Publishing 10th International Conference on Vibrations in Rotating Machinery

11-13 September 2012, Imeche London, UK
1. Auflage 2012
ISBN: 978-0-85709-453-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 840 Seiten

This book presents the papers from the 10th International Conference on Vibrations in Rotating Machinery. This conference, first held in 1976, has defined and redefined the state-of-the-art in the many aspects of vibration encountered in rotating machinery. Distinguished by an excellent mix of industrial and academic participation achieved, these papers present the latest methods of theoretical, experimental and computational rotordynamics, alongside the current issues of concern in the further development of rotating machines. Topics are aimed at propelling forward the standards of excellence in the design and operation of rotating machines. - Presents latest methods of theoretical, experimental and computational rotordynamics - Covers current issues of concern in the further development of rotating machines

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Modelling, dynamic behaviour and diagnostics of cracked rotors
P. Pennacchi*,     Politecnico di Milano, Dept. of Mechanical Engineering, Italy ABSTRACT
One of the most common incipient losses of structural integrity in mechanical structures is the development and propagation of a crack. This happens also in rotating machinery and a very rich, but also in some way confusing, literature about cracked rotors has appeared in the last 30 years.In the paper, a general and wide overview about the behaviour of cracked rotors will be presented, covering several aspects of this topic. In particular modelling, dynamic behaviour and diagnostics of cracks will be analyzed in detail, introducing also practical examples and cases of industrial machinery, including some topics barely documented in literature such as helical and annular crack development. 1 INTRODUCTION
The propagation of shaft cracks is one of the most dangerous faults that can occur in rotating machinery [1]. If this fault is not early detected, it can cause serious damage as well as long outages and expensive maintenance actions. Transverse cracks that propagate in the shafts of rotating machines cause a change of the local flexural stiffness of the rotor. Moreover, in horizontal machine-trains, the shaft weight causes rotor bending that gives rise to tensile axial stresses, acting in the lower area of the faulty cross-section, which tend to open the crack. Conversely, the axial compressive stresses generated in upper area of the cracked cross-section tend to cause the contact between the crack surfaces. Therefore, during a complete revolution of the shaft, a transverse crack can be subjected to a breathing phenomenon, that is a periodic closure and opening of the crack. Transverse cracks can propagate in the cross-section of the shafts of rotating machines as a consequence of very high fatigue stresses. In general, the stress distribution in a rotor is rather complex, being the consequence of many concomitant static and dynamic loads. Owing to some of the most important of them, the cracked section is subjected to time-varying axial stresses generated by different causes, in operating condition, both at low and high rotational speeds. Some of these stresses are periodic as they depend on mechanical phenomena that are influenced by the shaft angular position. Other important axial stresses are aperiodic as they are caused by quasi-static loads. With regard to this, the stresses caused, for instance, by changes of the machine thermal condition can be very important. In general, these thermal transients occur within time intervals that are considerably longer than the revolution period of the shaft. Therefore, the corresponding axial stresses gradually increase, or decrease, during sufficiently long time intervals. Detailed analyses of thermal effects on breathing mechanism are presented in [1][2]. In the case of very thin cracks, having planar surfaces, the time-varying axial stresses acting on the cracked section of the shaft can cause a temporary opening, or closure, of the crack. Negative compressive stresses can cause contact between the fracture surfaces, whereas positive tensile stresses tend to open the crack. This phenomenon is often called crack breathing. 2 CRACK BREATHING MECHANISM
The breathing mechanism has been analysed in literature by several authors: Mayes and Davies [3] proposed the so called “switching crack”, in which the crack is either completely closed or open and passes abruptly from one state to the other. Nelson and Nataraj [4] proposed a switching criterion based on the curvature of the shaft, while Wauer [5] based his criterion on the sign of the total axial strain at extreme fiber of the cross section of the shaft. Mayes and Davies [6] improved their model by introducing a smoothing trigonometric function of the rotation angle of the section. However, all these approaches, along with similar ones [7][8], suppose that the area of the cracks that is in contact for various rotation angles is a priori known as observed by Andrieux and Varé [9]. Functions that rule the breathing of the crack are still used in the literature, as in Sinou and Lees [10], Patel and Darpe [11], Sawicki et al. [12], Ishida et al. [13][14] and Sinou and Faverjon [15]. A different approach was proposed by Dimarogonas and Papadopoulos [16] by considering shaft elasticity and stress intensity factors, with a typical approach of Solid Mechanics. Starting from that and by the introduction of the concept of crack closure line by Darpe et al. [17][18] and of other improvements by Papadopoulos [19]. This method has been widely used (see for instance Papadopoulos [20] and Sekhar [21][22][23]). However, some limitations of this approach have been highlighted, such as the limitation to consider cracks deeper than 50% of the diameter [19] or thermal stresses or crack closure effects [1]. Other approaches, able to manage the previously mentioned limitations have been proposed in [9][24] and [25]. However, owing to the interest in defining the true breathing behaviour of cracked shafts, some tests have been performed in the laboratory of the Dept. of Mechanical Engineering of Politecnico di Milano, as described in [1], to measure the strains in different points of a cracked shaft, with 70 mm diameter, under different load conditions. To this aim, a series of strain-gauges have been applied close to the crack and also directly across the crack lips (see figure 1). Figure 1 Detail of strain-gauge positions close and across the crack, from [1]. The horizontal cracked specimen was subjected to different stationary loads and rotated in different angular positions in order to excite the breathing of the crack. Two different non trivial effects have been observed: the crack closure effect and the local contact conditions of the crack lips in closed crack configuration, which are now discussed: a) Crack closure effect: Small loads, generating small bending moments in the locality of the crack, were not able to open the crack. The crack closure effect generates an internal bending moment that holds the crack closed. Only when the external bending moment overcomes the internal bending moment, then the lips of the crack start to open. b) Local contact conditions of the crack lips: When the crack is closed, with an external bending moment that sums up to the internal bending moment, the measured compressive strain is much higher than the theoretical strain calculated assuming a linear compressive stress distribution over the cracked section. This can be explained by assuming that the contact is not spread over all the cracked area when the crack is closed, but it occurs only on a smaller portion of the cracked surface, or on the crack lips only, determining higher strains associated also to stress intensity factors. This aspect is also related to the crack closure effect. Despite the fact that crack closure effects have been studied by several researchers (see for instance [26]), their influence, on the breathing behaviour of rotating shafts, has never been modelled suitably, to the author’s knowledge. Despite the highly non-linear stress and strain distribution in the cracked area and the non-linear breathing behaviour, the overall load-strain behaviour results are quite linear. The overall load vs. deflection law can therefore easily be represented by a linear model like the FLEX model presented in [1]. Cracked shaft diameter was larger than 70 mm when the crack had been initiated by means of a small notch generated by electro-erosion and had propagated roughly up to half way the shaft cross section by applying a constant bending load to the rotating shaft. The specimen was then machined and the diameter was reduced to the final one by turning, so that the initial slot was removed. The final cracked section has the shape shown in figure 2, as determined from ultrasonic test measurements. The cracked shaft has been clamped at one end but can be rotated around his axis by steps of 15° each. A vertical load has been applied at the other end and has been increased by steps. Theoretical stresses and strains are calculated assuming no crack. Figure 2 Shape of the crack and strain-gauge positions. Strain-gauges from A1 to A11 were applied each 15°, as close as possible to the crack. Strain-gauges from A12 to A15 were applied to an integer section, which was sufficiently far away from the crack to be not influenced by its breathing behaviour and their measurements are used as reference signals. Strain-gauges from A16 to A28 were applied at 15° in correspondence of the crack, but on the integer part opposite to the crack. Strain-gauges from B31 to B39 were applied across the lips, as it is shown in figure 2. Typical results are shown in figure 3 for measuring point A6, in which the maximum effect of the...