Exploring damped and undamped frequencies in beam structures with viscoelastic supports using GFEM and state-space formulation

Abstract

This study introduces a novel analytical and numerical framework for determining the damped and undamped frequencies of elastically restrained Euler–Bernoulli (EB) and shear beams (SB) supported by two-parameter (visco-Winkler) and three-parameter (visco-Pasternak) viscoelastic foundations (VF). The scientific novelty lies in extending the classical separation of variables approach and coupling it with eigenvalue-based dispersion relations to derive an innovative spatial matrix formulation for displacements, slopes, and their derivatives. This method provides enhanced accuracy and robustness, especially in modeling complex vibrational behavior in the presence of damping and shear effects, a challenge often encountered in conventional studies. The research further integrates the Galerkin finite element method (GFEM) to offer a shear locking-free solution, demonstrating convergence to exact results, and thereby addressing critical limitations in previous methods. Additionally, the study introduces the application of state-space formulations combined with the Runge–Kutta method (RK4) to precisely analyze the response of damped systems, which adds significant value in exploring complex beam dynamics. Through a comprehensive comparison of analytical and finite element methods (FEM), the findings are validated and visualized under varying damping conditions, providing practical insights for the design and optimization of structures with viscoelastic supports. The contributions of this work include not only a deeper understanding of the interaction between damping, foundation stiffness, and structural dynamics but also the development of a versatile and scalable approach that broadens the applicability of beam models in advanced engineering applications.