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Chapter 3

Gabor Transform-Based Order Tracking

LabVIEW Order Analysis Toolset User Manual

3-4

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processed by a Gabor transform, the index of the

n

th

order is given by the

following equation.

,

where

RPM

is the averaged instantaneous rotational speed in the time

interval,

N

is the number of frequency bins, and

f

s

is the sampling

frequency. In the LabVIEW Order Analysis Toolset, the number of
frequency bins

N

equals the length of the window.

Each order component in the order domain, as well as each harmonic in
the frequency domain, has a side band. Thus, in the joint time-frequency
domain, each order component contains the coefficients along the order
curve and some coefficients in the neighborhood. You must include the
coefficients in the neighborhood when selecting the order component.

Constant frequency bandwidth and constant order bandwidth are two ways
to define the bandwidth of the neighborhood for an order curve. Figure 3-3
illustrates constant frequency bandwidth and constant order bandwidth.

Figure 3-3.

Constant Frequency Bandwidth and Constant Order Bandwidth

Figure 3-3(a) illustrates constant frequency bandwidth. The neighborhood

is considered as the region between

and

, where

∆

f

is a constant frequency. The frequency of the bandwidth

∆

f

remains

constant over time.

index

round

RPM

60

-------------

N

f

s

----

n

×

×









=

500–

400–

300–

200–

100–

0–

0

5

25

30

35

10

15

20

39

Time (s)

(a) Constant Frequency Bandwidth

0

5

25

30

35

10

15

20

39

Time (s)

(b) Constant Order Bandwidth

F

requency (Hz)

500–

400–

300–

200–

100–

0–

F

requency (Hz)

n

RPM

60

-------------

∆

f

2

-----

+

n

RPM

60

-------------

∆

f

2

-----

–

«
...
32
33
34
35
36
...
»

Содержание LabVIEW

Страница 1: ...LabVIEW TM Order Analysis Toolset User Manual LabVIEW Order Analysis Toolset User Manual August 2003 Edition Part Number 322879B 01 ...

Страница 2: ...ermany 49 0 89 741 31 30 Greece 30 2 10 42 96 427 India 91 80 51190000 Israel 972 0 3 6393737 Italy 39 02 413091 Japan 81 3 5472 2970 Korea 82 02 3451 3400 Malaysia 603 9131 0918 Mexico 001 800 010 0793 Netherlands 31 0 348 433 466 New Zealand 0800 553 322 Norway 47 0 66 90 76 60 Poland 48 0 22 3390 150 Portugal 351 210 311 210 Russia 7 095 783 68 51 Singapore 65 6226 5886 Slovenia 386 3 425 4200 ...

Страница 3: ...accident actions of third parties or other events outside reasonable control Copyright Under the copyright laws this publication may not be reproduced or transmitted in any form electronic or mechanical including photocopying recording storing in an information retrieval system or translating in whole or in part without the prior written consent of National Instruments Corporation Trademarks LabVI...

Страница 4: ...ysis of Rotating Machinery 1 2 System Requirements 1 3 Installation 1 3 Example VIs 1 4 Acquiring Data for Example VIs 1 4 Configuring DAQ Hardware Used with Examples 1 4 Acquire Data Analog Tach VI 1 5 Acquire Data with PXI 4472 and TIO VI 1 6 Chapter 2 Order Analysis Order Analysis Definition and Application 2 1 Order Analysis Basics 2 1 Effect of Rotational Speed on Order Identification 2 4 Con...

Страница 5: ...3 8 Calculating Waveform Magnitude 3 10 Chapter 4 Resampling Based Order Analysis LabVIEW Order Analysis Toolset Resampling Method 4 1 Determining the Time Instance for Resampling 4 2 Resampling Vibration Data 4 4 Slow Roll Compensation 4 5 Chapter 5 Calculating Rotational Speed Digital Differentiator Method 5 1 Averaging Pulses 5 3 Appendix A Gabor Expansion and Gabor Transform Appendix B Referen...

Страница 6: ...o learn about how to use the LabVIEW Order Analysis Toolset Refer to Chapter 1 Introduction to the LabVIEW Order Analysis Toolset for information about the example VIs If you want to learn more about the algorithm used in the LabVIEW Order Analysis Toolset refer to Chapter 3 Gabor Transform Based Order Tracking and Chapter 4 Resampling Based Order Analysis For information about individual VIs refe...

Страница 7: ...cters that you should enter from the keyboard sections of code programming examples and syntax examples This font is also used for the proper names of disk drives paths directories programs subprograms subroutines device names functions operations variables filenames and extensions and code excerpts Platform Text in this font denotes a specific platform and indicates that the text following it app...

Страница 8: ...ither time or rotational speed Extraction of the order components from the original noise or vibration signal Measurement of the magnitude and phase of any order component as a function of rotational speed Presentation of data in a waterfall orbit or polar plot The LabVIEW Order Analysis Toolset includes easy and advanced VIs Use the easy VIs to perform simple tasks in just a few steps The advance...

Страница 9: ...ns for the Analysis of Rotating Machinery Order analysis is a powerful tool for analyzing rotating machinery when the rotational speed might change over time However to successfully use the LabVIEW Order Analysis Toolset you must observe the following condition and restriction Provide a signal directly related to the position of the shaft such as a pulse train from a tachometer or key phasor Altho...

Страница 10: ...Disable any automatic virus detection programs before you install After installation check your hard disk for viruses and enable any virus detection programs you disabled Windows 2000 NT XP Complete the following steps to install the LabVIEW Order Analysis Toolset 1 Log on as an administrator or as a user with administrator privileges 2 Insert the LabVIEW Order Analysis Toolset 2 0 installation CD...

Страница 11: ... have a Boolean control named Data Source The Data Source control has two choices Example and DAQ When you choose Example the VI uses prerecorded data generated during a fan run up as the data source When you choose DAQ the VI uses data you acquire with DAQ hardware When acquiring data through DAQ hardware National Instruments recommends you follow the following guidelines Use an anti aliasing fil...

Страница 12: ...bout the Acquire Data with PXI 4472 and TIO VI Acquire Data Analog Tach VI In analog tachometer examples setting Data Source to DAQ and clicking the Run button opens the Acquire Data Analog Tach VI The Acquire Data Analog Tach VI helps you acquire vibration data with a digital tachometer signal Figure 1 1 shows the Configuration tab of the Acquire Data Analog Tach VI Figure 1 1 Acquire Data Analog...

Страница 13: ... Figure 1 2 Figure 1 2 Acquire Data Analog Tach VI Acquisition Tab The Acquisition tab shown in Figure 1 2 allows you to acquire and observe data Click the Acquire button to acquire data Continue to configure the data acquisition and acquire data until you acquire the data you want Click the OK button to return to the front panel of the example VI to analyze the data Acquire Data with PXI 4472 and...

Страница 14: ...re the DAQ devices Use one of the counters on a TIO device to receive TTL compatible tachometer pulses Use the controls in the TIO Board Setting Digital Tach Signal section of the Configuration tab to configure the TIO device Use an NI PXI 4472 to acquire the data from the sound or vibration sensor Use the controls in the DSA Board Setting S V Signal and Input Settings sections of the Configure ta...

Страница 15: ...ith PXI 4472 and TIO VI Acquisition Tab The Acquisition tab shown in Figure 1 4 allows you to acquire and observe data Click the Acquire button to acquire data Continue to configure the data acquisition and acquire data until you acquire the data you want Click the OK button to return to the front panel of the example VI to analyze the data ...

Страница 16: ...f diagnosing heart problems You can use order analysis to study design and monitor rotating machinery By measuring and analyzing sound or vibration signals generated by a system with rotational components you can gain a better understanding of the system associate features of noise and vibration with the physical characteristics of the system and identify system characteristics that change with ti...

Страница 17: ...lysis for rotating machinery Figure 2 1 shows the relationship between frequency and order spectra Figure 2 1 Order and Frequency Domain Display of a Shaft Rotating at 3 000 rpm The top graph in Figure 2 1 shows a vibration signal from a machine running at 3 000 revolutions per minute rpm The rotational speed is computed from the tachometer signal which is shown as the second graph in Figure 2 1 T...

Страница 18: ...ults in a spectral peak at the first order Misalignment or bending of the shaft generates a large second order Oil whirl might lead to strong fractional orders Gears belts and blades might enhance high orders Figure 2 2 shows the order spectrum of the vibration signal measured from a PC fan with seven blades and four coils Figure 2 2 Order Spectrum of a PC Fan with Seven Blades and Four Coils The ...

Страница 19: ...lysis is effective only when the rotational speed remains constant Effect of Rotational Speed on Order Identification The ability to make a reliable identification of individual orders from the conventional power spectrum depends on whether rotational speed remains constant or varies This section discusses the effect rotational speed has on the conventional power spectrum and discusses classical h...

Страница 20: ...ed on the fast Fourier transform FFT The upper right plot in Figure 2 3 shows the frequency time spectral map computed from the short time Fourier transform STFT with a 1 024 point Hanning window Because of the constant rotational speed of the fan during data acquisition you can identify several peaks in both the power spectrum and the frequency time spectral map The peaks indicate different order...

Страница 21: ...e change in the frequency and amplitude of the signal The faster the frequency changes the wider the overall frequency bandwidth becomes as measured from the power spectrum Figure 2 4 illustrates a conventional power spectrum on the left and the frequency time spectral map on the right for a signal with constant frequency and amplitude Figure 2 4 Constant Frequency In Figure 2 4 the overall freque...

Страница 22: ...he overall frequency bandwidth of the signal whose frequency increases with time in Figure 2 5 is much wider than that of the signal whose frequency is constant in Figure 2 4 When frequency or amplitude vary with time the corresponding overall frequency bandwidth as measured from the conventional power spectrum becomes wide When the frequency bandwidth of the fundamental component widens the bandw...

Страница 23: ...ional speed increases both the fundamental frequency bandwidth and the frequency bandwidths of related orders widen causing orders to overlap Whether you can separate nearby orders in a power spectrum depends on the rate of change in the rotational speed the window used and the highest order of interest In Figure 2 6 the power spectrum is measured over the entire duration of the fan run up and doe...

Страница 24: ...cannot control the rotational speed of the system well or the system is not allowed to run at the critical rotational speed for a sufficient length of time Order Analysis An important goal of order analysis is to uncover information about the orders that might become buried in the power spectrum due to a change in rotational speed While the orders are hidden in the overall power spectrum in Figure...

Страница 25: ...abor expansion you cannot reconstruct the general STFT using an inverse Fourier transform You can compute the Gabor transform by either STFT or windowed Fourier transform However to ensure reconstruction of the signal you have to carefully manage the ratio of the analysis window to the window shift step and the capture of information at signal edges Use the following methods to ensure reconstructi...

Страница 26: ...or expansion the reconstructed data is longer than the original data To avoid the reconstructed data being longer than the original data consider the signal as periodic and use wrap padding shown in Figure 2 7 b In wrap padding when the first few data blocks are analyzed data blocks at the end of the data samples are wrapped ahead to fill the windows You also can wrap data blocks backward from the...

Страница 27: ...bout the Gabor transform and the Gabor expansion Refer to Chapter 3 Gabor Transform Based Order Tracking for information about how the LabVIEW Order Analysis Toolset uses a method based on the Gabor transform for order tracking 0 4 0 2 0 0 0 2 0 4 384 300 200 100 128 8064 8200 8300 8400 8500 8576 Time a Zero Padding 0 4 0 2 0 0 0 2 0 4 256 100 0 200 7808 7900 8000 8100 8200 8320 Time b Wrap Paddin...

Страница 28: ...er or a multiplied tachometer signal to trigger the analog to digital conversion which ensures a sampling process spaced equally in the rotation angle The hardware solution requires additional hardware devices and a tracking anti aliasing filter Software programs designed to acquire angle samples complete the following steps to acquire the angle samples 1 Collects the measured signal and the tacho...

Страница 29: ...the spectrum of angle samples Figure 2 10 Spectrum of Angle Samples In Figure 2 10 the Angle Samples plot shows the signal sampled at a constant angle interval The Order Spectrum plot shows the order spectrum obtained by performing a FFT on the angle samples In the Order Spectrum plot significant peaks appear at each order The Order vs Rev plot shows the result of a STFT on the angle samples where...

Страница 30: ...ponents change as the rotational speed changes you can consider the rotational speed and the frequency of order components to remain constant in a relatively short time interval The adaptive filter method of order analysis filters out the desired order by using a bandpass filter whose passband frequency shifts according to the rotational speed The Vold Kalman Order Tracking Filter is an example of...

Страница 31: ...oise or vibration signal Reconstruct the desired order components in the time domain Overview of Gabor Order Analysis The Gabor transform can give the power distribution of the original signal as the function of both time and frequency Figure 3 1 shows a frequency time spectral map computed by performing a Gabor transform on a sample vibration signal from a rotating machine Figure 3 1 Frequency Ti...

Страница 32: ...white curves have magnitudes larger than the magnitudes in local neighborhoods off the curves The white curves indicate the order components and are referred to as order curves From the frequency time spectral map you can separate the order curves or any other part of the signal in which you are interested from the intact original signal You then can use Gabor expansion to reconstruct time wavefor...

Страница 33: ...y performing Gabor expansion f Calculate the waveform magnitude and phase Refer to the Important Considerations for the Analysis of Rotating Machinery section of Chapter 1 Introduction to the LabVIEW Order Analysis Toolset for information about a condition and restriction for using the LabVIEW Order Analysis Toolset to analyze rotating machinery Extracting the Order Components After generating the...

Страница 34: ...d some coefficients in the neighborhood You must include the coefficients in the neighborhood when selecting the order component Constant frequency bandwidth and constant order bandwidth are two ways to define the bandwidth of the neighborhood for an order curve Figure 3 3 illustrates constant frequency bandwidth and constant order bandwidth Figure 3 3 Constant Frequency Bandwidth and Constant Ord...

Страница 35: ...der curves select a subset of the initial Gabor coefficient array The subset of the initial Gabor coefficient array contains only the coefficients in the neighborhood of the selected order curves You can generate the subset by performing a mask operation on the initial Gabor coefficient array To perform the mask operation you must construct a mask array The mask array is a 2D array the same size a...

Страница 36: ... number of elements set to TRUE determines the bandwidth of the passband Extracting Orders Figure 3 4 illustrates the order extraction and signal reconstruction process Figure 3 4 Gabor Coefficients Figure 3 4 a shows a frequency time spectral map after the Gabor transform 500 400 300 200 100 0 0 2 4 6 8 10 12 15 0 2 4 6 8 10 12 15 0 2 4 6 8 10 12 15 0 2 4 6 8 10 12 15 Time s Time s Time s Time s ...

Страница 37: ... 3 5 Original Signal and Extracted Order Component Unlike the reverse discrete Fourier transform the Gabor expansion in general is not a one to one mapping A Gabor coefficient is the subspace of a two dimensional function An arbitrary two dimensional function such as the masked coefficient array might not have a corresponding time waveform Usually the Gabor coefficients of the reconstructed time w...

Страница 38: ... outside the passband the magnitude of the reconstructed signal is no longer zero Instead the magnitude does have some certain value However the value of the magnitude quickly decreases as the frequency leaves the passband Displaying Spectral Maps Before extracting order components from the joint time frequency domain you might want to identify the order components in which you are most interested...

Страница 39: ...lar to the rpm time function shown in Figure 3 7 b In Figure 3 7 a notice the horizontal white lines The horizontal white lines indicate the large power around the resonance frequencies The physical characteristics of the system containing the rotating machinery determine the resonance frequency The resonance frequency does not change as the rotational speed changes 5000 4000 3000 2000 1000 0 0 5 ...

Страница 40: ...tral map and the order rpm spectral map you can clearly and efficiently observe all the order components in the entire time and rpm ranges Calculating Waveform Magnitude The reconstructed time waveform of the selected order contains only a few frequency components in a relatively short time interval Therefore the reconstructed time waveform of the selected order displays like a sine waveform in wh...

Страница 41: ...u to complete the following tasks Resample even time space samples to even angle spaced samples Obtain magnitude and phase information for each order Figure 4 1 illustrates the resampling based order analysis process provided by the LabVIEW Order Analysis Toolset Figure 4 1 Resampling Based Order Analysis Diagram Tachometer Signal Step 1 Data Acquisition Vibration Signal Step 2 Determine the Time ...

Страница 42: ...er magnitudes and phases if necessary Refer to the Important Considerations for the Analysis of Rotating Machinery section of Chapter 1 Introduction to the LabVIEW Order Analysis Toolset for information about a condition and restriction for using the LabVIEW Order Analysis Toolset to analyze rotating machinery Determining the Time Instance for Resampling To software resample even time spaced sampl...

Страница 43: ...ime sequence for a smaller angle The transfer function of the CIC filter is given by the following equation where L is the interpolation factor and n is the order The CIC filter has the advantage of only using a few samples in the original time sequence to obtain a single resampled point while maintaining good accuracy when the original signal is a narrow band signal Rotational speed usually does ...

Страница 44: ...gle spaced samples you must be able to calculate the value of the vibration signal at any time instance According to the Nyquist sampling theorem you can exactly reconstruct the signal for all time instances using band limited interpolation if and only if the original signal is band limited to half of the sampling rate For example a continuous time signal x t that is band limited to fs 2 Hz and sa...

Страница 45: ... lower that the previous sampling rate The LabVIEW Order Analysis Toolset uses a digital adaptive interpolation filter to complete the entire resample process The bandwidth of the adaptive interpolation filter automatically changes according to the new sampling rate to prevent the aliasing phenomenon The stopband attenuation of the interpolation filter controls the accuracy of the resampling As th...

Страница 46: ...ensated signal Figure 4 4 Slow Roll Compensation The LabVIEW Order Analysis Toolset can perform the vector based slow roll compensation on order magnitudes and phases The toolset performs slow roll compensation subtracting a slow roll vector from the measured magnitude and phase Use the OAT Extract Slow Roll Vector VI to compute the slow roll vector from a slow roll signal Use the OAT Slow Roll Co...

Страница 47: ...al speed as a function of time during the data acquisition process The LabVIEW Order Analysis Toolset uses a digital differentiator to calculate the rotational speed The tachometer processing VIs buffer the input tachometer pulses Because the tachometer processing VIs buffer the input tachometer pulses they can use tachometer pulses occurring before and after each time instance when calculating ro...

Страница 48: ...ation you can present rotational speed as the first derivative of θ k While the time interval between pulses varies with the rotational speed the cumulative angle increment remains constant Calculating the first derivative of θ t usually involves step by step polynomial curve fitting which requires multiple computations in each iteration Instead of observing θ t directly the LabVIEW Order Analysis...

Страница 49: ...e rotational speed also increases The LabVIEW Order Analysis Toolset can automatically adjust the order of the differentiator according to the number of pulses in the buffer Automatically adjusting the order of the differentiator improves the accuracy of the calculated rotational speed when the buffer contains a large number of pulses and makes it possible to generate a result even if the buffer c...

Страница 50: ... In Figure 5 2 the dotted line represents the results of calculating the rotational speed with no averaging The solid line in Figure 5 2 represents the results of calculating the rotational speed averaged by five As Figure 5 2 shows the averaged rotational speed is smoother than the non averaged rotational speed 3059 3 3055 0 3050 0 3045 0 3040 0 19 2 19 3 19 4 19 5 19 6 Rotational Speed RPM Speed...

Страница 51: ...implementation was an open research topic until Bastiaans discovered the relationship between the Gabor expansion and the short time Fourier transform STFT in the early 1980s Over the years many different implementation schemes for the discrete Gabor expansion were proposed The LabVIEW Order Analysis Toolset uses an extension of the method originally developed by Wexler and Raz to implement the di...

Страница 52: ...form in Equation A 2 contains redundancy from a mathematical point of view However the redundancy in Equation A 2 provides freedom for the selection of better window functions h k and γ k Notice that the positions of the window functions h k and γ k are interchangeable In other words you can use either of the window functions as the synthesis or analysis window function Therefore h k and γ k are u...

Страница 53: ...E For the LMSE solution the Euclidean distance between the dual functions is minimum as given by the following equation where denotes the matrix form of Equation A 4 When the error is small and Equation A 2 becomes A 5 Equations A 5 and A 1 form an orthogonal like Gabor transform pair In the case of an orthogonal like Gabor transform pair the Gabor coefficients cm n are the projection of the signa...

Страница 54: ... exist For example the following equation represents a modified two dimensional function where Φm n denotes a binary mask function whose elements are either 0 or 1 Applying the Gabor expansion to the modified two dimensional function results in the following equation The following inequality results from Gabor expansion The Gabor coefficients of the reconstructed time waveform are not equal to the...

Страница 55: ...Gabor expansion from Equation A 1 and the Gabor transform from Equation A 2 in matrix form as shown in the following equations where H denotes the analysis matrix and G denotes the synthesis matrix Notice that for over sampling the following relationships are true Therefore in the case of over sampling the iterative process is described by the following equations C Gs s H T Gs H T G I GH T I C 1 Φ...

Страница 56: ...t in terms of the LMSE to the masked Gabor coefficients ΦC The masked Gabor coefficients are the desirable Gabor coefficients The case of γ k h k usually implies considerable over sampling which results in a huge amount of redundancy The amount of redundancy causes slow computation speed and huge memory consumption making γ k h k an impractical Gabor transform method Usually the LabVIEW Order Anal...

Страница 57: ...her Order Tracking Methods SAE Noise and Vibration Conference and Exposition Traverse City MI April 30 through May 3 2001 Gade S H Herlufsen H Konstantin Hansen and H Vold Characteristics of the Vold Kalman Order Tracking Filter Brüel Kjær Sound Vibration Measurement A S 1999 Hewlett Packard Application Note 243 1 Effective Machinery Measurements using Dynamic Signal Analyzers Qian Shie and Dapang...

Страница 58: ...sted Support Options Contact NI engineers and other measurement and automation professionals by visiting ni com support Our online system helps you define your question and connects you to the experts by phone discussion forum or email Training Visit ni com training for self paced tutorials videos and interactive CDs You also can register for instructor led hands on courses at locations around the...

Страница 59: ...uency less than half the sampling frequency See also anti aliasing filter and Nyquist frequency anti aliasing filter Analog lowpass filters used before analog to digital conversion to filter out the frequencies greater than half the sampling frequency See also aliasing and Nyquist frequency C critical sampling Occurs in Gabor transform when the window length equals the window shift step In critica...

Страница 60: ...oint time frequency transforms H harmonic Frequencies that are integer or fractional multiples of a fundamental frequency L LMSE Least mean square error N Nyquist frequency Half the sampling frequency Any analog frequency component above the Nyquist frequency will after sampling be converted or aliased to a frequency below the Nyquist frequency See also aliasing and anti aliasing filter O order A ...

Страница 61: ...h k over sampling Occurs in a Gabor transform when the window length is greater than the window shift step In over sampling the number of Gabor coefficients cm n is more than the number of original data samples s k R resampling A method to sample a time sequence at a different time interval RMS Root mean square RPM Revolutions per minute S STFT Short time Fourier transform W window shift step The ...

Страница 62: ...s 1 4 guidelines for 1 4 diagnostic resources C 1 displaying spectral maps 3 8 documentation conventions used in the manual vii getting started vii how to use this manual vii online library C 1 related documentation viii drivers instrument C 1 software C 1 E example code C 1 examples data acquisition 1 4 extracting order components using Gabor transform 3 3 G Gabor expansion A 1 equation for A 1 t...

Страница 63: ... Up Kit 1 1 Order Analysis Toolset using in analysis of rotating machinery 1 2 orders definition of 2 2 over sampling definition of A 2 P phone technical support C 1 processing tachometer signals 5 1 calculating rotational speed averaging pulses 5 3 professional services C 1 programming examples C 1 R references B 1 related documentation viii resampling 2 13 4 1 determining time instance 4 2 S sof...

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