Nonlinear vibration behavior of self-sustaining CNT nanobeams under thermo-magnetic fields: surface energy insights for advanced sports applications

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.