A novel pattern in a class of fractal models with the non-perturbative approach

This paper provides evidence on how the fractal vibration system can be converted into a damping system in a continuous space. The fractal Duffing vibration is considered an example. It is shown that the inertia force in the fractal space can be transformed into a combination of the inertia force and the damping force in a continuum space employing the fractional parameter. In addition, the velocity force can be converted to both the damping force and the stiffness force in the smooth space. The nonlinear system is solved using the non-perturbation method. The calculations showed that the approximate solution agrees with the corresponding numerical one when the fraction parameter becomes the unity value. The new approach is represented as the best one because it highlights the damping influence of fraction derivatives. This approach can help a large number of researchers interested in studying nonlinear fractal oscillations.