MULTIDIMENSIONAL COUPLED VIBRATIONS OF PIEZOELECTRIC VIBRATORS——COMPOSITE PIEZOELECTRIC VIBRATORS

This paper presents an analysis of the three-dimensional coupled vibrations of a compositepiezoelectric vibrator by using the apparent elasticity method. A very simple analytic formula for the resonant frequency of the vibrator is obtained and withit the variation of the frequency constant with the geometric size of the vibrator is calculated. A satisfactory agreement is obtained as compared with the published results calculated by thefinite element method. Because the formula is quite simple the resonant frequency of the vibrator can be quickly cal-culated by using a scientific calculator,without having to use a computer,and so this method ismore convenient than the finite element method for engineeing design and estirmation. In addition,a dispersion relation of the phase velocity of "longitudinal"waves in the finiteisotropic solid cylinder is given and a comparison between this relation and those in an infinitecylinder is made. This method may also be used to calculate the resonant frequencies of other     

A SUSCEPTANCE METHOD TO MEASURE PARAMETERS OF PIEZOELECTRIC VIBRATORS

By way of theoretical analyses of the admittance locus of piezoelectric vibrators,a susceptancemethod for measurement of piezelectric vibrators is suggested.By use of the method,we measuredthe relevant elastic,piezoelectric constants and electromechanical coupling coefficients of LiTaO_3crystal.     

Advantage analysis of PMN-PT material for free-flooded ring transducers

The acoustic radiation characteristics of free-flooded ring transducers made of PZT4 and PMN-PT materials are calculated and compared.First,the theoretical formulae for free-flooded ring transducers are studied.The resonant frequencies of a transducer made of PZT4 and PMN-PT materials are calculated.Then,the transmitting voltage responses of the free-flooded ring transducers are calculated using the finite element method.Finally,the acoustic radiation characteristics of the free-flooded ring transducers are calculated using the boundary element method.The calculated results show that the resonant frequencies of the free-flooded ring transducer made of PMN-PT are greatly reduced compared with those made of PZT4 with the same size.The transmitting voltage response of the transducer made of PMN-PT is much higher than that of the transducer made of PZT4.The calculated 3-dB beamwidth of the acoustic radiated far-field directivity of the free-flooded ring transducer made of PZT4 at the resonant frequency 1900 Hz is 63.6 and that of the transducer made of PMN-PT at the resonant frequency 1000 Hz is 64.6.The comparison results show that the free-flooded ring transducer made of PMN-PT material has many advantages over that made of PZT4.The PMN-PT is a promising material for improving the performance of free-flooded ring transducers.     

AMPLITUDE AND PHASE CHARACTERISTICS OF ELEMENTS CLOSELY ARRANGED ON THE CYLINDRICAL ARRAY

The theoretical calculations of the surface pressure amplitude and phase distribution are givenfor an infinitely long cylinder on which the piston elements are closely arranged.The analysis includesthe effects of sound scattering by the cylinder.A generalexpression is derived,which gives the directivityof an element and phase distributions on a circumference of the cylinder for various frequencies.The theoretical results by and large coincide with the measurements.Using the theoretical formulae inthis paper we can provide correct compensations of delay time or amplitude and phase weightingdesired for beam forming in the array design.     

Mechanical properties of silicon nanobeams with an undercut evaluated by combining the dynamic resonance test and finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications.A numerical experimental method of determining resonant frequencies and Young’s modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by using a laser Doppler vibrometer is presented in this paper.Silicon nanobeam test structures are fabricated from silicon-oninsulator wafers by using a standard lithography and anisotropic wet etching release process,which inevitably generates the undercut of the nanobeam clamping.In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut,dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value △L,which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data.By using a least-square fit expression including △L,we finally extract Young’s modulus from the measured resonance frequency versus effective length dependency and find that Young’s modulus of a silicon nanobeam with 200-nm thickness is close to that of bulk silicon.This result supports that the finite size effect due to the surface effect does not play a role in the mechanical elastic behaviour of silicon nanobeams with thickness larger than 200 nm.     

Mechanical properties an undercut evaluated resonance test and of silicon nanobeams with by combining the dynamic finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications. A numerical experimental method of determining resonant frequencies and Young＇s modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by using a laser Doppler vibrometer is presented in this paper. Silicon nanobeam test structures are fabricated from silicon-oninsulator wafers by using a standard lithography and anisotropic wet etching release process, which inevitably generates the undercut of the nanobeam clamping. In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut, dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value △L, which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data. By using a least-square fit expression including △L, we finally extract Young＇s modulus from the measured resonance frequency versus effective length dependency and find that Young＇s modulus of a silicon nanobeam with 200-nm thickness is close to that of bulk silicon. This result supports that the finite size effect due to the surface effect does not play a role in the mechanical elastic behaviour of silicon nanobeams with thickness larger than 200 nm.     

Mechanical properties of silicon nanobeams with undercut evaluated by combining dynamic resonance test and finite element analysis

Mechanical properties of silicon nanobeams are of prime importance in nanoelectromechanical system applications. A numerical experimental method of determining resonant frequencies and Young's modulus of nanobeams by combining finite element analysis and frequency response tests based on an electrostatic excitation and visual detection by laser Doppler vibrometer is presented in this paper. Silicon nanobeams test structures are fabricated from silicon-on-insulator wafers by using a standard lithography and anisotropic wet etching release process, which inevitably generates the undercut of the nanobeam clamping. In conjunction with three-dimensional finite element numerical simulations incorporating the geometric undercut, dynamic resonance tests reveal that the undercut significantly reduces resonant frequencies of nanobeams due to the fact that it effectively increases the nanobeam length by a correct value Δ L, which is a key parameter that is correlated with deviations in the resonant frequencies predicted from the ideal Euler-Bernoulli beam theory and experimentally measured data. By using a least-square fit expression including Δ L, we finally extract Young's modulus from the measured resonance frequency versus effective length dependency and find that Young's modulus of silicon nanobeam with 200-nm thickness is close to that of bulk silicon. This result supports that the finite size effect due to surface effect does not play a role in mechanical elastic behaviour of silicon nanobeams with the thickness larger than 200 nm.     

Local resonant characteristics of a layered cylinder embedded in the elastic medium

Three kinds of resonant modes of a single layered circular elastic cylinder embedded in the elastic medium are analysed by considering the oscillation of the scatter＇s core, based on the fact that the core moves as a rigid body when the shell material is very compliant. The resonant frequencies of the single resonator acquired by our method are in good agreement with those calculated by the local interaction simulation approach （LISA） for the local resonant phononic crystal. Therefore, the local resonant characteristics of a single layered circular elastic cylinder can be used to evaluate the resonant frequencies of the phononic crystal. The effects of the geometrical and physical parameters of the shell and the core are also studied in details. This work is significant for designing the locally resonant phononic crystal based on the local resonant characteristics of the single resonator, and the resonant frequencies can be tuned by selecting the geometrical sizes and the materials.     

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied.It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory.The governing equations of motion are derived by using the Rayleigh-Ritz method and transformed into Mathieu equations,which are formed to determine the stability criterion and stability regions for parametricallyexcited linear resonant beams.An improved stability criterion is obtained using periodic Lyapunov functions.The boundary points on the stable regions are determined by using a small parameter perturbation method.Numerical results and discussion are presented to highlight the effects of beam length,axial force and damped coefficient on the stability criterion and stability regions.While some stability rules are easy to anticipate,we draw some conclusions:with the increase of damped coefficient,stable regions arise;with the decrease of beam length,the conditions of the damped coefficient arise instead.These conclusions can provide a reference for the robust design of parametricallyexcited linear resonant sensors.     

Acoustic anechoic layers with singly periodic array of scatterers: Computational methods,absorption mechanisms,and optimal design

The acoustic properties of anechoic layers with a singly periodic array of cylindrical scatterers are investigated. A method combined plane wave expansion and finite element analysis is extended for out-of-plane incidence. The reflection characteristics of the anechoic layers with cavities and locally resonant scatterers are discussed. The backing is a steel plate followed by an air half space. Under this approximate zero transmission backing condition, the reflection reduction is induced by the absorption enhancement. The absorption mechanism is explained by the scattering/absorption cross section of the isolated scatterer. Three types of resonant modes which can induce efficient absorption are revealed. Due to the fact that the frequencies of the resonant modes are related to the size of the scatterers, anechoic layers with scatterers of mixed size can broaden the absorption band. A genetic optimization algorithm is adopted to design the anechoic layer with scatterers of mixed size at a desired frequency band from 2 kHz to l0 kHz for normal incidence, and the influence of the incident angle is also discussed.     

MEASUREMENT AND CALCULATION OF ELECTROMECHANICAL PARAMETERS OF PIEZOCERAMIC THIN CYLINDRICAL TUBE TRANSDUCERS

On the basis of the electrical admittance equation for the unloaded tangentially and radiallypolarized piezoceramic thin circular cylindrical tube,the method and formulas,and the prognm forcalculating the electromechanical parameters from the measured data are presented.If the size,density,capcitance at low frequency,frequencies at maximum and minimum admittances near theradial resonant frequency and the longitudinal resonant frequency,etc.are known from themeasurement,then the Young's modulus,Poisson's ratio,electromechanical coupling coefficents,permitivity and piezoelectric constants can be calculated with the formulas given here.Practicalmeasurements and calculations for several samples have been carried out with this method.     

RECONSTRUCTIONS OF PHASE SPECTRUM AND TRANSIENT RESPONSES OF SPHERICAL PIEZOELECTRIC TRANSDUCER

In this paper,a simple method deriving the transfer function of a spherical piezoelectric transducerhas been given.Transient responses have been obtained from the analytical expression of transferfunction by using FFT.Calculated results show that the impulse response of the spherical piezoelectrictransducer also exhibits the minimum phase behaviour and so the phase spectrum of its transfer func-tion can be reconstructed from the amplitude spectrum by using the Hilbert transform,the reconstruct-ed phase spectrum is in good agreement with that calculated from the analytical expression of transferfunction directly. Based on above-mentioned results,the phase spectrum of transfer function of a F42-c spherical pic-zoelectric transducer has been reconstructed from the measured values of its amplitude spectrum for thefirst time.The reconstructed phase spectrum is basically consistent with the measured values of phasespectrum. In addition,the impulse response and step response of F42-c spherical piezoelectric tra     

SCATTERING OF PLANE WAVE BY PIEZOELECTRIC CYLINDER——TRANSFER FUNCTION,TRANSIENT RESPONSES AND PHASE RECONSTRUCTION OF TRANSDUCER

In this paper,under Kirchhoff's assumptions,a theory of the thin shell of radially-Polarizedpiezoelectric ceramic cylinder has been developed.Using the methods of Green's function and separa-tion of variables the scattering of a plane wave by the cylinder has been solved.The normal modeseries expressions of the velocity distribution of the surface of the cylinder and of the total soundfield have been obtained and from those an exact analytic expression of the transfer function of thecylinder as a sound receiver has been give.Arbitrary transient response of the cylinder can be cal-culated from the transfer function by using FFT. For PZT-5 piezoelectric ceramic cylinder,the transfer function,impulse response and step responsehave been calculated.The results show that the impulse response displays a typical minmum phasebehaviour and the cylindrical piezoelectric receiver belongs to the minimum phase system.Therefore,the phase spectrum of transfer function of the receiver can be reconstructed from its amplit     

ANALYSIS OF THE LOWER MODES OF FREE VIBRATION OF A PIEZOELECTRIC CERAMIC DISK POLARIZED IN THE THICKNESS DIRECTION

By means of the adjoints method of solving the differential equations of two-point boundaryvalue problems,the equations of motion of a boundary stress-free ceramic disk vibrator polarizedin the thickness direction are solved by numerical integration.The resonance frequencies of the lowermodes of radial vibration under the influence of motion in the thickness direction and the particledisplacement distribution patterns are obtained for disks whose thickness-diameter ratios are less than0.45.Correction curves for thickness effect on fundamental frequencies are given.Curves showing.the effect of Poisson ratio on the relation between the second harmonic-fundamental ratio and thethickness-diameter ratio are also given. In the numerical calculations,piezoelectric ptoperties of the ceramic disks are fully taken intoconsideration and no restrictions are set for the dimensions of the disks.Experimental results agreewell with those obtained by calculation.     

Dynamic calibration of shock overpressure transducers

Piezoelectric transducers for dynamic overpressure measurements are commonly calibrated with static or quasistatic loads and the calibration is extrapolated to frequencies up to 30% of the resonant frequency of the piezoelectric crystal. Sinusoidal pressure generators are also used for dynamic calibration up to 500 Hz in the range of 3 MPa. This paper describes a method for dynamic calibration using transient overpressures, with rise time of 2 μsec and width 40 μsec, generated by exploding wires in air. The calibration is done in the range of 600 kPa.     

Band structure characteristics of T-square fractal phononic crystals

The T-square fractal two-dimensional phononic crystal model is presented in this article.A comprehensive study is performed for the Bragg scattering and locally resonant fractal phononic crystal.We find that the band structures of the fractal and non-fractal phononic crystals at the same filling ratio are quite different through using the finite element method.The fractal design has an important impact on the band structures of the two-dimensional phononic crystals.     

Novel cell parameter determination of a twisted-nematic liquid crystal display

In this paper a novel method is proposed to determine the cell parameters including the twist angle, optic retardation and rubbing direction of twisted-nematic liquid crystal displays （TNLCD） by rotating the TNLCD. It is a single-wavelength method. Because using subtraction equation of transmittance as curve fitting equation, the influence of the light from environment and the absorption by polarizer, the sample of TNLCD and analyser on the transmittance is eliminated. Accurate results can also be obtained in imperfect darkness. By large numbers of experiments, we found that not only the experimental setup is quite simple and can be easily adopted to be carried out, but also the results are accurate.     

Strained and Piezoelectric Characteristics of Nitride Quantum Dots

The deformation potential and piezoelectric field in nitride GaN/AIN quantum dots （QDs） are investigated in the framework of effective mass approximation （EMA） and finite element method （FEM）. The strained fields and piezoelectric characteristics are studied by using FEM for GaN/AIN QDs （GaN embedded in AIN） in the shape of truncated hexagonal pyramids. We presented the calculated results of the electronic states, wave functions, QD strain field distribution and piezoelectric effects in the QDs. Effects of spontaneous and piezoelectric polarization are taken into account in the calculation. The theoretical results are dependent on QD shapes and sizes. Some of them make the GaN/AIN QDs interesting candidates in optoelectronic applications.     

Laser generation of surface waves on cylinder and coated cylinder

Analytical models of acoustic field excited by a pulsed-laser line source on a cylinder and a coated cylinder were presented.Surface wave dispersive behaviors for a cylinder with a slow coating were analyzed and compared with that of a bare cylinder.Based on this analysis, the laser-generated transient response of the cylindrical Rayleigh wave on cylinder and the perturbed cylindrical Rayleigh wave on coated cylinder and the higher modes were calculated from the models using residue theory and FFT technique.The theoretical results from the superposed waveform of the cylindrical Rayleigh wave and higher modes for both cylinder and coated cylinder agree well with the waveforms obtained from experiment.The transient response of perturbed Raleigh wave on coated cylinder is quite different with cylindrical Rayleigh wave on cylinder because of the guide of surface coating.The results show that the model and numerical method provide a useful technique for quantitatively characterizing coating parameters of coated cylinder by the laser generated surface waves.     

Nonperturbative solutions to cylindrical resonant cavities with dissipative medium and imperfectly conducting walls

Cylindrical waveguides without end surfaces can serve as two-dimensional resonant cavities. In such cavities the electromagnetic oscillations corresponding to an eigenfrequency can always be taken as TM or TE modes even when the walls have a finite conductivity and the medium is absorptive. This paper obtains analytic solutions to the field equations when the cylinder has a circular cross section. Some nonperturbative conclusions are drawn from the eigenvalue equation. Approximate analytic results for the resonant frequencies are obtained when the absorption of the medium is small and the walls are good conductors. Stability of the eigen modes is discussed. Similar results for the coaxial line are presented.