Tacholess Time Synchronous Averaging for Gear Fault Diagnosis in Wind Turbine Gearboxes Using a Single Accelerometer
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe manuscript suggests a TSA method for wind turbine fault detection. I have the following comments for authors to revise the manuscript.
1. Wind turbines often work under varying speed conditions. The TSA method is usually for constant speed conditions. How does the TSA work with wind turbines in the present study? The experiment study is also the constant speed condition.
2. In Section 3.2, the f0, the frequency component with the highest amplitude in the TFR is taken as the GMF by default. This is NOT correct. The f0 can be the GMF, or its harmonics (as shown in Figs 4 and 5), or the instantaneous rotating frequency, the ghost frequency etc.
3. The definition of FDR in Table 3 should be provided.
Comments on the Quality of English LanguageGOOD
Author Response
Dear Reviewer 1,
Thank you for your review and feedback. We've carefully examined your comments and questions, addressing them thoroughly in the attached document. Your input has been invaluable in refining our work. Our responses provide clarity and enhance the quality of our submission.
Best regards,
Nguyen Trong Du
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper presents a method for performing diagnosis of wind turbine gearboxes where the accelerometer response signal is used both for deriving feature indicators (FIs) and also to provide information, normally requiring a tacho signal, to use time synchronous averaging (TSA) as one of the preprocessing tools.
One thing that is not made completely clear is whether the method can be used for wide speed variations (e.g. ± 30%), which are typical of many wind turbines, or just for nominally constant speed machines. The latter case is the only one used for demonstration in the paper. On the other hand, the main novelty in the paper is the use of the generalized Fourier transform (GFT) as a means of order tracking, to allow resampling at equal (rotation) angle intervals, and in the original reference [26] for GFT, examples are given of signals with widely varying frequency content.
A number of other methods have been used for this purpose, including the extraction of phase information from the signal, and are duly cited in the paper, and many can deal with the above-mentioned speed variations, but the main distinction claimed in the paper is whether they are based on using a “phase signal” or not. This is not very well explained, since for TSA, signals have to be sampled at equal intervals of phase, and the GFT uses the concept of x0(t), said to be the “signal's phase”. Later, in the Results section, it is perhaps implied that what is meant by having a “phase signal” is actually a “tacho signal”.
The paper is marred by a number of errors, and lack of detail, in the mathematical expressions and concepts, and their descriptions. A minor one, carried over from [13], is that the right-hand side (RHS) of Eq. (4) has both positive and negative values, so the modulus sign should be applied to both sides of the equation. More critical are Eqs. (5) and (6), with no differential to indicate the variable over which they are integrated.
For Eq. (5) it is obviously dt, but this means that the units of the resulting Fourier transform (FT) are Us, or U/Hz, where U are the units of x(t), in other words a spectral density. The square of the amplitude of X(f) has units U2 s2, or U2 s/Hz, in other words energy spectral density (ESD). The Fourier transform applies only to a transient, whose total energy over the infinite integral is finite. If x(t) is a truncated sinusoid (e.g. a GMF) the ESD can be converted to PSD (U2/Hz) by dividing it by T, the length of the truncated record. The power (U2) of the sinusoid can be found by integrating the PSD over the smooth smeared peak around the nominal frequency of the sinusoid (dominated by the FT of the rectangular window of length T, i.e. with 3 dB bandwidth 1/T).
Eq. (6) is supposed to be the equation for the short-time Fourier transform (STFT). It is not clear if this is what is meant by the term FSST used to describe X(t,f) in the paper. In any case, if the RHS of (6) is integrated wrt differential dt, then the LHS should be X(t,f). If it is desired to have X(t,f) with the same time axis as x(t), the variables on the RHS have to be reversed, so that the integration is over the dummy variable with differential dt. The window function g(t) should be symmetric around zero (so that the f of X(t,f) corresponds to the time t of x(t). The limits of the integral in Eq. (6) are wrong even if t is replaced by t. For infinite time records and windows such as Gaussian, the limits should be ±inf, but in practice the window must be limited to a length of Tw, such as a rectangle, Hanning or truncated Gaussian, and if the length of x(t) is T, the integration can be from 0 to T. However, the result (i.e. the spectrum of the windowed record) will only be valid when x(t) and g(t) overlap, i.e. from t = Tw/2 to T – Tw/2.
Eq. (8) is completely unclear, and does not seem to be used elsewhere in the paper. Superscript n is said to be “the number of zero crossings within the time interval 𝑡”, but zero crossings occur at phase intervals of p, not 2p. Is n meant to be the number of cycles of fmax at time t? However, each fmax only occurs over a short period of time around time t, not the whole time up to t.
The description of the GFT follows [26] but is less clearly described, with similarly named variables having completely different properties. For example, x0(t) (s0(t)in [26]) is a phaselike function, whereas x(t) is acceleration; f0(t) is first said to be df(t)/dt (i.e. the rate of change of frequency f(t), or angular acceleration) and then f0 is said to be “a constant time-invariant frequency” (i.e. a frequency that is not a function of time), which it must be in expressions such as Eq. (13) and f(t) = f0 + x0’(t).
The description of the Flowchart in Section 3.2 is confusing. The “time-frequency distribution” is said to be done using the “wavelet transform”, but so far the only time-frequency distribution described is the STFT, seemingly represented as TFR (time-frequency representation?), written wrongly as TRF in the heading of 3.1. There are multitudinous variants of the wavelet transform, so if one has genuinely been used it should be defined in detail. Most wavelet transforms have poor frequency resolution, typically 1/1-octave for orthogonal transforms such as harmonic wavelets, and are thus unsuited for estimating the peak value of a frequency component. The situation is further confused by the use in Ref. [26] of the wavelet packet transform (WPT), where for each level of scale the frequency steps are uniform rather than 1/n-octave, but the spacing of the levels is 1/n-octave. Ref. [26] explains this as being compatible with the fact that each tile is able to enclose a single component, whose frequency variation has been flattened by the GFT, but not “straightened”, as there is still some variation around the mean frequency. It is thus an advantage for the resolution not to be too fine; it just has to separate the different components into different tiles of the WPT. Using such a tiling would not allow the estimation of the variation of fmax within each tile, which is supposedly what this step is for. It can be done using the STFT, as the error in the estimate of the peak of the spectral ridge is less than the 3 dB bandwidth of the smooth peak (1/T).
Since the only example in the paper is for a machine at constant nominal speed, the estimates of the local instantaneous speed are presumably just random fluctuations around the mean speed, and the term “straightening” then refers to smoothing of these variations, rather than removing the curvature of the speed lines which would result from actual speed variations.
In the discussion of the experimental results, it is not made clear what the difference is between the six different cases for both healthy and faulty conditions. Can it be assumed that the fault is the same for all six faulty cases, but just measured at different times, possibly with different loads, even though the nominal speed is the same?
The results in Figs. 7 and 8 seem to indicate that the method actually applied works quite well, even if it has been poorly or even incorrectly described in the paper. In Fig. 8 it can be assumed that the results using a tacho can be aligned so as to identify the individual teeth on the faulty gear wrt to the phase of the tacho signal, but this would not be possible for the proposed method (in particular because the actual shaft speed component has been removed by a highpass filter). It should be explained how the two sets of results have been aligned in Fig. 8. The cases are all said to be case 16, but appear to correspond to the actual cases of Fig. 7. I suspect the alignment has been done using cross correlation, but this should be stated. Some of the misalignment visible in some of the diagrams might be attributable to displacement of the time axis of the STFT, X(t,f), as discussed above, if the window g(t) has not been centred on zero. It is unlikely that any of the FIs compared in Fig. 9 would be affected by displacement of the time axis of X(t,f).
It should be noted that many of the FIs used in the paper are only valid for measurements at one speed (because of the huge effect that fixed transfer functions can have on the various harmonics of the shaft speed (in particular gearmesh harmonics) at different speeds for the same fault, and many are also greatly affected by different loads (anything involving gearmesh harmonics). Thus, if the method is intended to be applied to measurements with varying speed and load, the choice of FIs must be restricted accordingly.
Since the problems with the paper seem to be more in the description, rather than the actual implementation of the method, it could be considered for publication if completely rewritten to remove the errors and inconsistencies.
The following additional language corrections should be made:
p.4 The heading of Section 2 should be “Literature Review”.
p.6 “where |𝑋(𝑡,𝑓)|2 represents the spectrogram of the signal 𝑥(𝑡)”.
Comments on the Quality of English LanguageIn general quite clear.
Author Response
Dear Reviewer 2,
Thank you for your review and feedback. We've carefully examined your comments and questions, addressing them thoroughly in the attached document. Your input has been invaluable in refining our work. Our responses provide clarity and enhance the quality of our submission.
Best Regard,
Nguyen Trong Du
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsIn this paper, aiming at the problem of gearbox fault diagnosis of wind power transmission system under variable speed. This paper proposes a new fault diagnosis method of wind power gearbox based on improved tacholess time-domain synchronous averaging (TSA) technology combined with time-domain feature extraction. The thesis mainly carried out the work in the following aspects:
1、 A vibration signal preprocessing technique without tachometer is proposed. The proposed technique eliminates the effect of rotational speed through generalized demodulation time-frequency analysis method (GDTFA), and completes the signal pre-processing with TSA. Based on the measured variable speed data, the difference of time-domain feature values which given in Section 2.2 before and after calculating the algorithm. The effectiveness of the proposed method to eliminate the effect of rotational speed is verified.
2、 By calculating the time-domain feature values given in Section 2.2, a statistical table of normalized values is given. According to the difference of value distribution between normal and fault state shown in the statistical table, it is concluded that the proposed method can be used for gear fault identification.
2. Existing problems
1、 The problem solved in this paper is the fault diagnosis of gear under time-varying speed, but the title does not highlight the key point of the problem to be solved, which makes the readability of the paper poor. It is suggested that the title of the paper should be revised to highlight the key issues and techniques used in the paper, so as to make the readers fresh and fresh.
2、 In order to solve the influence of rotational speed, this paper adopts TSA, GDTFA, vibration signal time-domain feature extraction and other technologies. However, the introduction only introduces the research background of wind power gearbox in large quantities, and lacks necessary overview of the technical means involved in this paper, so it is suggested to add.
3、 The paper is not innovative enough.
Looking at the paper, we can draw a conclusion. The innovation of this paper is as follows: by means of GDTFA, the effect of rotational speed is eliminated; The TSA is used to highlight the periodic components and realize tacholess fault diagnosis.
The main technical of this paper are classical TSA, GD proposed in reference 26 (2005) and time-domain feature index proposed in reference 25 (2020). As we all know, it is a very mature and effective method to realize fault diagnosis by calculating time domain characteristic index. In addition, the focus of this paper is the fault diagnosis of gears, but the paper only gives the normalized distribution values of multiple time domain feature indicators under normal and fault conditions, and the distribution situation is different, some are staggered distribution, and some have obvious boundary range. This paper lacks an accurate description of fault diagnosis based on time-domain characteristic indexes.
The proposed method is less innovative than the latest research results. In order to solve the influence of rotational speed, the time-frequency graph is obtained by SST, and the frequency components that do not change with time are obtained by GD. However, at present, the improvement methods based on SST are very rich, and the performance of instantaneous frequency extraction is also greatly improved. It is suggested that the author continue to improve the innovation of the algorithm.
4、 The effectiveness of the proposed method needs to be verified. In this paper, only a set of measured data is used for verification, and it is suggested to supplement the results of algorithm verification on public data sets.
Comments on the Quality of English LanguageThe language level of the article is average, with certain logic and readability, but there is still room for improvement
Author Response
Dear Reviewer 3,
Thank you for your review and feedback. We've carefully examined your comments and questions, addressing them thoroughly in the attached document. Your input has been invaluable in refining our work. Our responses provide clarity and enhance the quality of our submission.
Best regards,
Nguyen Trong Du
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsAs I have had to include equations in my reply, I have decided to upload a PDF version of my report
Comments for author File: Comments.pdf
A couple of suggestions have been made in the review
Author Response
Dear Reviewer,
We have addressed all the reviewers' comments in detail, as outlined in the attached point-by-point response document.
Thank you for being so considerate.
Yours Sincerely,
Trong-Du Nguyen
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors have addressed my concerns well. The revised topic can highlight the key point of time-varying speed; The newly added literature review makes up for the lack of time-frequency representation and tacholess technique; Additional experiments demonstrate the effectiveness of the proposed method in estimating frequency.
Author Response
Thank you the reviewer for accepting our paper to publish the Machines journal
