On the Effects of Structural Coupling on Piezoelectric Energy Harvesting Systems Subject to Random Base Excitation

Round 1

Reviewer 1 Report

This study proposes to investigate the novel approach of coupling separate energy harvesters in order to scavenge more power from a stochastic point of view. My concerns:

1. Page 3, Line 103. How is the resistor connected to two coupled piezoelectric beam in Figure 1 ? It should be given in this figure.
2. Page 3, Line 113. How are Equation(1) and Equation (2) derived from? Is there any reference?
3. Page 11, Line 243. How is Equation(42 ) derived from? More details should be given.
4. Which values can support the view of ‘The final optimized model demonstrated that the coupling of the structures lack the potential to significantly improve the energy harvesting under random excitation’. The power output of the optimal model configuration should be compared with other configurations to support the view above.
5. Check the indexes of the equations in this paper. There are a lot of errors of the indexes of the equations showed as reference source not found in this paper.
6. Also, some typos need to be corrected, for example

-Page 25, Line (516): (A-1) is mistyped as (A-2).

Author Response

The authors would like to sincerely thank the Reviewer for the time and efforts devoted to reviewing our paper. The comments were carefully investigated, and modifications were made based on the comments to improve the quality of the paper. The changes are highlighted in yellow in the manuscript and each point is specifically addressed below to expand on the rationale of the changes. We hope that the paper quality has reached a satisfactory level.

Reviewer #1:

This study proposes to investigate the novel approach of coupling separate energy harvesters in order to scavenge more power from a stochastic point of view. My concerns:

Our reply: We thank the reviewer for the valuable comments that have improved the original version of the paper.

• How is the resistor connected to two coupled piezoelectric beams in Figure 1? It should be given in this figure.

Our reply: We thank the reviewer for reminding us of this shortcoming. How the resistors are connected to the piezoelectric beams can now be seen in figure 1.

• How are Equation (1) and Equation (2) derived from? Is there any reference?

Our reply: These are equations of motion for a Euler beam. They are taken from the book by Rao: Vibration of Continuous systems. We have included this reference in the revised paper.

• How is Equation (42) derived from? More details should be given.

Our reply: Thank you for mentioning this point. To address this issue, we have added Appendix C and equation (42) is derived in this appendix

• Which values can support the view of ‘The final optimized model demonstrated that the coupling of the structures lack the potential to significantly improve the energy harvesting under random excitation? The power output of the optimal model configuration should be compared with other configurations to support the view above.

Our reply: Thank you very much for this important point. We have added the explanation below, in section 5, to show clearly that, considering the final optimized model, the coupling of the structures lacks the potential to improve the energy harvesting under random excitation:

‘’ At first glance, one may conclude that increasing the coupling by enhancing the spring stiffness from its lower band, 240 N/m, to the optimal value, 665.94 N/M, has increased the expected power obtained from the system, but this is not correct. If we compare the overall expected mean of the total power when k=240 N/mwith the expected power of the optimized system, we can see that the difference is negligible. This small difference is observed even when we put k=0 instead of 240 N/m. This issue comes from the inherent properties of the GA since this process is a zero-order optimization algorithm. In the cases that the optimal values occur at the vertices of the region, the GA often requires a large number of iterations and normally convergence is assumed once the variation in optimal values of objective function falls below a threshold.  As discussed above, the coupling does not have a positive effect on enhancing the total harvested power of the optimised system which can be observed from figure 12. This figure illustrates that when all of the other design variables are constant, increasing the coupling power decreases the mean expected harvested power of the system. ‘’

The optimization process covers all the range of possible configurations by trying different quantities for each of the design parameters.

• (A-1) is mistyped as (A-2) Check the indexes of the equations in this paper. There are a lot of errors of the indexes of the equations showed as reference source not found in this paper. For example-Page 25, Line (516): (A-1) is mistyped as (A-2)

Our reply:  Corrected.

The authors would like to sincerely thank the Reviewers for the time and efforts devoted to reviewing our paper. The comments were carefully investigated, and modifications were made based on the comments to improve the quality of the paper. The changes are highlighted in yellow in the manuscript and each point is specifically addressed below to expand on the rationale of the changes. We hope that the paper quality has reached a satisfactory level.

Author Response File: Author Response.docx

Reviewer 2 Report

please see the attached

Comments for author File: Comments.pdf

Author Response

The authors would like to sincerely thank the Reviewer for the time and efforts devoted to reviewing our paper. The comments were carefully investigated, and modifications were made based on the comments to improve the quality of the paper. The changes are highlighted in yellow in the manuscript and each point is specifically addressed below to expand on the rationale of the changes. We hope that the paper quality has reached a satisfactory level.

Reviewer #2:

In this paper, the authors investigate the harvestable power of a 2-DOF vibration energy harvester comprising of two cantilever beams connected via a rigid base and a linear spring. Analytical closedform expressions are derived based on linear modal analysis, leading to optimisation of the harvester. The paper is generally well-written, the topic is indeed timely, and the methods followed by the authors are rigorous and pertinent to this problem.:

Our reply: We thank the reviewer for the valuable comments that have improved the original version of the paper.

Part 1: Major comments

1. The main conclusion of this paper is that the linear coupling between the two beams does
   not significantly enhance the harvestable energy, if not having a negative effect (page 24,
   lines 461-462; Conclusions, lines 481-482). I am not sure there is enough evidence to
   support this, not least because a band-limited white noise is considered and not harmonic
   Figure 12 shows that the expected value of the total harvested energy is
   increased for reducing k (down to about 1000 N/m). However, the genetic optimization has
   found an optimal k at 665.94, well above the lower limit of 240.The stated conclusion is contradicting with the optimization results, which imply that there is
   some gain to include k at its optimal value. I think that the authors should look closer in the
   response when k<665.94, down to 0, in order to evaluate if there is a significant advantage
   from the coupling of the beams or not.

Our reply: The authors agree with the reviewer. As already mentioned in response to the first reviewer, we have added the explanation below to show clearly that, considering the final optimized model, the coupling of the structures lacks the potential to improve the energy harvesting under random excitation:

’ At first glance, one may conclude that increasing the coupling by enhancing the spring stiffness from its lower band, 240 N/m, to the optimal value, 665.94 N/M, has increased the expected power obtained from the system, but this is not correct. If we compare the overall expected mean of the total power when k=240 N/mwith the expected power of the optimized system, we can see that the difference is negligible. This small difference is observed even when we put k=0 instead of 240 N/m. This issue comes from the inherent properties of the genetic algorithm (GA) since this process is a zero-order optimization algorithm. In the cases that the optimal values occur at the vertices of the region, the GA often requires a large number of iterations and normally convergence is assumed once the variation in optimal values of objective function falls below a threshold.  As discussed above, the coupling does not have a positive effect on enhancing the total harvested power of the optimised system which can be observed from figure 12. This figure illustrates that when all of the other design variables are constant, increasing the coupling power decreases the mean expected harvested power of the system.  ‘

 Moreover, as it has already been mentioned in the paper the optimum solution seems to have beams with very different thicknesses and therefore very little coupling in the modes. Consequently, there is not a significant advantage in coupling the two beams via a linear spring.In lines 456-457, the authors state that “… a remarkable improvement of 639.88% in the

2. expected total harvested power would be achieved ...” by applying the optimization results
   in Table 8. However, there is no evidence of that, e.g. a graph or a table with numeric values
   of the power across different steps in the optimization process.

Our reply: The authors thank the reviewer for mentioning this important comment. The genetic algorithm is a multi-step parallel computer algorithm which carries out the optimization. Since this optimization problem considers all possible configurations, its high computational cost has left us only with the option of saving the results and not the intermediate objective function. However, in order to show the effect of each design parameter, individually, figures 10-15 have been plotted and the related results have been discussed in the paper.

3. The authors mention that a genetic algorithm was used to solve the optimization problem of
   (66) in Section 5. Please provide details of the algorithm or of the commercial software
   used and the relevant routines. .

Our reply: We have used the genetic algorithm of the MATLAB software for the optimization problem related to equation (66). We have not changed any of the MATLAB defaults of the Genetic Algorithm.

Part 2: Minor comments

1. Figure 1. L should be L_s

Our reply: The authors thank you for the detailed evaluation of the paper. The mentioned mistake is corrected in the revised version.

2. Line 132, “In appendix A” should be “In Appendix A” .

Our reply: Corrected.

3. Table 2 is cited in line 137, but not shown in the manuscript
   Our reply:
4. Include the copyright symbol after “Abaqus”
   Our reply:
5. Several citations are misprinted in the manuscript as “Error! Reference source not found.”
6. Our reply: Corrected
7. There are several inline math typesetting errors that need to be fixed, e.g. line 161
   Our reply:
8. hp is missing from Table 3 and Table 5. Although a parametric study is conducted later in
   Section 4, the results of Tables 6 and 7 depend on the used hp
   Our reply: hp has been added to table 3.
9. Figure 5. The legend was not properly printed.
   Our reply:
10. Line 237. “Table 1” should be “Table 6”.
    Our reply:
11. Lines 257 and 259. The phrase “base acceleration” in the figure captions should be replaced
    by “harmonic base acceleration”.
    Our reply:
12. Figure 10. Which value did you use for k? Are the rest of the parameters given in Tables 1
    and 3?
    Our reply: The authors are grateful for mentioning this crucial issue. Since the same confusion may arise for understanding the figures of section 4, we have added a paragraph, in the parametric study section, that can address the mentioned problem by the reviewer for all of the figures in this part:
    ‘’ In order to find the individual impact of the design parameters, we kept all of the physical parameter values of the whole system constant (consistent with the values in Tables 1, 3, 4 and 5) except the parameter that we intend to evaluate its sole effect on the system’’.
13. Line 357. The phrase “the relative maximum” should be “the local maximum”
    Our reply:
14. Figure 12. Linked to major comment 1, I think that these cases should be compared to the
    case of completely uncoupled beams (i.e. no spring and rigid base).
    Our reply: We are grateful for this important comment. This issue has been addressed in the section Design Optimization, as was mentioned before in addressing your first major comment.
    
    
15. Line 375. “is took place” should read “takes place”.
    Our reply:
16. Lines 379-380: There a formatting error (text in bold)
    Our reply:
17. Lines 424-441. Formatting error (centered text).
    Our reply:
18. Line 451: Delete the first occurrence of “the”.
    Our reply:
19. Lines 488-512. The authors did not process the fields Supplementary materials … Conflicts of
    Our reply: Corrected.
    
    

Author Response File: Author Response.docx

Reviewer 3 Report

In this paper, a multi-body system composed of two cantilever harvesters with two identical piezoelectric patches based on random excitation was modeled and simulated. This work is of some interest for designing other types of piezoelectric-based energy harvesters with complex structures undergoing random excitation, but I have some questions below:

1. There are many mistakes in the manuscript, which cause great trouble to read.
2. The focus of this paper was to model the device, but many equations had no reference and no reasoning process. Can the authors give some more step by step derivation?
3. Tables 2 and 6 were missing.
4. What is the difference between table 3 and table 5?
5. The annotation of the red line in figure 5 is missing.
6. The abscissa is not marked in Figure 13
7. The final optimized model demonstrated that the coupling of the structures lack the potential to significantly improve the energy harvesting under random excitation. This is a contrary point of view to what was previously demonstrated for harvesters under deterministic loads. Can the author explain the mean reason?
8. There is no reference about the modeling part and there is no experimental verification with numerical results.
9. This manuscript focused more on the influence of design parameters on the scavenged electrical power, and did not establish the relationship between random excitation and the scavenged electrical power. This is not good because the purpose of the manuscript is to model and simulate a multi-body system based on random excitation.
10. The quality of writing is bad and should be addressed.

Author Response

The authors would like to sincerely thank the Reviewer for the time and efforts devoted to reviewing our paper. The comments were carefully investigated, and modifications were made based on the comments to improve the quality of the paper. The changes are highlighted in yellow in the manuscript and each point is specifically addressed below to expand on the rationale of the changes. We hope that the paper quality has reached a satisfactory level.

Reviewer #3:

In this paper, a multi-body system composed of two cantilever harvesters with two identical piezoelectric patches based on random excitation was modeled and simulated. This work is of some interest for designing other types of piezoelectric-based energy harvesters with complex structures undergoing random excitation, but I have some questions below:

Our reply: The authors thank the reviewer for the detailed and clear comments that have improved the quality of revised paper.

1. There are many mistakes in the manuscript, which cause great trouble to read.
   The focus of this paper was to model the device, but many equations had no reference and no reasoning process.

Our reply: In the revised paper all of the mistakes mentioned by the reviewers are corrected

    2.Can the authors give some more step by step derivation?

Our reply: We have included the relevant reference that derives equations (1) and (2). We have added Appendix C in the revised version and equation (42) is derived in this appendix.Furthermore, equations (51) and (59) have been changed in the revised paper so that their step-by-step derivation make more sense for the reader.Another issue that may have led to confusion about the derivation of the equations is probably the removal of references due to technical issues. In the revised version of the paper, all the references related to derivation of mathematical equations have been corrected and this has made the paper significantly easier to follow.

    3.What is the difference between table 3 and table 5?                                                                               

Our reply: Table 3 illustrates the physical and geometrical properties of the piezoelectric materials used in this paper while table 5 shows those of the piezoelectric layers employed in a paper, ‘cantilevered piezoelectric energy harvesters’’, By Inman and Erturk      

    4.The annotation of the red line in figure 5 is missing.  

Our reply: Corrected

    5. The abscissa is not marked in Figure 13.  

Our reply: Corrected

    6. The final optimized model demonstrated that the coupling of the structures lacks the potential to significantly improve the energy harvesting under random excitation. This is a contrary point of view to what was previously demonstrated for harvesters under deterministic loads. Can the author explain the mean  reason?                          

Our reply: The authors are grateful for mentioning this crucial issue in the paper. In previous studies, it has shown that when mechanical systems undergo harmonic excitation, changing the coupling can tune one of the natural frequencies of the system to approach the excitation frequency. This effect can lead to harvesting more power out of the system. In contrast to previous research papers, here, we have studied the role of coupling while the mechanical system undergoes a wide-band random excitation and we could see that coupling had a negative effect on harvested power of the system.

We have included the above explanation to the design optimization part, section 5, of the revised paper.

    7. There is no reference about the modeling part and there is no experimental verification with numerical results.  

Our reply: In the Validation section, we have compared the dynamic behavior of the system related to this paper with the experimental results obtained in the paper ‘cantilevered piezoelectric energy harvesters’’, by Inman and Erturk. To this end, a harvester with zero coupling and mechanical properties as stated in Tables 4 and 5 are considered to make our model comparable with the model presented in ‘cantilevered piezoelectric energy harvesters’’.                       

    8. This manuscript focused more on the influence of design parameters on the scavenged electrical power and did not establish the relationship between random excitation and the scavenged electrical power. This is not good because the purpose of the manuscript is to model and simulate a multi-body system based on random excitation.     

Our reply: The focus of this paper is the influence of design parameters, specifically coupling, on the scavenged electrical power when the systems undergoes random excitation. The relationship between random excitation and the expected harvested electrical power has been presented by equation (61) and equation (62) analytically.      

   9. The quality of writing is bad and should be addressed.  

Our reply: Thanks for your comments. We have tried our best that the writing quality of the revised version be improved and easier to understand.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The authors have satisfactorily addressed my major and minor comments. The paper is recommended for publication.

Reviewer 3 Report

The authors had made sufficient modifications,so i recommend this work to be published in Aerospace.