5 Diagnostic techniques
5.1 Harmonics and sideband cursors
By adding harmonics and sideband cursors to the toolkit, the diagnostic capabilities of frequency analysis have been greatly enhanced. A harmonic cursor is a set of markers that indicate all members of a particular harmonic family and have a very fine resolution. The key to diagnostic capability is that harmonics are exact integer multiples of the fundamental frequency, so if the position of higher-order harmonics can be determined, the accuracy of harmonic spacing will increase in proportion to the harmonic order. In addition, when scaling in the high-frequency range, if N harmonics of the same frequency spacing can be selected at the same time, the accuracy will increase in proportion to the number of N. Although individual vernier lines may be limited by the spectral resolution of the Fast Fourier Transform (FFT) analysis, this means that the harmonic spacing must be adjusted more finely (by at least one more factor N) than the spectral resolution within the scaling band.
A sideband cursor similarly depicts a set of sidebands at a given spacing around a specified central "carrier" frequency. Accuracy differs from harmonic cursors in that the family is not limited to passing through zero frequencies, but the same accuracy gains can be obtained by simultaneously selecting multiple sidebands in the same family. Also, in some cases, the sideband is also a harmonic, such as in gear vibration, the tooth bite frequency is an integer multiple of the shaft speed (the number of teeth on that gear), while modulation is a multiple of the shaft speed, which means that the resulting sideband is also a harmonic. In this case, the harmonic cursor can be used to determine the sideband spacing more precisely. The center (carrier) frequency of the sideband family should be expressed with the same precision as the accuracy obtained from the harmonic cursor in the sideband vernier family (the carrier tends to be a harmonic) and not at the center frequency of the analysis line closest to the harmonic involved.
5.1.1 Cursor Application Examples
The application of the vernier can be illustrated by examples, such as the separation of the shaft speed harmonics from the harmonics of the main (line) frequency in the vibration of an induction motor, as shown in Figure 2.20 (Chapter 2), where very high accuracy can be achieved by using a harmonic cursor, as both are true harmonic series, as shown in the figure. Similarly, an example of using a sideband cursor is given in Figure 2.22 to confirm that the sideband spacing is the slip frequency multiplied by the number of poles. In this case, the slip frequency can be determined very precisely by applying a harmonic cursor to the axis velocity and (double) the dominant frequency component, and is used to set the sideband cursor. In fact, the frequency difference between the double dominant frequency and the nearest shaft harmonic will always represent the sideband spacing of the rotor fault, as shown in Figure 2.22.
A more powerful application applies to gears, as harmonic verniers can be used to blindly determine the number of teeth of a gear pair on the upper teeth, as long as this indicates a "tooth chasing" design. If these numbers are m and n, respectively, then the tracing design means that they have no common factor. This is considered good design practice because it means that every tooth on one gear is in contact with each tooth on the paired gear, for example, a failure on one tooth can have a smear effect on the other without causing a more localized effect. For example, if both m and n are divisible by 3, this means that the three teeth on one gear will form a composite "tooth" that will always bite with a similar composite three-tooth group on the other gear. As shown in Figure 5.1, for the tooth tracking design, the tooth bite frequency is the first common harmonic frequency of the rotational speed of the two gears. The degree of harmonic proximity in other places is closest to 1/(m × n). Figure 5.1 also shows the results of applying a harmonic cursor to the acceleration spectrum of a gear pair. This requires a machine speed to stabilize at about 1:20,000, but this is not uncommon, and in cases where this is not the case, sequential tracking (Section 3.6.1) can be used to achieve this degree of stability. It also requires the presence of a reasonable number of harmonics of two axial velocities (on the logarithmic amplitude or decibel scale) in the spectrum, specifically low-order harmonics and sidebands near the bite frequency, but again this is not unusual. The harmonic cursor is set sequentially on each axis, first on the lower order and then gradually adjusted by zooming to higher frequency bands.
This usually determines the fundamental frequency of the required accuracy. Then, if one compares the list of two harmonic series, the bite frequencies correspond to the degree to which they match more than 1:10,000. In Figure 5.1, this frequency is found to be about 3357.7 Hz with a matching degree of 1:33,000, indicating that the number of teeth is 34 and 135, respectively. In this case, the smallest substitution difference occurs at about 3258 Hz, with a difference of about 1:5000, which coincides with m × n. Note that in the diagram, the shaft speed is only accurate to three digits, but it can be accurate to five digits by dividing the highest adjusted harmonic frequency by the corresponding order.
5.2 Minimum entropy deconvolution
Many of the vibration signals measured on the machine from the source to the sensor via the transmission path are largely distorted. This is especially noticeable on pulsed-type signals, which are often the result of sharp internal shocks, such as localized spalling from gears and bearings. Many of the diagnostic tools described in subsequent chapters rely on being able to identify a series of response pulses generated by these shocks and using, for example, envelope analysis to determine their repetition rate. However, this can only be achieved if the length of the impulse response functions (IRFs) is shorter than the interval between them, which is not always true for high-speed machines.
The Least Entropy Deconvolution (MED) method aims to reduce the propagation of IRFs in order to obtain a signal closer to the original pulse that produced them. This method was originally proposed by Wiggins [1] to enhance the reflection of different subsurface layers in seismic analysis. The basic idea is to find a reverse filter that counteracts the effects of the transmission path, assuming that the original excitation is pulsed and therefore has a high kurtosis. The name comes from the fact that an increase in entropy corresponds to an increase in chaos, and that the pulse signal is very structured, requiring all significant frequency components to have zero phase at the same time at each impact. Thus, minimizing entropy maximizes the structure of the signal, which corresponds to maximizing the kurtosis of the inverse filter output (which corresponds to the original input of the system). This method can also be referred to as "maximum kurtosis deconvolution" because the criterion used to optimize the inverse filter coefficient is to maximize the kurtosis of the inverse filter output. The MED method was applied to gear diagnostics by Endo and Randall [2], and by Sawalhi et al. [3] to bearing diagnostics.
Figure 5.2 illustrates the basic idea. The forced signal x [n] passes through the structural filter h, and its output is mixed with the noise e [n] to obtain the measurement output v [n]. The reverse (MED) filter f produces the output y[n], which must be as close as possible to the original input x[n]. Of course, the input x [n] is unknown, but it is assumed to be pulsed as much as possible.
The filter f is modeled as a finite impulse response (FIR) filter with L coefficients, such that
where f[i] must reverse the impulse response of the system IRFH[i] such that
The delay LM is to make the inverse filter causal. It shifts the entire signal to lm, but does not change the pulse spacing.
The method adopted in [2] is the objective function method (OFM) given in [4], where the objective function to be maximized is the kurtosis of the output signal y [n], which is achieved by adjusting the coefficient of the filter f [l]. This kurtosis is given by the following formula
where the maximum value is determined by finding the value of f[l] which makes the derivative of the objective function zero
Lee and Nandi [4] describe how this can be achieved iteratively when the filter coefficient f [l] converges within a specified tolerance.
In [2] and [3], the MED process is combined with AR linear predictive filtering (Section 3.6.3), and the total process is referred to as "ARMED". The AR operation achieves significant whitening of the spectrum, but because it uses an autocorrelation function, it has no phase information, whereas the MED process maximizes the impact of the filtered signal through phase adjustment. [2] An example is shown in Figure 5.3.
In this case, AR linear prediction was used to remove the normal gear meshing signal, but retained the per-revolution pulse from the tooth flank spalling, which has now become visible in (b). Note that although the self-spectra of (b) and (c) (in (e) and (f), respectively) are almost identically white, the MED filter significantly enhances the fault pulses.
Another example applied to bearing diagnostics is shown in Figure 5.4. As reported in [3], the bearing under test is a high-speed bearing similar to that used in gas turbine engines, but mounted on a test rack, in which real tooth flank spalling is introduced. In this case, AR filtering, in addition to pre-whitening, is used to remove discrete frequency components related to the harmonics of the tester's axis speed.
Again, AR filtering makes the pulses visible, but due to high speed operation (12,000 rpm), the excitation impulse response length is comparable to the spacing between them and tends to overlap. In AR operation, the kurtosis only improved from -0.40 to 1.25, but increased to 38.6 after MED was applied. This fault is at a fairly advanced stage, but the use of MED means that it can be detected at an earlier stage.