A genetic algorithm based inverse problem approach was proposed by He et al. (2003) for the identification of pedestal looseness in the rotor-bearing system. The proposed approach considered the pedestal looseness identification as an inverse problem, and formulated this problem as a multi-parameter optimization problem by establishing a non-linear dynamics model of the rotor-bearing system with the pedestal looseness, and then utilizes a genetic algorithm to search for the solution. In addition, the non-linear dynamics model and the parameter sensitiveness of the system response were investigated. The responses of the system were obtained by using the fourth-order Runge-Kutta integration. The numerical experiments suggest that good identification of the pedestal looseness is possible and the proposed approach is feasible.
The complicated nonlinear phenomena of rotor system with pedestal looseness were analyzed by applying rotor dynamics and nonlinear dynamics theory by Li et al. (2005). By the bifurcation diagrams of the change of rotating speed, it was discovered that vibration of rotor system with pedestal looseness was violent in the sub-critical whirling speed. On the contrary, in the supercritical whirling speed, vibration of rotor was weak. But, under certain conditions, subharmonic resonance could occur and induce violent vibration. In addition, characteristics of vibration of the rotor with pedestal looseness were studied by frequency spectrums. The analysis result provided a theoretical reference for rotor fault diagnosis for rotating machinery.
Because of wrong setting or long-term running of rotating machinery, the looseness may occur in the bearing seats or bases. And also bring impact and rubbing of rotor-stator that is the looseness and rub-impact coupling fault. A mechanics model and a finite element model of a vertical dual-disk cantilever rotor- bearing system with coupling faults of looseness and rub-impact were set up by Lu et al. (2007). Based on the nonlinear finite element method and the contact theory, dynamical characteristics of the system under the influence of the looseness, and impact-rub clearance were studied. The results showed that the impact-rub of rotor-stator could reduce the low frequency vibration caused by looseness, and the impact-rub caused by looseness had obvious orientation. Also, the conclusion of diagnosing the looseness and rub-impact coupling faults was given at the end of the paper.
15.6 Shaft Cracks
The presence of various flaws (such as cracks, notches, slits etc.) in any structures and machineries may lead to catastrophic failures. They are particularly likely to occur in instances where shaft stresses are high and where machine has endured many operation cycles throughout its life, so that material failure has occurred as a consequence of fatigue. If the shaft cracks can be detected before catastrophic failure occurs then the machine can be temporarily taken out of service and repaired before situation gets out of hand. The presence of a transverse shaft crack sometimes is detected by monitoring changes in vibration characteristics of the machine. The shaft stiffness at the location of the crack is reduced, by an amount depending on the crack size. This in turn affects the machine natural frequencies, so that changes in natural frequencies may be symptomatic of a shaft crack. Therefore, it demands the detection and the diagnostic (i.e. its localization and sizing) of such flaws so that corrective action can be taken well before it grows critical. Such detection and diagnostic techniques should be practicable in terms of taking experimental measurements. But unfortunately the changes in natural frequency may not occur until the crack has reached a dangerously large size. For this reason most users of rotating machinery depend upon changes in vibration amplitudes, phase and frequency spectrum to detect shaft cracks, rather than on changes in natural frequency.
A transverse crack (breathing crack model) results in significant changes in both 1 x and 2 x rotational speed vibration components. The 1 x rotational speed component may change in both amplitude (which may either grow or decay) and phase, as a consequence of the change in rotor bending stiffness. A transverse crack also results in a bigger rotor bow due to a steady load (for example gravity) which may be detected under ‘slow roll’ conditions. The classical symptom of a cracked shaft is the occurrence of a vibration component at 2 x shaft rotational frequency as a consequence of the shaft asymmetry at the crack location in the presence of a steady load. The 2 x rotational frequency component is usually prevalent when the machine is running at a critical speed or ½ x any critical speed (similar to sub-critical speed due to gravity). Other types of crack, in particular those which originate from the center of the shaft, may not cause a change in the shaft bending stiffness which is significant enough to enable the crack to be detected by 2 x rotational speed components in the vibration spectrum.
There are plenty of literatures that deal with the crack (or the flaw) modeling, free and forced vibrations analysis of cracked beams, and detection, localization and severity estimation of cracks. Wauer (1990) presented a review of literatures in the field of dynamics of cracked rotors, including the modeling of the cracked part of structures and determination of different detection procedures to diagnose fracture damages. The review formed a basis for analysing dynamics of cracked beams and columns (i.e. non-rotating, cracked structural elements which is relevant to the cracked rotor problems). Gasch (1993) provided a survey of the stability behaviour of a rotating shaft with a crack and of forced vibrations due to imbalances. Dimarogonas (1996) reviewed the analytical, numerical and experimental investigations on the detection of structural flaws based on changes in dynamic characteristics. Salawu (1997) reviewed the use of natural frequencies as a diagnostic parameter in the structural assessment procedures using the vibration monitoring. The relationship between frequency changes and structural damages were discussed. Various methods proposed for detecting damages by using natural frequencies were reviewed. Factors (e.g. choice of measuring points, effects of ambient conditions on dynamic responses, and consistency and reliability of testing procedures, etc.) that could limit successful application of the vibration monitoring for the damage detection and structural assessment were also discussed.
Plethora of crack detection and diagnostic methods is available in literatures based on feature extractions of the free and forced responses, which becomes very complicated and difficult to use in practice. Doebling et al. (1998) provided an overview of methods to detect, locate, and characterize damages in the structural and mechanical systems by examining changes in measured vibration responses. The scope of this paper was limited to methods that use changes in modal properties (i.e. modal frequencies, modal damping ratios, and mode shapes) to infer changes in mechanical properties. The review included both methods that were based solely on changes in the measured data as well as those methods that used the finite element model (FEM) in the formulation. Doebling et al. (1998) classified the vibration based methods into various categories. The methods are broadly based on the linear and nonlinear effects of damage of structures. Another classification system given by Rytter (1993) for damage-identification methods defined four levels of damage identification, as follows: (i) Level 1: Determination of the presence of damage in the structure (ii) Level 2: Level 1 plus determination of the geometric location of the damage (iii) Level 3: Level 2 plus quantification of the severity of the damage and (iv) Level 4: Level 3 plus prediction of the remaining service life of the structure. Vibration-based damage identification methods that do not make use of some structural model primarily provide Level 1 and Level 2 damage identification. When vibration-based methods were coupled with a structural model, Level 3 damage identification could be obtained in some cases. Level 4 prediction was generally associated with the fields of fracture mechanics, fatigue-life analysis, or structural design assessment.
Sabnavis et al. (2004) presented a review of literatures published since 1990 and some classical papers on the crack detection and severity estimation in shafts. The review was based on three categories namely vibration-based methods, modal testing and non-traditional methods. They also discussed the types and causes of rotor cracks and fundamentals of the crack propagation.
In addition to aforementioned reviews, several authors contributed to the development of ‘cracked element’, analysis of cracked beams and crack detection using changes in natural frequencies. Dimarogonas and Paipetis (1983) showed a beam with a transverse crack, in general, can be modeled in the vicinity of the crack by way of a local flexibility (compliance) matrix, connecting the longitudinal force, bending moment and shear force and corresponding displacements. If torsion is also added, a 6×6 compliance matrix may result and it has off-diagonal terms that indicate coupling of the respective forces and displacements, and therefore coupling of respective motions. This matrix is a diagonal matrix in the absence of crack for the symmetrical beam cross-section. Gounaris and Dimarogonas (1988) developed the Euler–Bernoulli beam cracked element based on the fracture mechanics approach. Theoretically, coefficients of the compliance matrix are computed based on available expressions of the stress intensity factor (SIF) and associated expressions of the strain energy density function (SEDF) by using the linear fracture mechanics approach. Fernandez-saez et al. (1999) obtained an expression for the closed form fundamental natural frequency of a cracked Euler-Bernoulli beam by applying the Rayleigh method. This approach was applied to simply supported beams with a rectangular cross-section having a crack in any location of the beam span. Behzad and Bastami (2004) investigated the effect of axial forces on changes in natural frequencies of shafts.
Lee et al. (1992) proposed a switching crack model with two different stiffness states, depending upon whether the crack is open or close. The necessary conditions for the crack opening and closing were analytically derived from the simple rotor with the switching crack.