Round 1

Reviewer 1 Report

The article describes the iterative PIFEM method for estimating the instantaneous frequency of the non-stationary signal, which is essential in diagnosing damage to rotating machines. The article was written carefully, but some comments could improve it:

 1. In the description of the experiment it was written (line 254): A vibration signal is acquired by an eddy current sensor, the same description is in Figure 9. Is it not, however, a vibration sensor? Please explain it.

2. Please describe the experiment in more detail. How was AM-FM modulation of the vibration signal obtained? What was the shaft speed control function? Is the vibration amplitude recorded with the Eddy-current sensor in Fig. 10a?

 

3. Figure 4 shows the result of the time-frequency analysis using the IMSST method. Please describe this method briefly or at least expand on the abbreviation.

Author Response

Comment 1:

 In the description of the experiment it was written (line 254): A vibration signal is acquired by an eddy current sensor, the same description is in Figure 9. Is it not, however, a vibration sensor? Please explain it.

Response 1:

We thank you for your valuable comment. Eddy current sensors use the principle of eddy current formation to sense displacement. Eddy currents are formed when a moving or changing magnetic field intersects a conductor or vice versa. The relative motion causes a circulating flow of electrons, or currents, within the conductor. These circulating eddies of current create electromagnets with magnetic fields that oppose the effect of the applied magnetic field. The stronger the applied magnetic field, or greater the electrical conductivity of the conductor, or the greater the relative velocity of motion, the greater the currents developed and the greater the opposing field. Eddy current probes this formation of secondary fields to find out the distance between the probe and target material. Thus, the eddy current sensor is a vibration sensor in Figure 9, which can sense the displacement between the sensor and the rotor.

 

Comment 2:

 Please describe the experiment in more detail. How was AM-FM modulation of the vibration signal obtained? What was the shaft speed control function? Is the vibration amplitude recorded with the Eddy-current sensor in Fig. 10a?

Response 2:

We gratefully appreciate your valuable comment. Firstly, a change in the rotating speed of the rotor causes a change in the amplitude and frequency of the rotor vibration, we can thus obtain a vibration signal with AM-FM modulation by changing the speed of the rotor in the experiment. Secondly, the shaft speed control function is based on the pulse width modulator (PWM) by changing the duty cycle of the outgoing square wave causes the average current power on the load to vary from 0 to 100%  to change the motor speed. Then, as explained in Response 1 above, the eddy current sensor can sense the displacement between the sensor and the rotor, thus the vibration amplitude can be recorded with the eddy current sensor in Figure 10a. Finally, we re-describe the experiment in more detail as follows:

“In this part, a vibration signal collected from a rotor test rig under the time-varying rotating speed condition is considered. The rotor test rig is shown in Figure 9. The rotor is driven by a DC motor with a rated current of 1.95 A and maximum output power of 148 W, and the motor is controlled by a speed controller (DH5600) that changes the 220 V AC power into the PWM signal for motor speed control. The rotor with a diameter of 10 mm is supported by two bearing brackets and connected with the motor shaft by a coupling, and contains two mass disks with a weight of 500 g and a diameter of 75 mm. A vibration signal is acquired by an eddy current sensor (5E102) fixed on the sensor bracket, then transmitted to a dynamic data acquisition instrument (DH5922) for amplification and filter with a sampling frequency of 2000 Hz and a sampling duration of 10 s under the speed-up and speed-down process, finally pass to the computer for analysis and storage, which is shown in Figure 10a and its amplitude spectrum is shown in Figure 10b. The eddy current sensor [26] uses the principle of eddy current formation to sense displacement. It can be used to measure shaft displacement in rotating machinery and has been around for many years offering manufacturers high-linearity, high-speed measurements, and high resolution. A change in the rotating speed of the rotor causes a change in the amplitude and frequency of the rotor vibration.”

 

Comment 3:

 Figure 4 shows the result of the time-frequency analysis using the IMSST method. Please describe this method briefly or at least expand on the abbreviation.

Response 3:

We are extremely grateful to you for pointing out this problem. The IMSST is the abbreviation of the improved multisynchrosqueezing, which is an improved SST-based TFA postprocessing method. It has been revised in the manuscript as follows:

“The energy concentration of the TFR generated by the improved multisynchrosqueezing (IMSST) [1719] in Figure 4a4c is significantly higher than the STFT in Figure 3a4a, where the IMSST is an improved SST-based TFA postprocessing method, but the IF estimated by the IMSST is still inaccurate compared with the real IF in Figure 4b4d, where the MAPE is 1.55%.”

 

 

Reviewer 2 Report

Paper is written in a very good way and the scientific soundness and quality of presentation are very relevant. But I do not agree with your following affermation: "However, as typical linear TFA methods, both STFT and CWT cannot obtain the op- 51 timal time and frequency resolution simultaneously in the TFR. They consequently often 52 generate a blurry TFR and fail to extract nonlinear time-varying features for the nonsta- 53 tionary signal, especially when the law of the IF is nonlinear". I think that CWT is a powerful mean to obtain time and frequency resolution simultaneously in the TFR. At this proposal, I suggest these two references: "A Practical Guide to Wavelet Analysis", C. Torrence and G.P Compo, Bullettin of the American Meteorological Society, pp.61-78, 1998 ; "Wavelet Analysis in Volcanology: the case of the Phlegrean Fields", G. Pucciarelli, Journal of Environmental Science and Engineering, vol. A, No. 6 pp. 300-307, 2017.

Author Response

Comments:

 “Paper is written in a very good way and the scientific soundness and quality of presentation are very relevant. But I do not agree with your following affermation: "However, as typical linear TFA methods, both STFT and CWT cannot obtain the optimal time and frequency resolution simultaneously in the TFR. They consequently often generate a blurry TFR and fail to extract nonlinear time-varying features for the nonstationary signal, especially when the law of the IF is nonlinear". I think that CWT is a powerful mean to obtain time and frequency resolution simultaneously in the TFR. At this proposal, I suggest these two references: "A Practical Guide to Wavelet Analysis", C. Torrence and G.P Compo, Bullettin of the American Meteorological Society, pp.61-78, 1998 ; "Wavelet Analysis in Volcanology: the case of the Phlegrean Fields", G. Pucciarelli, Journal of Environmental Science and Engineering, vol. A, No. 6 pp. 300-307, 2017.”

Response:

Thank you for this valuable feedback.  We totally agree with your opinion that the CWT is a powerful mean to obtain time and frequency resolution simultaneously in the TFR. The manuscript has been revised as follows. According to your suggestion, we have properly cited these articles as Reference [10] and [11].

“The CWT is a powerful method to obtain time and frequency resolution simultaneously in the TFR [10,11], which uses a longer window to analyze the lower frequency that provides a higher frequency resolution and a lower time resolution, and a shorter window to analyze the higher frequency that provides a lower frequency resolution and a higher time resolution [1012]. However, as typical linear TFA methods, both STFT and CWT cannot obtain the optimal time and frequency resolution simultaneously in the TFR. They consequently often generate a blurry TFR and fail to extract nonlinear time-varying features for the nonstationary signal”

Reviewer 3 Report

The paper describes a proposal for a novel procedure for the estimation of the Instantaneous Frequency (IF) of vibration signals. This algorithm, named parameterized IF estimation method (PIFEM), is intended for the condition monitoring of rotatory machinery under time-varying operating speeds.

 The method relies on high-order polynomial functions. It is tested on both mono- and multi-component signals, including numerical and experimental data.

 Overall, the analyses seem to have been properly executed and well reported. The topic is of good interest to researchers and practitioners in the field of vibration-based condition and structural health monitoring. However, some issues are listed here below and should be addressed in order to achieve full acceptance.

 

1.      In the section dedicated to the numerically simulated data (Section 3), a 1:1 signal-to-noise ratio (0 dB) is used. This is indeed very noisy for vibration signals originating from rotatory machinery; this point is a good thing, as it indicates that the proposed approach can be applied even in extreme (quite unrealistic) cases. Nevertheless, more cases, with varying levels of signal-to-noise ratio, can be used to test the procedure’s reliability and practical limits (even with negative dB i.e. noise larger than signal).

2.      The method is compared to the results from STFT, which is however quite well-known to be not truly reliable and accurate, to a recently-developed algorithm (IMSST), and to the Polynomial Chirplet Transform (PCT). Apart from this latter one, the two other algorithms are, respectively, quite outdated and too new to be already considered well-established. Thus, it would be better to benchmark also the proposed PIFEM against other current state-of-the-art procedures for IF estimation, such as the first conditional spectral moment of the time-frequency distribution of the signal, the derivative of the phase of the analytic form of the signal (obtained by the Hilbert transform), or other well-established approaches.

3.      In the numerical case study, N = 8 and \epsilon = 0.1 are utilised. However, N = 10 is used instead for the experimental case study. A sensitivity analysis of the effects of these two parameters, especially N, would be a good addition to the paper.

4.      In their Introduction, the Authors (correctly) mention not only the Instantaneous Frequency (IF) but also other time-varying features utilised for condition monitoring such as the closely-related Instantaneous Phase and Instantaneous Amplitude. However, recently, the Instantaneous Spectral Entropy has been proposed as well, with applications for both the condition monitoring of wind turbines (https://doi.org/10.3390/app12031059) and the structural health monitoring of multi-storey frame structures (https://doi.org/10.3390/buildings12030310). These aspects could be added to the discussion.

5.      The most common use of IF is for the analysis of nonstationarity signals. This is clearly visible in both the numerical and experimental case studies. However, it would be interesting to evaluate the algorithm also in the case of abrupt frequency changes (as it may happen for several realistic conditions in rotatory machines). This can be tested, for instance, by slightly changing the numerical case study, as defined in Eq. 24 and 25, to include a piecewise formulation (or adding a further signal with these time-varying properties).

 

Author Response

see attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

The authors have adequately and diligently faced the main observations raised by this reviewer.

The content manuscript is now suitable for possible publication; only a few editorial points could still be improved:

  

1.      Ref 25 In the Reference List: please do not use full capitalized text for the title.

2.       Figures 8 and 9 could be centred on the page.

 

3.      The Algorithm on page 4 could optionally be numbered as Algorithm 1 (according to the Instructions for Authors).

Author Response

Dear Prof. Cardoso,

 RE: Revised Manuscript of Paper for Review

Thank you and all reviewers for taking your busy schedule to review our manuscript entitled “Parameterized instantaneous frequency estimation method for vibration signal with nonlinear frequency modulation” (Manuscript ID: machines-1879421). We have corrected our paper again according to your comments and professional advice. Any revisions to the manuscript have been marked up using the “Track Changes” function. We reply to each comment in a point-by-point fashion as follows:

Reply to Reviewer #1

We gratefully thank you for your time spent making constructive remarks and useful suggestions to raise the quality of our manuscript.

Comments:

 “The authors have adequately and diligently faced the main observations raised by this reviewer. The content manuscript is now suitable for possible publication; only a few editorial points could still be improved”

We discuss each of your comments individually along with our corresponding responses as follows. To facilitate this discussion, we first retype your comments in italic font and then present our responses to the comments.

Comment 1:

 Ref 25 In the Reference List: please do not use full capitalized text for the title.

Response 1:

We are extremely grateful to you for pointing out this problem. We have corrected Ref 25 in the Reference list using the correct format as follows:

Loughlin, P.; Cakrak, F.; Cohen, L. Conditional Moments Analysis of Transients with Application to Helicopter fault data. Mech. Syst. Signal Process. 2000, 14, 511–522, doi:10.1006/mssp.1999.1287.

Comment 2:

Figures 8 and 9 could be centred on the page.

Response 2:

Thank you for your careful check. We have corrected the format for Figures 8 and 9, which are now centered on the page in the revised manuscript.

Comment 3:

The Algorithm on page 4 could optionally be numbered as Algorithm 1 (according to the Instructions for Authors).

Response 3:

We are appreciative of your valuable comments. The Algorithm on page 4 has been numbered as Algorithm 1 in the revised manuscript.