**3.2 Blade (high) frequency range**

So-called blade spectral components, with frequencies given by Eqs.(3) to (6), are usually low in amplitude. Typically they fall into the frequency range from a few hundred hertz to about 1020 kilohertz. In vibration displacement spectra they are undistinguishable, so velocity or acceleration spectra have to be employed. Constant-percentage bandwidth (CPB) analysis is the most convenient tool; 23% CPB yields satisfactory results.

Technical condition of the individual fluid-flow system components, i.e. rotor stages and bladed diaphragms, influences the *<sup>k</sup>* coefficients in Eqs.(1) and (2) and hence the vibration amplitudes in relevant frequency bands. Blade components are, however, highly sensitive to control and interference. Influence of control may be seen as a competition between two mechanisms. First, with nozzle control typical for large steam turbines, there is an asymmetry of steam pressure distribution over the turbine cross-section that depends on the control valve opening. This asymmetry affects forces resulting from the steam flow thrust, again via the *k* coefficients. As turbine load increases and consecutive valves are opening, pressure distribution becomes more uniform. Second, with increasing turbine load and steam mass flow, the 0 coefficient also increases. As already mentioned, it may be expected that the former mechanism shall influence vibration patterns at points close to the control stage, as the asymmetry decreases as we move downstream the steam expansion path. The latter should be noticeable for last low-pressure turbine stages, with long blades and large cross-section area. In practice, influence of steam flow asymmetry on blade components is quite strong in points located at the high-pressure turbine; operation at extremely low loads2 may cause them to increase even by a few times. Steam mass flow influence is usually much weaker.

<sup>2</sup> Load minimum is usually imposed by the steam generator (boiler or nuclear reactor) stable operation considerations.

Unbalance 1*f*0 component in vertical and horizontal directions, constant

amplitude and phase, decreasing at low rotational speed

2 *f*0 component in vertical and horizontal directions, 'bananashaped' or flattened shaft orbits, high harmonic components in axial

1 *f*0 component in vertical and horizontal direction (also at low rotational speed), strong correlation between 1 *f*0 components in

Continuous changes of 1 *f*0 and 2 *f*0 components amplitudes and phases during steady-state operation, reduction of critical speeds and increase of vibration amplitudes on passing through them

*<sup>k</sup>* coefficients in Eqs.(1) and (2) and hence the vibration

*k*

Increase of sub-harmonic components (typically slightly below 0.5 *f*0), relative vibration increase, shaft orbits with loops, high and unstable amplitudes of higher harmonic components, sensitive to

Malfunction Typical symptoms

vertical and axial directions,

Table 1. Typical steam turbine malfunctions and their representation in low-frequency

So-called blade spectral components, with frequencies given by Eqs.(3) to (6), are usually low in amplitude. Typically they fall into the frequency range from a few hundred hertz to about 1020 kilohertz. In vibration displacement spectra they are undistinguishable, so velocity or acceleration spectra have to be employed. Constant-percentage bandwidth (CPB)

Technical condition of the individual fluid-flow system components, i.e. rotor stages and

amplitudes in relevant frequency bands. Blade components are, however, highly sensitive to control and interference. Influence of control may be seen as a competition between two mechanisms. First, with nozzle control typical for large steam turbines, there is an asymmetry of steam pressure distribution over the turbine cross-section that depends on the control valve opening. This asymmetry affects forces resulting from the steam flow thrust, again via the

coefficients. As turbine load increases and consecutive valves are opening, pressure distribution becomes more uniform. Second, with increasing turbine load and steam mass

former mechanism shall influence vibration patterns at points close to the control stage, as the asymmetry decreases as we move downstream the steam expansion path. The latter should be noticeable for last low-pressure turbine stages, with long blades and large cross-section area. In practice, influence of steam flow asymmetry on blade components is quite strong in points located at the high-pressure turbine; operation at extremely low loads2 may cause them to

2 Load minimum is usually imposed by the steam generator (boiler or nuclear reactor) stable operation

increase even by a few times. Steam mass flow influence is usually much weaker.

0 coefficient also increases. As already mentioned, it may be expected that the

bearing oil pressure

analysis is the most convenient tool; 23% CPB yields satisfactory results.

direction

Misalignment

Permanent rotor bow

Rotor crack

Bearing problems

vibration-based symptoms.

**3.2 Blade (high) frequency range** 

bladed diaphragms, influences the

flow, the

considerations.

Fig.5 shows relative standard deviation (/*Ŝ*, where *Ŝ* denotes mean value) plotted against mid-frequency of 23% CPB spectrum bands, determined for a 120 MW steam turbine. It is immediately seen that for the harmonic range /*Ŝ* is below 0.1, while in the blade range it may be as high as about 0.6 to 0.8. Similar analysis for other turbine types has yielded quantitatively comparable results (Gałka, 2011b). In such circumstances, a time history of a blade spectral component has to be considered a monotonic curve with large fluctuations imposed; an example is shown in Fig.6. Therefore the very occurrence of a high amplitude cannot be unanimously considered as indicative of a fluid-flow system failure. From the point of view of measurement data processing, values heavily influenced by control and/or interference have to be treated as outliers.

Fig. 5. Relative standard deviation vs. frequency: results for a 120 MW unit, low-pressure turbine casing rear/left side, horizontal direction; data obtained from 90 consecutive measurements (after Gałka, 2011b).

Fig. 6. Time history of the 2500 Hz component: 200 MW unit, low-pressure turbine casing front/right side, vertical direction

 = 

values of

function of

In both cases,

necessary repair.

/

> :

while Fréchet operator yields:

Analytically this may be expressed as

Pearson, 1933). In this particular case, it yields

usually yield *Sl* values differing just by a few percent.

Vibration-Based Diagnostics of Steam Turbines 327

be added that they conform to all relevant requirements (in particular, vertical asymptote at

*S S*

0 <sup>1</sup> ( ) ln 1 /

*S S <sup>b</sup>*

<sup>0</sup> ( ) ( ln / )

is the shape factor to be determined empirically and *S*0 = *S*(

In order to determine *Sl*, the concept of symptom reliability is introduced. Symptom reliability *R*(*S*) is defined (Cempel, Natke and Yao, 2000) as the probability that a machine classified as being in good condition (*S* < *Sl*) will remain in operation with the symptom value *S* < *Sbr*, where *Sbr* denotes value corresponding to breakdown. This may be written as

 *R*(*S*) P(*Sbr* > *S* | *S* < *Sl*) . (17)

*S R S p S dS* ( ) ( \*) \* 

where *p*(*S*) denotes the symptom probability density function. Determination of the limit value must involve some measure of acceptable operational risk. This may be accomplished by using the Neyman-Pearson rule, known from statistical decision theory (Neyman and

*l*

where *G* denotes the availability of the machine (or group of machines) and *A* is the acceptable probability of erroneous condition classification as 'faulty', i.e. performing an un-

For a given symptom operator, *p*(*S*) may be estimated from experimental data, providing that the available database is sufficiently large. In practice (Gałka, 1999) about 100 individual data points will allow for a reasonable estimation. Weibull and Fréchet operators

A set of limit values should be considered specific for a given turbine example; experience has shown that generalization of results over the entire type should be avoided. It has to be kept in mind that an overhaul often results in a considerable modification of vibration characteristics. This refers mainly to harmonic components, which are sensitive even to minor repairs or adjustments, while blade components are typically influenced only by

*S R S G G p S dS A* ( ) () 

*l*

 

*<sup>b</sup>*), while some other operators (e.g. Pareto or exponential) are valid only for small

 

*<sup>b</sup>*. Weibull operator results in the following expression for a symptom as a

*b*

1/

. (16)

. (18)

, (19)

1/

, (15)

= 0).

It has to be noted that in steam turbines there are sources other than the fluid-flow system that generate vibration components with frequencies in the same range. Typically this is the case with oil pump and governor, driven from turbine shaft via gears. If unexpectedly high amplitudes are encountered, additional narrow-band analysis provides conclusive data, as frequencies of these components may be easily calculated.

### **4. Quantitative diagnosis**

In short, qualitative diagnosis provides an answer to the question 'what', while quantitative diagnosis is expected to tell 'how much'. This problem is becoming particularly important when a turbine is operated beyond its design lifetime, which is by no means uncommon. It has to be kept in mind that many turbines still in operation had been designed a few dozen years ago, with much less knowledge of lifetime consumption mechanisms and therefore larger safety margins. Quantitative diagnosis is obviously mandatory if condition-based maintenance is to be introduced.

By necessity, for a quantitative condition assessment a reference scale of some kind has to be used. Such scale may be provided by three values: basic, limit and admissible. Basic value *Sb* corresponds to a new object with no malfunctions or faults present. Limit value *Sl* may be considered as determining the 'normal' operation range: if *S* > *Sl*, further operation is still possible, but the machine cannot fulfill all requirements (concerning e.g. reliability, economy, output, environmental impact etc.). Admissible value *Sa* is determined from safety considerations: *S* > *Sa* indicates high possibility of imminent breakdown and should result in machine tripping.

As *Sa* is in practice irrelevant to technical diagnostics and *Sb* may be determined in a rather straightforward manner, the *Sl* estimation is fundamental for quantitative diagnostics. A complex machine is characterized by a large number of symptoms, and obviously each of them may be assigned its specific limit value. An approach to this estimation is provided by the Energy Processor model and the concept of symptom reliability (for a comprehensive and detailed treatment, see Natke and Cempel, 1997). This approach is based on the fact that any energy-transforming object is a source of residual processes, such as vibration, noise, thermal radiation etc. The power of these processes *V* can be shown to depend on the object condition. In the simplest case the relation is given by

$$V = V\_0 \left(1 - \frac{\theta}{\theta\_b}\right)^{-1},\tag{14}$$

where *V*0 = *V*( = 0) and *<sup>b</sup>* denotes time to breakdown, determined by the time-invariant properties of the object. As Eq.(14) has been derived with quite restrictive assumptions, several modifications have been proposed, applicable for various types of diagnostic objects (see e.g. Gałka and Tabaszewski, 2011); they inevitably result in considerable complication of the mathematical description.

In practice *V* is usually non-measurable and accessible only via measurable symptoms. A symptom is related to *V* by so-called *symptom operator* . Several types of symptom operators have been proposed (see e.g. Natke and Cempel, 1997). In steam turbine applications, Weibull and Fréchet operators have been found particularly appropriate; it also has to be added that they conform to all relevant requirements (in particular, vertical asymptote at = *<sup>b</sup>*), while some other operators (e.g. Pareto or exponential) are valid only for small values of /*<sup>b</sup>*. Weibull operator results in the following expression for a symptom as a function of :

$$S(\theta) = S\_0 \left( \ln \frac{1}{1 - \theta \, / \, \theta\_b} \right)^{1/\gamma},\tag{15}$$

while Fréchet operator yields:

326 Mechanical Engineering

It has to be noted that in steam turbines there are sources other than the fluid-flow system that generate vibration components with frequencies in the same range. Typically this is the case with oil pump and governor, driven from turbine shaft via gears. If unexpectedly high amplitudes are encountered, additional narrow-band analysis provides conclusive data, as

In short, qualitative diagnosis provides an answer to the question 'what', while quantitative diagnosis is expected to tell 'how much'. This problem is becoming particularly important when a turbine is operated beyond its design lifetime, which is by no means uncommon. It has to be kept in mind that many turbines still in operation had been designed a few dozen years ago, with much less knowledge of lifetime consumption mechanisms and therefore larger safety margins. Quantitative diagnosis is obviously mandatory if condition-based

By necessity, for a quantitative condition assessment a reference scale of some kind has to be used. Such scale may be provided by three values: basic, limit and admissible. Basic value *Sb* corresponds to a new object with no malfunctions or faults present. Limit value *Sl* may be considered as determining the 'normal' operation range: if *S* > *Sl*, further operation is still possible, but the machine cannot fulfill all requirements (concerning e.g. reliability, economy, output, environmental impact etc.). Admissible value *Sa* is determined from safety considerations: *S* > *Sa* indicates high possibility of imminent breakdown and should result in

As *Sa* is in practice irrelevant to technical diagnostics and *Sb* may be determined in a rather straightforward manner, the *Sl* estimation is fundamental for quantitative diagnostics. A complex machine is characterized by a large number of symptoms, and obviously each of them may be assigned its specific limit value. An approach to this estimation is provided by the Energy Processor model and the concept of symptom reliability (for a comprehensive and detailed treatment, see Natke and Cempel, 1997). This approach is based on the fact that any energy-transforming object is a source of residual processes, such as vibration, noise, thermal radiation etc. The power of these processes *V* can be shown to depend on the object

*b*

1

*<sup>b</sup>* denotes time to breakdown, determined by the time-invariant

, (14)

. Several types of symptom

*V V*

<sup>0</sup> 1

 

properties of the object. As Eq.(14) has been derived with quite restrictive assumptions, several modifications have been proposed, applicable for various types of diagnostic objects (see e.g. Gałka and Tabaszewski, 2011); they inevitably result in considerable complication

In practice *V* is usually non-measurable and accessible only via measurable symptoms. A

operators have been proposed (see e.g. Natke and Cempel, 1997). In steam turbine applications, Weibull and Fréchet operators have been found particularly appropriate; it also has to

frequencies of these components may be easily calculated.

condition. In the simplest case the relation is given by

symptom is related to *V* by so-called *symptom operator*

**4. Quantitative diagnosis** 

maintenance is to be introduced.

machine tripping.

where *V*0 = *V*(

of the mathematical description.

= 0) and

$$S(\theta) = S\_0 (-\ln \theta \,/\, \theta\_b)^{-1/\gamma}. \tag{16}$$

In both cases, is the shape factor to be determined empirically and *S*0 = *S*(= 0).

In order to determine *Sl*, the concept of symptom reliability is introduced. Symptom reliability *R*(*S*) is defined (Cempel, Natke and Yao, 2000) as the probability that a machine classified as being in good condition (*S* < *Sl*) will remain in operation with the symptom value *S* < *Sbr*, where *Sbr* denotes value corresponding to breakdown. This may be written as

$$R(S) \equiv \mathcal{P}(S\_{\mathcal{H}} \ge S \mid S \le S\_{\mathcal{H}}) \quad . \tag{17}$$

Analytically this may be expressed as

$$R(S) = \bigcap\_{S}^{\circ} p(S^{\*})dS^{\*}\,. \tag{18}$$

where *p*(*S*) denotes the symptom probability density function. Determination of the limit value must involve some measure of acceptable operational risk. This may be accomplished by using the Neyman-Pearson rule, known from statistical decision theory (Neyman and Pearson, 1933). In this particular case, it yields

$$R(S\_l) \cdot G = G \bigcap\_{S\_l}^{\circ} p(S)dS = A \quad , \tag{19}$$

where *G* denotes the availability of the machine (or group of machines) and *A* is the acceptable probability of erroneous condition classification as 'faulty', i.e. performing an unnecessary repair.

For a given symptom operator, *p*(*S*) may be estimated from experimental data, providing that the available database is sufficiently large. In practice (Gałka, 1999) about 100 individual data points will allow for a reasonable estimation. Weibull and Fréchet operators usually yield *Sl* values differing just by a few percent.

A set of limit values should be considered specific for a given turbine example; experience has shown that generalization of results over the entire type should be avoided. It has to be kept in mind that an overhaul often results in a considerable modification of vibration characteristics. This refers mainly to harmonic components, which are sensitive even to minor repairs or adjustments, while blade components are typically influenced only by

Vibration-Based Diagnostics of Steam Turbines 329

type and on the turbine element involved, so the primary idea was to employ this approach in qualitative diagnostics. General guidelines for steam turbines are given in Table 2 (after

Vibration evolution assessment is, however, far more important for a quantitative diagnosis. If we limit our attention to Weibull and Fréchet operators, we may expand relevant ex-

> /

*SS A* <sup>0</sup> () 1

*b*

*<sup>b</sup>*. The constant *A* depends on the symptom operator and is given by

(23)

over 24 h deformation of casings

a few hours to a few days variations of natural

a few days to a few months deformation of casings

a few hours to a few weeks material creep effects

a period of a few seconds flutter, problems with control

(24)

minutes rubbing in labyrinth seals

 

*a a* 1/ 1 1 1 ln

a few minutes to a few hours

a few minutes to a few hours

Stepwise (discontinuous) a few seconds damage of blades,

a few to a few dozen

Cyclic or nearly cyclic variable soft rubbing in seals

Table 2. Failures and damages of steam turbines revealed in vibration evolution parameters

*A a a*

for the Fréchet operator. We may thus infer that, if lifetime consumption (given by

Rapid random bearing instability,

<sup>1</sup> (1/ 1) ( ln ) 

small, symptom time history will be well approximated by a straight line and its slope will

(1 ) 1

Evolution type Timescale Failure

*<sup>b</sup>* = *a*, wherein 0 < *a* << 1 (for mathematical

deformation of rotors, thermal unbalance (temporary)

thermal unbalance (temporary)

frequencies

and/or foundations

cracks of rotor elements

steam flow instability

/*<sup>b</sup>*) is

, (22)

) in the form of

Orłowski, 2001).

pressions for *S*(

which is valid for

 << 

Simple (linear or nearly linear increase)

Complex (usually fluctuations superimposed on an increasing trend)

> Exponential or nearly exponential

(after Orłowski, 2001)

for the Weibull operator and

) into Taylor series around

reasons, *a* = 0 is unacceptable). Truncating higher-order terms, we obtain *S*(

*A*

major overhauls that involve opening of turbine casings. Formally such overhaul is equivalent to creating a new object. Normalization of the influence of overhauls (which determine machine life cycles) is quite straightforward if *S*0 values are available, which is usually the case.

### **5. Evolutionary symptoms**

Insofar attention has been focused on vibration characteristics recorded at some given moment . Diagnostic information is obviously also contained in symptom time histories. Although state-of-the-art vibration monitoring systems facilitate so-called trending, i.e. plotting of *S* against , this is seldom used for diagnostic purposes. It has to be mentioned that this refers to steady-state operation data, not transients (startups or shutdowns). In general, any quantity pertaining to the *S*() time history may be evaluated in terms of diagnostic reasoning and treated as a symptom itself.

Time histories of vibration components, especially in the blade frequency range, are usually quite irregular. As already mentioned, symptom time history may be considered a monotonic trend with superimposed fluctuations resulting from control and interference (cf. Eq.(9)). If a fault develops fast and strongly influences vibration patterns, this trend is clearly visible (see Fig.3). On the other hand, if condition evolution is slow, it may be suppressed by control and interference to a point where it is barely distinguishable. The latter is often the case for the blade frequency range. Various data smoothing procedures have been proposed to extract the monotonic trend, including three-point averaging, wherein *k*th symptom reading *S*(*k*) is replaced with *S*(*<sup>k</sup>*) given by:

$$S\_i{}^{\prime}(\theta\_k) = \frac{1}{3} [S\_i(\theta\_{k-1}) + S\_i(\theta\_k) + S\_i(\theta\_{k+1})] \,. \tag{20}$$

Another option is peak trimming, which in fact consists in eliminating isolated outliers. This method is based on the assumption that if

$$S(\theta)/S(\theta\_{k^{\cdot}}) \succeq \mathfrak{c} \text{ and } S(\theta\_{k})/S(\theta\_{k^{\cdot}1}) \succeq \mathfrak{c},\tag{21}$$

then the *S*(*k*) value is suspicious and treated as an outlier; in such cases, *S*(*<sup>k</sup>*) is replaced by *S*(*k*) = [*S*(*k*-1) + *S*(*<sup>k</sup>*+1)]/2. For steam turbines *c* = 1.5 is reasonable.

Six basic types of vibration evolution can be distinguished for rotating machines in general (Morel, 1992), namely:


Moreover, each type is characterized by a 'timescale' ranging within broad limits, from seconds to years. Both evolution type and timescale depend on the malfunction or damage

major overhauls that involve opening of turbine casings. Formally such overhaul is equivalent to creating a new object. Normalization of the influence of overhauls (which determine machine life cycles) is quite straightforward if *S*0 values are available, which is

Insofar attention has been focused on vibration characteristics recorded at some given

Although state-of-the-art vibration monitoring systems facilitate so-called trending, i.e.

that this refers to steady-state operation data, not transients (startups or shutdowns). In

Time histories of vibration components, especially in the blade frequency range, are usually quite irregular. As already mentioned, symptom time history may be considered a monotonic trend with superimposed fluctuations resulting from control and interference (cf. Eq.(9)). If a fault develops fast and strongly influences vibration patterns, this trend is clearly visible (see Fig.3). On the other hand, if condition evolution is slow, it may be suppressed by control and interference to a point where it is barely distinguishable. The latter is often the case for the blade frequency range. Various data smoothing procedures have been proposed to extract the monotonic trend, including three-point averaging,

*k*) is replaced with *S*(

*S S SS i k ik ik ik* 1 1 <sup>1</sup> '( ) [ ( ) ( ) ( )] <sup>3</sup>

*k*-1) > *c* and *S*(

*k*) value is suspicious and treated as an outlier; in such cases, *S*(

*<sup>k</sup>*+1)]/2. For steam turbines *c* = 1.5 is reasonable.

Six basic types of vibration evolution can be distinguished for rotating machines in general


Moreover, each type is characterized by a 'timescale' ranging within broad limits, from seconds to years. Both evolution type and timescale depend on the malfunction or damage

Another option is peak trimming, which in fact consists in eliminating isolated outliers. This

 

*k*)/*S*(

  . Diagnostic information is obviously also contained in symptom time histories.

, this is seldom used for diagnostic purposes. It has to be mentioned

*<sup>k</sup>*) given by:

. (20)

*<sup>k</sup>*+1) > *c* , (21)

*<sup>k</sup>*) is replaced by

) time history may be evaluated in terms of

usually the case.

plotting of *S* against

moment

**5. Evolutionary symptoms** 

wherein *k*th symptom reading *S*(

 *S*(

(Morel, 1992), namely:

increasing curve),



then the *S*(

*S*(*k*) = [*S*(*k*-1) + *S*(

method is based on the assumption that if



general, any quantity pertaining to the *S*(

diagnostic reasoning and treated as a symptom itself.

*k*)/*S*( type and on the turbine element involved, so the primary idea was to employ this approach in qualitative diagnostics. General guidelines for steam turbines are given in Table 2 (after Orłowski, 2001).

Vibration evolution assessment is, however, far more important for a quantitative diagnosis. If we limit our attention to Weibull and Fréchet operators, we may expand relevant expressions for *S*() into Taylor series around /*<sup>b</sup>* = *a*, wherein 0 < *a* << 1 (for mathematical reasons, *a* = 0 is unacceptable). Truncating higher-order terms, we obtain *S*() in the form of

$$S(\theta) \cong S\_0 \left( 1 + A \frac{\theta}{\theta\_b} \right) \, \tag{22}$$

which is valid for << *<sup>b</sup>*. The constant *A* depends on the symptom operator and is given by


$$A = \frac{1}{\gamma(1-a)} \left( \ln \frac{1}{1-a} \right)^{1/\gamma - 1} \tag{23}$$

Table 2. Failures and damages of steam turbines revealed in vibration evolution parameters (after Orłowski, 2001)

for the Weibull operator and

$$A = \frac{1}{\gamma a} (-\ln a)^{-(1/\gamma + 1)}\tag{24}$$

for the Fréchet operator. We may thus infer that, if lifetime consumption (given by /*<sup>b</sup>*) is small, symptom time history will be well approximated by a straight line and its slope will

Vibration-Based Diagnostics of Steam Turbines 331

(cf. Fig.3). It has to be kept in mind, however, that in the harmonic range normalization of

Up to this point, the deterministic approach has been employed. It may be argued that this is not compliant with the statistical nature of vibration generation mechanisms. What is more important, however, is the fact that statistical approach allows for eliminating problems resulting from the influences of control and interference. The basic idea may be

The main assumption in the statistical approach is that a symptom is a random variable rather than a deterministic function of machine condition parameters. Parameters of this random variable also depend on object condition and thus may be themselves accepted as symptoms (sometimes they are referred to as meta-symptoms, in order to stress that they are not directly measurable physical quantities). The idea of determining such symptoms is

Fig. 8. The idea of statistical symptom determination: parameters pertaining to measured

Obviously elements of the control and interference vectors are also random variables. Moreover, for a given turbine at some fixed location, it is reasonable to assume that statistical parameters of these random variables do not change with time, so each of them is characterized by a time-invariant probability distribution. Now, let us assume that we determine probability distribution of a vibration-based symptom *S* (say, vibration velocity level in a given frequency band, measured in a given point) in a manner shown in Fig.8. If it can

.

symptom value distribution are determined within a time window

summed up as 'if we cannot get rid of it, then try to make use of it'.

life cycles is mandatory.

**6. Statistical symptoms 6.1 Dispersion measures** 

shown schematically in Fig.8.

be shown that

not change substantially with time. In fact this is compatible with the main mechanisms of lifetime consumption, i.e. fatigue and creep, for which linear approximations for << *<sup>b</sup>* are also valid. Such case is illustrated by an example shown in Fig.7a. On the other hand, for a considerable lifetime consumption, the slope will initially increase with time (Fig.7b) to a point wherein linear approximation is no longer acceptable and *S*() resembles an exponential curve. For close to *<sup>b</sup>* even exponential fit fails (Fig.7c).

Fig. 7. (a) Linear fit slope vs. time: 230 MW unit, front low-pressure turbine bearing, horizontal direction, 6.3 kHz band; (b) the same, 200 MW unit, low-pressure turbine casing rear/ right side, horizontal direction, 3.15 kHz band; (c) vibration velocity vs. time: 200 MW unit, high pressure/intermediate pressure bearing, vertical direction, 8 kHz band. Data smoothing: peak trimming at *c* = 1.5 followed by three-point averaging. Red line in (c) represents exponential fit.

Although quantitative assessment results should not be generalized over different turbine types, it has been estimated that, for components from the blade frequency range, vibration velocity vs. time plots with linear slope values below about (10 20) 10-6 (mm/s)/day are typical for normal lifetime consumption (natural damage) with substantially lower than *b*. For the harmonic frequency range, the value of 10-4 (mm/s)/day may be accepted as a very rough estimate. Accelerated damage may result in a value higher by an order of magnitude (cf. Fig.3). It has to be kept in mind, however, that in the harmonic range normalization of life cycles is mandatory.
