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//INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 559 ± [558±583/26] 29.10.2001 4:05PM For example: 6db 2 10 db 3:16 20 db 10 40 db 100 If the S/N ratio is less than 10 db, it becomes difficult to differentiate the periodic part of the spectrum from noise. Several types of analyzers exist today that allow a time-domain signal to be converted to a frequency-domain spectrum. The resulting spectrum of all spectrum analyzers is equivalent to the amplitude/frequency plot, which is obtained by passing the given signal across a set of constant bandwidth filters and noting the output of each filter at its center frequency. Unfortunately, such a simple procedure cannot be used because, for adequate resolution, each filter can cover only a very narrow frequency band, and because of the cost involved. In the so-called ``wave analyzer'' or ``tracking filter'' one filter is utilized by manually incrementing the filter across the time input to determine which frequencies exhibit a large ampli- tude. In time-compression real-time analyzers (RTA) the filter is swept electronically across the input. The term ``real time'' as applied here means the instrument takes the time-domain signal and converts it to a frequency domain while the event is actually taking place. In technical terms, real time is viewed when the rate of sampling is equal to or greater than the bandwidth of the filters taking the measurements. RTAs use an analog-to-digital converter and digital circuits to speed up the data signal effectively and improve the sweeping filter scan rate, thus creating an apparent time com- pression. Both of the previous analyzers are basically analog instruments and, because of the characteristics of analog filtering, may be quite slow at lower frequencies. The Fourier analyzer is a digital device based on the conversion of time- domain data to a frequency domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minicomputer to solve a set of simultaneous equations by matrix methods. Time domains and frequency domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a function of time may be decomposed into a series of oscillatory functions (each with a characteristic frequency), which when superpositioned or summed at each time, will equal the original expression of the variable. This Spectrum Analysis 559 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 560 ± [558±583/26] 29.10.2001 4:05PM process is shown graphically in Figure 16-1. Since each of the oscillatory signals has a characteristic frequency, the frequency domain reflects the amplitude of the oscillatory function at that corresponding frequency. The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F(t) with a period t, a Fourier series may be expressed as Ft a 0 2 I n1 a n cos n!t b n sin n!t16-2 Here a and b are amplitudes of the oscillatory functions cos (n!t) and sin (n!t), respectively. The value of ! is related to the characteristic frequency f by ! 2f 16-3 Figure 16-1. Decomposition of a time signal into a sum of oscillatory functions from which a spectrum can be obtained. 560 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 561 ± [558±583/26] 29.10.2001 4:05PM The previous function may also be written in a complex form as Ft I ÀI G!e i!t d! 16-4 where: G! 1 2 I ÀI Fte Ài!t dt 16-5 The function G(!) is the exponential Fourier transform of F(t) and is a function of the circular frequency !. In practice the function F(t) is not given over the entire time domain but is known from time zero to some finite time T, as shown in Figure 16-2. The time span T may be divided into K equal increments of Át each. For computational reasons, let K 2 p where p is an integer. Also, let the circular frequency span ! n be divided into N parts where N 2 q . (In practice, N is often set equal to K.) By setting f K=NT, the frequency interval Á! becomes Á! 2Áf 2K NT 16-6 Now, discrete equations analogous to Equations (16-3) and (16-4) may be defined Ft k Á! NÀ1 n0 G! n e i! n t k 16-7 and G! n Át 2 kÀ1 k0 Ft k e Ài! n t k 16-8 where the limits are set at 0 and N À 1 for computational reasons. By using Euler identities, Equations (16-6) and (16-7) can be written G! n real nÀ1 n0 Ft k cos! n t k 16-9 Spectrum Analysis 561 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 562 ± [558±583/26] 29.10.2001 4:05PM G! n imaginary nÀ1 n0 Ft k sin! n t k 16-10 Ft k Á! nÀ1 n0 G! n real cos! n t k G! n imaginary sin! n t k 16-11 Comparison of the previous equations with Equations (16-6) and (16-7) reveal that the Fourier transform is really just a Fourier series constructed over a finite interval. Figure 16-2. Discrete Fourier transform representation. 562 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 563 ± [558±583/26] 29.10.2001 4:05PM The equations may be rewritten in a simpler form by making the following definitions: " F k Ft k 16-12 G n G! n 16-13 ! n nÁ! 2nK NT 16-14 t K KÁt 16-15 so that Equations (16-6) and (16-7) become " F k Á! nÀ1 n0 G n e 2i=Nnk 16-16 G n T 2K KÀ1 K0 " F k e À2i=Nnk 16-17 If we further define F k T 2K " F k 16-18 and W e À2i=N we have G n KÀ1 k0 F nk 16-19 or in matrix form G n W nk F k n 0; 1; 2; FFF; N À 1 k 0; 1; 2; FFF; K À 1 16-20 Spectrum Analysis 563 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 564 ± [558±583/26] 29.10.2001 4:05PM The matrices [G]and[F ] are column matrices with row numbers n and k, respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of G n occur in complex conjugate pairs. In general, we may write G n a n ib n jG n je i n 16-21 where: jG n ja 2 n b 2 n 16-22 n tan À1 b n =a n 16-23 From the time function F(t) and the calculation of [W], the values of G n may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by N 2 q . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. Finding the values of G n allows the determination of the frequency- domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G( f ), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. Vibration Measurement Successful measurement of machine vibration requires more than a trans- ducer randomly selected, installed, and a piece of wire to carry the signal to the analyzer. When the decision to monitor vibration is made, three choices of measurement are available: (1) displacement, (2) velocity, and 564 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 565 ± [558±583/26] 29.10.2001 4:05PM (3) acceleration. These three measurement types emphasize different parts of the spectrum. To understand this peculiarity, it is necessary to consider the differences in the characteristics of each. Consider a simple harmonic vibration. The displacement, x, is given by x A sin !t Successive differentiation gives the expressions for velocity ( x) and accel- eration ( x) x A sin !t x A! cos !t x ÀA! 2 sin !t In actual practice, these are specified Displacement: peak-to-peak measure 2A Velocity: maximum measure A! Acceleration: maximum measure A! 2 It can be observed that displacement is independent of frequency, velocity is proportional to frequency, and acceleration is proportional to the square of the frequency. If the displacement and frequency are known, the velocity and acceleration can be calculated. To measure any of the signals, a vibration transducer is used. A trans- ducer is a device that translates some aspect of machine vibration into a time-varying voltage output that can be analyzed. The frequency range to be analyzed should be carefully considered before selecting a transducer. It should be kept in mind, however, that there is no one best sensor, and several kinds may be needed to analyze a given machine. Also, in many cases signal conditioning of the transducer signal may be required prior to analysis. Displacement Transducers Eddy-current proximity probes are primarily used as displacement trans- ducers. Eddy probes generate an eddy-current field, which is absorbed by a conducting material at a rate proportional to the distance between the probe and the surface. They are often used to sense shaft motion relative to a bearing (by mounting them within the bearing itself) or to measure thrust Spectrum Analysis 565 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 566 ± [558±583/26] 29.10.2001 4:05PM motions. They are generally indifferent to hostile environments, including temperatures up to 250 F (121 C) and are not expensive. One drawback is that shaft surface conditions and electrical runout can result in false signals. Also, the smallest displacement that can be successfully measured is limited by the S/N ratio of the system. In practice, it is difficult to measure values less than 0.0001 of an inch. If shaft displacement is being measured, the shaft runout (measured with the same pickup) should be less than the smallest measurable value. To achieve the proper shaft runout, it is necessary that the shaft be precision ground, polished, and demagnetized. Velocity Transducers Usual types of velocity transducers are made up of an armature mounted in a magnet. The motion of the armature in the magnet creates a voltage output proportional to the velocity of the armature. Usually, the forces being measured must be relatively great to cause a signal output. However, the signal is quite strong when mounted on the machine bearings, and amplification is usually not needed. They are very rugged but are also large and cost roughly 10 times as much as a proximity probe. Because of damping, transfer function characteristics of the armature- magnet construction generally limit the low-frequency response to approxi- mately 10 Hz. At the high end of the frequency range, the resonant peak of the pickup itself is the limiting factor. Thus, the useful linear bandwidth is limited. The main advantage of the velocity pickup is that it is a high-output/ low-impedance device, and hence, it provides an excellent S/N ratioÐeven under less than ideal conditions. The major disadvantage of the velocity pickup is its sensitivity to placement. The probe is directional so that if the same force is applied horizontally or vertically, the probe will give different readings. Acceleration Transducers Most accelerometers consist of some small mass mounted on a piezo- electric crystal. A voltage is produced when accelerations acting on the mass create a force acting on the crystal. Accelerometers have a wide frequency response and are not excessively costly. They also are temperature resistant. Accelerometers have two main limitations. First, they are extremely low- output/high-impedance devices requiring loading impedances of at least 1M. Such requirements rule out the use of long cables. One solution has been to have an amplifier built into the pickup to provide a low-impedance/ amplified signal. A power supply is required, and the weight is increased. 566 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 567 ± [558±583/26] 29.10.2001 4:05PM The second limitation of this pickup is illustrated by an example. Acceler- ation of one g at 0:5H z represents a displacement of 100 inches. It is obvious that in spite of its wide-band response (sometimes 0:1À15 kH z ), it is severely limited at the low end by a poor S/N ratio. The transducer type used should be matched to the machine being analyzed. A knowledge of the types of problems normally encountered will benefit this selection. For instance, the noncontacting shaft displacement probe helps to correct misalignment and balancing problems but is inappro- priate in analyzing gear mesh problems and blade passage frequencies. Also, if signal integration or double integration is to be carried out, the lowpass filters used to attenuate high-frequency spectra also have a highpass filter, which effectively creates a lower frequency limit (often as high as five Hz). As mentioned before, one main criterion in deciding which transducer to use is the frequency range to be analyzed. Figure 16-3 shows the frequency limitations placed on the three types of transducers discussed previously. Dynamic Pressure Transducers The use of dynamic pressure transducers gives early warning of problems in the compressor. The very high pressure in most of the advanced gas turbines cause these compressors to have a very narrow operating range between surge and choke. Thus, these units are very susceptible to dirt and Figure 16-3. Limitations on machinery vibrations analysis systems and transducers. Spectrum Analysis 567 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 568 ± [558±583/26] 29.10.2001 4:05PM blade vane angles. Dynamic pressure transducers are used to obtain a spectrum where the blade and vane passing frequency are monitored. As the compressor approaches surge, the second order of the blade passing frequency (2 Ânumber of blades  running speed) approaches the magnitude of the first order of the blade passing frequency. The early warning provided by the use of dynamic pressure measurement at the compressor exit can save major problems encountered due to tip stall and surge phenomenon. The use of dynamic pressure transducer in the combustor section, espe- cially in the low NO x combustors ensures that each combustor can is burn- ing evenly. This is achieved by controlling the fuel flow to each combustor can till the spectrums obtained from each combustor can are close to being identical. The dynamic pressure transducers when used in this application must be mounted so that the probes are not exposed to the full combustor temperatures. This can be done by the use of buffer gases. This technique has been used and found to be very effective and ensures smooth operation of the turbine. Taping Data For many reasons, it may be inconvenient to take the spectrum analyzer to the field each time an analysis is to be made. Often, several machines are to be analyzed at various locations. Also, a hostile environment may exist at the test site, which might result in damage to the analyzer. A way of over- coming these problems is offered by data taping. With a tape, a permanent record is made. Since each channel of the tape offers a place for data to be stored, this record may be a condensation of several inputs either from different transducers or from the same transducer at various locations. A continuous tape monitor is very beneficial. In the event of machine failure, an analysis of the playback will help diagnose the problem. The choice of what kind of tape recorder to use is an important decision. AM tape recorders are much less expensive than FM recorders and usually have a voltage saturation limit of 20 or more volts. An FM recorder may be saturated by as little as one volt. A drawback to AM recorders is a rather high roll-off frequency of about 50 Hz (3000 rpm). Data below the roll-off frequency is attenuated and appears to be lessened in magnitude. An FM recorder has no lower frequency limit; however, it may require careful signal conditioning (attenuation or amplification) to prevent tape saturation. Usually, if the problems lie at the high frequencies, an AM recorder is the best selection. Regardless of the recorder type, a calibration of input signals is recommended using a known oscillating signal and is usually best done by following manufacturer's instructions. 568 Gas Turbine Engineering Handbook [...]... //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 580 ± [5 58 583 /26] 29.10.2001 4:06PM 580 Gas Turbine Engineering Handbook Figure 16-13 Gear box signature (high-frequency end) Figure 16-14 Axial-flow compressor spectrum showing blade passing frequency //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 581 ± [5 58 583 /26] 29.10.2001 4:06PM Spectrum Analysis 581 Figure 16-15 Jet engine acoustic signature... //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 586 ± [ 584 ±604/21] 29.10.2001 4:06PM 586 Gas Turbine Engineering Handbook Figure 17-1 Typical phase lag between force and vibration amplitude chart Figure 17-2 Distribution of unbalance in a rotor //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 587 ± [ 584 ±604/21] 29.10.2001 4:06PM Balancing 587 The existence of unbalance in a rotor system may be... determining the amount and location of the unbalance and (2) installing a mass or masses equal to Figure 17-3 Balanced rotor //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 588 ± [ 584 ±604/21] 29.10.2001 4:06PM 588 Gas Turbine Engineering Handbook the unbalance to counteract its effects or removing the mass of the unbalance exactly at its location Static techniques to determine unbalance can be performed... once-per-revolution component at power is less than the norm, indicating better plot shows a third engine with a fan damaged by //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 582 ± [5 58 583 /26] 29.10.2001 4:06PM 582 Gas Turbine Engineering Handbook Figure 16-16 Machinery analyses showing comparison of baseline signature to signature before overhaul ingesting a bird on takeoff The damaged fan has a large... shows a plot obtained from a casing-mounted pickup and the classical, high twice-per-revolution radial vibration A high axial //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 5 78 ± [5 58 583 /26] 29.10.2001 4:05PM 5 78 Gas Turbine Engineering Handbook Figure 16-10 A typical misalignment signature plot Figure 16-11 Real-time plot for a compressor shows details of critical frequencies vibration also exists... (0.005 08 mm) subsynchronous component at 9000 rpm Using the analyzer in the continuous real-time mode, this //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 575 ± [5 58 583 /26] 29.10.2001 4:05PM Spectrum Analysis Figure 16-5 Vibration spectrum (rpm = 20,000, Pd = 1200 psig) Figure 16-6 Vibration spectrum (rpm = 20,000, Pd = 1250 psig) 575 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 576 ± [5 58 583 /26]... (26-10-01)/CHAPTER 17.3D ± 589 ± [ 584 ±604/21] 29.10.2001 4:06PM Balancing 589 FPO Figure 17-4 Evacuation chamber for a high-speed balancing rig (Courtesy of Transamerica Delaval, Inc.) FPO Figure 17-5 Control room for high-speed balancing rig (Courtesy of Transamerica Delaval, Inc.) //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 590 ± [ 584 ±604/21] 29.10.2001 4:06PM 590 3 4 5 6 Gas Turbine Engineering Handbook... analyzing machinery problems; spectra in both subharmonic and high frequencies are needed to evaluate machinery problems fully //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 583 ± [5 58 583 /26] 29.10.2001 4:06PM Spectrum Analysis 583 Figure 16-17 Machinery analyses showing the comparison of baseline signature to signature before overhaul Bibliography Bickel, H.J., ``Calibrated Frequency Domain Measurements... India, 19 78, pp 6±32 Lang, G.F., ``The Fourier Transform What It is and What It Does,'' Informal Nicolet Scientific Corporation Monograph, December 1973 Lubkin, Y.J., ``Lost in the Forest of Noise,'' Sound and Vibration Magazine, November 19 68 Mitchell, H.D., and Lynch, G.A., ``Origins of Noise,'' Machine Design Magazine, May 1969 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 584 ± [ 584 ±604/21]... speeds using modern balancing techniques; however, in most cases they are basic problems that must be initially corrected before any balancing can be 584 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 17.3D ± 585 ± [ 584 ±604/21] 29.10.2001 4:06PM Balancing 585 done Rotor mass unbalance from dissymmetry, nonhomogeneous material, distortion, and eccentricity can be corrected so that the rotor can run without . by following manufacturer's instructions. 5 68 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 569 ± [5 58 583 /26] 29.10.2001 4:05PM The use of computers. Discrete Fourier transform representation. 562 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 563 ± [5 58 583 /26] 29.10.2001 4:05PM The equations may be rewritten. (1) displacement, (2) velocity, and 564 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 16.3D ± 565 ± [5 58 583 /26] 29.10.2001 4:05PM (3) acceleration. These three
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