Analyzing the Effect of Rotary Inertia and Elastic Constraints on a Beam Supported by a Wrinkle Elastic Foundation: A Numerical Investigation

Abstract

This article presents a modal analysis of an elastically constrained Rayleigh beam that is placed on an elastic Winkler foundation. The study of beams plays a crucial role in building construction, providing essential support and stability to the structure. The objective of this investigation is to examine how the vibrational frequencies of the Rayleigh beam are affected by the elastic foundation parameter and the rotational inertia. The results obtained from analytical and numerical methods are presented and compared with the configuration of the Euler–Bernoulli beam. The analytic approach employs the technique of separation of variable and root finding, while the numerical approach involves using the Galerkin finite element method to calculate the eigenfrequencies and mode functions. The study explains the dispersive behavior of natural frequencies and mode shapes for the initial modes of frequency. The article provides an accurate and efficient numerical scheme for both Rayleigh and Euler–Bernoulli beams, which demonstrate excellent agreement with analytical results. It is important to note that this scheme has the highest accuracy for eigenfrequencies and eigenmodes compared to other existing tools for these types of problems. The study reveals that Rayleigh beam eigenvalues depend on geometry, rotational inertia minimally affects the fundamental frequency mode, and linear spring stiffness has a more significant impact on vibration frequencies and mode shapes than rotary spring stiffness. Further, the finite element scheme used provides the most accurate results for obtaining mode shapes of beam structures. The numerical scheme developed is suitable for calculating optimal solutions for complex beam structures with multi-parameter foundations.