Nonlinear Dynamics Study of Giant Magnetostrictive Actuators with Fractional Damping

Round 1

Reviewer 1 Report

The manuscript may be accepted if the following errors are corrected:

1. Line 227 needs to be changed to correct

2. Line 229 needs to be changed to correct

3. Line 232 needs to be changed to correct

4. Line 252 needs to be changed to correct

5. Equation (20): The article should clearly state where the nonlinearity is?

6. To enrich the introduction section, the authors should refer to some of the following related publications:

Because the article studies vibrations of structures, it is necessary to cite some articles on vibrations of structures, for example.

- https://doi.org/10.1155/2021/1586388

-  https://doi.org/10.3390/ma12030404

Author Response

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Reviewer 2 Report

The intrinsic mechanism and dynamical characteristics of the GMA dynamical system was investigated, and the dynamical equations of the nonlinear GMA system containing fractional damping terms were established and analyzed using the averaging method. The numerical results show that the fractional order has a greater influence on the bifurcation and chaotic behavior of the system. The study is interesting. However, the bifurcation diagrams, such as, Fig.16, Fig.18, Fig.21 are not so clear. Moreover, the writing need be polished again. The references can be updated.

Author Response

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Reviewer 3 Report

The paper introduces a simple ordinary differential equation model of a so-called GMA, and then adds a fractional derivative term to this ODE. The results are then analysed both in terms of abstract dynamical systems, as well as at least some comparisons with experimentally measured values. In principle the work is interesting, and relevant to researchers in this area, as well as possibly to other researchers using fractional derivatives in other contexts. There are many places though where the presentation is confusing, so I would urge the authors to do a major rewrite to try to present things more clearly. As just a few of many examples of things that should be clarified:

1. The captions for figures 2 and 3 certainly need to explain the figures far more clearly! Basically, for every figure, ask yourself whether the caption is really describing the figure clearly, and not just to someone like you who already knows it well, but to someone who is seeing it for the first time.

2. In Eq (1) there is a variable T_3 that I don't think is properly defined. Similarly I wasn't always clear about what some of the other variables throughout the paper actually meant. So again, go through the entire paper, identify every variable you introduce, and double-check that you really have properly defined it, explained what it is, etc.

3. I did not understand at all where Eq (9) comes from, or later on Eq (13) with the fractional derivative. How do these relate to the ODEs Eqs (8) without fractional derivatives and (14) with fractional derivatives??

4. Also, why does Eq (13) come before Eq (14)?? Eq (14) after all is the fractional derivative ODE, whereas Eq (13) is presumably something to do with solutions to that equation. How can you discuss aspects of the solutions first, and only present the equation afterwards?

5. On page 7, lines 225/226 it says "the weak nonlinear factor epsilon in the system is crowned before..." What does "crowned" mean here?? There were a number of places throughout the paper where I think the English just isn't quite good enough, and I wasn't entirely clear on what the sentence was trying to say. So at some point anyway some of the English should be tidied up and clarified a bit.

6. At the bottom of page 8 it says "To solve the numerical solution... Eq (13) is ..." Don't you mean Eq (14) here, which is after all the fractional ODE to be solved? There were also a number of other places where I suspect the equation numbers listed don't actually correspond to what you want them to correspond to. So as with some of the other comments above, please go through the entire paper and double-check that every equation number really does refer to the equation that you intend it to. Having mistakes in which equations you are referring back to makes it incredibly difficult and frustrating for readers to try to follow!

7. Eqs (21)-(23) is another of those cases where I can't understand why you are presenting the material in this order. My understanding anyway is that you are discussing detailed diagnostics related to the solutions of your ODE. So why are you presenting this BEFORE you talk about how you are going to solve the ODE at all? First explain how you will numerically solve your ODE, then explain what further diagnostic quantities you might want to obtain once you have x(t)...

8. On page 8 an algorithm is mentioned for how you solved your fractional ODE Eq (14). I am not familiar with this particular algorithm, but that is all right.. I have a question though about how initial conditions work here! In particular, you say here that you use initial conditions such as (0, 0, 0.5), presumably corresponding to x, y, and z. However, if I look at the original ODE Eq (14), its highest derivative is the second derivative on x. So I would expect this ODE to only require TWO initial conditions, on x and dx/dt. So how/why does this particular fractional algorithm now require three initial conditions??  In terms of the results later on, for the vast majority of them initial conditions are irrelevant anyway, since they are the equilibrated solutions after all initial transients have died away. So it won't change the results, but readers might appreciate a little more discussion of this algorithm, along with a few words about initial conditions both in the original ODE and in this version reformulated for this algorithm.

The results section after that is fine. Some of them were maybe a bit technical, and not all readers would necessarily be interested in all of them, but as solutions of an interesting fractional ODE, they were carefully done and thoroughly presented. So please just do a major rewrite of all the introductory sections where you introduce and explain the basic fractional model at all...

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Previous comments have been adequately addressed, so I believe this paper is now acceptable for publication.