Relationship Between Support Geometry and Natural Frequencies of Cantilevered Square Plate: Design Consideration

Abstract Purpose Incorporating the natural modes of plate structures into the design process is essential for optimizing their vibrational performance and ensuring structural integrity. As the support geometry varies, the natural frequencies of attached plate shift and replace one another in a stair-like mechanism. This study explores how this shifting mechanism can be exploited in the design of cantilevered square plates, addressing a significant gap in existing research on designing such plates based on their dynamic behavior. The aim of this work is to develop design tools and guidelines that take into consideration the effects of geometric parameters, specifically support thickness and length, on the plate’s vibrational characteristics. Methodology Through a systematic parametric analysis using Galerkin-based finite element analysis, key points are identified where the plate achieves its lowest six non-dimensional natural frequencies: 0.001594, 0.003907, 0.009792, 0.012479, 0.014215, and 0.024771, as support stiffness increases with thickness. Results Critical loss points associated with decreasing support stiffness as length increases are discovered. Novel threshold lines and mathematical expressions are proposed based on these findings, defining the conditions under which support geometry significantly impacts vibrational behavior and emphasizing the necessity of considering support geometry in engineering design. The results are applicable to plate thicknesses $$\:\left({\text{t}}_{\text{p}}\right)<0.05$$ m and widths (b) > 0.1 m, highlighting their generalizability across broader ranges of plate geometry. This work provides valuable insights for making informed design decisions in various plate configurations. Recommendations Future research should investigate more complex geometries using advanced theories like Mindlin-Reissner plate theory to overcome the limitations of classical thin plate theory, particularly in higher vibrational modes.