Contents
Chapter 5 Signal features based monitoring of UTS
5.5 Fractal theory
Signal features based monitoring of UTS
The time-frequency distribution of the amplitude in Eq. 5.26 is called the Hilbert transform (Huang et al., 1971). When the Hilbert transform is applied on , the magnitude is kept unchanged but the phase of all frequency components is shifted by .
In the analysis of non-stationary signals, HHT found appreciable importance than other signal processing techniques (Prah and Okine, 2008). Even though, the shortcoming with HHT is that, the first IMF usually covers broad frequency range such that mono component conditions may be difficult to achieve. Once it occurs in actual signal processing this will lead to redundant signal features not correlating to actual process information. In the present study a modified HHT with WPT is proposed and presented in section 5.7 which is free from the aforementioned limitation.
Chapter 5
another method for computing the FD of waveforms (1995).This method gives a quick estimate of the FD as compared to other methods.
5.5.1 Higuchi’s algorithm
This algorithm was proposed by Higuchi (1988). This algorithm is best suited for finite length time sequence. Let consider a finite length time series sampled at regular interval as .
From the given time series a new time series is constructed as follows:
(5.27) where, denotes the gauss’ notation and both and are integers and and indicate the initial and interval time, respectively. For a time interval , we get sets of new time series. Now the length of the new curve is determined as follows:
(5.28) The term represents the normalization factor for the curve length of subset time series.
Let the length of the curve for the time interval is , as the average value over sets of . If , then the curve has a fractal dimension of . In the curve of versus , the slope of the least square linear best fit is the estimate of the FD.
5.5.2 Katz’s algorithm
Katz (1988) proposed another algorithm for computing fractal dimension of waveform. Although this algorithm is slower than the other methods but the advantage is that it is derived from the original waveform, and eliminates the preprocessing step of creating a binary sequence as required for the algorithm as proposed by Petrosian (1995).
The FD of a curve can be defined as:
(5.29)
Signal features based monitoring of UTS
where, is the total length of the curve or sum of distances between successive points, and is the diameter which defines the distance between the first point in the sequence and the point in the sequence which gives the farthest distance. Mathematically d can be expressed as
(5.30) The FDs computed in this fashion depends on the measurement units used, so if the units are different then so as the FDs. Katz’s method solves this issue by introducing a normalized distance parameter which is the average distance between successive points. Thus FD of a curve now becomes
(5.31)
where, is the average distance between successive points. Again defining a parameter as then FD of a curve can be expressed as which is the Katz’s FD of a waveform.
(5.32)
5.5.3 Validation of fractal dimension codes
The algorithms described above are used in this study to compute fractal dimension of signals acquired during FSW process. Code for each algorithm is developed using MATALB. All the developed codes are then tested for known waveform whose FD is known to check the proper functioning. In testing of the developed codes, fractional Brownian motion (with H = 0.5, 0.7, 0.95 and number of data points, N = 1024) waveforms as shown in Fig. 5.2 is generated in MATLAB and the waveforms are used for estimating the FD using the developed codes. The comparison results for the computed fractal dimensions using the developed codes are listed in Table 5.1. It is observed that percentage error between the theoretical and computed fractal dimensions are reasonable and the codes can be implemented for real time applications.
Chapter 5
(a) (b) (c)
Fig. 5.2 Fractional Brownian motion curves with Hurst parameter (a) H = 0.5 (b) H= 0.7 and (c) H = 0.95
Table 5.1 Comparison between theoretical and estimated fractal dimensions
Algorithm
H = 0.5 H = 0.7 H = 0.95
Theoretical FD = 1.5 Theoretical FD = 1.3 Theoretical FD = 1.05 Estimated % Error Estimated % Error Estimated % Error
Higuchi 1.51 -0.67 1.32 -1.53 1.05 0
Katz 1.49 0.67 1.28 1.53 1.06 -0.95
The signal processing techniques discussed in the aforementioned sections and subsections are utilized for processing of main spindle motor and welding motor current signals, vertical force and torque signals and tool rotational speed signal acquired during the welding experiments. The signals are processed in time domain and time frequency domain. Frequency domain analysis of the signals does not offer any intelligible information regarding the process behaviour. In the present study a total of five signals as mentioned above and four signal processing techniques are decided to implement for effective processing of signals. This results in twenty possible combinations for analysis of signals. However, it is not feasible that each signal processing techniques would result in effective processing of signals and the features extracted would be efficient in representing the process behaviour. Owing to this signals with different characteristics are processed with most suitable signal processing technique. Once the signal features are extracted those are fed to ANN and SVR models for fusion of features and for prediction of UTS of the joints.