Analysis of Free Vibrations of Homogeneous Rectangular Thin Plates Using the Finite Difference Method (FDM)

Ghaniullah Safi; Noorullah Noori; Abdul Wakil Baidar

doi:10.62810/jnsr.v4i1.303

Authors

Ghaniullah Safi Department of Mathematics, Faculty of Education, Nuristan University, Nuristan, Afghanistan
Noorullah Noori Department of Mathematics, Faculty of Mathematics, Kabul University, Kabul, Afghanistan
Abdul Wakil Baidar Department of Mathematics, Faculty of Mathematics, Kabul University, Kabul, Afghanistan

DOI:

https://doi.org/10.62810/jnsr.v4i1.303

Keywords:

Boundary conditions, FDM, Free vibration, Kirchhoff–Love plate theory, Rectangular plates

Abstract

This study presents a numerical investigation of the free vibration behavior of homogeneous rectangular thin plates, formulated within the Kirchhoff–Love plate theory using the Finite Difference Method (FDM). Dimensionless natural frequencies for the first three vibration modes were computed under various classical boundary conditions—fully clamped (CCCC), supported (SSSS), and mixed (CSCS)—across multiple grid discretizations. The analysis focuses on the convergence of natural frequencies with grid refinement and the influence of boundary constraints on vibration characteristics. The proposed FDM framework employs central difference schemes for derivative approximations, ensuring high accuracy, numerical stability, and rapid convergence, with minimal change in computed frequencies beyond moderate grid sizes. Comparative results with existing studies confirm the approach's reliability and effectiveness. The findings reveal that boundary conditions significantly influence both mode shapes and frequencies. Fully clamped plates exhibit the greatest stiffness, producing the highest natural frequencies, while supported configurations yield lower frequency responses. Mixed boundary conditions produce intermediate behaviors, demonstrating the sensitivity of vibration characteristics to edge constraints. Overall, the findings provide essential insights into the structural design, optimization, and stability assessment of plate structures in engineering applications.

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