The Swinging Sticks Pendulum: Small Perturbations Analysis
, Marcelo F. Ciappina 1,2,3,* and Sergio Elaskar 4
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors\begin{center} Report on \end{center}
\begin{center}\emph{The swinging sticks pendulum: small perturbations analysis}\end{center}
\begin{center} by Y. Li, R. Tang, B. Kumar Das, M F Ciappina, and S. Elaskar. \end{center}
\textbf{Summary.} The primary objective of this paper is to investigate the dynamics of the swinging sticks pendulum using the Lagrangian formalism. The study numerically solves the equations of motion derived from the Euler-Lagrange equations by employing the Bulirsch-Stoer method-an accurate extrapolation technique based on the modified midpoint method. A key contribution of this work is the analysis of the system's behavior near a stable equilibrium point, where the pendulum exhibits small-amplitude oscillations. In this regime, nonlinear terms in the equations of motion can be neglected, enabling a simplified analysis using methods from nonlinear dynamics and Fourier analysis.
Furthermore, multiple trajectories are generated and analyzed to explore frequency interactions and uncover the emergence of complex dynamical behaviors. The proposed approach offers a robust framework for examining the consistency between numerical, analytical, and Fast Fourier Transform (FFT) results, with a high degree of agreement observed across all tests. The findings are reinforced by rigorous mathematical analysis, and all key research questions are thoroughly addressed.
\textbf{Recommendation.}
The paper is promising and suitable for publication in its present form.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis manuscript provides a detailed and rigorous analysis of the dynamical behavior of the swinging sticks pendulum under small perturbations. The authors derive the equations of motion using the Lagrangian formalism, identify the equilibrium configurations, and analyze the system's local dynamics through normal mode analysis. The use of both numerical and analytical approaches - including Bulirsch-Stoer integration, Fast Fourier Transform, and phase-space diagrams - demonstrates a thorough and well-executed study of the low-energy regime. However, this work seems as a simple example of classical mechanics, not furnishing anything new in techniques or impressive results. I do not think that it is suitable to this journal at the present form. Please, make clear to me and to the readers why it is important to solve this particular problem.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper aims to study how small initial disturbances affect the periodic motion of a nonlinear double pendulum system and validate analytical predictions with numerical simulations and frequency analysis techniques. This is my comment
a. Kindly add calculation value in abstract
b. background study about pendulum is very low in introduction, many literature discuss about this topic
c. Novelty is very low, compared to previous work on coupled pendulum systems, particularly referencing double pendulum chaos studies.
d. it will great if author validated with experiment
e. The authors should provide more discussion on how FFT resolution and windowing parameters were selected for frequency extraction
f. kindly add organized of this paper in the end of introduction
g. The explanation of fixed point stability via eigenvalues lacks a clear physical interpretation. What does center vs. unstable mean for system behavior?
h. The transition between linear and nonlinear regimes is not quantitatively defined. Authors should estimate an energy threshold where linearization breaks down.
i. While the paper references chaos theory, no chaotic behavior is demonstrated or quantified. Lyapunov exponents or phase space plots for high-energy conditions would add value.
j. How author get the parameter value? can you make sure that parameter obtain in experimental validation
k. The Poincaré sections are mentioned but not fully developed, what were the cross-section conditions, and how were they computed?
l. Figures 2–11 lack informative captions. Authors should describe initial conditions, simulation duration, and key observed behavior in captions.
m. For numerical vs analytical comparisons, i suggest author include error analysis or uncertainty bounds
n. Compare your results with previous studies on double pendulums or swinging sticks and discuss how your findings
o. Check typo and grammatical error
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors clarified that their analysis presents novelty in the study of this particular dynamical system. Since the paper is well-written, now more with the new modifications, I recommend publication.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsMy comment still not solve
a. Calculation of performance of this work is still not include in abstract
b. author still not discuss about previous study about pendulum. Author just say "While the dynamics of complex pendulums have been studied previously [4-12]" It is general statement, what is different your work with literature [4]-[12] ?
d. validation as experiment is not verified
i. verified chaotic system is very important
j. author claim "We have a physical swinging sticks pendulum in our workplace". Kindly mention your Refs that your parameter refer to physical experiment
k. Poincare section should be draw using MATLAB or other software
Author Response
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Author Response File: Author Response.pdf
Round 3
Reviewer 3 Report
Comments and Suggestions for AuthorsThis is my comment: Please add calculation value in abstract, you can add like equilibrium Energy and limit energy. Check typo and grammatical error
Author Response
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Author Response File: Author Response.pdf
