Outline signal processing techniques utilised

3.5.1 Time Synchronous Averaging

Time Synchronous Average has been utilised extensively in the application of vibration monitoring to rotating machines (Vachtsevanos et al, 2006; Tavner et al, 2008; Ha et al, 2015) and has a natural application in this area. The algorithms utilised are often simple and seek to characterise the underlying transient artefacts in a signal taken from rotational machinery. The general process of TSA can be considered to consist of three phrases:

1. Data Capture in the time-domain,

2. Re-sampling to fixed locations in the displacement domain, and

3. Averaging over multiple machinery rotations at each (re)sample location. The effect of averaging at specific sample points for multiple rotations has the effect of reducing noise in the signal and highlights any underlying characteristic fluctuations over a single turbine rotation.

Although TSA is a promising technique there are shortcomings and caveats to be noted. The re-sampling exercise and the true representation of the signal being measured at the defined displacement indexes are highly dependent on the interpolation scheme utilised and

70 the rotation velocity of the machine. The rotational velocity has an impact on the required time-domain sampling rate which must be,

𝑓_𝑠 = 2 ∙ 𝑋 ∙ 𝜔_𝑚𝑎𝑥 (3.11)

Where X is the number of displacement points over which the interpolation is conducted. The effectiveness of the interpolation scheme is highly dependent on the statistical nature of the signal being measured and the method assumes that the data related to each interpolation displacement point is normally distributed.

In this work encoder measurements were available for the experimental data and the position of the turbine at each calculation time step was used in the case of the simulations. In this case the process of conducting TSA is simplified. Figure 3.3 shows the overall process undertaken, and was adapted from the work of Ha et al (2015). The algorithm for conducting TSA is illustrated via the following pseudo code.

1. Use encoder data to identify zero crossing sample number.

2. Split the measured data at the zero crossing samples, right inclusion (Each zero sample belongs to the proceeding rotation.

3. Resample, M, measured data points with time index to, N, data points with fixed displacement index using cubic spline interpolation.

4. Calculate ensemble average.

The effectiveness of the TSA process is highly dependent on the number of rotations used to create the ensemble average. The noise reduction given by applying the TSA process has been quoted as having been effectively modelled by (Ha et al, 2015), 1/sqrt(number of rotations). As the specific noise reduction requirements will vary depending on the specific application, averaging over various numbers of rotations will be conducted to find the most appropriate noise reduction factor whilst minimising the number of rotations averaged.

71

Figure 3.3: Schematic of the TSA process (Ha et al, 2015).

3.5.2 Discrete Time Fourier Transform

Feature extraction for rotating machines has for many years been conducted utilising frequency domain representations of time-domain signals (Vachtsevanos et al, 2006; Tavner et al, 2008). The frequency characteristic extraction processes adopted for such feature extractions are thus numerous and are related to the characteristics of the signal being considered. The most basic method is the Fourier transform. This has formed the foundations of many transformations of time series data into the frequency domain. As the data utilised throughout this thesis has been sampled or synthesised through discrete modelling processes the discrete counter-part of the Fourier transform, namely the Discrete Fourier transform (DFT) is deployed here. Both the discrete and continuous Fourier transforms are taken by considering the projection of a time domain signal onto the orthogonal basis of sinusoidal functions (Osgood, 2007). This process is represented mathematically by the following discrete case formula (Smith, 1997):

𝑋(𝑓) = ∑ 𝑥(𝑛𝑇)𝑒−2𝜋𝑗𝑓𝑇𝑛 ∞ 𝑛= −∞ (3.12) 1. Divide and Resample 2. Ensemble average.

72 The DFT is calculated throughout this research utilising the Matlab implementation of the FFT (Mathwork, 2016). The implementation is based on the open source FFTW library and allows the calculation of 3.12 with n log n calculations rather than the required n2 if calculating the transform directly via 3.12 (Frigo and Johnson, 2005).

3.5.3 Short-Time-Fourier-Transform and the Spectrogram

Spectrograms are produced by windowing time-series (or indeed displacement series data) into segments and taking the Fourier transform of each segment. This gives an estimation of the change in frequency content of a given signal over time. The Fourier transform can be, in simple terms, considered as a projection of a data set onto a basis of sine and cosine (complex exponential) terms at increasing frequencies highlighting the frequency content of a given dataset. The resolution of the Fourier transform is given as the reciprocal of the data set length or indeed -the window length when producing a spectrogram. The maximum observable frequency is then given by the half the reciprocal of the sample period. The windowing process is given mathematically by multiplying the signal x(t) by a window function w(τ-t) and observing the Fourier transform. The windowing function is then progressed through the dataset with a given overlap between transforms. This is known as the Short Time Fourier Transform (STFT) (Feng et al, 2013) and is given by:

𝑆𝑇𝐹𝑇_𝑥(𝑡, 𝑓) = ∫ 𝑥(𝜏)𝑤(𝜏 − 𝑡)EXP(−𝑗2𝜋𝑓𝜏) 𝑑𝜏 ∞

−∞

(3.13)

Within the process of conducting the STFT it is assumed the signal is stationary at the scale of the window length used to segment the dataset. This along with the fixed nature of the time frequency resolution for a given window length and the unresolvable issues associated with the Heisenberg uncertainty principle means that the transform is best suited

73 for analysing quasi-static (stationary at the scale of the windowing function) signals (Feng et al, 2013).

The application of the STFT results in a Spectrogram as the output. Throughout the course of the research Spectrograms were utilised to inspect the results of applying the STFT signal processing to time series of interest. As such the author has included a brief guide to the format and interpretation of the spectrogram figures presented throughout the thesis. To present the correct interpretation of a spectrogram and indeed, to show that the process of calculating the spectrograms throughout the thesis was correct a spectrogram of a known signal was created. The following signal was generated using Matlab:

𝑥(𝑡) = 5 ∙ sin(200 ∙ 2 ∙ 𝜋 ∙ 𝑡) + 1 ∙ sin (0.25 ∙ 2 ∙ 𝜋 ∙ 𝑡3_{) (3.14)}

The signal was created to harbour content of both fixed and varying frequencies. One can note in equation (3.14) that the signal is made up of two sinusoids the first with an amplitude of 5 and a fixed frequency of 200 Hz; the second sinusoid has an amplitude of 1 and has a frequency that increases as a function of time. Figure 3.4 shows the spectrogram of the above signal generated via the STFT in Equation (3.13). The interpretation of the spectrogram can be readily understood by noting that the x-axis refers to time, the y-axis frequency and lastly the colour of spectrogram to amplitude. It can be seen that the fixed frequency sinusoid appears as a straight line through the spectrogram and the time-varying frequency can be observed as polynomial varying with time. Furthermore is can be noted that the fixed frequency content sinusoid has the lightest colour as it has the highest amplitude of 5 and the time varying sinusoid has a slight darker colour corresponding to an amplitude of 1. Throughout the thesis the spectrograms calculated, via both the STFT and the Hilbert-Huang Transform, are presented to give the reader a qualitative view of the observed changes in spectrums for differing turbine rotor imbalance settings. The amplitudes of interest are extracted as time-series, as outlined in Sections 3.6.4 to 3.6.6, and then used to train and test

74 a naïve Bayesian classifier, as presented in Section 3.6.7 – this given a quantitative indicator of the performance of each monitoring approach.

Figure 3.4: Spectrogram of the test signal defined by Equation (3.14).

3.5.4 Empirical Mode Decomposition

This section considers the use of Empirical Mode Decomposition (EMD) for fault feature extraction. EMD is considered as a strong candidate for feature extraction as the selection of a prior basis on which to decompose the given signal is not required. The process generates a basis empirically adapting to the specific data set at hand. EMD was developed originally as the initial step required during the Hilbert-Huang transformation and generates a series of monotonic signal components to which the Hilbert transform can be applied in order to accurately estimate instantaneous frequency of the monotonic components, although just the EMD process is studied here.

EMD utilises the intuition of local high frequency artefacts in a signal. These can be estimated by considering, for example, two subsequent extrema. Utilising this intuition and considering the overall ‘trend’ or structure of the time series local high frequency artefacts can be extracted from the overall time series in an iterative manner (Rilling et al, 2003). EMD is utilised to represent the signal as a sum of Intrinsic Mode Functions (IMFs) which adhere to the mono-component requirement needed for suitable estimate of the

75 instantaneous frequency for a given signal. An IMF function must satisfy two stipulations (Yang et al, 2008b):

1. The number of extrema and zero crossings in the IMF must be equal of vary at most by 1.

2. The mean at any point in the signal of the maxima envelope and the minima envelope must be zero.

The EMD process involves IMF sifting, an iterative process for extracting IMFs by cubic spline interpolation of local minima and maxima for envelope estimation, a detailed algorithm can be found in (Huang et al, 1998). Essentially the output from the EMD process is the signal as a summation of IMFs with an associated residue which gives the underlying slow moving changes in the signal. This is written as:

𝑥(𝑡) = ∑ 𝑐𝑖(𝑡) + 𝑟𝑛(𝑡) 𝑛

𝑖=1

(3.5)

where ci(t) is the ith IMF and rn(t) is the process residual.

3.5.5 Hilbert Huang-Transform.

The extension of the EMD process outlined in the section above to the full Hilbert Huang transform involves taking the Hilbert transform of each of the developed IMFs acquired from application of the EMD procedure developed in the previous section. The instantaneous frequency of the IMF can then be estimated by calculating the derivative of the phase information resultant from the application of the Hilbert transform with respect to time. The summation in equation can then be expressed as a generalised Fourier decomposition of the IMFs as such:

𝑥(𝑡) = ∑ 𝑎_𝑖(𝑡)𝑒𝑥𝑝(𝑗 ∫ 2𝜋𝑓_𝑛(𝑡)𝑑𝑡)

𝑛

𝑖=1