Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative

In this paper, the subharmonic resonance of Duffing oscillator with fractional-order derivative is investigated using the averaging method. First, the approximately analytical solution and the amplitude–frequency equation are obtained. The existence condition for subharmonic resonance based on the approximately analytical solution is then presented, and the corresponding stability condition based on Lyapunov theory is also obtained. Finally, a comparison between the fractional-order and the traditional integer-order of Duffing oscillators is made using numerical simulation. The influences of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability are also investigated.

Homotopy Perturbation Method for Analysis Nonlinear Vibration of Double-Walled Carbon Nanotubes

To obtain the approximately analytical solution of double-walled carbon nanotubes (DWNTs) nonlinear vibration. In this study, homotopy perturbation method (HPM) was used to solve nonlinear vibration equation of DWNTs. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison of the result obtained by the HPM with exact solutions reveals that only the first or second order approximation of the HPM leads to higher accurate solution.