LabVIEW TM Order Analysis Toolset User Manual LabVIEW Order Analysis Toolset User Manual August 2003 Edition Part Number 322879B-01
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Contents About This Manual How to Use This Manual ...............................................................................................vii Conventions ...................................................................................................................vii Related Documentation..................................................................................................
Contents Chapter 3 Gabor Transform-Based Order Tracking Overview of Gabor Order Analysis............................................................................... 3-1 Extracting the Order Components ................................................................................. 3-3 Masking ......................................................................................................................... 3-5 Extracting Orders .....................................................................
About This Manual This manual provides information about the LabVIEW Order Analysis Toolset, including system requirements, installation, and suggestions for getting started with order analysis and the toolset. The manual also provides a brief discussion of the order analysis process and the algorithm used by the LabVIEW Order Analysis Toolset.
About This Manual bold Bold text denotes items that you must select or click in the software, such as menu items and dialog box options. Bold text also denotes the names of parameters, dialog boxes, sections of dialog boxes, windows, menus, palettes, and front panel controls and buttons. italic Italic text denotes variables or cross references. monospace Text in this font denotes text or characters that you should enter from the keyboard, sections of code, programming examples, and syntax examples.
Introduction to the LabVIEW Order Analysis Toolset 1 This chapter introduces the LabVIEW Order Analysis Toolset and the Order Analysis Start-Up Kit, outlines system requirements, and gives installation instructions. Overview of the LabVIEW Order Analysis Toolset The LabVIEW Order Analysis Toolset is a collection of virtual instruments (VIs) for LabVIEW.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset application is built with components found in the LabVIEW Order Analysis Toolset. The order analysis application provides an example of how the LabVIEW Order Analysis Toolset can help you successfully complete analysis projects.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset System Requirements You must have LabVIEW 6.1 or later Full Development System or Professional Development System installed to run the LabVIEW Order Analysis Toolset. Refer to the LabVIEW Release Notes for the required system configuration for LabVIEW. Note Order analysis is a memory-intensive task, especially when you display spectral maps. Increasing the amount of RAM in your system can significantly increase system performance.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset Example VIs If you have experience with order analysis, the example VIs, located in the examples\Order Analysis directory, can help you learn how to use the LabVIEW Order Analysis Toolset.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset set to DAQ and you click the Run button on the front panel of the example VI, one of the following VIs opens: • Acquire Data (Analog Tach) VI • Acquire Data with PXI 4472 and TIO VI Refer to the Acquire Data (Analog Tach) VI section for information about the Acquire Data (Analog Tach) VI and the Acquire Data with PXI 4472 and TIO VI section for information about the Acquire Data with PXI 4472 and TIO VI.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset tachometer signal and S&V for the sound or vibration sensor. After choosing data acquisition settings, enter the number of pulses you want the tachometer to generate per revolution in the Tach Pulse/Rev text box. Use the controls in the Channel Info section of the Configuration tab to specify the channel information for the sound or vibration sensor. After configuring the DAQ device, click the Acquisition tab, shown in Figure 1-2. Figure 1-2.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset Figure 1-3. Acquire Data with PXI 4472 and TIO VI Configuration Tab Use the Configuration tab of the Acquire Data with PXI 4472 and TIO VI, shown in Figure 1-3, to configure 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.
Chapter 1 Introduction to the LabVIEW Order Analysis Toolset Figure 1-4. Acquire Data with 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. LabVIEW Order Analysis Toolset User Manual 1-8 ni.
2 Order Analysis This chapter gives brief descriptions of the need for order analysis, the basic concepts of order analysis, the effect of rotational speed on order identification, and the different order analysis methods. Order Analysis Definition and Application When it is impossible or undesirable to physically open up a system and study it, you often can gain knowledge about the system by measuring and analyzing signals associated with the system.
Chapter 2 Order Analysis vibration signals. The frequency-domain representations of noise and vibration behave as harmonics of the machine rotational speed. In many industries, the harmonics related to the rotational speed are referred to as orders. The corresponding harmonic analysis is called order analysis. The harmonic at the same frequency as that of the rotational speed is the first order; the harmonic at twice the frequency of the rotational speed is the second order and so on.
Chapter 2 Order Analysis Assuming that speed remains constant during data acquisition, you can use the following equations to switch between the frequency domain and the order domain. RPM Frequency = ------------- × Order 60 60 Order = Frequency × ------------RPM Orders often reflect the physical characteristics of rotating machines.
Chapter 2 Order Analysis Like classical harmonic analysis, order analysis is a powerful tool for gaining a better understanding of the condition of rotating machinery. However, compared to harmonic analysis, order analysis is more effective for the analysis of rotating machinery because you can use order analysis when a machine runs at a constant speed and when the rotational speed varies.
Chapter 2 Spectrum Order Analysis STFT 570 500 450 Frequency (Hz) 400 350 300 250 200 150 100 50 0 31.9 31.9 31.9 Figure 2-3. PC Fan Running at Constant Speed The bottom plot in Figure 2-3 depicts the tachometer pulses and the signal from an accelerometer mounted on the PC fan. The plot on the left in Figure 2-3 illustrates a conventional power spectrum based on the fast Fourier transform (FFT).
Chapter 2 Order Analysis Variable Rotational Speed In addition to testing rotating machinery running at a constant speed, researchers often perform tests involving run-up and run-down. Like a swept-sine stimulus, testing run-up and run-down provides a stimulus over a wide range of frequencies. According to Fourier analysis theory, the frequency bandwidth of a signal is proportional to the change in the frequency and amplitude of the signal.
Chapter 2 Order Analysis time, the overall frequency bandwidth, as measured from the conventional power spectrum, is the minimum. Figure 2-5 depicts a signal whose frequency changes as a function of time. 0.5 Spectrum Frequency-Time Spectral Map Frequency 0.4 Overall Bandwidth Becomes Wide as the Frequency Changes 0.3 0.2 0.1 0.0 Time Waveform 1.0 0.5 0.0 –0.5 –1.0 0 50 100 150 200 255 Time Figure 2-5.
Chapter 2 Order Analysis harmonics overlap in the conventional power spectrum, you are unable to identify different harmonics by using the conventional power spectrum. Figure 2-6 shows the signal from the same PC fan as the one depicted in Figure 2-3. However, in Figure 2-6, the rotational speed of the fan increases with time. Spectrum STFT 1500 Frequency (Hz) 1250 1000 750 500 250 0 31.9 31.9 31.9 Time (sec.) Figure 2-6.
Chapter 2 Order Analysis orders overlap, you are unable to derive meaningful information about the individual orders. Harmonic Analysis Harmonic analysis is suitable for the analysis of rotating machinery only when the rotational speed remains constant. In classical harmonic analysis, the fundamental frequency does not change over time. Although the phases and amplitudes of the individual harmonics can vary over time, the center frequencies of all the harmonics remain constant.
Chapter 2 Order Analysis Currently, the following methods generally are used to perform order analysis: • Gabor transform • Resampling • Adaptive filter The LabVIEW Order Analysis Toolset provides the Gabor-transform-based method and the resampling method of order analysis. Gabor Transform The Gabor transform is one of the invertible joint time-frequency transforms.
Chapter 2 Order Analysis Windows Data Blocks Zeros 1 2 n–1 n Zeros (a) Zero Padding Windows Data Blocks n–1 n 1 2 n–1 n 1 (b) Wrap Padding Figure 2-7. Padding Schemes In zero padding, shown in Figure 2-7(a), the first window and the last window only cover one data block each. The remaining area in the range of the window is filled with zeros. Thus, extra data samples are added. After Gabor expansion, the reconstructed data is longer than the original data.
Chapter 2 Order Analysis Figure 2-8 shows two examples of padding. Amplitude 0.4 0.2 0.0 –0.2 –0.4 –384 –300 –200 –100 128 8064 Time (a) Zero Padding 8200 8300 8400 8500 8576 Amplitude 0.4 0.2 0.0 –0.2 –0.4 –256 –100 0 100 200 256 7808 7900 Time (b) Wrap Padding 8000 8100 8200 8320 Figure 2-8. Padding Examples Refer to Appendix A, Gabor Expansion and Gabor Transform, for more information about the Gabor transform and the Gabor expansion.
Chapter 2 Order Analysis Resampling Resampling is a widely used method of order analysis. Figure 2-9 illustrates the resampling method. Sample at Constant Time Frequency Sample at Constant Angle Order Figure 2-9. Resampling In the resampling method, time-samples are converted to angle samples. The time-samples are samples of the physical signal that are equally spaced in time. The angle samples are samples that are equally spaced in the rotation angle.
Chapter 2 Order Analysis After acquiring angle samples with either hardware or software, you can perform a Fourier transform on the angle samples to analyze them. Because the time domain has changed into the angle domain, the frequency domain now becomes the order domain. You can separate and observe the order components through either the STFT spectrum or the order spectrum, which is actually the FFT spectrum. Figure 2-10 shows the spectrum of angle samples. 10 Order Spectrum Order vs.
Chapter 2 Order Analysis After the resampling process takes place, recovering a time waveform at a specific order might be difficult. Refer to Chapter 4, Resampling-Based Order Analysis, for information about how the LabVIEW Order Analysis Toolset uses software-based resampling for order analysis.
Gabor Transform-Based Order Tracking 3 This chapter discusses a new order analysis method based on the Gabor transform and provided by the LabVIEW Order Analysis Toolset that enables you to complete the following tasks: • Analyze the order components of a noise 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.
Chapter 3 Gabor Transform-Based Order Tracking In Figure 3-1, the magnitudes of coefficients are shown as gray scale, with full white indicating a maximal magnitude and full black indicating a minimal magnitude. Because the rotational speed changes little in each time portion of the frequency-time spectral map, the spectrum of each order is clearly distinguishable. As the rotational speed varies over time, the frequency of one certain order component changes.
Chapter 3 Gabor Transform-Based Order Tracking Complete the following steps to perform Gabor order analysis. 1. Acquire data samples from the tachometer and noise or vibration sensors synchronously at some constant sample rate. 2. Use the LabVIEW Order Analysis Toolset VIs to complete the following steps. a. Perform a Gabor transform on the noise or vibration samples to produce an initial Gabor coefficient array. b. Calculate the rotational speed from the tachometer signal.
Chapter 3 Gabor Transform-Based Order Tracking processed by a Gabor transform, the index of the nth order is given by the following equation. RPM N index = round ------------- × ---- × n , 60 fs where RPM is the averaged instantaneous rotational speed in the time interval, N is the number of frequency bins, and fs is the sampling frequency. In the LabVIEW Order Analysis Toolset, the number of frequency bins N equals the length of the window.
Chapter 3 Gabor Transform-Based Order Tracking Figure 3-3(b) illustrates constant order bandwidth. The neighborhood is RPM RPM considered as the region between n + --k- × ------------- and n – --k- × ------------- . 60 60 2 2 RPM While the frequency bandwidth k ------------- varies in time, the order bandwidth 60 k remains constant. You can use the Gabor Order Analysis VIs to determine the frequency bandwidth or order bandwidth automatically or in response to your input.
Chapter 3 Gabor Transform-Based Order Tracking VI to perform the mask operation. Refer to the LabVIEW Order Analysis Toolset Help for information about the Template to Mask VI. You can consider the mask operation a time-variant bandpass filter in the joint time-frequency domain. The center frequency of the passband equals the frequency of the order curve. The number of elements set to TRUE determines the bandwidth of the passband.
Chapter 3 Gabor Transform-Based Order Tracking Figure 3-4(b) shows a mask array around the fourth order with a constant frequency bandwidth. The TRUE values comprise the white area, while the FALSE values comprise the black areas. Figure 3-4(c) shows the masked coefficients. The black areas indicate values set to zero and correspond to the FALSE values in the mask array. The white area contains values copied from the original coefficient array and corresponds to the TRUE values in the mask array.
Chapter 3 Gabor Transform-Based Order Tracking time-variant filters. Figure 3-6 shows an original signal and the spectrum of the reconstructed signal. dB –49.1– Original Signal –100.0– Reconstructed Order –150.0– Expected Pass Band –179.0– 0 50 100 150 200 250 Frequency (Hz) 300 350 408 Figure 3-6. Spectrum of Reconstructed Signal In Figure 3-6, one row is selected from the joint time-frequency coefficient array.
5000 3000 4000 2500 2000 1500 1000 1000 500 0 0 0 5 10 15 20 25 30 36 0.0 5.0 Time (s) (a) Frequency-Time Spectral Map 10.0 15.0 20.0 25.0 30.0 36.0 Time (s) (b) RPM vs.
Chapter 3 Gabor Transform-Based Order Tracking Figure 3-7(c) shows the frequency-rpm spectral map. The horizontal axis represents rpm, or rotational speed. You can map the rpm axis from the time axis in the frequency-time spectral map by the rpm-time function. The frequency of each order component is calculated by the following equation.
4 Resampling-Based Order Analysis This chapter describes the resampling method provided by the LabVIEW Order Analysis Toolset, determining the time instance for resampling, resampling vibration data, and slow roll compensation.
Chapter 4 Resampling-Based Order Analysis Complete the following steps to perform resampling-based order analysis. 1. Acquire data samples from tachometer and noise or vibration sensors synchronously at some constant sample rate. 2. Use the LabVIEW Order Analysis Toolset VIs to complete the following steps: a. Determine the pulse edges from the tachometer signal and interpolate the pulse edges to get the time instance for resampling. b.
Chapter 4 Resampling-Based Order Analysis a larger margin of samples for analysis and need to resample 2.5K samples in one revolution. Most of the time, a tachometer does not produce enough pulses in one revolution. Therefore, you need to multiply the number of pulses generated by the tachometer, that is, interpolate the time sequence for a smaller angle interval. When you interpolate the time sequence for a smaller angle interval, you need a constant rate integer factor interpolation filter.
Chapter 4 Resampling-Based Order Analysis Figure 4-2. Original Time Sequence and CIC Interpolation Filter with an Interpolation Factor of Eight Resampling Vibration Data The resampling operation converts a vibration signal from the time domain into the angle domain. To resample vibration data into even angle spaced samples, you must be able to calculate the value of the vibration signal at any time instance.
Chapter 4 Resampling-Based Order Analysis According to the Nyquist sampling theorem, you can exactly reconstruct the time signal x(t) from samples x(nTs) with the following equation. x̂ ( t ) = ∑ x ( nT )h ( nT s s s – t) (4-1) n sin (π f s t) where hs is a sinc function defined by h s ( t ) = sinc ( f s t) = ---------------------. π fs t Figure 4-3 shows the plot of hs(t) with fs = 1. Figure 4-3.
Chapter 4 Resampling-Based Order Analysis When monitoring shaft vibration with a proximity probe, the acquired signal contains not only the vibration of the shaft but also the shaft runout, such as a nonconcentric shaft condition at the probe measurement plane. You can measure the shaft runout at a slow roll speed and can compensate for it in the raw proximity probe signal. Figure 4-4 shows the raw signal acquired from a nonconcentric shaft and the compensated signal.
Calculating Rotational Speed 5 This chapter describes the digital differentiator method used by the LabVIEW Order Analysis Toolset to calculate rotational speed and describes averaging pulses to smooth noisy calculated rotational speed results. Digital Differentiator Method To successfully extract order components, create spectral maps, and calculate waveform magnitudes, you need rotational speed as a function of time during the data acquisition process.
Chapter 5 Calculating Rotational Speed t1 t2 t3 t4 t (a) Tachometer Pulse ϖ3 θ4 ∆θ θ3 ∆θ θ2 θ1 ∆θ t t1 t2 t3 t4 (b) Rotation Angle vs. Time Figure 5-1. Tachometer Signal and Rotation Angle The arrival time tk of the pulses P(k) in the tachometer signal correspond to the cumulative rotation angle θ(tk). The cumulative rotation angle function θ(t) increases by a fixed angle θ in the time interval between times tk and tk+1.
Chapter 5 Calculating Rotational Speed of an equally spaced sequence requires fewer computations than calculating the first derivative of an unequally spaced sequence. For the evenly separated function series tk, you can use a digital differentiator to calculate the first derivative. The digital differentiator is implemented as a special FIR filter with the following equation. dt----dθ 1 ≈ ------∆θ θ = θk M ∑h t i k–i , (5-1) i = –M where hi represents the differentiator coefficients.
Chapter 5 Calculating Rotational Speed Figure 5-2 shows an example of the rotational speed calculated from the tachometer pulses. Rotational Speed (RPM) 3059.3 3055.0 Speed (No Average) 3050.0 Speed (Averaged by 5) 3045.0 3040.0 19.2 19.3 19.4 Time (s) 19.5 19.6 Figure 5-2. Cumulative Revolution and Rotational Speed In Figure 5-2, the dotted line represents the results of calculating the rotational speed with no averaging.
A Gabor Expansion and Gabor Transform This appendix presents an overview of the Gabor expansion and the Gabor transform methods used in the LabVIEW Order Analysis Toolset. This appendix also describes application issues associated with using the discrete Gabor-expansion-based time-varying filter. Gabor Expansion and Gabor Transform Basics The Gabor expansion characterizes a signal jointly in the time and frequency domains.
Appendix A Gabor Expansion and Gabor Transform The sampled STFT is also known as the Gabor transform and is represented by the following equation. N–1 c m, n = ∑ s̃ [ k ]γ∗ [ k – m∆M ]e – j2πnk ⁄ N (A-2) n=0 where ∆M represents the time sampling interval and N represents the total number of frequency bins. The ratio between N and ∆M determines the Gabor sampling rate. For numerical stability, the Gabor sampling rate must be greater than or equal to one. Critical sampling occurs when N = ∆M.
Appendix A Gabor Expansion and Gabor Transform corresponding dual window function from ∆M independent linear systems with the following equation. L- – 1 -------∆M δ[q] ∑ ã [ k + p∆M + qN ]γ∗ [ k + p∆M ] = ---------N 0 ≤ k < ∆M (A-4) p=0 where L denotes the window length. ã [ k ] denotes a periodic auxiliary function1 given by the following equation.
Appendix A Gabor Expansion and Gabor Transform Discrete Gabor-Expansion-Based Time-Varying Filter Initially, discrete Gabor expansion seems to provide a feasible method for converting an arbitrary signal from the time domain into the joint time-frequency domain or vice versa. However, discrete Gabor expansion is effective for converting an arbitrary signal from the time domain into the joint time-frequency domain or vice versa only in the case of critical sampling, ∆M = N.
Appendix A Gabor Expansion and Gabor Transform 4. Compute the new Gabor coefficients after you obtain the time waveform. 5. Repeat steps 1 through 4 until the time waveforms converge. Without a loss of generality, rewrite the Gabor expansion from Equation A-1 and the Gabor transform from Equation A-2 in matrix form, as shown in the following equations. C = Gs T s = H Gs where H denotes the analysis matrix and G denotes the synthesis matrix.
Appendix A Gabor Expansion and Gabor Transform k k k k k Ck and s converge. C = ΦC with k → ∞ . For C = ΦC with k → ∞ , the support of Ck in the time-frequency domain is inside the masked area if and only if the following equation is true. L- – 1 --N ∑ γ∗ [ iN + k ]h [ iN + k + m∆M ] = i=0 L- – 1 --N (A-6) ∑ h∗ [ iN + k ]γ [ iN + k + m∆M ] i=0 L for 0 ≤ k < N and 0 ≤ m < --------- . Two trivial cases for Equation A-6 are ∆M critical sampling and when γ[k] = h[k]. In critical sampling, N = ∆M.
B References This appendix lists the reference material used for the LabVIEW Order Analysis Toolset. • Albright, Michael, and Shie Qian. Comparison of the New Proposed Gabor Order Tracking Technique vs. Other 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.
Technical Support and Professional Services C Visit the following sections of the National Instruments Web site at ni.com for technical support and professional services: • Support—Online technical support resources include the following: – Self-Help Resources—For immediate answers and solutions, visit our extensive library of technical support resources available in English, Japanese, and Spanish at ni.com/support.
Glossary Numbers/Symbols ∞ Infinity. 2D Two-dimensional. A adaptive filter A type of filter that operates on a recursive algorithm to achieve the goal of optimum. It is self-designing and suitable for tracking time variations of the input signal, but complete knowledge of the relevant signal characteristics is not available.
Glossary F FFT Fast Fourier transform. fundamental component Portion of a signal whose frequency is at the fundamental frequency. G Gabor coefficient The result of Gabor transform. Gabor expansion The inverse Gabor transform used on Gabor coefficients to recover a time domain input signal. Gabor transform One of the invertible joint time-frequency transforms. H harmonic Frequencies that are integer or fractional multiples of a fundamental frequency. L LMSE Least mean square error.
Glossary order curves The high power density curves in a spectral map which indicate order components. orthogonal-like Gabor transform pair The pair is composed of a Gabor transform and a Gabor expansion when the Gabor coefficients cm, n found by each method are the projection of the signal on the synthesis window function h[k]. over sampling Occurs in a Gabor transform when the window length is greater than the window shift step.
Index A drivers instrument, C-1 software, C-1 adaptive filter, 2-15 analysis of rotating machinery, using the toolset, 1-2 applying an iteration to reconstruct a time waveform, A-4 E example code, C-1 examples, data acquisition, 1-4 extracting order components using Gabor transform, 3-3 C calculating rotational speed, 5-1 averaging pulses, 5-3 digital differentiator, 5-1 calculating waveform magnitude, 3-10 contacting National Instruments, C-1 conventions used in the manual, vii critical sampling, defin
Index K professional services, C-1 programming examples, C-1 KnowledgeBase, C-1 R M references, B-1 related documentation, viii resampling, 2-13, 4-1 determining time instance, 4-2 masking, 3-5 extracting orders, 3-6 reconstructing a signal, 3-7 N S National Instruments customer education, C-1 professional services, C-1 system integration services, C-1 technical support, C-1 worldwide offices, C-1 software drivers, C-1 spectral maps, displaying, 3-8 support, technical, C-1 system integration serv
