Research on the Nonlinear Vibration of Carbon Nanotube Embedded in Fractal Medium

The vibration of carbon nanotubes embedded in elastic medium is a hot topic and many research findings are available. But when the medium is a fractal medium such as chips, ceramics, etc., its vibration characteristics are difficult to explain with the existing theories. So in this paper, we mainly study the nonlinear vibration of carbon nanotube embedded in a fractal medium. The fractal variational principle of the problem is successfully established, and is proved to have a strong minimum condition according to the He–Weierstrass Theorem. Then the Hamiltonian theory combined with the residual equation is implemented to find the solution of the fractal nonlinear vibration. Finally, the impacts of different fractal orders on the vibration characteristics are also presented.