Analytical and Computational Analysis of Damped Frequency of Beams Supported by Elastic Constraints and Visco-Winkler Foundations

PurposeThis study investigates the vibrational behavior of Euler-Bernoulli beams (EBB) with elastic constraints, with and without two parametric foundations (Visco-Winkler foundations). The aim is to determine the damped and undamped natural frequencies and analyze the effects of damping, linear, and rotational springs on system dynamics, providing insights for structural and vibrational engineering.MethodsThe characteristic equation is derived using the separation of variables, and the roots are extracted using root-finding techniques. Analytical results are compared with finite element method (FEM) simulations developed via the Galerkin finite element method (GFEM) in MATLAB. Damped equations of motion are solved using state-space formulation, and the system response is evaluated using the Runge-Kutta 4th order (RK4) method.ResultsThe study demonstrates that damping effects are most prominent in reducing frequencies of the initial vibrational modes. Among the parameters analyzed, linear spring constants exert a greater influence on natural frequencies compared to rotational spring constants. The finite element method (FEM) results show excellent agreement with analytical predictions, confirming the reliability of both approaches. Additionally, the elastic foundation's constant parameter significantly affects the system's natural frequencies, highlighting its role in vibrational behavior.ConclusionsThe study confirms the validity of the analytical and FEM approaches for evaluating damped and undamped frequencies in beams with elastic constraints. These findings have practical implications for optimizing structural designs with damped supports, contributing to advanced applications in vibrational engineering.