Thickness-shear vibration of an AT-cut quartz resonator with a hyperbolic contour

A theoretical analysis is performed on thicknessshear vibrations of a contoured AT-cut quartz resonator with a hyperbolic thickness variation using the Legendre equation and hypergeometric function. Based on the solution, resonant frequencies and modes are calculated. Strong energy trapping of the modes is observed. The effects of the parameters of the hyperbolic contour on resonant frequencies and modes are examined. A comparison with the conventional contoured resonator in the literature with a quadratic thickness variation is made. The behaviors of the two types of resonators are qualitatively similar, with small but significant quantitative differences.     

Modelling of anisotropic superconducting waveguide as equivalent transmission line

This article presents a method of linking full-wave field analysis to distributed-circuit analysis by modelling anisotropic superconducting waveguides, consisting of two or more parallel superconductors, as equivalent transmission lines. As an application of this method, a full-wave solution is first found for an anisotropic superconducting planar waveguide by solving Maxwell's equations and London's equations modified to account for the effect of both anisotropy and normal electron conduction in anisotropic superconductors. Based on the full-wave solution, a complete set of equivalent transmission line parameters is then derived analytically to model the anisotropic superconducting planar waveguide. Numerical results are given to show quantitatively how the effect of anisotropy influences the transmission properties of the anisotropic superconducting planar waveguide at high frequencies. The results are compared with those for an isotropic superconducting planar waveguide as well as with some well-known results from a quasi-static approximation. It is shown that the full set of equivalent transmission line parameters based on the full-wave analysis is desired to model properly some strongly anisotropic superconducting waveguides at millimeter-wave frequencies.     

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.     

Spectral analysis of vibrational harmonic motion by use of a continuous-wave CO2 Doppler lidar

Vibrational motion of a harmonic oscillator was investigated with a focused continuous-wave (cw) CO2 Doppler lidar at 9.1-microm wavelength. A continuum of frequencies along with many discrete, equally spaced, resonant frequency modes was observed. The frequency modes are similar in structure to the oscillatory longitudinal modes of a laser cavity and arise because of interference of the natural resonant frequency of the oscillator with specific frequencies within the continuum. Each consecutive resonant frequency mode occurred for a movement of the oscillator much less than the wavelength of incident lidar radiation. For vigorous vibration of the oscillator, the observed spectra may be indicating nonlinear motion.     

Modal Coupling in Micromechanical Vibratory Rate Gyroscopes

The authors present modeling approaches to describe the coupling of modes in a resonant vibratory rate gyroscope. Modal coupling due to off-diagonal stiffness and damping terms is considered. Three analytical modeling approaches are presented in the context of a$z$-axis micromechanical vibratory rate gyroscope fabricated in an integrated polysilicon surface-micromachining process. The first approach is based on frequency-response analysis of the gyroscope output. The second approach takes the route of state-space-based system identification to identify the modal-coupling parameters. A third approach based on measured vibration data identifies the coupling parameters due to stiffness and damping. These three methods are then applied to predict the extent of displacement and force coupling between the drive and the sense axes of an existing device as a function of varying degrees of matching between the resonant frequencies associated with the drive and the sense modes. Experimental data show that as the resonant frequencies of the drive and sense modes are brought closer together, an improvement in overall resolution and scale factor of the device is obtained at the expense of an enhanced coupling of forces to displacements between the two axes and the onset of instability for an open-loop sensing implementation.     

Using anisotropic dispersive materials for high-precision determination of the energy of charged particles

Vavilov-Cherenkov radiation (VCR) generated by a charged particle moving in a waveguide filled with an anisotropic dispersive material is considered. It is shown that, for a certain character of dispersion and anisotropy, which can be realized using artificial metamaterials, it is possible to obtain a pronounced dependence of the frequencies of excited modes on the Lorentz factor in a narrow interval of this parameter. This circumstance can be used for determining the energies of particles and beams with an error that is smaller than the error of measurement of the VCR frequencies. It is also possible to create a “subthreshold” Cherenkov detector, which detects all particles moving with velocities below a certain preset value.     

A coupled-mode theory for periodic piezoelectric composites

A coupled oscillator model to calculate the resonance spectrum of a one-dimensional piezoelectric composite plate, used in ultrasonic transducers, is proposed. Two resonant modes, one produced by the elastic wave reflection on the plate boundaries (thickness resonance) and the other by the reflection on the periodic discontinuities (lateral resonance) are considered. A Kronig-Penney model is used to calculate the lateral resonances. The thickness resonance is obtained with an effective medium model. The coupling of these two modes is described by a biquadratic equation whose solutions are the resonant frequencies of the piezoelectric composite plate. A criterion for a distribution of phases to keep the spurious lateral resonances away from the thickness resonance vicinity is obtained.     

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.     

The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

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.     

Thickness-shear and thickness-twist vibrations of an AT-cut quartz mesa resonator

We study thickness-shear and thickness-twist vibrations of an AT-cut quartz plate mesa resonator with stepped thickness. The equations of anisotropic elasticity are used with the omission of the small elastic constant c(56). An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and energy trapping are calculated and examined. The solution shows that a mesa resonator exhibits strong energy trapping of thickness-shear and thickness-twist modes, and that the trapping is sensitive to some of the structural parameters of the resonator.     

Identifying,Resolving, and Quantifying Anisotropy in ReS2 Nanomechanical Resonators

As an emerging two-dimensional semiconductor, rhenium disulfide (ReS₂) is renowned for its strong in-plane anisotropy in electrical, optical, and thermal properties. In contrast to the electrical, optical, optoelectrical, and thermal anisotropies that are extensively studied in ReS₂, experimental characterization of mechanical properties has largely remained elusive. Here, it is demonstrated that the dynamic response in ReS₂ nanomechanical resonators can be leveraged to unambiguously resolve such disputes. Using anisotropic modal analysis, the parameter space for ReS₂ resonators in which mechanical anisotropy is best manifested in resonant responses is determined. By measuring their dynamic response in both spectral and spatial domains using resonant nanomechanical spectromicroscopy, it is clearly shown that ReS₂ crystal is mechanically anisotropic. Through fitting numerical models to experimental results, it is quantitatively determined that the in-plane Young's moduli are 127 and 201 GPa along the two orthogonal mechanical axes. In combination with polarized reflectance measurements, it is shown that the mechanical soft axis aligns with the Re-Re chain in the ReS₂ crystal. These results demonstrate that dynamic responses in nanomechanical devices can offer important insights into intrinsic properties in 2D crystals and provide design guidelines for future nanodevices with anisotropic resonant responses.     

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.     

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.     

Analysis of microwave ovens loaded with lossy process materials using the transmission-line-matrix method

The transmission-line-matrix (t.l.m.) numerical analysis technique is used in this paper to evaluate resonant frequencies and field parameters of the generated modes in microwave ovens loaded with lossy process materials. Analysis is made for the variation of resonant frequencies with the loss factor of the loading material. Temperature rise profile through the load is predicted from the calculated results of field parameters of sustained modes. The demonstrated data are for the dimensional parameters of a rectangular cavity of a commercial microwave oven operating at the standard heating frequency 915 MHz.     

Development of a novel ultrasonic motor resonator using topology optimization

The use of topology optimization in the design of a novel stator for an ultrasonic motor (USM) is investigated. The design challenge is to produce a stator, with two resonant modes whose frequencies are in a ratio of 1:2. When driven together, these modes result in a contact point trajectory in a figure of eight shape. As a result, only one electronic amplifier is required to drive the proposed device. In contrast traditional travelling wave USM, with elliptical contact point trajectories, require two modes with equal resonant frequencies to be driven 90° out of phase, and therefore require two amplifiers, one for each mode. To achieve a suitable stator design, a slightly unconventional topology optimization problem formulation is proposed, in which the objective function is to minimize the amount of material with intermediate density, while satisfying a constraint related to the frequency ratio of selected resonant modes. The planar design produced using the optimization procedure was refined using a detailed three dimensional finite element analysis. A prototype of the proposed stator design was manufactured and experimentally characterized. Scanning laser vibrometry measurements from two positions were used to measure the figure of-eight motion. Finally, the stator was fitted with a preloaded slider to form a simple linear motor demonstrator which was characterized experimentally. The prototype motor produced a slider speed of 14 mm/s reversibly and a maximum force of 50 mN.     

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.     

Anisotropic MoS2 Nanosheets Grown on Self‐Organized Nanopatterned Substrates

Manipulating the anisotropy in 2D nanosheets is a promising way to tune or trigger functional properties at the nanoscale. Here, a novel approach is presented to introduce a one‐directional anisotropy in MoS₂ nanosheets via chemical vapor deposition (CVD) onto rippled patterns prepared on ion‐sputtered SiO₂/Si substrates. The optoelectronic properties of MoS₂ are dramatically affected by the rippled MoS₂ morphology both at the macro‐ and the nanoscale. In particular, strongly anisotropic phonon modes are observed depending on the polarization orientation with respect to the ripple axis. Moreover, the rippled morphology induces localization of strain and charge doping at the nanoscale, thus causing substantial redshifts of the phonon mode frequencies and a topography‐dependent modulation of the MoS₂ workfunction, respectively. This study paves the way to a controllable tuning of the anisotropy via substrate pattern engineering in CVD‐grown 2D nanosheets.     

Study of the bending modes in circular quartz resonators

An experimental and theoretical study of bending modes in a partially electroded circular piezoelectric quartz (AT-cut) with free edge is presented. The quartz is excited by a voltage pulse applied on the electrodes, and its surface is scanned by a laser vibrometer that measures the out-of-plane displacements. The classical theory of bending of thin disks is used to describe the flexural modes at frequencies lower than the first thickness shear resonance (6 MHz). A fairly good agreement is found between experimental and theoretical results for the forced mode shapes and for the resonance frequencies. However, it appears that the two springs used to maintain the disk in position introduce extra clamping conditions. Several source shapes were studied, among which a collection of an arbitrary number of forces is particularly useful. The two-dimensional wavenumber representation shows the presence of anisotropy related to the crystallographic axes at higher frequencies, which is not predicted by the model. The experimental phase velocities are compared to those given by the classical theory of disks and to those of Lamb A(0) mode. This study confirms the correspondence at low frequencies between the A(0) mode and the bending eigenmodes of a disk with finite size.     

Directed growth of horizontally aligned gallium nitride nanowires for nanoelectromechanical resonator arrays

We report the growth of horizontally aligned arrays and networks of GaN nanowires (NWs) as resonant components in nanoelectromechanical systems (NEMS). A combination of top-down selective area growth (SAG) and bottom-up vapor-liquid-solid (VLS) synthesis enables flexible fabrication of highly ordered nanowire arrays in situ with no postgrowth dispersion. Mechanical resonance of free-standing nanowires are measured, with quality factors (Q) ranging from 400 to 1000. We obtained a Young's modulus (E) of approximately 338 GPa from an array of NWs with varying diameters and lengths. The measurement allows detection of nanowire motion with a rotating frame and reveals dual fundamental resonant modes in two orthogonal planes. A universal ratio between the resonant frequencies of these two fundamental modes, irrespective of their dimensions, is observed and attributed to an isosceles cross section of GaN NWs.