International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 12 Issue: 08 | Aug 2025
p-ISSN: 2395-0072
www.irjet.net
Free Vibration of a Square Plate with Bi- Dimensional Circular Varying Thickness and Thermal Effect Manju Bala1, Dr Ashish Kumar2 1Research Scholar, Department of Mathematics, Arni University, Kangra (H.P.) 2Associate Professor, Department of Mathematics, Arni University, Kangra (H.P.)
---------------------------------------------------------------------***--------------------------------------------------------------------Abstract Visco-elastic plates play a crucial role in numerous engineering applications, including mechanical structures, aircraft components, and industrial systems. Accurate analysis of their behavior and strength is essential for optimal design and material utilization. Plates with variable thickness are particularly important in advanced technologies such as nuclear reactors, aerospace structures, naval vessels, submarines, and earthquake-resistant designs. This study presents a mathematical model to analyze the vibrational behavior of a square visco-elastic plate with linearly varying thickness in both directions, subjected to clamped boundary conditions on all four edges. The influence of thermal gradients—linear in one direction and parabolic in the other—on the plate’s vibration is examined. An approximate frequency equation is derived using the Rayleigh–Ritz method with a two-term deflection function. The natural frequencies for different taper parameters and thermal gradients are computed using MATLAB, and the results are illustrated through graphical representations. This work provides valuable insights for engineers and researchers working in high-temperature and advanced structural applications.
Keywords: Square Plate, Vibration, Frequency. Introduction In modern engineering and technology, increasing attention is being directed toward the influence of elevated temperatures on non-homogeneous plates with variable thickness, owing to their widespread application in fields such as nuclear power plants, aeronautics, chemical industries, and energy systems. Materials like metals and their alloys often exhibit visco-elastic behavior under such high-temperature conditions. During thermal exposure, especially under intense heat fluxes, material properties undergo significant changes, making it imperative to consider thermal effects in structural analysis. These changes cannot be neglected, as they have a direct impact on the mechanical performance and vibration behavior of plate structures. Numerous studies have shown that material non-homogeneity significantly affects vibration characteristics. This nonhomogeneity may be inherent or introduced through engineering design, as seen in materials like plywood, delta wood, and fiber-reinforced plastics, which are commonly used to enhance structural integrity. The combined effect of visco-elastic material behavior and thickness variation is particularly relevant in advanced technological applications, including aerospace structures, ocean engineering, and precision instruments in electronics and optics. These configurations are known to offer improved strength-to-weight ratios and resilience under harsh environmental conditions. A review of the current literature reveals limited research on the vibration analysis of non-homogeneous visco-elastic plates with variable thickness subjected to thermal gradients, particularly in square or circular geometries. The present study aims to fill this gap by analyzing the effect of thermal gradients on the vibration behavior of a visco-elastic square plate with thickness varying linearly in both in-plane directions. The plate is assumed to be clamped along all four edges, with temperature distribution taken as linear in one direction and parabolic in the other. The material non-homogeneity is considered in terms of a temperature-dependent modulus of elasticity. Using modern computational tools (MATLAB), the natural frequencies for the first two modes of vibration are calculated for various combinations of taper parameters and thermal gradients. The results are presented graphically to provide insights into the dynamic response of such plate systems under thermal and structural variations.
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