Logic developments of three novel improved order tracking approaches

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Chapter 2 Logic developments of three novel improved order tracking approaches

In this chapter, three novel improved order tracking approaches are developed based upon available order tracking methods, namely Vold-Kalman filter order tracking and computed order tracking, Intrinsic mode function and Vold-Kalman filter order tracking and intrinsic cycle re-sampling. The logic of the discussions  on three improved approaches is outlined below to simplify understanding of the thesis:
1 The discussions on the Vold-Kalman filter and computed order tracking (VKC-OT) method emphasize the benefits that each order tracking method (VKF-OT and COT) brings to the subsequent Fourier analysis so that the method can provide clearer order
2 The discussions on the intrinsic mode function and Vold-Kalman filter order tracking (IVK-OT) method emphasize the relationship between an intrinsic mode function and an order wave so that the sequential use of the two methods is developed to distinguish useful information in an IMF in terms of rotational speed.
3 The discussions on intrinsic cycle re-sampling (ICR) method emphasize the logic of exclusion frequency variation effects in an IMF and the interpretation of the resultant reconstructed IMF through empirical re-sampling. So that the rationale of the method to approximate order tracking effects as well as how to use ICR spectra results can be clarified.

Vold-Kalman filter and computed order tracking

The first technique is a novel technique that combines the use of Vold-Kalman filter order tracking and computed order tracking to improve the subsequent Fourier analysis and therefore to achieve a clear and focused order spectrum. It is called Vold-Kalman filter and computed order tracking (VKC-OT). Combining  the use of the two order tracking methods to improve the subsequent Fourier analysis requires an understanding of the nature of these techniques and how their characteristics affect the Fourier analysis. Therefore, in the following each order tracking method will be discussed in terms of its characteristics for the subsequent Fourier analysis.

Discussions on Vold-Kalman filter order tracking

Herlufsen et al. (1999) describe order tracking as the art and science of extracting the sinusoidal content of measurements, with the sinusoidal content or orders/harmonics at frequencies that are multiples of the fundamental rotational frequency. To this end, VKF-OT relies on two equations to complete the filtering, namely the data equation and the structural  equation.  These  equations  define local constraints, which ensure that the unknown phase assigned orders are smooth and that the sum of the orders should approximate the total measured signal. This implies that the order components extracted from the Vold-Kalman filter should be harmonic and smooth waves. To explore the  reason that  these  arguments are valid, the analytical form of data and structural equations described by Tůma (2005) are considered here for discussion.

Discussions on computed order tracking

Computed order tracking is a very commonly performed and effective order tracking technique. Although inevitably errors will be introduced during the re-sampling process and its artificial assumptions (Fyfe and Munck, 1997), the technique still renders very useful results, and effectively transforms non-stationary time domain data to stationary angular domain data for rotating machinery. Blough (2003) uses a graphic representation to explain this transformation process on a simple sine wave. This is illustrated in chapter 1  Figure 1.3. It clearly demonstrates that the re-sampled data has the  same  properties as a stationary frequency sine wave sampled at uniform time intervals. This uniformly spaced re-sampled data or stationary re-sampled data can be effectively processed by using traditional Fourier transform to obtain clear estimates of the orders of interest. This implies a clearer analysis of the signal using the Fourier transform. However, COT does not address the quality of the  raw data. Imperfections, such as distorted harmonic waves and noise, continue to exist. Besides, COT can only deal with the raw data as a whole and  therefore  loses the ability to separate each different order signal from the raw signal.

Development of Vold-Kalman filter and computed order tracking

The main ideas from above discussions about two techniques may be summarized as follows:

• Equation (2.1) indicates that the order components from the Vold-Kalman filter are clearly harmonic in nature.
• Equations (2.1) and (2.2) show that these order components may be harmonic waves of varying frequency due to the possibility of the varying fundamental frequency w(i)
• Equation (2.3) can be further demonstrated that the filtered order components from the Vold-Kalman filter are smooth waves.
• It can be seen from discussion of computed order tracking that the re-sampling process can transform varying frequency harmonic waves to stationary frequency harmonic waves. A Fourier analysis is then used to transform the re-sampled time domain data to the order domain.

Based upon the discussion of Vold-Kalman filter order tracking above, it is argued that the Vold-Kalman filter enforces the smoothness as well as the harmonic nature of the filtered data. The harmonic nature does not, however, ensure a stationary harmonic wave, although the re-sampling process can transform data from a non-stationary harmonic wave to a stationary harmonic wave in frequency. This suggests the possibility of using a Vold-Kalman filter to obtain smooth but possibly varying frequency harmonic waves and then transforming them to become stationary in frequency by using the re-sampling process of computed order tracking.
Therefore, if data are obtained from a non-stationary and noisy real machinery system and the data are then processed through a Vold-Kalman filter followed by the re-sampling process of COT, one may obtain order waves that are smooth, stationary frequency harmonic waves. Under these conditions the stringent requirements of Fourier analysis are largely satisfied. One may therefore expect clear and focused order spectra by means of this process. Based upon the above reasoning it follows that if the two order tracking methods are applied in sequence (VKF-OT and then COT), the restrictions of Fourier analysis can be largely satisfied to render clean order spectra. This combined use of order tracking techniques may be referred to as Vold-Kalman filter and computed order tracking (VKC-OT). Figure 2.1 describes graphically the logic of the combined use of the two order tracking techniques in sequence.

Intrinsic mode function and Vold-Kalman filter order tracking

The literature indicates that both IMFs and order tracking techniques are effective in diagnosing faults in rotating machinery, e.g. Eggers et al. (2007); Gao et al. (2008); Wu et al. (2009). This suggests investigating the relationship  between IMFs and order waves. However, this has not been explored further in the literature. To this end, the following will firstly exploit the relationship between  an intrinsic mode function (IMF) and an order wave in rotating machinery and then develop the Intrinsic mode function and Vold-Kalman filter order tracking (IVK-OT) technique of combining abilities of two kinds of methods.

Chapter 1 Problem statements and literature survey
1.1 Introduction
1.2 Towards the improvement of order tracking analysis
1.2.1 Computed order tracking
1.2.2 Vold-Kalman filter order tracking
1.2.3 Intrinsic mode functions through empirical mode decomposition
1.2.4 Improved order tracking
1.3 A review of three basic order tracking methods
1.3.1 Computed order tracking
1.3.2 Vold-Kalman filter order tracking
1.3.3 Intrinsic mode functions through empirical mode decomposition
1.4 Scope of work
Chapter 2 Logic developments of three novel improved order tracking approaches
2.1 Vold-Kalman filter and computed order tracking
2.1.1 Discussions on Vold-Kalman filter order tracking
2.1.2 Discussions on computed order tracking
2.1.3 Development of Vold-Kalman filter and computed order tracking
2.2 Intrinsic mode function and Vold-Kalman filter order tracking
2.2.1 Discussions on the relationship between an intrinsic mode function and an order wave in time domain
2.2.2 Discussions on the relationship between an intrinsic mode function and an order wave in order domain
2.2.3 Discussions on the resolution of an IMF
2.2.4 Combined use of empirical mode decomposition and Vold-Kalman filter order tracking
2.3 Intrinsic cycle re-sampling
2.3.1 Development of intrinsic cycle re-sampling
2.3.2 Interpretation on the reconstructed intrinsic mode function result
2.3.3 Discussions on intrinsic cycle re-sampling in terms of rotating machine vibration signals
2.4 Summary
Chapter 3 Simulation studies
3.1 Single-degree-of-freedom rotor model simulation analysis
3.1.1 Single-degree-of-freedom rotor modelling
3.1.2 Equations of motion for the single-degree-of-freedom system
3.1.3 Single-degree-of-freedom system analysis
3.1.3.1 Application of Vold-Kalman filter and computed order tracking
3.1.3.2 Application of intrinsic mode function and Vold-Kalman filter order tracking
3.2 Simplified gear mesh model simulation analysis
3.2.1 Simplified gear mesh modelling
3.2.2 Application of intrinsic cycle re-sampling method
3.3 Summary
Chapter 4 Experimental studies
4.1 Automotive alternator set-up data analysis
4.1.1 Experimental automotive alternator set-up
4.1.2 Experimental fault description
4.1.3 Application of VKC-OT and IVK-OT techniques on alternator experimental set-up
4.1.3.1 Application of Vold-Kalman filter and computed order tracking
4.1.3.2 Application of intrinsic mode function and Vold-Kalman filter order tracking
4.2 Transmission gear box set-up data analysis
4.2.1 Experimental gear box set-up
4.2.2 Experimental fault description
4.2.3 Application of intrinsic cycle re-sampling
4.3 Summary
Chapter 5 Conclusions
5.1 Contributions of the research
5.1.1 A review of the development of three improved order tracking approaches
5.1.2 Contributions of each improved order tracking approach
5.1.3 Contributions of the three improved order tracking approaches as a whole in order related vibration signals
5.2 Future work
Appendix 
Reference 
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