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<p>This study presents exact analytical solutions for the local and nonlocal transverse vibration of Euler–Bernoulli nanobeams resting on Winkler–Pasternak elastic foundations. Incorporating Eringen’s nonlocal elasticity theory, the model captures small-scale effects, while rotational and translational springs represent general boundary conditions. For the first time, exact and general frequency equations are derived for nanobeams with arbitrary boundary conditions, elastic foundations, and tip masses. These equations are numerically solved to obtain precise natural frequencies, showcasing the accuracy and versatility of the proposed framework. The influence of key parameters such as nonlocal effects, boundary flexibility, foundation stiffness, tip masses, and slenderness ratio on the first three natural frequencies and mode shapes is systematically analyzed. Results reveal the significant role these factors play in shaping vibrational behavior, providing critical insights into the dynamic response of nanostructures. Presented in graphical and tabular formats, the findings offer benchmark solutions for validating future models. This work advances the understanding of nanobeam dynamics and supports the design and optimization of advanced nanoscale systems embedded in elastic media with flexible supports and tip masses.</p>

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Exact analysis of local/nonlocal vibration in nanobeams on elastic foundations with arbitrary supports and tip masses

  • 期刊论文

  • 发表日期：

    2025年7月21日

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  • Mohsen Bambaeechee

摘要

This study presents exact analytical solutions for the local and nonlocal transverse vibration of Euler–Bernoulli nanobeams resting on Winkler–Pasternak elastic foundations. Incorporating Eringen’s nonlocal elasticity theory, the model captures small-scale effects, while rotational and translational springs represent general boundary conditions. For the first time, exact and general frequency equations are derived for nanobeams with arbitrary boundary conditions, elastic foundations, and tip masses. These equations are numerically solved to obtain precise natural frequencies, showcasing the accuracy and versatility of the proposed framework. The influence of key parameters such as nonlocal effects, boundary flexibility, foundation stiffness, tip masses, and slenderness ratio on the first three natural frequencies and mode shapes is systematically analyzed. Results reveal the significant role these factors play in shaping vibrational behavior, providing critical insights into the dynamic response of nanostructures. Presented in graphical and tabular formats, the findings offer benchmark solutions for validating future models. This work advances the understanding of nanobeam dynamics and supports the design and optimization of advanced nanoscale systems embedded in elastic media with flexible supports and tip masses.