Frequency formula for a class of fractal vibration system

  • Yi Tian College of Data Science and Application, Inner Mongolia University of Technology, Hohhot, China


Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.


Anjum, N., & He, J.H. (2019). Laplace transform: Making the variational iteration method easier. Applied Mathematics Letters, 92, 134-138.

Anjum, N., & He, J.H. (2020a). Higher-order homotopy perturbation method for conservative nonlinear oscillators generally and microelectromechanical systems' oscillators particularly. International Journal of Modern Physics B, 34(32), 2050313.

Anjum, N., &He, J.H. (2020b) Homotopy perturbation method for N/MEMS oscillators. Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.6583.

He, C.H., Shen, Y., Ji, F.Y., & He, J.H. (2020). Taylor series solution for fractal Bratu-type equation arising in electrospinning process. Fractals, 28(1), 2050011.

He, J.H. (1999). Variational iteration method–a kind of non-linear analytical technique: Some examples. International Journal of Non-Linear Mechanics, 34(4), 699-708.

He, J.H. (2003). Homotopy perturbation method: a new nonlinear analytical technique. Applied mathematics and computation, 135(1), 73-79.

He, J.H. (2013). Exp-function method for fractional differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 14(6), 363-366.

He, J.H. (2014). Homotopy perturbation method with two expanding parameters. Indian Journal of Physics, 88(2), 193-196.

He, J.H. (2018). Fractal calculus and its geometrical explanation. Results in Physics, 10, 272-276.

He, J.H. (2019). A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes. Journal of Electroanalytical Chemistry, 854, 113565.

He, J.H. (2020a). Taylor series solution for a third order boundary value problem arising in architectural engineering. Ain Shams Engineering Journal, 11(4), 1411-1414.

He, J.H. (2020b).Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation. Results in Physics, 17, 103031.

He, J.H. (2020c).Generalized variational principles for buckling analysis of circular cylinders. Acta Mechanica, 231(3), 899-906.

He, J.H. (2021). On the fractal variational principle for the telegraph equation. Fractals, 29(1), 2150022.

He, J.H., & Ain, Q.T. (2020). New promises and future challenges of fractal calculus: From two-scale Thermodynamics to fractal variational principle. Thermal Science, 24(2A), 659-681.

He, J.H., & El-Dib, Y. O. (2020). Homotopy perturbation method for Fangzhu oscillator. Journal of Mathematical Chemistry, 58(10), 2245–2253.

He, J.H., Yang, Q., He, C.H., & Khan, Y. (2021a). A simple frequency formulation for the tangent oscillator. Axioms, 10,320.

He, J.H., Hou, W.F.,Qie, N.,Gepreel, K. A., Shirazi, A.H., & Mohammad-Sedighi,H. (2021b). Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators. Facta Universitatis, Series: Mechanical Engineering, 19(2), 199-208.

He, J.H., El-Dib, Y. O., & Mady, A.A. (2021c). Homotopy perturbation method for the fractal Toda oscillator. Fractal and Fractional, 5, 93.

He, J.H., & Ji, F.Y. (2019a). Taylor series solution for Lane-Emden equation. Journal of Mathematical Chemistry, 57(8), 1932-1934.

He, J.H., & Ji, F.Y. (2019b).Two-scale mathematics and fractional calculus for thermodynamics. Thermal Science, 23(4), 2131-2133.

He, J.H., & Wu, X.H. (2006). Exp-function method for nonlinear wave equations. Chaos Solitons & Fractals, 30(3), 700-708.

Ren, Z.F., Yao, S.W., & He, J.H. (2019). He’s multiple scales method for nonlinear vibrations. Journal of Low Frequency Noise Vibration and Active Control, 38(3-4), 1708-1712.

Song, H.Y. (2020). A thermodynamic model for a packing dynamical system. Thermal Science, 24(4), 2331-2335.

Yu, D.N., He, J.H., & Garcıa, A. G. (2019). Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators. Journal of Low Frequency Noise Vibration and Active Control, 38(3-4), 1540-1554.

How to Cite
Tian, Y. (2022). Frequency formula for a class of fractal vibration system. Reports in Mechanical Engineering, 3(1), 55-61.