**1. Introduction**

314 Mechanical Engineering

Ozgumus, O. & Kaya, M. O. (2010). Vibration analysis of rotating tapered Timoshenko beam

Rossi R.E. (2007). *Introducción al análisis de Vibraciones con el Método de Elementos Finitos*.

Rossi, R. E.; Gutiérrez R. H. & Laura P. A. A. (1991). Transverse vibrations of a Timoshenko

Singh, A.P.; Mani, V. & Ganguli, R. (2007). Genetic programming metamodel for rotating

Shu, C. (2000). *Differential Quadrature and Its Application in Engineering*, Springer-Verlag,

Shu, C. & Chen, W. (1999). On optimal selection of interior points for applying discretized

Vinod, K. G., Gopalakrishnan, S. & Ganguli, R. (2007), Free vibration and wave propagation

Yang, J. B.; Jiang, L. J. & Chen, D. CH. (2004). Dynamic modelling and control of a rotating

*and Vibration*. Vol.222, No.2, pp. 239-257, ISSN 0022-460X

concentrated mass at the other. *J. Acoust. Soc. Am,* Vol.89, pp.2456-2458. Seon, M. H.; Benaroya, H. & Wei, T. (1999). Dynamics of transversely vibrating beams using

EdiUNS, Universidad Nacional del Sur, IBSN 978-987-1171-71-2, Bahía Blanca,

beam of nonuniform cross section elastically restrained at one end and carrying a

four engineering theories. *Journal of Sound and Vibration.* Vol.225, pp.35-988, ISSN

beams, *CMES - Computer modelling in Engineering and Sciences*, Vol.21. No.2, pp.

boundary conditions in DQ vibration analysis of beams and plates. *Journal of Sound* 

analysis of uniform and tapered rotating beams using spectrally formulated finite elements. *International Journal of Solids and Structures*; Vol.44, pp. 5875-5893, ISSN

Euler–Bernoulli beam. *Journal of Sound and Vibration*. Vol.274, pp. 863-875, ISSN

using DTM. Meccanica. Vol. 45, pp. 33-42, ISSN 0025-6455

Argentina.

0022-460X

133-148.

0020-7683

0022-460X

ISBN 1852332093, London, England

Of three general maintenance strategies – run-to-break, preventive maintenance and predictive maintenance – the latter, also referred to as condition-based maintenance, is becoming widely recognized as the most effective one (see e.g. Randall, 2011). To exploit its potential to the full, however, it has to be based on reliable condition assessment methods and procedures. This is particularly important for critical machines, characterized by high unit cost and serious consequences of a potential failure. Steam turbines provide here a good example.

In general, technical diagnostics may be defined as determining technical condition on the basis of objective methods and measures. The objectivity implies that technical condition assessment is based on measurable physical quantities. These quantities are sources of diagnostic symptoms. For any given class of objects, the development of technical diagnostics essentially involves four principal stages (Crocker, 2003), namely:


At the *measurement* stage we are able to measure physical quantities relevant to the object technical condition. On the basis of measurement data, at the *qualitative diagnostics* stage faults and malfunctions are identified and located with the aid of an appropriate diagnostic model. *Quantitative diagnostics* consists in estimating damage degree (advancement), for which a reference scale is necessary. Finally, *prognosis* is an estimation of the period remaining until an intervention is needed. Qualitative diagnostics may be viewed as being aimed at detecting hard (random) failures, while the aim of the quantitative diagnosis is to trace the soft (natural) fault evolution (Martin, 1994).

Complex objects, like steam turbines, are characterized by a number of residual processes (such as vibration, noise, heat radiation etc.) that accompany the basic process of energy transformation, and hence a number of condition symptom types. For all rotating machines, vibration-based symptoms are the most important ones for technical condition assessment, due to at least three reasons:


Vibration-Based Diagnostics of Steam Turbines 317

very low frequencies (a few hertz) may be indicative of cracks in turbine casings and other

Individual components from the blade frequency range are produced as a result of interaction between steam flow and the fluid-flow system, and hence may be considered specific to steam turbines. There are three basic phenomena involved (Orłowski, 2001;


First of these can be described in the following way: discharge edges of stationary and rotating blades introduce local interruptions of steam flow, thus reducing its thrust on a rotating blade and causing an instantaneous force of the opposite direction. Resulting force

respectively, *n* is number of blades in a stage (stationary or rotating) under consideration

for the second phenomenon, it results from the fact that manufacture of blades and their assembly into rotor stages or bladed diaphragms are not perfect, so for each blade the corresponding discharge cross-section is slightly different from the other ones. Resulting

The third phenomenon is related to turbine control and shall be dealt with a little later. It should be mentioned, however, that – unlike the first two – the influence of control valves opening is usually limited to the vicinity of the control stage and diminishes as we move along the steam expansion path. Frequencies of basic spectral components resulting from interaction between steam flow and the fluid-flow system can be, on the basis of above

where *l* and *b* denote numbers of blades in rotor stages and bladed diaphragms, respectively. Components given by Eqs.(5) and (6) result from interactions between rotor stages and adjacent bladed diaphragms. Each turbine stage is thus in general characterized

 = 2

force has a form of a pulse generated once per rotation and thus may be expressed by

0 + *k* cos *k*(*t* + 

denotes angular frequency. This force can thus be expressed as a series of harmonic

*t* + 

*<sup>k</sup>*) (1)

*<sup>k</sup>*) . (2)

*<sup>k</sup>* are amplitude and phase of the *k*-th component,

*knu*, where *u* is the rotational speed in s-1. As

 *u* (3)

 *u* (4)

 *u*/2 (5)

 *u*/2 (6)


0 + *k* cos *k*(*n*

*k* and 

non-rotating elements.

Orłowski and Gałka, 1998), namely:


*q*1 is thus periodic and can be expressed by

 *q*1 =

0 is time-averaged thrust,

components with frequencies equal to *kn*

 *q*2 =

 *fw* = *l* 

 *fk* = *b* 

 *f*(*w*+*<sup>k</sup>*)/2 = (*l* + *b*)

 *f*(*w*-*<sup>k</sup>*)/2 = (*l* - *b*)

by as many as six individual vibration components.

considerations, expressed by

where 

and 

Of all vibration-based symptom types (see e.g. Morel, 1992; Orłowski, 2001), three are of particular importance for steam turbine diagnostics:


These symptoms form the basis of diagnostic reasoning in both permanent (*on-line*) and intermittent (*off-line*) monitoring systems.
