ArticleNo Access

FRACTAL OSCILLATION AND ITS FREQUENCY-AMPLITUDE PROPERTY

School of Science, Xi’an University of Architecture and Technology, Xi’an, P. R. China

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, P. R. China

National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, P. R. China

E-mail Address: hejihuan@suda.edu.cn

Corresponding author.

Search for more papers by this author

SHUAI-JIA KOU

School of Science, Xi’an University of Architecture and Technology, Xi’an, P. R. China

Search for more papers by this author

CHUN-HUI HE

School of Civil Engineering, Xi’an University of Architecture & Technology, Xi’an 710055, P. R. China

E-mail Address: Mathew_He@yahoo.com

Corresponding author.

Search for more papers by this author

ZUO-WEI ZHANG

School of Automation, Northwestern Polytechnical University, Xi’an, P. R. China

Search for more papers by this author

, and

KHALED A. GEPREEL

Math. Depart. Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Mathematics Department, Faculty of Science, Zagazig University, Egypt

Search for more papers by this author

https://doi.org/10.1142/S0218348X2150105XCited by:90 (Source: Crossref)

Abstract

When an oscillator vibrates in a porous medium or along an unsmooth surface, a fractal model can be adopted using the two-scale fractal derivative. This paper elucidates a simple but effective method for fast and exclusive insight into the asymptotically periodic property at the initial stage and the extremely low frequency property when time tends to infinity. Some examples are illustrated to show the one-step solution process and the accurate results.

Keywords:

References

1. Z. L. Tao, G. H. Chen and Y. H. Chen , Variational iteration method with matrix Lagrange multiplier for nonlinear oscillators, J. Low Freq. Noise Vib. Act. Control 38(3–4) (2019) 984–991. Web of Science, Google Scholar
2. X. Yao and J. H. He , On fabrication of nanoscale non-smooth fibers with high geometric potential and nanoparticle’s non-linear vibration, Thermal Sci. 24(4) (2020) 2491–2497. Web of Science, Google Scholar
3. J. H. He and Y. O. El-Dib , Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation, Results Phys. 19 (2020) 103345, https://doi.org/10.1016/j.rinp.2020.103345. Web of Science, Google Scholar
4. J. H. He , A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals. 28(2) (2020) 20500243. Link, Web of Science, Google Scholar
5. J. H. He, N. Qie and C.-H. He , Solitary waves travelling along an unsmooth boundary, Results Phys. 24 (2021) 104104. Web of Science, Google Scholar
6. K. L. Wang and S. W. Yao , Fractal variational theory for Chaplygin-He gas in a microgravity condition, Comput. Meth. Appl. Mech. Eng. 6(SI) (2020) 1606–1612. Google Scholar
7. Y. Wang and J. Y. An , Amplitude–frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion, J. Low Frequency Noise Vib. Active Control 38(3–4) (2018) 1008–1012. Web of Science, Google Scholar
8. J. H. He and Q. T. Ain , New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle, Thermal Sci. 24(2A) (2020) 659–681. Web of Science, Google Scholar
9. Q. T. Ain and J. H. He , On two-scale dimension and its applications, Thermal Sci. 23(3) (2019) 1707–1712. Web of Science, Google Scholar
10. J. H. He , Fractal calculus and its geometrical explanation, Results Phys. 10 (2018) 272–276. Web of Science, Google Scholar
11. N. Anjum and J. H. He , Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions, Alex. Eng. J. 59(6) (2020) 4343–4352. Web of Science, Google Scholar
12. D. Tian, Q. T. Ain and N. Anjum , Fractal N/MEMS: From pull-in instability to pull-in stability, Fractals (2020), https://doi.org/10.1142/S0218348X21500304. Web of Science, Google Scholar
13. J. H. He and Y. O. El-Dib , Homotopy perturbation method with three expansions, J. Math. Chem. 59 (2021) 1139–1150. Web of Science, Google Scholar
14. J. H. He and H. Latifizadeh , A general numerical algorithm for nonlinear differential equations by the variational iteration method, Int. J. Numer. Meth. Heat Fluid Flow 30(11) (2020) 4797–4810. Web of Science, Google Scholar
15. J. H. He and Y. O. El-Dib , The reducing rank method to solve third-order Duffing equation with the homotopy perturbation, Numer. Methods. Partial Differential Equations (2020), https://doi.org/10.1002/num.22609. Web of Science, Google Scholar
16. J. H. He and X. Jin , A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube, Math. Methods. Appl. Sci. (2020), https://doi.org/10.1002/mma.6321. Web of Science, Google Scholar
17. J. H. He , A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, Int. J. Numer. Methods. Heat Fluid Flow 30(11) (2020) 4933–4943. Web of Science, Google Scholar
18. J. H. He , A simple approach to Volterra-Fredholm integral equations, J. Appl. Comput. Mech. 6(SI) (2020) 1184–1186. Google Scholar
19. J. H. He , Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20(10) (2006) 1141–1199. Link, Web of Science, Google Scholar
20. J. H. He , The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, J. Low Freq. Noise Vib. Act. Control 38(3–4) (2019) 1252–1260. Web of Science, Google Scholar
21. J. H. He , The simplest approach to nonlinear oscillators, Results Phys. 15 (2019) 102546, https://doi.org/10.1016/j.rinp.2019.102546. Web of Science, Google Scholar
22. J. H. He , A tutorial review fractal spacetime and fractional calculus, Int. J. Theor. Phys. 53(11) (2014) 3698–3718. Web of Science, Google Scholar
23. J. H. He , Fractal calculus and its geometrical explanation, Results Phys. 10 (2018) 272–276. Web of Science, Google Scholar
24. K. L. Wang and K. J. Wang , A modification of the reduced differential transform method for fractional calculus, Thermal Sci. 22(4) (2018) 1871–1875. Web of Science, Google Scholar
25. Y. T. Zuo and H. J. Liu , A Fractal Rheological model for SiC paste using a fractal derivative, J. Appl. Comput. Mech. 6 (SI) (2020) 1434–1439. Google Scholar
26. Y. T. Zuo , Effect of SiC particles on viscosity of 3D print paste: A fractal rheological model and experimental verification, Thermal Sci. 25(3) (2021), 2403–2407, https://doi.org/10.2298/TSCI200710131Z. Web of Science, Google Scholar
27. L. Y. Xu, Y. Li, X. X. Li and J. H. He , Detection of cigarette smoke using a fiber membrane filmed with carbon nanoparticles and a fractal current law, Thermal Sci. 24(4) (2020) 2469–2474. Web of Science, Google Scholar
28. J. H. He , On the fractal variational principle for the telegraph equation, Fractals 29(1) (2021) 2150022, https://doi.org/10.1142/S0218348X21500225. Link, Web of Science, Google Scholar
29. F. Y. Ji et al., A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Appl. Math. Model. 82 (2020) 437–448. Web of Science, Google Scholar
30. C. H. He et al., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28 (1) (2020) 2050011. Link, Web of Science, Google Scholar
31. L. Lin and S. W. Yao , Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface, J. Eng. Fibers Fabrics. 14 (2019) 1558925019895643. Web of Science, Google Scholar
32. X. J. Li, Z. Liu and J. H. He , A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals 28(5) (2020) 2050093, https://doi.org/10.1142/S0218348X20500930. Link, Web of Science, Google Scholar
33. J. H. He , A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes, J. Electroanal. Chem. 854 (2019) 113565. Web of Science, Google Scholar
34. N. Anjum and Q. T. Ain , Application of He’s fractional derivative and fractional complex transform for time fractional Camassa–Holm equation, Thermal Sci. 24(5A) (2020) 3023–3030. Web of Science, Google Scholar
35. F. J. Liu, X. J. Zhang and X. Li , Silkworm (Bombyx mori) cocoon vs. wild cocoon multi-layer structure and performance characterization, Thermal Sci. 23(4) (2019) 2135–2142. Web of Science, Google Scholar
36. Y. Wang, S. W. Yao and H. W. Yang , A fractal derivative model for snow’s thermal insulation property, Thermal Sci. 23(4) (2019) 2351–2354. Web of Science, Google Scholar
37. J. Fan, X. Yang and Y. Liu , Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism, J. Eng. Fibers Fabrics. 14 (2019) https://doi.org/10.1177/1558925019872200. Web of Science, Google Scholar
38. Q. L. Wang et al., Eratum: Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals 27(5) (2019) 1992001. Link, Web of Science, Google Scholar
39. Q. L. Wang et al., Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals 26(6) (2018) 1850086. Link, Web of Science, Google Scholar
40. Q. T. Ain et al., The fractional complex transform: A novel approach to the time-fractional Schrodinger equation, Fractals 28 (2020) 2050141. Link, Web of Science, Google Scholar
41. L. Cveticanin, M. Zukovic and D. Cveticanin , Influence of nonlinear subunits on the resonance frequency band gaps of acoustic metamaterial, Nonlinear Dyn. 93(3) (2018) 1341–1351. Web of Science, Google Scholar
42. D. Tian and Z. Liu , Period/frequency estimation of a nonlinear oscillator, J. Low Freq. Noise Vib. Act. Control 38(3–4) (2019) 1629–1634. Web of Science, Google Scholar
43. C. H. He, J. H. Wang and S. W. Yao , A complement to period/frequency estimation of a nonlinear oscillator, J. Low Freq. Noise Vib. Act. Control 38(3–4) (2019) 992–995. Web of Science, Google Scholar
44. C. X. Liu , A short remark on He’s frequency formulation, J. Low Freq. Noise Vib. Act. Control (2020) https://doi.org/10.1177/1461348420926331. Web of Science, Google Scholar
45. E. Z. Alex, M. P. P. Lius, H. J. C. Isaac, M. R. Oscar and O. T. Daniel , He’s frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions, J. Low Freq. Noise Vib. Act. Control (2020) https://doi.org/10.1177/1461348420972820. Web of Science, Google Scholar
46. A. Elias-Zuniga, L. M. Palacios-Pineda, I. H. Jiménez-Cedeño, O. Martínez-Romero and D. Olvera-Trejo , A fractal model for current generation in porous electrodes, J. Electroanal. Chem. 880 (2021) 114883. Web of Science, Google Scholar
47. C. H. He, C. Liu, J. H. He, A. H. Shirazi and H. Mohammad-Sedighi , Passive atmospheric water harvesting utilizing an ancient Chinese ink slab and its possible applications in modern architecture, Facta Univ. Mech. Eng. (2021), https://doi.org/10.22190/FUME123456001A. Web of Science, Google Scholar
48. Y. Shen , The Lagrange interpolation for He’s frequency formulation, J. Low Freq. Noise Vib. Act. Control (2021), https://doi.org/10.1177/1461348420972824. Web of Science, Google Scholar