Elastic Triangular Plate Dynamics on Unilateral Winkler Foundation: Analysis Using Chebyshev Polynomial Expansion for Forced Vibrations

Abstract

This research delves into the intricate dynamic and static characteristics of an elastic triangular plate supported by a unilateral Winkler foundation, with a specific focus on forced vibrations. The study considers the triangular plate under both uniformly distributed load and eccentrically applied concentrated load scenarios. The governing equation governing the plate's behavior is established through a meticulous analysis of its static and dynamic responses. Utilizing a series of Chebyshev polynomials to represent admissible displacement functions and employing Lagrange equations of motion, we derive a comprehensive understanding of the system's dynamics. Due to the non-linear nature of the unilateral Winkler foundation, an iterative numerical solution methodology is devised. Detailed numerical investigations shed light on the static behavior of the plate under concentrated loads, exploring a broad spectrum of parameters encompassing plate geometry and foundation stiffness. Transitioning to dynamic analyses in the time domain, we adopt a stepwise time variation approach for concentrated loads, employing constant acceleration procedures to solve the governing differential equations. Through visual representations, we offer insights into the time variations of contact regions and plate displacements across various foundation and plate parameters, emphasizing the non-linear effects arising from plate lift-off phenomena. Extensive exploration of parameter and loading effects underscores the profound influence of unilateral foundation properties on both static and dynamic triangular plate behaviors.