Damping Identification Sensitivity in Flutter Speed Estimation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper introduces the application of two existing modal parameter estimation methods, namely Fast Relaxed Vector Fitting (FRVF) and the Loewner Framework (LF), to GVT laboratory data measured on Flexible Wing test rig, and this was for flutter speed estimation purpose. Another 3rd existing modal parameter estimation method, N4SID, is used as a reference to compare the FRVF and the LF methods. The conclusion stated that both methods gave a comparable acceptable result.  Some remarks/questions that authors should consider to make the paper easier to follow.  

• Line 25 / abstract: typo “LFinformed”?
• In the introduction (Line 46-48), the authors wrote “In the frequency domain, ill-conditioning in the fitting process complicates the identification of modal parameters, especially for datasets exhibiting high noise or complexity”. This is not a true statement on frequency-domain Id techniques. There are some frequency domain techniques that are well-conditioned and can be used with high orders system and for a wide range of frequency without any numerical issues (e.g., the poly-reference linear least squares complex frequency domain estimator, in short pLSCF). Also, the using of orthogonal polynomials enabled the usage of the frequency-domain techniques in a well-conditioned way. So, authors should carefully revise this statement.
• Typo on line 113 “fo”.
• In lines 167-168, it is written “…applicable to various input types”. What is meant here by "various input types"? Does it mean periodic /random,..?
• In equation (3), Why does A come with a minus signe? Residue should be ( 1/2 i m w_d). Then, if one multiplies the residue with i/i , one gets -i/2mw_d which is a negative imaginary value that is an important characteristic of the residue of a driving point.
• In equation (8), What is N? And do you consider here complex conjugate poles or real-valued poles or both?
• What is the fact behind those two terms (d+ s e) to model the higher frequencies contributions? In modal theory, the lower and the higher frequencies contributions are modeled with so-called lower and upper residual terms. For instance, for displacement / force transfer function, the lower residual term is LR/s^2 , and the upper residual term is UR where LR and UR are real-valued constants.  Those two terms can easily be deduced by evaluating (using transfer function equation of a SDOF system) the behavior of a mode located at far below / far above the frequency band of interest.  It is not clear how those two terms d and s e are derived to model higher frequencies?
• How do you set the initial values of the poles? What is the criterion adopted to stop this iterative estimation process? I understood that initial poles set is considered. Then, the residues with d and e term are estimated. Then, the initial poles are updated, which is not clear btw from the text how this is done? The estimation process needs to be clarified more. Also, is there any grantee that the estimated poles are stable (i.e., with a negative real part)? How is the number of estimated modes (model size) set?
• For LF framework, it is not clear how is the optimal model size (i.e., the number of states) set? Also, is the stability of the estimated model always granted?
• Line 368, there is an incomplete sentence “the aeroelastic model and the wing of interest”?
• The instability is clear from figure 4 where the damping becomes a negative damping. What is then the need for figure 5?
• Figure 4-d: this plot is difficult to be interpreted? Are you plotting the imaginary parts and their complex conjugates? What are the objective of plotting the real and imaginary parts of the poles (i.e. figure 4 b & d)?

Author Response

Please see the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

First of all, thanks for your contribution to the scientific community. Advances in this field are scarce and are more than welcome.
The paper is very well structured and written, with a hard work behind. However I have some minor comments, and a very important one (the last).
1. Row 45. You mention that "SI is well stablished as a tool". My understanding is that SI is a compendium of tools and techniques, a generic term, not a specific tool. If this is not the case and there is a specific technique called this way, please clarify in row 37 with references. Note. Reading further I believe you mean to mention a specific technique that is good for calculating large-scale structures, but instead you wrote "SI". Could this be a typo? If not I would like to know why you state the computationally demanding and ill-conditioned problems as global problems for all techniques.
2. Rows 46 and 47. You mention that "in the frequency domain, ill conditioning in the fitting process complicates the identification of modal parameters..." and then you provide a reference. Reading further I understand that you refer to techniques other than FRVF (reference 4 mentions FRVF. Initially I thought you were referring only to FRVF). Can you please mention which techniques face this problem? Is it a general problem or specific to some techniques?
3. Rows 48 and 49. I don´t disagree with the statement, but could you specify if this is a general problem for time-domain techniques or particular for N4SID? Can you give more examples?
4. Row 78. You mention damping ratios as a generic term (defined in row 36). Which damping model are you considering? Structural? Viscous?... Please refer to the appropriate paragraph.
5. Rows 170 to 173. I understand that the caveats you provide for time domain and frequency domain techniques are limited to the formulation of FRVF employed. If this is the case, please specify so. As written it seems that time domain techniques in general are limited to the frequency ranges of interest, while this argument can be used (indeed is more appropriate) for frequency domain techniques (again, as referring to generic techniques).
6. Paragraph in row 309 imposes a very strong limitation. Only the first mode of flutter is analyzed. What about other modes? Mode 1 bending vs. mode 2 torsion for example? Is this limitation applicable to the whole method or only to this paper? Can other combinations be easily calculated?
7. Equation 7 and 37 mix terms. In equation 7 c is viscous damping, while in 37 is aerodynamic stiffness
8. Row 327 and 329. d = structural damping... but it is employed as viscous damping. Can you clarify?
9. I understand that structural terms and mathematical terms repeat letters (c for damping and cord, s for span and Laplace transform variable...) and in this case are easily differentiatable. However, please review and change the letters that can be changed (c is always cord and viscus damping coefficient, so changing c to something else will probably create more confusion to the reader, but the span is not always referred to as s)

10. The conclusions are very poor. No limitations are specified. For example, this study is limited to small UAVs (below 7 Kg), but this is only mentioned in the abstract. The Reynolds number is not mentioned in section 3, and this is important if the technique is intended to be extrapolated to bigger aircraft, because the airspeed is very low (50 kts, although probably fast for a 7 Kg UAV... is it? say it). Note that to prepare a GVT for a 7 Kg UAV (instrumentation, sw licenses...) is probably more expensive than manufacturing 10 test specimens and reaching the actual flutter speed. This is a limitation, but a future line of work could be  (must be if you ask me) to extrapolate the results to bigger UAVs. How can it be done? I miss a deep discussion on the results, limitations and further improvements.

In my opinion, the conclusions section is the only that needs to be rewritten. The rest (but for minor comments) is very good.

Author Response

See attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper, titled "Structural Damping Identification Sensitivity in Flutter Speed Estimation," focuses on using advanced frequency-domain system identification methods to predict the flutter onset speed of a flexible wing model. The study applies techniques such as Fast Relaxed Vector Fitting (FRVF) and the Loewner Framework (LF) to extract modal parameters from ground vibration testing data, comparing these results with a benchmark method (N4SID). The identified modal parameters—natural frequencies and damping ratios—are then used to construct a simplified two-degree-of-freedom reduced order model for estimating flutter speeds. The research highlights that even small errors in damping ratio identification can significantly impact the predicted flutter speed, thereby underlining the sensitivity of flutter predictions to the accuracy of these modal parameters.

 

Overall, while the paper presents a novel approach and seems to be scientifically sound, major revisions are required before I can recommend the publication of this manuscript in Vibration.

 

• In the abstract, the term "UAV" appears without its full expansion; it would be beneficial to spell it out upon first mention. A general check for all abbreviations would be necessary.
  • The concluding paragraph of the introduction seems ambiguous; perhaps a complete rewrite or restructuring would enhance clarity.
  • Renaming Section 2 to "Methodology" could provide a clearer indication of its content. But the authors should feel free to find a better and more descriptive title.
  • There is an issue with Figure 2—the y-axis looks distorted, and a white background might improve its visual appeal.
  • Adding a representation of the cross-section of the tested wing could significantly aid reader comprehension.
  • Including an image of the experimental setup—with details on sensor arrangement, shaker table configuration, and calibration—could further strengthen the paper.
  • Adding pictures of each step to the schematic diagram (Figure 1) that illustrates the entire workflow—from data acquisition through to flutter speed estimation—could significantly aid reader understanding.
  • The methodology section currently contains results, which may disrupt the flow; moving these findings to the results section would improve structure.
  • Some assumptions appear at the beginning of the results section; these might be more appropriately placed in the methodology section.
  • Several figures like Figures 4 and 5 have too small texts. Adjusting the text font might also help make the figures more legible.
  • A discussion on the limitations of the reduced-order model assumptions would be necessary to help the reader understand where this paper stands in the literature.
  • It might be helpful to address how experimental noise and uncertainties could impact the identification of damping ratios and the subsequent flutter speed predictions.
  • A comparative discussion on the computational efficiency and scalability of the FRVF, LF, and N4SID methods might add further depth.
  • The conclusion could benefit from a stronger focus on method-related outcomes, with clearer future work suggestions.
  
  

Author Response

See attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

No more comments to the authors. The paper was adapted according to the comments sent in the first review.  

Author Response

Thank you very much for the time and effort spent reviewing our manuscript.

We are happy that the revised manuscript met your expectations.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors.

Thanks for your kind reply.

I concur with all your corrections but two of them, comments 4 and 8. I finally decided to recommend a minor review, but as you will see below I spent some time trying to decide if major or minor.

4:

Row 78. You mention damping ratios as a 
generic term (defined in row 36). Which 
damping model are you considering? 
Structural? Viscous?... Please refer to the 
appropriate paragraph 
This is outlined in Section 2.4, lines 382-387 on 
page 12. 
As the modal parameters considered here are 
obtained from wind-oA GVTs, all modal damping 
ratios are considered structural only. Such that 
viscous  structural. This is a standard approach in 
aeroelastic modelling, as once the flight speed 
increases aerodynamic damping becomes 
dominant (Wright & Cooper, 2014) and the 
structural damping alterations can be neglected. 
This consideration is added in the introduction as 
well, see lines 86-89 on page 2. 

8:

Row 327 and 329. d = structural damping... 
but it is employed as viscous damping. Can 
you clarify? 
Thank you for pointing this out. As reported in 
literature (Section 10.5.6, Wright & Cooper, 2014), 
it is a standard approach to consider that the 
damping identified from wind-oA GVT tests is 
viscous ≈ structural due to the dominance of 
aerodynamic eAects on the system onset and the 
assumptions that structural-only components 
aAect the GVT results (particularly true for small 
amplitudes – no aerodynamic damping). We 
acknowledge that damping might change with 
frequency; however, this change is mostly 
influenced by aerodynamic eAects, as widely 
accepted in the aeroelastic community. These are 
now clarified in the text, see lines 86-89 on page 2 
and lines 382-387 on page 12.

 

The reference you cite, Wright and Cooper, is a generic textbook on aeroelasticity (very good indeed, I cited it myself in my Thesis) and doesn´t dig deep on differences between both damping models, but nevertheless they are important. The onset of both kinds of damping is different and the coefficients are also different. Citing Soroka: "For low values of damping, up to g = 0.2, the two concepts give nearly the same results when g is taken as twice the critical viscous damping ratio" (Note on the Relations Between Viscous and Structural Damping Coefficients | Journal of the Aeronautical Sciences)

I am not Soroka, by the way :)

Both structural and viscous damping models are commonly used for aeroelastic analysis depending on the process followed, the flutter onset considered, dynamic pressure (not applicable for GVT of course), the structure itself... but this process will return a number and this number needs to be compared to a flutter damping threshold, which can lead to a catastrophic event if the analyst, tester and validator compare different numbers because they talk about different concepts.

I am not questioning the use of one model or another, just requesting to be clear on the model used and the coefficients returned. Probably the solution is so simple as to change "structural" for "viscous" in the paper, but using both terms indistinctly is not acceptable.

 

 

 

Author Response

Dear Reviewer #2,

First of all, many, many thanks for your highly professional work and your very diligent review. All your comments helped us to greatly improve the final manuscript. We sincerely appreciate this.

Regarding your specific point, this is due to a potentially inappropriate choice of terms on our side.

We intended to highlight that we considered the damping as a property of the structure itself only – from that the term structural. We did not intend to refer to a structural damping model or a direct hysteretic damping. We confirm that the damping considered in this work arises from the viscous damping model, which, however, as explained in your comments and Soroka's work, can have a similar result to structural damping for low-amplitude GVTs. To back this argument up, we have added the following references to our manuscript:

• P. Coleman, "Damping formulas and experimental values of damping in flutter models," NACA Technical Note, no. 751, National Advisory Committee for Aeronautics, Washington, DC, USA, Feb. 1940. [Online]. Available: http://hdl.handle.net/2060/19930081534
• W. Soroka, ‘Note on the Relations Between Viscous and Structural Damping Coefficients’, Journal of the Aeronautical Sciences, vol. 16, no. 7. American Institute of Aeronautics and Astronautics (AIAA), pp. 409–410, Jul. 1949. doi: 10.2514/8.11822.

To better clarify this point in the revised manuscript, we have removed any mention of structural damping within the main text and title. Specifically, we have extended our explanation of the damping assumption used in this work in lines 381-389 on page 12.

Thank you again for your feedback.

General Remarks:

While reviewing the flutter code for open access publication, we discovered an inconsistency in the way the reduced frequency was handled in the unsteady aerodynamics term. It was treated as constant across all frequency values, which, while a feasible approach, is not as rigorous as updating it iteratively with respect to the pitching frequency. Nevertheless, the new predicted flutter speed values deviate by less than 2% from the original ones.

Reviewer 3 Report

Comments and Suggestions for Authors

I can now recommend publication of this manuscript as all my comments are addressed.

 

Author Response

Thank you for your time and effort reviewing our manuscript.

We are happy that the revised manuscript can now be recommended for publication.