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O r d e r A n a l y s i s U s i n g Z o o m F F T 012-81

Order Analysis Using Zoom FFT by Henrik Her/ufsen, Briiel & Kjaer

i

Introduction In the analysis of vibration signals

from rotating machines w i th varying speed, order analysis using an FFT analyzer can be a useful tool , parti­cularly when analysing the vibra­tions during a run-up or a run-down of a machine. Of particular interest in that case is the analysis of struc­tural resonances excited by the fundamental or the harmonics of the rotational frequencies in the me­chanical system. Determination of the critical speeds, where the nor­mal modes of the rotating shaft are excited, is very important on large machines such as turbines and gen­erators (turbogenerators).

Use of an FFT analyzer in the nor­mal internal sampling mode { i .e . non-tracking) and plotting of the spectrum at certain fixed steps in ro­tation speed of the machine gives the so-called Campbell diagram. This is a 3-D waterfal l type of plot, where vibration levels as a function of frequency are plotted against rota­tion speed of the machine (plotted vertically). This means that the har­monic components appear on radial lines through the point (0 Hz, 0 speed) whi le structural resonances appear on vertical straight lines (constant frequency lines). This plot can thus be very useful. The smear­ing of the components, which ap­pears because the t ime window

used for the individual spectra re­presents a certain sweep in the speed, is however a disadvantage. The power of the components be­comes spread over several lines. In particular, high frequency compo­nents in the spectrum, such as toothmesh frequencies, might be smeared so much that details in sideband structures are lost in the analysis. This is the main reason why order analysis is used instead.

In general one can say that order analysis, by use of an FFT analyzer, is an analysis by which the har­monic pattern of the vibration signal from a rotating machine is stabi­lized in certain lines independent of speed variations. This means that all the power of a certain harmonic is concentrated in one line and the smearing that would result in nor­mal analysis is avoided.

A fundamental requirement of the FFT analyzer is that it has an exter­nal sampling facility, but provided this is available an order analysis can be performed. In the baseband mode ( i .e . calculation of all the lines available according to the sam­pling rate), however, speed varia­tions wi l l cause the so-called antial i­asing filter to limit the analysis range of the analyzer. This wi l l be explained later. By changing manu­

ally the antialiasing fi lter during the run-up or run-down, a fast and real­t ime order analysis can be per­formed, but only the lowest lines give correct analysis (how many lines depends on the step in cut-off frequency from one filter to the next).

The zoom facility in the High Reso­lution Signal Analyzer Type 2 0 3 3 provides a solution to this problem as it permits alias-free order analy­sis over a wide range of speed. Or­der analysis of the fundamental and lower harmonics is possible, with­out distortion from the antialiasing filter, over a speed variation of 1 to 15 (for example from 4 0 0 r.p.m. to 6 0 0 0 r.p.m.). High resolution order analysis around some higher fre­quency components can be per­formed without distortion from the filter over a reduced speed variation and can, as will be shown later, be very useful.

The intention of this Application. Note is, f irstly, to explain the princi­ple of order analysis (tracking) by use of an FFT analyzer wi th zoom fa­cility and, secondly, to give an exam­ple of the application of this analy­sis as performed on a large turbo­generator system in a power gene­rating plant.

Principle and Instrumentation A typical instrument set-up is

shown in Fig. 1 . The basic idea in or­der analysis is to pick up a pulse train synchronous wi th the rota­tional frequency of the machine and use this as a reference to generate the internal clock in the analyzer. The rotation frequency is, as shown, detected by a tachometer probe, such as the Magnetic Pick­up Type MM 0002 or the Photoelec­tric Probe Type MM 0012 . The ana­lyzer is provided with a multiple of this frequency which becomes the synchronizing frequency for the an­alog-to-digital converter, i.e. the ex­ternal sampling rate. The frequency multiplication (with factor N) is car­ried out by the Tracking Frequency Multiplier Type 1 9 0 1 , see Fig.1.

This means that the analyzer wil l sample the signal a certain number of times for each rotation of the ma­chine instead of at a fixed sampling rate. This principle is illustrated in Fig.2. A sweep of the fundamental component of some hypothetical sig­nal is shown in Fig.2(a) (sweep rate heavily exaggerated). In the normal internal sampling mode the content in the time memory of the analyzer and calculated spectrum wi l l be as shown in Fig.2(b). The sampling fre­quency fs = 2,56 x fmax , where f max. is the selected full scale fre­quency of the analyzer. This is a normal Fourier analysis.

With external sampling, however, the component wil l be sampled a fixed number of times for each pe­riod. With a multiplication factor N = 8 we wil l get 8 samples per pe­riod of the fundamental, as shown in Fig.2(c). The analyzer wi l l inter­pret this fundamental sweeping fre­quency as a fixed frequency and it wi l l show up in one spectrum line, without any smearing. With a 1K (K = 1024) time memory (giving a 400 line spectrum) it wi l l show up in line number 1 0 2 4 / N = 1 0 2 4 / 8 = 128, because the sampling fre­quency corresponds to line 1024 . A detailed discussion of the FFT princi­ple wi l l be found in Ref. 1. Likewise the harmonics wi l l show up at mult i­ples of this line number.

Fig.3 shows the analysis of the first 4 harmonics of a 50 Hz to 45 Hz sweep of a distorted sine-

Accelerometer 4368

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Ext. Sampling

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F i g . 1 . Instrumental set-up for order analysis

a)

Original signal, fundamental component

b)

1 L i m JlilL m HL I r» 1 I 1 Sample

number

A

nternal sampling freq

U = 2,56 ■ f max

fmax. frequency

c)

f " " " t <

o 1 1 1 1 1 1 1 1 o I n Sample number

A

External sampling freq

fs - 8 f FUND

128 400 Line number

810110

Fig.2. Principle in order analysis a) original signal, fundamenta l component b) samples and calculated spectrum wi th internal sampling frequency fs

c) samples and calculated spectrum wi th external sampling frequency f s

2 , 5 6 f max 8f FUND

2

wave a) with internal sampling and b) with external sampling (tracking). With tracking, all the power of the individual harmonics is concen­trated in a particular line. Thus the fundamental in Fig.3(a) is smeared over more than 6 lines and the fund­amental at line 8 0 in Fig.3(b) is seen to be 8dB higher. At the 4. harmonic the smearing is 4 times wider and the component is 8 + 6 = 14dB higher with tracking than without tracking. In this example a 4 0 0 line baseband analysis and a multiplication factor of 12,8 was used, placing the fundamental in line number 1 0 2 4 / 1 2 , 8 = 80 .

In any sampling process, i .e . con­verting a continuous t ime signal to a discrete signal, aliasing of fre­quencies has to be considered. Af­ter the sampling process it is impos­sible to distinguish between low fre­quencies and frequencies above half the sampling frequency f s / 2 , called the Nyquist frequency fN. This is the principle behind the use of the stroboscope for example. High frequencies become aliased down to low frequencies which the eye can follow and interpret. In or­der to avoid this ambiguity it is ne­cessary to lowpass filter the analog signal before sampling. This is done by an antialiasing filter whose cut­off frequency can be changed in dis­crete steps. To get a correct spec­trum the filter should be chosen rel­ative to the sampling frequency, as shown in Fig.4(a). In the normal baseband mode (non-zoomed) 4 0 0 lines are calculated up the cut-off frequency. The aliasing, having the

folding the spectrum is indicated by the dot-

Change of the rotation speed of the machine, i.e. change of the sampling frequency, will, however, cause the filter to reduce the analysis range and dynamic range, as shown in Fig.4(b) and (c). In (b) the speed of the machine has been increased relative to that in (a) and it is seen that the filter cut-off has entered the 4 0 0 line analysis range. In (c) the speed has been dec­reased relative to (a) and the compo­nents with frequencies above fN , which come through the filter, are folded back into the analysis giving wrong results.

effect of around fN

ted lines.

In the baseband mode one is thus restricted to very limited speed varia-

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t ions in order to retain a correct 400- l ine order spectrum.

By use of zoom, however, this problem can be solved. Zoom is an analysis of only a part of the total baseband range. The High Resolu­tion Signal Analyzer Type 2 0 3 3 is in fact a 4000- l ine analyzer calcu­lating and displaying only 4 0 0 lines at a t ime. For a detailed discussion

of zoom FFT see Ref. 2. Zooming at the lower 4 0 lines in the spectrum in Fig.4(a), for example, wi l l give a 4 0 0 line analysis, as shown in Fig.4(d). From this situation the rota­tion speed obviously can be in­creased a factor of ten before distor­tion in the 4 0 0 line spectrum from the filter occurs (as in (b)). Further­more, the speed can be decreased a factor of 1 ,5 before aliased compo-

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Fig.4. Analysis ranges w i t h external sampling fs in di f ferent si tuat ion a) Baseband analysis w i t h correct f s

b) Baseband analysis w i t h too high fs

c) Baseband analysis w i t h too low fs

d) + e) zoom analysis, w i thou t d istor t ion f r o m fitter or aliasing

nents enter the zoom analysis range (as in (c)).

An order analysis of the funda­mental and its harmonics (i. e. lower 4 0 0 lines) is thus possible wi thout changing the fi lter whi le the speed is changing a factor of 15. In practice this means that the important part of a run-up or run­down of the machine can be covered. The lower speed limit might here be set by the Tracking Frequency Multipl ier Type 1901 which cannot accept less than J> Hz (300 r.p.m.) at the input.

For a zoom analysis around some higher harmonic (e .g . a toothmesh frequency) as in Fig.4(e) the al low­able speed range for correct analy­sis wi l l be reduced and it might be necessary to change the fi lter dur­ing the run-up or run-down in order to cover the whole speed range of interest.

Thus before performing an order analysis two parameters have to be determined: 1) The cut-off fre­quency of the antialiasing fi lter and 2) The frequency multiplication fac­tor N. The cut-off frequency has to be equal or higher than the fre­quency, at ful l rotation speed, of the highest harmonic of interest. That antialiasing filter wi th the low­est cut-off frequency fulf i l l ing this requirement should be chosen. The frequency mult ipl ication factor N de­termines the line numbers in which the harmonics wi l l appear on the an­alyzer. In the zoom mode, the High Resolution Signal Analyzer Type 2 0 3 3 is using the 10 K t ime me­mory for calculation of selected 4 0 0 lines out of the 4 0 0 0 lines (zoom factor 10). The sampling frequency thus corresponds to line 1 0 2 4 0 . From the desired arrangement of the fundamental and harmonics the multipl ication factor N is now given by the relation: N * LINE No. OF FUNDAMENTAL = 10 2 4 0 .

The analysis is performed in the zoom mode, requiring the 10 K time record, and is not in real-t ime. The calculation t ime in the zoom mode is one second. Therefore it is only applicable to large machines which speed up and slow down very slowly, so that real-t ime analy­sis is not necessary and the ampli­tude variations (amplitude smear­

ing) along the 10 K record is negligi­ble even for relatively high Q reson­ances. In practice, the run through the speed range of interest should last more than, say, 5 minutes.

Practical Measurements

Run-down of a large turbogenera­tor:

The vibration signals (accelera­tion) at the bearings of the genera­tor and the turbine were recorded at the same time as a tacho signal from a Photoelectric Tachometer Probe Type M M 0 0 1 2 giving the ro­tation frequency of the shaft.

The run-down lasted nearly 15 minutes and was therefore suitable for order analysis (zoom tracking). Firstly, the lowest 10 harmonics of the vibration signal are analysed during the run-down. This is done by arranging for the fundamental to be in line 40 of the lowest 4 0 0 lines of a zoom analysis.

Choice of multipl ication factor N and antialiasing filter is as fol lows:

In zoom mode the sampling fre­quency corresponds to line 10 2 4 0 , which means that

N = 1 0 2 4 0

LINE No. OF FUNDAMENTAL

Thus N = 1 0 2 4 0 40

= 256

Max. rotation speed is 50 Hz, meaning that for the analysis of the first 1 0 harmonics the 5 0 0 Hz antia-iasing filter should be chosen.

Aliasfree tracking would then be possible down to 3,2 Hz rotation speed without changing the fi lter, but the Tracking Frequency Mul t i ­plier Type 1 901 limits the order an­alysis to a minimum of 5 Hz rotation speed. For each 2 Hz change in rota­tion speed the order spectrum was manually stored on the Digital Cas­sette Recorder Type 7400(see Fig. 1).

A 3-dimensional plot of the stored spectra of the vertical vibra­tion signal at the generator bear­ings is shown in Fig.5. A baseline at 65 dB is inserted in order to get rid of the visually disturbing random noise and provide a reference le­vel. The first 3 harmonics are signi­ficant in level and show characteri­stic resonances. For example, a re­sonance is seen in the fundamental between 16 — 18 Hz rotation speed which evidently is also ex­cited by the second harmonic be­tween 8 — 10 Hz rotation speed. The constant frequency compo­nents, presumably vibrations from other machines transmitted through the foundations, show up on hyper­bolic curves in the rotation speed— harmonic order plane. The curves are given by: c x n = f, where c is speed, n is harmonic order and f is the frequency. Fig.6 shows some ex­amples of constant frequency curves. The curve for the 150 Hz

Ace. 3V on generator

Rot. speed: 50 Hz - 6 Hz, step 2 Hz

Hann. weighting Exp. average 4

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50

Machine Rotation

Speed Hz

40 i

80 120 160 200 240 280 320 360 400

1 i

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Harmonic Order 810112

Fig.5. Order analysis of lower harmonics of generator vibration signal during rundown

Fig.6. Constant frequency curves

component is drawn in Fig.5. No­tice the smearing of the constant frequency components. As an over­all view of all the related compo­nents this plot is very useful. A more detailed analysis of the indivi­dual harmonics can be made by plot­ting the level of a particular line ver­sus rotation speed on the X-Y Recor­der Type 2 3 0 8 . Fig.7a) and b) show the fundamental and second har­monic versus rotation speed and ac­tual frequency for the same signal. The X deflection is a DC voltage pro­portional to frequency from the Tracking Frequency Mult ipl ier Type 1 9 0 1 . The resonances can now be

seen in more detail. It is seen that the resonance between 16 and 18 Hz actually has its peak at 17,5 Hz and is excited by both the fundamental and second harmonic. The broad resonance shape around 29 Hz in the fundamental is also dis­cernible in the second harmonic. The increased level at 45 Hz in the fundamental corresponds to a crit i­cal speed stated by the manufac­turer to be at 45 ,8 Hz. This reson­ance is not seen in the second har­monic. In the 5 0 H z — 100Hz range of the second harmonic at least four resonance peaks are seen. Some of these might be com-

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- — * ■ — .■ j x p * . - . ■ . ■ . ■ J _ .-.■.■.■.■ j _ j - **---

TO 15 20 25 30 35 40 45 50 Hz H

actual freq. = rotation speed)

810106

Briiel & Kjarr

Fu l l Scale Level:,

F.S. Frequency: .

Weighting:

Time Funct ion Start: seconds End: second i

120

Average Mode: Exp.

No. of Spectra:

Com merits:

acc, 3 V

2. harmonic of generator vib. signal

line 80 N = 256

h " •

Record fin.:

Pate:

Sign.:

GP 1002

dB Y" I

10 'r~~ y*

20

30

40

60

70

20 40 80 -I ] L J c_3

0

- * *t

^ ^ _ l Jys-rs ^ ^ ^ ^ ^

10

Not Expanded: Q Expanded: _

10 20

-J,

30 20 40 50

30 60 70

40 rot. speed Hz 50 80 90 100

actual freq. Hz Measuring Object: 810103

Fig.7. a) Fundamental of generator vibration signal during rundown b) 2. harmonic of generator vibration signal during rundown

binations of more than one reson­ance.

As mentioned, acceleration was measured in this example. Accelera­tion is the vibration parameter which puts emphasis on the high frequencies and is thu$ preferable if it is wanted to raise the higher har­monics relative to the dominating first harmonics. If, however, rather a measure of the energy in the vi­bration is wanted the velocity should be measured (by integration of acceleration), as the kinetic en­ergy is proportional to velocity squared.

The first two harmonics in Fig.7a) and b) are shown on a linear fre­quency scale. Structural resonances are normally analyzed on a logarith­mic frequency scale, as constant Q-factor (amplification ratio of reso­nant peaks) corresponds to constant percentage bandwidth. In this exam­ple, with analysis over only one de­cade, a logarithmic frequency scale is however not essential, but could have been chosen just as well.

As well as order analysis of the lower harmonics, order analysis of some higher frequency range can be of great interest. This is the case when, for example, gears, impel­lers, etc. which are mounted on the shaft give rise to high frequency components.

i

Fig.8 shows a zoomed order ana­lysis centred on the 39th harmonic of the vibration signal of the turbine during run-down. The 39th har­monic is the toothmeshing fre­quency of a gear on the turbine shaft driving an oil pump. The mult i­plication factor N was set to 512 placing the fundamental in line 20. The 39th harmonic then appears in line 780 .

The spectrum was stored on the digital cassette recorder for each 2 Hz change in rotation speed. The 3-dimensional plot in Fig.8 shows 200 lines of the 400 line zoomed spectrum for rotation speeds from 50 Hz to 24 Hz. The 39th harmonic is 1950 Hz at full speed and the 5 kHz fitter was chosen. With this filter no aliased components wil l en­ter the analysis range before the ro­tation speed is below ~ 1 7 Hz. If an­alysis below that speed were re-

6

Brijel & Kja?r

i

quired, the filter could be changed to 2 kHz at that speed, and the ana­lysis would still be aliasfree.

It is seen that the dominant fam­ily of sidebands comes from modula­tion by the oil pump shaft rotation frequency. This shaft rotates at - 2 3 , 5 Hz at full speed. The lines showing the positions of this side­band family are drawn on the 3-D plot. Some modulation from the main shaft rotation frequency is seen as well as sidebands with a

spacing of 20 lines. A baseline at 54 dB has been inserted. Flat weighting is used and this is why components appear as single lines. The flat {or rectangular) time weight­ing function has the advantage of giving a narrow filter bandwidth equal to the line spacing, while the smooth Hanning window gives a bandwidth of 1,5 times the line spacing. The flat weighting should, however, only be used when the time window contains an integer number of periods (see Ref. 1).

It is interesting to see that the toothmesh component excites reson­ances at rotation speeds of 38 Hz and 28 Hz. It is also seen how first the left sideband frequencies, then the toothmesh frequency, and f i ­nally the right sideband frequencies are excited during the run-down. This high resolution analysis can give useful information for diagnos­tic purposes.

In order to see the detailed behav­iour of the toothmesh component during the run-down, line No. 7 8 0 was recorded versus rotation speed (and also calibrated in terms of ac­tual frequency) on an X-Y Recorder Type 2 3 0 8 , as was done for the fundamental and second harmonic of Figs.7a) and b). The result is shown in Fig.9. The filter was here set to 2 kHz and analysis down to 8 Hz rotation speed performed.

Separating this component from the sidebands by means of an an­alog tracking filter would require less than 1 % filter bandwidth, and to obtain the same signal to noise ratio would require a fi lter band­width as low as 0,1 3%.

Apart from the significant reson­ances already seen in Fig.8, the an­alysis in Fig.9 reveals several lower peaks in the toothmesh component.

The vibration signal at the genera­tor bearing contained among other components a significant 37th har­monic and harmonics of this. A 3-di-mensional plot of a 4 0 0 line zoomed order analysis around this 37th harmonic during run-down is shown in Fig.10. Mult ipl ication fac­tor N was 5 1 2 , placing the 37th harmonic in line No. 7 4 0 . This com­ponent was found to be caused by a fan wi th 37 blades in the generator cooling system. Some peaks are eas­ily seen. No sideband structure is seen around this component, indi­cating that it is a rather pure blade-passing frequency wi thout modula­t ion. The fundamental in line 7 4 0 is plotted versus rotation speed in Fig.11a. Peaks showed up in the higher harmonics at nearly the same rotation speeds as in the fund­amental. Thd second harmonic of the blade-passing frequency is shown in Fig.1 1 b). This indicates that the increases are not due to structural resonances, but might be

Ace. 1 v on turbine

Rot. speed: 50 - 24 Hz, step 2 Hz

Flat weighting Exp. average 8

N = 512 54 dB baseline Zoomtracking at line 780

Left sidebands from oilpump shaft modulation

Toothmesh Right sidebands from oilpump shaft modulation

nine lotation Speed

Hz

680 700 720 740 760 780 800 820 840 860 880 Line N° 810111

Fig.8. 2 0 0 line zoomed order analysis around toothmesh component

Bruel & Kjair

Full Scale Level:

F.S. Frequency . Weighting:

Time Function Start: seconds End: seconds

90 dB ? " " i

Not Expanded: — Expanded: _

u j - ■■ J J J J J - - - i - j

Hann. Average Mode: Exp.

No. of Spectra: 8 10 ^ -, j - - - - y - . -

Comments:

Ace, 1 V 20> J—>

ne78Q

T o o t h m e s h

c o m p o n e n t 30 r ^ ; ^ r r r '

of o i l p u m p gear

du r ing r u n d o w n 40 - -

50 - *

60

Record No.:. Datej

Sign.:

70 >

^ ^ r r ; r v r

f- r r ; ^ ^

__ ;

-i

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20 40 80 -[_J 5 3 TIZ

0

195 i .■.■.■.■

390

10

585

15

780 ^ -

20

975

25

1170

30

1365

35

1560 --^^^^ L- ^ ^

40

QP 1002

1755 * 1950

Actual freq. Hz 45 50

Rot. Speed Hz Measuring Object:

810105

Fig.9. Toothmesh component of oil pump gear during rundown

caused by increased turbulence in the blower at different speeds.

Conclusion By use of the external sampling

facility and zoom mode in the High Resolution Signal Analyzer Type 2033 aliasfree order analysis (track­ing) has become possible in the ana­lysis of vibrations during run-ups or run-downs of machines.

The analysis is performed in the zoom mode, requiring the long time record, and is not in real t ime, which means that the method is only applicable to machines that speed up and slow down slowly. The run through the speed range of interest should last more than, say, 5 minutes. The limitation is set by the one second calculation time and the amplitude variation along the time record. The 10 K time record contains a certain speed range of the machine and it is an average amplitude in that range which is cal­culated. This is however only a l imi­tation for very high Q resonances and caused no problems in the ex­ample showed in this note.

The great advantage of order ana­lysis is that all the components of in­terest can be analysed simultane­ously, and that all the power of the different harmonic or sideband com­ponents is concentrated in single lines, avoiding the smearing effect.

Apart from analysis of the lower harmonics, which of course are of main interest in determining reson­ances and critical speeds of large machines, zoomed order analysis around higher frequencies is possi-

Acc. 3V on generator

I

l l fcUB

I Rot. speed: 50 Hz - 28 Hz, step 2 Hz

Hann. weighting Exp. average 8

N = 512

Zoom tracking at line 740

70

Machine Rotation Speed

Hz 540 640 740 840 940

Line N 810113

Fig.10. 4 0 0 line zoomed order analysis around the 37 . harmonic (line 740) during rundown

ble. The high resolution analysis thus obtained gives the possibility of diagnosis which might have been impossible without use of tracking.

References Ref. 1:IM.Thrane:The discrete Fou­

rier Transform and FFT Analyzers. B & K Technical Review No. 1 -1979.

Ref. 2:N.Thrane:Zoom FFT. B & K Technical Review No. 2 -1980 .

Bri* l & K ] * r

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1 4 1

n*t*- < s * fr ^ E t + 4 + i

* 2 1

- . L I ■

Sign.: _ L 4i 4 i 4- 4 r ■ i M d m¥t J • 1

4- " 1 - ?

70 « y *

I C ^ I ■ J - V L ' J I . - J I I . 1 J*IIS I I ■■ _ ■ .- I

* 4-■ .■

* n i . - i i . - i i . - * . - I . - J J I T I I i i J p - i n i

i J I Mi r -. :

! ' 1B50

l_) ca C_D Actual f req. Hz 0 5 10 IB 20 ^ J 25 I X 35 4 0 4 5 5 0 - * -

OF tooz MuHinng Obpct:

' Rot . Speed H i OF tooz MuHinng Obpct:

5/ortw

Bnjtl * K|*r

FulJ Scale LCTEI:

F S FriK|ufrTKYn fc

Weighting:

Time Function Start-

no

No. df Sp_h.tr.i_ 8

Commentj:

Acc, 3 V

2. harmonic

o f bladt- passing

freq- o f ' an

H4t*rd No.:

S*gn_:

OP 1002

_dB f

| ^J "J-.- r.- r.-i .-i irr.- ir.-

20

J - -

4 0 :■•*■■-r-m ir .■ .■ + - r r ir

j - - r - -

_

^iLP v■■■■■■■ ■■■JI-"j■ ■■I -V IUI I ■

D U H --' r r i r ^JI .■

7 & J - - - - - - 4 - -

20 40 80 s j-CD IK] C71

0

l«cOndi End: s«a>nd_i

3 0 | . . . r - . - T .

II.-J .-J .-i jr..-i.-ji

r -T.-I+- ir tiV r ■■■ r.- i

■ .-J.VI J .-J -

N&t l-Kpandei-l _ _Expandefl. _

f >

r i+r.-n

i-.- r i rnr. - i - + i.- i ir.- i- -pi-' ir.-n

r.-ir.-n.-iiJ i .■ 3J11.■ Ji.vi .■ I . , J I . , I . V . V r.-ji.-L.-i .-J

to

t

IS

Menunng Object:

3700 . ■ . .■ .■ .■

Actual f rq .Hz J5 5 0 - ^

Rot .«Wed Hz

81QW7

Fig.11- Fundamental a) and second harmonic b) of blade-passing frequency of fan during run­down

8

Y

.

:>

Bruel & Kjaer Instruments, Inc. 185 Forest Street, Marlborough, Massachusetts 01752/(617) 481-7000 TWX 710/347-1187

Printed in Denmark by Nagrum Offset