Cepstrum

Many practitioners regard Cepstrum as a "novel" technique even though it dates back to the early 1960s and predates the FFT. One of the coauthors of the paper which introduced Cepstrum was J.W. Tukey who two years later was coauthor of the FFT algorithm (Cooley and Tukey, 1965). This first paper (Bogert et al, 1963) defines the Cepstrum as "the power spectrum of the logarithm of the power spectrum." The history of the technique has been admirably described by Randall (2013).

In mathematical terms the Cepstrum is given by

where Fxx(f) is the power spectrum, and F'1 denotes the inverse Fourier transform. Despite this apparently contorted definition, the logarithmic form is extremely useful. In the study of gears, for instance, the logarithmic scale reveals sideband structure which is often not apparent in the direct calculation on a linear scale. The Cepstrum gives insight into sideband structure and this has proved particularly useful in the analysis of gearing problems. Other notable studies have analyzed steam turbines with damages blades.

Another significant feature of the Cepstrum is that, because it is logarithmic, the force terms and the impulse response properties are additive rather than multiplicative, and this can be used to simplify some analyses substantially. Hence in general we may write

where p(t) is the applied force and h(t) is the impulse response and * denotes the convolution operation. Taking the Fourier transform and using the convolution theorem give

where X, P, and H are the respective Fourier transforms of x, p, and It respectively. Hence taking logs

This means an effective separation of force and system terms, a feature of considerable help in identifying sources.

Although most engineers working in the area of machine condition monitoring are aware of cepstrum, it remains something of a specialty. It is an extremely powerful tool that, perhaps, warrants wider usage.

Cyclostationary Methods

The development of cyclostationary analysis is rather more recent than many methods in current use. The approach is a sophisticated signal analysis approach to the study of systems which are not exactly periodic, yet exhibit some cyclic behavior. A good example of this, indeed a problem to which it has been successfully applied, is the motion of rolling element bearings. Naively one might expect that such a bearing would show truly periodic behavior, but this is not the case for a variety of reasons. Random slipping between the elements, possible speed fluctuations, and variations of axial and radial loads impose slight variations on the cycle time. Although these variations may be only slight, Antoni (2007) shows that such a variation can destroy the harmonic nature of the response.

The problem is resolved by observing that the variations in frequency are statistical and the variance of the speed fluctuations is periodic, albeit at an as yet unknown frequency. Denoting the Fourier transform of x(t) as X(co) then

where L is the interval over which the signal is sampled. Sx(co; a) is called the cyclic power spectrum. Maxima of this quantity indicate the frequency content of the original signal.

This technique involves some sophisticated signal processing, but when applied to appropriate problems has proved capable of revealing spectral features not apparent using more conventional approaches. Once the precise frequencies have been established in this way, any modulation caused by the variation from true periodicity can be corrected.

Higher-Order Spectra

A variety of techniques are described in this text, most of which treat the vibration spectra in various ways in order to detect and/or diagnose the machine's condition. As shown elsewhere, in many cases substantial progress can be made by assuming the system under study is linear even though it is known that this is rarely the case in any strict sense. Indeed, it can often be a valuable step in diagnosis to attempt an evaluation of the extent of nonlinear effects and there are several ways of approaching this, one being the use of so-called higher-order spectra (HOS). In examining

HOS, the analyst seeks to evaluate not only the distribution of frequency components (as in most studies), but also the relationships between them as this can yield considerable insight.

As a starting point, consider the power spectral density of a time series signal r. This is computed as

and gives a measure of the energy in a frequency component. This notion may be generalized to give the bispectrum which indicates the relationship between two frequency components, and their sum, for instance fk, //, and fk+i. The bispectrum is defined mathematically as

A bicoherence may also be defined (Fackerell et al., 1995) as

The bicoherence at any frequency pair fk, fi can be interpreted as the fraction of power at fk+i that is phase coupled to the two components.

Of course, this just defines the first term of a set of HOS, but to some extent the first few of the set are the most valuable. The next in the series is the trispectrum and this is given by

For the effective use of these tools there are a range of considerations with regard to sampling rate, sample size, and windowing techniques that are discussed by Fackerell et al. (1995). Here attention is focused on why these techniques are useful. A complete discussion would be ambitious indeed, but it is relatively straightforward to give an indication based on physical principles. Collis et al. (1998) also give some useful insight.

Consider a typical dynamic equation describing the behavior of a machine. In matrix formulation this may be written as

Here a nonlinear stiffness term has been included, K„. There is a slight difficulty here in rigorously defining the meaning of the squared term in a vector sense, but it can be taken as individual components for the purposes of the present discussion. Now assume that the forcing term comprises two components with angular speeds wx,co2, and suppose that, in some sense, the nonlinear terms are small, then the problem of solving 8.56 may be approached by first obtaining a linear solution and then using this to apply a correction. In this way an iterative procedure can be developed, but only the first-order correction is required for the current argument. Now let x = Xi + x„, for the linear

which may be written as

The second-order term in the solution can now be calculated using Equation 8.58 to give

The essence of the argument is that the forcing terms of the nonlinear correction will contain terms (Ae>‘0>t + which may be expanded to

give ae2'"''1 + /le2i"Jl1 +уе^ы 1+ft>2)f and so it is clear that components of the response at frequency cox+co2 are produced as a direct consequence of the system nonlinearity. This can be utilized as a test for system nonlinearity and in many instances this in itself is a good condition indicator.

The simple physical result is that if a linear system is excited simultaneously with forcing at frequencies fx and f2, then there will be excitation at these two frequencies and only these two frequencies. However, if the system has some nonlinearity, there will also be response at (f ±/2); hence, this provides an indicator of nonlinear behavior. Similarly, excitation at a single frequency will produce a range of harmonics. This is closely related to the discussion of Section 5.5 and the rather complicated behavior can be presented graphically by plotting the bispectrum against to frequency ranges.

Yunusa-Kaltungo and Sinha (2014) have discussed the use of bispectrum and trispectrum for the diagnosis of faults in rotating machine and simulated faults on a laboratory rig. They studied misalignment, shaft cracks, and shaft rub in their tests, and different patterns emerge for each category. This is physically reasonable since the type of nonlinearity is somewhat different for each case. Their summary results suggest considerable potential for condition monitoring. The paper emphasizes the need for both bispectrum and trispectrum results to distinguish between different faults. Further work is needed in terms of both experimental work and analytic studies on the types of nonlinearity arising in different faults on a range of machines. Nevertheless, this is a developing area with some potential.

 
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