Dynamics behavior of rotating bladed discs: A finite element formulation for the study of second and higher order harmonics

Annular finite elements for the computation of second and higher order harmonics modes of bladed rotating discs are developed. The elements take into account gyroscopic effect and stiffening due to centrifugal and thermal stresses (the latter not present in arrays of blades). The displacement field is expressed by a truncated Fourier series along the angle and by polynomial shape functions in the radial direction. This paper is the generalization of a previous study limited to zero- and first-order harmonics and deals only with second and higher order harmonics modes that are uncoupled from the modes involving the behavior of the rotor as a whole. Several cases have been studied to verify the accuracy of the disc and array of blades elements.     

A finite element analysis for the vibration modes of a bladed disc

The coupled vibration modes of a rotating blade-disc system are calculated by a finite element method. It is assumed that a large number of identical blades are present, such that the resulting blade loadings on the disc can be considered continuously distributed around the rim of the disc. The disc may have arbitrary profile, and the blades may be tapered and twisted, thus closely representing practical axial flow turbomachine configurations. The effects of rotation, thermal stress, and transverse shear and rotatory inertia in discs of moderately thick profile are readily incorporated into the finite element model. Calculated values of frequencies are compared with experimental data obtained on non-rotating models, and the convergence of the solution is examined by comparison with exact solutions, which can be obtained for configurations of simple geometry. Excellent agreement with experimental data is obtained when using remarkably few elements in the mathematical model, and convergence of the solution is extremely rapid.     

Vibration frequency of a curved blade with weighted edge

The natural vibration frequencies and mode shapes of a curved cylindrical blade with a weighted edge are investigated. A finite element method is used, in which curved cylindrical shell finite elements are utilized to model the blade. The weighted edge is modelled as a beam with its stiffness and mass added into the stiffness and mass of the blade. Vibration frequencies and mode shapes for blades with different boundary conditions and with different radii of curvature are obtained. Finite element results are compared with experimental results.     

Torsional vibrations of non-uniform rotating blades with attachment flexibility

The torsional vibrations of non-uniform pretwisted rotating blades are studied by using finite element methods based on both the Rayleigh-Ritz and Galerkin formulations. The apparent differences between the matrices obtained from these formulations are explained and, as obtained by using three different orders of elements, results are presented for blades with flexibly attached roots and for a non-uniform blade representative of a bearingless rotor. A parametric study is carried out to resolve a controversy regarding the relative importance of certain terms in the equations of motion of pretwisted rotating blades. In Appendix I, an exact solution is presented for the torsional vibrations of flexibly attached rotating blades with piecewise constant inertia and elastic properties, which serves as a benchmark solution for the finite element results.     

Vibration and stress analysis of thin rotating discs using annular finite elements

The finite element method is applied to the stress and vibration analysis of thin rotating discs. By making use of the axisymmetric properties, annular finite elements are derived which describe the bending and stretching of such discs and are characterized by having only four degrees of freedom. These elements incorporate the desired number of diametral nodes in their dynamic deflection functions, and allow for any specified thickness variation in the radial direction. The resulting mathematical model is thus particularly efficient for numerical computation. The accuracy and convergence of the method is demonstrated by numerical comparison with both exact and experimental data.     

Vibration frequency of a curved beam with tip mass

In flexible blade auto cooling fans, the first vibration frequency is of fundamental importance. These fan blades are usually curved and have a tip mass in the form of a strip along one edge. For the first frequence, the blade can be modelled as a curved beam with a tip mass. This paper reports on an investigation of the vibration frequency of a curved beam with a tip mass, in which both theoretical finite element and experimental methods were used. In the finite element methods, both the normal and tangential displacements are approximated by cubic polynomials to ensure that rigid body displacements are closely represented. The effect of the tip mass is incorporated into the mass matrix. The results show that the curvature has a slight effect on the first mode natural frequencies but has great influence on the higher frequencies, and that the coupling effect between the tip mass and the curvature is insignificant.     

Quasi-3D Navier–Stokes Model for a Rotating Airfoil

A quasi-3D model of the unsteady Navier–Stokes equations in a rotating frame of reference has been developed. The equations governing the flow past a rotating blade are approximated using an order of magnitude analysis on the spanwise derivatives. The model takes into account rotational effects and spanwise outflow at computing expenses in the order of what is typical for similar 2D calculations. Results are presented for both laminar and turbulent flows past blades in pure rotation. In the turbulent case the influence of small-scale turbulence is modelled by the one-equation Baldwin–Barth turbulence model. The computations demonstrate that the main influence of rotation is to increase the maximum lift.     

Noise Sources of Aerodynamic Origin in Air Blowers

The paper presents the results of theoretical and experimental studies of the occurrence and locations of aerodynamic noise sources in air blowers related to air flow around stationary and moving elements inside a machine body. These studies were based on basic research by Lighthill and Curle and used a developed method for measuring pressure pulsations on rotating blades and stationary elements of a machine body. The most significant sources of discrete and broadband components of aerodynamic noise were revealed. The role of blades in an impeller in the emission of discrete noise components was studied. It was established that broadband peaks in the emitted noise are associated with acoustic resonances of the internal volume of the air blower. It was shown that the turbulence and velocity of the incoming flow influence the intensity of aerodynamic sources inside the body. Our studies spurred both deeper research into the nature of aerodynamic noise sources that form in air blowers and recommendations for reducing the noise produced by these sources.     

Free vibration of rotating non-uniform discs: Spline interpolation technique calculations

Stress distributions and flexural vibration of rotating annular discs with radially varying thickness are calculated by means of a spline interpolation technique. For this purpose, the disc is divided into many ring-shaped elements and the radial displacement is expressed as a cubic spline function, which satisfies the equation of equilibrium of force at all the knots and also satisfies boundary conditions at both edges. Centrifugal stress distributions are calculated from the radial displacement. The transverse deflection of the disc is expressed as a quintic spline function. The frequency equation is derived from the conditions that this function satisfies the differential equation governing the flexural vibration of the disc at the knots and also satisfies the edge conditions. The method is applied to free-clamped rotating discs with linearly, parabolically and exponentially varying thickness, the natural frequencies and the mode shapes are calculated numerically, and the effects of rotating velocity and variable thickness are discussed.     

Finite element dynamic analysis of practical discs

A semi-analytical annular finite element is developed for the dynamic analysis of non-rotating, rotating or pre-stressed discs having varying thickness in the radial direction. The element is based on the Mindlin thick plate theory. It has 2 nodes, 12 degrees of freedom, parabolic thickness, and is capable of representing all the geometric and natural boundary conditions of thick plates. The element is applied to the dynamic analysis of non-rotating and rotating uniform discs, and to practical turbine discs. The predicted natural frequencies of the discs are compared with analytical, experimental and other finite element solutions.     

The acoustic simulation and analysis of complicated reciprocating compressor piping systems,I: Analysis technique and parameter matrices of acoustic elements

This paper describes the mathematical formulation, equations, and procedures employed in the development of a comprehensive digital computer program for acoustic simulation and analysis of large and complicated piping systems. The analysis technique used is the transfer matrix method in which the piping system, with or without multiple inputs and outputs, is represented by a combination of discrete acoustic elements interconnected to one another at two stations such that the acoustic pressure and volume velocity at one station are uniquely related to those at the other by a two-by-two parameter matrix. Parameter matrices of 19 acoustic elements are included in this paper. By making use of these parameter matrices and the analysis technique, any complicated practical reciprocating compressor piping system can be modelled or analyzed.     

A coincidence criterion for effective sound radiation from a resonant free running circular saw blade

A study has been made of the pure tone noise emitted from spinning circular discs vibrating in modes with nodal diameters only.The main application is idling circular saw blades at high speeds. It is shown that, above a certain rotational speed, which again is a function of the material parameters and the geometry, a disc will have a high radiation efficiency and therefore be an effective radiator of sound.     

Dynamic analysis of practical blades with shear center effect

Shear center effects on the natural frequencies and mode shapes of rotating and non-rotating practical blades are considered. An 18 degrees of freedom thick beam finite element is developed. Bending and shear force displacements and slopes, and torsional displacements are taken as degrees of freedom at both ends of the element. Total blade deflection slopes are considered as composed of bending and shear force deflection slopes in calculations of blade strain and kinetic energy. This element is compared with the existing thin and thick beam finite elements, and theoretical models. Results obtained for the vibration characteristics of rotating and non-rotating non-uniform aerofoil cross-sectioned blades are compared with the available calculated and experimental values. In all cases considered the element exhibits good convergence characteristics and produces accurate results.     

Analyses of contact forces and vibration response for a defective rolling element bearing using an explicit dynamics finite element model

This paper provides insights into the physical mechanism by which defect-related impulsive forces, and consequently, vibrations are generated in defective rolling element bearings. A dynamic nonlinear finite element model of a rolling element bearing with an outer raceway defect was numerically solved using the explicit dynamics finite element software package, LS-DYNA. A hypothesis was developed to explain the numerical noise observed in the predicted vibrations and contact forces, and the noise frequencies were analytically estimated. In-depth analyses of the numerically estimated dynamic contact forces between the rolling elements and the raceways of a bearing, which are not measured in practice, and have not been reported previously, are presented in this paper. Several events associated with the traverse of the rolling elements through the outer raceway defect are elaborated, and the impulsive force generating mechanism is explained. It was found that the re-stressing of the rolling elements that occurs near the end of a raceway defect generates a burst of multiple short-duration force impulses. The modelling results also highlight that much higher contact forces and accelerations are generated on the exit of the rolling elements out of defect compared to when they strike the defective surface. A bearing with a machined outer raceway defect was tested in a controlled experiment; the measured acceleration response compared favourably with the numerically modelled acceleration results, thereby, validating the low- and high-frequency characteristics of the de-stressing and re-stressing of the rolling elements, respectively.     

Post-growth tailoring of the optical properties of GaAs/AlGaAs quantum well structures

In this paper we describe a method for performing post-growth bandgap engineering in the GaAs/AlGaAs quantum well system. The method used is impurity-free vacancy diffusion. Using both single and multiple quantum well data we show that this allows blue shifts in the optical properties, while retaining both their distinctive excitonic and electrical characteristics. The electrical response is modelled and no comparative degradation of the quantum confined Stark effect is predicted, and this is confirmed experimentally. Possible applications of this technique are mentioned.     

离心压气机叶轮弯曲母线造型及叶片弯曲减少二次流损失机理初步探讨

在离心叶轮在母线几何设计方法基础上发展了叶轮弯曲母线造型方法。使用这种方法不仅可对现有叶轮进行精确模拟，还可在设计中利用叶片弯曲控制离心叶轮槽道内二砍流动，达成减少损失的目的。使用三元粘性数值模拟技术对这一机理进行了初步探讨．     

安徽琅琊山铜矿植物和土壤重元素同步辐射X射线荧光分析

利用同步辐射 X射线荧光分析方法研究了生长在金属矿山上的植物体内重元素的组成与含量 ,以及植物对重元素的富集特性。对采自安徽琅琊山铜矿的梓木草 (Lithospermum Erythrorhizon)和山慈菇(Tulipa Edulis) 2种植物的重元素分析结果表明 ,在锌木草各营养器官之间 ,各种重元素的分布极不均匀 ,Mn、Fe、Cu、Zn等主要富集在多年生根中。山慈菇叶片中的重元素的含量 ,比其他地方的同种植物较高。与一般植物体内重元素检测方法相比 (如火焰原子吸收光谱法 ) ,该法的灵敏度高 ,样品处理简便。     

单叶片桨噪声相移叠加法预报多叶片船舶螺旋桨非空化低频噪声

单个桨叶噪声预报螺旋桨非空化噪声可显著降低计算耗时,本文联合URANS和声类比方程对该方法进行验证,并对螺旋桨噪声的时域特征进行分析。首先计算均匀流中E779A螺旋桨的声辐射,揭示了桨叶上声压分布及测点声压信号的典型周期特征,采用单个叶片噪声相移叠加(简称"单叶片方法")预报螺旋桨噪声的结果,与对螺旋桨所有叶片积分的计算结果吻合良好,验证了均匀流中单叶片方法的可行性。将该方法应用于潜艇伴流场中无侧斜和大侧斜螺旋桨噪声辐射计算,预报结果与所有叶片积分的结果吻合较好,验证了非均匀流场中单叶片方法的可行性,说明单叶片相移叠加法预报螺旋桨普遍进流条件下的辐射噪声是可行的。研究结论也可为对转桨、泵喷等噪声预报提供参考。      

Vibration of collecting electrodes in electrostatic precipitators – Modelling,measurements and simulation tests

The paper presents a numerical model for simulating the vibration of collecting electrodes in an electrostatic precipitator. The method of finite elements (FEM) was used to describe the shell elements of the collecting electrodes. The remaining elements of the system were modelled with the application of the rigid finite elements method (RFEM). The results of measuring validation and testing calculations are discussed.