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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]

In this paper, the approximate frequency-amplitude response of the fractal equation that describes the non-Gaussian polymer chains' motion is derived using a two-scale dimension transform and the ancient Chinese algorithm Ying Bu Zu Shu. Theoretical predictions show that the oscillation amplitude of the polymer chains varies since the response curves shift to the left ( α < 1 ) and to the right ( α > 1 ) from those curves plotted for a fractal parameter value α = 1 . This system response suggests that the magnitude of the molecular chain stretching is modified during motion due to the polymer fractal molecular structure, which indicates the mobility and flexibility of the polymer molecules that unveil possible molecular structure defects and intermolecular cohesion. [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]