of correlated domain wall motion. In contrast, manganese (Mn) doping reduces the contribution of mobile domain walls to the dielectric and piezoelectric responses without changing the cluster area for correlated motion. In both Nb and Mn doped films, the cluster area increases and the cluster density drops as the film thickness increases from 250 to 1250 nm. This is evident in spatial maps generated from the analysis of irreversible to reversible ratios of the Rayleigh coefficients.

Tuning the primary resonance of vibrating beam micro-gyroscopes based on piezoelectric actuation and nonlinearity in curvature and inertia

Tuning the primary resonance of vibrating beam micro-gyroscopes based on piezoelectric actuation and nonlinearity in curvature and inertia

Experimental characterization of piezoelectric and elastic operational field-dependent non-linearities of multilayer benders for resonant driving of bone-conduction hearing aids via a dual-actuators side-by-side device

A soft piezoceramic multilayer (ML) bending thin and thick dual-actuators side-by-side device, for use in bone-conduction hearing aids (BCHA), was characterized experimentally. For this purpose, first, quasi-static (0.1 Hz square wave) transverse displacements (deflections) of each ML bender were measured using a laser vibrometer (LV) under unimorph and bimorph drivings at various non-amplified peak (p) voltages (1Vp–13Vp). The resulting peak displacement–voltage curves were found clearly nonlinear, indicating a pronounced piezoelectric (PE) field-dependent nonlinearity (FDNL). Then, experimental electric impedance analyses were conducted for each ML bender, using an impedance analyzer, under 1Vp unimorph driving for the whole audio frequency range (20 Hz–20 kHz). The identified three modes’ superposed impedance magnitudes and phases showed their increasing deviation to the frequency scale left with increasing the modes’ order, indicating a possible softening elastic FDNL, as confirmed later by harmonic frequency response LV measurements at 1Vp. Finally, the harmonic frequency responses, for the above audio frequency range, were measured for each ML bender using the LV under unimorph and bimorph drivings at various non-amplified voltages (1Vp–10Vp). Here, beside the PE FDNL, the frequency response functions showed clearly a pronounced softening elastic FDNL. Consequently, with increasing input voltages, the identified two modes’ displacement amplitudes were nonlinearly increasing, while resonant frequencies and corresponding quality factors were nonlinearly decreasing. The three parameters-voltage curves were found to fit well (R2 ≈ 1) with high-order polynomials. These experimental results contradict the widely assumed displacement–potential linearity of middle ear or BCHA implantable PE vibrators of open literature available types (with or without shim bimorphs, unimorphs or diaphragms, and stacks) under similar targeted driving voltage and frequency ranges. The ML benders type quasi-static, frequency response and electric impedance characterizations, as well as the dual actuators side-by-side device, are proposed here for the first time for hearing aid applications.

High-Level Vibration for Single-Frequency and Multi-Frequency Excitation in Macro-Composite Piezoelectric (MFC) Energy Harvesters, Nonlinearity, and Higher Harmonics.

This paper presents an extensive experimental investigation to identify the influence of signal parameters on a piezoelectric harvester's performance. A macro-fibre composite energy harvester was studied as an advanced, flexible, high-performance energy material. Gaussian white noise, and single-frequency and multi-frequency excitation were used to investigate nonlinearity and multiple-frequency interactions. Using single low and high frequencies, we identified the nonlinearity of the harvester's vibration. Multi-frequency excitation with a series of low-to-high-frequency harmonics mimicked the practical vibration signal. Under such multi-frequency excitation, the harvester's nonlinear behaviour was studied. Finally, the interaction effects among multiple frequencies were identified. The results show that under pure resonant excitation, high-level vibration led to high-level mechanical strain, which caused nonlinear vibration behaviour. Moreover, it was shown that the different harmonics excited the various structure bending modes, which caused the nonlinearity of multi-frequency excitation. The first four harmonics of the real-time signal were important. The experimental results emphasise the resonant nonlinearity and interactions of multi-frequency excitation effects.

Statistical linearization for random vibration energy harvesting with piezoelectric material nonlinearity

Due to piezoelectric softening and dissipative nonlinearities, the piezoelectric cantilever energy harvester exhibits nonlinear hysteresis when subjected to large excitation. These nonlinearities have brought significant challenges to the modeling and response prediction of randomly excited piezoelectric energy harvesting systems. In this study, the voltage responses of the nonlinear piezoelectric cantilever energy harvester under random excitation are initially assumed to follow the Gaussian distribution which is experimentally validated later. The equivalent linear transfer function are derived from the approximate linearization of the stiffness and damping using the statistical linearization (SL) technique. The mathematical expectations of the voltage responses are calculated from the multivariate normal distributions. Frequency sweep experiments are conducted to a cantilever energy harvester to identify the nonlinear piezoelectric material properties. The statistically linearized model was experimentally validated under random base acceleration excitation by comparing the probability density function of the predicted voltage responses and average power against the experimental measurements. The advantage of the SL technique lies in allowing one to use an iterative procedure to estimate the equivalent linear terms while the analytical expressions are unattainable because of the complex nonlinearity in the governing equations. The results show that the prediction of the SL model to the random base acceleration excitation agrees with experimental measurements with a broadband frequency range, although only the fundamental mode of the beam is considered.

Power characteristics of vibration-based piezoelectric energy harvesters: the effect of piezoelectric material nonlinearity

Piezoelectric energy harvesting has received tremendous interests in the past two decades as a viable solution to self-powered electronics and devices. Recently, significant emphasis has been given to nonlinear energy harvesters driven by the desire for broadband, high-performance energy harvesting. Numerous efforts have been devoted to the understanding and modeling of the electromechanical coupling and the effect of nonlinearities introduced by mechanical and electrical aspects of the system. However, very few works in the literature considered the effect of piezoelectric material nonlinearity on the system power performance. Nevertheless, it has been found that piezoelectric nonlinearity is significant even at low to moderate excitation level. This paper is motivated to study the power behavior of piezoelectric energy harvesters with piezoelectric nonlinearity, most importantly, the power limit and electromechanical coupling. For this purpose, an approximate model is developed from the nonlinear model in the literature to derive the closed-form expressions of important power characteristics. Analytical analysis shows that the effect of piezoelectric material nonlinearity results in a nonlinear damping term and a nonlinear stiffness term in the approximate model. The approximate solutions of optimal load resistance, maximum power, power limit, and critical electromechanical coupling are obtained and validated by numerical simulations first. The induced nonlinear damping reduces the power limit of the system compared to its linear counterpart. Interestingly, a harvester that exhibits strong electromechanical coupling under small excitation could become weakly coupled under large excitation. The analytical analysis and numerical results are validated by experiments.

Nonlinearity of Piezoelectric Micromachined Ultrasonic Transducer Using AlN Thin Film

We present the nonlinearity of Aluminum Nitride (AlN)-based Piezoelectric Micromachined Ultrasonic Transducer (PMUT) utilizing Laser Doppler Vibrometer (LDV) technique. The PMUT working at resonant frequency excite the piezoelectric layer into strong nonlinear region. The nonlinear phenomena are observed, such as frequency shift and nonlinear out-of-plane displacement magnitude. A mathematic model of piezoelectric nonlinearity is employed for analyzing the nonlinear behavior, and the second order piezoelectric coefficient is obtained subsequently. Approximately 120 harmonics, which are generated by PMUT nonlinearity, are obtained experimentally under a single-tone AC signal of a relatively high-level voltage. In addition, the number of harmonics can be controlled meticulously. Three different applications are developed to utilize harmonic generations in acoustic-optical hybrid microsystem and Radio Frequency (RF) field. The observation and analysis of AlN piezoelectric nonlinearity could benefit a further understanding of PMUT based on AlN thin film. We believe the generated harmonics can be potentially used in a wide variety of applications in signal processing and modulation.

Directional dependence of nonlinear piezoelectric surface acoustic waves in lithium niobate

A model for the propagation of finite-amplitude surface acoustic waves (SAWs) in a piezoelectric material [Hamilton et al., J. Acoust. Soc. Am. 100, 2567(A) (1996)] is used to investigate the dependence of SAW nonlinearity on crystal cut and propagation direction in lithium niobate (LiNbO3). The model is based on Hamiltonian formalism, in which the boundary conditions at the free surface need only be evaluated at linear order, thus simplifying computation of the nonlinear problem. In contrast with the scalar coefficient of nonlinearity that appears in evolution equations for bulk planewaves, nonlinearity of SAWs is described by a matrix for harmonic interaction due to the variation of depth dependence of the SAW field with frequency. Components of the nonlinearity matrix are generally complex-valued as a result of material anisotropy and piezoelectricity. Variation with propagation direction of the (1,1) component of the nonlinearity matrix, which governs second-harmonic generation, is presented for LiNbO3 crystals that are X-cut, Y-cut, and Z-cut. Relative contribution of each nonlinear mechanism (nonlinear elasticity, nonlinear piezoelectricity, electrostriction, and nonlinear dielectricity) to the nonlinearity matrix is examined. Distortion of an initially sinusoidal waveform during propagation is demonstrated by integrating coupled spectral evolution equations for the harmonic amplitudes.

Parameter Identification of the Nonlinear Piezoelectric Shear d15 Coefficient of a Smart Composite Actuator

The objective of this work is to characterize the nonlinear dependence of the piezoelectric d15 shear coefficient of a composite actuator on the static electric field and include this effect in finite element (FE) simulations. The Levenberg-Marquardt nonlinear least squares optimization algorithm implemented in MATLAB was applied to acquire the piezoelectric shear coefficient parameters. The nonlinear piezoelectric d15 shear constant of the composite actuator integrated with piezoceramic d15 patches was obtained to be 732 pC/N at 198 V. The experimental benchmark was simulated using a three-dimensional piezoelectric FE model by taking piezoelectric nonlinearity into consideration. The results revealed that the piezoelectric shear d15 coefficient increased nonlinearly under static applied electric fields over 0.5 kV/cm. A comparison between the generated transverse deflections of the linear and nonlinear FE models was also performed.

Nonlinear electromechanical modeling and robustness of soft robotic fish-like energy harvester: insights and possible issues

This work investigates the possible integration of an energy harvester in a bioinspired fish-like aquatic unmanned vehicle. The defined fish-like system utilizes a reduced complexity prescribed motion as the representation for energy harvester to be subjected to. Nonlinear electromechanical modeling is performed by considering the geometric and piezoelectric nonlinearities. A convergence analysis is carried out in order to determine the required modes in the Galerkin discretization due to the presence of nonlinear interactions between the prescribed and relative motions. The utilization of higher-order modeling for the strain and material leads to the identification of impactful prescribed motions terms that can activate the nonlinearities in the system, results in more harmonics to consider, and leads to the presence parametric excitation terms. Considering a reduced-complex model by decreasing the value of the quadratic constraint envelope that the fish-like system would be forced with, the soft-robotic system behaves more with a base excitation characteristic. Small damping would allow this prescribed motion with reduced quadratic envelope forcing still induces a hardening behavior, but the other harmonics and parametric resonance seen are greatly reduced. Considering this reduced complexity system, the interaction between the prescribed and base excitations is also investigated to demonstrate that when the two excitations are of similar nature constructive and destructive build of the response waveform can occur when looking at near the first natural resonance. It is shown that the quenching phenomenon can take place which may result in a destructive response of the piezoelectric energy harvester. The results show that the robustness of the fish-like robot is directly dependent on the design parameters including the damping of the structure, importance of the undulatory motion, and activation of the resonances.

Selecting nonlinear piezoelectricity for fully autonomous self-sensing synchronized switch damping on inductor technique

Recently, the importance of nonlinear piezoelectricity in smart piezoelectric devices has been receiving attention. Considering two sophisticated models of nonlinear piezoelectricity, we proposed a systematic methodology to select a model that is suitable for a piezoelectric smart attenuation device employing a fully autonomous self-sensing synchronized switch damping on inductor (SSDI) circuit. We derived the electromechanical equation based on the two models by considering a practical structure where the piezoelectric materials are attached using an adhesive and cover the structure partially. Then, the nonlinear parameters were determined using harmonic balance method (HBM) and nonlinear data fitting. We selected a suitable model by comparing the numerical analysis and experimental results of three datasets for the open, short, and SSDI circuits. It is difficult to distinguish the two models clearly from the frequency response peak values of the displacement or piezoelectric voltage. However, the difference between the two models was observed in the backbone curve properties of the peak frequency and the shape of frequency response curves, particularly for the short circuit. The methodology established in this work allows us to systematically select a suitable model of nonlinear piezoelectricity for a smart attenuation device.

A polar oxyhalogen-vanadate compound (C5NH13Cl)2VOCl4 with optical and two-staged dielectric switch behavior.

Polar organic-inorganic hybrid materials have been applied in ferroelectricity, as well as in devices based on nonlinear optical and piezoelectricity properties. Here, we used a rectangular pyramid structure of [VOCl4]2- to construct a polar compound (C5NH13Cl)2VOCl4 (1). Compound 1 crystallized in the monoclinic P21 space group. Coexistence of nonlinear optical switching behavior (space-group change from P21 to P21/n) and two-staged thermosensitive dielectric switching properties could be achieved under the stimulus of temperature. Our findings provide an effective approach for construction of polar materials.