Fractal equation of motion of a non-Gaussian polymer chain: investigating its dynamic fractal response using an ancient Chinese algorithm.

Elı́as-Zúñiga, Alex; Martínez-Romero, Óscar; Trejo, Daniel Olvera; Palacios-Pineda, Luis Manuel

doi:10.1007/s10910-021-01310-x

Cited by 14 publications

(11 citation statements)

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“…The mathematical treatment of Equation (23) can be done considering the solutions procedures discussed in [ 57 , 58 , 59 , 60 ]. Recently, other fractal theory definitions [ 61 , 62 ], and solution methods have been proposed [ 63 , 64 , 65 , 66 ]. However, the discussion of our findings using fractal material constitutive equations shall be addressed in a forthcoming paper to be published somewhere else.…”

Section: Discussionmentioning

confidence: 99%

Soft Tissue Hybrid Model for Real-Time Simulations

et al. 2022

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In this article, a recent formulation for real-time simulation is developed combining the strain energy density of the Spring Mass Model (SMM) with the equivalent representation of the Strain Energy Density Function (SEDF). The resulting Equivalent Energy Spring Model (EESM) is expected to provide information in real-time about the mechanical response of soft tissue when subjected to uniaxial deformations. The proposed model represents a variation of the SMM and can be used to predict the mechanical behavior of biological tissues not only during loading but also during unloading deformation states. To assess the accuracy achieved by the EESM, experimental data was collected from liver porcine samples via uniaxial loading and unloading tensile tests. Validation of the model through numerical predictions achieved a refresh rate of 31 fps (31.49 ms of computation time for each frame), achieving a coefficient of determination R2 from 93.23% to 99.94% when compared to experimental data. The proposed hybrid formulation to characterize soft tissue mechanical behavior is fast enough for real-time simulation and captures the soft material nonlinear virgin and stress-softened effects with high accuracy.

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Section: Discussionmentioning

confidence: 99%

Soft Tissue Hybrid Model for Real-Time Simulations

et al. 2022

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“…Substitution of y 1 (t) and y 2 (t) into equation (4) leads to the residuals R 1 (t) and R 2 (t) which can be used to obtain average trail residuals R1 ðtÞ and R2 ðtÞ needed to find the frequency-amplitude expression from Ji-Huan He's formula. Thus, in accordance with the ancient Chinese algorithm, [69][70][71][72][73] the steady-state trial residual functions y 1 and y 2 are assumed to be of the form…”

Section: The Two-scale Fractal Transformmentioning

confidence: 99%

“…It is also assumed that ω 1 = ω f and ω 2 = ω f . [69][70][71][72][73][74][75] Expanding equation (6) gives…”

Section: The Two-scale Fractal Transformmentioning

confidence: 99%

“…where T = 2π/ω i . [69][70][71][72][73][74][75] To simplify the computation of the integral (8) first, the transformation S = t 1/α is introduced. Hence, equations ( 7) and ( 8) become…”

Section: The Two-scale Fractal Transformmentioning

confidence: 99%

“…As usual, ω 1 = 1 and ω 2 = 2. [69][70][71][72][73][74][75] Considering the nature of the Helmholtz-Duffing anharmonic oscillator, it is further assumed that A 1 = A and A 2 = βA where β is determined fitting the approximate backbone curve…”

Section: The Two-scale Fractal Transformmentioning

confidence: 99%

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New analytical solution of the fractal anharmonic oscillator using an ancient Chinese algorithm: Investigating how plasma frequency changes with fractal parameter values

Elı́as-Zúñiga

Martínez-Romero

Olvera-Trejo

2022

Journal of Low Frequency Noise, Vibration and Active Control

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This paper uses the two-scale fractal dimension transform and He’s formula derived from the ancient Chinese algorithm Ying Bu Zu Shu to find the approximate frequency–amplitude expression of the fractal and forced anharmonic oscillator that can be used to study the nonlinear oscillations produced by the plasma physics fractal structures. The results show how the electron frequency and wavelength change as a function of the plasma physics fractal structure. In fact, if the value of the fractal parameter is decreased, the wavelength increases, and consequently, the system frequency decreases. The introduced solution procedure sheds a bright light on the easy-to-follow steps to obtain an accurate steady-state analytical solution of fractal anharmonic nonlinear oscillators.

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A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

El-Dib

2024

Arch Appl Mech

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Drawing on the principles of fractal properties and nonlinear vibration analysis, this paper delves into the investigation of a moving bead on a vertically rotated parabola. The dynamical nonlinear equation of motion, incorporating fractal derivatives, transforms traditional derivatives within continuous space. Consequently, the equation of motion takes the form of the Duffing-Van der Pol oscillator. Utilizing a non-perturbative approach, the nonlinear oscillator is systematically transformed into a linear one, boasting an exact solution. The analytical solution yields two valid formulas governing the frequency-amplitude relationships. Numerical solutions affirm that these proposed formulas offer highly satisfactory approximations to the analytical solution. Leveraging fractal properties through Galerkin’s method, the paper successfully determines the fractalness parameter of the medium, shedding light on the intricate dynamics of the system.

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Fractal equation of motion of a non-Gaussian polymer chain: investigating its dynamic fractal response using an ancient Chinese algorithm.

Cited by 14 publications

References 50 publications

Soft Tissue Hybrid Model for Real-Time Simulations

Soft Tissue Hybrid Model for Real-Time Simulations

New analytical solution of the fractal anharmonic oscillator using an ancient Chinese algorithm: Investigating how plasma frequency changes with fractal parameter values

A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

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