Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady-state response of a geared rotor system with slant cracked shaft and time-varying mesh stiffness

The dynamic behavior of geared rotor system with defects is helpful for the failure diagnosis and state detecting of the system. Extensive efforts have been devoted to study the dynamic behaviors of geared systems with tooth root cracks. When surface cracks (especially for slant cracks) appear on the transmission shaft, the dynamic characteristics of the system have not gained sufficient attentions. Due to the parametric excitations induced by slant crack breathing and time-varying mesh stiffness, the steady-state response of the cracked geared rotor system differs distinctly from that of the uncracked system. Thus, utilizing the direct spectral method (DSM), the forced response spectra of a geared rotor system with slant cracked shaft and time-varying mesh stiffness under transmission error, unbalance force and torsional excitations are, respectively, obtained and discussed in detail. The effects of crack types (straight or slant crack) and crack depth on the forced response spectra of the system without and with torsional excitation are considered in the analysis. In addition, how the frequency response characteristics change after considering the crack is also investigated. It is shown that the torsional excitations have significant influence on the forced response spectra of slant cracked system. Sub-critical resonances are also found in the frequency response curves. The results could be used for shaft crack detection in geared rotor system.     

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.     

Implementation of a cohesive zone model for crack closure analysis of rotors

The presence of a crack in a rotor introduces a local flexibility which affects its dynamic response. Moreover, the crack may open and close during the vibration period. The crack status is a function of time and also depends on the rotational speed and the vibration amplitude of the rotor. This nonlinear case is still a challenging research topic especially in the field of closing crack in the rotating shaft. A cohesive zone model is developed in order to analyze the stiffness of a crack in a rotating shaft. The proposed expression will be compared to three different crack models, namely, a breathing crack model, a switching crack model and an open crack model. Moreover, a cohesive law to predict and to analyse the stress at the crack tip is presented. The numerical model is implemented using a finite element formulation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration analysis of rotating systems with open and breathing cracks

This paper is concerned with vibration analysis of rotating systems containing cracks. The flexibility matrix of cracked element is calculated with modified integration limits which is more accurate than conventional methods. The effect of this modification on the coefficients of flexibility matrix is presented for a simple rotor system containing open crack. To model the crack breathing behavior, a new finite element approach is introduced and implemented. Then, the dynamic response of a rotor with a breathing crack is evaluated by using the frequency/time domain approach (short time Fourier transform). The ability of short time Fourier transform to detect small cracks is investigated and compared with the transient response. The results provide a possible basis for an on-line monitoring system.     

Identification of crack in a rotor system based on wavelet finite element method

The dynamics and diagnosis of cracked rotor have been gaining importance in recent years. In the present study a model-based crack identification method is proposed for estimating crack location and size in shafts. The rotor system has been modeled using finite element method of B-spline wavelet on the interval (FEM BSWI), while the crack is considered through local stiffness change. Based on Rayleigh beam theory, the influences of rotatory inertia on the flexural vibrations of the rotor system are examined to construct BSWI Rayleigh beam element. The slender shaft and stiffness disc are modeled by BSWI Rayleigh–Euler beam element and BSWI Rayleigh–Timoshenko beam element, respectively. Then the crack identification forward and inverse problems are solved by using surface-fitting technique and contour-plotting method. The experimental examples are given to verify the validity of the BSWI beam element for crack identification in a rotor system. From experimental results, the new method can be applied to prognosis and quantitative diagnosis of crack in a rotor system.     

Stability analysis of an open cracked rotor with the anisotropic rotational damping in rotating operation

The damping effects with the distinction of stationary damping and the anisotropic rotating damping on the dynamic stability of the rotating rotor with an open crack on the surface of the shaft is studied. The motion equations of the cracked rotor system are formed by Lagranges principal. Different from previous studies, the anisotropic system with the multi periodical varied coefficients is simplified by the moving frame method such that the stability analysis based on the root locus method can be applied. The corresponding Campbell diagram, decay rate plot and roots locus plot are derived to prove the destabilizing influence of both the rotational damping and the varied anisotropy ratio of the rotating damping. The effects of anisotropy of stiffness on the decisions of the critical range are also presented. The result with theoretical precision would not only generally provide practical applicability to crack detection and instability control of the heavy loading turbo-machinery system, but also give the suggestion that, the increased proportion and the aggravated anisotropy of the rotational damping due to the crack of the fatigue rotor should been taken into consideration on the modeling of cracked rotor system.     

Nonlinear response and dynamic stability of a cracked rotor

Establishment of a new approach for analyzing the nonlinear behavior of a cracked rotor system is the main goal of the present research. Nonlinear governing equations of motion are developed for the cracked rotor system with asymmetrical viscoelastic supports. In establishing the approach, the masses of the rotational shaft and a disc mounted on the shaft, geometric nonlinearity of the shaft, and the rotor’s extra displacements due to the existence of the crack are all taken into account. On the basis of the governing equations, the nonlinear behavior of the rotor system is analyzed numerically with considerations of the effects of the crack depth, the crack location, the locations of the disc, and the shaft’s rotational speed. The effects of the crack and the other system parameters on the dynamic stability of the rotor system are also investigated.     

飞行器内裂纹转子系统的非线性动力学特性研究

在Jeffcott转子的开闭裂纹及方波模型基础上,建立了飞行器内裂纹转子系统的运动模型．数值研究表明:当飞行器以不同的等速度飞行时,转子轴与水平面之间夹角的变化将造成重力分量的变化,从而使转子运动在周期解、拟周期或浑沌状态之间变化,而且出现非线性现象的转速比、刚度变化比等参数的范围、进入和退出浑沌的路径、响应中的频率成份也会发生变化．飞行器的飞行速度变化还会改变裂纹转子响应的稳定性．飞行器等速飞行后的加速过程将引起转子振幅的突升及其后的下降,而且会使裂纹转子系统响应可能在不同的非线性状态下交替改变．     

Crack nucleation sensitivity analysis

A simple analytical expression for crack nucleation sensitivity analysis is proposed relying on the concept of topological derivative and applied within a two‐dimensional linear elastic fracture mechanics theory (LEFM). In particular, the topological asymptotic expansion of the total potential energy together with a Griffith‐type energy of an elastic cracked body is calculated. As a main result, we derive a crack nucleation criterion based on the topological derivative and a criterion for determining the direction of crack growth based on the topological gradient. The proposed methodology leads to an axiomatic approach of crack nucleation sensitivity analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Substructuring FE–XFE approaches applied to three-dimensional crack propagation

Two substructuring methods are investigated in order to allow for the use of the eXtended Finite Element Method (X-FEM) within commercial finite element (FE) codes without need for modifying their kernel. The global FE problem is decomposed into two subdomains, the safe domain and the cracked domain based on the value of the level sets representing the crack. The safe domain is treated by the host FE software while the cracked domain is treated by an independent XFE code. The first substructuring method consists of calculating the Schur matrix of a cracked super-element with the XFE code. The second technique introduces the finite element tearing and interconnecting method (FETI) which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains. The stiffness matrices and nodal forces are provided by the XFE and FE codes for the cracked and safe subdomains, respectively. The solutions obtained with these two techniques are rigorously equivalent to those computed with the stand-alone XFE code. First, the computational efficiency of the two approaches is demonstrated. Second, a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3D cracked geometries. Finally, this contribution presents an application of the FE–XFE–FETI method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure.     

Finite element method for the vibration of cracked beams with varying cross section

Vibration analysis of cracked beams having linearly varying cross-sections both in thickness and width was investigated. A computer program using the finite element method has been written to find the dynamical characteristics (natural frequencies and mode shapes) of the cracked beam. The cracked section in the beam has been modeled by a massless spring whose flexibility depens on the local flexibility induced by the crack. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expression. Some examples have been given to explain the proposed method and investigate the effects of the depth and location of cracks on the natural frequencies and mode shapes. The results of current study and those in the literature are compared and good agreements have been found. Consequently it is showed that proposed method is reliable and simple. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

集中力作用下的半平面边缘裂纹问题

讨论了半平面中的倾斜裂纹问题。集中力作用于裂纹表面上,或作用于开裂半平面的边界上。利用有理保角映像函数方法求解这个问题,同时得到了闭合形式的解。最后,给出了二个数值例子和计算结果。     

Dynamic analysis and experiment investigation of a cracked dual-disc bearing-rotor system based on orbit morphological characteristics

The paper is aimed to examine dynamic behaviors of a dual-disc bearing-rotor system in multi-fault state, and the crack detection based on the orbit morphological characteristics and vibration responses is proposed. Dynamic response and vibration signal analysis are two significant studies in rotor system. Most researchers have simulated the nonlinear dynamics and analyzed the fault signal using various methods separately. However, the fault feature from vibration signal is tightly connected with the dynamic mechanism in the rotor system, especially in rotor system with coupling multi-fault. In the paper, the dynamic model of the dual-disc bearing-rotor system is established, which takes into account the effects of crack, rub-impact and nonlinear oil-film forces. The vibration responses and the effect of crack on dual-disc rotor system with multi faults are investigated. The existence of crack and the coupling effect of multi faults enrich dynamic behavior of the dual-disc bearing-rotor system, and the response near the 1/2 subcritical speed provides a criterion for crack detection. Experiment investigation is attempted for the first time, which is based on the changes of crack depth and rotation speed for multi-fault dual-disc rotor system. The analysis of the dynamic response and the orbit morphological characteristics from experiment can effectively detect the crack information.     

考虑表面效应时孔边均布径向多裂纹Ⅲ型断裂力学分析

理论研究了纳米尺度孔边均布径向多裂纹的Ⅲ型断裂性能.基于Gurtin-Murdoch表面弹性理论和保角映射技术,获得了孔和裂纹应力场的解析解,给出了裂纹尖端应力强度因子的闭合解.基于解答分析了应力强度因子的尺寸效应,讨论了裂纹数量、裂纹/孔径比和缺陷表面性能对应力强度因子的影响.结果表明:当孔和裂纹尺寸在纳米量级时,无量纲应力强度因子具有显著的尺寸效应;应力强度因子随裂纹数量的变化规律受裂纹/孔径比的影响;裂纹/孔径比对应力强度因子的影响受到缺陷表面性能的制约,同时表面性能对应力强度因子的影响也受限于裂纹/孔径比;表面效应对应力强度因子的影响与裂纹数量无关.     

A study of the perturbed stress state of a closed cylindrical shell with a system of part-through transverse cracks

We study the elastic equilibrium of a closed infinite circular cylindrical shell with a system of surface cracks of identical length and depth. We use the method of singular integral equations together with the modeling of solid matter in the plane of a part-through crack by irregularly distributed “line springs”. We conduct a numerical analysis of the variation of the relative stress intensity factor at the center of a crack as a function of the parameters of a crack and the number of cracks. We study cracks located on both the interior and exterior surface of the shell. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 63–65.     

A BE based finite element for crack propagation processes

A boundary element based finite macro element for the simulation of 3D crack propagation in the framework of linear elastic fracture mechanics is presented. While the major part of the numerical model is discretized with finite elements, a small domain containing the crack is meshed with boundary elements. By means of the Symmetric Galerkin BEM a stiffness formulation for the cracked BE domain is obtained which enables a direct FEM/BEM coupling. All necessary operations for the crack propagation are carried out within this boundary element based finite macro element and exploit the potential of the boundary integral formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

偏心裂纹板在裂纹面受两对反平面点力的弹塑性解析解

采用线场分析方法对理想弹塑性材料偏心裂纹板在裂纹面受两对反平面点力的情形进行弹塑性分析,分析不受小范围屈服条件的限制,求得了裂纹线附近应力场和位移场的弹塑性解析解、裂纹线上的塑性区长度随外荷载的变化规律及有限宽板具有偏心裂纹的承载力.