Nonlinear frequency resonance analysis of double-walled carbon nanotubes in thermal–magnetic fields for advanced sports equipment

This study investigates the nonlinear primary resonance behavior of harmonically excited nonlocal double-walled carbon nanotubes (DWCNTs) embedded in a viscoelastic medium, with potential applications in high-performance sports equipment. The analysis employs the nonlocal nonlinear Euler–Bernoulli beam theory and a curvature-dependent van der Waals (vdW) interaction model to capture interlayer forces accurately. The coupled governing equations incorporate nonlinear geometric effects along with axial thermal and magnetic fields, simulating the dynamic response under real-world conditions. Using the Galerkin approach with sinusoidal shape functions, the system is reduced to ordinary differential equations, which are then analyzed via the multiple time scales perturbation method. The study provides insights into the frequency response and instability conditions of DWCNTs, offering valuable guidance for the design of next-generation lightweight, high-strength sports materials with enhanced vibration control and durability.