Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections

 In this paper shear correction factors for arbitrary shaped beam cross-sections are calculated. Based on the equations of linear elasticity and further assumptions for the stress field the boundary value problem and a variational formulation are developed. The shear stresses are obtained from derivatives of the warping function. The developed element formulation can easily be implemented in a standard finite element program. Continuity conditions which occur for multiple connected domains are automatically fulfilled.     

Eddy-current power loss in toroidal cores with rectangular crosssection

In this paper, we investigate eddy-current power loss in toroidal cores with a rectangular cross section. An analytical method is used in which the field variables are expressed in terms of products of Bessel functions and trigonometric functions. A closed-form expression is derived for the power loss. The influence of various parameters such as skin depth and core dimensions over a wide range is discussed. Simplified expressions for the power loss at special cases such as very small and very large skin depths are provided. A comparison with measured values is presented     

Force-frequency coefficient of symmetrical incomplete circular quartz crystal resonator

The changes in the resonance frequencies of the thickness-shear vibration of symmetrical incomplete circular AT-cut quartz crystal resonators (QXRs), which are used as sensing elements in digital force sensors or pressure transducers, by the application of diametrical forces are discussed by considering the piezoelectric effect and the anistropic characteristics of crystal plates. Two-dimensional motion equations for predicting the frequency changes are derived from three-dimensional piezoelectricity equations, and equations deduced by Janiaud for solving the stress distribution in crystal plates are used to calculate the force-frequency coefficients of incomplete circular resonators. The results show that the piezoelectric effect of crystal plates decreases the force-frequency coefficient of the azimuth angle 0 degrees by 7% as compared with no piezoelectricity assumption. The incomplete circular shape can be applied to a larger load than the circular disc because the flat regions of the incomplete shape provide a distributed load application without the problems of stress concentration of a point force inherent to a circular disc. The incomplete shape gives an increased force sensitivity when compared with the complete disc near azimuth angle 0 degrees. The theoretical curves agree well with Ratajski's experimental results.     

Fast three-step method for shear moduli calculation from quartz crystal resonator measurements

Quartz crystal resonator measurements can be used for polymer material characterization. The non-gravimetric regime of these resonators is exploited: the electrical response of polymer-coated quartz resonators depends on the polymer shear modulus. Previously reported methods employ an electrical admittance analysis together with difficult and time-consuming data fitting procedures to calculate the film shear modulus. This contribution presents a fast and accurate three-step method for the calculation of complex shear moduli of polymer films from quartz crystal resonator measurements. In the first step, the acoustic load impedance is calculated from the electrical admittance of the quartz crystal. The key point of this method is the application of a family of approximations for the calculation of the shear modulus from the acoustic load impedance in the second step. In the third step, the best approximation is improved further in an iterative procedure.     

Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model

The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoulli's classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)theta cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation and this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared.     

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.     

Temperature coefficient of sensitivity dependence of themeasurement error of quartz crystal resonators

This paper discusses the measurement error of quartz crystal resonators (QXRs) vs. the temperature coefficient of force sensitivity. Based on the analysis of the difference between the temperature coefficient of frequency and the temperature coefficient of force sensitivity, the measurement error formula is deduced. According to the error formula, the physical significance of the temperature coefficient of force sensitivity is explained, and the method of selecting a scale temperature is given. The measurement error formula has been verified recently by experimental results. To expand the selectable range of cut modes and to compare the performances of QXRs, the selecting principle of the best group of the angles of cut of quartz crystal plate, &thetas;, ϑ, and azimuth angle of applied force, ψ is given simultaneously. According to the measurement error formula and the selecting principle of the best group of the angles, the best azimuth angle of AT-cut used as force sensitive resonators is found through the analysis of the measurement error. This paper provides a theoretical basis on which the measurement accuracy of QXR force sensors can be improved     

Design and fabrication of a mode rectangular X-cut quartz resonator with zero temperature coefficient

Rectangular X-cut quartz crystal resonators with cut angles /spl theta/ > 5.0/spl deg/ and aspect ratios R/sub zy/ (= width 2z/sub 0//length 2y/sub 0/) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle /spl theta/ and R/sub zy/ diagram. We proved that a fabricated resonator with R/sub zy/ = 0.400 and /spl theta/ = 16.0/spl deg/, whose design parameter is located in the area of the contour, had a zero temperature coefficient.     

Design and fabrication of a length-extensional mode rectangular X-cut quartz resonator with zero temperature coefficient.

Rectangular X-cut quartz crystal resonators with cut angles theta > 5.0 degrees and aspect ratios Rzy (= width 2z0/length 2y0) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle theta and Rzy diagram. We proved that a fabricated resonator with Rzy = 0.400 and theta = 16.0 degrees, whose design parameter is located in the area of the contour, had a zero temperature coefficient.     

Analysis of capacitance ratio of a rectangular X-cut length-extensional mode quartz crystal resonator by stresses

The electrical properties of an X-cut, length-extensional mode quartz crystal resonator of a cut angle theta around the X-axis were calculated by a variational method using stresses as trial functions. Analytical expressions of stresses were estimated by a linear regression on a cut angle best-fit to the results of finite-element method. The calculated dependence of the capacitance ratio on the cut angle was consistent with the measured results.     

A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.     

Free vibration of axially loaded rectangular composite beams using refined shear deformation theory

Free vibration of axially loaded rectangular composite beams with arbitrary lay-ups using refined shear deformation theory is presented. It accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply axial-flexural coupled vibration. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for rectangular composite beams to investigate effects of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes.     

The determination of the optimal length of crystal blanks in quartz crystal resonators

It is well understood that the strong coupling of thickness-shear and flexural vibrations in piezoelectric crystal plates only occurs at specific length at which the vibration mode conversion, like the flexural mode gradually converting to thickness-shear mode while the thickness-shear mode converting to higher-order flexural mode, happens. It is important to avoid the strong coupling of modes in a crystal resonator that uses thickness-shear vibrations to enhance the energy trapping. To achieve such a design goal, the length of a crystal blank should be carefully chosen such that the coupling is at its weakest, which usually is in the middle of two strong coupling points. Through a closer examination of the frequency spectra, or the frequency-length relationship in this study, we can see that the strong coupling points appear periodically. This implies that we can find exact locations with the plate theory that predicts the resonance frequency. Based on this observation, we first use the first-order Mindlin plate theory with the precise thickness-shear frequency, which is normalized to one, to find corresponding wavenumbers. Then the length as a variable is solved from the coupled frequency equation for exact coupling points in a crystal plate of AT-cut quartz. The optimal length of a crystal blank in the simplest resonator model is calculated for the coupled thickness-shear, flexural, and extensional vibrations. The solutions and the method will be important in the determination of optimal length of a crystal blank in the resonator design process.     

Proximity effect in quartz crystal microbalance

When an object approaches a vibrating quartz crystal microbalance (QCM) the resonant frequency changes. This "proximity effect" was seen at the distance of 10 mm in air and became more pronounced as the distance decreased. This effect depends on the quality factor (Q-factor) value of a QCM, conductivity of the object, and electrical connection of the object to QCM electrodes. A special setup was constructed to test the impact of the proximity effect on a QCM. Damping fluid was placed on one side of QCM, to change the Q-factor. A conducting metal disk was brought close to the other side of the QCM exposed to air. By varying the distance between the QCM and an object (metal disk), a shift in frequency was observed. This proximity effect was largest (>200 Hz for 10 MHz QCM) when the Q-factor was low and a conducting metal disk (e.g., Cu) was electrically shorted to the proximal (nearest) QCM electrode. The finite element modeling showed that the proximity effect was likely due to interaction of the object with the fringing electromagnetic field of the QCM. A simple modified Butterworth Van-Dyke model was used to describe this effect. It must be recognized that this effect may lead to large experimental artifacts in a variety of analytical QCM applications where the Q-factor changes. Therefore, in order to avoid artifacts, QCM and similar mass acoustic devices should not operate in the low Q-factor (<1000) regime.     

The refined theory of magnetoelastic rectangular beams

Summary. The problem of deducing a one-dimensional theory from a three-dimensional theory for a soft ferromagnetic elastic isotropic body is investigated. Based on the linear magnetoelasticity, the refined theory of magnetoelastic beams is presented by using the general solution for the soft ferromagnetic elastic solids and the Lure method. Based on the refined theory of magnetoelastic beams, the exact equations and solutions for the homogeneous beams are derived and the equations can be decomposed into three governing differential equations: the fourth-order equation, the transcendental equation and the magnetic equation. Moreover, the approximate equations and solutions for the beam under transverse loadings and magnetic field perturbations are derived directly from the refined beam theory. By omitting higher order terms and coupling effects, the refined beam theory can be degenerated into other well-known elastic and magnetoelastic theoretical models.     

The modulational method of quartz crystal oscillator frequency stabilization

A new trend in self-contained (without the use of quantum discriminators) frequency stabilization of an oven controlled crystal oscillator (OCXO) and frequency standard is proposed and discussed. The method developing the trend is called a modulational method and is based on the use of the reference properties of crystal resonator natural bulk vibrations (double-frequency and multi-frequency oscillators are not used in this case). The concept is based on dynamic modulation characteristics of an oscillator, and basic relationships are found for their calculation. The construction principles of the frequency control systems are formulated substantiating mathematically the essence of the method. Basic ratios of modulating signals are determined, the solution of which shows only a slight influence of the modulation signal on the Allan variance and spectral density of an OCXO. The results of the method's practical use are considered. Their subject is the OCXO with the oven system adapted to the ambient temperature and crystal frequency standard with aging rate compensation.     

Isolated electrodeless high-frequency quartz crystal microbalance for immunosensors

This paper presents a contactless technique to measure shear bulk wave resonance frequencies of an isolated quartz crystal in a flow cell. The line antenna placed outside the cell generates and detects the resonance frequencies in a wireless-electrodeless manner. It is revealed that this mechanism relies on the quasistatic electric field. A 0.3-mm-thick AT-cut quartz was used, and its overtone resonance frequencies up to 80 MHz were measured in liquids. Exact vibrational analysis was carried out for a triple-layered resonator system consisting of the adsorbed material layer, the electrode film, and the quartz plate. It predicts higher frequency sensitivity to the adsorbed material at higher modes when the electrode layer is removed. The 13th overtone (72-MHz resonance frequency) was used to detect human immunoglobulin G with concentrations between 0.1 and 20 microg/mL captured by protein A immobilized on one side of the crystal. The real-time measurement of the frequency response yielded the equilibrium constant KA=5.21 x 10(7) M(-1).