Exploring Nonlinear Dynamics and Stability of Embedded Carbon Nanotubes in Mechanical Engineering

Abstract This study investigates the nonlinear free vibration of an embedded single-walled carbon nanotube using a continuum mechanics framework and an elastic beam model. The analysis incorporates the effects of rippling deformation, midplane stretching, and interactions with the surrounding elastic medium on the nonlinear dynamics of the system. The Khater method is used to derive exact analytical solutions, revealing novel soliton structures, including dark, bright, and kink soliton solutions, which characterize the amplitude-modulated wave behavior of the embedded carbon nanotube. A comprehensive bifurcation analysis uncovers distinct dynamical regimes that identify critical parameters such as rippling amplitude and elastic medium stiffness that dominantly influence nonlinear free vibration. We explore the chaotic analysis to demonstrate chaotic behavior and visualized the Poincaré maps. To enhance the study, we create Poincaré maps and Lyapunov exponents that illustrate the temporal evolution of trajectories in phase space. This makes it easier to see how change occurs between different dynamical regimes. Graphical illustrations highlight geometric nonlinearities, environmental constraints, and intrinsic instabilities, offering insights into the vibrational resilience and energy dissipation mechanisms of embedded carbon nanotube. In addition, we conducted a stability study of the examined model under various initial conditions. This work advances the understanding of nanoscale mechanical systems by bridging nonlinear dynamics, stability analysis, and advanced computational techniques, with implications for nano-resonator design and nanomaterial-based technologies. Graphical Abstract