Influence of surface micro-beams with large deflection on the resonance frequency of a quartz crystal resonator in thickness-shear mode vibrations

We study the dynamic behavior of a quartz crystal resonator(QCR)in thickness-shear vibrations with the upper surface covered by an array of micro-beams(MBs)under large deflection.Through taking into account the continuous conditions of shear force and bending moment at the interface of MBs/resonator,dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations.It is found that the frequency shift produces a little right(left)translation for increasing elastic modulus(length/radius ratio)of MBs.Moreover,the frequency right(left)translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one.     

Effect of strain-gradients of surface micro-beams on frequency-shift of a quartz crystal resonator under thickness-shear vibrations

With introduction of the first-order strain-gradient of surface micro-beams into the energy density function,we developed a two-dimensional dynamic model for a compound quartz crystal resonator(QCR) system,consisting of a QCR and surface micro-beam arrays.The frequency shift that was induced by micro-beams with consideration of strain-gradients is discussed in detail and some useful results are obtained,which have important significance in resonator design and applications.     

Electric admittance analysis of quartz crystal resonator in thickness-shear mode induced by array of surface viscoelastic micro-beams

The electric admittance of a compound system composed of a thicknessshear mode(TSM) quartz crystal resonator(QCR) and an array of surface viscoelastic micro-beams(MBs) is studied.The governing equations of the MBs are derived from the Timoshenko-beam theory in consideration of shear deformation.The electrical admittance is described directly in terms of the physical properties of the surface epoxy resin(SU-8) MBs from an electrically forced vibration analysis.It is found that both the inertia effect and the constraint effect of the MBs produce competitive influence on the resonant frequency and admittance of the compound QCR system.By further comparing the numerical results calculated from the Timoshenko-beam model with those from the Euler-beam model,the shear deformation is found to lead to some deviation of an admittance spectrum.The deviations are revealed to be evident around the admittance peak(s) and reach the maximum when a natural frequency of the MBs is identical to the fundamental frequency of the QCR.Besides,a higher order vibration mode of the MBs corresponds to a larger deviation at the resonance.     

Coupled vibrations and frequency shift of compound system consisting of quartz crystal resonator in thickness-shear motions and micro-beam array immersed in liquid

The dynamic characteristics of a quartz crystal resonator(QCR) in thicknessshear modes(TSM) with the upper surface covered by an array of micro-beams immersed in liquid are studied. The liquid is assumed to be inviscid and incompressible for simplicity. Dynamic equations of the coupled system are established. The added mass effect of liquid on micro-beams is discussed in detail. Characteristics of frequency shift are clarified for different liquid depths. Modal analysis shows that a drag effect of liquid has resulted in the change of phase of interaction(surface shear force), thus changing the system resonant frequency. The obtained results are useful in resonator design and applications.     

Novel mode-coupling vibrations of AlN thin film bulk acoustic resonator operating with thickness-extensional mode

The dispersion curves of bulk waves propagating in both Al N and Zn O film bulk acoustic resonators(FBARs) are presented to illustrate the mode flip of the thickness-extensional(TE) and 2nd thickness-shear(TSh2) modes. The frequency spectrum quantitative prediction(FSQP) method is used to solve the frequency spectra for predicting the coupling strength among the eigen-modes in Al N and Zn O FBARs. The results elaborate that the flip of the TE and TSh2 branches results in novel self-coupling vibr...     

Axisymmetric vibrations of a cylindrical resonator measured by holographic interferometry

A circular, cylindrical, ultrasonic resonator excited at one of its resonant frequencies is studied by holographic interferometry. Displacement distributions associated with the axisymmetric oscillations of the resonator are measured with the aid of time-average holograms, and are compared with a simple one-dimensional theory of rod vibrations, corrected for radial inertia. Analysis shows the overall error bounds on measured displacements to be ±9 percent of the maximum displacement at the resonator tip. Although the accuracy of measurements could be increased by refinements in experimental techniques, the work reported here represents substantial improvement in measuring the vibratory motion characteristics of ultrasonic devices over the point-by-point technique used heretofore.     

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear(TS) and thickness-twist(TT) vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone families of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acoustic wave resonators and filters.     

On plane self-excited vibrations of a cantilever suspended wheel

Torsional vibrations of a wheel about its leg axis arising in the carriage rectilinear motion were dubbed the shimmy phenomenon. Because of insufficient understanding of dry friction laws in the case of point contact, the causes of the shimmy phenomenon were explained by specific features of tyre deformation [1–3].     

Influence of the resonator angular displacements in a hemispherical resonator gyro on the coupling between the working and beam-type vibrations

In [1], we have studied how the coupling between the working and beam type vibrations of the resonator affects the hemispherical resonator gyro operation. We show that such a coupling arises if the resonator is unbalanced. The beam type vibrations are considered as translational displacements of the resonator hemisphere in the plane orthogonal to the symmetry axis. In the present paper, we take into account the fact that the hemisphere translational displacement is accompanied by its rotation about the axis perpendicular to the displacement line. We show that in this case a more accurate balancing of the resonator is required to eliminate the coupling between the two vibration modes.     

Nonlinear dynamic analysis of a photonic crystal nanocavity resonator

A nonlinear dynamic model of a one-dimensional photonic crystal nanocavity resonator is presented. It considers the internal tensile stress and the geometric characteristics of a photonic crystal with rectangular(and circular) holes. The solution of the dynamic model shows that the internal tensile stress can suppress the hardening and softening behaviors of the resonator. However, the stress can reduce the amplitude, which is not conducive to an improvement of the sensitivity of the sensor. It is demonstrated that with an optimized beam length, the normalized frequency drift of the beam can be stabilized within 1% when the optical power increases from 2 mW to 6 mW. When the hole size of the resonator beam is close to the beam width, its increase can lead to a sharp rise of the resonant frequency and the promotion of hardening behavior. Moreover,the increase in the optical power initially leads to the softening behavior of the resonator followed by an intensification of the hardening behavior. These theoretical and numerical results are helpful in understanding the intrinsic mechanism of the nonlinear response of an optomechanical resonator, with the objective of avoiding the nonlinear phenomena by optimizing key parameters.     

Effect of small-amplitude vibrations on viscoelastic flows in a plane slider bearing

The qualitative changes of dynamic lift and friction forces caused by small-amplitude harmonic vibrations superimposed on flows in a plane slider bearing are considered for simple viscous and viscoelastic lubricating fluids. Low- and high-frequency disturbances are analysed in greater detail and the most beneficial situations discussed.     

基于二维质点振动模型的半球谐振陀螺谐振子进动分析

半球谐振陀螺是基于固体波的进动效应进行角速度检测的一种全固态陀螺。为分析半球谐振子中固体波的进动效应,基于虚功原理,建立了理想条件下半球谐振陀螺谐振子中固体波进动的动力学模型。针对进动波的动力学模型过于复杂、难于分析的缺陷,提出采用相似系统的分析方法,将该动力学模型等效为二维空间质点的简谐振动模型;采用快变量与慢变量分离的方法,对等效模型进行了参数分析,通过与实际的半球谐振陀螺信号处理系统采集的李沙育图形进行对比,验证了等效简谐振动模型的有效性和正确性。     

Investigation of the vibrational characteristics of a Helmholtz resonator with vibrations of finite amplitude

An investigation has been made of the elastic and dissipative characteristics of a Helmholtz resonator. On the basis of an analysis of the equation of the free vibrations in the resonator, and of an experimental investigation of the damping of the vibrations, it is shown that the elastic characteristic of the resonator can be regarded as linear and the dissipative characteristic, as quadratic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 185–186, September–October, 1976.     

Effect of streamline curvature on flow field of a turbulent plane jet in cross-flow

The effects of the streamline curvature and finite edge velocity on the flow field of a turbulent plane jet in cross-flow are studied numerically by incorporating the curvature effect in the k–ε turbulence model. The improvement in the predictions by the streamline curvature model is assessed by comparing its prediction with those by the standard k–ε model. The predictions by both the models are compared with available experimental data. It has been observed that the performance of the k–ε model with streamline curvature modification is superior to the standard k–ε model.     

Possibilities of a refined method of calculating plane vibrations of lamellar bodies

Institute of Mechanics, Academy of sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 11, pp. 69–77, November, 1991.