New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.     

The vibration signature of chordal cracks in a rotor system including uncertainties

The aim of this paper is to investigate the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence of a crack in rotating system. The random dynamic response of the cracked rotor is evaluated by expanding the changing stiffness of the crack (i.e. the breathing mechanism) as a random truncated Fourier series. To avoid the use of the Monte Carlo simulations (MCS), an alternative procedure that is based on a combination of the Harmonic Balance Method and the Stochastic Finite Element Method (SFEM) using the Polynomial Chaos Expansion (PCE) is proposed. So the response of the Fourier components of the cracked rotor is expanded in the polynomial chaoses. The random dynamic response obtained by applying this procedure is compared with that evaluated through numerical integration based on the Harmonic Balance Method and the Monte Carlo simulations.     

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.     

The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor

By analyzing the limitations of weight dominance and by taking the complicated whirl of the rotor into account, general equations of motion have been developed in case of a Jeffcott rotor with a transverse crack. The angle between the crack direction and the shaft deformation direction is used to determine the closing and opening of the crack, allowing one to study the dynamic response without assuming weight dominance. Using the new equations, the dynamic response of a cracked rotor near its critical speed has been computed via a numerical method to investigate the influence of nonlinear breathing of the crack and that of the imbalance orientation angle β on the stability, critical speed and peak response of the rotor. The results show that nonlinear breathing can improve the stability of a rotor in contrast to a rotor with an open crack, and, with a reversed imbalance (70°<β<270°), that it can reduce the vibration response in contrast to an uncracked rotor. The basic characteristics of a cracked rotor near its critical speed are similar to those of an uncracked rotor. The critical speed can be determined by measuring the rotation of the center of gravity. The critical speed of a cracked rotor is located between the natural frequencies of the fully open crack and those of the fully closed crack and depends on the imbalance orientation angle. Its value is lowest at β≈90° and highest at β≈270°. The peak in the response at the critical speed is mainly determined by the imbalance orientation angle. At β≈0° and 180°, the peak corresponds to the maximum and minimum response, respectively.     

Dynamic analysis of a geared rotor system considering a slant crack on the shaft

The vibration problems associated with geared systems have been the focus of research in recent years. As the torque is mainly transmitted by the geared system, a slant crack is more likely to appear on the gear shaft. Due to the slant crack and its breathing mechanism, the dynamic behavior of cracked geared system would differ distinctly with that of uncracked system. Relatively less work is reported on slant crack in the geared rotor system during the past research. Thus, the dynamic analysis of a geared rotor-bearing system with a breathing slant crack is performed in the paper. The finite element model of a geared rotor with slant crack is presented. Based on fracture mechanics, the flexibility matrix for the slant crack is derived that accounts for the additional stress intensity factors. Three methods for whirling analysis, parametric instability analysis and steady-state response analysis are introduced. Then, by taking a widely used one-stage geared rotor-bearing system as an example, the whirling frequencies of the equivalent time-invariant system, two types of instability regions and steady-state response under the excitations of unbalance forces and tooth transmission errors, are computed numerically. The effects of crack depth, position and type (transverse or slant) on the system dynamic behaviors are considered in the discussion. The comparative study with slant cracked geared rotor is carried out to explore distinctive features in their modal, parametric instability and frequency response behaviors.     

Dynamic behavior analysis of cracked rotor

In the present study, the additional slope is used to consider the crack breathing, and is expressed explicitly in the equation of motion as one of the inputs to produce the bending moment at the crack position. Inversely, the additional slope is calculated by integrating on the crack region based on a fracture mechanics concept. The response of a cracked rotor is formulated based on the transfer matrix method. The transient behavior due to the crack breathing is considered by introducing a ‘moving’ Fourier-series expansion concept to the additional slope. The time-varying harmonic components of the additional slope are used to calculate the harmonic responses. The application considered is a general rotor model composed of multiple shafts, disks and cracks, and resilient bearings at both ends. Verification analysis is carried out for a simple rotor model similar to those found in the literature. Using the additional slope, the cracked rotor behavior is explained by the crack depth and rotation speed increase. It is shown that region on the crack front line having the dominant stress intensity factor value moves from the central area to both ends, as the crack depth increases. The result matches well with the crack propagation pattern shown in a bench mark test in the literature. Whirl orbits near the critical and sub-critical speed ranges of the rotor are discussed. It is shown that there exists some speed range near the critical speed, where the temporary whirl direction reversal and phase shift exist. When an unbalance is applied, the peculiar features, such as the whirl direction reversal and phase shift, disappear.     

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.     

A continuous vibration theory for rotors with an open edge crack

In this paper a new continuous model for flexural vibration of rotors with an open edge crack has been developed. The cracked rotor is considered in the rotating coordinate system attached to it. Therefore, the rotor bending can be decomposed in two perpendicular directions. Two quasi-linear displacement fields are assumed for these two directions and the strain and stress fields are calculated in each direction. Then the final displacement and stress fields are obtained by composing the displacement and stress fields in the two directions. The governing equation of motion for the rotor has been obtained using the Hamilton principle and solved using a modified Galerkin method. The free vibration has been analyzed and the critical speeds have been calculated. Results are compared with the finite element results and an excellent agreement is observed.     

Detecting cracked rotors using auxiliary harmonic excitation

Cracked rotors are not only important from a practical and economic viewpoint, they also exhibit interesting dynamics. This paper investigates the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. After reviewing the modelling of cracked rotors, the paper analyses the use of auxiliary excitation of the shaft, often implemented using active magnetic bearings to detect cracks. Applying a sinusoidal excitation generates response frequencies that are combinations of the rotor spin speed and excitation frequency. Previously this system was analysed using multiple scales analysis; this paper suggests an alternative approach based on the harmonic balance method, and validates this approach using simulated and experimental results. Consideration is also given to some issues to enable this approach to become a robust condition monitoring technique for cracked shafts.     

The use of roving discs and orthogonal natural frequencies for crack identification and location in rotors

A variety of approaches that have been developed for the identification and localisation of cracks in a rotor system, which exploit natural frequencies, require a finite element model to obtain the natural frequencies of the intact rotor as baseline data. In fact, such approaches can give erroneous results about the location and depth of a crack if an inaccurate finite element model is used to represent an uncracked model. A new approach for the identification and localisation of cracks in rotor systems, which does not require the use of the natural frequencies of an intact rotor as a baseline data, is presented in this paper. The approach, named orthogonal natural frequencies (ONFs), is based only on the natural frequencies of the non-rotating cracked rotor in the two lateral bending vibration x–z and y–z planes. The approach uses the cracked natural frequencies in the horizontal x–z plane as the reference data instead of the intact natural frequencies. Also, a roving disc is traversed along the rotor in order to enhance the dynamics of the rotor at the cracked locations. At each spatial location of the roving disc, the two ONFs of the rotor–disc system are determined from which the corresponding ONF ratio is computed. The ONF ratios are normalised by the maximum ONF ratio to obtain normalised orthogonal natural frequency curves (NONFCs). The non-rotating cracked rotor is simulated by the finite element method using the Bernoulli–Euler beam theory. The unique characteristics of the proposed approach are the sharp, notched peaks at the crack locations but rounded peaks at non-cracked locations. These features facilitate the unambiguous identification and locations of cracks in rotors. The effects of crack depth, crack location, and mass of a roving disc are investigated. The results show that the proposed method has a great potential in the identification and localisation of cracks in a non-rotating cracked rotor.     

Dynamic analysis of asymmetric bladed-rotors supported by anisotropic stator

For the mathematical convenience of conventional rotor dynamic analysis, the components of the entire rotor system are often classified into two parts: the stationary parts and the rotating parts, depending upon whether or not the corresponding components rotate with respect to their axes of rotation. Even for bladed-rotors, the rotor blades have been treated along the same lines as the ordinary rotating components such as the rotor disk and the shaft. The distinct dynamic nature of the blades, therefore, has not been thoroughly taken into account in the conventional rotor dynamic analysis. In this paper, the rotating parts of a bladed-rotor system are further subdivided into the rotor blade-group and the other rotating components. The equation-of-motion for the bladed-rotor system is then developed, by modifying the conventional general rotor system equation to adopt the blade-group dynamics without loss of generality. Complex modal solutions to the bladed-rotor system are investigated based on a new modulated coordinate transformation approach, yielding newly defined directional frequency response functions that characterize the nature of asymmetry present in the rotating blade-array. Finally, the effects of the blade-group asymmetry on the rotor system dynamics are demonstrated with a pertinent numerical example.     

Vibration of cracked shafts in bending

Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.     

Dynamic response of cracked rotor-bearing system under time-dependent base movements

Dynamic response of cracked rotor-bearing system under time-dependent base movements is studied in this paper. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Both the open and breathing transverse cracks are considered in the analysis. The finite element model is established for the base excited rotor-bearing system with open or breathing cracks. Considering the time-varying base movements and transverse cracks, the second-order differential equations of the system will not only have time-periodic gyroscopic and stiffness coefficients, but also the multi-frequency external excitations. An improved harmonic balance method is introduced to obtain the steady-state response of the system under both base and unbalance excitations. The response spectra, orbits of shaft center and frequency response characteristics, are analyzed accordingly. The effects of various base angular motions, frequency and amplitude of base excitations, and crack depths on the system dynamic behaviors are considered in the discussions.     

风力发电机叶轮的数值优化设计法

１引言目前，在风力机叶轮的气动设计方面，还没有其系统的设计模型和方法，只有一些针对某一方面的模型，这些模型还无法规纳成一套可靠的系统的设计模型［１］，Ｇｏｕｒｉｅｒｅｓ在他的风力机设计理论的书中也只能介绍几种简化的有关设计模型［２］。所用的设计方法主要还是以经验设计为主，不是全面系统地考虑叶轮的各种影响因素，如Ｈａｂａｌｉ［３］在他的设计工作中主要考虑了叶片翼型对叶轮运行的气动作用，ＶＯｕｔｓｉｎａｓ［４］只是对偏航工况下叶轮的气动特性及操作进行了分析研究，文献［５］也仅对叶轮功率等气动特性进行…     

ANALYSIS OF NON-LINEAR DYNAMIC STABILITY FOR A ROTATING SHAFT-DISK WITH A TRANSVERSE CRACK

In this paper, the nonlinear dynamic stability of a rotating shaft-disk with a transverse crack is studied. The crack and the disk are located in arbitrary positions of the shaft respectively. Using the equivalent line-spring model, the deflections of the system with a crack are constructed by adding a deflection to the deflections of the uncracked system. The unstable regions are confirmed by Runge-Kutta method and the Floquet theory. The effects of crack depth, crack position, disk position, disk thickness and rotating speed on the principal unstable regions are discussed. The numerical results are compared with available data.     

An analytical study of rotor dynamics coupled with thermal effect for a continuous rotor shaft

This paper presents an analytical analysis of a continuous rotor shaft subjected to universal temperature gradients. To this end, an analytical model is derived to investigate the generic thermal vibrations of rotor structures. The analytical solutions are obtained in a rotating frame and include parameters related with both the thermal environment and the rotor dynamic structures. This provides an insight into the mechanisms for the rotor thermal vibration. Furthermore, numerical results based on the analytical solutions are given. An index denoting the temperature gradients is proposed for the occasions with nonlinear cross-sectional temperature distributions. Finally, the factors influencing the thermal vibrations are analyzed. The results show that the thermal vibration is affected by many factors including the shaft size, rotational speeds, heating locations, critical speed, etc. Moreover, it is investigated how the convection coefficient and the heat conductivity influence the thermal vibrations in order to provide an insight into the management of thermal vibrations from the perspective of thermal aspects.     

The dynamics of a cracked rotor with an active magnetic bearing

The dynamic characteristics of a cracked rotor with an active magnetic bearing (AMB) are theoretically analyzed in this paper. The effects of using optimal controller parameters on the dynamic characteristics of the cracked rotor and the effect of a crack on the stability of the active control system are discussed. It is shown that the dynamic characteristics of the cracked rotor with AMBs are clearly more complex than that of the traditional cracked rotor system. Adaptive control with AMBs may hide the fault characteristics of the cracked rotor, rather than helping to diagnose a crack; this will depend on the controller strategy used. It is very difficult to detect a crack in the rotor with an AMB support system when the vibration of the rotor system is fully controlled. Monitoring the super-harmonic components of 2× and 3× revolution in the sub-critical speed region can be used as an index to detect a crack in the rotor with an AMB system. If the effect of the crack is not taken into account at the design stage of the controller, then the rotor-AMB system will lose its stability in some cases when cracks appear.     

Damage detection of cracked thick rotating blades by a spatial wavelet based approach

This paper presents a technique for blade damage detection based on spatial wavelet analysis. The wavelet transform is used to analyze spatially distributed signals (e.g. mode shape) of cracked thick rotating blades. First, a finite element model is applied to the vibration of a thick rotating blade with a single edge crack. The effects of transverse shear deformation and rotatory inertia are taken into account. Then the mode shapes of the cracked rotating blade are analyzed by wavelet transformation. The effects of crack locations and sizes on the wavelet coefficients are studied. It is found that the distributions of the wavelet coefficients can identify the crack position of the rotating blades by showing a peak at the position of the crack. Then the signals are analyzed by wavelet transform. It is found that the distributions of the wavelet coefficients can identify the crack position. Assumed measurement errors are added to nth 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.     

Soil influence on unbalance response and stability of a simple rotor-foundation system

The equations of motion are set up for a simple rotor (Jeffcott or Laval rotor) on a rigid foundation mass resting on an elastic half space (soil). The unbalance response and the stability limit against self-excited vibrations caused by the internal damping of the rotating shaft are calculated. The numerical results presented as response diagrams and stability graphs show that the damping effect of the soil on the system, due to radiation of energy, may have a very positive influence on the smooth running of the rotors.     

传动系统同轴度测量方法的改进

针对现有轴-轴同轴度测量方法的缺陷,通过建立一套数学模型,得出了测量设备安装偏差量、设备测量值和传动轴旋转角度之间的关系方程,从而实现了只需在小角度范围内旋转,即可得出不含安装误差的轴-轴同轴度偏差量.另外,针对测量设备的特点提出了测量孔-孔同轴度的方法,从而实现了一套设备可以同时测量轴-轴、孔-孔同轴度.