Vibration analysis of SWCNTs resting on an elastic foundation using the initial values method

Single-walled carbon nanotubes (SWCNTs) possess exceptional mechanical and dynamic properties, making them indispensable for diverse applications such as nanoelectromechanical systems, sensor technologies, energy storage devices and composite material design. However, a more accurate, efficient and systematic analysis of their vibrational behaviour under conditions involving elastic foundation stiffness and nanoscale effects remains essential for tailoring their applicability to specific purposes. This study introduces a novel methodological approach to analyse the vibrational behaviour of SWCNTs resting on an elastic foundation, leveraging nonlocal elasticity theory to incorporate small-scale effects. The proposed framework combines the initial value method with the approximate transfer matrix technique to efficiently compute natural frequencies. Vibrational analyses were performed under various boundary conditions, including simply supported, clamped–clamped and clamped–simply supported. Parametric studies revealed that increases in the nonlocal parameter significantly reduce natural frequencies, whereas a higher elastic foundation stiffness enhances the system rigidity and increases vibrational frequencies. This framework offers substantial computational efficiency and accuracy, presenting a robust tool for modelling SWCNT dynamics. The results provide valuable insights for designing effective SWCNT-based structures, contributing to their applications in nano-engineering and other engineering domains requiring precise vibrational analysis.