10.7 Overlapped Processing

Back

Digital Fourier analysis is performed on blocks of sampled data (e.g., 210 ¼ 1024 samples at a time). In

overlapped processing, each data block is made to include part of the previous data block that was

analyzed. After completing a computation, the overlapped data at the end of the computed block is

moved to the beginning of the block, and the leading vacancy is filled with new data so that the end data

in one block is identical to the beginning data in the next block, in the overlapped region. In other words,

the overlapped portions of each data block (the two end portions) are processed twice. It follows that if

there is 50% (or more) overlapping then the entire data block is processed twice. Three main reasons can

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© 2005 by Taylor & Francis Group, LLC

be given for using overlapped processing in digital Fourier analysis:

1. It is an effective means of averaging spectral results.

2. It reduces the waiting time for assembling the data buffer.

3. It reduces the error caused by the end shaping effect of time windows (when a window other than

the rectangle window is used).

From reasons 1 and 2, it is clear that, due to overlapping, the statistical error of computations is

reduced for the same speed of computation, and the computing power is more efficiently used. To explain

reason 3, let us examine Figure 10.20. This example shows a 50% overlap in data. It is seen that the

window function can be assumed to be relatively flat, at least over 50% of the window length (record

length). Then the entire data block will correspond to the flat part of the window in three successive

analyses. Consequently, the shaping error (or the error due to increased analysis bandwidth) that is

caused by a nonrectangular time window is virtually eliminated by overlapped processing. The flatness of

a time window is determined by its effective noise bandwidth Be: The effective record length Te is

defined as

Te ¼

1

Be ð10:57Þ

Box 10.3

USEFUL RELATIONS FOR DIGITAL SPECTRAL

COMPUTATIONS

F yðtÞ

DFT

ðFFTÞ ! Y ð f Þ Fourier spectrum

1

T

Y p ð f ÞY ð fÞ ¼Power spectral density (PSD)

Power spectrum ¼ B £ Power spectral density ¼

B

T

Y p ð f ÞY ð f Þ

Energy spectrum ¼ T £ Power spectrum ¼ BY p ð f ÞY ð f Þ

Energy spectral density ¼

1

B £ Energy spectrum ¼ Y p ð f ÞY ð f Þ

RMS spectra

(always shown for

þve frequencies only)

¼

􀀑 2

B

Ðf þB

f lY ð f 0Þl2 df 0

􀀜1=2

p one sided

p like lY ðf Þl but smoother

p no phase information

p increase B ! high

bandwidth

Note:

T ¼ Record length

B ¼ Bandwidth of digital analysis (minimum frequency for which meaningful results are obtained) ! includes

window effect

Periodic or stationary signals

(infinite energy)

Use power spectra

Transient signals (finite energy) Energy spectra can be used

One-sided spectrum ¼ 2 £ (þve frequency part of two-sided spectrum)

Coherent output power ¼ coherence g2

uy £ output power ˆ could be power spectrum (spectrum or spectral density)

or PSD of the output

Vibration Signal Analysis 10-29

© 2005 by Taylor & Francis Group, LLC

which provides a measure for the flat segment of the window. The percentage effective record length is

given by Te as a percentage of the actual record length T: The degree of overlapping is chosen using the

relation

%overlap ¼ 100 1 2

Te

T

􀀏 􀀐

ð10:58Þ

Example 10.3

For a Hamming window, Be ¼ 1:4=T: Hence, a typical value for the percentage overlap is

100 1 2

1

1:4

􀀏 􀀐

¼ 29%

We might want to use a conservative overlap and even go up to 50% in this case because the window is

not quite flat.

10.7.1 Order Analysis

Speed related vibrations in rotation machinery may be analyzed through order analysis. Machinery

vibrations under start-up (accelerating) and shut-down (decelerating) conditions are analyzed in this

manner. Orders represent the rotating-speed-related frequency components in a response signal. The

ratio of the response frequency to the rotating speed is termed “order.”

Order analysis is done essentially through digital Fourier analysis of a rotating-speed-related response

signal. Practically, this may be accomplished in many ways. The format in which the spectral results are

presented will depend on the procedure used in order analysis. Some of the typical formats of data

presentation are given below.

10.7.1.1 Speed-Spectral Map

As the rotating speed of a machine is changing in a given range, the Fourier spectrum of the response

signal is determined for equal increments of speed. The results are presented as a speed spectral map

which is a three-dimensional cascade diagram (or waterfall display). The two base axes of the plot are

spectral frequency and rotating speed. The third axis gives the spectral magnitude (see Figure 10.21).

These types of plots are useful in identifying order-related components during start-up or coast-down

conditions. Note that for each speed the frequency band of digital Fourier analysis is kept the same (i.e.,

fixed sampling rate). Each distinct crest trace denotes an order-related resonance. The fact that these

traces are almost straight lines indicates the significance of order (the ratio, frequency/rotating speed) in

exciting these resonances.

FIGURE 10.20 Overlapped processing of windowed signals.

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© 2005 by Taylor & Francis Group, LLC

10.7.1.2 Time-Spectral Map

Under variable speed conditions (not necessarily accelerating or decelerating) the response signal is

Fourier analyzed at equal increments of time. The results are plotted in a cascade diagram, with frequency

and time as the base axes. The third axis again represents the magnitude of the Fourier spectrum (see

Figure 10.22). In this case, the crest traces are not necessarily straight, and can change their orientation

arbitrarily. This variation in crest orientation is determined by the degree of speed variation.

1000

2000

3000

4000

Speed

(rpm)

1.0 400.0

Frequency (Hz)

Spectrum

Magnitude

(dB)

0.0

40.0

Calibration: 100 mV/g

FIGURE 10.21 A speed-spectral map obtained from order analysis.

0.0

5.0

10.0

15.0

Time

(s)

0.5 200.0

Frequency (Hz)

Spectrum

Magnitude

(dB)

0.0

40.0

Calibration: 100 mV/g

FIGURE 10.22 A time-spectral map obtained from order analysis.

Vibration Signal Analysis 10-31

© 2005 by Taylor & Francis Group, LLC

10.7.1.3 Order Tracking

In order tracking, a “tracking frequency multiplier” monitors the rotating speed of the machine (as for a

speed-spectral map). But, in the present case, the sampling rate of the response signal (for Fourier

analysis) is changed in proportion to the rotating speed. Note that, in this manner, the maximum useful

frequency (approximately 400/512 £ Nyquist Frequency) is increased as the rotating speed increases, so

that the aliasing effects are reduced. If the same sampling rate is used for high speeds (as in the Speed-

Spectral Map discussed above), aliasing error can be significant at high rotating speeds.

In presenting order tracking spectral results, the frequency axis is typically calibrated in orders. Both

speed-spectral maps and time-spectral maps may be presented in this manner. Other types of data

presentation may be used as well in order analysis. For example, instead of the Fourier spectrum of the

response signal, power spectrum or composite power spectrum (in which the total signal power is

computed in specified frequency bands and presented as a function of the rotating speed) may be used in

the schemes described in this section.

Order analysis provides information on most severe operating speeds with respect to vibration (and

dynamic stress). For example, suppose that, for a given speed of operation, two major resonances occur,

one at 10 Hz and the other at 80 Hz. Then, the structure of the system (rotating machine and its support

fixtures) should be modified to change and preferably damp out these resonances. Furthermore, the most

desirable operating speed can be chosen in terms of the lowest resonant peaks by observing a speedspectral

map.

Bibliography

Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, Wiley-

Interscience, New York.

Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ.

Broch, J.T. 1980. Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum.

de Silva, C.W. 1983. Dynamic Testing and Seismic Qualification Practice, D.C. Heath and Co., Lexington,

MA.

de Silva, C.W., Optimal estimation of the response of internally damped beams to random loads in the

presence of measurement noise, J. Sound Vib., 47, 4, 485 – 493, 1976.

de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib. Dig.,

18, 10, 3 – 10, 1986.

de Silva, C.W. 2000. Vibration — Fundamentals and Practice, CRC Press, Boca Raton, FL.

de Silva, C.W. 2004. Mechatronics — An Integrated Approach, CRC Press, Boca Raton, FL.

Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd, Letchworth.

MATLAB Control Systems Toolbox, The Math Works, Inc., Natick, MA, 2004.

Meirovitch, L. 1980. Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville,

MD.

Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum.

10-32 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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