Fractal Duffing Oscillators with Two Degrees of Freedom and Cubic–Quintic Nonlinear Stiffness

The harmonic equivalent method is a non-perturbative approach to nonlinear vibration issues, aiming to create linearly coupled systems from coupled vibrations. However, there is still much to be discovered about managing interconnected nonlinear components. This paper examines the nonlinear components of a fractal-connected system and offers suggestions. This paper explores insights into the principles and uses of nonlinear systems in science and engineering by investigating the dynamic behavior of a connected cubic–quintic damping fractal system analytically using an innovative approach to analytical examination. A two-scale transformation and reformulation of the system into fractal form simplify its governing equations for dynamic and stability analysis. Two analytical scopes are presented: one decouples nonlinear systems using weighted averaging functions, and the other converts even nonlinearities into odd terms using El-Dib’s frequency formulas for linear representation, enabling an equivalent linear representation of the system. The resilience of the decoupled system is verified by numerical simulations using Mathematica, which shows high agreement and minimal relative errors. It also accurately reflects dynamic behavior. Additionally, the work uses the bridging techniques of El-Dib and Elgazery to convert a linear damping fractal coupled system into a classical continuous-space form. A scaling fractal factor is made possible by re-expressing the fractal structure using pseudo-dimensional parameters. The linearly linked damping system has an exact analytical solution. The paper provides valuable insights into the design and control of coupled nonlinear oscillatory systems by validating analytical solutions through numerical simulations using Mathematica.