Vibration and stability of an Euler–Bernoulli beam with up to three-step changes in cross-section and in axial force

The paper treats the vibration of beams with up to three-step changes in cross-section and in which the axial force in each portion is constant but different. The system parameters are the step positions, the flexural rigidity, the mass per unit length and the axial force in the beam portions—all of which were normalized. The frequency equation for 16 combinations of classical boundary conditions are expressed as fourth-order determinants equated to zero. The first three frequency parameters are tabulated for sets of system parameters (arbitrarily chosen and which includes a stepped beams under tensile or compressive axial end force). Critical compressive end force which causes a stepped beam to buckle are tabulated. Buckling under a system of axial forces, one of which is critical is discussed and several critical combinations of the system parameters are tabulated. Beams of constant depth and step change in breadth, of constant breadth and step change in depth and shafts with step change in diameter are considered. It is shown that stepped shafts are inferior machine elements if dynamic properties are the prime consideration.     

Transverse vibrations of an Euler–Bernoulli uniform beam carrying several particles

The Euler–Bernoulli uniform beam considered in this paper consist of n portions and carry (n+1) particles, two of which are at the beam ends. For the classical beam eigen-value technique developed here, n co-ordinate systems are chosen with origins at the particle locations. The mode shape of the jth portion of the beam is expressed in the form Y_j(X_j)=AU_j(X_j)+BV_j(X_j) in which U_j(X_j) and V_j(X_j) are ‘modified’ mode shape functions applicable to that portion but the constants A and B are common to all the portions. From the boundary conditions at the right end, the frequency equation was expressed in closed form as a second-order determinant equated to zero. Schemes are presented to compute the four elements of the determinant (from a recurrence relationship) and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes. The first three natural frequency parameters of 16 combinations of the classical boundary conditions are tabulated for beams with three and up to nine portions for selected particle location and mass parameters. Frequency parameters of beams with one and up to 500 equi-spaced, equi-mass systems are also tabulated. The approaches in previous publications include those based on various approximate methods like finite element, Rayleigh–Ritz, Galerkin, transfer matrix, etc. The results in the present paper may be used to judge the accuracy of values obtained by approximate methods.     

Transverse vibrations of an Euler–Bernoulli uniform beam carrying two particles in-span

Vibrations of beams carrying different combinations of particles, heavy bodies and spring-mass systems which are located on or off resilient supports have been tackled by several researchers. Most of the approaches were based mainly on various approximate methods. In this paper an analytical solution based on the classical beam eigenvalue technique is presented for the vibrations of a beam carrying two particles. For purpose of analysis, the beam was divided into a portion from one end to the first particle, a portion between particles and a portion from the second particle to the other end. The frequency equation is expressed in closed form as a 2nd order determinant equated to zero. Schemes are presented to compute the 4 elements of the determinant and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes. The first three natural frequency parameters are tabulated for 16 combinations of the classical boundary conditions and several combinations of the location and mass of the particles. The beam mode shape is the juxtaposition of the mode shapes of the three portions of the beam. Some examples of normalised beam mode shapes and location/s of node/s are also presented. The results may be used to judge the accuracy of values obtained by approximate methods.     

The dynamic performance of a new ultra-fast response Coriolis flow meter

The paper identifies the requirements for both a Coriolis flow-tube and the associated flow-transmitter, to achieve an ultra-fast dynamic response. A ‘new’ meter was assembled using a commercially available straight flow-tube, selected for its high drive (fundamental resonant) frequency of about 750 Hz (water-filled) and this was interfaced with a newly extended version of Oxford’s digital transmitter technology. This new transmitter gives measurement updates that are calculated every half drive-cycle, at 1.5 kHz, and are output via a high precision frequency pulse signal. Dynamic response tests in the laboratory and during a field trial have shown the ‘new’ meter to have a response time at least an order of magnitude faster than was reported previously [Clark C, Cheesewright R. Experimental determination of the dynamic response of Coriolis mass flow meters. Flow Measurement and Instrumentation 2006;17:39–47] from tests on the fastest response commercially available Coriolis meters. The ‘new’ meter shows a typical delay of 3.8 ms between a change in fluid flow rate and the corresponding change in the frequency output. Suggested further developments of the new transmitter technology indicate a route that could be followed to produce transmitters suitable for the newly emerging micro-machined Coriolis flow-tubes, which have fundamental resonant frequencies in the range 10–30 kHz.     

Development of a Wittrick–Williams algorithm for the spectral element model of elastic–piezoelectric two-layer active beams

In this paper, a Wittrick–Williams algorithm is developed for the elastic–piezoelectric two-layer active beams. The exact dynamic stiffness matrix (or spectral element matrix) is used for the development. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the sign change of the determinant of spectral element matrix via infinity instead of via zero. The uniform and partially patched active beams are considered as the illustrative examples to confirm the present algorithm.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

固定边界阶梯圆盘的设计及其振动特性计算

利用薄板的弯曲振动理论,推出了固定边界条件下带有一个阶梯的阶梯圆盘的频率方程,提出了求解频率方程的数值计算方法,编制了计算程序,由此可计算出阶梯圆盘的半径和节线位置,设计出完整的阶梯圆盘。并用有限元仿真软件对阶梯圆盘的振型和位移分布进行了计算,两者计算结果基本吻合,为这种条件下阶梯圆盘的设计提供了方法。     

阶梯轴弯曲固有振动分析及应用

给出阶梯轴弯曲固有振协精确的分析方法,该法亦可用于工程中阶梯梁弯曲固有振动分析。我们用该方法计算了硅压阻式加速度传感顺五阶梯硅梁固频率和振型。     

求解任意梁的普遍化方法

介绍了一种求解任意弹性梁的新方法。该方法利用奇异函数与拉普拉斯变换相结合的方法导出弹性梁弯曲变形的普遍表达式,并利用边界条件和约束处的变形协调条件确定约束反力和变形常数(左端面的挠度和转角),对由固定和活动铰链支座、径向和角度弹性支承以及固定端等支承形式任意组合而成的,具有任意支承沉降的,承受任意载荷(集中力、集中力偶和均布力)的,具有任意阶梯形状的静定或超静定弹性梁具有普遍的适用性。该方法可以方便、准确地确定任意梁在支座处的约束反力以及任一截面的挠度和转角等参数,可用于复杂梁的计算机分析、优化设计和计算机辅助设计。     

Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four- digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3-D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler–Bernoulli beam theory.     

Sensing of small signals on CRT-displayed medical radiographs after A/D conversion

The paper describes an experimental evaluation, by ROC analysis, of displayed CRT images after digitisation of analogue medical radiographs. Two types of images were used in these studies: (1) images of 2 mm diameter beads, and (2) images of 2 mm diameter cavities. The psychophysical experiment was carried out by obtaining ROC curves based on signal detection theory. Detectability was evaluated as the ‘area under the ROC curve’. It was found that the detectabilities of images on the CRT were significantly lower than those of the original images. As for the images of bead signals, the ‘areas under the ROC curve’ were 0.91 and 0.70 for the original images and the digital images respectively. However, the ‘area under the ROC curve’ could be improved to 0.84 or 0.80 by a multi-cycled display going through 2 or 4 cycles of contrast enhancement in the studies of bead images. The ‘area under the ROC curve’ of the cavity images was better than that of the beads.     

Calibration of flowmeters in superheated and wet steam

The vapour mass fraction (dryness fraction) of wet-steam flows such as those encountered in the steam main of a power or process plant will rarely be less than 90%. In flowmetering of high-quality saturated steam, the moisture content in the steam is allowed for by correcting the mass flow indicated by the steam flowmeter. Although there are a number of different ways of correcting the mass flow for water content, there is still doubt as to the methods used. In the work reported in this paper, three types of flowmeter were calibrated: first in superheated steam and then in an equilibrium wet-steam mixture to determine their wet-steam correction factors. The correction factors were found to be dependent on the types of flowmeter. For consistency with recommended practice, the method recommended by the ‘Shell Flowmetering Engineering Handbook’ (Ed. G. W. A. Danen, McGraw-Hill, 1985) can be used universally with reasonable accuracy provided a high-efficiency separator is placed upstream of the steam meter to increase the dryness fraction to above 95%.     

Vibrations of rotating,pretwisted and tapered blades

In this paper, turbomachinery blades are idealised as rotating pretwisted and tapered cantilever beams with rectangular cross-section. Expressions for potential and kinetic energy of such beams are derived and the Lagrangian function obtained thereupon is minimised according to the Ritz process. The resulting linear, homogenous algebraic equations are expressed in matrix form and it is solved for the first three natural frequencies. Pretwist, rotation and decreasing width taper are shown to have considerable influence on the flexural frequencies of the beams considered.     

Modified inverse iteration method using the side condition and the step length

An efficient numerical method which can calculate the natural frequencies and mode shapes for very large structural systems is presented. This method applies the accelerated Newton-Raphson technique to eigenproblems. If eigenvalues are not multiple, this method can calculate the natural frequencies and mode shapes without a numerical instability which may be often encountered in the inverse iteration method with shift. The efficiency of this method is verified by comparing convergence and solution time for numerical examples with those of the well-known methods such as, the subspace iteration method and the determinant search method.     

利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。     

Vibration of stepped thickness plates

The vibration characteristics of stepped thickness plates of rectangular geometry with simply supported edges are investigated through an exact analysis. To illustrate the effect of sudden variation of thickness in the step form on the dynamic characteristics of plates, numerical calculations have been made for various combinations of thickness step, step-length ratio and side ratio. Natural frequencies for the first eight to nine modes are presented in the form of tables. Nodal lines are also plotted for a few selected cases. Several interesting features such as that of the thickness step splitting the degenerate frequencies of uniform rectangular plates into distinct ones, and the “frequency crossing” of some of the modes are observed.     

Adaptive vibration control of micro-cantilever beam with piezoelectric actuator in MEMS

This paper proposes a dynamical model and the governing equations of motion of the micro-cantilever beams based MEMS with piezoelectric actuator (PZT). The Rayleigh–Ritz method is used to reduce the order of the system and the state equations are presented in modal space. The first ten mode frequencies and mode shapes of the micro-cantilever beam with and without PZT are studied. The effects of PZT on the modal frequencies and shapes of the beam system can be ignored for the reason that the beam holds larger nature frequencies and Q values in micro-scale. A rational linearizing feedback controller with a high gain observer is designed to eliminate the unwanted deflection of the micro-cantilever beam system. The open-loop step response and the effects of situated places of PZT on the frequency responses of the system are discussed. Various frequency responses of the beam system, subject to different applied control voltages and feedback gains, are illustrated. The four resonances are well controlled, while the anti-resonance has little change. Computer simulations are provided to demonstrate the performance of the designed control scheme.     

Development in laser interferometry for position sensing

Interferometric developments which improve the performance of the laser interferometer as a position sensing device are described. The double pass attachment is an optical accessory which doubles the resolution of the Hewlett-Packard linear and plane mirror interferometers. Unlike previous attempts to extend resolution optically, this device does not fold just one of the two interfering beams and hence does not introduce error due to its own motion. Simple addition of a quarter wave plate to the attachment gives a differential version of the plane mirror interferometer. Various configurations of this differential interferometer, and their application to machine tools and to integrated circuit lithographic and inspection equipment, are discussed. A new ‘wavelength tracking’ device, based on differential interferometry, directly and precisely monitors changes in laser wavelength inside a highly stable mechanical cavity. Easily incorporated as an additional interferometer axis, the device improves compensation for changes in laser wavelength.     

On the dynamic response of beams with multiple geometric or material discontinuities

An analytic framework is developed for determining closed form expressions for the natural frequencies, mode shapes, and frequency response function for Euler–Bernoulli beams with any number of geometric or material discontinuities. The procedure uses a convenient matrix formulation to generalize the single discontinuity beam problem to beams with multiple step changes. Specifically, the multiple discontinuity beam problem is solved by analyzing the total structure as a series of distinct Euler–Bernoulli elements with continuity and compatibility enforced at separation locations. The method yields each respective section's eigenmode which may then be superpositioned to give the entire beam's mode shape and derivation of the frequency response function follows. Although the Euler–Bernoulli beam problem is demonstrated, any one-dimensional continuous structure is amenable to the prescribed analysis. Theoretical predictions are experimentally validated as well.