Vibration damage detection of a Timoshenko beam by spatial wavelet based approach

This paper presents a technique for structure damage detection based on spatial wavelet analysis. The wavelet transform is used to analyze the mode shape of a Timoshenko beam. First, the mode shapes of the Timoshenko beam containing a transverse crack are obtained. The crack is represented as a rotational spring. Then these spatially distributed signals are analyzed by wavelet transformation. It is observed that distributions of the wavelet coefficients can identify the crack position of Timoshenko beam by showing a peak at the position of the crack. It is also demonstrated that the crack position can be detected by this method even though the crack is very small. Assumed measurement errors are added to the mode shape for evaluating the effect of measurement errors on the capability of detecting crack position. The moving average method is used to process the data with assumed measurement errors. The crack positions can also be identified when there exist assumed measurement errors.     

Variation in the stability of a rotating blade disk with a local crack defect

The effect of a near root local blade crack on the stability of a grouped blade disk is investigated in this paper. A bladed disk comprised of periodically shrouded blades is used to simulate the coupled periodic structure. The blade crack is modeled using the local flexibility with coupling terms. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations in the rotating blades has also been considered. Using the Galerkin's method, the imperturbation equations of a bladed disk in which one of the blades is cracked, subject to fluctuations in the rotation speed, can be derived. Employing the multiple scales method, the boundaries of the instability zones in the mistuned turbo blade system are approximated. Numerical results indicate that an additional unstable zone is introduced near the localization frequency and the regions of unstable zones are varied with the crack size and fluctuations in disk speed.     

Damage detection of a rectangular plate by spatial wavelet based approach

This paper presents a technique for structure damage detection based on spatial wavelet analysis. Many damage detection methods require modal properties before and after damage. This method only needs the spatially distributed signals (e.g. the displacements or mode shapes) of the rectangular plate after damage. First, spatially distributed signals of the rectangular plate with damage are obtained by finite element method. The damaged region is represented as the elements with reduced stiffness. Then these spatially distributed signals are analyzed by wavelet transformation. It is observed that distributions of the wavelet coefficients can identify the damage position of rectangular plate by showing a peak at the position of the damage. It is also demonstrated that this method is very sensitivity to the damage size.     

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.     

Crack detection in beams by wavelet analysis of transient flexural waves

In this paper, a method of crack detection in beam is provided by wavelet analysis of transient flexural wave. Firstly, we introduce a rotational spring in the cracked-beam model, and calculate the transient flexural wave propagation by the Reverberation Matrix Method. Secondly, at any point in this beam, the arrival time of waves with different group velocities can be identified by means of analysis of wavelet transform. From the signal of mid-frequency flexural wave extracted by wavelet transform, we can exactly determine the existence and position of crack in the beam. Here, a similar experiment is also produced to test the transient strain signal in beam, and the result after wavelet transform analysis is shown in accord with the theoretical one. Therefore, a powerful method is proved by the theoretical and experimental studies on the cracked beam, which can detect the crack accurately utilizing the wavelet analysis of the mid-frequency flexural wave.     

Mode shapes analysis of a cracked beam and its application for crack detection

In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.     

运用小波变换对遥感FTIR光谱进行信息提取

主要描述了如何利用小波变换技术,对一些强度较弱和混叠干扰的遥感FTIR光谱图进行信息提取。采用了墨西哥帽函数(Mexican hat function)小波,对三氯甲烷、丙酮的标准谱图及其两者混合物的遥感谱图信号进行了连续小波变换。结果发现,尺度越小,小波系数模的极大值点与突变点位置的对应就越准确。但小尺度下,小波系数由于受噪声影响,往往只凭一个尺度不能定位突变点的位置。相反,在大尺度下,对噪声进行了一定的平滑,极值点相对稳定,但由于平滑又会使定位产生偏差。因此,在用小波变换模极大值法判断信号突变点时,需要把多尺度结合起来综合观察。总之,小波分析技术能够较准确地、稳定地定位信号突变点,提取信息,并且具有平滑和放大有用信号的作用。对于非严重重叠的混合遥感光谱,根据对不同尺度下的小波变换系数的模极大值及其定位的分析,也可较好地识别谱图。     

Effect of unbalanced rotor whirl on blade vibrations

In this paper, vibration of a bladed unbalanced flexible rotor is studied. The blade that is attached to the disk is considered as a fixed-free Euler-Bernoulli beam. Position of the blade with respect to the eccentric mass is taken into consideration. Coupled equations of the motion of unbalanced rotor and the blades are obtained through Lagrange equations. The dynamic equations have time variant periodic coefficients. Transient vibration analysis showed that rotor acceleration excites the blade vibration with its own natural frequency. While the rotor passes through its own natural frequency (critical speed) the blade vibration is again excited but this time with the rotor natural frequency. Modal behavior of the blades are different for subcritical, supercritical and for critical speed of the rotor. In the subcritical run of the rotor, blades located from 0° to 180° with respect to the eccentric mass are deflected in the negative direction while the rest are deflected in the positive direction. For supercritical run of the rotor, modal behavior of the blades is just the opposite. For critical speed of the rotor, blades located 90° to 270° from the eccentric mass are deflected in the positive direction while the rest of the blades are deflected in the negative direction. Blades have also different deflections. When the deflections of the blades are plotted with respect to their position angle, distribution of the blade deflections has a sinusoidal shape.     

利用频闪结构光测量旋转叶片的三维面形

在对快速甚至高速运动物体进行研究时,需要对运动物体各个时刻的三维面形和形变量进行数字化描述。提出了一种利用频闪光作为结构照明光源,基于傅里叶变换轮廓术对旋转叶片每个时刻的三维面形和变形量进行动态测量的光学方法。使用自行设计的同步控制单元对风扇叶片旋转位置进行检测,用该检测信号同步驱动频闪结构光源的发光和图像采集系统“冻结”记录下旋转叶片表面变形条纹的瞬时图像,再运用傅里叶变换轮廓术计算出每一瞬间时刻旋转叶片的三维面形,分析这些三维面形数字化结果可以进一步得到旋转叶片的变形量。通过对家用电风扇的实验,验证了该方法的合理性和可行性。该方法可望在高速运动物体的面形和变形研究上有广泛运用。     

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.     

Damage identification for beams in noisy conditions based on Teager energy operator-wavelet transform modal curvature

Modal curvatures have been widely used in the detection of structural damage. Attractive features of modal curvature include great sensitivity to damage and instant determination of damage location. However, an intrinsic deficiency in a modal curvature is its susceptibility to the measurement noise present in the displacement mode shape that produces the modal curvature, likely obscuring the features of damage. To address this deficiency, the Teager energy operator together with wavelet transform is tactically utilized to treat modal curvature, producing a new modal curvature, termed the Teager energy operator-wavelet transform modal curvature. This new modal curvature features distinct capabilities of suppressing noise, canceling global trends, and intensifying the singular feature caused by damage for a measured mode shape involving noise. These features maximize the sensitivity to damage and accuracy of damage localization. The proposed modal curvature is demonstrated in several analytical cases of cracked pinned–pinned, clamped–free and clamped–clamped beams, with emphasis on characterizing damage in noisy conditions, and it is further validated by an experimental program using a scanning laser vibrometer to acquire mode shapes of a cracked aluminum beam. The Teager energy operator-wavelet transform modal curvature essentially overcomes the deficiency of conventional modal curvature, providing a new dynamic feature well suited for damage characterization in noisy environments. (The Matlab code for implementing Teager energy operator-wavelet transform modal curvature can be provided by the corresponding author on request.)     

二维小波变换及在青藏高原地形重构中的应用

计算和检验了4种不同小波基底和不同阶小波基底对单脉冲和尖峰两个信号的分解及重构的性质,结果表明双正交小波的精度最高,Coifman小波的精度最低.通过设阈值对小波和快速Fourier变换进行截断后再重构,说明小波具有重构精度高、所用非零系数少以及误差被局域化等优点,这就削弱了"Gibbs现象",限制了它的影响范围.在此基础上,用二维快速Fourier和二维离散小波变换对实际青藏高原及其附近地形做了分解、重构及截断后重构等,说明二维小波变换能以较之FFT更少的非零系数和误差局域化等重构大地形,表明了二维小波变换在未来大气环流数值模式中的应用潜力.     

An insight into the breathing mechanism of a crack in a rotating shaft

The time history of local flexibilities associated with a breathing crack in a rotating shaft is the concern of this paper. Considering quasi-static approximation, the deflections of a circular cross-section beam presenting a crack of different depths, due to bending or torsion loads are analyzed with the aid of a refined nonlinear contact-finite element procedure in order to predict accurately the time-variant flexibility of the fractured shaft. This method predicts the partial contact of crack surfaces, and it is appropriate to evaluate the instantaneous crack flexibilities. The bending load is applied in several aperture angles, in order to simulate a rotating load on a fixed beam. Results obtained for the rotating beam can then be used for the analysis of cracked, horizontal axis rotors. The effect of friction is also considered in the cracked area. Portions of crack surfaces in contact are predicted, the direct and the cross-coupled flexibility coefficients are calculated by applying energy principles. The numerical results compared with relevant previously published results, show high consistency.     

小波变换在载频条纹相位分析法中的应用研究

为了克服在非平稳信号分析中傅里叶变换的全局性缺陷,以及窗口傅里叶分析的单一分辨率和伸缩窗口傅里叶分析的尺度不确定性问题,采用伽博解析小波变换技术对空间载频光栅条纹进行相位分析,有效地提取出相对于载频条纹基频的完整相位信息,从本质上解决了上述问题。以三维轮廓术为例,与傅里叶分析进行了对比研究,给出了小波分析应用在空间载频条纹相位分析中详细而完整的理论推导证明、计算机模拟以及实验验证结果。     

Denoising of sensor signals for the flange thickness measurement based on wavelet analysis

A railway wheelset is subject to normal wear due to large part to friction contact between the wheelset and the rail. Because the wear of wheelset will bring the hidden security troubles to the operation of the railway, it is very important to measure the wheelset's geometrical parameters, especially the flange thickness. The optoelectronic method is proposed and can dynamically measure the flange thickness on line. Fast Fourier transform and wavelet analysis methods are used to denoise the sensor's signals. It is found that the wavelet transform produces a much better way of denoising of the signals compared with the fast Fourier transform. Comparisons of the flange thickness measurement with the wheelset creeping and the optoelectronic system are presented. The root-mean-square errors of the flange thickness with the manual measurement with the wheelset creeping and the optoelectronic method measurement with the wavelet analysis are 0.22 and 0.18, respectively. The changing range of manual measurement is much larger than that of the optoelectronic method because of the difference between every operator's measuring standard. Measurement results of the optoelectronic method show that the system has better repeatability and reliability compared to the manual measurement.     

直升机旋翼桨叶气动力学参数飞行测试系统

根据某型号直升机定型试飞的需要,研制出了动态测量旋翼桨叶气动力学参数的光电测试系统。其中运动参数(挥舞角、摆振角和变距角)由二维位置敏感传感器响应桨叶表贴发光点的位置变化测得,挥舞应变由光纤光栅传感器测得。利用自主搭建的桨叶运动姿态仿真平台开展了相关测试实验。结果表明:各角度值与对应输出电压间有很好的线性关系,测量范围内的角度测量误差小于1%;光纤光栅与电阻应变片对比测试的一致性较好,线性相关系数达到了0.999;光纤光栅应变测量的复现性很好,两次应变测量的相对偏差仅为1.22%。系统具有较高的测试精度。     

Mitigation of edgewise vibrations in wind turbine blades by means of roller dampers

Edgewise vibrations in wind turbine blades are lightly damped, and large amplitude vibrations induced by the turbulence may significantly shorten the fatigue life of the blade. This paper investigates the performance of roller dampers for mitigation of edgewise vibrations in rotating wind turbine blades. Normally, the centrifugal acceleration of the rotating blade can reach to a magnitude of 7–8g, which makes it possible to use this kind of damper with a relatively small mass ratio for suppressing edgewise vibrations effectively. The parameters of the damper to be optimized are the mass ratio, the frequency ratio, the coefficient of rolling friction and the position of the damper in the blade. The optimization of these parameters has been carried out on a reduced 2-DOF nonlinear model of the rotating wind turbine blade equipped with a roller damper in terms of a ball or a cylinder, ignoring the coupling with other degrees of freedom of the wind turbine. The edgewise modal loading on the blade has been calculated from a more sophisticated 13-DOF aeroelastic wind turbine model with due consideration to the indicated couplings, the turbulence and the aerodynamic damping. Various turbulence intensities and mean wind speeds have been considered to evaluate the effectiveness of the roller damper in reducing edgewise vibrations when the working conditions of the wind turbine are changed. Further, the optimized roller damper is incorporated into the 13-DOF wind turbine model to verify the application of the decoupled optimization. The results indicate that the proposed damper can effectively improve the structural response of wind turbine blades.     

Three-dimensional analysis of blade force and sound generation for an annular cascade in distorted flows

An unsteady lifting-surface theory for a rotating subsonic annular cascade has been developed to predict the unsteady blade forces and the acoustic power generation caused by interaction of blades with inlet distortions or wakes. Disturbance pressure and velocity fields induced by the rotor blades with fluctuating blade force are expressed in terms of the blade force distribution and kernel functions. The spanwise distribution of the blade force is given as a sum of blade force modes, and the kernel functions are resolved into the corresponding modal components. The sound pressure and intensity are expressed as a sum of acoustic modes, the modal components of which are given in terms of the blade force mode components.Numerical computations have been conducted .for interaction with the external disturbance flows that are sinusoidal in the circumferential direction, but possess a phase skewing in the radial direction. Correlations among the acoustic modes, the blade force modes and the flow patterns of the external disturbance have been investigated. When the predominant acoustic mode is subresonant, the blade force amplitude is reduced by the three-dimensional effect, which is lessened as the frequency increases. At deeply superresonant states, however, the three-dimensional effect upon the spanwise average of the blade force amplitude is small. The generated sound power is effectively reduced by increasing the radial non-uniformity of the external disturbance.     

Experimental investigations of the response of a cracked rotor to periodic axial excitation

In this paper, the coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses is investigated experimentally. The cracked rotor is excited axially using an electrodynamic exciter at a frequency equal to its bending natural frequency in both non-rotating and rotating conditions. The resulting time domain and frequency domain signals of the cracked rotor are studied. Spectral response of the cracked rotor with and without axial excitation is found to be distinctively different. When excited axially, it shows prominent presence of rotor bending natural frequency. However for an uncracked rotor, the response is similar with or without axial excitation. It is thus proposed that the response of the rotor to axial impulse excitation could be used for more reliable diagnosis of rotor cracks.     

Analytical periodic motions in a parametrically excited, nonlinear rotating blade

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated.