**3.1.3 Rotor bow**

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


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

$$\mathbf{r} = r\_e e^{j\delta} + \frac{Mr\_e o^2 e^{j\delta}}{\left[K - Moo^2 + jD(1 - \lambda)o\right]} \,\tag{12}$$

where *M* denotes unbalance mass, shifted at the distance *re* in the direction determined by the angle . *K* and *D* are stiffness and damping coefficients, respectively; denotes rotor

Vibration-Based Diagnostics of Steam Turbines 323

velocity ratio. It is therefore obvious that a suitable stability margin should be provided by

Bearing instability is nicely demonstrated with laboratory-scale model rotor systems. For large steam turbines in power industry, operated at a fixed rotational speed, this is a rare occurrence. Most frequently it results from bearing vertical displacement, due to thermal deformation and/or foundation distortion. Downward displacement reduces bearing load

In such circumstances, instability is unavoidable. Most typical symptom of this malfunction is the increase of sub-harmonic spectral components, often of the 'hump' shape centered slightly below 0.5 *f*0. Shaft orbits typically exhibit loops. Strong instability results in high relative vibration that leads to bearing damage. Proper adjustment of bearing positions is the primary action to be taken; sometimes reduction of the bearing size (length), in order to increase specific load, is necessary for a permanent remedy (Orłowski and Gałka, 1995).

Due to strong non-linearity, journal bearings generate higher harmonic components which may be very sensitive to bearing condition, clearances and oil pressure. An example is shown in Fig.4, in the form of a time history of the 3 *f*0 absolute horizontal vibration component. Initially very low, it increased dramatically following a minor bearing damage and remained at a high level, exhibiting considerable variations that suggest a resonance nature of the phenomenon. Permanent improvement was achieved only after a major overhaul. It has to be noted that such behavior is to a large extent influenced by design features; therefore care has to be taken when generalizing the results over other turbine types. In any case,

Typical malfunctions which have their representations in the low frequency range and their corresponding symptoms have been listed in Table 1, which summarizes this subsection.

Fig. 4. Time history of the 3 *f*0 component: 200 MW unit, rear intermediate-pressure turbine bearing, horizontal direction. Arrows: 1, bearing damage; 2, bearing position and

proper design and operation, which influence all three quantities that determine

*th* may become lower than the nominal rotational speed.

is the oil circumferential

*th*.

where *K* and *M* denote stiffness and mass, respectively, and

sensitivity of spectral components to oil pressure is decisive.

clearances adjustments; 3, major overhaul

and causes *K* to decrease, so that

angular velocity and is the fluid circumferential velocity ratio ( = /, where denotes average fluid angular velocity). First term describes the low-speed response (which, as mentioned earlier, is basically absent with 'plain' unbalance), while the second one refers to the dynamic synchronous response. It can be seen that for >> *r* (where *<sup>r</sup>* is the resonance angular speed), when the first and the third term in the denominator can be neglected, rotor response is close to zero. This is a feature characteristic for this malfunction (colloquially speaking, the rotor 'balances itself out'), but in large steam turbines with heavy flexible rotors the >> *<sup>r</sup>* condition is seldom fulfilled.

It has been shown (Gałka, 2009b) that permanent rotor bow causes simultaneous increase of the 1 *f*0 component in vertical and axial directions, so that a developing bow should result in strong correlation between these components (see also Section 6). Available data seem to confirm this conclusion, in fact based on quite simple model considerations.

### **3.1.4 Rotor crack**

As a very serious fault with potentially catastrophic consequences, rotor crack has received considerable attention (for perhaps the most comprehensive available review, see Bachschmid, Pennacchi and Tanzi, 2010). In general, crack reduces shaft stiffness and thus causes resonance to shift to a lower rotational speed. As a result, the 1 *f*0 component amplitude during steady-state operation will either increase or decrease. In large steam turbines, operated above the first critical speed, the latter may be the case. This effect may be combined with that of increasing rotor bow due to reduced bending stiffness. As a result of asymmetry introduced by a crack, the 2 *f*0 component may also increase substantially.

It is generally recognized that considerable continuous changes of first two harmonic components amplitudes (not necessarily both increasing!) and phases during steady-state operation indicate that a shaft crack is possibly present. Rates of these changes vary within broad limits, from the order of months to days or even hours – in the latter case, a catastrophic failure is most probably imminent. Such evolution of vibration patterns should serve as an alert. Presence of a crack may be confirmed by monitoring absolute and relative vibration during transients – typically after a turbine trip. Time histories of the 1 *f*0 and 2 *f*0 components, obtained in such manner, may be compared with reference data recorded after unit commissioning or a major overhaul. Significant reduction of critical speeds and increase of vibration amplitudes on passing through them are indicative of this malfunction, as well as is high overall relative vibration amplitude; the latter will sometimes render the startup impossible to complete, as the unit shall be tripped automatically below nominal rotational speed.

#### **3.1.5 Bearing problems**

A problem specific to shaft journal bearings is oil film instability that induces so-called selfexcited vibrations. This issue has attracted considerable attention and detailed theoretical models have been developed (Bently and Hatch, 2002; Kiciński, 2006). It can be shown that threshold rotational speed for the onset of instability *th* is given by

$$
\Omega\_{\rm th} = \frac{1}{\mathcal{A}} \sqrt{\frac{K}{M}}\ ,\tag{13}
$$

 = /, where

*r* (where

 >>  *<sup>r</sup>* is the resonance

denotes

is the fluid circumferential velocity ratio (

average fluid angular velocity). First term describes the low-speed response (which, as mentioned earlier, is basically absent with 'plain' unbalance), while the second one refers to the

angular speed), when the first and the third term in the denominator can be neglected, rotor response is close to zero. This is a feature characteristic for this malfunction (colloquially speaking, the rotor 'balances itself out'), but in large steam turbines with heavy flexible

It has been shown (Gałka, 2009b) that permanent rotor bow causes simultaneous increase of the 1 *f*0 component in vertical and axial directions, so that a developing bow should result in strong correlation between these components (see also Section 6). Available data seem to

As a very serious fault with potentially catastrophic consequences, rotor crack has received considerable attention (for perhaps the most comprehensive available review, see Bachschmid, Pennacchi and Tanzi, 2010). In general, crack reduces shaft stiffness and thus causes resonance to shift to a lower rotational speed. As a result, the 1 *f*0 component amplitude during steady-state operation will either increase or decrease. In large steam turbines, operated above the first critical speed, the latter may be the case. This effect may be combined with that of increasing rotor bow due to reduced bending stiffness. As a result of asymmetry

It is generally recognized that considerable continuous changes of first two harmonic components amplitudes (not necessarily both increasing!) and phases during steady-state operation indicate that a shaft crack is possibly present. Rates of these changes vary within broad limits, from the order of months to days or even hours – in the latter case, a catastrophic failure is most probably imminent. Such evolution of vibration patterns should serve as an alert. Presence of a crack may be confirmed by monitoring absolute and relative vibration during transients – typically after a turbine trip. Time histories of the 1 *f*0 and 2 *f*0 components, obtained in such manner, may be compared with reference data recorded after unit commissioning or a major overhaul. Significant reduction of critical speeds and increase of vibration amplitudes on passing through them are indicative of this malfunction, as well as is high overall relative vibration amplitude; the latter will sometimes render the startup impossible to complete, as the unit shall be tripped automatically below nominal

A problem specific to shaft journal bearings is oil film instability that induces so-called selfexcited vibrations. This issue has attracted considerable attention and detailed theoretical models have been developed (Bently and Hatch, 2002; Kiciński, 2006). It can be shown that

> 1

*th*

*K M* *th* is given by

, (13)

angular velocity and

 >> 

**3.1.4 Rotor crack** 

rotational speed.

**3.1.5 Bearing problems** 

threshold rotational speed for the onset of instability

rotors the

dynamic synchronous response. It can be seen that for

*<sup>r</sup>* condition is seldom fulfilled.

confirm this conclusion, in fact based on quite simple model considerations.

introduced by a crack, the 2 *f*0 component may also increase substantially.

where *K* and *M* denote stiffness and mass, respectively, and is the oil circumferential velocity ratio. It is therefore obvious that a suitable stability margin should be provided by proper design and operation, which influence all three quantities that determine *th*.

Bearing instability is nicely demonstrated with laboratory-scale model rotor systems. For large steam turbines in power industry, operated at a fixed rotational speed, this is a rare occurrence. Most frequently it results from bearing vertical displacement, due to thermal deformation and/or foundation distortion. Downward displacement reduces bearing load and causes *K* to decrease, so that *th* may become lower than the nominal rotational speed. In such circumstances, instability is unavoidable. Most typical symptom of this malfunction is the increase of sub-harmonic spectral components, often of the 'hump' shape centered slightly below 0.5 *f*0. Shaft orbits typically exhibit loops. Strong instability results in high relative vibration that leads to bearing damage. Proper adjustment of bearing positions is the primary action to be taken; sometimes reduction of the bearing size (length), in order to increase specific load, is necessary for a permanent remedy (Orłowski and Gałka, 1995).

Due to strong non-linearity, journal bearings generate higher harmonic components which may be very sensitive to bearing condition, clearances and oil pressure. An example is shown in Fig.4, in the form of a time history of the 3 *f*0 absolute horizontal vibration component. Initially very low, it increased dramatically following a minor bearing damage and remained at a high level, exhibiting considerable variations that suggest a resonance nature of the phenomenon. Permanent improvement was achieved only after a major overhaul. It has to be noted that such behavior is to a large extent influenced by design features; therefore care has to be taken when generalizing the results over other turbine types. In any case, sensitivity of spectral components to oil pressure is decisive.

Typical malfunctions which have their representations in the low frequency range and their corresponding symptoms have been listed in Table 1, which summarizes this subsection.

Fig. 4. Time history of the 3 *f*0 component: 200 MW unit, rear intermediate-pressure turbine bearing, horizontal direction. Arrows: 1, bearing damage; 2, bearing position and clearances adjustments; 3, major overhaul

Vibration-Based Diagnostics of Steam Turbines 325

mid-frequency of 23% CPB spectrum bands, determined for a 120 MW steam turbine. It is

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

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 measu-

Fig. 6. Time history of the 2500 Hz component: 200 MW unit, low-pressure turbine casing

/*Ŝ*, where *Ŝ* denotes mean value) plotted against

/*Ŝ* is below 0.1, while in the blade range it

Fig.5 shows relative standard deviation (

interference have to be treated as outliers.

rements (after Gałka, 2011b).

front/right side, vertical direction

immediately seen that for the harmonic range


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