Modeling and Stability Analysis for the Vibrating Motion of Three Degrees-of-Freedom Dynamical System Near Resonance

Round 1

Reviewer 1 Report

The Paper is good contains sound results. I recommend for publication with suggestion:

1. Update Abstract.
2.  Update introduction and clearify motivation by citing recent work.
3. Improve presentation and remove typos mistakes.
4. Revise conclusion.

Author Response

For the remarks and questions of Reviewer #1:

Authors would like to thank the reviewers for their great efforts in reviewing the manuscript and for the valuable comments that enrich the work

 

1. Update Abstract.

We appreciate your good note. Abstract has been updated.

2. Update introduction and clearify motivation by citing recent work.

Thanks for your good comment. The introduction has been already updated with seven recent works in year (2021) and one paper in (2022).

3. Improve presentation and remove typos mistakes.

Thanks and gratitude for your good observation. The presentation has been improved with the help of specialists, in addition to the corrections of typos mistakes. All corrections in the paper are presented in red color.

4. Revise conclusion.

We appreciate your good note. The conclusion has been revised.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this manuscript, the authors investigated the dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum,  in which its hanged point follows an elliptic route. The emerging cases of resonance are categorized according to the solvability requirements. The stability areas and the instability ones are examined utilizing the Routh-Hurwitz criteria and analyzed in line with solutions at the steady-state. The results presented in this manuscript are new and interesting. However, the following problems should be revised carefully: (1) why the subsystem that produces a swing motion? A comparsion is needed for the two system in [20] (2) What is the tau_n in differential operator on Line 170. (3) To avoid the possible conflict and further support this manuscript, the following recent related references should be included: [S. Djebali, A.G. Aoun, Resonant fractional differential equations with multi-point boundary conditions on , J. Nonlinear Funct. Anal. 2019 (2019), Article ID 21], [S. Reich, A. J. Zaslavski, Asymptotic behavior of a dynamical system on a metric space, J. Nonlinear Var. Anal. 3 (2019), 79-85]. (4) The English presentation should be improved and there are also many typos. They should be corrected point by point. 

Author Response

For the remarks and questions of Reviewer #2:

Authors would like to thank the reviewers for their great efforts in reviewing the manuscript and for the valuable comments that enrich the work

(1) Why the subsystem that produces a swing motion? A comparison is needed for the two system in [20].

Thanks for your good comment. In [20], the impact of shifting the absorber's position in relation to the fundamental system has been investigated. The efficiency of the first system has been improved when it is shifted away from its center of gravity, while for the second system it has been found that when the absorber is placed in or near the suspension point, the absorber has a zero or slight quenching effect.

In our work, the attached absorber has a good effect on the investigated dynamical system without any consideration of the position of the system’s center of mass and its relationship with the absorber.

(2) What is the tau_n in differential operator on Line 170.

Thanks for your good comment. τ_n denote the different time scales relating with τ by the relation τ_n = εⁿ τ which is stated according to the procedure of the approach of multiple scales.

(3) To avoid the possible conflict and further support this manuscript, the following recent related references should be included:

[S. Djebali, A.G. Aoun, Resonant fractional differential equations with multi-point boundary conditions on (0,+ꝏ) , J. Nonlinear Funct. Anal. 2019 (2019), Article ID 21].

[S. Reich, A. J. Zaslavski, Asymptotic behavior of a dynamical system on a metric space, J. Nonlinear Var. Anal. 3 (2019), 79-85].

Thanks and gratitude for your great note. These works have been described and included in the introduction section.

(4) The English presentation should be improved and there are also many typos. They should be corrected point by point.

We appreciate your good note. The language has been improved with the help of specialists, in addition to punctuation marks, and the grammatical mistakes. All corrections in the paper are presented in red color.

Author Response File: Author Response.pdf

Reviewer 3 Report

The article is devoted to the study of the behavior of an oscillatory system consisting of a nonlinear damping pendulum-spring associated with an absorber with linear damping, in which it is subjected to a force in the transverse direction of the spring. As noted by the authors, such models play an important role in engineering applications such as. construction, bridges, and spinning machines

Using the Lagrange equations, the equations of motion of this system were constructed, which were then solved using the approach of multiple scales. Also, a study was carried out on the fixed points of the analysis of the stability of solutions in the steady state. This made it possible to study nonlinear stability and determine the zones of stability and instability.

In general, the study is a detailed analysis of the sensitivity of the resulting system of equations to various parameters.

Figures 12-14 look exactly the same. It is not entirely clear from the text whether it follows from this fact that the asymptotic solutions Z, Phi, eta do not depend on the parameters C1, C2, C3.

Figures 15e and 15f show a phase shift, while Figures 16e and 16f do not. What could be the reason?

Line 85 most likely meant “damper” rather than “damped”.

Author Response

For the remarks and questions of Reviewer #3:

Authors would like to thank the reviewers for their great efforts in reviewing the manuscript and for the valuable comments that enrich the work

• Figures 12-14 look exactly the same. It is not entirely clear from the text whether it follows from this fact that the asymptotic solutions Z, Phi, eta do not depend on the parameters C1, C2, C3.

Thanks and gratitude for your good observation. The reason is due to that the approximate solutions of the variables Z, Φ,  and η don’t depend on c₁,c₂, and c₃ explicitly. Therefore, the variations seem to be slightly. These meaning have been included in the text with red color.

• Figures 15e and 15f show a phase shift, while Figures 16e and 16f do not. What could be the reason?

Thanks for your good comment. The reason is due to that, the curves of Fig. (15) are examined when ω₁ has different values while the plotted curves in Fig. (16) are graphed when ω₂ has different values.

• Line 85 most likely meant “damper” rather than “damped”.

We appreciate your good note. It has been corrected.

Author Response File: Author Response.pdf