This article discusses the modal behavior of elastically constrained beams under various types of foundations and provides insights into the effects of different factors on the eigenfrequencies of beams. Numerical and analytical techniques, specifically the Galerkin finite element method (GFM) and the separation of variables, are utilized to determine the eigenfrequencies and mode shapes of beams. Modal analysis of Timoshenko, shear, Rayleigh, and Euler-Bernoulli beams that are elastically constrained and resting on Winkler, Pasternak, and Hetényi foundations, considering non-classical boundary conditions, is included in the study. The effects of factors such as flexural rigidity, transverse modulus, and Winkler foundation constant on natural frequencies of different beam models are investigated. The proposed method efficiently converges to the exact solution without shear locking in the stiffness element. The results demonstrate that the natural frequencies of the beam rise because of the shear layer, flexural rigidity, and foundation constant. Furthermore, the Hetényi elastic foundation affects the natural frequency of the beam, depending on the relative values of beam stiffness and foundation stiffness. Additionally, incorporating both shear deformation and rotary inertia has a greater impact on the eigenfrequencies of Euler-Bernoulli beams compared to incorporating only one of these effects. The findings of this work provide valuable insights into the behavior of beams under different foundation conditions and have potential applications in the design and optimization of structures incorporating beams, thereby enhancing the understanding of beam analysis.

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.

This study investigates the free vibrations of elastically constrained shear and Rayleigh beams placed on the Pasternak foundation. Of particular interest, it is aimed to analyze the influence of shear strain, rotational inertia, elastic stiffness, and shear layer on the natural frequencies and eigenmodes of beam vibrations. For this purpose, the eigenfrequencies and eigenmodes are determined using analytical and numerical techniques. A finite element scheme is developed employing quadratic and cubic polynomials for slope and transverse displacement, respectively. The efficiency and accuracy of the finite element method are illustrated by comparing it with the analytical results for generalized and special cases. The underlying model analysis justifies that the natural frequencies of the beam vibration depend only on the geometry of the Rayleigh beam, while these frequencies depend on the physical and geometric properties of the shear beam. However, the natural frequencies of the Euler-Bernoulli depend solely on the geometric conditions of the beam.