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

The Okun Misery Index in the European Union Countries from 2000 to 2009

The study is composed of four main parts and a summary. The first part, introduction, discusses various measures of the economic system's efficiency that are used in practice. Part two emphasises that the GDP per capita according to purchasing power parity still remains the most popular among those measures. Further, it presents the ranking of the European Union countries taking that measure into account, the research period being 1999-2009. Part three points out that it is also the level of poverty (misery) that determines the economic system's efficiency. That level can be measured by means of various indicators, among others, the so called HPI-2 index calculated by the UN. It will be the Okun misery index, however, computed as the sum of inflation and unemployment rates that will be presented as an alternative being of interest from the macroeconomic point of view. The ranking of the European Union member states according to that measure in the 2000-2004 and 2005-2009 periods will be provided in part four. The article will end in a summary containing synthetic conclusions drawn from earlier observations.Opracowanie składa się z czterech części zasadniczych i podsumowania. W punkcie pierwszym omówiono różnorodne mierniki sprawności systemu gospodarczego wykorzystywane w praktyce. W części drugiej podkreślono, iż nadal najpopularniejszym z nich jest PKB per capita według parytetu siły nabywczej. Zgodnie z tym miernikiem przedstawiono ranking państw Unii Europejskiej w latach 1999-2009. W punkcie trzecim podkreślono, że o sprawności systemu gospodarczego decyduje także poziom ubóstwa. Może być on mierzony różnymi wskaźnikami, m.in. tzw. indeksem HPI-2 obliczanym przez ONZ. Jako ciekawą z makroekonomicznego punktu widzenia alternatywę ukazano jednak miarę wskaźnika ubóstwa Okuna obliczanego poprzez zsumowanie stopy inflacji i stopy bezrobocia. Ranking państw Unii Europejskiej według tej miary w okresach 2000-2004 oraz 2005-2009 zaprezentowano w części czwartej. Całość zamknięto podsumowaniem, w którym zawarto syntetyczne wnioski z przeprowadzonych obserwacji

Damage Evolution in Quasi-Brittle Materials: Experimental Analysis by AE and Numerical Simulation

This work investigates the extension of a total-collapse prediction method to include local failures in quasi-brittle materials as they undergo damage processes. The analysis is experimentally conducted with acoustic emission data from a basalt specimen under a prescribed displacement loading test. The proposed failure index is compared with the well-established b-value to evaluate its usefulness; the simulation results are also used to further investigations. In particular, the simulations show that the parameter calculation can be carried out by indirectly estimating the elastic energy released within the system throughout the damage process, which cannot be measured directly. It is concluded that the proposed method is valid, consistently outperforming the b-value as a failure precursor throughout the experimental studies

Crack Detectability in Vertical Axis Cooling Pumps During Operation

The problem which is faced in this paper is the analysis of the effects of a transverse propagating crack on the vibrational behaviour of a vertical axis cooling pump. The crack is assumed to develop in a section between the impeller and a seal, which prevents the hot water to flow upwards along the rotor shaft. The pressurized seal is fed with an injection of cold water. Crack initiation may be due to a thermal striping phenomenon. Afterwards, crack growth could be driven by a combination of thermal and mechanical loads, causing alternate cyclic stress in the shaft. Cracking instances of this type have been reported worldwide in several machines of similar design. In this paper, the fact is emphasized that the crack behavior is likely to be influenced by the thermal field and by the water pressure in the cracked area. A dynamical lineshaft model, integrated by an original representation of the crack, has been developed to investigate the possible vibratory symptoms related to a crack propagation. The vibrations are generally measured in correspondence of a rigid coupling which connects the motor shaft to the pump shaft, in position which is rather far away from crack. 1x rev., 2x rev and 3x rev. vibration components, which are generally displayed by the machine condition monitoring system and are the most significative symptoms of the presence of a transverse crack in a rotating shaft, are calculated

Truss-like Discrete Element Method Applied to Damage Process Simulation in Quasi-Brittle Materials

This paper discusses the combined application of the lattice discrete element method (LDEM) and the acoustic emission (AE) technique to analyze damage in quasi-brittle materials. These methods were used to study the damage in a concrete slab under pure-shear stress and a pre-fissured sandstone beam subjected to three-point bending. The first test was restricted to simulation results, whereas the second included experimental data. The discrete element method was used to perform the simulations for both tests, whereas the corresponding results and the information from the experiments were assessed using AE analysis tools. It was shown that the synergistic use of these two methods led to a comprehensive understanding of the two analyzed cases and offered an effective, generalizable approach for assessing damage processes in quasi-brittle materials