THICKNESS-SHEAR VIBRATION OF A RECTANGULAR QUARTZ PLATE WITH PARTIAL ELECTRODES

We study free vibration of a thickness-shear mode crystal resonator of AT-cut quartz. The resonator is a rectangular plate partially and symmetrically electroded at the center with rectangular electrodes. A single-mode, three-dimensional equation governing the thickness-shear displacement is used. A Fourier series solution is obtained. Numerical results calculated from the series show that there exist trapped thickness-shear modes whose vibration is mainly under the electrodes and decays rapidly outside the electrodes. The effects of the electrode size and thickness on the trapped modes are examined.     

ENERGY TRAPPING OF THICKNESS-SHEAR AND THICKNESS-TWIST MODES IN A PARTIALLY ELECTRODED AT-CUT QUARTZ RESONATOR

The thickness-shear and thickness-twist vibrations of a finite and partially electroded AT-cut quartz resonator are investigated. 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 free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.     

THICKNESS-SHEAR AND THICKNESS-TWIST VIBRATIONS OF CIRCULAR AT-CUT QUARTZ RESONATORS

Exact solutions for free vibration frequencies and modes are obtained for thickness-shear and thickness-twist vibrations of unelectroded circular AT-cut quartz plates governed by the two-dimensional scalar differential equation derived by Tiersten and Smythe. Comparisons are made with experimental results and the widely-used perturbation solution by Tiersten and Smythe under the assumption of weak in-plane anisotropy. Our solution is found to be much closer to the experimental results than the perturbation solution. For the frequency of the fundamental thickness-shear mode, the error of the perturbation method is 0.4549%, significant in resonator applications.     

An elastic electrode model for wave propagation analysis in piezoelectric layered structures of film bulk acoustic resonators

Wave propagation in a piezoelectric layered structure of a film bulk acoustic resonator(FBAR) is studied. The accurate results of dispersion relation are calculated using the proposed elastic electrode model for both electroded and unelectroded layered plates. The differences of calculated cut-off frequencies between the current elastic electrode model and the simplified inertial electrode model(often used in the quartz resonator analysis) are illustrated in detail, which shows that an elastic electrode model is indeed needed for the accurate analysis of FBAR. These results can be used as an accurate criterion to calibrate the 2-D theoretical model for a real finite-size structure of FBAR.     

Approximate theory of vibration of crystal plates at high frequencies

The series expansion of displacement in terms of simple thickness modes is used to obtain approximate two-dimensional equations of motion for crystal plates from the three-dimensional theory of elasticity. Approximate theories from the first to the fourth order are presented. Dispersion curves for AT-cut quartz plate are explored and compared with the solution of the three-dimensional equations for an infinite plate.     

Coupled extension and thickness-twist vibrations of lateral field excited AT-cut quartz plates简

In this paper, the coupled extension and thickness- twist vibrations are studied for AT-cut quartz plates under Lateral Field Excitation （LFE） with variations along the x1- direction. Mindlin＇s two-dimensional equations are used for anisotropic crystal plates. Both free and electrically forced vibrations are considered. Important vibration characteristics are obtained, including dispersion relations, frequency spectra, and motional capacitances. It is shown that, to avoid the effects of the couplings between extension and thickness-twist vibrations, a series of discrete values of the length/thickness ratio of the crystal plate need to be excluded. The results are of fundamental significance for the design of LFE resonators and sensors.     

Drive level dependency in quartz resonators

Common piezoelectric resonators such as quartz resonators have a very high Q and ultra stable resonant frequency. However, due to small material nonlinearities in the quartz crystal, the resonator is drive level dependent, that is, the resonator level of activity and its frequency are dependent on the driving, or excitation, voltage. The size of these resonators will be reduced to one fourth of their current sizes in the next few years, but the electrical power which is applied will not be reduced as much. Hence, the applied power to resonator size ratio will be larger, and the drive level dependency may play a role in the resonator designs.We study this phenomenon using the Lagrangian nonlinear stress equations of motion and Piola–Kirchhoff stress tensor of the second kind. Solutions are obtained using COMSOL for the AT-cut, BT-cut, SC-cut and other doubly rotated cut quartz resonators and the results compared well with experimental data. The phenomenon of the drive level dependence is discussed in terms of the voltage drive, electric field, power density and current density. It is found that the drive level dependency is best described in terms of the power density. Experimental results for the AT-, BT- and SC-cut resonators in comparison with our model results are presented. Results for new doubly rotated cuts are presented. The effects of spurious modes, quality factor and air damping on DLD are presented.     

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.     

PIV studies of coherent structures generated at the end of a stack of parallel plates in a standing wave acoustic field

Oscillating flow near the end of a stack of parallel plates placed in a standing wave resonator is investigated using particle image velocimetry (PIV). The Reynolds number, Re _d , based on the plate thickness and the velocity amplitude at the entrance to the stack, is controlled by varying the acoustic excitation (so-called drive ratio) and by using two configurations of the stacks. As the Reynolds number changes, a range of distinct flow patterns is reported for the fluid being ejected from the stack. Symmetrical and asymmetrical vortex shedding phenomena are shown and two distinct modes of generating “vortex streets” are identified.     

EFFECTS OF NONLINEARITY ON TRANSIENT PROCESSES IN AT-CUT QUARTZ THICKNESS-SHEAR RESONATORS

We study the effects of mechanical nonlinearity arising from large thickness-shear deformation on the transient process of an AT-cut quartz plate resonator. Mindlin's two-dimensional plate equation is used, from which a system of first-order nonlinear differential equations governing the evolution of the vibration amplitude is obtained. Numerical solutions by the Runge-Kutta method show that in common operating conditions of quartz resonators the nonlinear effect varies from noticeable to significant. As resonators are to be made smaller and thinner in the future with about the same power requirement, nonlinear effects will become more important and need more understanding and consideration in resonator design.     

Dependence of the natural frequencies and mode shapes of vibrations of an ideal gas on the acoustic resonance parameters

The problem of plane wave propagation through a circular hole is studied in the framework of long-wave approximation. The constructive notion of “apparent mass of holes” (Rayleigh; Fok) is used to construct a mathematical model of gas vibrations in an acoustic resonator and determine and analyze the natural frequencies and mode shapes for the velocity potential depending on the relative geometric parameters of the system. The high-precision calculations of the boundary value problem for the natural frequencies and mode shapes in the parametric approximation to the cross-section are based on a numerical-analytical accelerated convergence method. Two models are analyzed and compared, and the basic qualitative properties of gas vibrations are revealed depending on the basic parameters such as the mode number, relative size of the hole, and the dividing wall location.     

The nonlinear analysis of elastic wave of piezoelectric crystal plate with perturbation method

When investigating or designing acoustic wave sensors, the behavior of piezoelectric devices is supposed to be linear. However, if the sensors are subjected to a strong elastic field, the amplitude of the elastic strain induced in the piezoelectric material is so large that the nonlinearity, which affects the stability and performance of the piezoelectric sensors, can no longer be ignored. In this paper, we perform a theoretical analysis on nonlinear anti-symmetric thickness vibration of thin-film acoustic wave resonators made from quartz. Using Green’s identity, under the usual approximation of neglecting higher time harmonics, a perturbation analysis is performed from which the resonator frequency–amplitude (A–F) relation is obtained. Numerical calculations are made. Furthermore, the validity of the method is examined.     

Measurement of acoustic velocity in the stack of a thermoacoustic refrigerator using particle image velocimetry

Thermoacoustic refrigeration systems generate cooling power from a high-amplitude acoustic standing wave. There has recently been a growing interest in this technology because of its simple and robust architecture and its use of environmentally safe gases. With the prospect of commercialization, it is necessary to enhance the efficiency of thermoacoustic cooling systems and more particularly of some of their components such as the heat exchangers. The characterization of the flow field at the end of the stack plates is a crucial step for the understanding and optimization of heat transfer between the stack and the heat exchangers. In this study, a specific particle image velocimetry measurement is performed inside a thermoacoustic refrigerator. Acoustic velocity is measured using synchronization and phase-averaging. The measurement method is validated inside a void resonator by successfully comparing experimental data with an acoustic plane wave model. Velocity is measured inside the oscillating boundary layers, between the plates of the stack, and compared to a linear model. The flow behind the stack is characterized, and it shows the generation of symmetric pairs of counter-rotating vortices at the end of the stack plates at low acoustic pressure level. As the acoustic pressure level increases, detachment of the vortices and symmetry breaking are observed.     

Hamiltonian systems of propagation of elastic waves and localized vibrations in the strip plate

In this paper, based on Lagrange–Germanian theory of elastic thin plates, applying the method in Hamiltonian state space, the elastic waves and vibrations when the boundary of the two lateral sides of the strip plate are free of traction are investigated, and the process of analysis and solution are proposed. The existence of all kinds of vibration modes and wave propagation modes is also analyzed. By using eigenfunction expansion method, the dispersion relations of waveguide modes in the strip plate are derived, and the comparisons with the dispersion relations directly obtained by the classical theory of thin plates are also presented. At last, the results are analyzed and discussed.     

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.     

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.     

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.     

Flexural Vibrations and Heating of a Circular Bimorph Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

The flexural vibrations and dissipative heating of a circular bimorph piezoceramic plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces. The viscoelastic behavior of piezoceramics is described in terms of temperature-dependent complex moduli. The nonlinear coupled problem of thermoviscoelasticity is solved by step-by-step integration in time, using the discrete-orthogonalization method to solve the mechanics equations and the finite-differences method to solve the heat-conduction equations. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of the nonuniform electroding on the resonant frequency, amplitude, and modes of flexural vibrations and the amplitude- and temperature-frequency characteristics of the plate __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 94–100, September 2005.     

Modal acoustic impedance force on a spherical source near a rigid interface

The modal acoustic radiation load on a spherical surface undergoing angularly periodic axisymmetric harmonic vibrations while immersed in an acoustic halfspace with a rigid (infinite impedance) planar boundary is analyzed in an exact fashion using the classical technique of separation of variables. The formulation utilizes the appropriate wave field expansions, the classical method of images and the appropriate translational addition theorem to simulate the relevant boundary conditions for the given configuration. The associated acoustic field quantities such as the modal impedance matrix and the modal acoustic radiation force acting on the spherical surface are determined. The analytical results are illustrated with a numerical example in which the spherical surface, excited in vibrational modes of various orders, is immersed near an impervious rigid wall. The presented solution could eventually be used to validate those obtained by numerical approximation techniques.     

The effect of laminations on the vibrational modes of circular annular plates

The purpose of this article is to present an experimental study of the effect of laminations on the vibrations of circular annular plates. To obtain a basis for comparison with experimental data, the natural frequencies and mode shapes of a series of solid circular annular plates were calculated using the finite element method. An extensive range of experiments were performed on both a series of solid models and a series of laminated models under a range of normal clamping pressures. Based on the analytical and experimental results, it was found that the vibrational behavior of the laminated plates was dominated by that of the individual plate of which they were composed and that the effects of the laminations on vibrations were mode type dependent. The effects on the transverse vibrational modes were dependent on both the normal clamping pressure and the number of plates. The amplitude of the frequency response function for these modes reduced quickly, and the resonant frequency of such modes shifted higher as the clamping pressure or the number of plates increased. For the in-plane vibrational modes, the amplitude of the frequency response function reduced slightly as the number of plates increased; the resonant frequency of such modes could be considered to be a constant and independent of both the clamping pressure and the number of plates.