Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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.     

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.     

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.     

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.     

Interlaminar Crack Propagation Analysis of ENF Specimens Based on the Cohesive Zone Model北大核心CSCD

Based on the classical laminated plate theory and the cohesive zone model, a theoretical model for general delamination cracked laminates was established for crack propagation of pure mode Ⅱ ENF specimens. Compared with the conventional beam theory, the proposed model fully considered the softening process of the cohesive zone and introduced the nonlinear behavior of ENF specimens before failure. The predicted failure load is smaller than that under the beam theory and closer to the experimental data in literatures. Compared with the beam theory with only fracture toughness considered, the proposed model can simultaneously analyze the influences of the interface strength, the fracture toughness and the initial interface stiffness on the load-displacement curves in ENF tests. The results show that, the interface strength mainly affects the mechanical behavior of specimens before failure, but has no influence on crack propagation. The fracture toughness is the main parameter affecting crack propagation, and the initial interface stiffness only affects the linear elastic loading stage. The cohesive zone length increases with the fracture toughness and decreases with the interface strength. The effect of the interface strength on the cohesive zone length is more obvious than that of the fracture toughness. When the adhesive zone tip reaches the half length of the specimen, the adhesive zone length will decrease to a certain extent. Copyright ©2022 Applied Mathematics and Mechanics. All rights reserved.     

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.     

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．     

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.     

On the oil-whipping of a rotor-bearing system by a continuum model

The nonlinear dynamic behavior of a rotor-bearing system is analyzed based on a continuum model. The finite element method is adopted in the analysis. Emphasis is placed on the so-called “oil-whip phenomena” which might lead to the failure of the rotor system. The dynamic response of the system in unbalanced conditions is approached by a direct integration method. It is found that a typical “oil-whip phenomenon” is successfully simulated, and the effect of the refinement of the finite element mesh is also checked. Furthermore, the bifurcation behavior of the oil-whip phenomenon that is of much concern in recent nonlinear dynamics research is analyzed. The rotor-bearing system is also examined by a simple discrete model. Significant differences are found between these two models. It is suggested that a careful examination should be made in modeling the nonlinear dynamic behavior of a rotor system.     

半无穷大裂纹端部粘聚力分析

准脆性材料裂纹端部断裂过程区粘聚力是导致非线性断裂特性的重要原因,根据准脆性材料的断裂特性,对存在粘聚力分布的半无穷大裂纹力学分析模型,由变形叠加原理得到以该粘聚应力分布为未知函数的积分方程,通过对积分方程的分析推证,得到了该分布函数解的数学结构和级数型表达式;提出了由实际裂纹张开位移,确定裂纹端部粘聚力分布函数的两种方法:其一由连续的裂纹张开位移通过积分变换求解未知函数级数展开项的系数,其二是由离散的裂纹张开位移数据通过最小二乘法确定该函数;推导出了相应方法求解未知量的代数方程,并且给出了适当的算例和讨论。     

Simulation of crack growth using cohesive crack method

We present an effective cohesive discrete crack method in the context of the Reproducing Kernel Particle Method (RKPM) in order to study fracture of concrete structures. The discrete crack approach is based on the visibility method and a simple node splitting scheme. We also present an effective implementation of the visibility method and an iteration free algorithm by including the cohesive force term directly into the stiffness equations. The crack is represented by straight-line segments and the cohesive zone model is employed to model the post-localization behavior of concrete. The method is applied to several examples involving mode I and mixed-mode fracture. These results are compared to experimental data and show good agreement.     

A generalized cohesive element technique for arbitrary crack motion

A computational method for arbitrary crack motion through a finite element mesh, termed as the generalized cohesive element technique, is presented. In this method, an element with an internal discontinuity is replaced by two superimposed elements with a combination of original and imaginary nodes. Conventional cohesive zone modeling, limited to crack propagation along the edges of the elements, is extended to incorporate the intra-element mixed-mode crack propagation. Proposed numerical technique has been shown to be quite accurate, robust and mesh insensitive provided the cohesive zone ahead of the crack tip is resolved adequately. A series of numerical examples is presented to demonstrate the validity and applicability of the proposed method.     

Subcritical and critical states of a crack with failure zones

The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction, which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account.The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness. The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity.The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system.In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws.     

Finite Element Modelling of Cracks Based on the Partition of Unity Method

In this paper a finite element model for the analysis of brittle materials in the post cracking regime is presented. The model allows the representation of failure zones several times smaller than the structure itself using relatively coarse finite element meshes. The formulation is based on the partition of unity method. Discontinuous shape functions are used to enrich the continuous approximation of the displacement field where a crack has opened [2]. The magnitude of the displacement jump is determined by extra degrees of freedom at existing nodes. The crack path is completely independent of the structure of the mesh and is continuous across element boundaries. To model inelastic deformations around the crack tip a cohesive crack model is used. A representative numerical example illustrates the performance of the proposed model.     

Structural Crack Identification Based on the Variational Mode Decomposition北大核心CSCD

In order to enrich the bridge damage detection method and further improve the accuracy of bridge damage identification, a detection method for simply supported beams with cracks under dynamic loads was proposed not based on the complete finite element model. Under the premise of not blocking traffic, the method only needs to analyze and deal with the acceleration responses of the simply supported beam span, which reduces the mounting, dismounting and maintenance of sensors in practical engineering. At the same time, based on the model, an analytical formula of the acceleration at the midspan of the simply supported cracked beam was derived. Based on the theoretical derivation, the instantaneous energy and the mean energy difference were constructed through the variational mode decomposition and the Hilbert transform, and these 2 crack identification indexes were used to effectively identify small cracks with a crack depth ratio of only 5%. Then the influences of different wheel loads, environmental noises and damage degrees on detection results were studied. The results show that: ① the instantaneous frequency has a better recognition effect for crack positions; ② the mean energy difference is sensitive to crack depth ratio δ and the wheel load magnitude; ③ this method has strong noise robustness. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Investigation of the local behavior of different cohesive zone models

Cohesive interface elements are well suited for three-dimensional crack propagation analyses as long as the crack path is known. This is the case e.g. in delamination analyses of laminated composite structures or failure analyses of adhesively bonded joints. Actually, they are widely used in such applications for both brittle and ductile systems. As long as the strength and fracture toughness of the material are accurately captured it is generally accepted that the shape of the cohesive law has little to no influence on the mechanical behavior of the investigated structures. However, when having a look on the local behavior of different cohesive zone models, such as stress distribution in the fracture process zone, the results exhibit certain differences. These will be studied in the present contribution. Especially the local stress distribution will be investigated and the effect on the computational efficiency will be pointed out. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration analysis of cracked postbuckled plate

In this paper, a methodology is introduced to address the free vibration analysis of cracked plate subjected to a uniaxial inplane compressive load for the first time. The crack, assumed to be open and at the edge is modeled by a massless linear rotational spring. The governing differential equations are derived using the Mindlin theory, taking into account the effect of initial imperfection. The response is assumed to be consisting of static and dynamic parts. For the static part, differential equations are discretized using the differential quadrature element method and resulting nonlinear algebraic equations are solved by an arc-length strategy. Assuming small amplitude vibrations of the plate about its buckled state and exploiting the static solution in the linearized vibration equations, the dynamic equations are converted into a non-standard eigenvalue problem. Finally, natural frequencies and modal shapes of the cracked buckled plate are obtained by solving this eigenvalue problem. To ensure the validity of the suggested approach an experimental setup and a numerical finite element model have been made to analyze the vibration of a cracked square plate with simply supported boundary conditions. Also, several case-studies of cracked buckled plate problem have been solved utilizing the proposed method, and effects of selected parameters have been studied. The results show that the applied load and geometric imperfection as well as the position, size and depth of the crack have different impact on natural frequencies of the plate.     

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.