Practical Measurements of Vibration Using the Moiré Effect^†

Round 1

Reviewer 1 Report

1.Enhance the clarity of key formulas and their derivations in Section 2. For instance, Equation (1) for the moiré pattern period TM is mentioned but not explicitly presented in the text, which may confuse readers. Adding the full formula and a brief explanation of each parameter’s physical meaning (e.g., T1 and T2 as the periods of the two grids) would improve readability .
2.Provide more details on the experimental repeatability in Section 3. The paper mentions successful measurements of small displacements (e.g., 0.5 mm) at various distances, but lacks information on how many times each experiment was repeated and the standard deviation of results. Including such data would strengthen the reliability of the sensitivity threshold claims .
3.Improve the annotation of critical figures, such as Figure 15 (measurements of simulated vibrations). The figure shows results for 0.5, 1, and 2 pixels but lacks direct labels for the error margins (mentioned as <3%). Adding error bars or inset text indicating the deviation would make the experimental validation more intuitive .
3.Expand the discussion on limitations in Section 6. While noise is briefly noted as a negative factor, the paper should elaborate on specific constraints, such as how extreme lighting conditions (e.g., direct sunlight) or grid wear affect measurement accuracy. This would help readers better assess the method’s practical applicability .

1.Enhance the clarity of key formulas and their derivations in Section 2. For instance, Equation (1) for the moiré pattern period TM is mentioned but not explicitly presented in the text, which may confuse readers. Adding the full formula and a brief explanation of each parameter’s physical meaning (e.g., T1 and T2 as the periods of the two grids) would improve readability .
2.Provide more details on the experimental repeatability in Section 3. The paper mentions successful measurements of small displacements (e.g., 0.5 mm) at various distances, but lacks information on how many times each experiment was repeated and the standard deviation of results. Including such data would strengthen the reliability of the sensitivity threshold claims .
3.Improve the annotation of critical figures, such as Figure 15 (measurements of simulated vibrations). The figure shows results for 0.5, 1, and 2 pixels but lacks direct labels for the error margins (mentioned as <3%). Adding error bars or inset text indicating the deviation would make the experimental validation more intuitive .
3.Expand the discussion on limitations in Section 6. While noise is briefly noted as a negative factor, the paper should elaborate on specific constraints, such as how extreme lighting conditions (e.g., direct sunlight) or grid wear affect measurement accuracy. This would help readers better assess the method’s practical applicability .

Author Response

Comment 1a: Enhance the clarity of key formulas and their derivations in Section 2.        

Reply 1a: Two formulas and extra explanations were added to Sec. 2.1 (lines 128-160).

 

Comment 1b: For instance, Equation (1) for the moiré pattern period TM is mentioned but not explicitly presented in the text, which may confuse readers. Adding the full formula and a brief explanation of each parameter’s physical meaning (e.g., T1 and T2 as the periods of the two grids) would improve readability.

Reply 1b: We use the magnification instead of the period, see added explanations around this formula and below it in Sec. 2.1 (lines 128-137, 140-141, 148-160).

 

Comment 2: Provide more details on the experimental repeatability in Section 3. The paper mentions successful measurements of small displacements (e.g., 0.5 mm) at various distances, but lacks information on how many times each experiment was repeated and the standard deviation of results. Including such data would strengthen the reliability of the sensitivity threshold claims.

Reply 2: Improved statistics: each experiment now shows the number of repetitions;
for most experiments, the deviation is shown.

 

Comment 3: Improve the annotation of critical figures, such as Figure 15 (measurements of simulated vibrations). The figure shows results for 0.5, 1, and 2 pixels but lacks direct labels for the error margins (mentioned as <3%). Adding error bars or inset text indicating the deviation would make the experimental validation more intuitive.            

Reply 3: Many captions were improved (Figs. 2, 12, 14, 18, 19, 23).  Figure 14 was redrawn; the size bar was added.

 

Comment 4: Expand the discussion on limitations in Section 6. While noise is briefly noted as a negative factor, the paper should elaborate on specific constraints, such as how extreme lighting conditions (e.g., direct sunlight) or grids wear affect measurement accuracy. This would help readers better assess the method’s practical applicability.

Reply 4: Advantages and limitations are discussed in Sec. 6 (lines 693-702).  For noise effects, see Secs. 2.5, 6 (lines 328-329, 661-668). The lighting conditions are described in Sec. 2.4 (lines 271-274). For a grid mounting, refer to Sec. 2.4 (283-285).

 

Comment 5: The English could be improved to more clearly express the research.

Reply 5: Done. See multiple corrections of sentence structure, separate words, and punctuation across the whole text.

Author Response File: Author Response.docx

Reviewer 2 Report

Section 6. Discussion is very short and does not discuss the results of the paper but only it points future directions of the research (in particular, the addition of noise and weather effects in the results obtained).

Section 7. Conclusion is also very short and must be significantly expanded by account of the results obtained.

(Line 324): “Table” should be replaced by “Table 1”.

(Line 420): Abbreviation LVDT should be deciphered.

(Line 443): Where is the “red arrow” in Fig. 21?

Author Response

Comment 1: Section 6. Discussion is very short and does not discuss the results of the paper but only it points future directions of the research (in particular, the addition of noise and weather effects in the results obtained).

Reply1: Results are discussed in Sec. 6 (lines 703-710).

 

Comment 2: Section 7. Conclusion is also very short and must be significantly expanded by account of the results obtained.

Reply 2: Results are included in Sec. 7 (lines 722-726).

 

Comment 3: (Line 324): “Table” should be replaced by “Table 1”.

Reply 3: Done (line 387).

 

Comment 4: (Line 420): Abbreviation LVDT should be deciphered.

Reply 4: Done (line 745).

 

Comment 5: (Line 443): Where is the “red arrow” in Fig. 21?        

Reply 5: Done. (The explanation was removed together with Figure).

 

Comment 6: This research paper (no review paper) is overloaded with different (often one-type) figures. There are 34 figures in 24 pages of the paper. The number of the figures could be reduced by one-third without losing the results of the article. For example, the following figures are uninformative: Figs. 12, 16, 21 and 22 (at presence of upper part of Fig. 18).  

Reply 6: The four mentioned Figs. were removed, one redrawn.

 

Author Response File: Author Response.docx

Reviewer 3 Report

The article presents a study on the practical measurement of vibrations using the moiré effect, which allows a significant increase in the sensitivity of non-contact measurements. It explains that the moiré effect occurs when two similar periodic gratings overlap, generating patterns with longer periods that amplify small, even sub-pixel, displacements—thus enabling precise measurement of minute movements using digital cameras.

The results demonstrate sensitivity of up to 0.1 mm at distances of up to 20 meters, with errors under 3% compared to traditional sensors. The system enables simultaneous bidirectional measurements and is robust under adverse external conditions.

The technique is promising for structural monitoring and public safety, offering a precise, non-invasive optical solution for vibration evaluation in infrastructures.

 

To strengthen the work for publication, it is recommended to expand experimental validation under diverse environmental conditions—analyzing the effects of noise, variable lighting, and external vibrations—to demonstrate the moiré method’s robustness in public safety contexts.

Additionally, broaden comparisons with other optical and non-optical methods. A deeper statistical analysis of sensitivity and detection threshold should be conducted to evaluate error and repeatability.

Exploring applications in other types of infrastructure and optimizing automated data processing would enhance its applicability.

Finally, a parametric study of grating design and complementary numerical simulations would support the results and provide practical recommendations.

Author Response

Comment 1: To strengthen the work for publication, it is recommended to expand experimental validation under diverse environmental conditions—analyzing the effects of noise, variable lighting, and external vibrations—to demonstrate the moiré method’s robustness in public safety contexts.

Reply 1: Environment effects are discussed in Sec. 6 (lines 645-651).  Noise is discussed in Secs. 2.5, 6 (lines 328-329, 661-668). The lighting conditions and vibration measures are described in Sec. 2.4 (lines 271-274, 283-285).

 

Comment 2: Additionally, broaden comparisons with other optical and non-optical methods.               

Reply 2: The comparison with other methods is added to Sec. 6 (lines 669-692).

 

Comment 3: A deeper statistical analysis of sensitivity and detection threshold should be conducted to evaluate error and repeatability.

Reply 3: Improved statistics: each experiment now shows the number of repetitions;
for most experiments, the deviation is shown.

 

Comment 4: Exploring applications in other types of infrastructure and optimizing automated data processing would enhance its applicability.

Reply 4: Other applications and planned improvements are added to Sec. 6 (lines 711-717).

 

Comment 5: Finally, a parametric study of grating design and complementary numerical simulations would support the results and provide practical recommendations.

Reply 5: Various grids are discussed in Sec. 6 (lines 656-663). The numerical estimation of the angular misalignment is given in Sec. 2.4 (277-282).

 

Comment 6: The English could be improved to more clearly express the research.

Reply 6: Done. See multiple corrections of sentence structure, separate words, and punctuation across the whole text.

 

Author Response File: Author Response.docx

Reviewer 4 Report

Please see my comments in attached file.

Please see my comments in attached file.

Comments for author File: Comments.pdf

Author Response

Comment 1: Provide detailed protocols for grid preparation, alignment and mounting. Specify tolerances for parallelism, period uniformity and angular alignment, as even small alignment errors can significantly affect moiré sensitivity and measurement accuracy.

Reply 1: Grid making, alignment, and mount are discussed in Sec. 2.4 (lines 275-282). The accuracy in phase measurements is mentioned in Sec. 6 (lines 653-655). Another factor strongly affecting the accuracy is wrapping; it is briefly described in the added Sec. 2.5 (lines 313-329).

 

Comment 2: Describe the calibration procedure for both the moiré device and the camera system. Specify how the spatial scale is determined (e.g. using a calibrated ruler or a reference object in the field of view) and how the magnification coefficient is experimentally verified.              

Reply 2: In our method, no calibration is needed; see the last paragraph of Sec. 2.3 (lines 257-259).

 

Comment 3: Clearly specify the camera model, sensor size, resolution, lens focal length and image processing settings (e.g. exposure time, ISO, light intensity). These factors have a direct influence on sensitivity and noise.

Reply 3: Camera models are listed in the beginning of Sec. 2.4 (lines 263-270).

 

Comment 4: Standardise and document the lighting conditions during experiments, as lighting can affect the visibility and contrast of moiré patterns.

Reply 4: The lighting conditions are included in Sec. 2.4 (lines 271-274).

 

Comment 5: For laboratory experiments, describe how external vibrations and environmental disturbances are minimised or measured.

Reply 5: Issues related to vibrations are briefly described in the beginning of Sec. 2.4 (lines 283-285).

 

Comment 6: Quantify sources of error, including camera noise, grid imperfections, and environmental factors. Provide uncertainty estimates for measured displacements and explain how noise is filtered or attenuated.

Reply 6: Noise effects are discussed in Secs. 2.5, 6 (lines 328-329, 661-668). Various grids and environment effects are discussed in Sec. 6 (lines 656-660, 645-651). Improved statistics: each experiment now shows the number of repetitions; for most experiments, the deviation is shown.

 

Comment 7: For comparisons with LVDT sensors or accelerometers, state the calibration and traceability of these reference devices. Ensure synchronised data acquisition and provide quantitative comparison metrics (e.g. mean square error, correlation coefficients).

Reply 7: Specifications of LVTD and accelerometer are added to the beginning of Secs. 4.1, 4.2
(lines 482-485, 504). The synchronization of sensors is mentioned in Sec. 4 (lines 476-477).

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The authors have introduced necessary revision and the paper may be accepted in this form.

No.

Reviewer 3 Report

It is hereby stated that the manuscript entitled Practical measurements of vibration using the moiré effect has been accepted for publication, once the author satisfactorily addressed the corrections requested during the review process.

The document had been previously evaluated, and observations were issued to improve its academic and scientific quality. After the submission of the revised version, it was verified that the modifications were carried out in accordance with the recommendations, meeting the criteria of clarity, methodological rigor, and relevance of the content.

Consequently, the manuscript is now ready to be integrated into the final stage of the editorial process, ensuring its proper presentation and contribution to the corresponding field of study.

Final decision: Accepted with corrections satisfactorily completed.

It is hereby stated that the manuscript entitled Practical measurements of vibration using the moiré effect has been accepted for publication, once the author satisfactorily addressed the corrections requested during the review process.

The document had been previously evaluated, and observations were issued to improve its academic and scientific quality. After the submission of the revised version, it was verified that the modifications were carried out in accordance with the recommendations, meeting the criteria of clarity, methodological rigor, and relevance of the content.

Consequently, the manuscript is now ready to be integrated into the final stage of the editorial process, ensuring its proper presentation and contribution to the corresponding field of study.

Final decision: Accepted with corrections satisfactorily completed.

Reviewer 4 Report

Now, it is ready to be published.

Now, it is ready to be published.