Abstract
This research examines the nonlinear dynamics of carbon nanotube-based nanobeams subjected to harmonic forcing, aiming to advance the design of high-performance athletic gear. The analysis accounts for geometric nonlinearities, damping from a viscoelastic substrate, and surface effects essential to creating lightweight yet robust components for items such as tennis rackets, golf clubs, and protective equipment. Given the elevated surface-to-volume ratio in nanomaterials, the study evaluates surface elasticity and residual stresses to improve shock absorption and energy dissipation. Equations of motion are formulated based on Euler–Bernoulli beam theory, discretized through the Galerkin approach employing trigonometric modes, and resolved via the method of multiple scales. Critical factors, including viscoelastic damping factors, crystal directions ([100] and [111]), and nonlinear geometry, are assessed for their impact on the primary resonance curve. Findings reveal that strategic adjustment of these variables can profoundly modify the frequency-amplitude behavior, allowing customized rigidity and attenuation suited to sporting needs. This work establishes a basis for engineering advanced nanomaterials in sports, harnessing nonlinear vibrations to enhance functionality, longevity, and user protection.
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Funding
This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).
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R.H.L. and M.S. prepared the initial draft; K.H and L.M. built models and collected data; S.I.A., R.H.L. and H.R. prepared figures; L.M., W.A. and R.K reviewed and modified the manuscript; K.H. and R.K. supervised the project; All authors reviewed the manuscript.
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Hadj Lajimi, R., Hajlaoui, K., Mostafa, L. et al. Nonlinear vibration behavior of self-sustaining CNT nanobeams under thermo-magnetic fields: surface energy insights for advanced sports applications. Sci Rep (2026). https://doi.org/10.1038/s41598-026-45044-9
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DOI: https://doi.org/10.1038/s41598-026-45044-9
