Model Of Piezoelectric Beam Research Articles

Finite element analysis of the electro-mechanical coupling effects in piezoelectric laminated structures

This study utilized the COMSOL Multiphysics software to conduct finite element analysis on the piezoelectric coupling effect of the composite layer structure. The article begins by introducing the basic principles and history of the piezoelectric effect, emphasizing the importance and widespread application of piezoelectric materials in the field of smart materials. Subsequently, it defines in detail the models of bimorph piezoelectric beams and multilayer composite piezoelectric plates, and conducts potential verification. The bimorph piezoelectric beam model uses PVDF material, and the potential distribution is monitored by boundary probes, confirming the influence of the quantity and distribution of sensor electrodes on voltage distribution. The multilayer composite piezoelectric plate model combines a Si non-piezoelectric semiconductor layer with a PZT-4 piezoelectric material layer, demonstrating the in-plane distribution of potential under lateral mechanical loads. This study not only verifies the reliability of COMSOL software in piezoelectric coupling analysis but also provides an in-depth understanding of the behavior and performance of piezoelectric materials in complex environments.

Exponential stability of piezoelectric beams with delay and second sound

AbstractIn this paper, we consider a fully‐dynamic piezoelectric beam model subjected to a magnetic effect, where the heat flux is given by Cattaneo's law. It is well known that, in the absence of delay, the dissipation produced by the heat conduction is strong enough to make the piezoelectric beams exponentially stable. However, time delay effects may destroy this behavior. Here, we show the existence and uniqueness of solutions through the semigroup theory. Furthermore, under a smallness condition on the delay, we prove an exponential stability result via establishing the appropriate Lyapunov functional. Finally, we numerically illustrate the asymptotic behavior of the solution.

On the energy decay of a viscoelastic piezoelectric beam model with nonlinear internal forcing terms and a nonlinear feedback

In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature.

Existence and robustness results of attractors for partially-damped piezoelectric beams

This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $.

Boundary feedback stabilization of a novel bilinear and extensible piezoelectric beam model

Boundary feedback stabilization of a novel bilinear and extensible piezoelectric beam model

Global attractors for a novel nonlinear piezoelectric beam model with dynamic electromagnetic effects and viscoelastic memory

Global attractors for a novel nonlinear piezoelectric beam model with dynamic electromagnetic effects and viscoelastic memory

Longtime dynamics for a novel piezoelectric beam model with creep and thermo-viscoelastic effects

In this work, a novel fully-dynamic piezoelectric beam model is considered. Electromagnetic and thermal effects are taken into consideration by Maxwell’s equations and the Coleman–Gurtin law (instead of the Fourier’s law), respectively. Our model accounts also for thermal and electromagnetic (creep) past histories, which are in line with the time response of PVDF at the applied stress in the longitudinal direction. Under suitable assumptions, the existence and uniqueness of solutions are proved by the semigroup theory. The main purpose of this paper is to establish the longtime dynamics of the model. Therefore, the quasi-stability property of the model and the existence of smooth global attractors with finite fractal dimension are obtained. The existence of exponential attractors for the associated dynamical system is also proved.

Active vibration optimal control of piezoelectric cantilever beam with uncertainties

Considering the stiffness characteristics of piezoelectric layer, the bending stiffness of piezoelectric cantilever beam is obtained by applying the first-order shear deformation theory. The finite element model of piezoelectric cantilever beam is established by Hamilton variation principle, and the modal superposition method is employed to reduce the order of the finite element model. At the maximum strain point, the sensors/actuators are equipped in pairs. Based on the uncertain dynamic model of piezoelectric cantilever beam, the independent modal space control method based on LQR (linear quadratic regulator) control is employed for the active control of the smart beam structure, and the weighted matrices Q and R are selected according to the energy criterion. The numerical simulations and experiments verify the effectiveness of the proposed finite element model and the active vibration optimal control.

The stationary response of piezoelectric cantilever beam model excited by colored noise

This article provides a method to predict the steady-state response of a piezoelectric cantilever under the excitation of Gaussian colored noise which is a better approximation to the ambient random excitation than the white noise. In this paper, a cantilever energy harvester model, which is fully-paved with piezoelectric ceramic and excited by Gaussian colored noise, is built by the Lagrange’s equations and is nondimensionalized. Then the dynamic equation and piezoelectric equation are transformed into a pair of stochastic differential equations about transient equivalent amplitude and transient phase. After that, the stochastic averaging method is applied to simplify the stochastic differential equations into an Itô type equation about the equivalent amplitude, the drift coefficient and the diffusion coefficient are obtained. This method provides another approach which is different from the stochastic linearization method and moment method. On this basis, some crucial outputs such as the steady-state probability density function (PDF) of the equivalent amplitude, the beam’s displacement, the transient output voltage, the joint probability density function of displacement and velocity as well as the mean square value of output voltage are all obtained. With different noise intensity values and delay factors, the Monte Carlo numerical simulation has verified the correctness of the theoretical prediction.

Numerical study on gust energy harvesting with an efficient modal based fluid-structure interaction method

Gust disturbance during flight causes elastic vibrations and rigid body motions of an aircraft. A macro fiber composite (MFC)-based energy harvester can be used to obtain energy from gust-induced vibrations. This paper aims to present an efficient fluid-structure interaction (FSI) method for fast and accurate calculation of gust response and harvested energy of a fly wing aircraft half model. In this method, computational fluid dynamics (CFD) and computational structural dynamics (CSD) are coupled by embedding a modal-superposition-based structural solver into the CFD solver. Parallel data exchange and interpolation strategy on the fluid-structure interface is introduced to improve the computational efficiency. MFC harvester is modeled by a piezoelectric beam model, and the displacement of the piezoelectric beam is obtained by interpolation from FSI results. The numerical results of gust response and MFC voltage output are in good accordance with wind tunnel test results, while the calculation time is only 1/8 of the time consuming of a direct CFD/CSD coupling method. The influences of gust frequency, gust amplitude, freestream velocity, and the number of retained modes on energy harvesting are investigated. Results indicate that the accuracy of the calculation can be guaranteed by the superposition of several retained modes, and the additional suction provided by leading-edge vortices causes a nonlinear increase in energy harvesting near the stalling angle of attack.

Forced vibration of piezoelectric and flexoelectric Euler–Bernoulli beams by dynamic Green’s functions

This work presents the dynamic response of piezoelectric and flexoelectric Euler–Bernoulli beams subjected to mechanical loads by means of Green’s functions. Exact solutions in closed form for the electromechanical coupling behaviors are derived. The present solutions will reduce to those of classical piezoelectric beam models by neglecting the flexoelectric effect. Numerical results for a $${\text {BaTiO}}_{3}$$ cantilever show that the tip deflection of the beam will be separated when the beam thickness reduces to one micrometer; therefore, pure mechanical experiments can be used to determine the flexoelectric effect by a bending test under different electrical boundary conditions. In addition, the dynamic output voltage under open-circuit condition and the dynamic output charge under short-circuit condition are dominated by the flexoelectric effect. The dynamic output voltage and charge become negligible when the beam thickness reduces to one micrometer. By reducing the length–thickness ratio or introducing the effective piezoelectric coefficient, the electric response can be more easily detected at microscale. These results demonstrate that the flexoelectric effect can be strong enough to dominate the electromechanical coupling behavior, coordinating with the ‘ever green’ trend to miniaturization, and the flexoelectric effect can be applicable for energy harvesters, sensing, and others.

A piezoelectric beam model with geometric, material and damping nonlinearities for energy harvesting

To predict electrical generation in piezoelectric small-scale beam energy harvesting devices, it is important to have a complete mathematical model that captures the different associated phenomena. In the literature, some authors propose several alternatives of non-linear mathematical formulations, with non-linearities coming from different physical aspects. All these formulations present good aptitudes to predict the nonlinear behavior of the system under different values of accelerations, geometry and boundary conditions. At the same time, they do not represent a unified general proposal for modeling multimodal energy harvesting devices of any type of mode generation and boundary conditions at large excitations. In this sense, this paper presents a mathematical description of inextensional nonlinear Euler-Bernoulli piezoelectric beams that combines the best contributions of the literature to the voltage generation of multimodal nonlinear piezoelectric energy harvesters (geometric, material and damping non-linearities). The developed analytical model yields a total set of N+ 1 ordinary differential equations for the first N modes and for the output voltage. However, direct solution of this ordinary nonlinear differential system of N equations is computationally costly. Instead, a reduced algebraic system of 2() algebraic equations is proposed applying the method of averaging. Its main advantage is that it makes more suitable and computationally economical for the implementation of a parameter identification process involving any number of piezoelectric inserts (unimorph or bimorph) and mode of generation (d33 or d31). Two types of validations are presented for some selected physical systems to test the validity of the assumptions: a numerical one, by the direct integration of the equations of motion and an experimental one. A final comparison between the results demonstrates the importance of the having a unified nonlinear model to predict the generated voltage in multimodal energy harvesters.

Development of accurate piezoelectric beam models based on Boley’s method

The solution of the bi-potential equation gives analytical formulas for two-dimensional isotropic problems. In 1954, Boley published a fast-converging, iterative method for rectangular cross-section elastic beams, in order to calculate in-plane stresses and deflections. In this contribution, Boley’s iterative method is extended to piezoceramic materials. After defining potential functions for the in-plane stresses and electric fields, two coupled partial differential equations for these two potential functions are obtained. Extending the Boley’s iterative method, by including the piezoelectric effect, it is shown that the solution of the first iteration exactly yields the analytical results for thin piezoelectric beams that are already known in the open literature. The first iteration also shows that the distribution of the electric potential through the beam thickness is a second-order polynomial with respect to the thickness coordinate. The second and third iterations are found to be higher-order terms, which depend on the thickness-to-length ratio of the beam. They can be interpreted as correction terms to obtain more accurate results for moderately thick beams. In this work, analytical and numerical results for a hinged-hinged bimorph under a sinusoidal force (sensing problem) or under a sinusoidal voltage load (actuation problem) are compared to analytical 2D plane stress results. It is shown that for higher values of the thickness-to-length ratio (i.e. moderately thick or very thick beams), where the Bernoulli-Euler beam bending theory yields unsatisfactory results, the latter are modified by the higher order Boley’s method iterations such that the results converge to analytical 2D results.

Study on Fundamental Technology of Multi Directional Vibration Piezoelectric Power Generation

The problem of low vibration energy collection efficiency in the environment is the focus of the current piezoelectric power generation technology. In order to improve the collection efficiency of vibration energy in the environment, a key technology of piezoelectric power generation is studied, which can collect the vibration energy of multiple directions in the environment. Firstly, the mathematical model of piezoelectric cantilever beam is established by using the basic theory of piezoelectric power generation technology, and the vibration mechanics analysis of the piezoelectric cantilever beam is carried out. Then, the finite element simulation of the piezoelectric cantilever beam is carried out by using ANSYS, and the natural frequency is consistent with the ambient vibration/environment frequency; finally, multi-directional piezoelectric power generation device is made, and the theoretical analysis and experimental test are carried out. The experimental results show that the key technology of multidirectional vibration piezoelectric power generation can effectively improve the collection efficiency of vibration energy in the environment.

Size effect of tip mass on performance of cantilevered piezoelectric energy harvester with a dynamic magnifier

A distributed-parameter model of cantilevered piezoelectric beam with a dynamic magnifier has been proposed for the efficient analysis of a piezoelectric energy harvester, but there appears no beam model suitable for a piezoelectric energy harvester with tip mass offset and a dynamic magnifier. To deal with tip mass offset, the size effect of tip mass offset on the kinetic equation and boundary condition has been considered. A modified model of cantilevered piezoelectric energy harvester with tip mass offset and a dynamic magnifier has been developed by using the generalized Hamilton’s principle. Analytical formulation of the eigenfunction and natural frequency of the modified model have been presented. The modified model has been demonstrated by parametric studies. The results obtained show that the harvesting power can be dramatically enhanced with proper selection of the design parameters of the dynamic magnifier and tip mass offset. The tip mass offset significantly affects the accuracy of the analysis. It is observed that even a small change in tip mass geometry results in a substantial change of energy harvester performance, not only to change the resonant frequency but also to affect the strain distribution along the energy harvester length. The modified model is more suitable for the harvester with tip mass offset and dynamic magnifier.

Design and analysis of a 3-link micro-manipulator actuated by piezoelectric layers

The purpose of this paper is to design and analyze a 3-link micro-device proposed as a micro-manipulator. This micro-manipulator includes 3 micro-beams as links connected to one another with no conventional or flexural joints. While the structure of the micro-manipulator is monolithic, end-effector workspace is achieved through deflection of links which is actuated by piezoelectric layers. By combining static analysis of the links through a multilayer piezoelectric beam model and kinematic analysis of the micro-manipulator, inverse kinematic has been solved utilizing the Taylor series expansion technique and the perturbation method. The obtained results through the present model reveal that the end-effector can provide a workspace of several microns with nano-accuracy. The required piezoelectric voltage for certain end-effector displacement was evaluated, and the error of inverse kinematic was investigated. A finite element model was also constructed to evaluate micro-manipulator behavior. The results show there is a good agreement between the analytical model and the finite element method. Avoiding complex geometry, the micro-manipulator offers a simple structure with a more feasible manufacturing process while it provides acceptable workspace for the end-effector with high accuracy.

Modified Continuum Mechanics Modeling on Size-Dependent Properties of Piezoelectric Nanomaterials: A Review.

Piezoelectric nanomaterials (PNs) are attractive for applications including sensing, actuating, energy harvesting, among others in nano-electro-mechanical-systems (NEMS) because of their excellent electromechanical coupling, mechanical and physical properties. However, the properties of PNs do not coincide with their bulk counterparts and depend on the particular size. A large amount of efforts have been devoted to studying the size-dependent properties of PNs by using experimental characterization, atomistic simulation and continuum mechanics modeling with the consideration of the scale features of the nanomaterials. This paper reviews the recent progresses and achievements in the research on the continuum mechanics modeling of the size-dependent mechanical and physical properties of PNs. We start from the fundamentals of the modified continuum mechanics models for PNs, including the theories of surface piezoelectricity, flexoelectricity and non-local piezoelectricity, with the introduction of the modified piezoelectric beam and plate models particularly for nanostructured piezoelectric materials with certain configurations. Then, we give a review on the investigation of the size-dependent properties of PNs by using the modified continuum mechanics models, such as the electromechanical coupling, bending, vibration, buckling, wave propagation and dynamic characteristics. Finally, analytical modeling and analysis of nanoscale actuators and energy harvesters based on piezoelectric nanostructures are presented.

On magnetic plucking configurations for frequency up-converting mechanical energy harvesters

Magnetic plucking applies the strategy of frequency up-conversion in inertial energy harvesting when the energy source, such as human motion, provides excitations with very low and irregular frequencies. In a typical implementation, a slower moving inertial mass magnetically plucks a piezoelectric cantilever beam which converts mechanical energy to electrical energy at a higher frequency. We categorize several feasible magnet configurations to achieve plucking. We classify these as either in-plane (the beam is deflected in the plane of proof mass motion) or out-of-plane (the beam is deflected orthogonal to the plane of proof mass motion). Whereas in-plane plucking induces a clean ring down due to its inherent jump phenomenon, out-of-plane plucking enables the capability of fabricating multiple piezoelectric beams on a single substrate. This paper presents an analysis of three different out-of-plane plucking configurations along with the in-plane repulsive configuration based on a three-dimensional analytical cube permanent magnet model. We derive a magnetically plucked piezoelectric beam model to investigate the dynamic characteristic for different plucking configurations. After validating the model with experimental results we extend the simulation into a larger driving frequency domain to compare two types of magnet configurations in terms of power generation.

A rail-borne piezoelectric transducer for energy harvesting of railway vibration

This paper investigates design, modelling, and test issues related to piezoelectric energy transducer. The model analyzes a rail-borne “seismic” energy harvester that is designed to generate electrical energy from local variations in rail acceleration. The energy harvester analyzed in this model consists of a piezoelectric PZT film clamped at one end to the rail with a tip mass mounted on its other end. It includes two sub-models in this paper: a vehicle-track interaction model considering vehicle travelling load; and a cantilevered piezoelectric beam model for the visualization of voltage and power profile and frequency response. Four rail irregularities (American 6th grade track spectrum, Chinese track spectrum, German high and low-disturbance track spectrum) are compared and implemented into the calculation script. The calculated results indicate a rail displacement of 0.2 mm to 0.8 mm. Vibration tests of the proposed rail-borne device are conducted; a hydraulic driven system with excitation force up to 140 kN is exploited to generate the realistic wheel-rail interaction force. The proposed rail-borne energy harvester is capable of energy harvesting at low-frequency (5 Hz to 7 Hz) and small railway vibration (0.2 mm to 0.4 mm rail displacement). The output power of 4.9 mW with a load impedance of 100 kOhm is achieved. The open circuit peak-peak voltage reaches 24.4 V at 0.2 mm/7 Hz/5 g wheel-rail excitation. A DC-DC buck converter is designed, which works at the resonance frequency of 23 Hz/5 g on a lab vibration rig, providing a 3.3 VDC output.

Generation characteristics of piezoelectric vibrator driven by groove cam: Simulation and experimental analysis

In this paper, a structure for a piezoelectric beam vibrator driven by a groove cam is analysed. The vibrator takes the simplified form of a piezoelectric beam model, where one end of the beam is clamped and the other end is simply supported, and is thus named the clamped/simply supported piezoelectric beam model (CSPBM). Mathematical models of the damped forced vibration and electromechanical energy conversion processes of the CSPBM are established based on the harmonic displacement excitation of the simply supported end. Factors that affect the energy generated by the CSPBM are analyzed theoretically and are simulated separately using both the MATLAB software and ANSYS software. Theoretical analysis results indicate that there is an optimal value of the ratio of the base plate thickness to the beam thickness (α) at which the energy generated by the CSPBM is maximum. In addition, the optimal α value of a unimorph beam is about 0.3 and is irrelevant to the material parameters of the beam. The voltage and energy generated by the CSPBM are measured on an experimental bench and the results show that the maximum generated voltage increases with increasing the first natural frequency when the α value of piezoelectric beam and the amplitude of displacement excitation are constant. In addition, the theoretical results of the generated voltage are coincident with experimental results, which confirms the validity of the theoretical model. The vibrator driven by a groove cam provides a practical form of the piezoelectric beam excited by the displacement.