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 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.     

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.     

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.     

Rigid finite element model of a cracked rotor

The article introduces a new mathematical model for the cracked rotating shaft. The model is based on the rigid finite element (RFE) method, which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. In this article, the RFE method is extended and adopted for the modeling of rotating machines. An original concept of crack modeling utilizing the RFE method is developed. The crack is presented as a set of spring–damping elements of variable stiffness connecting two sections of the shaft. An alternative approach for approximating the breathing mechanism of the crack is introduced. The approach is simple and allows one to intuitively and systematically prepare and analyze the model of a cracked rotor.The proposed method is illustrated with numerical and experimental results. The experiments conducted for the uncracked free–free rotor as well as the numerical results obtained with other software confirm the accuracy of the RFE model. The numerical analysis conducted for a set of cracked rotors has shown that, depending on the eccentricity and its angular location, the breathing behavior of the crack may take different forms. In spite of this, the frequency spectra for different cracks are almost identical.Due to its simplicity and numerous advantages, the proposed approach may be useful for rotor crack detection, especially if methods utilizing the mathematical model of the rotor are applied.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

THE DYNAMIC STIFFNESS MATRIX METHOD IN FORCED VIBRATION ANALYSIS OF MULTIPLE-CRACKED BEAM

The dynamic behaviour of a beam with numerous transverse cracks is studied. Based on the equivalent rotational spring model of crack and the transfer matrix for beam, the dynamic stiffness matrix method has been developed for spectral analysis of forced vibration of a multiple cracked beam. As a particular case, when the excitation frequency is close to zero, the solution for static response of beam with an arbitrary number of cracks has been obtained exactly in an analytical form. In general case, the effect of crack number and depth on the dynamic response of beam was analyzed numerically.     

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.     

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.     

Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions

Vibration measurements offer an effective, inexpensive and fast means of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model with emphasis on changes in modal parameters and secondly, the breathing crack model focusing on nonlinear response characteristics. The open crack model based on linear response characteristics can identify the crack only at an advanced stage. Researchers have shown that a structure with a breathing crack behaves more like a nonlinear system, similar to that of a bilinear oscillator and the nonlinear response characteristics can very well be investigated to identify the presence of the crack. In the present study, the bilinear restoring force is approximated by a polynomial series and a nonlinear dynamic model of the cracked structure is developed using higher order frequency response functions. The effect of crack severity on the response harmonic amplitudes are investigated and a new procedure is suggested whereby the crack severity can be estimated through measurement of the first and second harmonic amplitudes.     

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.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh-Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys^©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.     

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.     

Nonlinear acoustic response through minute surface cracks: FEM simulation and experimentation

The second harmonic of a Rayleigh wave passing through a minute surface crack has been numerically analyzed by semi-explicit FEM including special elements which account for a nonlinear stress-strain relation at crack surfaces. Minute cracks perpendicular to a free, flat surface close under compressive stress when width of the crack opening is less than the longitudinal amplitude of the Rayleigh wave. Thereafter, compressive and shear stresses are partially transmitted through the closed cracks, whereas tensile and shear stresses are not transmitted through cracks that remain open. This leads to marked nonlinear ultrasonic response. Calculation was performed for an aluminum block having a surface crack. The transverse component of the Rayleigh wave propagating through the cracks shows distorted waveforms, making the second harmonic amplitude clearly noticeable. In an experiment, the second harmonic component of the leaky Rayleigh wave was detected for a simple crack model consisting of two aluminum blocks, by use of a PVDF line-focused transducer. The results of the experiment show that the second harmonic amplitude is a second-order function of the fundamental wave amplitude, and is more pronounced for low compressive stress applied to close the crack surfaces.     

Low-cycle fatigue small crack initiation and propagation behaviour of cast magnesium alloys based on in-situ SEM observations

In-situ observations on the initiation and propagation behaviour of low-cycle fatigue small cracks in cast magnesium–aluminium alloys (AM50 and AM60B) were carried out with scanning electron microscopy (SEM) to elucidate the resistance to fatigue cracking and to evaluate the fatigue small crack growth rate accurately and quantitatively. The results indicate that the fatigue small cracks formed preferentially on β-phase (Mg₁₇Al₁₂) boundaries at room temperature. In addition, the effects of the parameters of stress levels in low-cycle fatigue and temperatures as well as microstructure on fatigue small crack propagation behaviour are revealed. The variation of crack open displacement (COD) with stress levels and cycles at elevated temperature shows that it is unsuitable to estimate the fatigue small crack growth rate of cast magnesium alloys using conventional measurement methods such as the plastic-replica technique due to the obvious difference between microscopic cracks in the open and closed states. Stabilized crack propagation behaviour is limited to cases where the physical crack length is less than 1?mm in low-cycle fatigue.     

A method for determining the location and depth of cracks in double-cracked beams

The influence of two transverse open cracks on the antiresonances of a double cracked cantilever beam is investigated both analytically and experimentally. It is shown that there is a shift in the antiresonances of the cracked beam depending on the location and size of the cracks. These antiresonance changes, complementary with natural frequency changes, can be used as additional information carrier for crack identification in double cracked beams. Experimental results from tests on plexiglas beams damaged at different locations and different magnitudes are found to be in good agreement with theoretical predictions. Based on the results of the present work, an efficient prediction scheme for crack localization and characterization in double cracked beams is proposed.     

Dynamic caustics and its applications

For the study of elastodynamic problems of propagating cracks it is necessary to evaluate the dynamic stress intensity factor K^d_I which depends on the form of expressions for the stress components existing at the running crack tip at any instant of the propagation of the crack and the corresponding dynamic mechanical and optical properties of the material of the specimen under identical loading conditions. In this paper the distortion of the form of the corresponding reflected caustic from the lateral faces of a dynamically loaded transparent and optically inert specimen containing a transverse crack running under constant velocity was studied on the basis of complex potential elasticity theory and the influence of this form on the value of the dynamic stress intensity factor was given. The method was applied to the study of a propagating Mode I crack in a PMMA specimen under various propagation velocities and the corresponding dynamic stress intensity factor K^d_I evaluated. Also, crack propagation behaviour of notched composites in dynamic loading modes are reviewed and evaluated. A relatively large data base using metal-epoxy particulates, rubber-toughened poly(methyl methacrylate), and Sandwich plates are given. In all cases, a combination of high-speed photography and the optical method of dynamic caustics has been used. Results on the dynamic crack propagation mode, fracture toughness and crack propagation velocities of several rubber-modified composite models are presented. The composite models studied include specimens with one and/or two ‘complex’ two-stage inclusions, i.e. PMMA round inclusions surrounded by concentric rubber rings, one and/or press-fifting inclusions without rubber interface, all under dynamic loading. In all cases both qualitative and quantitative results were obtained. Also, results on crack propagation mode, crack propagation velocity, stress intensity factors and on the influence of the sandwich phases on crack propagation mode are presented.     

DYNAMIC RESPONSE OF A BEAM WITH A CRACK SUBJECT TO A MOVING MASS

An iterative modal analysis approach is developed to determine the effect of transverse cracks on the dynamic behavior of simply supported undamped Bernoulli-Euler beams subject to a moving mass. The presence of crack results in higher deflections and alters the beam response patterns. In particular, the largest deflection in the beam for a given speed takes longer to build up, and a discontinuity appears in the slope of the beam deflected shape at the crack location. Crack effects become more noticeable as crack depth increases. The effect of the inertia force due to the moving mass is, in general, qualitatively similar and additive to the effect of the crack. The exact effect of crack and mass depends on the speed, time, crack size, crack location, and the moving mass level. Other approximate methods, namely a stationary mass model and a single iteration technique, are also evaluated. The stationary mass approach is useful for light moving masses (<20% of beam mass) and cracks at mid-span. For other cases, the errors can be unacceptably large. The results of the single-iteration approximation are quite close to the iterative modal analysis approach, which indicates that this approximate solution is an excellent tool for the analysis of the moving mass problem.     

Image-based sizing of surface-breaking cracks by SH-wave array ultrasonic testing

This paper presents calibration-free crack sizing techniques based on ultrasonic imaging. The techniques are intended for 2D (line) surface-breaking cracks with the size of the incident wavelength or greater. The probing wave mode is the anti-plane shear wave (SH-wave). Two methods are employed for the ultrasonic imaging. One is a synthetic aperture focusing technique (SAFT) and the other is a computed time-reversal focusing technique (TRFT). In this paper, those methods are modified so that crack tips are located directly from measured A-scope waveforms without any calibration experiments. The results are shown as a peak in the ultrasonic image created by the respective methods. Reasonable accuracies of the proposed techniques are demonstrated first for the sizing of slits with known depths. The techniques are applied thereafter to the sizing of fatigue cracks. Since fatigue cracks may not be open without an external load, ultrasonic measurements are taken with and without external loads. The results of the imaging show that the depths of open cracks can be estimated accurately. It is also shown that crack opening (closing) behavior can be deduced by observing appearance (disappearance) of the peak in the images indicating the crack tip.