Dynamic analysis of horizontal axis wind turbine by thin-walled beam theory

A mixed flexible-rigid multi-body mathematical model is applied to predict the dynamic performance of a wind turbine system. Since the tower and rotor are both flexible thin-walled structures, a consistent expression for their deformations is applied, which employs a successive series of transformations to locate any point on the blade and tower relative to an inertial coordinate system. The kinetic and potential energy terms of each flexible body and rigid body are derived for use in the Lagrange approach to formulate the wind turbine system's governing equation. The mode shapes are then obtained from the free vibration solution, while the distributions of dynamic stress and displacement of the tower and rotor are computed from the forced vibration response analysis. Using this dynamic model, the influence of the tower's stiffness on the blade tip deformation is studied. From the analysis, it is evident that the proposed model not only inherits the simplicity of the traditional 1-D beam element, but also able to provide detailed information about the tower and rotor response due to the incorporation of the flexible thin-walled beam theory.     

Influence of Operating Parameters on Unbalance in Rotating Machinery Using Response Surface Method

Wide range of rotating machinery contains an inherent amount of unbalance which leads to increase in the vibration level and related faults. In this work, the effect of different operating conditions viz. the unbalanced weight, radius, speed and position of the rotor disc on the unbalance in rotating machine are studied experimentally and analyzed by using Response Surface Methodology (RSM). RSM is a technique which consists of mathematical and statistical methods to develop the relationship between the inputs and outputs of a system by distinct functions. L27 Orthogonal Array (OA) was developed by using Design of Experiments (DOE) according to which experimentation has been carried out. Three accelerometer sensors were mounted to record the vibration responses (accelerations) in radially vertical, horizontal and axial directions. The responses recorded as root mean square values are then analysed using RSM. The relationship between response and operating factors has been established by developing a second order, non-linear mathematical model. Analysis of variance (ANOVA) has been performed for verification of the developed mathematical models. Results obtained from the analysis show that the unbalance weight and speed are most significant operating conditions that contribute the most to the effect the unbalance has on the rotating spindle.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

Detecting magnetic field direction by a micro beam operating in different vibration modes

A new method to detect the magnetic field direction by using a silicon structure is presented in this paper. The structure includes a micro beam and an in-plane coil electrode. When the electrode under a magnetic field is applied with an alternating current, the micro beam is actuated under the effect of the Lorentz forces. Magnetic fields of different directions cause different vibration profiles. The direction of the magnetic field is obtained by measuring the vibration amplitudes of the micro beam, which is driven to work at first- and second-order resonant modes. A micro structure has been fabricated using the bulk micromachined silicon process. A laser Doppler vibrometer system is implemented to measure the vibration amplitudes. The experimental results show that the amplitude of the structure, which depends on the different modes, is a sine or cosine function of the angle of the magnetic field. It agrees well with the simulation result. Currently a resolution of 10° for the magnetic field direction measurement can be obtained using the detecting principle.     

Theoretical and experimental research on a disk-type non-contact ultrasonic motor

We developed a disk-type non-contact ultrasonic motor based on B22 vibration mode. The rotors of SU-8 photoresist are fabricated by the UV-LIGA process to control their shapes and thicknesses. So the structures of them are optimized by the experiments. It is found that the revolution speed of disk-type non-contact ultrasonic motor not only depends on the vibration amplitude of the stator, but also the weight and construction of the rotors. The maximum revolution speed of the optimal rotor is 3569 rpm at the input voltage of 20 V and the driving frequency of 45.6 kHz. The exciting principle of traveling wave is presented with theoretical equations. The electric signals applied to the piezoelectric ceramic are designed by the principle. The natural frequency and corresponding vibration mode are calculated and analyzed using finite element method. It is shown that experimental results are in good agreement with simulation, which verifies the effectiveness of the finite element model. Moreover, the levitation distance between the stator and rotor is measured by a CCD laser displacement transducer.     

Contact-impact analysis of geared rotor systems

In this paper, a finite element formulation, used to analyze the contact-impact behavior of geared rotor systems coupled with the rotational, lateral, and axial vibrations between gears at high rotational speeds, has been developed. A gear impact element to model the contact-impact behavior between gears has been developed and its numerical method is discussed. A relative displacement measurement idea has been proposed to measure vibration parameter for contrast experiment in high rotational geared system. The equations of motion are derived and solved iteratively during each time increment until the unbalanced force decrease to an acceptable tolerance level. Based on the proposed method, an analysis program, GEARS, has been developed. The contact-impact behavior of geared rotor systems is analyzed especially under high rotational speed condition as numerical examples, which are demonstrated to show the effectiveness of the proposed method.     

A miniaturization of the multi-degree-of-freedom ultrasonic actuator using a small cylinder fixed on a substrate

Multi-degree-of-freedom ultrasonic actuator has been studied for robot arms and multidimensional precision table and so on because of its simple structure, silent operation, and holding force. In this study, we aim to miniaturize multi-degree-of-freedom ultrasonic actuator for fabrication on a substrate. This actuator consists of a stainless steel cylinder and a PZT ring. The cylinder is fixed on a substrate and the PZT ring is glued to the substrate near the cylinder. The 1st longitudinal vibration and the 2nd bending vibration are simultaneously excited in the cylinder to make elliptical motion at the top of the cylinder and a ball rotor placed on the cylinder rotates because of the friction force. Length of the cylinder was decided so as to tune the resonance frequency of the 1st longitudinal vibration to the 2nd bending one. Actuator performances are evaluated experimentally using a 14 mm height and 7 mm diameter stainless steel cylinder with a 0.5 mm thickness PZT ring. The rotation about the cylinder axis is tested using the two orthogonal bending vibrations with 90 degrees phase difference. Also, the rotation about horizontal axes were investigated using the combination of the longitudinal vibration and one of two bending vibrations. We measured the rotation speed of a steel ball and obtained 15.8 rps using a 6 mm diameter ball rotor.     

Numerical analysis of the hybrid transducer ultrasonic motor: comparison of characteristics calculated by transmission-line and lumped-element models

In this paper, a hybrid transducer ultrasonic motor is numerically analyzed by using two equivalent electrical circuit models. A transmission-line model for the torsional vibration in the stator, which can model any torsional vibration mode and their combinations, was introduced and compared with a lumped-element model, which modeled the fundamental torsional resonance mode in the stator. The calculation result by using the transmission-line model demonstrated that the second harmonic torsional vibration increased either with the static spring force by which the rotor was pressed to the stator or with the load torque placed on the rotor. The difference in the calculated motor performance between the two models appeared when the second harmonic torsional vibration became large at a sufficient static spring force.     

PLANE ELASTICITY PROBLEM OF TWO-DIMENSIONAL OCTAGONAL QUASICRYSTALS AND CRACK PROBLEM

The plane elasticity theory of two-dimensional octagonal quasicrystals is developed in this paper. The plane elasticity problem of quasicrystals is reduced to a single higher-order partial differential equation by introducing a displacement function. As an example, the exact analytic solution of a Mode I Griffith crack in the material is obtained by using the Fourier transform and dual integral equations theory, then the displacement and stress fields, stress intensity factor and strain energy release rate can be calculated. The physical significance of the results relative to the phason and the difference between the mechanical behaviours of the crack problem in crystals and quasicrystals are figured out. These provide important information for studying the deformation and fracture of the new solid phase.     

二维振动方向变换器的耦合振动及其等效电路

二维振动方向变换器集功率合成和二维超声输出的功能于一体,在功率超声技术中具有重要的应用价值。然而,对于二维振动方向变换器的设计分析只有一种较为复杂的波动方程法。为此,本文研究了二维振动方向变换器的另外一种简明的设计分析方法——等效电路法。通过引入二维机械耦合系数和纵向力转换系数,利用力电类比原理建立了二维振动方向变换器的同相及反相二维耦合振动的统一等效电路模型。利用本文提出的设计方法,计算了两种不同材料的二维振动方向变换器的谐振频率,与有限元计算结果及实验测试结果一致,为该类超声振动系统的工程应用提供了一种简洁直观的设计分析方法。      

Two-dimensional coupled vibration and equivalent circuit of longitudinal-longitudinal orthogonal composite vibration direction converter

With the advantages of transmitting energy from multiple directions to one direction and transforming vibration from one source to multiple directions, the two-dimensional vibra?tion direction converter has important applications in power ultrasonic technology. However, for the complexity of using the wave equation to design and analysis the two-dimensional vibration direction converter, a concise equivalent circuit for the converter is investigated. By introducing the two-dimensional mechanical coupling coefficient and the longitudinal force transform coef?ficient, and using the electromechanical analogy principle, the equivalent circuit and resonance frequency equation of the two-dimensional vibration direction converter vibrating in anti-phase and in-phase two-dimensional coupled vibration are deduced. The resonance frequencies of the vibration direction converters of two different materials are calculated by using the proposed frequency equation, which are in agreement with the results from the finite element method and the experimental test. It provides a concise method for the design and applications of such ultrasonic vibration system.     

Study on the step-type circular ring ultrasonic concentrator in radial vibration

In this paper, the plane radial vibration of an isotropic metal thin circular rings is studied and its equivalent circuit model is obtained. Based on the equivalent circuit model, the step-type circular ring concentrator consisting of two metal thin circular rings in radial vibration is analyzed. Its compound equivalent circuit is derived and the resonance frequency equation is obtained. The relationship between the resonance frequency, the radial displacement amplitude magnification and the geometrical dimensions is analyzed. The resonance frequency of the step-type radial concentrator is calculated based on the resonance frequency equation. For comparison, the resonance frequency of the step-type radial concentrator is also obtained by using numerical method. It is illustrated that the resonance frequencies from these two methods are in a good agreement with each other.     

Three-dimensional vibration analysis of a torus with circular cross section

The free vibration characteristics of a torus with a circular cross section are studied by using the three-dimensional, small-strain, elasticity theory. A set of three-dimensional orthogonal coordinates system, comprising the polar coordinate (r, theta) at each circular cross section and the circumferential coordinate phi around the ring, is developed. Each of the displacement components u(r), v(theta), and w(phi) in the r, theta, and phi directions, respectively, is taken as a product of the Chebyshev polynomials in the r direction and the trigonometric functions in the theta and phi directions. Eigenfrequencies and vibration mode shapes have been obtained via a three-dimensional displacement-based extremum energy principle. Upper bound convergence of the first seven eigenfrequencies accurate to at least six significant figures is obtained by using only a few terms of the admissible functions. The eigenfrequency responses due to variation of the ratio of the radius of the ring centroidal axis to the cross-sectional radius are investigated in detail. Very accurate eigenfrequencies and deformed mode shapes of the three-dimensional vibration are presented. All major modes such as flexural thickness-shear modes, in-plane stretching modes, and torsional modes are included in the analysis. The results may serve as a benchmark reference for validating other computational techniques for the problem.     

QUASI-PERIODIC VIBRATION OF CRACKED ROTOR ON FLEXIBLE BEARINGS

A cracked rotor on flexible bearings is studied in this paper. The vibration of such a system has many complexities because of the crack and bearing flexibility. However, if the properties of the bearings are known, the system can be simplified by supposing that, the vibration due to weight is dominant. Equations of motion are derived, and a linear system in which the crack has been considered as an external disturbance described by a series of trigonometric functions is obtained. Consequently, the quasi-periodic vibrations of the rotor and bearings are established by harmonic balance method and approximate values of the vibration determined by truncating the higher order terms. It is believed that the simulated results will be useful for crack detection in the case of weight-dominant rotors.     

Study on Combined Method Based on 3-D ESPI

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Study on Combined Method Based on 3-D ESPI

The finite element method (FEM),whether the calculation is accurate or not,depends closely on object boundary condition.If the three dimensional displacement of the object obtained in experiment is regarded as its boundary condition,a new method combining the results of experiment and calculation,called combined method (CM),is formed.The combined method possess advantages of experiment and calculation.It can correct calculation and improve the accuracy of FEM.Accordingly it has more practicability.In this paper,the three dimensional displacement fields of a typical beam loaded at three points are tested by using 3-D electric speckle pattern interferometry (ESPI).Using the experimental results as boundary condition the whole three-dimensional displacement fields can be calculated by FEM.The beam′s three-dimensional displacement fields obtained by FEM agree very well with those obtained by experiment.This proves that the combined method is effective and practicable.     

In-plane vibration of thin elliptic plates submitted to uniform pulsed microwave irradiations

The technique of acoustic generation by microwave excitation in structures is applied here to study the in-plane vibration of full or hollowed elliptic plates. The absorption of pulsed microwave irradiations by a material causes a sudden rise of its temperature and the generation of an acoustic wave by thermoelastic effect. A semi-analytic theoretical model is developed to predict the in-plane displacement fields in elliptic thin plates submitted to a uniform temperature rise. It is assumed that the isotropic and viscoelastic plate constitutive material is submitted to a thermoelastic excitation under a plane stress state. The wave equations that govern the Helmholtz displacement potentials are resolved in an elliptic cylindrical system of coordinates by means of infinite angular and radial Mathieu functions series. The displacement field is finally obtained by taking into account the zero stress conditions on the boundaries of the plates. The comparison between the theoretical and the experimental responses of full and hollowed elliptic plates shows a good agreement that permits the validation of the developed model.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

Low-frequency harmonic vibration analysis with temporal speckle pattern interferometry

In this paper, the time sequence phase method (TSPM) has been applied to measure the displacement caused by low-frequency vibration in temporal speckle pattern interferometry (TSPI). The principle is that by capturing the temporal speckle patterns related to the object vibration, the whole-field displacement responses (amplitude and phase) of the vibrating object can be calculated through scanning these fluctuations. Thus, quantitative measurement can be carried out using a conventional ESPI system without a camera synchronized to the object vibration or a phase shifting system. The elaboration on the method is given and experimental results are presented.