**2. Vibration generation and vibrodiagnostic symptoms**

Just like all rotating machines, steam turbines generate broadband vibration, so that power density spectra typically contain a number of distinct components. Due to different vibration generation mechanisms involved, it is convenient to divide the entire frequency range under consideration (typically from a few hertz up to some 10 to 20 kilohertz) into two sub-ranges, commonly referred to as the *harmonic* (or 'low') and *blade* (or 'high') frequency ranges, respectively. Sometimes the sub-harmonic range (below the fundamental frequency *f*0 resulting from rotational speed) is also distinguished. This division is shown schematically in Fig.1.

Fig. 1. Schematic representation of dividing the entire power density spectrum frequency range into sub-harmonic, harmonic and blade frequency ranges (after Gałka, 2009a).

Components from the harmonic range result directly from the rotary motion of turbine shaft and are related to malfunctions common to all rotating machines, such as:


Components that fall into the sub-harmonic range are typically determined mainly by the stability of the oil film in shaft bearings (Bently and Hatch, 2002; Kiciński, 2006). Those of

Of all vibration-based symptom types (see e.g. Morel, 1992; Orłowski, 2001), three are of

These symptoms form the basis of diagnostic reasoning in both permanent (*on-line*) and

Just like all rotating machines, steam turbines generate broadband vibration, so that power density spectra typically contain a number of distinct components. Due to different vibration generation mechanisms involved, it is convenient to divide the entire frequency range under consideration (typically from a few hertz up to some 10 to 20 kilohertz) into two sub-ranges, commonly referred to as the *harmonic* (or 'low') and *blade* (or 'high') frequency ranges, respectively. Sometimes the sub-harmonic range (below the fundamental frequency *f*0 resulting from rotational speed) is also distinguished. This division is shown

Fig. 1. Schematic representation of dividing the entire power density spectrum frequency range into sub-harmonic, harmonic and blade frequency ranges (after Gałka, 2009a).

and are related to malfunctions common to all rotating machines, such as:

Components from the harmonic range result directly from the rotary motion of turbine shaft

Components that fall into the sub-harmonic range are typically determined mainly by the stability of the oil film in shaft bearings (Bently and Hatch, 2002; Kiciński, 2006). Those of

particular importance for steam turbine diagnostics:

**2. Vibration generation and vibrodiagnostic symptoms** 


intermittent (*off-line*) monitoring systems.


schematically in Fig.1.




very low frequencies (a few hertz) may be indicative of cracks in turbine casings and other non-rotating elements.

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; Orłowski and Gałka, 1998), namely:


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

$$q\_1 = \zeta\_0 + \Sigma \zeta\_k \cos \ k (nat + \Psi\_k^\*) \tag{1}$$

where 0 is time-averaged thrust, *k* and *<sup>k</sup>* are amplitude and phase of the *k*-th component, respectively, *n* is number of blades in a stage (stationary or rotating) under consideration and denotes angular frequency. This force can thus be expressed as a series of harmonic components with frequencies equal to *kn* = 2*knu*, where *u* is the rotational speed in s-1. As 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 force has a form of a pulse generated once per rotation and thus may be expressed by

$$q\_2 = \zeta\_0 + \Sigma \zeta\_k \cos \ k (\alpha t + \Psi\_k') \,. \tag{2}$$

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 considerations, expressed by

$$f\_w = 1 \cdot u \tag{3}$$

$$f\_k = b \cdot u \tag{4}$$

$$f\_{(w\ast k)/2} = (1+b)\cdot u/2\tag{5}$$

$$f\_{\{\mathbf{u}\sim\mathbf{b}\}/2} = \left(l\mathbf{l}\cdot\mathbf{b}\right)\cdot\mathbf{u}/2\tag{6}$$

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 by as many as six individual vibration components.

2000)

efficiency. Assuming that

pressure. In fact, *ri* and *po* are the **R**(

cannot thus be a single-valued one.

**3. Qualitative diagnosis** 

Randall, 2011).

Vibration-Based Diagnostics of Steam Turbines 319

Let us assume that the influence of interference may be reduced to a point wherein it can be neglected. As control parameters are, at any given moment, known, there is obviously a possibility of symptom normalization with respect to them, either model-based or empirical. It has to be kept in mind, however, that normalization with respect to *Pu*, which seems most straightforward, in practice may be only approximate. *Pu* can be expressed as (Traupel,

controlled by changing *i* (qualitative control), *dm*/*dt* (quantitative control), or both. The latter method (known as group or nozzle control) is typically used in large steam turbines. Each control valve supplies steam to its own control stage section; the number of these valves in large steam turbines is usually from three to six and they are opened in a specific sequence. At the rated power the last valve is only partly open, or even almost closed, as it provides a reserve in a case of a sudden drop of steam parameters. Furthermore, *i* depends also on condenser vacuum, which for a given unit may change within certain limits depending on overall condenser condition, cooling water temperature, weather etc. Thus

 *Pu* = *f*(*r*1, *r2*, …, *rk*, *po*) , (11) where *ri* denotes *i*th valve opening, *k* is the number of valves and *po* is the condenser

may yield the same value of *Pu*. In view of Eqs.(9) and (11), any *Si*(*Pu*) function (*Si* **S**)

Some attention has been paid to developing experimental relations of the *Si* = *f*(*Pu*) type (see e.g. Gałka, 2001), bearing in mind that they are approximate and applicable to a given turbine type only. Such relations turn out to be strongly non-linear and differences between individual symptoms are considerable. In general, within the load range given by roughly *Pu* = (0.85 1.0)*Pn*, where *Pn* is rated power, variations are quite small; thus, when dealing with large sets of data, the simplest approach is to reject those acquired at extremely low or high loads. It has to be added that the fact of vibration-based symptoms dependence on control parameters and interference may serve as a basis for developing certain diagnostic

As already mentioned, qualitative diagnosis consists in determining what malfunctions or damages are present and localizing them. In this Section the influences of control and interference shall be neglected, i.e. it shall be assumed that symptoms under consideration are

For obvious reasons, the following review does not claim to be exhaustive and is concentrated on issues relevant to steam turbine applications. For comprehensive and detailed treatment the reader is referred e.g. to (Morel, 1992; Bently and Hatch, 2002;

*<sup>t</sup>* remains constant (which is only an approximation), *Pu* may be

) vector parameters, various combinations of which

*<sup>t</sup>* , (10)

*<sup>t</sup>* is the turbine

 *Pu = (dm*/*dt*)*i*

procedures; this issue shall be dealt with in Section 6.

deterministic functions of condition parameters *Xi* **X**.

where *dm*/*dt* denotes steam mass flow, *i* is the enthalpy drop and

Vibration signal that can be effectively measured in an accessible point of a turbine is influenced not only by relevant generation mechanisms, but also by its propagation to this point, as well as by operational parameters and interference (see e.g. Radkowski, 1995; Gałka, 2011b). In general terms it may be expressed as (Radkowski, 1995)

$$z(r,t) = h\_{\mathcal{V}}(r,t) \* u\_w(r,t) + \eta(r,t) \quad , \tag{7}$$

where *z* denotes measured diagnostic signal, *hp* is the response function for signal propagation from its origin to the measuring point and denotes uncorrelated noise; all these quantities are functions of the spatial variable *r* and dynamic time *t. uw*(*r*,*t*) is given by

$$\ln \mu\_w(r, t) = \sum\_{i=1}^{n} h\_i(r, D\_i, t) \* \ge (t) + h(r, t) \* \ge (t) \tag{8}$$

where *Di* describes development of the *i*th defect, *hi* is the response function pertaining to this defect and *h* is the response function with no defect present; *x*(*t*) is the input signal, generated by an elementary vibroacoustic signal source. This model is shown schematically in Fig.2.

Fig. 2. Model of vibroacoustic signal generation and propagation (after Radkowski, 1995).

An alternative general relation, in a vector form, is provided by (Orłowski, 2001)

$$\mathbf{S}(\boldsymbol{\theta}) = \mathbf{S}[\mathbf{X}(\boldsymbol{\theta}), \mathbf{R}(\boldsymbol{\theta}), \mathbf{Z}(\boldsymbol{\theta})] \; , \tag{9}$$

where **S**, **X**, **R** and **Z** denote vectors of symptoms, condition parameters, control parameters and interference, respectively, all varying with time .1 Control parameters may be defined as resulting from object operator purposeful action, aimed at obtaining demanded performance (Gałka, 2011b). In steam turbines, usually (at least in power industry) the 'demanded performance' means demanded output power; active load *Pu* can thus be treated as a scalar measure of the vector **R**. As for the interference, two types can be distinguished: *external interference* (the source is outside the object) and *internal interference* (the source is within the object). With some reservations, the former can be identified with measurement errors, while the latter refers to all other contributions to the uncorrelated noise (*t*) in Eq.(7).

<sup>1</sup> The reason for using *t* and symbols for denoting time is that the former refers to the 'dynamic' time (e.g. that enters equations of motion), while the latter is for the 'operational' time – the argument of equations pertaining to technical condition evolution.

Vibration signal that can be effectively measured in an accessible point of a turbine is influenced not only by relevant generation mechanisms, but also by its propagation to this point, as well as by operational parameters and interference (see e.g. Radkowski, 1995;

where *z* denotes measured diagnostic signal, *hp* is the response function for signal

these quantities are functions of the spatial variable *r* and dynamic time *t. uw*(*r*,*t*) is given by

*u rt h rD t xt hrt xt*

(,) (, ,) () (,) ()

where *Di* describes development of the *i*th defect, *hi* is the response function pertaining to this defect and *h* is the response function with no defect present; *x*(*t*) is the input signal, generated by an elementary vibroacoustic signal source. This model is shown schematically

Fig. 2. Model of vibroacoustic signal generation and propagation (after Radkowski, 1995).

where **S**, **X**, **R** and **Z** denote vectors of symptoms, condition parameters, control parameters and

resulting from object operator purposeful action, aimed at obtaining demanded performance (Gałka, 2011b). In steam turbines, usually (at least in power industry) the 'demanded performance' means demanded output power; active load *Pu* can thus be treated as a scalar measure of the vector **R**. As for the interference, two types can be distinguished: *external interference* (the source is outside the object) and *internal interference* (the source is within the object). With some reservations, the former can be identified with measurement errors, while the

(e.g. that enters equations of motion), while the latter is for the 'operational' time – the argument of

symbols for denoting time is that the former refers to the 'dynamic' time

(*t*) in Eq.(7).

An alternative general relation, in a vector form, is provided by (Orłowski, 2001)

) = **S**[**X**(), **R**(), **Z**(

 **S**(

1 The reason for using *t* and

interference, respectively, all varying with time

latter refers to all other contributions to the uncorrelated noise

equations pertaining to technical condition evolution.

, (8)

(*r*,*t*) , (7)

)] , (9)

.1 Control parameters may be defined as

denotes uncorrelated noise; all

Gałka, 2011b). In general terms it may be expressed as (Radkowski, 1995)

*n w ii i*

1

 *z*(*r*,*t*) = *hp*(*r*,*t*) *uw*(*r*,*t*) +

propagation from its origin to the measuring point and

in Fig.2.

Let us assume that the influence of interference may be reduced to a point wherein it can be neglected. As control parameters are, at any given moment, known, there is obviously a possibility of symptom normalization with respect to them, either model-based or empirical. It has to be kept in mind, however, that normalization with respect to *Pu*, which seems most straightforward, in practice may be only approximate. *Pu* can be expressed as (Traupel, 2000)

$$P\_u = (dm/dt)\Delta\text{i}\,\eta\_{\parallel} \tag{10}$$

where *dm*/*dt* denotes steam mass flow, *i* is the enthalpy drop and *<sup>t</sup>* is the turbine efficiency. Assuming that *<sup>t</sup>* remains constant (which is only an approximation), *Pu* may be controlled by changing *i* (qualitative control), *dm*/*dt* (quantitative control), or both. The latter method (known as group or nozzle control) is typically used in large steam turbines. Each control valve supplies steam to its own control stage section; the number of these valves in large steam turbines is usually from three to six and they are opened in a specific sequence. At the rated power the last valve is only partly open, or even almost closed, as it provides a reserve in a case of a sudden drop of steam parameters. Furthermore, *i* depends also on condenser vacuum, which for a given unit may change within certain limits depending on overall condenser condition, cooling water temperature, weather etc. Thus

$$P\_u = f(r\_{1\_\ell}, r\_{2\_\ell}, \dots, r\_{k\_\ell}, p\_0) \tag{11}$$

where *ri* denotes *i*th valve opening, *k* is the number of valves and *po* is the condenser pressure. In fact, *ri* and *po* are the **R**() vector parameters, various combinations of which may yield the same value of *Pu*. In view of Eqs.(9) and (11), any *Si*(*Pu*) function (*Si* **S**) cannot thus be a single-valued one.

Some attention has been paid to developing experimental relations of the *Si* = *f*(*Pu*) type (see e.g. Gałka, 2001), bearing in mind that they are approximate and applicable to a given turbine type only. Such relations turn out to be strongly non-linear and differences between individual symptoms are considerable. In general, within the load range given by roughly *Pu* = (0.85 1.0)*Pn*, where *Pn* is rated power, variations are quite small; thus, when dealing with large sets of data, the simplest approach is to reject those acquired at extremely low or high loads. It has to be added that the fact of vibration-based symptoms dependence on control parameters and interference may serve as a basis for developing certain diagnostic procedures; this issue shall be dealt with in Section 6.

#### **3. Qualitative diagnosis**

As already mentioned, qualitative diagnosis consists in determining what malfunctions or damages are present and localizing them. In this Section the influences of control and interference shall be neglected, i.e. it shall be assumed that symptoms under consideration are deterministic functions of condition parameters *Xi* **X**.

For obvious reasons, the following review does not claim to be exhaustive and is concentrated on issues relevant to steam turbine applications. For comprehensive and detailed treatment the reader is referred e.g. to (Morel, 1992; Bently and Hatch, 2002; Randall, 2011).

Vibration-Based Diagnostics of Steam Turbines 321

seen that balancing results in a considerable decrease of the 1 *f*0 component, but the improvement is only temporary. If this component is comparatively high at low rotational speed, coupling problem (offset rotor axles) is a possible root cause, especially in turbines

Ideally the entire turbine-generator unit shaft line (with overall length approaching 70 m in large units in nuclear power plants) should be a continuous and smooth curve; a departure from such condition is referred to as misalignment. The shape of this line is determined by shaft supports (journal bearings). As they displace during the transition from 'cold' to 'hot' condition, due to changing temperature field (this process may take even a few days to complete), at the assembly stage care has to be taken to ensure that the proper shape is maintained during normal operation. Relative vertical displacements may be even of the

Misalignment modifies distribution of load between individual shaft bearings and therefore affects shaft orbits. With increasing misalignment magnitude they typically evolve from elongated elliptical shape through bent ('banana') and finally to highly flattened one (Bently and Hatch, 2002). High misalignment may lead to oil film instability, but in large steam turbines (especially modern ones, with only one bearing per coupling) this is a very rare occurrence. As for absolute vibration, 2 *f*0 component in directions perpendicular to the turbine axis is generally recognized as the basic misalignment symptom. Care, however, has to be taken when dealing with the turbine-generator coupling, as this component may be dominated by the influence of the generator (asymmetric position of rotor with respect to the stator electromagnetic field); in the latter case, dependence on the excitation current is usually conclusive. Marked misalignment is often accompanied with relatively high amplitudes of harmonic components in axial direction, but this symptom can by no means



Permanent bow is obviously the most serious one. As it causes the center of gravity to move off from the shaft centerline, it basically produces an unbalance (cf. Fig.3). In general, rotor

*K M jD*

where *M* denotes unbalance mass, shifted at the distance *re* in the direction determined by

*Mr e*

2 <sup>2</sup> [ (1 ) ]

 

**r** , (12)

denotes rotor

*<sup>j</sup> <sup>j</sup> <sup>e</sup> <sup>e</sup>*

. *K* and *D* are stiffness and damping coefficients, respectively;

In general, three types of turbine rotor bow can be distinguished, namely:

response vector may be expressed as (Bently and Hatch, 2002):

*r e*

with rigidly coupled rotors.

order of millimeters (Gałka, 2009a).

**3.1.2 Misalignment** 

be considered specific.

properties, and


**3.1.3 Rotor bow** 

mation).

the angle
