The dynamic behaviour of heavy, horizontal axis, turbogenerator units 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 turbogenerator unit 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.
A very special case of damage in mechanical structures is the development and propagation of transverse cracks in rotating shafts. Transverse cracks are cracks in which the crack surface is orthogonal to the rotation axis of the shaft. The obvious difficulty in inspecting a rotating shaft during the operation of the machines makes the detection of cracks in these structures much more difficult than in static (non-rotating) structures. Therefore, symptoms are needed that can easily be measured (which are typically the vibrations) and that are able to indicate clearly the presence of a crack in a rotating shaft. If these symptoms arise in a machine, the machine can be stopped, the shaft can be removed and inspected with standard procedures and catastrophic failures of the complete set can be avoided. The accurate modelling of cracked shaft dynamical behaviour allows simulating the vibrations of a shaft affected by a transverse crack in different positions and with different depths. More in detail, the vibrations can be calculated in correspondence to the bearings of the machine, where they are measured in real machines, and its severity allows to predict the possibility of detecting the presence of the crack. A crack in rotating shafts is most likely to appear in correspondence to sharp changes of diameter or of the geometry of the shaft (presence of holes, slots for keys, threads and so on) in regions of high stress concentration.
Thermal stresses that develop in thermal machines, like steam turbines, and thermal shocks are also responsible for generating high local stress intensity factors that can cause the starting of a crack and its propagation.
In rotating shafts, the cracks propagate generally in a plane perpendicular to the shaft axis, when the axial bending stresses are prevailing, generating a transverse crack. The propagation velocity in a rotating shaft may change from case to case. Very frequently the crack moves by steps, alternating progresses to stops: both can be seen on the cracked surface pattern where rest lines called beach marks are recognizable. Generally, when the crack is approaching to a dangerous depth, it propagates more quickly, with a propagation velocity that increases exponentially, as can be deduced also by the so-called Paris law [1]. The final growth up to a critical dangerous depth takes sometimes only few days of operation. Cracks in power station and industrial plant machinery, steam turbines, generators and pumps have been discovered and documented in many European power plants as well as in the Far East and in the USA [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and [22]. They have generally been discovered by analysing the monitored vibrations and the machines have been stopped before the occurrence of a catastrophic failure, but in some cases the symptoms have not been recognized in time and the machines burst. As far as the authors know only one published paper [23] deals with the possibility of discovering from vibration measurements the presence of cracks in 900 MW turboset units, for a given position of the crack and for different crack depths and circumferential extensions.
In the following the effects of transverse cracks with rectilinear tips and different depths (in this case the circumferential extensions are defined directly from the depth) that have developed in any position of a shaft-line are analysed.
The excitation of vibrations in correspondence to the bearings of cracked turbogenerator units, where generally shaft vibrations are measured, is analysed. All the positions of the crack along the shaft-line are considered in order to evaluate the effects on the dynamic response at rated speed and during run-down transients.
Summarizing, the following results emerge from the analysis:
i
crack forces depend on depth and are proportional to the static bending moment divided by the third power of the shaft diameter;
ii
the excitation of maximum vibrations at a given rotating speed per unit crack force occurs when the corresponding mode shape presents high curvature in correspondence to the crack position;
iii
the maximum response occurs obviously in resonant conditions with 1X, 2X and 3X critical speeds.
Difficulties in detecting small cracks in rated speed conditions are highlighted. The vibration trend at the increasing crack depths is derived, and the advantages of considering run-down transients instead of rated speed conditions are emphasized. These results could contribute consistently to an assessment of crack-excited vibrations and to the possibilities of detecting cracks in turbogenerator units from standard vibration measurement and monitoring systems.
