A Unified Analytical Framework for Modal Analysis of Beams with Elastic Supports and End Masses

Ahmed Muaiqeeb Zakwan
GameAbove College of Engineering and Technology, Eastern Michigan University, Ypsilanti, MI 48197, USA
Ibrahim M. Ashqar
GameAbove College of Engineering and Technology, Eastern Michigan University, Ypsilanti, MI 48197, USA
Mohamad S. Qatu
GameAbove College of Engineering and Technology, Eastern Michigan University, Ypsilanti, MI 48197, USA

Article ID: 4473

Keywords: Euler-Bernoulli beam; free vibration; elastic boundary restraints; lumped end masses; modal analysis; finite element validation

Abstract

Predicting vibration in beamlike structures with non-ideal supports remain a significant engineering challenge. Classical beam models usually assume perfectly rigid boundaries. Real structures, however, contain flexible supports and attached end components that modify the dynamic response. This study develops a unified analytical framework for the free vibration analysis of Euler–Bernoulli beams with elastic restraints and lumped end masses. The formulation incorporates translational stiffness, rotational stiffness, and rotary inertia within a compact and physically transparent model. Frequency-dependent boundary conditions are assembled into a concise 4 × 4 characteristic matrix. Classical free–free, clamped–free, and clamped–clamped conditions emerge naturally as limiting cases. MATLAB is employed to extract the natural frequencies and mode shapes. Validation is performed through three-dimensional (3D) ANSYS simulations using quadratic HEX20 solid elements. The results demonstrate strong agreement between analytical and finite element predictions across a wide range of stiffness and mass parameters. Mesh convergence studies confirm stable and reliable solutions for the first six bending modes. The findings clearly illustrate that boundary flexibility and end inertia substantially influence modal characteristics and vibration behavior. In addition, the proposed formulation maintains analytical simplicity while capturing realistic structural behavior with high accuracy. The framework also provides a reliable benchmark for validating advanced numerical models and future experimental investigations. Overall, the proposed approach offers an accurate, computationally efficient, and practical tool for vibration prediction and engineering design of realistic beam structures.

Published

2026-10-28

How to Cite

Ahmed Muaiqeeb Zakwan, Ibrahim M. Ashqar, & Mohamad S. Qatu. (2026). A Unified Analytical Framework for Modal Analysis of Beams with Elastic Supports and End Masses. Mechanical Engineering Advances, 4(3). https://doi.org/10.59400/mea4473

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Issue

Vol. 4 No. 3 (2026)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

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