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This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He’s formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values.

Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation