Deflections and Strains in Cracked Shafts due to Rotating Loads: A Numerical and Experimental Analysis

In this article the deflections of a circular cross-section beam presenting a transverse crack of varying depths caused by various loads (bending, torsion, shear, and axial loads) are analyzed with the aid of a rather refined three-dimensional model that takes into account the nonlinear contact forces in the cracked area. The bending and shear loads are applied in several different angular positions in order to simulate a rotating load on a fixed beam or, by changing the reference system, a fixed load on a rotating beam. Torsion and axial loads are fixed with respect to the beam.Results obtained for the rotating beam can then be used for the analysis of cracked horizontal-axis heavy rotors in which the torsion is combined with the bending load. The effect of friction is also considered in the cracked area. The characteristic "breathing" behavior of the cracked area was analyzed and compared to that obtained with a rather simple one-dimensional model. The differences in results with respect to those based on fracture mechanics are emphasized. In order to highlight the effect of the presence of the crack, the deflections of the uncracked beam loaded with the same loads were subtracted from the deflections of the cracked beam.Finally, a cracked specimen was extensively analyzed by means of several strain gauges to study the strain distribution on the outer surface around the crack in various loading conditions. Consistent pre-stresses were found, and they influence the breathing behavior. The experimental results were compared with those obtained using the onedimensional linear model

Light and short arc rubs in rotating machines: Experimental tests and modelling

Rotor-to-stator rub is a non-linear phenomenon which has been analyzed many times in rotordynamics literature, but very often these studies are devoted simply to highlight non-linearities, using very simple rotors, rather than to present reliable models. However, rotor-to-stator rub is actually one of the most common faults during the operation of rotating machinery. The frequency of its occurrence is increasing due to the trend of reducing the radial clearance between the seal and the rotor in modern turbine units, pumps and compressors in order to increase efficiency. Often the rub occurs between rotor and seals and the analysis of the phenomenon cannot set aside the consideration of the different relative stiffness. This paper presents some experimental results obtained by means of a test rig in which rub conditions of real machines are reproduced. In particular shortarcrubs are considered and the shaft is stiffer than the obstacle. Then a model, suitable to be employed for real rotating machinery, is presented and the simulations obtained are compared with the experimental results. The model is able to reproduce the behaviour of the test rig

A sensitivity analysis of vibrations in cracked turbogenerator units versus crack position and depth

The dynamic behaviour of heavy, horizontal axis, turbogeneratorunits affected by transverse cracks can be analysed in the frequency domain by means of a quasi linear approach, using a simplified breathing crack model applied to a traditional finite element model of the shaft-line. This allows to perform a series of analyses with affordable computational efforts. Modal analysis combined to a simplified approach for simulating the dynamical behaviour allows to predict the severity of the crack-excited vibrations, resolving the old-age question on how deep a crack must be to be detected by means of vibration measurements of the machine during normal operating conditions. The model of a 320 MW turbogeneratorunit has been used to perform a numerical sensitivity analysis, in which the vibrations of the shaft-line, and more in detail the vibrations of the shaft in correspondence to the bearings, have been calculated for all possible positions of the crack along the shaft-line, and for several different values of the depth of the crack

Cracked Rotors, A Survey on Static and Dynamic Behaviour Including Modelling and Diagnosis

Cracks can develop in rotating shafts and can propagate to relevant depths without affecting consistently the normal operating conditions of the shaft. In order to avoid catastrophic failures, accurate vibration analyses have to be performed for crack detection. The identification of the crack location and depth is possible by means of a model based diagnostic approach, provided that the model of the crack and the model of the cracked shaft dynamical behavior are accurate and reliable. This monograph shows the typical dynamical behavior of cracked shafts and presents tests for detecting cracks. The book describes how to model cracks, how to simulate the dynamical behavior of cracked shaft, and compares the corresponding numerical with experimental results. All effects of cracks on the vibrations of rotating shafts are analyzed, and some results of a numerical sensitivity analysis of the vibrations to the presence and severity of the crack are shown. Finally the book describes some crack identification procedures and shows some results in model based crack identification in position and depth. The book is useful for higher university courses in mechanical and energetic engineering, but also for skilled technical people employed in power generation industries

Transverse crack modelling and validation in rotor systems including thermal effects

In this paper a model is described that allows to simulate the static behaviour of a transversal crack in a horizontal rotor, under the action of the weight and other possible static loads and the dynamical behaviour of the rotating cracked shaft. The crack “breaths”, i.e. the mechanism of opening and closing of the crack is ruled by the stress acting on the cracked section due to the external loads; in a rotor the stress are time depending with a period equal to the period of rotation, thus the crack “periodically breaths”. An original simplified model is described that allows cracks of different shape to be modelled and thermal stresses to be taken into account, since they may influence the opening and closing mechanism. The proposed method has been validated using two criteria. At first the crack “breathing” mechanism, simulated with the model has been compared with the results obtained by a non-linear 3D FEM calculation and a good agreement in the results has been observed. Then, the proposed model allows the development of the equivalent cracked beam. The results of this model are compared with those obtained by the above said 3D FEM. Also in this case, there is a good agreement in the results. Therefore the proposed crack model and equivalent beam model can be inserted in the finite beam element model used for the rotor dynamical behavior simulation: the obtained equations have time depending coefficients, but they can be integrated in the frequency domain by using the harmonic balance method. The model is suitable for finite beam elements with 6 degrees of freedom per node, in order to account also for torsion vibrations and coupling between torsion and flexural vibrations

Identification of transverse crack position and depth in rotor systems

This paper introduces a method for the identification of the position and the depth of a transverse crack in a rotor system, by using vibration measurements. As it is reported in literature and from field experience, a transverse crack modifies the dynamic behaviour of the rotor, generating in a horizontal axis shaft periodical vibrations with 1x, 2x and 3x rev. components. A model based diagnostic approach and a least squares identification method in the frequency domain are used for the crack localisation along the rotor. The crack depth is calculated by comparing the static bending moment, due to the rotor weight and to the bearing alignment conditions, to the identified “equivalent” periodical bending moment, which simulates the crack. Finally, the validation of the proposed method is carried out statically and dynamically by means of experimental results obtained on a test rig

Observation of subdiffusion of a disordered interacting system

We study the transport dynamics of matter-waves in the presence of disorder and nonlinearity. An atomic Bose-Einstein condensate that is localized in a quasiperiodic lattice in the absence of atom-atom interaction shows instead a slow expansion with a subdiffusive behavior when a controlled repulsive interaction is added. The measured features of the subdiffusion are compared to numerical simulations and a heuristic model. The observations confirm the nature of subdiffusion as interaction-assisted hopping between localized states and highlight a role of the spatial correlation of the disorder.Comment: 8 pages, to be published on Physical Review Letter