基于区间B样条小波有限元的转子裂纹定量识别

研究一种基于区间B样条小波有限元的转子横向裂纹定量识别方法.构造包含转动惯量影响的区间B样条小波Rayleigh梁单元,高精度求解裂纹转子前三阶固有频率,获得裂纹相对位置和相对深度作为变量的固有频率解曲面.然后将实测的裂纹转子前三阶固有频率作为裂纹识别问题的输入,利用三条等高线的交点定量识别出裂纹存在的相对位置和相对深度.数值仿真和试验研究结果表明,该方法鲁棒性强,单元数量少,辨识精度和效率高,为转子系统裂纹定量识别提供了新方法.     

Modified breathing mechanism model and phase waterfall plot diagnostic method for cracked rotors

Transverse breathing cracks are a primary damage mode in rotor systems and seriously influence the safety and reliability of equipment operation. The vibration characteristics exhibited by cracked rotors when passing through critical speeds serve as important evidence in the diagnosis of cracks, and a breathing mechanism model without weight dominance is needed to study these resonant characteristics. In this work, a restoring force modified model is proposed for studying the breathing mechanism of cracked Jeffcott rotors without weight dominance. Furthermore, a novel phase waterfall plot method that can identify frequency components with weak amplitudes is proposed to analyze the vibration response characteristics of cracked Jeffcott rotors. Numerical and experimental studies indicate that the phase waterfall plots effectively recognize the weak characteristic frequencies of cracked rotors. This study can also provide references for the crack monitoring of rotor systems.     

受弯梁中开裂纹的位置识别与分析

利用有限元计算判定受弯梁中开裂纹的位置 ,从中得出 :同正常梁相比 ,裂纹梁的固有频率与振型的变化不但与裂纹深度而且与裂纹位置有关 ,因而 ,通过裂纹梁低阶固有频率及振型的变化情况可以判定裂纹的位置。对于裂纹较浅的情况 ,直接利用振型与固有频率的变化很难判定裂纹的位置 ,必须借用一些特征参数来提高识别的敏感性 ,这样 ,裂纹梁中早期裂纹的识别也是可行的     

Rotor crack detection based on high-precision modal parameter identification method and wavelet finite element model

In this paper, a new method based on high-precision modal parameter identification method and wavelet finite element (WFE) model is presented to determine the depth and location of a transverse surface crack in a rotor system. The rotor system is modeled using finite element method of B-spline wavelet on the interval (FEM BSWI), while the crack is equivalent as a weightless rotational spring. Additionally, a novel method based on empirical mode decomposition (EMD) and Laplace wavelet is proposed to acquire modal parameters with high precision, which is implemented to improve the precision of crack identification. By providing the first three natural frequencies, contours for the specified natural frequency are plotted in the same coordinate, and the intersection of the three curves predicts the crack location and size. The experimental results indicate that the proposed method can accurately identify the position and depth of different cracks. The effectiveness and reliability of the proposed method is verified.     

基于区间B样条小波有限元的裂纹故障定量诊断

研究基于模型的结构裂纹故障诊断中的正反问题,即求解含裂纹参数结构的固有频率以及利用实测固有频率,定量识别裂纹参数。构造用于求解正问题的一维区间B样条小波裂纹单元,通过求解裂纹结构有限元模型,绘制以裂纹等效刚度与裂纹位置为变量的三阶频响函数解曲线,将实际测出的系统前三阶固有频率作为输入,根据曲线的交点定量预示出裂纹的位置和深度。实验研究表明,文中构造的区间B样条小波裂纹单元有效克服了传统有限元分析在求解裂纹奇异性问题时存在的效率低、精度差甚至难以收敛到正确解的缺陷,同时具有足够的辨识精度,为早期裂纹故障定量诊断提供新方法。     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

Crack detection in a beam with an arbitrary number of transverse cracks using genetic algorithms

In this paper, a crack detection approach is presented for detecting depth and location of cracks in beam-like structures. For this purpose, a new beam element with an arbitrary number of embedded transverse edge cracks, in arbitrary positions of beam element with any depth, is derived. The components of the stiffness matrix for the cracked element are computed using the conjugate beam concept and Betti’s theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain precise natural frequencies and mode shapes of the beam knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results and available experimental measurements. Moreover, by knowing the natural frequencies and mode shapes, an inverse problem is established in which the location and depth of cracks are determined. In the inverse approach, an optimization problem based on the new finite element and genetic algorithms (GAs) is solved to search the solution. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam. The proposed approach is verified through a cracked beam containing various cracks with different depths.     

Model-based identification of two cracks in a rotor system

The dynamics and diagnostics of cracked rotor have been gaining importance in recent years. Relatively few authors have addressed the problem of multi crack assessment for rotors. In the present study a model-based method is proposed for the on-line identification of two cracks in a rotor. The fault-induced change of the rotor system is taken into account by equivalent loads in the mathematical model. The equivalent loads are virtual forces and moments acting on the linear undamaged system to generate a dynamic behaviour identical to the measured one of the damaged system. The rotor has been modelled using finite element method, while the cracks are considered through local flexibility changes. The cracks have been identified for their depths and locations on the shaft. The nature and symptoms of the fault, that is crack, are ascertained using the FFT.     

Dynamic response of a cracked rotor with an unbalance influenced breathing mechanism

The vast majority of studies on cracked rotors assume that the breathing response of a fatigue crack is weight-dominant i.e. the effect of dynamic forces on the breathing response is negligible. In this study, the assumption of weight-dominance is removed and the coupling effect of unbalance angle and magnitude on the breathing behaviour of a crack is examined. The proposed breathing model is shown to be greatly influenced by unbalance eccentricity and rotor speed, whereas weight-dominant breathing models are unaffected by these factors. A significant difference in the vibration behaviour of a weight-dominant model and the proposed model was particularly seen around the critical speed of deeply cracked rotors. High unbalance eccentricity and a 180° placement of the unbalanced mass resulted in the disappearance of 2X and/or 3X harmonic components at one-half and one-third of the rotor critical speed when the vibration was predicted using the proposed model. This result suggests careful placement and size of the unbalance mass may allow for the isolation of rotor cracks from other rotor faults in the frequency domain by negating the effects of the crack breathing.     

Three-steps-meshing based multiple crack identification for structures and its experimental studies

Multiple crack identification plays an important role in vibration-based crack identification of structures. Traditional crack detection method of single crack is difficult to be used in multiple crack diagnosis. A three-step-meshing method for the multiple cracks identification in structures is presented. Firstly, the changes in natural frequency of a structure with various crack locations and depth are accurately obtained by means of wavelet finite element method, and then the damage coefficient method is used to determine the number and the region of cracks. Secondly, different regions in the cracked structure are divided into meshes with different scales, and then the small unit containing cracks in the damaged area is gradually located by iterative computation. Lastly, by finding the points of intersection of three frequency contour lines in the small unit, the crack location and depth are identified. In order to verify the effectiveness of the presented method, a multiple cracks identification experiment is carried out. The diagnostic tests on a cantilever beam under two working conditions show the accuracy of the proposed method: with a maximum error of crack location identification 2.7% and of depth identification 5.2%. The method is able to detect multiple crack of beam with less subdivision and higher precision, and can be developed as a multiple crack detection approach for complicated structures.     

Crack fault quantitative diagnosis based on finite element of B-spline wavelet on the interval

The model-based forward and inverse problems in the diagnosis of structural crack faults were studied. The forward problem is to solve the natural frequencies through a cracked structural model and the inverse problem is to quantitatively determine the crack parameters using the experimental testing frequencies. Then, the one-dimensional crack element of B-spline wavelet on the interval (BSWI) was built to solve the forward problem. Contour plots of normalized crack location versus normalized crack size were plotted by using the first three natural frequencies as the inputs. The intersection of the three curves predicted the normalized crack location and size. The experimental study verified the validity of the wavelet-based crack element in solving crack singular problems to overcome the disadvantages of the traditional finite element method (FEM), such as low efficiency, insufficient accuracy, slow convergence to correct solutions, etc. At the same time, it had adequate identification precision. The new method can be applied to prognosis and quantitative diagnosis of incipient crack. __________ Translated from Journal of Mechanical Strength, 2005, 27 (2) (in Chinese)     

含裂纹转子的振动特性计算用于裂纹在线检测的研究

本文研究含裂纹转子的振动特性。文章首先根据裂纹结构的应力强度因子，得到含裂纹单元体的刚度矩阵，进而得到裂纹转子的整体刚度矩阵和质量矩阵，列写出裂纹转子的运动微分方程，导出裂纹引起转子系统刚度变化与系统振动特性参数变化间的关系，从而可由实时在线测得的振动参数判断裂纹的位置。通过对模拟转子的计算和实测，表明本文提出的方法是一种可用于大型旋转机械转子轴裂纹在线检测的极为有效的方法．     

基于有限元位移模式的含裂纹梁结构动力学模型

针对裂纹的存在将降低梁的刚度的实际情形,首先根据断裂力学理论,引入裂纹梁因裂纹扩展而释放的应变能表达式,然后根据金属材料的特点,运用有限元位移法建立裂纹梁单元的动力学模型,再在梁单元模型的基础上应用有限元位移法建立裂纹梁结构的动力学方程。研究表明：基于有限元位移模式所建立的动力学方程较好地体现了裂纹梁动态性能与其结构参数和裂纹参数之间的内在关系,反映了裂纹的位置及长度对含裂纹梁结构动态性能的影响,为建立含裂纹梁结构动力学模型提供了一种新的有效方法。最后通过实例对理论分析结果进行了验证。     

Crack detection in beams using experimental modal data and finite element model

In this paper, an analytical, as well as experimental approach to the crack detection in cantilever beams by vibration analysis is established. An experimental setup is designed in which a cracked cantilever beam is excited by a hammer and the response is obtained using an accelerometer attached to the beam. To avoid non-linearity, it is assumed that the crack is always open. To identify the crack, contours of the normalized frequency in terms of the normalized crack depth and location are plotted. The intersection of contours with the constant modal natural frequency planes is used to relate the crack location and depth. A minimization approach is employed for identifying the cracked element within the cantilever beam. The proposed method is based on measured frequencies and mode shapes of the beam.     

含多条裂纹梁的模态与振动疲劳寿命分析

基于Paris公式,提出了一种含多条裂纹梁疲劳寿命预估的方法。在模态分析中,基于传递矩阵方法,利用无质量的弯曲弹簧等效裂纹,提出一种求解含有多条裂纹梁固有振型的方法,分析裂纹数目、裂纹位置、裂纹深度对裂纹梁固有频率的影响。在振动疲劳分析中,研究了在简谐激励作用下裂纹数目对裂纹尖端应力强度因子的影响。通过Paris疲劳裂纹扩展方程和同步分析法,考虑裂纹梁振动与裂纹扩展的相互作用,分析了裂纹数目和裂纹位置对裂纹梁疲劳寿命的影响。结果表明,裂纹数量、裂纹位置和深度对梁的模态参数和疲劳寿命有重要影响。     

Free vibration analysis of Euler-Bernoulli beam with double cracks

In this paper, the influence of two open cracks on the dynamic behavior of a double cracked simply supported beam is investigated both analytically and experimentally. The equation of motion is derived by using the Hamilton’s principle and analyzed by numerical method. The simply supported beam is modeled by the Euler-Bemoulli beam theory. The crack sections are represented by a local flexibility matrix connecting three undamaged beam segments. The influences of the crack depth and the position of each crack on the vibration mode and the natural frequencies of a simply supported beam are analytically clarified for the single and double cracked simply supported beam. The theoretical results are also validated by a comparison with experimental measurements.     

基于小波有限元技术的单螺杆压缩机螺杆轴的裂纹识别

为了能够有效地对单螺杆压缩机螺杆轴进行裂纹识别,以小波有限元技术为支撑,提出了单螺杆压缩机螺杆轴的裂纹识别方法.首先,结合有限元理论和Daubechies小波提出了适用于单螺杆压缩机螺杆轴裂纹识别的小波有限单元;然后,建立了单螺杆压缩机螺杆轴的振动小波有限元方程,可以对含有不同裂纹的螺杆轴进行固有频率的计算;接着,生成了裂纹识别数据库;最后,通过对4种含有不同裂纹的螺杆轴进行裂纹识别,验证了该方法的有效性.     

Crack localisation and sizing in a beam based on the free and forced response measurements

In the present paper, a method of the crack localisation and sizing in a beam from the free and forced response measurements is developed. The method gives crack flexibility coefficients as a by-product. Timoshenko beam theory is used in the beam modelling for transverse vibrations. The finite element method (FEM) is used for the cracked beam free and forced vibration analysis. An open transverse surface crack is considered for the crack model. The effect of the proportionate damping has been included. A harmonic imbalance force of known amplitude and frequency is used to dynamically excite the beam with the help of an independent exiting unit. The crack localisation and sizing algorithm is iterative in nature. The iteration starts with an initial guess for the crack depth ratio and iteratively estimates the crack location and the crack depth until getting the desired convergence for both the crack location and the crack depth. For estimation of bounded flexibility coefficients, a regularisation technique has been adopted. The method has been illustrated through numerical examples. The prediction of the crack location and size are in good agreement even in the presence of the measurement error and noise.     

Vibration analysis of non-uniform beams having multiple edge cracks along the beam's height

Bending vibration of non-uniform rectangular beams with multiple edge cracks along the beam's height is investigated. These cracks are called height-edge cracks in this paper. The energy based method is used for defining the vibration of height-edge cracked beams. The opening form of the height-edge crack is determined when the external moment is assumed to be applied for stretching the beam's width. Strain energy increase is obtained by calculating the strain change at the stretched surface by taking into account the effect of angular displacement of the beam due to the bending. The Rayleigh-Ritz approximation method is used in the analysis. The cases of multiple cracks are analysed in the method by using the approach based on the definition of strain disturbance variation along the beam. Examples are presented on a fixed-fixed beam and several cantilever beams having different taper factors. When the results are compared with the results of a commercial finite element program, good agreement is obtained. The effects of taper factors, boundaries and positions of cracks on the natural frequency ratios are presented in graphics.     

悬臂梁裂纹参数的识别方法

以梁振动理论作为基础 ,将含裂纹梁的振动问题转化为由弹性铰联接两个弹性梁系统的振动问题 ,得到理论计算含裂纹梁振动频率的特征方程。由此特征方程计算得到裂纹深度参数和位置参数变化时悬臂梁振动固有频率的变化规律。利用计算裂纹悬臂梁振动固有频率的特征方程 ,提出一种辩识裂纹深度和位置参数的数值计算方法。并通过对模拟悬臂梁裂纹的分析说明文中方法的有效性。