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This paper elucidates how the two-scale fractal dimension transform, and a transformation method can be applied to replace the Warner–Gent equation that models the fractal dynamic response of porous viscohyperelastic materials with an equivalent power-form equation. Furthermore, this research work elucidates the advantages of modeling viscohyperlastic materials using the fractal Warner–Gent's model since the values of the fractal dimension parameter unveil how the global molecular structure of viscohyperelastic materials varies as a function of the vibration frequency wavelength. Compared to the original one, the accuracy attained from the Warner–Gent power-form equivalent equation is examined by plotting the frequency–amplitude and time–amplitude curves obtained from the corresponding numerical integration solutions. It is found that both numerical integration solutions agree well since the root-mean-square-error (RMSE) values remain small. [ABSTRACT FROM AUTHOR]

This paper investigates the applicability of the ancient Chinese algorithm jointly with the two-scale fractal dimension transform to find the frequency–amplitude relationship of fractal equations of motion with and without damping terms. Analytical results show that for a fractal equation of motion without damping terms, the oscillation amplitudes do not exhibit decaying effects. However, when damping terms are included, the fractal parameter tends to shift the decaying oscillation amplitudes that decrease faster with time for fractal values less than one. This paper provides an efficient tool for finding the amplitude–frequency relationship of damped fractal oscillators. To illustrate the solution process, the steady-state solution of the fractal equation of motion that arises in plasma physics is derived. The proposed approach elucidates the applicability of He's formulation jointly with the two-scale fractal calculus to find the frequency–amplitude of fractal systems with and without damping terms. [ABSTRACT FROM AUTHOR]