Numerical calculation of the temperature difference between the extremities of a thermoacoustic stack plate

When a plate is placed in a thermoacoustic resonator, one extremity of the plate heats up while the other cools down due to the thermoacoustic effect. In the present work the temperature difference between the extremities of the plate is calculated numerically. The computed temperature difference is compared to the one predicted by the linear theory. Some discrepancies are found even at low acoustic Mach numbers. These discrepancies can not be attributed to non-linear effects, rather they exist because of thermal effects. In particular the mean temperature in the fluid and in the plate are not equal. The discrepancies can be eliminated using corrective coefficients. Some deviations between the linear theory predictions and numerical results also occur at higher Mach numbers and are due to non-linearities, and especially temperature non-linearities that are generated in the vicinity of the plate.     

四边简支钢筋混凝土矩形薄板的热屈曲

基于刚性板和小挠度理论,考虑混凝土的材料非线性,推导了热载作用下钢筋混凝土矩形薄板的平衡方程和稳定方程。给出了四边简支钢筋混凝土矩形薄板在横向变温和温度均匀变化时临界屈曲温度变化的封闭解,并对工程中常见的钢筋混凝土矩形薄板在温度均匀变化时的临界屈曲温度变化进行了计算,讨论了板的材料常数、长宽比和相对厚度对临界屈曲温度变化的影响,从而为工程结构中钢筋混凝土矩形薄板的临界屈曲温度变化的计算提供了理论计算依据。     

Investigation of thermoelastic damping in rectangular microplate resonator using modified couple stress theory

Thermoelastic damping is a significant energy lost mechanism at room temperature in micro-scale resonators. Prediction of thermoelastic damping (TED) is crucial in the design of high quality MEMS resonators. In this study the governing equations of motion and the thermal couple equation of a microplate with an arbitrary rectangular shape are derived using the modified version of the couple stress theory. Analytical expressions are presented for calculating the quality factor (QF) of TED in a rectangular microplate considering the plane stress and plane strain conditions. As a case study, a rectangular microplate resonator is considered with material property of gold that has a considerably high value of length-scale parameter in comparison with silicon and the effect of the length-scale parameter on the QF of TED is discussed in detail. The relation between QF and temperature increment for microplates with clamped boundary conditions based on plane stress and plane strain models are studied and results obtained by considering classical and modified couple stress theory (MCST) are compared. The effect of thickness of the plate on the rigidity ratio is studied and the critical thickness which is an important design parameter is obtained using the MCST for three boundary conditions. Variations of TED versus the plate thickness for various boundary conditions according to the classical and the modified couple stress theories are investigated.     

Piezoelectric ceramic rectangular transducers in flexural vibration

Based on the equivalent elastic method and coupled vibration theory, an analytic method is presented to study the flexural vibration of rectangular transducers consisting of piezoelectric ceramic thin plates. By introducing a mechanical coupling coefficient, the flexural vibration of the piezoelectric ceramic rectangular thin plate is reduced to two simple, one-dimensional flexural vibrations of narrow piezoelectric ceramic strips. The resonance frequency equations for the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration are derived under the free and simply supported boundary conditions analytically. The relationship between the resonance frequency and the flexural vibrational order, the geometrical shape, and the dimensions of the piezoelectric ceramic rectangular thin-plate transducer is analyzed. It is demonstrated that the one-dimensional vibrational theory for the flexural vibration of a narrow piezoelectric ceramic strip and the stripe-mode flexural vibrational theory for the piezoelectric ceramic rectangular thin plate can be derived directly from the theory obtained in this paper. Experimental results show that the measured resonance frequencies of the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration under free-boundary conditions are in good agreement with the calculated results. The method presented in this paper can be used in the resonance frequency analysis of vibrating systems in coupled vibration.     

Thickness-shear mode shapes and mass-frequency influence surface of a circular and electroded AT-cut quartz resonator

Finite-element solutions for the fundamental thickness shear mode and the second-anharmonic overtone of a circular, 1.87-MHz AT-cut quartz plate with no electrodes are presented and compared with previously obtained results for a rectangular plate of similar properties. The edge flexural mode in circular plates, a vibration mode not seen in the rectangular plate is also presented. A 5-MHz circular and electroded AT-cut quartz plate is studied. A portion of the frequency spectrum is constructed in the neighborhood of the fundamental thickness-shear mode. A convergence study is also presented for the electroded 5-MHz plate. A new two-dimensional (2-D) technique for visualizing the vibration mode solutions is presented. This method departs substantially from the three-dimensional (3-D) ;wire-frame' plots presented in the previous analysis. The 2-D images can be manipulated to produce nodal line diagrams and can be color coded to illustrate mode shapes and energy trapping phenomenon. A contour plot of the mass-frequency influence surface for the plated 5-MHz resonator is presented. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied to the resonator surface.     

Thickness-shear and thickness-twist vibrations of rectangular quartz crystal plates with nonuniform thickness

An analysis was performed on the vibration of a rectangular contoured quartz plate resonator. The surface of the resonator has different curvatures in the two in-plane directions. The analysis was based on the equation by Tiersten and Smythe, which has variable coefficients. The power series method was used. Thickness-shear and thickness-twist vibration frequencies and modes were obtained. The effects of the curvatures of the resonator surface were examined. Two types of boundary conditions were considered from which the proper size of the resonator was determined. The results are useful in the understanding of these resonators and their design.     

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.     

Generation of induced Smith-Purcell radiation: free-electron laser in open system

We have used the framework of the dispersion equation to study coherent Smith–Purcell (SP) radiation induced by a relativistic magnetized electron beam in the absence of a resonator. As an important example of the application of the obtained results of our previous paper JMO v.57, 2060, (2010) the growth rate of SP FEL in the case with a rectangular grating was calculated. The growth rate of the instability is proportional to the square root of the electron beam current. The calculated results are consistent with the experimental data obtained by Urata et al. [Phys. Rev. Lett. 80, 516 (1998)].     

Lam e-mode miniaturized quartz temperature sensors

Lam e-mode is very useful for realization of a miniaturized quartz crystal resonator because its resonant frequency principally depends only on the contour dimensions. Because the heat capacitance for the miniaturized quartz crystal resonator is small and the frequency response versus temperature is very rapid, the quartz crystal resonator is useful for application in temperature sensors. In addition, because a Lam e-mode quartz crystal resonator has zero temperature coefficients, designated LQ(1) cut and LQ(2) cut, and, particularly, the resonator for LQ(1) cut has a comparatively large value of the second-order temperature coefficient beta, a Lam e-mode quartz crystal resonator can be obtained with the large first-order temperature coefficient or when beta=0. In this paper, when cut angles phi=45 degrees and theta=45 degrees , alpha has a value of 44.6x10(-6)/ degrees C in the calculation and 39.9x10(-6)/ degrees C in the experiments with beta=0; when phi=51.5 degrees and theta=45 degrees , alpha=68.1x10(-6)/ degrees C in the calculation and 62.0x10(-6)/ degrees C in the experiments with a value of beta larger than that of phi=45 degrees and theta=45 degrees . For both cut angles, the calculated frequency change vs. temperature is found to be sufficiently large and slightly larger than the measured one.     

矩形共振腔截面纵横比对其共振频率影响研究

亥姆赫兹共振腔具有良好的低频消声性能,被广泛地应用于低频消声方面.在实际应用中,共振腔的使用往往受到安装空间的限制,有时候只能安装矩形腔.利用定体积定长度时的矩形共振腔模型,从理论上探讨了腔体纵横比对共振腔共振频率的影响.同时设计了发动机进气消声器性能测试专用实验台,从实验角度研究了腔体截面纵横比与共振频率之间的关系.理论与实验初步结果表明,在算例条件下截面纵横比对共振频率影响不大,在应用时可以不考虑纵横比问题.所讨论内容对其它条件下的研究有借鉴作用.     

波浪与弹性板作用的数值模拟

该文采用高阶有限元和边界元联合的方法求解波浪与弹性板的相互作用。其中流场采用边界元法求解,结构弹性响应方程采用基于Mindlin板理论的有限元方法求解,通过模态叠加技术实现了弹性板变形与流场相互作用的解耦。通过对一矩形板的计算,验证了该文方法与他人试验结果和数值模拟结果都吻合良好。利用这一模型进一步分析了波浪与弹性圆形板的作用问题,并对圆形板运动响应的收敛性进行了分析。     

Doubly rotated contoured quartz resonators

Doubly rotated contoured quartz resonators are used in the design of temperature-compensated stable clocks and dual-mode sensors for simultaneous measurements of pressure and temperature. The design of these devices is facilitated by models that can predict frequency spectra associated with the three thickness modes and temperature and stress-induced frequency changes as a function of crystalline orientation. The Stevens-Tiersten technique for the analysis of the C-mode of a doubly rotated contoured quartz resonator is extended to include the other two thickness modes. Computational results for harmonic and anharmonic overtones of all three thickness modes of such resonators help in optimizing the radius of curvature of the contour and electrode shape for suppression of unwanted modes and prevention of activity dips. The temperature and stress-induced changes in thickness-mode resonator frequencies are calculated from a perturbation technique for small dynamic fields superposed on a static bias. The static bias refers to either a temperature or stress-induced static deformation of the resonator plate. Phenomenological models are also used for calculating the temperature and stress-induced changes in resonant frequencies as a function of crystalline orientation. Results for the SBTC-cut quartz plate with a spherical convex contour of 260 mm indicate that normal trapping occurs for the third (n=3) and fifth (n=5) harmonic of the A-mode, the fundamental (n=1) and third (n=3) harmonic of the B-mode, and the fundamental (n=1) and fifth (n=5) harmonic of the C-mode     

Temperature stresses in a heat-sensitive ceramic plate

The quasistatic thermoelastic state of a rectangular ceramic plate freely supported around the perimeter is discussed. The problem is reduced to the solution of a system of nonlinear inhomogeneous integrodifferential equations consisting of a nonlinear differential heat-conduction equation with allowance for the temperature dependence of the thermophysical characteristics of the material and an integrobiharmonic equation characterizing the plate's deflection. The expression for the components of the temperature stresses caused by a nonlinear temperature field are derived in the form of ternary series. The thermoelastic state is calculated for a rectangular backing-type plate produced from the commercial ceramic VK 94-1.Translated from Problemy Pochnosti, No. 10, pp. 88–93, October, 1990.     

Comparison of Talbot and 1-to- N-way phase-locked array resonators

Phased-array resonators provide an important basis for achieving high output powers from arrays of low-power elements. We have recently proposed a novel form of 1-to-N-way phased-array resonator based on the beam splitting and regeneration characteristics of rectangular sectioned multimode waveguides. We compare its performance with that of the widely used, yet problematic, Talbot resonator. Our design is found to have significant advantages over the Talbot resonator in terms of improved modal stability, unique photon-mixing characteristics, and near- and far-field outputs of quasi-Gaussian form.     

High performance distributed Bragg reflector microwave resonator

A new resonator device structure is described that achieves Q-factors well above those currently realizable for conventional room temperature microwave structures. The new structure consists of a microwave cavity, for which the enclosure walls consist of distributed Bragg reflectors (DBR) made of low-loss sapphire. For this configuration, most of the resonant field resides in empty space, with small field strengths in the thin layers of sapphire which comprise the DBR structure. The physical structure takes the form of interpenetrating concentric rings and plates of low-loss sapphire contained in a cylindrical metal enclosure. The theoretical analysis of the DBR resonant structure allows the positions and dimensions of the component rings and plates to be precisely determined for a specified resonant frequency. The resonator Q can be accurately calculated, and plots of the resonant fields clearly show the physical mechanism leading to the observed efficiency of this resonator structure. Experimental results are given for resonators designed at 9.0 and 13.2 GHz. The measured unloaded Q's at room temperature are over 650000 and 450000, respectively.     

Theory of 1- N-way phase-locked resonators

A 1-N-way resonator based on beam splitting and beam combining effects in rectangular cross-sectional multimode waveguides was recently proposed. Such a resonator structure offers a valuable way in which N low-power laser elements may be combined in a coherent fashion. We examine the case of passive 1-N-way resonators. We develop a theory of these 1-N-way structures to show that there is only one possible mode of these resonators. The theory is used to give a scaling law for the design tolerances of the beam splitting and beam combining region of the resonator.     

混合边界矩形板的自由振动分析

应用一块板的一般解析解来求解混合边界矩形板的自由振动问题。一块板的一般解析解适用于求解任意边界条件矩形板的振动问题。混合边界矩形板的求解过程是将整块板分解为若干块板,每块板仅有均匀的边界,每两块相连的板边则具有连续性条件。利用全部边界条件和连续性条件即可求解各阶固有频率和振型。对几种具有简支边、平夹边或自由边混合边界的板做了计算,并与其它文献结果进行了对比。     

Experimental study on the characteristic of the NS-GT cut quartz crystal resonator oscillating in the sub-resonant frequency

We previously reported that the dynamic photo-elastic method was a very effective measuring technique for the stress distribution of vibrating quartz crystal resonators. The existence of a twisted asymmetrical vibration mode has been verified experimentally when the NS-GT cut quartz crystal resonator was vibrating in the main resonant frequency (MRF). A MRF and a sub-resonant frequency (SRF) of the NS-GT cut quartz resonator were defined as follows. If a mechanical standing wave was in the x' or y' direction of the resonator, the former was MRF vibration and the latter was SRF vibration, respectively. In this paper, stress distributions of two samples of the NS-GT cut quartz crystal resonator, one of which had a thickness of 80 mum and the other 150 mum, were measured by the dynamic photo-elastic method when the resonators were vibrating in each SRF. Thereafter, vibration modes of those resonators were estimated by the experimental data of stress distributions. We find that the vibration mode of the 80-mum resonator had a simple mechanical standing wave on the y' direction and the vibration mode of the 150-mum resonator was combined with a shearing mode in the SRF vibration. From the experiment, we decided that vibration modes of the NS-GT cut quartz crystal resonator were composed of the longitudinal stress T(3)' belonging to the z' direction of the plate and of the shearing stress T(5)' when the plate thickness was thickened and the resonator was oscillating in the SRF.     

Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators

We study straight-crested waves and vibration modes with variations along the x(3) direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.