recording parameters of torsional vibrations of turbogenerator shafts

4
It is proposed that the mechanical state of turbogenerators be regularly monitored by photoelectric chronometric recording of low-frequency torsional vibrations of their shafts. The vibrations are caused by small sudden changes which are continuously occurring in the operating parameters of the turbine unit. It is known that a qualitatively new level of experimental study could be reached in involving the operation of machines and mechanisms if the relative error made in measuring the speed of rotation of shafts could be reduced to 10 –4 %. In this article, we present the results obtained from an attempt to precisely monitor the rotation of a turbogenerator shaft (TS) by performing measurements with a photoelectric system. The system has a relative error of 5·10 –4 % at industrial frequen- cies, which means that it could reduce the absolute error made in measuring the time of a single revolution T (T = 0.02 sec at a frequency of 50 Hz) to 10 –7 sec [1]. A built-in quartz generator is used to ensure such accuracy when a small number of measurements is being made, while a highly stable external oscillator is connected to the system when observations are made over a fairly long period of time. When the operation of cyclic mechanisms is monitored by successively recording the duration of each cycle and multiple cycles or characteristic stages, the result is a sequence of numbers – a time series and its main form of representa- tion: the chronogram. A more appropriate term would be the “periodogram,” which has already been used in Schuster’s math- ematical theory of time series [2]. The variations of the terms of the series contain all of the information on the given mech- anism that can be obtained by mathematical analysis with the level of accuracy that has been achieved to date. An effective method of analyzing time series is modern digital spectrum analysis [3, 4]. However, the use of this approach requires consideration of the so-called “echo” [5] or “window” effect [3, 4] that occurs due to the unavoidable lim- itation on the duration of the time interval chosen to perform the measurements. In digital spectrum analysis, this effect can cause weak spectrum lines to be be masked by the side lobes of stronger lines. (In the same way, the effect reduces the aper- ture of the optical system, i.e., it narrows the band of “space frequencies,” and it lowers the resolution of the system – which until the creation of the diffraction theory of optical systems led to various incongruities, including false openings). In par- ticular, unsubstantiated use of the least accurate type of window – a “rectangular” window – to study cycles can create the false impression that quasisteady cycles generally cannot be monitored chronometrically by using their steady-state frequen- cies. In fact, the Fourier transform of a rectangular window whose duration is equal to the period of one cycle T has the form where f is a frequency that vanishes at frequency of the cycle f = 1/ T and its integral multiples. However, the situation changes radically if we change over from a spectral representation to a coordinate-time representation. The effect of such a change can be demonstrated by using the study of the regime of rotation of a TG shaft as an example: if technical chronometry could Uf T fT fT j fT ( ) sin , = π π e Measurement Techniques,Vol. 43, No. 12, 2000 RECORDING PARAMETERS OF TORSIONAL VIBRATIONS OF TURBOGENERATOR SHAFTS M. I. Kiselev, N. V. Novik, and V. I. Pronyakin UDC 531.7:681.2.082/.083 Translated from Izmeritel’naya Tekhnika, No. 12, pp. 34–36, December, 2000. Original article submitted July 11, 2000. 0543-1972/00/4312-1066$25.00 © 2000 Plenum Publishing Corporation 1066

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It is proposed that the mechanical state of turbogenerators be regularly monitored by photoelectric

chronometric recording of low-frequency torsional vibrations of their shafts. The vibrations are caused by

small sudden changes which are continuously occurring in the operating parameters of the turbine unit.

It is known that a qualitatively new level of experimental study could be reached in involving the operation of

machines and mechanisms if the relative error made in measuring the speed of rotation of shafts could be reduced to 10–4%.

In this article, we present the results obtained from an attempt to precisely monitor the rotation of a turbogenerator shaft (TS)

by performing measurements with a photoelectric system. The system has a relative error of 5·10–4% at industrial frequen-

cies, which means that it could reduce the absolute error made in measuring the time of a single revolution T (T = 0.02 sec

at a frequency of 50 Hz) to 10–7 sec [1]. A built-in quartz generator is used to ensure such accuracy when a small number of

measurements is being made, while a highly stable external oscillator is connected to the system when observations are made

over a fairly long period of time.

When the operation of cyclic mechanisms is monitored by successively recording the duration of each cycle and

multiple cycles or characteristic stages, the result is a sequence of numbers – a time series and its main form of representa-

tion: the chronogram. A more appropriate term would be the “periodogram,” which has already been used in Schuster’s math-

ematical theory of time series [2]. The variations of the terms of the series contain all of the information on the given mech-

anism that can be obtained by mathematical analysis with the level of accuracy that has been achieved to date.

An effective method of analyzing time series is modern digital spectrum analysis [3, 4]. However, the use of this

approach requires consideration of the so-called “echo” [5] or “window” effect [3, 4] that occurs due to the unavoidable lim-

itation on the duration of the time interval chosen to perform the measurements. In digital spectrum analysis, this effect can

cause weak spectrum lines to be be masked by the side lobes of stronger lines. (In the same way, the effect reduces the aper-

ture of the optical system, i.e., it narrows the band of “space frequencies,” and it lowers the resolution of the system – which

until the creation of the diffraction theory of optical systems led to various incongruities, including false openings). In par-

ticular, unsubstantiated use of the least accurate type of window – a “rectangular” window – to study cycles can create the

false impression that quasisteady cycles generally cannot be monitored chronometrically by using their steady-state frequen-

cies. In fact, the Fourier transform of a rectangular window whose duration is equal to the period of one cycle T has the form

where f is a frequency that vanishes at frequency of the cycle f = 1/T and its integral multiples. However, the situation changes

radically if we change over from a spectral representation to a coordinate-time representation. The effect of such a change

can be demonstrated by using the study of the regime of rotation of a TG shaft as an example: if technical chronometry could

U f TfT

fTj fT( )

sin,= π

π

− πe

Measurement Techniques, Vol. 43, No. 12, 2000

RECORDING PARAMETERS OF TORSIONAL

VIBRATIONS OF TURBOGENERATOR SHAFTS

M. I. Kiselev, N. V. Novik,and V. I. Pronyakin

UDC 531.7:681.2.082/.083

Translated from Izmeritel’naya Tekhnika, No. 12, pp. 34–36, December, 2000. Original article submitted July 11, 2000.

0543-1972/00/4312-1066$25.00 ©2000 Plenum Publishing Corporation1066

reduce the relative error in measuring the period of rotation of the shaft to 5·10–4%, it would open up new possibilities for

studying the operation of TGs and other cyclic machines [1].

The problem of precisely measuring time intervals during turbogenerator operation in the machine room of a heat-

ing and power plant amounts to solving the traditional problem of determining the moment of the signal is received when

interference is present. To do this,the circuit that generates the reference signal is closed when the groove in the measure-

ment disk mounted on the TG shaft reaches the position specified for it relative to the light beam which is modeled by the

groove and is received by the first lens in the optical channel. In the ideal case when noise is absent,the front of the photo-

electric pulse has a characteristic point of inflection that marks the reference point. Under actual conditions,when there is

open-channel noise (dust,bias lighting, etc.), vibrational noise, and unstable kinematic parameters characterizing the motion

of the shaft,the parameters of the initial pulse are not reproducible. It thus becomes necessary to provide a means of auto-

matically compensating for disturbances of the position of the characteristic reference point.

According to [1],with the level of accuracy achieved thus far, it has been determined that the rotation of the shaft is

nonuniform and the control-and-measuring equipment in the electrical system cannot measure the variation: even two suc-

cessive revolutions turn out to be of different duration, nonuniformity is manifest over the course of a single revolution,and

the shaft also undergoes torsional vibrations. The nonuniformity is caused by instability of the pressure head in the turbine

and the parameters of the electrical load. Figure 1 shows a typical chronogram describing the rotation of a shaft 46 m long

and about 100 tons in weight.

Chronometric monitoring of the regime of rotation of TG shafts entails regular measurement of their time of revo-

lution – the continuously fluctuating period or its fractions. Mathematical analysis of the time series which are obtained

makes it possible to calculate the spectral characteristics of motion. However, the space-time structure of the motion is of

equal interest. Direct digital spectrum analysis of long-period torsional vibrations of the rotor of a generator is ineffective

not only due to possible interference (the side lobes of adjacent spectral lines mentioned at the beginning of the article), but

also because of the inadequate resolution. The main reason for its ineffectiveness is the brief time that even the perturbations

with the longest periods persist due to the damping system and, thus,the low intensity of the corresponding spectral lines.

We will show how information on low-frequency torsional vibrations of shafts can be obtained on the basis of precision

chronograms.

Let the angle of rotation of a rotating shaft within the test section of the shaft be specified by the expression

ψ(t) = ωnt + ϕ(t), (1)

where ωn is the nominal angular velocity; ϕ(t) is the perturbation of the uniform motion. In addition to systematic mea-

surements of each period, we measure its n multiples. Thus,n is the number of grooves in the measurement disk. Examining

the set of all time readings {tik}, where i is the number of the period and k (1 ≤ k ≤ n) is the number of the reading within

1067

Fig. 1. Chronogram of the period of revolution of a TG shaft,where T is the period

of one revolution and t is time.

each period, we can identify n subsequences of periods {Tik} whose elements are shifted by the intervals ∆ti

k. Here, the series

of intervals ∆tik ≈ (1/n)Ti is obtained by measurement.

On the measured time interval corresponding to N revolutions,each subsequence corresponds to its own values of

the nominal period

(2)

and the nominal angular velocity ωkn = 2π/Tn

k. Here, the boundaries of the current periods are represented in the form

tik = iTn

k + δtik.

The following is valid for each revolution:

(3)2 1

1

1 1π = = − + + − − +[ ]−

∫ − −˙ ( ) ( ) ( ) ( ) .ψ ω δ δ ϕ δ ϕ δt dt t t iT t i T t

t

tk

ik

ik k

ik k

ik

ik

ik

n n n

TN

t tkik

ik

i

N

n = −+=∑1

11

( )

1068

Fig. 2. Chronograms of a series {δtik} of torsional vibrations of a TG shaft.

Fig. 3. Graph of the angle of deflection of a shaft from its nominal position.

At δt << Tik, this relation reduces to the form

(4)

where 1 < i < N.

It is necessary to determine ϕ(iTnk) in the resulting system of equations,while the values of the quantities Tn

k, ωkn,

and δtik are determined from the measurements. If one element of the set {δti

k} vanishes (δti0k0 = 0), then the following is

simultaneously valid: ϕ(i0Tnk0) = 0,and at k = k0 the continuous chain of equations (4) decomposes into separate equations

The position of the shaft at i = i0, k = k0 corresponds to the groove with the number k and can be taken as the refer-

ence point for the circular vibrations of the shaft.

For long-period perturbations,whenϕ̇(iTnk) << ωk

n for all i, in the first approximation we obtain

and the chronogram is transformed into the law of motion of a shaft with a time step equal to the nominal (mean) period of

rotation.

Additional measurements are necessary when ϕ̇ ≥ ωn. To analyze actual chronograms, instead of the equality

δti0k0 = 0 it is sufficient to take δ ti0

k0 ≤ ∆t, where ∆t is the absolute error of measurement of the time intervals.

The given approach was realized in analyzing chronograms describing the rotation of the shaft of a TG-250/300 tur-

bogenerator at heating and power plant No. 26 in Moscow. The initial data was converted into chronograms representing the

sequences {δtik}. We chose to examine the sections on which there were abrupt changes in the variations of the periods due

to sudden changes in the operating regime of the TG. Figure 2 shows fragments of chronograms corresponding to torsional

vibrations of the shaft.

At ωn = 314 rad·sec–1, the variations of the period amounting to roughly 1.2·10–6 sec in Fig. 2 (top chronogram)

correspond to torsional vibrations with an initial amplitude of 0.4′. The period of those vibrations,corresponding to rough-

ly 60 revolutions,turns out to be equal to 1.2 sec.

All of the graphs show the excitation of high-frequency torsional vibrations with a period equal to several nominal

periods of revolution. Vibrations of this type can easily be filtered, such as by the sliding-mean method (Fig. 3).

Thus,precision chronograms obtained under factory conditions have been used to obtain a space-time scan which

is equivalent to an oscillogram and describes the low-frequency torsional vibrations of a rotating turbogenerator shaft. In the

scan,the nominal frequency of revolution of the shaft plays the role of a carrier frequency modulated by the low-frequency

signal. Use of the chronograms to obtain such scans opens up additional possibilities for systematic chronometric monitor-

ing of the condition of turbogenerators during service, since the period and decrement of the low-frequency torsional vibra-

tions of their shafts depend on the turbogenerators’ design parameters.

REFERENCES

1. M. I. Kiselev et al.,Izmer. Tekh., No. 12,28 (1996).

2. A. Schuster, Proc. R. Soc. London Ser. A, 77, 136 (1905).

3. S. L. Marple Jr., Digital Spectrum Analysis and Its Uses[Russian translation], Mir, Moscow (1990).

4. D. Brillinger, Time Series. Data Analysis and Theory [Russian translation], Mir, Moscow (1980).

5. M. Kendall,Time Series [Russian translation] (with foreward by Yu. P. Lukashin),Financy i Statistika, Moscow

(1981).

ϕ ω δ( ) ,iT tk kik

n n≈ −

ω ϕ δ ϕn n nk k

ik kiT t iT+[ ] + =˙ ( ) ( ) .0

ω ϕ δ ϕ ω ϕ δ ϕn n n n n nk k

ik k k k

ik ki T t i T iT t iT+ −( )[ ] + −( ) = +[ ] +−˙ ( ) ( ) ˙ ( ) ( ),1 11

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