Theoretical analysis and experimental measurement for resonant vibration of piezoceramic circular plates

Based on the electroelastic theory for piezoelectric plates, the vibration characteristics of piezoceramic disks with free-boundary conditions are investigated in this work by theoretical analysis, numerical simulation, and experimental measurement. The resonance of thin piezoceramic disks is classified into three types of vibration modes: transverse, tangential, and radial extensional modes. All of these modes are investigated in detail. Two optical techniques, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are used to validate the theoretical analysis. Because the clear fringe patterns are shown only at resonant frequencies, both the resonant frequencies and the corresponding mode shapes are obtained experimentally at the same time by the proposed AF-ESPI method. Good quality of the interferometric fringe patterns for both the transverse and extensional vibration mode shapes are demonstrated. The resonant frequencies of the piezoceramic disk also are measured by the conventional impedance analysis. Both theoretical and experimental results indicate that the transverse and tangential vibration modes cannot be measured by the impedance analysis, and only the resonant frequencies of extensional vibration modes can be obtained. Numerical calculations based on the finite element method also are performed, and the results are compared with the theoretical analysis and experimental measurements. It is shown that the finite element method (FEM) calculations and the experimental results agree fairly well for the resonant frequencies and mode shapes. The resonant frequencies and mode shapes predicted by theoretical analysis and calculated by finite element method are in good agreement, and the difference of resonant frequencies for both results with the thickness-to-diameter (h/D) ratios, ranging from 0.01 to 0.1, are presented.     

Free vibration analysis of the piezoceramic bimorph with theoretical and experimental investigation

This paper investigates the vibration characteristics of an asymmetric, three-layered piezoceramic circular bimorph under traction-free boundary conditions by applying the electroelasticity and Kirchhoff plate theory. The asymmetric, three-layered bimorph consists of an isotropic shim layer and two piezoceramic layers of equal thickness and same polarization. Two optical techniques, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are used to validate the theoretical analysis. The resonant frequencies of the piezoceramic bimorph also are measured by an impedance analyzer. Both theoretical and experimental results indicate that the transverse vibration modes cannot be measured by impedance analysis, and only resonant frequencies of extensional vibration modes are present. However, transverse vibration modes of the piezoceramic bimorph can be obtained by the AF-ESPI and LDV measurements. The numerical calculations also are obtained using the finite-element method (FEM), and the results agree comparatively well with the theoretical analysis and experimental measurements. According to the theoretical calculation, the variations in resonant frequencies and effective coupling factors versus the various layer-thickness ratios also are investigated in this work.     

Transverse vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants.

The transverse vibration of piezoceramic rectangular thin plates is investigated theoretically and experimentally using the Ritz's method incorporated with the defined equivalent constants. The equivalent constants are derived by comparing the characteristic equations of transverse resonant frequencies between isotropic and piezoceramic disks. By replacing the Poisson's ratio and flexural rigidity with the equivalent constants, the well-known Ritz's method can be used to investigate the transverse vibration of piezoceramic rectangular plates. Two different types of boundary conditions-clamped-free-free-free (CFFF) and clamped-free-clamped-free (CFCF)-are analyzed in this paper. For the experimental measurement, two optical techniques-amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV)-are used to validate the analytical results. Both the transverse vibration modes and resonant frequencies of piezoceramic rectangular plates are obtained by the AF-ESPI method. Numerical calculations using the finite-element method (FEM) are performed, and the results are compared with the theoretical analysis and experimental measurements. Excellent agreements are obtained for results of both resonant frequencies and mode shapes. According to the theoretical calculations with different equivalent Poisson's ratios, resonant frequency variations versus aspect ratios ranging from 0.1 to 10 also are discussed for the first several modes in the work.     

Vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants

The transverse vibration of piezoceramic rectangular thin plates is investigated theoretically and experimentally using the Ritz's method incorporated with the defined equivalent constants. The equivalent constants are derived by comparing the characteristic equations of transverse resonant frequencies between isotropic and piezoceramic disks. By replacing the Poisson's ratio and flexural rigidity with the equivalent constants, the well-known Ritz's method can be used to investigate the transverse vibration of piezoceramic rectangular plates. Two different types of boundary conditions - clamped-free-free-free (CFFF) and clamped-free-clamped-free (CFCF) - are analyzed in this paper. For the experimental measurement, two optical techniques - amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV) - are used to validate the analytical results. Both the transverse vibration modes and resonant frequencies of piezoceramic rectangular plates are obtained by the AF-ESPI method. Numerical calculations using the finite-element method (FEM) are performed, and the results are compared with the theoretical analysis and experimental measurements. Excellent agreements are obtained for results of both resonant frequencies and mode shapes. According to the theoretical calculations with different equivalent Poisson's ratios, resonant frequency variations versus aspect ratios ranging from 0.1 to 10 also are discussed for the first several modes in the work.     

Investigations of the transverse vibration characteristics of piezoceramic circular plates with V-notches

This article pertains to experimental validation of the study presented in a previous paper by the same authors , in which the transverse vibration characteristics of piezoceramic circular plates with V-notches were investigated theoretically using Ritz’s method incorporating defined equivalent constants. For experimental measurement, two optical methods – amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometry (LDV) – were employed to conceptualize the analytical results. The AF-ESPI method demonstrates the advantages of combining high fringe sensitivity and noise reduction, which are achieved using the time-averaged and subtraction electronic speckle pattern interferometry methods, respectively. Both vibration mode shapes and resonant frequencies of V-notched piezoceramic plates were obtained using the AF-ESPI method; in addition, the resonant frequencies were also determined quickly using the LDV method. By varying the corner angles and notch–depth ratios, various types of piezoceramic plate (V-notched, sectorial, semicircular, segmented, and sharp-notched) were used for experimental measurements. Numerical calculations were performed using the finite element method, and the results were compared with the experimental measurements and theoretical predictions. It was shown that the resonant frequencies and mode shapes obtained using the two experimental methods agreed fairly closely with the theoretical and numerical results.     

The influence of electrode designs on the resonant vibrations for square piezoceramic plates

In this paper, the resonant vibrations of square piezoceramic plates with four different electrode designs are investigated. Two experimental techniques, the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and the impedance analysis, are used to access the influence of the electrode arrangement on the resonant characteristics of square piezoceramic plates. Both the out-of-plane and in-plane resonant frequencies and full-field mode shapes of piezoceramic plates with various electrode designs are obtained from the AF-ESPI method. The impedance analyzer is used to measure the resonant and antiresonant frequencies of piezoceramic plates. The dynamic electromechanical coupling coefficient (EMCC), which relates to the ability of conversion between mechanical and electrical energy, is determined from the measured values of resonant and antiresonant frequencies. Experimental results of the resonant vibration characteristics of the square piezoceramic plates are verified by numerical computations based on the finite-element method. Excellent agreement between the experimental and numerical results is found in resonant frequencies and corresponding mode shapes. It is found that the electrode design has important influence on the resonant characteristics of piezoceramic plates. The effect of different designs of electrode is more significant in the in-plane modes than that in the out-of-plane modes.     

Vibration characteristics of composite piezoceramic plates at resonant frequencies: experiments and numerical calculations

The experimental measurement of the resonant frequencies for the piezoceramic material is generally performed by impedance analysis. In this paper, we employ an optical interferometry method called the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) to investigate the vibration characteristics of piezoceramic/aluminum laminated plates. The AF-ESPI is a powerful tool for the full-field, noncontact, and real-time measurement method of surface displacement for vibrating bodies. As compared with the conventional film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Because the clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding vibration mode shapes are obtained experimentally at the same time. Excellent quality of the interferometric fringe patterns for both the in-plane and out-of-plane vibration mode shapes are demonstrated. Two different configurations of piezoceramic/aluminum laminated plates, which exhibit different vibration characteristics because of the polarization direction, are investigated in detail. From experimental results, we find that some of the out-of-plane vibration modes (Type A) with lower resonant frequencies cannot be measured by the impedance analysis; however, all of the vibration modes of piezoceramic/aluminum laminated plates can be obtained by the AF-ESPI method. Finally, the numerical finite element calculations are also performed, and the results are compared with the experimental measurements. Excellent agreements of the resonant frequencies and mode shapes are obtained for both results.     

Resonant vibration investigations for piezoceramic disks and annuli by using the equivalent constant method

This paper presents the equivalent constant method to investigate the transverse vibration of piezoceramic disks and annuli. By comparing the characteristic equations of resonant frequencies between isotropic and piezoceramic disks, the named equivalent Poisson's ratio v is derived, then the transverse vibration characteristic equation can be expressed as a single formulation for these two materials. To verify this method, characteristic equations of transverse vibration for piezoceramic disks and annuli with many different boundary conditions are discussed and calculated for resonant frequencies. Numerical calculations based on the finite-element method (FEM) also are performed, and the results agree rather well with the theoretical predictions. With the aid of the relations between frequency parameter and equivalent Poisson's ratio in explicit form, the other application of equivalent Poisson's ratio is the inverse evaluation of material constants. The laser Doppler vibrometer (LDV) and impedance analyzer are used to experimentally obtain the resonant frequencies of transverse and radial extensional vibrations, respectively. By the experimental results for the traction-free piezoceramic disk, the planar Poisson's ratio v(p) and planar electromechanical coupling coefficient k(p) are determined.     

Experimental measurement and numerical analysis on resonant characteristics of cantilever plates for piezoceramic bimorphs

Three experimental techniques are used in this study to access the resonant characteristics of piezoceramic bimorphs in parallel and series connections. These experimental methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), laser Doppler vibrometer-dynamic signal analyzer (LDV-DSA), and impedance analysis, are based on the measurement of full-field displacement, point-wise displacement, and electric impedance, respectively. Because the clear fringe patterns will be shown only at resonant frequencies, both the resonant frequencies and the corresponding vibration mode shapes are successfully obtained at the same time by the AF-ESPI method. LDV-DSA is used to determine the resonant frequencies of the vibration mode for out-of-plane motion. The impedance analysis is used to measure the resonant and antiresonant frequencies for in-plane motion. Although the out-of-plane mode is the dominant motion of piezoceramic bimorphs, it is found in this study that the amount of displacement for the in-plane motion in parallel connection is large enough to be measured by AF-ESPI and impedance. It is interesting to note that resonant frequencies of the specimen in parallel connection for the out-of-plane motion determined by LDV-DSA are the same as that for the in-plane motion obtained by impedance. Furthermore, both in-plane and out-of-plane mode shapes for the specimen in parallel connection are obtained in the same resonant frequency from the AF-ESPI method. It is concluded in this study that the particle motions of piezoceramic bimorphs for parallel connection in resonance are essentially three-dimensional. However, it is found that only out-of-plane vibration modes can be excited for the specimen in series connection. Numerical computations based on the finite-element method are presented, and the theoretical predicted results are compared with the experimental measurements. Good agreements between the experimental measured data and numerical calculated results are found for resonant frequencies and mode shapes of the piezoceramic bimorph.     

Experimental measurement and numerical analysis on resonant characteristics of piezoelectric disks with partial electrode designs

Three experimental techniques are used in this study to access the influence of the electrode arrangement on the resonant characteristics of piezoceramic disks. These methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), laser Doppler vibrometer-dynamic signal analyzer (LDV-DSA), and impedance analysis, are based on the measurement of full-field displacement, pointwise displacement, and electric impedance, respectively. In this study, one full electrode design and three nonsymmetrical partial electrode designs of piezoelectric disks are investigated. Because the clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and the corresponding vibration mode shapes are successfully obtained at the same time for out-of-plane and in-plane motions. The second experimental method is the impedance analysis, which is used to measure the resonant and antiresonant frequencies. In addition to these experimental methods, LDV-DSA is used to determine the resonant frequencies of the vibration mode with out-of-plane motion. From the experimental results, the dependence of electrode design on the vibration frequencies and mode shapes is addressed. Numerical computations based on the finite element method are presented, and the results are compared with the experimental measurements. The effect of different designs of electrode is more significant in the in-plane modes than that in the out-of-plane modes.     

Experimental measurements and finite element analysis of the coupled vibrational characteristics of piezoelectric shells

Piezoelectric plates can provide low-frequency transverse vibrational displacements and high-frequency planar vibrational displacements, which are usually uncoupled. However, piezoelectric shells can induce three-dimensional coupled vibrational displacements over a large frequency range. In this study, three-dimensional coupled vibrational characteristics of piezoelectric shells with free boundary conditions are investigated using three different experimental methods and finite element numerical modeling. For the experimental measurements, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) is used to obtain resonant frequencies and radial, lateral, and angular mode shapes. This optical technique utilizes a real-time, full-field, non-contact optical system that measures both the natural frequency and corresponding vibration mode shape simultaneously. The second experimental technique used, laser Doppler vibrometry (LDV), is a pointwise displacement measurement method that determines the resonant frequencies of the piezoelectric shell. An impedance analyzer is also used to determine the resonant frequencies of the piezoelectric shell. The experimental results of the resonant frequencies and mode shapes for the piezoelectric shell are verified with a numerical finite element model. Excellent agreement between the experimental and numerical results is found for the three-dimensional coupled vibrational characteristics of the piezoelectric shell. It is noted in this study that there is no coupled phenomenon at low frequencies over which radial modes dominate. However, three-dimensional coupled vibrational modes do occur at high resonant frequencies over which lateral or angular modes dominate.     

Vibration analysis of angle-ply laminated composite plates with an embedded piezoceramic layer

An optical full-field technique, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), is used in this study to investigate the force-induced transverse vibration of an angle-ply laminated composite embedded with a piezoceramic layer (piezolaminated plates). The piezolaminated plates are excited by applying time-harmonic voltages to the embedded piezoceramic layer. Because clear fringe patterns will appear only at resonant frequencies, both the resonant frequencies and mode shapes of the vibrating piezolaminated plates with five different fiber orientation angles are obtained by the proposed AF-ESPI method. A laser Doppler vibrometer (LDV) system that has the advantage of high resolution and broad dynamic range also is applied to measure the frequency response of piezolaminated plates. In addition to the two proposed optical techniques, numerical computations based on a commercial finite element package are presented for comparison with the experimental results. Three different numerical formulations are used to evaluate the vibration characteristics of piezolaminated plates. Good agreements of the measured data by the optical method and the numerical results predicted by the finite element method (FEM) demonstrate that the proposed methodology in this study is a powerful tool for the vibration analysis of piezolaminated plates.     

Dynamic in-plane resonant characteristics of piezoceramic and piezolaminated composite plates.

Piezolaminated composite plates have received considerable attention in various industrial applications due to their intelligent characteristics. In this investigation, two experimental measurement techniques are used to determine the in-plane resonant vibration of angle-ply laminated composites embedded with a piezoceramic layer (piezolaminated plates) for different stacking angles. The first method is a full-field optical technique, which is called the AF-ESPI (amplitude-fluctuation electronic speckle pattern interferometry). This is the major experimental method. The AF-ESPI method is used to determine the in-plane resonant frequency and corresponding mode shape of a single-layer piezoceramic plate and piezolaminated plates with five different stacking angles. The second experimental technique, the impedance analyzer, is employed to determine the in-plane resonant frequency. Finally, numerical computations based on the finite element analysis are presented for comparison of the two experimental results. Excellent agreement between the experimentally measured data and the numerically calculated results are found for in-plane resonant frequencies and mode shapes. This study indicates that the dynamic characteristics of inplane resonant vibrations for piezolaminated plates with different stacking angles are quite different.     

Electromechanical coupling efficiency of transverse vibration in piezoelectric plates according to electrode configuration

This study proposes a method of configuring electrodes for piezoelectric plates to improve the electromechanical coupling efficiency of transverse (out-of-plane) vibration. Two optical techniques were employed to investigate the transverse vibration. We first used a non-contact, full-field measurement technique called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) to determine resonant frequencies and mode shapes in real time. Second, a point-wise measurement device called a laser Doppler vibrometer was used to obtain the spectrum for out-of-plane frequencies. Four electrode configurations with completely free boundary conditions were used to study the electromechanical coupling efficiency of piezoelectric plates. We conclude that the design of the electrodes clearly influences the electromechanical coupling efficiency of piezoelectric plates. The experimental results related to resonant frequency and mode shape were verified using the finite element method (FEM). The numerical calculations are in good agreement with experimental results. Furthermore, according to FEM results, the distribution of electric potential gradient is similar to the mode shape measured by AF-ESPI, making it applicable to the configuration of electrodes in optimizing the excitation of piezoelectric plates. This study proposes the determination of the electromechanical coupling factor to predict the efficiency of transverse vibration in piezoelectric plates.     

High-frequency resonant characteristics of triple-layered piezoceramic bimorphs determined using experimental measurements and theoretical analysis

This is an experimental, theoretical, and numerical investigation of vibration characteristics in high-frequency resonance, which are studied for parallel- and series-type piezoelectric bimorphs. In the experimental measurements, the full-field optical technique known as electronic speckle pattern interferometry (ESPI) is used to measure the transverse (out-of-plane) and planar (in-plane) resonant frequencies and corresponding mode shapes for piezoelectric bimorphs. In addition, in-plane resonant frequencies are obtained from impedance analysis and the response curves of the frequency spectra show different vibration characteristics of the piezoelectric bimorphs with different electrical connections. Piezoelectric bimorphs with normal connections have three-dimensional coupled vibration characteristics and the out-of-plane vibration dominates the motion. However, only in-plane vibration motions can be excited in the high-frequency range for abnormal connections, and the resonant characteristics are similar to the single-layered piezoelectric plate. The triple-layered piezoelectric bimorphs with abnormal connection are also analyzed using theoretical analysis. The resonant frequencies, mode shapes, and normalized displacements are calculated based on the analytical solution. The experimental results and the theoretical analysis are in good agreement with the numerical calculations using the finite element method. From the discussion of the results for the parallel- and series-type piezoelectric bimorphs with normal and abnormal connections, the vibration characteristics at high frequencies are completely analyzed in this study.     

The investigation of three-dimensional vibration for piezoelectric rectangular parallelepipeds using the AF-ESPI method

Electronic speckle pattern interferometry (ESPI) is a powerful tool for the full-field measurement of a deformed body. In this paper, a three-dimensional vibrating block that couples the out-of-plane and in-plane motions is investigated using the amplitude-fluctuation ESPI (AF-ESPI). This method demonstrates the advantages of combining high processing speed, such as in the subtraction method, with high fringe sensitivity, such as in the time-averaged method. The optical system for AF-ESPI is then employed to analyze the volume vibration of piezoelectric material for a rectangular parallelepiped configuration. Based on the fact that fringe patterns measured by the AF-ESPI method appear as a clear picture only at the resonant frequency, both the natural frequencies and the out-of-plane and in-plane vibration mode shapes are successfully obtained in this study. Finally, the impedance analysis as well as the finite element method (FEM) with three-dimensional model are also conducted to compare with the result obtained by AF-ESPI. It is shown that the numerical calculation and the experimental result agree fairly well for both the resonant frequency and the mode shape in three-dimensional configurations     

Vibration analysis of piezoelectric materials by optical methods

This study provides two noncontact and realtime optical measurement methods to assess the displacement, natural frequencies, and mode shapes of a vibrating piezoelectric material. The methods are carried out using amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), which are full-field and point-wise displacement measurement, respectively. Because the fringe patterns measured by AF-ESPI appear as a clear picture at the natural frequency, both natural frequencies and mode shapes of the vibrating material can be successfully obtained. In the LDV system, a swept-sine excitation signal from the function generator to the beam can result in a corresponding peak in frequency response curve at natural frequencies. From the frequency response curve, the natural frequencies are thus acquired. Measured results by both methods are seen to be in good agreement with theoretical predictions by the Galerkin method and finite element method.     

Identification of elastic, dielectric, and piezoelectric constants in piezoceramic disks

Three-dimensional modeling of piezoelectric devices requires a precise knowledge of piezoelectric material parameters. The commonly used piezoelectric materials belong to the 6mm symmetry class, which have ten independent constants. In this work, a methodology to obtain precise material constants over a wide frequency band through finite element analysis of a piezoceramic disk is presented. Given an experimental electrical impedance curve and a first estimate for the piezoelectric material properties, the objective is to find the material properties that minimize the difference between the electrical impedance calculated by the finite element method and that obtained experimentally by an electrical impedance analyzer. The methodology consists of four basic steps: experimental measurement, identification of vibration modes and their sensitivity to material constants, a preliminary identification algorithm, and final refinement of the material constants using an optimization algorithm. The application of the methodology is exemplified using a hard lead zirconate titanate piezoceramic. The same methodology is applied to a soft piezoceramic. The errors in the identification of each parameter are statistically estimated in both cases, and are less than 0.6% for elastic constants, and less than 6.3% for dielectric and piezoelectric constants.     

An approximated 3-D model of cylinder-shaped piezoceramic elements for transducer design

In this paper an approximated 3-D model of cylinder shaped piezoceramics is described. In the hypothesis of axial symmetry, the element vibration in the extensional and radial directions is described by two coupled differential wave equations. The model is obtained choosing, as solution of these equations, two orthogonal wave functions, each depending only on one axis, corresponding to the propagation direction. The mechanical boundary conditions are applied imposing continuity between the stresses and the external forces on the surfaces of the element in an integral way, while, as far as the electrical boundary condition is concerned, two possibilities are explored: to neglect the piezoelectric constant in the transverse direction and to impose an integral condition also for the electric field. Comparisons with experimental results show this last approach to give better results. The model predicts with sufficient accuracy only the first radial and the first thickness modes of the cylinder-shaped piezoceramic element of arbitrary aspect ratio; but, for these modes, it is able to compute all the relations between the input applied voltage and the output forces and velocities on every external surface. Because only these two modes are of relevance in the practical applications of piezoceramic elements as ultrasonic transducers, the model can be used as a simple and useful tool in transducer design and optimization. Experimental validations of the model are also shown in the work.     

Theoretical analysis of a ceramic plate thickness-shear mode piezoelectric transformer

We perform a theoretical analysis on a ceramic plate piezoelectric transformer operating with thickness-shear modes. Mindlin's first-order theory of piezoelectric plates is employed, and a forced vibration solution is obtained. Transforming ratio, resonant frequencies, and vibration mode shapes are calculated, and the effects of plate thickness and electrode dimension are examined.