Determining Rotor Blade Multi-Mode Vibration Components^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript deals with a well-known technqiue that is blade tip timing. The applied methodological approach is not new, since it is clearly always referred to [11], which is an own previous work of the authors, and is repeated in the appendix. The only new thing is the experimental data that is reported. However, no reference data is given. No comparison of different evaluation approaches is given. And: Vibration analysis (spectral analyses) of turbines/compressors by means of blade time timing are definitly not new (see my comment w.r.t. the literature reference, that previous works from this topic are not considered). As a result, degree of novelty is too low for an original research paper, and the interconnection with existing research works is not well worked out. Therefore, I must recommend to reject the submitted manuscript.

Author Response

Thank you very much for your comments and suggestions.

 

The manuscript deals with a well-known technqiue that is blade tip timing. The applied methodological approach is not new, since it is clearly always referred to [11], which is an own previous work of the authors, and is repeated in the appendix.

 [11] does indeed explain the tip-timing least squares method, but it does not show how to find the blade vibration frequencies in the flow and whether there are coupled modes.

 

The new section was added. Validation of the algorithm by numerical simulation

The only new thing is the experimental data that is reported.

However, no reference data is given.

The data cannot be shared because it concerns a military aircraft.

 

The new section was added. Validation of the algorithm by numerical simulation

 

No comparison of different evaluation approaches is given. And: Vibration analysis (spectral analyses) of turbines/compressors by means of blade time timing are definitly not new (see my comment w.r.t. the literature reference, that previous works from this topic are not considered).

The article presents a multi-mode blade analysis and I tried to present only such literature.

 

I added literature items

 

 

 

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

Szczepanik R., Experimental Investigations of Aircraft Engine Rotor Blade Dynamics, Air Force Institute of Technology, Warsaw 2013.

Barnett, V., Lewis, T. Outliers in Statistical Data (3rd ed.). Wiley., 1994

 

 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors!

Sincere congrats for the research work You've done. With all my consideration regarding Your work, I suggest a basic reorganizing of Your paper. In this stage, I consider that a more detailed description, with added explications, will raise as well as the clarity of the presentation but also the number of   interrested readers. In my humble opinion, an article must be written for students and young researchers too. That is why I kindly recommend the revision. See my  observations in the attachment. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

In some places, not  especially English grammar, but the definitions must be revised. see my comments in the attachment.

Author Response

Rew2

Thank you very much for the very comprehensive review, many kind comments and language corrections

 

Next idea is “The number of harmonics is determined on the assumption that the

vibration velocity has only one mode”. In my humble opinion, consistent and detailed

explications can be added here. Why this assumption?

In our numerical calculations, we noticed that by maintaining the single mode (Eq 2) whilst varying the ω_w and using the algorithm presented in the Appendix and [11], we can obtain local extremes: multiples of the rotation speed frequencies Ω, 0.5Ω, 2Ω, 3Ω, 4Ω, 5Ω,…, ω_{w =} ω_ie+ j Ω , j= ± 0, 1, 2, …,  ω_{w =} j Ω - ω_ie, j= 1, 2, …. . For us, the ω_ie frequencies are important. By assuming a single-mode solution (i.e. Eq.(2)), we can obtain the amplitude calculated as a first approximation.

 

With ω_ie and Eq. (1), the same algorithm (Appendix) can be used to calculate the blade amplitude and phase for each mode.  If the amplitudes for ω_ie are of a similar order, they show the mode coupling. This is a novel approach, which is demonstrated later on in this article

In our numerical calculations, we noticed that by maintaining the single mode (Eq 2) and varying ??  we can obtain local extreme: 

This is a novel approach, which is demonstrated later in our article

 

The quantity ?? present in equation (2) is not defined.

This is because this value is variable.

The last paragraph, where another angular frequency Ω is invoked, is very

hard to understand. Personally I consider that the quality of the presentation must be seriously

improved.

 

I agree that this may have caused a misunderstanding. However, we hope the above explanation of the local extreme: 

I changed Eq.(1) and Eq.(2):

                                             k

                                  V(t_v) =  Σ (a_i sin(2pω_ie t_v) + b_i cos(2pω_ie t_v))                             (1)

     :                                i=1                             

 

                                V(t_v) =  a sin(2pω_w t_v) + b cos(2pω_w t_v)                               (2)

 

 

 

 

 

First of all, the Appendix must be deleted from the end of the paper, and included

A list of symbols

A list of symbols was added

And quantities at the beginning of the paper could essentially help for a more easier lecturing. I

am sure that the authors, top skilled specialists in the field, are very familiarized with the

problematic presented here, but, in my humble opinion, a scientific article must realize a

positive impact on a larger mass: Msc students, PhD students, manufacturing engineers (they

control the manufacturing technology) and so on. The ‘Introduction’ chapter announces a

novel method. This novel method must be presented in the largest possible extenson in this

second section.

the reference [11] contains a detailed description of the procedure of finding blade amplitude and phase, and assumes the number of harmonics, but it does not show a method of determining which harmonics. This article, on the other hand, demonstrates how to determine the harmonics.

 

We will extend the description of the procedure in [11] in the Appendix, but not in the article to avoid the accusation of self-plagiarism.

I added the new section

3. Validation of the algorithm by numerical simulation

 

Section 3, entitled ‘Experimental results of the 1st stage SO-3 engine compressor during run-

up’ is structured on two paragraphs and two figures. Paragraph 1 presents the experimental

equipment and the tested engine. The SAD-2 and SNDL systems, which were developed, are

mentioned in the literature. Despite of this, I consider that a short presentation, emphasizing

the roles and the differences between this systems would be benefic for the better understand

of what deals this paper with.

OK

The first of these devices, developed in 1986, was SAD [ ], which measured blade tip displacement in relation to the blade root with two sensors placed respectively at the blade tip and root. On the stand, the sensors were permanently fixed to the compressor casing or on a special steel frame in the turbine inlet. The system was designed to measure of rotor blades in an SO-3 1^st stage compressor. The number of blades in this stage was 28. This system can monitor and record the displacements of up to 32 blades at various rotation speeds. This technique was next upgraded to the SNDŁ system, which was used in Iskra trainer aircraft to measure blade vibrations during flights. The SNDŁ system uses two sensors in the compressor casing and indication lamps in the cockpit that switch on when rotor blade

vibration amplitude exceeds 2 mm. In 1988, the SNDŁ system measured 1^st compressor stage blade vibrations and stress in 250 turbojet engines during flight, warning pilots of excessive rotor blade amplitudes.

 

The second paragraph offers information on the data represented

in figure 2. Here the vibration amplitudes over rotational speed (primed in rotation/min) are

presented for all 28 blades. I consider that in line 84 is a typing error “Figure 2 shows blade

displacement, with  distans 1 mm representing a 3 mm displacement.”. I suppose it is 1mm on the

graph equals 0,3 mm in the reality.

the distance between two adjacent blades on the x axis  (Fig.6 ) corresponds to 3 mm of blade displacement. This distance is the same between all the adjacent blades.

 

I recommend also to avoid expression 2EO, and add instead an explication of what engine

ordered forced vibrations mean in this case.

It was added

 

The Engine Order refers to the frequency of oscillations that is a direct multiple of the engine’s rotation speed and is always presented in the Campbell Diagram.

 

The next section in misnumbered: it must be the 4th section.

Yes, thank you

It is entitled “Numerical analysis”. The computing was preformed using a FEM[14]. Here also I recommend to describe more detailed the computing’s subject, the method used, the goal of computing and the hypotheses, initial data and so on.

        

                This chapter presents a numerical analysis of blade vibrations in the 1^st stage compressor of a one-pass SO-3 ISKRA engine. In several cases, blade failure caused engine damage and plane crashes. In order to obtain a full understanding of the measurement results, it was necessary to carry out numerical analyses using the Finite Element Method (FEM).

The blade was modelled using Solid 45, isoparametric elements with eight nodes [16, 17]. The mesh was as follows: one element for the thickness, 13 elements for the blade chord, and 10 levels for the blade length. This mesh was sufficient to obtain convergence with the experimental measurements [17].

                The blade was made of 18H2N2 (structural chromium-nickel alloy steel for carburizing). In the calculations it was fixed in the blade root. Its natural frequencies were compared with experimental ones using strain-gauges and tip-timing analysis [17].

 

The rotor blade frequencies in the 1^st stage compressor varied due to manufacture tolerances, e.g. 535, 512, 525, 524, 525, 525, 516, 525, 524, 515, 527, 518, 519, 544, 522, 517, 529, 518, 509, 523, 529, 509, 517, 518, 524, 507, 524, 525 Hz. These frequencies correspond to the first bending blade frequency.

 

For Ω=12128 rpm, 464, 399, 408, 406, 405, 402, 392, 408, 404, 391, 396, 396, 393, 413, 400, 390, 395, 395, 395, 395, 407, 402, 398, 406, 401, 397, 395 Hz.

Figures 3-6 present the distribution of amplitudes over frequencies, for the same constant

value Ω = 250 Hz. It is not clearly emphasized – or defined – what value is Ω representing?

If we admit that for Ω = 250 Hz the number of rotations per minute is 15000 that is OK.

But in the trigonometric functions in the Fourier series there are used the circular frequencies,

? = 2 ⋅ ? ⋅ ?

60 That means that between the vibration frequency and the pulsation frequency,

related to the circle, must make the difference.

 

OK   Yes I agree so  I have changed Eq.(1) and Eq.(2) added 2p

 

 

                                             k

                                  V(t_v) =  Σ (a_i sin(2pω_ie t_v) + b_i cos(2pω_ie t_v))                             (1)

     :                                i=1                             

 

                                V(t_v) =  a sin(2pω_w t_v) + b cos(2pω_w t_v)                               (2)

 

 

whole section is too concise as presentation, Often, the same phrases are written, For

example, “Which frequency is the most important for multi-mode coupling will be shown

when the amplitude and phase are calculated using the algorithm in the Appendix.” Is the

same in lines138-140 and 190-191.

 

It was changed

The final result, the important amplitudes and phases are

missing.

15000 rpm

In the case of four blade vibration components k (Eq. 1), only the first one has a maximal amplitude of 0.27 mm, resulting from 2EO. For frequency 1396 Hz blade amplitude is 0.05 mm, 1944 Hz (0.02mm) and 3073 Hz (0.01). This shows that the rotor blades vibrate predominately with one component. Tab.3 shows that the higher the number of harmonics, the greater the accuracy of the calculations, though the improvement is only slight.

12138 rpm

In the case of blades vibrating with three components, k=3 (Eq. 1), the first, at 464 Hz, has a maximal amplitude of 0.81 mm. The amplitude is higher than for the nominal condition (Tab. 3) as a result of rotation stall [17]. For 1470 Hz, it is 0.10 mm, and for 1902 Hz it is 0.09mm, again higher than for nominal conditions due to rotation stall. This shows that in the case of non-nominal conditions, the differences between blade amplitude components are smaller, which is evidence of multi-mode coupling.

 

Amplitude and phase are presented in Tab 3 and 4

Table 1 synthetize the vibration amplitudes and frequencies for the condition of 1, 2, 3 and 4

vibration modes. The table contains both amplitudes ??? and ??? The lecture at this point

becomes very difficult.

the algorithm presented in the appendix is ​​based on the amplitude of the blade vibration velocities, not on the blade vibration displacements. Hence, the tables contain the vibration velocity amplitudes that have been converted to vibration displacements using the EQ.(A8)

 

Paper ends with the section “Conclusions.” Here are emphasized the results presented in the

section before. Here I recommend to emphasize the theoretical and the practical significance

of the results of the performed research

Finally, I consider the scientific value of the article acceptable, but it must be supposed to

major revisions.Also, in some sentences, the quality of English must be seriously improved.

Suggestions

Line 11-13: You write ‘SO-3 engine 1st stage compressor rotor blades were analysed with a

tip-timing the Least Squares algorithm for nominal 15000 rpm and non-

nominal 12130 regimes.’ I suggest a reformulation here. I understand that tip

timing data were processed using the least squares algorithm. Thus, I suggest

“SO-3 engine 1st stage compressor rotor blades were analysed for the nominal 15000 rpm and the non-nominal 12130 regimes using the propose Least Squares algorithm over the tip-timing method/data collection/procedure.”

It was changed

Line 13: You write ‘OPR-sensor’. Her, in the introduction, in my humble opinion, due to the

fact that not all reader are specialists in the narrow field, it would be much

easier the lecture if You could add the whole name’once per revolution’

sensor, and in parentheses the abbreviation.

Line 24: You wrote ‘…that each blades vibrated…’. I recommend, “that each blade vibrated”.

It was changed

 

Line 26: I suggest to correct ‘auto based methods’ in “autoregressive based methods”

It was changed

 

https://www.researchgate.net/publication/26488409_A_Class_of_Methods_f

or_the_Analysis_of_Blade_Tip_Timing_Data_from_Bladed_Assemblies_U

ndergoing_Simultaneous_ResonancesAAAPart_I_Theoretical_Development

Line 35: Instead ‘…from the tip-timing velocity time of arrival…’ try to find another

formulation. It is difficult to understand

tip-timing based on the blade vibration velocity

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper proposes a new method for determining the frequencies and amplitudes of coupled modes in rotor blade vibration. The academic value and practicality of the paper can be further enhanced through the following improvements:
1. The paper should provide a comparison between the new method and traditional methods (such as single-mode analysis and other multi-mode analysis methods) to verify its accuracy and advantages.
2. It is recommended to use simulated data or experimental data with known results to verify the reliability of the method.
3. A detailed description of the setup, including the positions of sensors, sampling frequency, and data preprocessing steps, is suggested to be more explicit.
4. Discuss the impact of data quality on the results and how to handle noise and outliers.
5. It is recommended to elaborate on the application principle of the least squares method in multi-mode analysis in the theoretical part, as well as the reasons for choosing this method, and provide mathematical derivations or algorithmic processes.
6. Conduct an in-depth discussion on the multi-mode analysis results, explaining the reasons for the changes in blade vibration modes at different speeds.
7. Analyze the possible influencing factors, such as blade material, geometry, and speed variations, and their impact on vibration modes.
8. Point out the limitations of the method, such as applicable conditions and requirements for sensor accuracy, and propose possible directions for improvement.
9. Highlight the position and innovation of the method in the current research context, discuss its connections and differences with other related works, and further emphasize its uniqueness and contributions.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

 

 

Rew3

Thank you very much for your comments and suggestions.

 

 

This paper proposes a new method for determining the frequencies and amplitudes of coupled modes in rotor blade vibration. The academic value and practicality of the paper can be further enhanced through the following improvements:
1. The paper should provide a comparison between the new method and traditional methods (such as single-mode analysis and other multi-mode analysis methods) to verify its accuracy and advantages.

 

I added new position of Reference and description in the Introduction:

 

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

The comparison between the new method and single-mode analysis is presented in new section:

Validation of the algorithm by numerical simulation .

If we assume Eq( 2) we have one-mode analysis. In  one modal analysis the multimode coupling can not be analysed.

In our method, we

                                              k

                                  V(t_v) =  Σ (a_i sin(2pω_ie t_v) + b_i cos(2pω_ie t_v))

                                             I=1                           

The main difference between our method and literature method is:

It is written in the paper:

In all the above papers, the multimode analysis assumed the coupling of successive n modes, and only then a numerical analysis indicated which ones were important.

This paper presents, for the first time, a technique to determine the frequencies and amplitudes of coupled modes in blade vibrations

2. 
   It is recommended to use simulated data or experimental data with known results to verify the reliability of the method.

It was done. New Section 3  Validation of the algorithm by numerical simulation

 

3. 
   A detailed description of the setup, including the positions of sensors, sampling frequency, and data preprocessing steps, is suggested to be more explicit.

It was done in Section 4

SAD-2 [17], developed in 1986, measured blade tip displacement in relation to the blade root with two sensors placed respectively at the blade tip and root. The sensors on the stand were permanently fixed to the compressor casing or on a special steel frame in the turbine inlet. The system was designed to measure of rotor blades in an SO-3 1^st stage compressor. The number of blades in this stage was 28. This system could monitor and record the displacements of up to 32 blades at various rotation speeds. This technique was next upgraded to the SNDŁ system, which is used in ISKRA trainer aircraft to measure blade vibrations during flights. The SNDŁ system uses two sensors in the compressor casing and indication lamps in the cockpit that switch on when rotor blade vibration amplitude exceeds 2 mm. In 1988, the SNDŁ system measured 1^st compressor stage blade vibrations in 250 turbojet engines during flight, warning pilots of excessive rotor blade amplitudes.

 

 

 

Each sensor’s signal frequency range was 0.025-20kHz. The casing sensors were installed at 12.9 deg

Sampling frequency:  1/1µs
Time step  Δt=1µs

4. 
   Discuss the impact of data quality on the results and how to handle noise and outliers.

Fig. 2 shows the measured displacements of each of the 28 compressor rotor blades rotating from 7500 to 16000 rpm  after de-noising using Median Absolute Deviation [18].

 

Fig.  A  the results with the noise

 

Fig.  A  the results with the noise

Fig.  6  the results with the de- noise using Median Absolute Deviation [18]

 

Figure 6. Vibration of the 28 1^st stage rotor blades rotating from 7500 to 16000 rpm

 

 

The manuscript deals with a well-known technqiue that is blade tip timing. The applied methodological approach is not new, since it is clearly always referred to [11], which is an own previous work of the authors, and is repeated in the appendix.

This paper presents, for the first time, a technique to determine the frequencies and amplitudes of coupled modes in blade vibrations based on the algorithm presented in [11].

5. It is recommended to elaborate on the application principle of the least squares method in multi-mode analysis in the theoretical part, as well as the reasons for choosing this method, and provide mathematical derivations or algorithmic processes.

the reasons for choosing this method

 The least squares method is one of the forms of approximation when we have a large number of measurement results

 

I added description in Appendix.

The least squares method is only tool for multi-mode analysis see Eq. (A12) in Appendix.

 

We assumed the multi-mode form of blade vibration  which consist of k mode Eq.(A7).

Next

 

Using the functional F in the Least Squares Technique for various t_vm    (Huffel [14], Huffel, Lammering [15]):  

                                                                                               M

            F =  Σ(V(t_vm) - v(t_vm) )²                                    (A12)

                      m=1

where: V(t_vm) is the vibration velocity of blade tip (Eq. A7) and v(t_vm) is the mesured velocity of blade tip (Eq. A11), matrix a will be found.

From  Eq.A9 and taking into account that the first derivative of functional F(a) is equal to 0:

                                           M                                  M

                                  a = ( Σ  s₍t_vm)^Ts₍t_vm))^-1  (Σ s₍t_vm)^T v₍t_vm))                                       (A13)

                                          m=1                                 m=1

 By integrating Eq.A8 :

                                     X(t_vm)=C + s_x(t_vm) a                                                                (A14)

where:

s_X=s_X(t_vm) = [-(2pω₁)^-1cos(2pω₁ t_vm), (2pω₁)^-1sin(2pω₁ t_vm), -(2pω ₂ )^-1cos(2pω₂ t_vm),

(2pω₂)^-1sin(2pω₂ t_vm),..., -(2pω_k)^-1cos(2pω_k t_vm), (2pω_k)^-1sin(2pω_k t_vm)]          (A15)

 

6. 
   Conduct an in-depth discussion on the multi-mode analysis results, explaining the reasons for the changes in blade vibration modes at different speeds.

In the first case  250 Hz we have Forced vibration when only one mode is predominant .

In second case we have rotating stall (nonsynchronous vibration) where we have two modes coupling

7. 
   Analyze the possible influencing factors, such as blade material, geometry, and speed variations, and their impact on vibration modes.

It was done in [16,17] in form of Campbell diagram and calculation of natural frequencies for in dependence of rotation speed.

I added in paper

This chapter presents a numerical analysis of blade vibrations in the 1^st stage compressor of a one-pass SO-3 ISKRA engine. In several cases, blade failure caused engine damage and plane crashes. In order to obtain a full understanding of the measurement results, it was necessary to carry out numerical analyses using the Finite Element Method (FEM).

The blade was modelled using Solid 45, isoparametric elements with eight nodes [16, 17]. The mesh was as follows: one element for the thickness, 13 elements for the blade chord, and 10 levels for the blade length. This mesh was sufficient to obtain convergence with the experimental measurements [17].

The blade was made of 18H2N2 (structural chromium-nickel alloy steel for carburizing). In the calculations it was fixed in the blade root. Its natural frequencies were compared with experimental ones using strain-gauges and tip-timing analysis [17].

 

he free vibration of the rotor blade was calculated using an FEM [16, 17]. The first four natural frequencies at 15000 rpm are 509.17, 1527.7, 1863.1, and 3127.8 Hz. From [17], it was known that the forced and non-synchronous (caused by rotating stall) blade vibration frequencies would be close to their natural frequencies. The rotor blade frequencies in the 1^st stage compressor varied due to manufacture tolerances, e.g. 535, 512, 525, 524, 525, 525, 516, 525, 524, 515, 527, 518, 519, 544, 522, 517, 529, 518, 509, 523, 529, 509, 517, 518, 524, 507, 524, 525 Hz. These frequencies correspond to the first bending blade frequency.

For Ω=12128 rpm, 464, 399, 408, 406, 405, 402, 392, 408, 404, 391, 396, 396, 393, 413, 400, 390, 395, 395, 395, 395, 407, 402, 398, 406, 401, 397, 395 Hz.

and speed variations, and their impact on vibration modes.

 

It is seen in the Campbell diagram [16], [17]

8. 
   Point out the limitations of the method, such as applicable conditions and requirements for sensor accuracy, and propose possible directions for improvement.

The limitation of the method is the amount of data. The time step must be appropriate see [11] in our case is Δt=1µs (it is written in the text) 

Each sensor’s signal frequency range was 0.025-20kHz

The accuracy of the calculation was done in the new section 3. Validation of the algorithm by numerical simulation

 

9. 
   Highlight the position and innovation of the method in the current research context, discuss its connections and differences with other related works, and further emphasize its uniqueness and contributions.

The new position of literature was added

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

 

It is written in Introduction

In all the above papers, the multimode analysis assumed the coupling of successive n modes, and only then a numerical analysis indicated which ones were important.This paper presents, for the first time, a technique to determine the frequencies and amplitudes of coupled modes in blade vibrations

 

The English could be improved to more clearly express the research.

 

I changed description          

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The paper applies the Least Squares Technique to real-world experimental data from an SO-3 engine, demonstrating practical relevance in rotor blade vibration analysis. The use of two casing sensors and OPR measurements under both nominal (15,000 rpm) and non-nominal (12,130 rpm) conditions provides empirical validation of the algorithm. FEM-based natural frequency calculations corroborate experimental results.

The algorithm appears to be an incremental extension of the authors’ prior work , with no significant theoretical or methodological breakthroughs. The use of two sensors and Least Squares is well-established in tip-timing literature.

The paper does not clearly articulate how the proposed algorithm improves upon existing multi-mode analysis techniques (e.g., sparse reconstruction, non-uniform Fourier transform).

Critical parameters (e.g., sensor specifications, sampling rates) and system descriptions (SAD-2/SNDL) are lacking, reducing reproducibility.

Grammatical errors and awkward phrasing (e.g., “analysed with a tip-timing the Least Squares algorithm”).

Recent advancements in tip-timing (post-2020) are not addressed, weakening the contextualization of the work.

In nominal conditions (15,000 rpm, Ω = 250 Hz), the dominant vibration frequency is 535 Hz with a maximum amplitude of 0.27 mm, with other frequencies (1,396 Hz, 1,944 Hz, 3,073 Hz) having lower amplitudes.

In non-nominal conditions (12,130 rpm, Ω = 202.3 Hz), frequencies are 464 Hz, 1,470 Hz, and 1,902 Hz, with higher amplitudes (0.81 mm, 0.10 mm, 0.09 mm) indicating multi-mode coupling, possibly due to rotating stall. These frequencies align with FEM-calculated natural frequencies, validating the approach.

Suggestions:

Explicitly state how their algorithm advances beyond prior methods (e.g., computational efficiency, reduced sensor requirements).

Include sensor specifications, FEM model validation steps, and algorithmic improvements over [11].

Correct grammatical errors, ensure consistent terminology, and provide all figures/tables.

Incorporate recent studies on multi-mode blade vibration analysis (e.g., machine learning applications, advanced sparse techniques).

move the controller to main body since thats the the framework you are utilizing to build upon your work. if you move the main controller ( previous work) to your appendix then whats ur novelty in the current paper?

addresss the future work 

Comments on the Quality of English Language

okay

Author Response

Thank you very much for your comments and suggestions.

 

 

Rew 4

The paper applies the Least Squares Technique to real-world experimental data from an SO-3 engine, demonstrating practical relevance in rotor blade vibration analysis. The use of two casing sensors and OPR measurements under both nominal (15,000 rpm) and non-nominal (12,130 rpm) conditions provides empirical validation of the algorithm. FEM-based natural frequency calculations corroborate experimental results.

The algorithm appears to be an incremental extension of the authors’ prior work , with no significant theoretical or methodological breakthroughs. The use of two sensors and Least Squares is well-established in tip-timing literature.

The paper does not clearly articulate how the proposed algorithm improves upon existing multi-mode analysis techniques (e.g., sparse reconstruction, non-uniform Fourier transform).

Introduction

( [8] Fourier transform, [9], [10] sparse reconstruction)

In all the above papers, the multimode analysis assumed the coupling of successive n modes, and only then a numerical analysis indicate which ones were important

This paper proposes a new method for determining the frequencies and amplitudes of coupled modes in rotor blade vibration. The academic value and practicality of the paper can be further enhanced through the following improvements:

I added new position of Reference and description in the Introduction:

 

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

 

 

Critical parameters (e.g., sensor specifications, sampling rates) and system descriptions (SAD-2/SNDL) are lacking, reducing reproducibility

It was added:

Each sensor’s signal frequency range was 0.025-20kHz. The casing sensors were installed at 12.9 deg

Sampling frequency:  1/1µs
Time step  Δt=1µs

It was done in Section 4

SAD-2 [17], developed in 1986, measured blade tip displacement in relation to the blade root with two sensors placed respectively at the blade tip and root. The sensors on the stand were permanently fixed to the compressor casing or on a special steel frame in the turbine inlet. The system was designed to measure of rotor blades in an SO-3 1^st stage compressor. The number of blades in this stage was 28. This system could monitor and record the displacements of up to 32 blades at various rotation speeds. This technique was next upgraded to the SNDŁ system, which is used in ISKRA trainer aircraft to measure blade vibrations during flights. The SNDŁ system uses two sensors in the compressor casing and indication lamps in the cockpit that switch on when rotor blade vibration amplitude exceeds 2 mm. In 1988, the SNDŁ system measured 1^st compressor stage blade vibrations in 250 turbojet engines during flight, warning pilots of excessive rotor blade amplitudes.

 

Grammatical errors and awkward phrasing (e.g., “analysed with a tip-timing the Least Squares algorithm”).

It was changed

Recent advancements in tip-timing (post-2020) are not addressed, weakening the contextualization of the work.

Literature is concentrate on multi-mode tip-timing analysis.

I added new position of Reference and description in the Introduction:

 

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

Zhu et el. [12] also used blade tip-timing vibration velocity obtained from the Taylor series expansion with compressive sensing-based methods in multi-mode blade synchronous vibrations. Five sensors in the casing and a one per revolution sensor were used. This method cannot be applied in engineering because of its low accuracy, inefficiency and difficult parameter selection.

Zhu et el. [13] proposed an acceleration-based blade tip-timing method for synchronous vibration. This method captures rapid changes of vibration, with three sensors in the casing without an OPR sensor. 

 

 

Suggestions:

Explicitly state how their algorithm advances beyond prior methods (e.g., computational efficiency, reduced sensor requirements).

Computational efficiency:

In all the above papers, the multimode analysis assumed the coupling of successive n modes, and only then a numerical analysis indicated which ones were important.

 This paper presents, for the first time, a technique to determine the frequencies and amplitudes of coupled modes in blade vibrations

Our calculation are done only for coupled modes  not for all successive n modes, which reduce the time of calculation.

Include sensor specifications, FEM model validation steps, and algorithmic improvements over [11].

Each sensor’s signal frequency range was 0.025-20kHz.

Correct grammatical errors, ensure consistent terminology, and provide all figures/tables.

It was done

Incorporate recent studies on multi-mode blade vibration analysis (e.g., machine learning applications, advanced sparse techniques).

I added new position of Reference and description in the Introduction:

 

Zhu Y., Wang Y., Qiao B., Liu M. Chen X.,  Blade tip timing for multi-mode identification based on the blade vibration velocity, Mechanical System and Signal Proc., 209, 111092, 2024.

Qiao Y. Da. B. Wang, Z. Yang, M. Liu, X. Chen, Improved non-contact vibration measurement via acceleration-based blade tip timing. Mechanical Syst. Signal Process.152, 109373, 2024.

 

 

 

move the controller to main body since thats the the framework you are utilizing to build upon your work. if you move the main controller ( previous work) to your appendix then whats ur novelty in the current paper?

In all the above papers, the multimode analysis assumed the coupling of successive n modes, and only then a numerical analysis indicated which ones were important

This paper proposes a new method for determining the frequencies and amplitudes of coupled modes in rotor blade vibration. The academic value and practicality of the paper can be further enhanced through the following improvements

the reference [11] contains a detailed description of the procedure of finding blade amplitude and phase, and assumes the number of harmonics, but it does not show a method of determining which harmonics. This article, on the other hand, demonstrates how to determine the harmonics.

 

We will extend the description of the procedure in [11] in the Appendix, but not in the article to avoid the accusation of self-plagiarism.

I added the new section

3. 3. Validation of the algorithm by numerical simulation

 

addresss the future work 

The analysis must be done for gas turbine to find the flutter parameter and flow-induced vibration

 

 

 

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

This paper proposes a novel algorithm based on blade tip-timing (BTT) and the least squares technique to determine multi-mode vibration components of rotor blades. Experimental analysis of the 1st-stage compressor rotor blades in an SO-3 engine under nominal (15,000 rpm) and non-nominal (12,130 rpm) conditions revealed that blades predominantly vibrate in a single mode under nominal operation, while a second mode becomes significant under non-nominal conditions (e.g., rotating stall), with frequencies close to the blades’ natural frequencies. The algorithm, validated using two casing sensors and an OPR sensor, demonstrates effectiveness in both synchronous and asynchronous vibration monitoring, offering a promising approach for turbomachinery diagnostics.

1.The advantages of the new algorithm compared with the existing method are not clearly explained in the introduction, and it is suggested to supplement the advantages of the new algorithm used in the paper, such as the calculation time and the number of sensors required, and other advantages of the existing method.

2.Figure Clarification: Enhance annotations in figures (e.g., Fig. 7-10) to clarify axis labels and highlight critical frequency components.

3.Sample Justification: Justify the focus on Blade 1 and discuss the generalizability of results to all 28 blades.

4.Terminology Definition: Define acronyms (e.g., BTT, OPR) upon first mention to improve readability.

5.Conclusion Expansion: Elaborate on the algorithm’s potential applications in gas/steam turbines and suggest future research directions (e.g., real-time monitoring).

6.Language Polishing: Correct typos (e.g., “Last Squares” → “Least Squares”) and refine ambiguous phrasing for clarity.

Author Response

I upload file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have improved the manuscript quality, but there are still major issue that should be solved before a positive recommendation of the manuscript appears to be possible:

• I strongly recommend to add a sharp explanation and definition of multi-mode analysis of coupled-modes and a respective sketch on it.
• Still, the state of the art appears to be not complete. In particular, multi-frequency analysis is state of the art, and that is not well represented in the introduction. As an example, the comparison of two different sensing approaches can be found here: https://doi.org/10.1016/j.ymssp.2015.04.026, but there are many more. So what is really novel, is it the ongoing development of the approach presented in [11]? You should put in context also with the works of others and should be very precise about what the actual novelty is. I would assume that in particular a revision of the introduction according to my frist bullet point would help here.
• Abstract: It is mentioned that the accuracy is determined, but the result is not mentioned. So the abstract is not complete and not focused on the results, yet.
• Where is the measurement setup und principle explained? (Positioning of the sensors?: I recommend a sketch in Section 2 in addition to the very general one in the appendix) Where  is A_ie.

 

 

Author Response

 

    

 

 

Rew1

Thank you very much for your comments and suggestions.

• I strongly recommend to add a sharp explanation and definition of multi-mode analysis of coupled-modes and a respective sketch on it.

If the blade vibrates simultaneously with several modes for a given rotational speed, we are dealing with a multi-mode analysis.

 

    Such coupling can be easily determined for forced vibrations using a Campbell diagram.

In this paper, we analyse blade multi-mode coupling at 15000 rpm and 12300 rpm, see Tab 1, 2, 3 and 4.

 

• Still, the state of the art appears to be not complete. In particular, multi-frequency analysis is state of the art, and that is not well represented in the introduction.

As an example, the comparison of two different sensing approaches can be found here: https://doi.org/10.1016/j.ymssp.2015.04.026, but there are many more.

 

• Paper: A laser-optical sensor system for blade vibration detection of high-speed compressors by Neumann et al.

 

In this paper, blade tip clearance and blade vibration measurements in a transonic compressor were presented for rotating stall and flutter. The multi-mode vibration of the blade is not analysed, so I did not include this article in the literature review

•  
• In my opinion, the most important items are included in the literature review.
•  
• However, if the Reviewer suggests additional important items, I will gladly add them to the literature review
•  

 

• So what is really novel, is it the ongoing development of the approach presented in [11]? You should put in context also with the works of others and should be very precise about what the actual novelty is. I would assume that in particular a revision of the introduction according to my frist bullet point would help here.
•  
• The essence of article[11] is to search for all harmonic components (amplitudes, frequencies and phases ) of the rotor blades using the time of blade arrival measured by sensors in the casing.

              But the number of harmonics k in Eq.(1) must be assumed in advance and their number must be high               enough to ensure a small calculation error.

•  
•  
•                                                          k
• V(t_v) =  Σ (a_i sin(2pω_ie t_v) + b_i cos(2pω_ie t_v))                             (1)
•                                          i=1     

          In this paper, a technique to determine the frequencies and amplitudes of  coupled blade vibration              modes based on the algorithm in  [11] is shown. This allows us to determine the precise number of            coupled blade vibration harmonics without having to make a prior assumption as to their number.             Therefore, the calculation time is considerably shorter. This is the novelty. 

  So we can find, for example, that only the first and second modes are coupled, and thus not have to analyse many components: k in Eq(1)

 

As presented in EQ(1) and in the Appendix, the developed method of identifying the blade vibration velocity amplitudes A_i, blade displacements A_xi and phase angles φ_i requires the assumption of vibration frequencies that correspond to the number of harmonics.

• In the simulation, it was assumed that the blade vibrated with 4 harmonics for the parameters given in Table 1.
• We used the algorithm (Appendix) to calculate the number of harmonics and compared the results with those assumed in Table 1.

 

 

In paper:

Introduction

The essence of article[11] is to search for all harmonic components (amplitudes, frequencies and phases ) of the rotor blades using the time of blade arrival measured by sensors in the casing. But the number of harmonics k  must be assumed in advance and their number must be high enough to ensure a small calculation error.

 In this paper, a technique to determine the frequencies and amplitudes of coupled blade vibration modes based on the algorithm in [11] is shown. This allows us to determine the precise number of coupled blade vibration harmonics without having to make a prior assumption as to their number. Therefore, the calculation time is considerably shorter. This is the novelty.

 

• Abstract: It is mentioned that the accuracy is determined, but the result is not mentioned. So the abstract is not complete and not focused on the results, yet.

 

I have now included in the Abstract:

The validation of the algorithm is presented in a numerical simulation, which assumes the blade vibration parameters. This shows the accuracy of the calculated vibration velocity amplitude and phase, as well as the good agreement between the calculated and assumed velocities. The accuracy of the calculations increased with the number of rotations up to N=50. Therefore, N=50 was used in further calculations

            In the paper we added :

                          The accuracy of the vibration velocity amplitude e_iA, phase e_iφ, and accuracy σ between

                the calculated and assumed velocities (see Eq.A18) for N=25 ( σ= 0.79 m/s) and N= 50

               (σ=0.79 m/s) are higher than for N=20 (σ=0.23 m/s). Therefore, N=50 will be used in further

                           calculations.

 

• Where is the measurement setup und principle explained?
•  
• The measurement setup is in the Air Force Institute of Technology in Warsaw. The tip-timing method used to find blade vibration parameters is described in [17].

             In the paper:

• Rotor blade vibrations in the 1^st stage compressor of an ISKRA trainer aircraft SO-3 one-pass engine were measured at the Air Force Institute of Technology in Warsaw [16] and [17] using a OPR inductive sensor and two inductive sensors in the casing. Each sensor’s signal frequency range was 0.025-20kHz. The casing sensors were installed at 12.9 deg. (see Fig. A1, ϰ _l+1 -ϰ _l =9 deg.) over the 1^st stage. The SO-3 engine stand consisted of the compressor with seven stages, combustion chamber and one stage turbine (Fig. 5). Two tip-timing signal processing systems were used to analysed blade vibration: the SAD-2 [16-17] system for laboratory tests and an SNDŁ [16-17] system for on-line blade vibration measurement in trainer aircraft. These systems were developed at the Air Force Institute of Technology in Warsaw [16-17].
•  
• (Positioning of the sensors?: I recommend a sketch in Section 2 in addition to the very general one in the appendix)

see section 4, line 4 below the title

The casing sensors were installed at 12.9 deg (see Fig. A1, ϰ _l+1 -ϰ _l =12.9 deg.) over the 1^st stage. The SO-3 engine stand consisted of the compressor with seven stages, combustion chamber and one stage turbine (Fig. 5).

I did not include a sketch in Section 2, but it is fully explained in Appendix  in the   

description of Fig. A1 :

 

Fig. A1 schematically presents the disc,  two blades, and two sensors, l and l+1, above the blades. The angle between sensors l and l+1 is  ϰ _l+1 -ϰ _l 

 

 

• Where  is Aie?

Thank you for drawing my attention to this. I have added a full explanation to the article in red type.

 

List of symbol

A_ie             blade velocity amplitudes in simulation

A_i             blade velocity amplitudes

ω_ie                       blade frequency in simulation

ω_i                       blade frequency

φ_ie            velocity phase shift angles in simulation

φ_i            velocity phase shift angles

 

• Section 3
•  
• The purpose of this simulation is to verify the algorithm presented in this paper. It was assumed that the rotor speed was Ω=15000, rpm=250 Hz, and the rotor blade vibrated with the assumed parameters in Table 1, i.e., with 4 harmonics, the velocity amplitudes A_ie, frequencies ω_ie, and phase shift angles φ_ie.
• Having assumed the blade vibration parameters (Tab. 1), the blade time of arrival was calculated by two sensors mounted at 20 deg. angles in the compressor casing and a blade tip radius R=0.314 m.
• First to be determined were the blade velocity, based on times of arrival t_1n, t_2n , … then t_v1,n, t _v2,n, ... (Eq.A11) and velocities v_1,n, v_2,n,… (Eq.A11). The above data were determined with the time step Δt=1µs (Eq.A3). These were used to determine the blade vibration parameters a (Eq.A10), velocity amplitudes A_i, displacement amplitudes A_xi, phase angles φ_i (Eq.A7), velocity V (Eq.A9), and displacement X (Eq.A14).
• The velocity amplitudes A_i, frequencies ω_i, and phase shift angles φ_i  are compared with A_ie, ω_ie, φ_ie assumed in Table 1 to validate the algorithm.
•  
• Next, we can calculate the accuracy of the vibration velocity amplitude:
•                                                                         ε_iA =( A_ie –A_i)/A_ie                                             (3)
•                                        where: A_ie is the blade amplitude from the simulation (Tab. 1), A_i is the
• calculated blade amplitude using the algorithm in the Appendix.
•  
• Phase accuracy:
•                                              ε_iφ =( j_ie –j_i)/j_ie                                                          (4)
• where: j_ie is the phase from the simulation (Tab. 1), j_i is the calculated phase using  the algorithm in the Appendix.
•  
• As presented in EQ(1) and in the Appendix, the developed method of identifying the blade vibration velocity amplitudes A_i, blade displacements A_xi and phase angles φ_i requires the assumption of vibration frequencies that correspond to the number of harmonics.
• In the simulation, it was assumed that the blade vibrated with 4 harmonics for the parameters given in Table 1.
• We used the algorithm (Appendix) to calculate the number of harmonics and compared the results with those assumed in Table 1.

 

Appendix

•  
• Amplitude accuracy is ε_Ai :               
•                                                     ε_iA =( A_ie –A_i)/A_ie                                                           (A16)
•                                     where: A_ie is the assumed in simulation blade amplitude  (see Tab. 1), A_i is the calculated blade amplitude 
•     
• Phase accuracy:
•      ε_iφ =( j_ie –j_i)/j_ie                                                                                           (A17)
• where: j_ie is the phase from the simulation (see Tab. 1), j_i is the
• calculated phase 
• Sigma error of calculated velocity V(t_vm) and experimental velocity v(t_vm)
•                      (A18)
• where: V(t_vm) is the vibration velocity of the blade tip (Eq. A7) and v(t_vm)
• is the simulated velocity of the blade tip (Eq. A11).
•  

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors, this second version of Your paper is much better organized.

The presentation was significantly improved. The structure of the chapters remains the same, but the explications given are much clearer.

The figures were improved.

The Appendix was enlarged and now it contains sufficient information to constitute a good support for understanding.

A lot of congratulations!

 I propose now the 'accept as it is'.

Author Response

Thank you very much

 

Reviewer 3 Report

Comments and Suggestions for Authors

The changes are quite good and could be accepted now.

Author Response

 Thank you very much

Reviewer 4 Report

Comments and Suggestions for Authors

publishable

Author Response

Thank you very much

 

Reviewer 5 Report

Comments and Suggestions for Authors

Suggest accept the revised manuscript.

Author Response

Thank you very much