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This study aims at investigating the nonlinear dynamic behavior of rotating blade with transverse crack. A novel nonlinear rotating cracked blade model (NRCBM), which contains the spinning softening, centrifugal stiffening, Coriolis force, and crack closing effects, is developed based on continuous beam theory and strain energy release rate method. The rotating blade is considered as a cantilever beam fixed on the rigid hub with high rotating speed, and the crack is deemed to be open and close continuously in a trigonometric function way with the blade vibration. It is verified by the comparison with a finite element-based contact crack model and bilinear model that the proposed NRCBM can well capture the dynamic characteristics of the rotating blade with breathing crack. The dynamic behavior of rotating cracked blade is then investigated with NRCBM, and the nonlinear damage indicator (NDI) is introduced to characterize the non-linearity caused by blade crack. The results show that NDI is a distinguishable indicator for the severity level estimation of the crack in rotating blade. It is found that severe crack (i.e., a closer crack position to blade root as well as larger crack depth) is expected to heavily reduce the stiffness of rotating blade and apparently result in a lower resonant frequency. Meanwhile, the super-harmonic resonances are verified to be distinguishable indicators for diagnosing the crack existence, and the third-order super-harmonic resonances can serve as an indicator for the presence of severe crack since it only distinctly appears when the crack is severe.

Blade tip timing (BTT) is a noncontact blade vibration measurement method for rotating blade condition monitoring. However, it is difficult to install a once-per-revolution (OPR) probe due to the internal environment limitation of an engine. To solve this problem, an OPR-free BTT method is proposed in this article. Compared with the existing OPR-free methods that are based on blade displacement, this method is based on blade spacing change (BSC). In addition, a probe layout selection method based on minimum redundancy linear array (MRLA) is introduced to reduce the redundancy of the acquired signal. Simulations are conducted to prove that the proposed probe layout selection method has a stronger signal extraction capability and better robustness. The experiment is conducted to show that BSC is an alternative way for rotating blade condition monitoring and the technical route of this article is feasible.

Blade tip timing (BTT) is an effective noncontact measurement technology for rotating blade health monitoring. However, due to the mismatching between the high-speed rotation and the limited amount of probes, the signal collected from the BTT system is severely undersampled, which induces the difficulty in feature extraction. Multiple signal classification (MUSIC) has the potential to overcome the undersampled problem once the probes are properly placed. Whereas, if traditional MUSIC is directly used in BTT, the accuracy of frequency identification cannot be high enough and the identified number of frequency components is also severely restrained. To address these two problems, an improved MUSIC is proposed as an alternative methodology to extract the blade vibration frequency for BTT. Based on the orthogonality of the signal subspace and the noise subspace from undersampled signal, the presented method can effectively identify the vibration frequency components from the undersampled signal of BTT.

Vibration signal analysis has become one of the important methods for machinery fault diagnosis. The extraction of weak fault features from vibration signals with heavy background noise remains a challenging problem. In this article, we first introduce the idea of algorithm-aware sparsity-assisted methods for fault feature enhancement, which extends model-aware sparsity-assisted fault diagnosis and allows a more flexible and convenient algorithm design. In the framework of algorithm-aware methods, we define the generalized structured shrinkage operators and construct the generalized structured shrinkage algorithm (GSSA) to overcome the disadvantages of $l_{1} $ -norm regularization-based fault feature enhancement methods. We then perform a series of simulation studies and two experimental cases to verify the effectiveness of the proposed method. In addition, comparisons with model-aware methods, including basis pursuit denoising and windowed-group-lasso, and fast kurtogram further verify the advantages of GSSA for weak fault feature enhancement.

Blade tip timing (BTT) methods have been increasingly implemented for blade health monitoring (BHM). However, there are two drawbacks of current signal analysis methods preventing them from applying to online monitoring: first, current online monitoring requires the manual judgment of the resonance region, which is time-consuming. Second, existing BTT resonance signal analysis methods are not suitable for online monitoring. The spectral-analysis-based method presents spectral aliasing, while the computational complexity of the sparse-based method is usually high. In this article, we propose an adaptive online BHM method that includes two steps: automatic resonance region recognition and parameters identification in resonance area. For the former step, we demonstrate different methods for synchronous and asynchronous resonances, use the cross correlation to judge the occurrence of synchronous vibration, and use the linear estimation to determine the appearance of asynchronous vibration. For the latter step, an iterative adaptive least-squares periodogram is adopted for its tradeoff between spectral aliasing and computational complexity. The effectiveness of the above steps is first verified using different simulation data separately. Then, the laboratory data are used to test the effectiveness of the whole method. Finally, the online monitoring function of the proposed method is verified by the engineering data with both synchronous and asynchronous resonances.

Rotating blade is one of the most important components for turbomachinery. Blade crack is one of the most common and dangerous failure modes for rotating blade. Therefore, the fault mechanism and feature extraction of blade crack are vital for the safety assurance of turbomachinery. This study is aimed at the nonlinear dynamic model of rotating blade with transverse crack and the prior feature extraction of blade crack faults based on the vibration responses. First and foremost, a high-fidelity breathing crack model (HFBCM) for rotating blade is proposed on the basis of criterion for stress states at crack section. Since HFBCM is physically deduced from the perspective of energy dissipation and the coupling between centrifugal stress and bending stress is considered, the physical interpretability and the accuracy of the crack model are enhanced comparing with conventional models. The validity of the proposed HFBCM is verified through the comparison study among HFBCM, conventional crack models, and finite element-based contact crack model (FECCM). It is suggested that HFBCM behaves best among the analytical models and matches well with FECCM. With the proposed HFBCM, the nonlinear vibration responses are investigated, and four types of blade crack detection indicators for rotating blade and their quantification method are presented. The numerical study manifests that all these indicators can well characterize the occurrence and severity of crack faults for rotating blade. It is indicated that these indicators can serve as the crack-monitoring indexes.

Blade Tip Timing (BTT) method is increasingly implemented for rotating blade health monitoring for its non-contact property. However, the BTT data is usually highly undersampled as only a few sensors could be installed on the case. Due to the limited number of sensors installed, the arrangement can have a significant impact on BTT data quality. Different from the exhaustive method used in previous researches, in this paper, a mathematical model guided by optimizing objective is proposed for sensor configuration. Considering the dimensional characteristics of the sampling matrix, this paper no longer uses a single matrix coherence value as the optimization target, but adopts the Equiangular tight frame matrix as the goal of the whole sampling matrix. Moreover, this paper first considers the physical constraints of the actual installation and modal prior on the sensor arrangement. The physical constraints fused equiangular tight frame method is then solved by an alternating minimization approach. In addition, considering the disadvantages of the previous vibration parameter identification algorithms in terms of amplitude recovery and noise filtering, an iterative reweighted L1-norm based parameter identification method is applied to obtain the vibration parameters from the highly undersampled BTT data with better amplitude reconstruction accuracy. Both the simulation and experiment results are given to verify the effectiveness of the developed methods.

Bearing faults are one of the most common inducements for machine failures. Therefore, it is very important to perform bearing fault diagnosis reliably and rapidly. However, it is fundamental but difficult to extract impulses buried in heavy background noise for bearing fault diagnosis. In this paper, a novel adaptive enhanced sparse period-group lasso (AdaESPGL) algorithm for bearing fault diagnosis is proposed. The algorithm is based on the proposed enhanced sparse group lasso penalty, which promotes the sparsity within and across groups of the impulsive feature of bearing faults. Moreover, a periodic prior is embedded and updated dynamically through each iteration of the optimization procedure. Additionally, we formed a deterministic rule about how to set the parameters adaptively. The main advantage over conventional sparse representation methods is that AdaESPGL is parameter free (forming a deterministic rule) and rapid (extracting the impulsive information directly from the time domain). Finally, the performance of AdaESPGL is verified through a series of numerical simulations and the diagnosis of a motor bearing. Results demonstrate its superiority in extracting periodic impulses in comparison to other state-of-the-art methods.

This study aims at the systematical improvement and comparative analysis of analytical crack models for the rotating blade. Part I of this study focuses on analytical modeling, model modification, and model validation of transverse crack for the rotating blade. The most widely applied analytical crack models for the rotating blade are reviewed and compared, and then their limitations are discussed. It is indicated that the conventional analytical crack models suffer from low physical interpretability and vibration prediction accuracy. By considering these limitations of conventional analytical crack models, model modification is performed to enhance the physical meaning and improve the accuracy. First, the stress-based breathing crack model is modified by direct calculation of the breathing function based on the theory of linear elastic fracture mechanics and resetting the correction factor of centrifugal stiffening stiffness. Second, the vibration-based breathing crack models, including bilinear breathing crack model and cosine breathing crack model, are modified by introducing the coupling effect between bending stress and centrifugal stress based on the stress state at the blade crack section. The additional bending moment induced by the blade part outside the crack section is considered in all analytical models. The modified crack models’ validity is verified by comparing vibration responses obtained by the modified crack models, the finite element contact crack model, and the conventional crack models. The comparative results suggest that the modified models promote the physical interpretability and improve the vibration prediction accuracy of analytical crack models.

Compressor blades often operate in high temper-ature, high pressure, and heavy load environments. They are in direct contact with the outside air, which causes their life to be severely limited. Blade vibration is an important factor that reflecting the state of the blade. Therefore, effective blade vibration monitoring methods are needed. Blade tip timing (BTT) is a non-contact blade vibration measurement method that has the potential to achieve real-time condition monitoring. However, one of the biggest obstacles to real-time condition monitoring is the once per revolution (OPR) sensor in BTT. Due to the internal environment limitations of aero-engines, installing the OPR sensor in an actual engine is extremely expensive, and it is almost impossible to install the OPR sensor. To solve this problem, the relationship between the blade spacing change (BSC) and the blade vibration is analyzed and the measurement principle of BSC is presented in this study. Based on BSC, we proposed a novel BTT measurement method that does not need the OPR signal. The simulation and experiment are conducted to show the rationality of the proposed method.

The harsh working environment of the compressor hinders the condition monitoring of rotating blades. Blade faults are mostly induced by foreign object damage or high cycle failure due to initial defects. Detecting these failures before blades being broken will not only ensure engine safety but largely reduce repair costs. As a non-contact measurement technique, Blade Tip Timing(BTT) becomes a more popular method recently for health monitoring and fault diagnosis. By analyzing BTT data, many blade vibration parameters could be obtained. In this paper, we discuss how to extract first bend natural frequency, amplitude and static offset from BTT data. Then the change rule of these parameters around blade fault is illustrated. Afterward, a clustering method is employed to combine these parameters. In addition, a test with two faulty blades is introduced and used to verify the effectiveness of the proposed method.

Blade Tip Timing (BTT) methods have been increasingly implemented for blade health monitoring. However, most of them are based on the prior knowledge that the resonance’s location is known. Since real BTT test usually takes more than ten hours, it’s unrealistic to Figure out every resonance from the measured raw data manually. Furthermore, BTT data analysis suffers from its inherent under-sampled and non-uniform shortcoming. In this paper, we present a simple yet effective method for BTT-based blade health monitoring. The method starts with an automatic resonance recognition, where cross-correlation and Savitzky Golay filter are used to locate the resonance region. Then an adaptively reweighted least-squares periodogram algorithm is designed to identify parameters of the resonant vibration. The effectiveness of the algorithm was tested using both simulation and real test data.

Blade tip timing (BTT) is one of the most important non-contact monitoring methods for blade vibration estimation. BTT predominantly consists of two steps: 1) acquiring the original pulse signal generated by the rotating blade through optical probes. 2) Obtaining the arrival time of the original pulses through a high-precision counter and then transforming it to deflection. Multiple noise is involved in BTT measurement, which is further complicated by the variable operating environment owing to the complexity and multiplicity of blade vibration and the transmission path. With the introduction of prior knowledge, sparse representation has proved a promising tool for the reconstruction of blade vibration features. However, the classical sparse representation model applied in BTT, is mostly formulated and conducted based on the simple assumption that the noise follows a Gaussian distribution. The assumption, too idealized for real practices, restricts the performance promotion of sparse representation in BTT. To address this problem and to represent the unknown noise, a robust sparse representation model based on a mixture of Gaussians (MoG) is proposed in this work. The solution algorithm of the proposed model is then derived from the perspective of the expectation maximization (EM) algorithm. To validate the effectiveness of the present method, the performance of the developed methodology is discussed in terms of different regular items.

The harsh working environment of compressor threats the integrality of bladed disk. Crack is one of the most fatal faults of rotating blades. Detecting the crack of blades at early stage could ensure the safe operation of gas turbine and save economic costs. Blade tip timing (BTT) is a noncontact blade vibration measurement technique, which is widely used in recent years. In this article, a procedure of crack detection using BTT data is proposed. First, some typical parameters are taken into consideration, including resonance frequency, amplitude change of natural frequency, amplitude of blade tip, phase, and interblade spacing. It has been found that the frequency of the blade will decrease with the increment of crack length and reduction of crack height. In addition, the resonance frequency of the bladed disk will split when the blade cracks. The amplitude of different orders of resonance frequency varies with cracks. The amplitude of the first order changes very slightly, whereas the second and third orders are relatively obvious. The blade tip vibration amplitude and phase change with cracks without obvious regularity. The interblade distance will change due to the appearance of cracks. According to the above rules, the frequency and the interblade distance are adopted as crack diagnosis indicators. First, we use the Lomb–Scargle periodogram to quickly determine whether the crack appears and then use the blade frequency and interblade distance to locate the number of the blade having crack. In the experimental test, BTT data measured on a titanium bladed disk is used to verify the effectiveness of our method.

It is a challenging problem to extract periodic impulses submerged in the heavy background noise for fault diagnosis of rotating machinery. Thus, in this paper, we propose a novel algorithm named tunable Q-factor wavelet transform(TQWT)-based multi-scale dictionary learning for dealing with this problem. The algorithm exploits TQWT to decompose the measured vibration signal into different scales, and then it adopts K-SVD which can also be replaced with other more efficient dictionary learning algorithm to learn dictionaries at different scales. Once done, it employs a global maximum a posteriori estimator and inverse TQWT to extract feature signal. By comparison with TQWT-denoising and K-SVD-denoising, the proposed algorithm enjoys two main advantages: 1) the dictionaries learnt by our algorithm have the multi-scale characteristic which is essential to deal with non-stationary signal. 2) the dictionaries are learnt from noisy signals itself and thus are adaptive to different types of feature information. Effectiveness of our proposed algorithm is demonstrated by numerical simulation and fault diagnosis of motor bearing.

Rolling element bearing is an important component. As it is usually used in a complex environment, there are many failures occur on them. How to find the fault has become a pressing problem to be solved. The vibration signals generated by bearings are usually containing a variety of noise. The general diagnosis is divided into two stages: feature extraction and classification. Unlike conventional methods, there is no need to have a specific fault feature extraction step for sparse representation method. One of the dictionary learning methods which called the K-SVD is an algorithm does not need a defined dictionary but whose output is an over-complete dictionary studied by signals. The method that iteratively updating the K-SVD-trained dictionary based on the outcome of a linear classifier usually leads to the local minima. In order to train a dictionary that works well both in representation and classification, we use the Discriminative K-SVD which the labels are directly embedded in the dictionary learning step. Discriminant K-SVD can find all parameters of global optimum at the same time. The complexity of Discriminative K-SVD is related to that of K-SVD. Finally, the proposed method is applied in the practical bearing experiments, the results not only confirmed the accuracy of the proposed method for finding the fault types of bearings but also identified the damage degrees of bearings.