An explicit solution for the vibration of a carbon chain inside carbon nanotubes (CNTs) was obtained using continuum modeling of the van der Waals (vdW) interactions between them. The effect of the initial tensile force and the amplitude of the carbon chain as well as the radii of the CNTs on the vibration frequency were analyzed in detail, respectively. Our analytical results show that the vibration frequency of the carbon chain in a (5,5) CNT could be around two orders of magnitude higher than that of an independent carbon chain without initial tensile force. For a given CNT radius, the vibration frequency nonlinearly increases with increasing amplitude and initial tensile force. The obtained analytical cohesive energy and vibration frequency are reasonable by comparison of present molecular dynamics (MD) simulations. These findings will be a great help towards understanding the vibration property of a nanowire in nanotubes, and designing nanoelectromechanical devices.

Fiber Bragg grating (FBG) sensors are widely used in aerospace monitoring and intelligent manufacturing due to their high sensitivity, yet their deployment relies on manual assembly, limiting precision, integration density, and conformality under extreme conditions. Here, we propose a programmable direct-FBG-patterning (DFP) assembly paradigm for one-step integration of intrinsically brittle multiplexed FBG arrays onto arbitrary curved and non-developable surfaces. A mechanics-optics coupled framework identifies three minimum bending radii governed by interfacial debonding, fiber fracture, and optical attenuation, defining the achievable feature size and multidirectional sensing capability. Guided by this framework, cross-fiber routing and conformal assembly strategies enable high-density sensor networks along a single continuous fiber beyond the limits of manual assembly. We further demonstrate robust and versatile FBG sensor integration in applications including structural displacement reconstruction, phonation, and gesture monitoring, establishing a general strategy for quasi-distributed sensing beyond the constraints of manual assembly.