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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]

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]

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]

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]

This paper focuses on predicting the shape, duration, and peak magnitude of the displacement, velocity, and acceleration curves while dropping weight over a viscoelastic fractal cushioning packaging system. Furthermore, to capture high-frequency harmonic components observed during the impact time span, the approximate frequency–amplitude expression of the governing equation of motion will be obtained via an ancient Chinese algorithm and He's formulation by assuming initial trial solutions based on Jacobi elliptic functions. Numerical simulations confirmed the ability of the Jacobi elliptic functions to capture high-frequency harmonics observed during the cushioning packaging system dynamic response with a root mean square error (RMSE) value that does not exceed 0.0218, which is an indication of the great accuracy attained from our derived solution when compared to the exact numerical one. Furthermore, our derived approximate solution predicts that when a weight is dropped over the cushioning packaging, the cushion material with smaller porosity will absorb the produced kinetic energy faster. [ABSTRACT FROM AUTHOR]

In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He's two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution. [ABSTRACT FROM AUTHOR]

This paper aims to study the dynamic response of the fractal Fangzhu's forced device to elucidate how the harvester device's surface fractal patterned influences water molecules' motion due to the increase or reduction of the wave velocity. It is also investigated under which conditions the oscillation amplitude favors the complete water removal from Fangzhu fractal surface device. Based on the approximate fractal frequency–amplitude relation derived using an ancient Chinese algorithm, and by varying the value of the fractal dimension parameter linked to the material surface morphology, it was found that the Fangzhu water harvester device can efficiently capture microscopic water droplets if the surface morphology is such that the fractal dimension α is less than one. This condition can be used to develop superhydrophobic surfaces mimicking fractal patterned surface structures. [ABSTRACT FROM AUTHOR]

In this paper, the ancient Chinese algorithm is applied to derive fractal microgravity forced oscillator's frequency–amplitude response curves using the two-scale fractal transform derivative. Depending on the fractal parameter values, it is shown that the system restoring forces exhibit hardening or softening behavior. Furthermore, the proposed solution approach is simple, efficient, and accurate for obtaining the approximate steady-state solution of a fractal microgravity nonlinear oscillator with a purely nonlinear power-form restoring force. [ABSTRACT FROM AUTHOR]

In this paper, the steady-state solution of Duffing-type oscillators with fractal-order derivative is obtained. First, the two-scale fractal-order derivative transform is used to write the fractal differential equation of motion as an ordinary differential non-homogenous equation of motion. Then the ancient Chinese algorithm and He's formulation are used to find the approximate frequency-amplitude response curve. The results show that the steady-state response diagrams exhibit hardening or softening behavior depending on the fractal parameter value. [ABSTRACT FROM AUTHOR]

In this work, the fractal cubic–quintic Duffing's equation analytical solution is obtained using the two-scale transform and elliptic functions. Then, the analytical solution is used to study wave propagation in a fractal medium. Since the value of the fractal parameter adjusts the pulse frequency and wavelength propagation velocity, depending upon the fractal medium physical properties, it is found that the information contained in the pulse can be carried out faster over long distances without distortion or loss of intensities. This paper offers a new light on the applicability of the two-scale transform of fractal theory to comprehend natural phenomena. [ABSTRACT FROM AUTHOR]

This paper aims to derive the equivalent power-form representation for the Toda oscillator, which describes the intensity fluctuations of Nd:YAG lasers. A two-scale dimension transform is introduced to study the transient fractal response of Toda oscillator. Numerical simulations indicate that for increasing values of the fractal exponent α , the frequency of the Toda oscillator increases. [ABSTRACT FROM AUTHOR]

In this paper, an equivalent power-form transformation method with a weighted function is applied for solving the one-dimensional fractal Bratu's boundary value equation. Numerical integration solutions obtained from the equivalent Bratu's equation as well as those from its approximate Taylor's series solution reveal that the proposed methodology yields highly accurate solutions. Therefore, it is believed that by applying the power-form transformation, various fractal differential equations in which two scales are needed because of the physical laws involved in modeling the observed phenomena, can be solved by treating the nonlinear terms as equivalent power-form terms. [ABSTRACT FROM AUTHOR]

In this work, the fractal nature of Selective Laser Melting (SLM) additive manufacturing process (AM) is elucidated. Fractal dimension and lacunarity of metallic powders are calculated from Scanning Electron Microscopy (SEM) images adapted from literature. The complexity and homogeneity of the textures of the powder beds are also studied through the comparison of fractal dimension and lacunarity. It is found that better densification results are obtained when the powder bed's fractal dimension is closer to the golden mean number of 1.618. Furthermore, this finding is extended to expressions for predicting the component's bulk density produced via SLM by setting the α exponent equal to the golden mean value and finding the proportionality constant, C , using a nonlinear least squares method. The proposed approach works well since theoretical prediction and experimental data compare well with root-mean-square-error (RMSE) values that do not exceed 7 × 1 0 − 0 8 . This work sheds new light on enhancing additive manufacturing technologies considering the fractal nature of SLM since its process mathematical models are constructed around Euclidian space-time with continuous smooth assumptions that should be adapted to include the fractal nature of the manufacturing process aiming to improve their precision. The underlying interweaving of SLM, as a fractal process, and the golden mean number is revealed. [ABSTRACT FROM AUTHOR]