Piezoelectric Rectangular Plate Research Articles

Theoretical study on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates

This paper carried out a theoretical research on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates. Firstly, the perturbation theory was used to study the bending problem of functionally-graded lead-free piezoelectric rectangular thin plates, and the nonlinear governing equations for this bending problem were established. Then, the three piezoelectric coefficients reflecting the piezoelectric property were taken as the perturbation parameters, and the nonlinear governing equations were transformed into multiple sets of linear equations by the multi-parameter perturbation method. Finally, the solutions of these linear equations were solved by the semi-inverse solving method. Based on the finite element simulations, the effectiveness of the perturbation solutions obtained here was verified. The research results showed that it was reasonable to choose three piezoelectric coefficients as perturbation parameters, which was accord with the basic idea of perturbation theory. The analysis results indicated that there was a cross transfer relationship between external loads and perturbation solutions of every order, which could be used to simplify the perturbation expansion. Moreover, the deformation of the functionally-graded lead-free piezoelectric thin plate was less than that of the functionally-graded thin plate, that is, piezoelectric properties in lead-free piezoelectric materials had a piezoelectric enhancement effect.

Nonlinear transient coupled thermo-electromechanical dynamic responses of the piezoelectric structure with the high-order temperature-dependent material properties

This work aims to study the nonlinear transient coupled thermo-electromechanical dynamic responses of the piezoelectric structure with the high-order temperature-dependent material properties. The multi-field coupling nonlinear model is established based on the non-Fourier thermoelasticity theories by introducing the function of high-order temperature dependency of the arbitrary material property. The virtual work principle and the nonlinear finite element approach are proposed to investigate the two-dimensional case of the orthotropic rectangular piezoelectric plate of the crystal class mm2 subjected to the zonal transient heating loads. The influences of the high-order temperature-dependent material parameters on the structural responses and wave propagations are discussed.

Nonlinear combination resonance analysis of parametric-forced excitation for an axially moving piezoelectric rectangular thin plate in thermal- electromechanical coupling field

In this paper, the combination resonance of parametric-forced excitation characteristics for an axially moving rectangular piezoelectric plate under a thermal-electromechanical field is studied. Based on Kirchhoff-Love plate theory and Von Karman theory, the transverse vibration governing equations are derived from Hamilton's principle. The equations are discretized to ordinary differential equations by the Galerkin method. Then, the multiple scales method is applied to solve the system combination resonance equation, two different resonance states and corresponding amplitude-frequency response equations are obtained by eliminating the secular term, respectively. Additionally, the stability of the steady-state responses is analyzed by Lyapunov stability. Based on the numerical analysis, the influence of axial velocity, external voltage, central temperature difference, structural damping, and other parameters on nonlinear combination resonance response are investigated. The effect of parameter variation on period-doubling bifurcation and the chaotic motion of the system are also discussed by the system global bifurcation diagram.

Analysis on Thermo-Electrical Principal Parametric Resonance of an Axially Moving Piezoelectric Thin Plate

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton’s principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin’s method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion.

Accurate Dynamic Electromechanical Solution for Rectangular Piezoelectric Plate Based on Modified FSDT

In this study, the modified first-order shear deformable theory (FSDT) is applied to analyze the resonant characteristics of piezoelectric (PZT) moderately thick plate vibrators with different thicknesses. A simple and straightforward simplification method between mechanical and electrical functions is also proposed based on the modified FSDT. The natural frequencies and associated mode shapes of piezoelectric plates with completely free boundary are analyzed. The displacement fields of the flexural modes are presented based on the superposition method. This analytical method provides a strong-form novel solution for the flexural-dominated motion’s out-of-plane and in-plane coupling displacement fields. The solutions of displacement functions, which are obtained from the superposition method, show excellent convergence for numerical calculation. To verify the validity of the theoretical solution, the resonant frequencies and the corresponding mode shapes are compared with that obtained by the finite element method (FEM) and experimental measurements using the amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI) technique. According to the electromechanical formula and the derived theoretical solution, four types of electrode are designed to excite the moderately thick PZT specimen for examining the application feasibility of the proposed method. It is shown in this study that the theoretical analysis, numerical calculation, and experimental measurement are fully compared, and excellent agreements of the piezoelectric moderately thick plate are also obtained.

Concept and electromechanical-coupling modeling of a torsional vibration excitation method

The sandwiched type torsional piezoelectric actuators have widely been employed in the fields of ultrasonic cutting, ultrasonic drilling, ultrasonic welding, ultrasonic wire drawing, ultrasonic spraying, and microdroplet generation, due to their unique vibration form. However, the circular piezoelectric ceramic plate polarized in the circumferential direction holds a complicated manufacturing process, a long processing period, and a high price. It can only be used for the excitation of the torsional vibration of circular rods, and cannot be used for square cross-section beams, severely restricting its applications. In order to solve the above problems, a novel excitation method is proposed for generating torsional vibration in the square cross-section beam in this study, which uses the d15 vibration mode of rectangular piezoelectric ceramic plates to produce the torsional vibration, and a novel transfer matrix model of square torsional piezoelectric element is created. Then a case study is conducted in this paper to confirm the proposed excitation method and the developed model, in which two novel sandwiched type torsional piezoelectric actuators are designed. Moreover, electromechanical coupling models of these two torsional piezoelectric actuators are developed using the transfer matrix method, in order to reveal their dynamics behaviors. Finally, the finite element method and the experimental investigations are adopted to analyze and test the frequency response characteristics and vibration shapes of the proposed piezoelectric actuators. Both simulation analysis and experimental tests verify the effectiveness of the proposed excitation method and the correctness of the developed electromechanical coupling dynamic models. The proposed excitation method provides a new idea for the design of sandwiched type torsional piezoelectric actuators, and the developed theoretical model lays a theoretical foundation for the further research of torsional piezoelectric actuators.

ON THE INFLUENCE OF THE PARAMETERS OF THE ELECTRIC EXCITATION PULSE OF A PIEZOELECTRIC PLATE OF A NORMAL PROBE ON THE CHARACTERISTICS OF THE FREQUENCY SPECTRUM AND PULSE OF THE RECORDED ECHO SIGNAL

On the basis of computer modeling of the electro acoustic channel of a normal probe for a reflector in the form of a bottom surface, the influence of some typical forms and corresponding frequency spectra of the electric pulse of excitation of the piezoelectric plate on the characteristics of the frequency spectrum and pulse of the recorded echo signal is considered. Calculations of signals using an exciting pulse in the form of a voltage jump (step), in the form of half a sine wave and a rectangular shape have shown the feasibility of using rectangular pulses with a duration equal to half of the oscillation period at the nominal frequency of the probe. The maximum of the pulse envelope of the recorded bottom echo for the excitation pulse of a rectangular shape piezoelectric plate is approximately twice as high as the maximum of the envelope when the piezoelectric plate is excited by a pulse in the form of a jump and a quarter more than the maximum when excited by a semi sinusoidal pulse for the nominal frequencies of the probes 2.5 and 5.0 MHz.

Design and experiment of a low frequency non-contact rotary piezoelectric energy harvester excited by magnetic coupling

In this paper, a low frequency non-contact rotary piezoelectric energy harvester excited by magnetic coupling (L-PEH) was proposed. Through the coupling between the magnet inside the rotor driven by rotating motion and the magnet on the piezoelectric plate, the process of rotating motion transforming into electrical energy was realized. The purpose of this paper was to improve the output performance of piezoelectric energy harvester at low frequency by optimizing the magnetic field conditions, so the influence of the number of magnets and the arrangement of magnets on the output performance was studied in detail. A series of theories, simulations and experiments proved that the alternative magnet arrangement (mode 1) could optimize the output performance of the L-PEH. When the arrangement of magnets was mode 1 and the number of magnets was 2, the peak-to-peak voltage of rectangular and circular piezoelectric plates was 182.5 V and 4.4 V at 250r/min. Under the above optimal parameter variables, the L-PEH could generate the maximum output power of 140.45 mW. In addition, the output power generated by the prototype could fully meet the power consumption of low-power electronic equipment such as light-emitting-diodes (LEDs). In general, by changing the number and arrangement of magnets, i.e. optimizing the magnetic field conditions, L-PEH could effectively collect rotational energy and had a good application prospect in the power supply of micro electronic equipment.

Analysis on vibration characteristics of large-size rectangular piezoelectric composite plate based on quasi-periodic phononic crystal structure

Based on the theory of composite materials and phononic crystals (PCs), a large-size rectangular piezoelectric composite plate with the quasi-periodic PC structure composed of PZT-4 and epoxy is proposed in this paper. This PC structure can suppress the transverse vibration of the piezoelectric composite plate so that the thickness mode is purer and the thickness vibration amplitude is more uniform. Firstly, the vibration of the model is analyzed theoretically, the electromechanical equivalent circuit diagram of three-dimensional coupled vibration is established, and the resonance frequency equation is derived. The effects of the length, width, and thickness of the piezoelectric composite plate at the resonant frequency are obtained by the analytical method and the finite element method, the effective electromechanical coupling coefficient is also analyzed. The results show that the resonant frequency can be changed regularly and the electromechanical conversion can be improved by adjusting the size of the rectangular piezoelectric plate. The effect of the volume fraction of the scatterer on the resonant frequency in the thickness direction is studied by the finite element method. The band gap in X and Y directions of large-size rectangular piezoelectric plate with quasi-periodic PC structures are calculated. The results show that the theoretical results are in good agreement with the simulation results. When the resonance frequency is in the band gap, the decoupling phenomenon occurs, and then the vibration mode in the thickness direction is purer.

Flexoelectric effect on thickness-shear vibration of a rectangular piezoelectric crystal plate

Thickness-shear (TSh) vibration of a rectangular piezoelectric crystal plate is studied with the consideration of flexoelectric effect in this paper. The developed theoretical model is based on the assumed displacement function which includes the anti-symmetric mode through thickness and symmetric mode in length. The constitutive equation with flexoelectricity, governing equations and boundary conditions are derived from the Gibbs energy density function and variational principle. For the effect of flexoelectricity, we only consider the shear strain gradient in the thickness direction so as to simply the mathematical model. Thus, two flexoelectric coefficients are used in the present model. The electric potential functions are also obtained for different electric boundary conditions. The present results clearly show that the flexoelectric effect has significant effect on vibration frequencies of thickness-shear modes of thin piezoelectric crystal plate. It is also found that the flexoelectric coefficients and length to thickness ratio have influence on the thickness-shear modes. The results tell that flexoelectricity cannot be neglected for design of small size piezoelectric resonators.

Singularity Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure with 1 : 2 Internal Resonance

This study investigates the dynamical behavior of the composite laminated piezoelectric rectangular plate with 1 : 2 internal resonance near the singularity using the extended singularity theory method. Based on the previous four-dimensional averaged equations of polar coordinates where the partial derivative terms are not equal to zero, the universal unfolding with codimension 3 of the proposed system is given. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3. In addition, the plots of the transition set in three bifurcation parameters space are discussed. These numerical results can show that the stability near the singularity of the proposed system is better when period ratio is less than zero.

Active control of vibrations of piezoelectric‎ rectangular nanocomposite micro plates reinforced with graphene platelet in thermal ambient considering the structural damping

This paper investigates on the active control ‎of vibrations of rectangular nanocomposite micro plates reinforced with graphene platelet bonded with piezoelectric‎ layers in thermal environment regarding the structural damping. The micro plate is subjected to a transverse load. The thermo-mechanical features of the reinforced layers are determined employing the Halpin-Tsai micromechanical model. The first order shear deformation theory is accompanied by the modified couple stress theory to derive the motion equations of the micro plate. The Ritz technique is applied to the governing equations to discretize the equations of motion. In order to active control ‎of vibrations of the micro plate, a closed-loop PD-controller is proposed. For the sake of determination of the transient response, the Newmark- direct integration technique is applied to the discretized governing equations of motion ensuing from the Ritz technique. The present findings are validated with the available data in the literature. A parametric study is established to shed light on the impression of the material length scale parameter, the boundary condition type, the graphene platelet distribution pattern and its weight fraction, and the control gain values on the dynamic response of the micro plate when it is subjected to a pulse-uniformly distributed transverse force. The results illustrate the more effectiveness of the designed PD controller for X-GPL micro plates with four clamped boundaries especially when the size dependency is incorporated into the formulation. Moreover, the PD controller performance boosts up at higher temperatures.

Efficient excitation of transverse vibrational modes using improved configurations of PFCs connected to an isotropic plate

In this work, we investigate efficient excitations of transverse (out-of-plane) vibrations of isotropic rectangular plates using piezoelectric fiber composites (PFCs) by experimental measurement. Four PFCs are separately bonded on a rectangular plate with four configured electrode connections to excite transverse vibration of the plate. The proposed configurations of the PFCs, including placements and electrode connections, are inspired by efficient transverse excitations of piezoelectric rectangular plates, whose mode shapes can be divided into four groups and whose electromechanical coupling efficiency can be largely improved by designs of partial electrodes. To visually demonstrate the excitation efficiency, an amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) measurement technique is set up to obtain full-field transverse mode shapes. Other than AF-ESPI, a laser Doppler vibrometer (LDV) displacement sensing system is employed to investigate the excitation performances of the PFCs according to resonant frequencies with emphasis on the influences of the measured nodal lines of the plate. The experimental results presented in this study indicate that the excitation efficiency of the transverse vibrations in flexible structures can be improved by bonding distributed PFCs with configured electrode connections. In addition, we show that the vibrations of isotropic rectangular plates can be related to those of piezoelectric rectangular plates, from the experimental modal excitation point of view.

Bending behaviors of the in-plane bidirectional functionally graded piezoelectric material plates

The transverse bending behaviors of in-plane bidirectional functionally graded piezoelectric material (FGPM) plates are semi-analytically investigated by the scaled boundary finite element method (SBFEM) in association with the precise integration method (PIM). The proposed scheme is able to explore the structural characteristics of FGPM plates with the material coefficients obeying arbitrary form of mathematical functions according to the in-plane coordinates. The present methodology selects only four quantities consisting of three translational displacement components and the electric potential as the fundamental unknowns. Additionally, variations of the four primary variables across the thickness direction are expressed as an analytical exponential matrix. In the developed approach, plates are regarded as a kind of three dimensional structure. But, an arbitrary in-plane surface discretized with two-dimensional spectral elements is set as the research domain. The practice is conducive to cut down the computational effort and increase the calculation efficiency. The SBFEM governing equations are formulated from the three-dimensional basic equations of piezoelectric materials without any assumptions on the plate kinematics and distributions of electromechanical components. By means of the scaled boundary coordinate system and the dual vector methodology, the key partial differential equations of piezoelectric materials are conveniently converted into the easily-solved first order ordinary differential SBFEM governing equation. The stiffness matrix is constructed from the analytical exponential matrix aided by the highly accurate PIM to predict the changing patterns of mechanical and electric quantities. To further improve the precision, the technology of dividing the plate into two parts with equal thickness is exploited. Finally, numerical exercises of square, rectangular and triangular piezoelectric plates are provided to validate the accuracy and fast convergence of the developed technique and reveal the effect of geometrical shapes, gradient functions, types of external loadings and thickness-to-span ratios on the static flexure of FGPM plates owning the in-plane bidirectional stiffness.

Parametric Vibrations of a Hinged Thermoviscoelastic Rectangular Piezoelectric Plate with Shear Strains and Dissipative Heating Taken Into Account*

We presented the model of parametric vibrations of the hinged rectangular thermoviscoelastic piezoelectric plate taking into account the shear strains and dissipative heating. A solution to this problem is reduced to the classic Mathieu equation. We studied the effect of the temperature of dissipative heating on the parametric vibrations of the plate.

Stability analysis and numerical simulation of a composite laminated piezoelectric rectangular plate system

This paper investigates the stability behavior in three-degree-freedom composite laminated piezoelectric rectangular plate system based on the Routh-Hurwitz criterion and the normal form theory. Firstly, the stability behavior of the composite laminated piezoelectric rectangular plate is determined by applying the Routh-Hurwitz criterion. Then, the normal forms of the composite laminated piezoelectric rectangular plate systems for the case of one non-semi-simple double zero and two pairs of pure imaginary eigenvalues are obtained by using the normal form theory. Finally, numerical simulations are carried out to illustrate the main results.

Bending analysis of thick functionally graded piezoelectric rectangular plates using higher-order shear and normal deformable plate theory

In this paper, bending-stretching analysis of thick functionally graded piezoelectric rectangular plates is studied using the higher-order shear and normal deformable plate theory. On the basis of this theory, Legendre polynomials are used for approximating the components of displacement field. Also, the effects of both normal and shear deformations are encountered in the theory. The governing equations are derived using the principle of virtual work and variational approach. It is assumed that plate is made of piezoelectric materials with functionally graded distribution of material properties. Hence, exponential function is used to modify mechanical and electrical properties through the thickness of the plate. Finally, the effect of material properties, electrical boundary conditions and dimensions are investigated on the static response of plate. Also, it is shown that results of the presented model are close to the three dimensional elasticity solutions.

Aspect ratio dependence of the effective electromechanical coupling coefficient Ke of the transverse vibration mode for a piezoelectric rectangular thin plate

In this paper, the apparent elasticity method is used to derive the transverse effective electromechanical coupling coefficient Ke of a piezoelectric rectangular thin plate with an arbitrary aspect ratio G (defined as the ratio of the two transverse dimensions). A unified formula Ke with regard to G is obtained, and the dependence of Ke on G is presented and discussed. The results show that, at G = 1 (square thin plate), Ke drops to 0 at the low-frequency branch, whereas it reaches its maximum value at the high-frequency branch. It is proposed that the maximum value of Ke corresponds to the planar vibration mode. In addition, thin plates made of the PZT-4 ceramic with different aspect ratios were fabricated and measured to check the validity of the above method. For the low-frequency branch of Ke, the experimental results were consistent with the trend of the theoretical values. For the high-frequency branch of Ke within the range of 0.2 ≤ G ≤ 0.55 and 1.82 ≤ G ≤ 5, however, the experimental results exhibited large deviations compared with the theoretical values owing to the negative influence of disturbing resonances. In contrast, the experimental results within the range of 0.6 ≤ G ≤ 1.67 were in good agreement with the theoretical values and reached a maximum value at G = 1. It is suggested that the piezoelectric plates with 0.6 ≤ G ≤ 1.67 should be preferred for practical applications due to their undisturbed high-frequency resonances with high values of Ke.

Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates

A model of piezoelectric rectangular thin plates with the consideration of the coupled thermo-piezoelectric-mechanical effect is established. Based on the von Karman large deflection theory, the nonlinear vibration governing equation is obtained by using Hamilton’s principle and the Rayleigh-Ritz method. The harmonic balance method (HBM) is used to analyze the first-order approximate response and obtain the frequency response function. The system shows non-linear phenomena such as hardening nonlinearity, multiple coexistence solutions, and jumps. The effects of the temperature difference, the damping coefficient, the plate thickness, the excited charge, and the mode on the primary resonance response are theoretically analyzed. With the increase in the temperature difference, the corresponding frequency jumping increases, while the resonant amplitude decreases gradually. Finally, numerical verifications are carried out by the Runge-Kutta method, and the results agree very well with the theoretical results.

Sound transmission loss of double-wall piezoelectric plate made of functionally graded materials via third-order shear deformation theory

According to the third-order shear deformation assumption, this paper analytically investigates the sound transmission loss (STL) through a rigidly baffled finite rectangular double wall piezoelectric plate structure made of functionally graded materials (FGMs) with enclosed acoustic cavity, under harmonic plane sound excitation and initial external electric voltage. The effective piezoelectric material properties of each plate are supposed to be continuously variable across the thickness direction using a power law model in terms of the volume fractions of the material phases. The coupled vibroacoustic governing equations are achieved utilizing Hamilton’s principle and then solved analytically by applying the sound velocity potential method in conjunction with double Fourier series expansions to determine STL equation. Numerical studies are performed to illustrate the effects of the acoustic cavity depth, initial external electric voltage, incident elevation angle, gas type used in acoustic cavity, and material gradient on the changes of STL curves of simply supported double wall FGM piezoelectric plate. The profound effect of these factors on sound isolation performance is clearly shown.