A multi-order probabilistic approach for Instantaneous Angular Speed tracking debriefing of the CMMNO׳14 diagnosis contest

TL;DR: A novel approach for the estimation of the Instantaneous Angular Speed (IAS) of rotating machines from vibration measurements is proposed, originated from the organisation of a contest during the conference CMMNO 2014.

About: This article is published in Mechanical Systems and Signal Processing.The article was published on 2016-12-15 and is currently open access. It has received 95 citations till now. The article focuses on the topics: Order tracking & Angular velocity.

Summary

1 Introduction

• In many aspects, the Instantaneous Angular Speed (IAS) is to the Condition Monitoring of rotating machines what the heartbeat signal is to medicine.
• Its use for order tracking is also discussed in the review paper [4].
• The IAS enters as an essential ingredient to several processing techniques dedicated to the analysis of machines operating under nonstationary regimes.
• One important contribution of the cyclononstationary framework is to extend the principles of order analysis to any type of signal, including both deterministic and random types.
• One difficulty actually stems from the multi-component nature of the signals where different families of harmonics may coexist, each being characterized by numerous components (several tens or hundreds of harmonics are commonplace in some applications).

2 CMMNO 2014 diagnosis contest

• The fourth International Conference on Condition Monitoring of Machinery in Non-Stationary Operations (CMMN0 2014) was held in Lyon on the 15th and 16th of December 2014.
• Several months before the event, a diagnosis contest was launched: a signal, as well as some kinematic data were provided to participants.
• The aim of the contest was to estimate the IAS of the rotating machine, and to propose a diagnosis of potential faults.
• 23 people registered to the contest, but only 8 contributions from 7 different countries have been received in due form by the Organization Committee.
• They provided the best estimation of the IAS, as compared to a reference that was directly measured with an angle encoder.

2.1 The case study: a wind turbine gearbox signal

• Though gearbox defects appear to be 50% less frequent than electric system faults, the main cause of overall wind turbine downtime is related to gearbox faults, as is can be seen in Fig. 1 and according to WindStats database analysis focused on more than 30000 wind turbines [60].
• Bearing or even gearbox replacement never exceeds a few days.
• Most of the works are focused on time based methods.
• Angular domain analysis will therefore not only focus on the observation on gearbox-kinematic dependent phenomena, but it will also blur all phenomena that are not steady in the angle domain as soon as the rotating speed varies.

2.2 Analysis of the contributions of the contestants

• Results were compared to the reference speed directly measured with an angle encoder.
• This is the reason why the general shape of the IAS obtained by the contestants is correct (efficient tracking of the instantaneous frequency of the selected harmonic), yet with a scaling error due to the association to a wrong mechanical component.
• Moreover, it is important to note that it is not strictly a multiple of the rotation frequency, which is probably due to imperfect coupling between the main mechanism and the mechanical system responsible of these few asynchronous components.
• This step was quite hazardous, when starting with a component that was not in the expected order chart.
• Finally, the extracted IAS profiles resulted from the association of this particular harmonic to various (but most of the time wrong) meshing orders of the main mechanism, which explains the different scaling errors in the results.

3 The multi-order probabilistic approach

• The originality of the proposed approach is to use the whole spectrum harmonics, as well as all known potential orders, at each time step, to build a probability function of the IAS.
• The basic idea is to use, besides a response sig- nal, all the information that is available about the kinematics of the machine.
• The knowledge (eventually partial) of the kinematics leads to a list of potential orders that are supposed to strongly contribute to the vibration response, typically meshing frequencies and their harmonics.
• Each of these orders is used to transform a short time instantaneous spectrum into a probability density function of the fundamental rotation frequency.
• These pdfs are then combined to obtain a global pdf, allowing the extraction of the most probable IAS.

3.1 Expected shape of the short time instantaneous response spectrum

• The response spectrum results from the combination of the structural response and the excitation spectrum.
• In the case of strongly periodical excitations, the effects of the structure and of the excitation are generally well separated: the former governs the general envelop of the spectrum, while the latter generates sharp peaks at integer multiples of the frequencies of periodic events constituting the excitation.
• The response of the structure is without interest considering the issue of identifying the IAS, and can actually be disturbing for the approach proposed in this work.
• A preliminary work therefore consists in suppressing, or at least attenuating the effect of the structure, by spectral whitening the signal.
• Several approaches can be used for spectral whitening –one of which being described in Sec. 4– yet this preliminary step is considered as out of the scope of this paper, and will not be developed further in this theoretical part.

3.2 Considering the instantaneous response spectrum as a probability density function of the IAS

• This function will be considered as constant between a left bound Ωmin and a right bound Ωmax.
• Several orders will lead to several pdfs, that have to be combined to obtain a meaningful result, i.e. an unimodal pdf (cf. Fig. 4).
• The spectrum of the response is indeed known up to one given frequency noted fmax.

3.3 A priori of continuity of the IAS

• The method described in the previous section allows the construction of a pdf of the IAS at one specific time step from a short time Fourier transform.
• In non-stationary conditions, this operation is realized independently for each consecutive time steps and the continuity of the most probable IAS value is thus not guaranteed, although it is an important assumption for any mechanical system because of the inertia of rotating shafts preventing too strong accelerations/decelerations.
• The variance of the Gaussian law (6) increases with k, which increases the smoothing effect of the convolution.
• The Gaussian is a Dirac distribution, and Eq. 7 becomes simply [Ωj]j = [Ωj].

4 Application to the wind turbine’s gearbox

• The proposed approach is now applied on the vibration signal of the CMMNO’14 contest.
• This averaging operation over the less energetic time steps (defined at each frequency) gives an estimation of the average wideband spectrum, and the whole time-frequency map is multiplied at each time step by the inverse of this spectrum.
• It is noteworthy that this parameter is not too sensitive and equivalent results are obtained in a range from 0.1 to 0.4Hz/s.
• The expected values of pdfs are extracted, at each time step, to obtain the evolution as a function of the time (Fig. 9, left).
• It gives an objective picture of what people actually use to solve one given problem.

Figures

Fig. 6. Time-frequency map of the vibration signal before (left) / after (right) whitening.

Fig. 1. Wind stat data base, 2008-2012 aggregated downtime per turbine system[60].

Table 1 Meshing frequencies of the gearbox.

Fig. 5. Estimation of the smoothed pdf at time j from pdfs at times j + k (k ∈ [−K...K])

Fig. 3. Results of the CMMNO 2014 contest. Left: estimated IASs compared to the reference (solid blue), right: relative error with respect to reference.

Fig. 2. CMMNO 2014 contest: Kinematic data of the gearbox.

Fig. 8. Pdf of the IAS at each time step, considering fundamental meshing orders and their 10 first harmonics, with smoothing.

Fig. 9. Left: measured (solid black) and identified (dotted red) IAS. Right: relative error with respect to the reference (%).

Fig. 4. Construction of the probability density functions of the rotation frequency, from the instantaneous spectrum and 3 orders of the fundamental rotation frequency. (1),(2): case 1 (fmax/Hi > Ωmax). (3): case 2 (fmax/Hi < Ωmax). (4): combined pdf.

Fig. 7. Pdf of the IAS at each time step, considering fundamental meshing orders (left) and considering fundamental meshing orders and their 10 first harmonics (right).