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The main objective of this study focuses on using, for the first time, the Cole–Cole semilogarithmic fractal plot to study the fractal rheological behavior of magnetorheological elastomer materials using the fractional Zener model. Experimental data and theoretical results show that the Cole–Cole semilogarithmic fractal plot predicts the Zener mathematical model's material parameters. This paper elucidates the benefits of using Cole–Cole semilogarithmic fractal plots to accurately predict material rheological properties of composite elastomer reinforced with nano or microiron particles observed during experimental tests. [ABSTRACT FROM AUTHOR]

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect's anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions. [ABSTRACT FROM AUTHOR]

The main goal of this work is to focus on using He's two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing's equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He's two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms. [ABSTRACT FROM AUTHOR]