Finite-element analysis of vibrational modes in piezoelectric ceramic disks

The natural vibrational modes of axially symmetric piezoelectric ceramic disks have been calculated by the finite-element method. The disks are of the type used as active elements in compressional wave ultrasonic transducers, and are electrically polarized in thickness with full electrodes on the disk's major faces. To optimize disk geometry for ultrasonic transducer application, the dependence of the vibrational modes on the disk diameter-to-thickness ratio for ratios from 0.2 (a tall cylinder) to 10.0 (a thin disk) has been studied. Series and parallel resonance frequencies for each of the modes are determined through an eigenfrequency analysis, and effective electromechanical coupling coefficients are calculated. The modal displacement fields in the disk are calculated to determine the physical nature of each mode. An analysis of the complete spectrum of piezoelectrically active modes as a function of diameter-thickness ratio is presented for the ceramic PZT-5H, including and identification of radial, edge, length expander, thickness shear, and thickness extensional vibrations. From this analysis, optimal diameter-to-thickness ratios for good transducer performance are discussed.     

Vibrations and static responses of asymmetric bimorph disks of piezoelectric ceramics

In an earlier article, the flexural vibrations in bimorph disks and extensional vibrations in homogeneous disks of piezoelectric ceramics were studied. In the present paper, the coupled flexural and extensional vibrations and static responses in an asymmetric bimorph disk, which is formed by bonding together two piezoelectric ceramic disks of unequal thickness and opposite polarization, are investigated. Governing equations of coupled motions for asymmetric bimorphs are deduced from the recently derived 2-D, first-order equations for piezoelectric crystal plates with thickness-graded material properties. Then, closed form solutions of these equations for circular disks are obtained for free vibrations, piezoelectrically forced vibrations, and responses under static voltage difference. Resonance frequencies, distribution of displacements and surface charges, impedances, and static responses are calculated for asymmetric bimorph disks of various thickness ratios and diameter-to-thickness ratios. Experimental data on resonances and impedances are obtained for asymmetric bimorph disks of PZT-857 for different thickness ratios. Comparisons of predicted and measured results show that the agreements are close.     

Development of a radial-torsional vibration hybrid type ultrasonic motor with a hollow and short cylindrical structure

A longitudinal-torsional hybrid-type ultrasonic motor has larger torque and lower revolution speed compared with other kinds of ultrasonic motors. It drives devices directly and precisely, so it is adaptable to many fields, especially aeronautics and astronautics, as a servo actuator. Due to the different sound propagation speeds of longitudinal and torsional vibrations in the stator, it is difficult to match resonant frequencies of longitudinal and torsional vibrations. In this paper, a new radial-torsional vibration hybrid-type ultrasonic motor is put forward, which utilizes longitudinal vibration derived from radial vibration by the Poisson effect. The short, hollow cylindrical structure easily makes resonant frequencies of first-order radial and torsional vibrations into degeneracy. First, the new structure of the motor is presented. Second, the principle of matching the resonant frequencies is developed, and the motor geometry is optimized by ANSYS software. Finally, a 60-mm diameter prototype is fabricated, which performs well. The no-load velocity and maximum torque are 25 r/min and 5 N·m, respectively. This kind of motor is small, light, and noiseless.     

A p‐version,first order shear deformation,finite element for geometrically non‐linear vibration of curved beams

A p‐version, hierarchical finite element for curved, moderately thick, elastic and isotropic beams is introduced. The convergence properties of the element are analysed and some results are compared with results published elsewhere or calculated using a commercial finite element package. It is verified that, with the proposed element, shear locking does not affect the computation of the natural frequencies and that low dimensional, accurate models are obtainable. Geometrically non‐linear vibrations due to finite deformations, which occur for harmonic excitations with frequencies close to the first three natural frequencies of vibration, are investigated using Newmark's method. The influence of the thickness, longitudinal inertia and curvature radius on the dynamic behaviour of curved beams are studied. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Study on a new type of radial composite piezoelectric ultrasonic transducers in radial vibration

In this paper, a new type of radial composite ultrasonic transducers in radial vibration is presented and studied. The radial composite ultrasonic transducer consists of a solid piezoelectric ceramic thin disk and a hollow metal thin, circular ring in radial vibration; and they are connected together in the radial direction. The radial vibrations of a piezoelectric ceramic thin disk and a hollow, metal, thin, circular ring are analyzed, respectively. Their radial electromechanical equivalent circuits are obtained. Based on the electromechanical equivalent circuits and using the boundary conditions between the piezoelectric ceramic thin disk and the hollow, metal, thin ring, the electromechanical equivalent circuit of the radial composite ultrasonic transducer is derived and the resonance frequency equation is obtained. The theoretical results from the resonance frequency equation are in good agreement with the measured radial resonance frequencies, and they also are in a good agreement with the results from the numerical method.     

Extensional,thickness-stretch and symmetric thickness-shear vibrations of piezoceramic disks

A set of two-dimensional (2-D), second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced previously. The new truncation procedure developed recently is used for it improves the accuracy of calculated dispersion curves. Closed-form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those obtained from the 3-D equations, and the predicted resonance frequencies are compared with experimental data. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes-i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thickness-shear (s.TSh)-is predominant.     

Theoretical, numerical, and experimental investigation on resonant vibrations of piezoceramic annular disks

In this study, vibration characteristics of thin piezoceramic annular disks with stress-free boundary conditions are investigated by theoretical analysis, numerical simulation, and experimental measurement. The nonaxisymmetric, out-of-plane (transverse), and axisymmetric in-plane (tangential and radial extensional) vibration modes are discussed in detail in terms of resonant frequencies, mode shapes, and electrical currents. Two optical techniques, amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), as well as the electrical impedance measurement are used to validate the analytical results. Both theoretical and experimental results indicate that the transverse and tangential vibration modes cannot be determined by the impedance analysis; hence, only resonant frequencies of extensional vibration modes are presented from the impedance analyzer. The LDV system is used to measure the resonant frequencies of transverse vibrations. However, both the transverse and extensional vibration modes and resonant frequencies of piezoceramic annular disks are obtained by the AF-ESPI method, and the interferometric fringes are produced instantly by a video recording system. Numerical results obtained by finite-element calculations are compared with those from theoretical analysis and experimental measurements. It is shown that the theoretical predictions of resonant frequencies and the corresponding mode shapes agree well with the experimental results. Good agreement between the predicted and measured electrical impedance also is found. The dependence of resonant frequencies and dynamic electromechanical coupling coefficients on the inner-to-outer radius ratio also is analyzed and discussed in this study.     

Radial vibration of the composite ultrasonic transducer of piezoelectric and metal rings

The radial composite ultrasonic transducer of a piezoelectric ceramic ring and a metal ring is studied. The radial vibrations of a piezoelectric ceramic ring polarized in the thickness direction and a metal ring are analyzed and their electromechanical equivalent circuits are obtained. On the basis of the electromechanical equivalent circuits of the piezoelectric and the metal ring and the radial boundary conditions, the total electromechanical equivalent circuit for the composite ultrasonic transducer is also obtained. The resonance frequency and anti-resonance frequency equations of the composite ultrasonic transducer are given. Some radial composite ultrasonic transducers are designed and manufactured and their resonance and anti-resonance frequencies are measured. Experiments show that the measured resonance frequencies are in a good agreement with the theoretical results.     

Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness

A thermo elastic analysis is presented for axisymmetric rotating disks made of functionally graded material (FGM) with variable thickness. Material properties are assumed to be temperature-dependent and graded in the radial direction according to a grading index power law distribution. The temperature field considered is assumed to be uniformly distributed over the disk surface and varied in the radial direction. Semi-analytical solutions for the displacement field are given for solid disk and annular disk under free-free and fixed-free boundary conditions. The effects of the thermal field, the material grading index and the geometry of the disk on the displacement and stress fields are investigated. Results of this study emphasize on the crucial role of the temperature-dependent properties in a high temperature environment. A comparison of these results with the reported ones in the literature that is temperature-dependent versus temperature-independent suggests that a functionally graded rotating disk with concave thickness profile can work more efficiently than the one with uniform thickness irrespective of whether the material properties are assumed to be temperature-dependent or temperature-independent.     

On the performance characterization of ultrasonic air transducerswith radiating membranes

An ultrasonic air transducer used for sensing essentially requires an efficient transduction into the surrounding medium. A transducer consisting of a metal membrane coupled with a resonating piezoelectric disk, the latter being driven in its radial mode of vibrations, has been found to be significantly applicable in this aspect. The present study reports the performance characterization at resonance and at higher frequencies of two such transducers with different dimensions of the housings. Vibrational amplitude characteristics are studied employing phase-locked laser interferometry technique that presents sufficiently high displacement amplitude of the membrane at resonance. A consequent high-acoustic pressure field is also obtained. The design aspects for such ultrasonic air transducers with efficient transduction capability and adjustable bandwidth are discussed     

大功率纵-弯超声振动系统的研究

本文研究了一种大功率纵－弯昨合模式超声振动系统。该系统由两个纵向大功率夹心式压电超声换能器及一个均匀截面直棒组成。文中给出了该振动系统的具体结构及其共振频率设计方程。根据均匀直棒的纵向及弯曲振动理论，导出了振动系统中纵向及弯曲振动同频共振的条件。     

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.     

Thickness-shear vibration of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness

An exact solution is obtained from the three-dimensional equations of linear piezoelectricity for thickness-shear vibrations of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness at its major faces. Effects of the shear stiffness of the electrodes on resonant frequencies are examined.     

3-D free vibration analysis of thick functionally graded annular sector plates on Pasternak elastic foundation via 2-D differential quadrature method

Early studies on annular sector plate vibrations were focused on two-dimensional theories, such as the classical plate theory and the first- and the higher-order shear deformation plate theories. These plate theories neglect transverse normal deformations and generally assume that a plane stress state of deformation prevails in the plate. These assumptions may be appropriate for thin plates. In this paper, free vibration of thick functionally graded annular sector plates with simply supported radial edges on a two-parameter elastic foundation, based on the three-dimensional theory of elasticity, using differential quadrature method for different circular edge conditions including simply supported-clamped, clamped–clamped, and free-clamped is investigated. A semi-analytical approach composed of differential quadrature method and series solution is adopted to solve the equations of motion. The material properties change continuously through the thickness of the plate, which can vary according to a power law, exponentially, or any other formulations in this direction. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The new results can be used as benchmark solutions for future research.     

Longitudinal vibrations of a silica fiber segment characterized using a fiber Bragg grating

A modulator assembly that excites longitudinal vibrations in a short (19 cm) silica fiber segment is described. A physical model of the modulator assembly is used to theoretically predict the behavior of longitudinal vibrations in the fiber segment. The longitudinal vibrations are experimentally characterized using an intrinsic fiber Bragg grating. Experimental results are compared to theoretical predictions.     

Vibration characteristics of a corrugated cylindrical shell piezoelectric transducer

We study coupled extensional and flexural cylindrical vibrations of a corrugated cylindrical shell piezoelectric transducer consisting of multiple pieces of circular cylindrical surfaces smoothly connected along their generatrices. Using the classical shell theory, a theoretical solution is obtained. Based on the solution, basic vibration characteristics of resonant frequencies, mode shapes, and internal forces are calculated and examined for the corrugated transducers, consisting of a few circular pieces each about 50 μm thick, with radius of 5 mm and span angle of 120°. For these transducers the first resonance is of the order of 10-100 Hz.     

Mode shapes and frequencies of radially symmetric vibrations of annular sandwich plates of variable thickness

An analysis and numerical results of radially symmetric vibrations of annular sandwich plates with core of linearly varying thickness are presented. The face sheets are treated as membranes of constant thickness, and the core is assumed to be solid as well as moderately thick. Due to linear thickness variation in the core, the face sheets take the shape of a truncated conical shell and because of this the face sheets membrane forces contribute to the bending and transverse shear of the core of the sandwich plate. Keeping this in view, the equations of motion for such a plate are developed by Hamilton’s energy principle. The frequency equations for three different combinations of boundary conditions, namely clamped at the inner edge and clamped or simply supported or free at the outer edge, are obtained by employing the differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as taper parameter, thickness of the core at the center, face thickness, and radii ratio on the natural frequencies has been analyzed. Three-dimensional mode shapes for a specified plate for all the three boundary conditions are illustrated. A comparison of results is presented.     

Lead zirconate titanate/poly(vinylidene fluoride-trifluoroethylene) 1-3 composites for ultrasonic transducer applications

A new procedure for preparing lead zirconate titanate (PZT)/poly(vinylidene fluoride-trifluoroethylene) (P(VDF-TrFE)) 1-3 composites with both phases piezoelectrically active is described. Sintered PZT rods are inserted into a prepoled copolymer matrix, and the composite is repoled under a lower electric field. Using this new procedure, the dipoles in the two phases are aligned in either the same or opposite directions. Composite disks, of 12.7-cm diameter and 0.33- to 0.60-mm thicknesses, have been fabricated with PZT rods of 0.8 or 1 mm diameter distributed in a square pattern with 3 mm center-to-center separation. The ceramic volume contents of the composite disks are 3.6 and 5.6%, respectively. The resonance characteristics of the composite disks consist of the resonance modes of the two constituent phases, but they are dominated by the coupled longitudinal thickness mode (H-mode) of the PZT rods. The coupled radial mode (L-mode) resonance of the PZT rods is significant only for thin disks. The observed resonance frequencies of the H- and L-modes agree well with the values calculated from the coupling theory. The thickness mode resonance of the copolymer matrix (T-mode) is present but hardly observable in thick disks. The composite disks have been fabricated into transducers with air-backing and with no front face matching layer, and their performance characteristics have been evaluated in water. The transmitting and receiving voltage responses of a PZT/P(VDF-TrFE) composite transducer are better than those of a PZT/epoxy composite transducer. The transmitting and receiving voltage responses are improved when the PZT rods and copolymer matrix are poled in opposite directions, especially when the resonance frequencies of the H- and T-modes are approximately equal. When the phases are poled in the same direction and the resonance peaks associated with the Hand T-modes just overlap, the bandwidth is improved. Using 0.33-mm thick composite disks, a transducer can be produced with three operating frequencies by poling the constituent phases in the same direction, or with two operating frequencies at equal efficiency by poling the constituent phases in opposite directions. The PZT/P(VDF-TrFE) 1-3 composite transducer, especially the one with multiple operating frequencies, should be very promising in the applications of medical ultrasonic imaging.     

喷涂阻尼厚度对车轮振动声辐射的影响

在列车车轮表面喷涂阻尼材料可以降低车轮振动声辐射,通过试验调查喷涂阻尼厚度对其减振降噪性能的影响。在半消声室进行对比试验,测试了斜曲型辐板车轮在无阻尼、喷涂1 mm和2 mm情况下的振动声辐射,和双S型辐板车轮在无阻尼、喷涂1 mm和4 mm阻尼下的声辐射。测试结果表明：对于斜曲型辐板车轮,2 mm阻尼层对车轮的减振区域和减振量均优于1 mm阻尼层,在径向和轴向激励下,1mm阻尼层降噪量分别为2.0 dB(A)和1.0 dB(A);对于双S型辐板车轮,在径向和轴向激励下,1 mm阻尼层降噪量分别为1.9 dB(A)和1.1 dB(A)。对于这两种辐板形式车轮,阻尼层增厚,降噪效果均增加。对于斜曲型车轮,在径向激励下阻尼具有更好的降噪效果,对于双S型车轮,在径向激励和轴向激励下阻尼降噪效果近似相同。