Thermo-Mechanical Vibrational Modelling of Trapezoidal Plate Subjected to Bi-Exponentially Alteration of Thickness and Heat Flow Gradient Under Mixed Edge Restrictions

  1. Urvashi Devi
  2. Manoj Kumar Dhiman

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Abstract

This paper demonstrates the wide – ranging contrast of analytical and numerical approaches to scrutinize the trapezoidal – shaped plates vibrational – pattern under the mixed edge restrictions, subjected to bi – exponentially alteration of thickness and heat flow gradient. These trapezoidal – shaped plates are often employed in numerous fields including space-technology, machinery domain, infrastructure and constructions, ocean engineering, heat transfer domain and energy system domain etc. The major intension of this paper is to observe how numerous alterations in edge restrictions and modelling strategy affects the predicted modal – frequencies and vibrational – patterns of the plates. We employ the Rayleigh – Ritz strategy to calculate the system of equations. At the end of this paper, I have compare my outcomes with those of existing study by scrutinizing the factors such as: frequency – pattern and non – uniformity constant and summarizing the evaluation in table for accuracy. The computational – based software “Maple” in order to employed model – frequency values to ensure the precision and efficiency in my analysis.

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  1. PREreview

    This Zenodo record is a permanently preserved version of a PREreview. You can view the complete PREreview at https://prereview.org/reviews/21898988.

    Summary

    This paper investigates the free vibration behavior of non-homogeneous trapezoidal plates with bi-exponential variations in both thickness and thermal distribution using the Rayleigh–Ritz method under CCCS boundary conditions. The authors study the effects of temperature gradients, thickness reduction parameters, and non-uniformity constants on the first two natural frequencies. The topic is relevant; however, several important issues related to model validation, numerical approximation, and mathematical notation should be addressed before the work can be considered for publication.

    Major Revision Comments

    1. Model Validation: The validation presented in Table 1 compares the current results with Ref. [16], but considerable differences can be observed. More importantly, Ref. [16] considers all-simply-supported (SSSS) boundary conditions, while the present study uses CCCS boundary conditions. Therefore, this comparison alone does not provide a sufficient validation of the current formulation. The authors should compare their model with established analytical or finite-element results using the same geometry, material distribution, and boundary conditions.

    2. Lack of Convergence Testing: The Rayleigh–Ritz solution uses a two-term deflection function to calculate the first two vibration modes. A convergence study should be provided using a larger number of terms, for example (N = 3, 4, 5) or higher, to demonstrate that the two-term approximation is sufficient and that the predicted natural frequencies are not significantly affected by additional terms.

    3. Truncation of Exponential Terms: The exact exponential expressions are replaced by a series truncated at the fifth-order term. Since Maple is used for the calculations, the reason for introducing this approximation should be clarified. If the truncated series is retained, the authors should quantify the corresponding approximation error over the parameter ranges considered in the study.

    4. Notation and Variable Definitions: Some variable definitions and mathematical notation are inconsistent throughout the manuscript. For example, quantities such as plate thickness and flexural rigidity appear to depend on both spatial coordinates, although they are sometimes written as functions of only one variable. The notation should be revised and all variables should be clearly and consistently defined.

    5. Section Title: Section 6 is titled "Experimental Modelling Approach," although no physical experiments are presented. The section mainly describes the Rayleigh–Ritz formulation and numerical calculations performed using Maple. A title such as "Analytical and Numerical Modelling Approach" would be more appropriate.

    Minor Revision Comments

    1. Language and Technical Writing: The manuscript would benefit from careful proofreading to correct grammatical errors and improve the academic writing style. For example, first-person expressions such as "I have compare" and "The plate I have employed" should be revised.

    2. Figure Presentation: Figures 1–7 should be improved for readability. Some legends, axis labels, and parameter values are difficult to read, and the figures currently appear similar to direct software output.

    3. Table 1 Data: The numerical values reported in Table 1 should be carefully checked. For example, the value (3342.22) appears inconsistent with the surrounding values and should be verified.

    4. Conclusion: The conclusion should be strengthened by clearly summarizing the main quantitative trends obtained from the study and briefly discussing the limitations of the current formulation.

    Competing interests

    The author declares that they have no competing interests.

    Use of Artificial Intelligence (AI)

    The author declares that they did not use generative AI to come up with new ideas for their review.

  2. Version published to 10.20944/preprints202607.1672.v1