# A Simple Frequency Formulation for the Tangent Oscillator

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Frequency Formulation

## 3. Tangent Oscillator

## 4. Hyperbolic Tangent Oscillator

## 5. Singular Oscillator

## 6. MEMS Oscillator

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Song, H.Y. A modification of homotopy perturbation method for a hyperbolic tangent oscillator arising in nonlinear packaging system. J. Low Freq. Noise Vib. Active Control
**2019**, 38, 914–917. [Google Scholar] [CrossRef] [Green Version] - Song, H.Y. A thermodynamics model for a packing dynamical system. Therm. Sci.
**2020**, 24, 2331–2335. [Google Scholar] [CrossRef] - Kuang, W.; Wang, J.; Huang, C.; Lu, L.; Gao, D.; Wang, Z.; Ge, C. Homotopy perturbation method with an auxiliary term for the optimal design of a tangent nonlinear packaging system. J. Low Freq. Noise Vib. Active Control
**2019**, 38, 1075–1080. [Google Scholar] [CrossRef] - Ji, Q.P.; Wang, J.; Lu, L.X.; Ge, C.F. Li-He’s modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system. J. Low Freq. Noise Vib. Active Control
**2021**, 40, 675–682. [Google Scholar] [CrossRef] - Clementi, F.; Gazzani, V.; Poiani, M.; Lenci, S. Assessment of seismic behaviour of heritage masonry buildings using numerical modelling. J. Build. Eng.
**2016**, 8, 29–47. [Google Scholar] [CrossRef] - Dertimanis, V.K.; Chatzi, E.N.; Masri, S.F. On the Active Vibration Control of Nonlinear Uncertain Structures. J. Appl. Comp. Mech.
**2021**, 7, 1183–1197. [Google Scholar] - El-Dib, Y.O.; Matoog, R.T. The Rank Upgrading Technique for a Harmonic Restoring Force of Nonlinear Oscillators. J. Appl. Comp. Mech.
**2021**, 7, 782–789. [Google Scholar] - Anjum, N.; He, J.H.; Ain, Q.T.; Tian, D. Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system. Facta Univ. Mech. Eng.
**2021**. [Google Scholar] [CrossRef] - Nawaz, Y.; Arif, M.S.; Bibi, M.; Naz, M.; Fayyaz, R. An effective modification of He’s variational approach to a nonlinear oscillator. J. Low Freq. Noise Vib. Active Control
**2019**, 38, 1013–1022. [Google Scholar] [CrossRef] - Ganji, D.D.; Gorji, M.; Soleimani, S.; Esmaeilpour, M. Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method. J. Zhejiang Univ.-Sci. A
**2009**, 10, 1263–1268. [Google Scholar] [CrossRef] - Ganji, S.S.; Barari, A.; Karimpour, S.; Domairry, G. Motion of a rigid rod rocking back and forth and cubic-quitic Duffing oscillators. J. Theor. Appl. Mech.
**2012**, 50, 215–229. [Google Scholar] - Beléndez, A.; Beléndez, T.; Martínez, F.J.; Pascual, C.; Alvarez, M.L.; Arribas, E. Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities. Nonlinear Dyn.
**2016**, 86, 1687–1700. [Google Scholar] [CrossRef] [Green Version] - Suleman, M.; Wu, Q.B. Comparative Solution of Nonlinear Quintic Cubic Oscillator Using Modified Homotopy Perturbation Method. Adv. Math. Phys.
**2015**, 932905. [Google Scholar] [CrossRef] - Razzak, M.A. An analytical approximate technique for solving cubic-quintic Duffing oscillator. Alex. Eng. J.
**2016**, 55, 2959–2965. [Google Scholar] [CrossRef] [Green Version] - Wang, K.J.; Wang, G.D. Gamma function method for the nonlinear cubic-quintic Duffing oscillators. J. Low Freq. Noise Vib. Active Control
**2021**. [Google Scholar] [CrossRef] - Wang, K.J. On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets. Math. Methods Appl. Sci.
**2021**. [Google Scholar] [CrossRef] - Wang, K.J. Generalized variational principle and periodic wave solution to the modified equal width-Burgers equation in nonlinear dispersion media. Phys. Lett. A
**2021**, 419, 127723. [Google Scholar] [CrossRef] - Wang, K.J.; Zhang, P.L. Investigation of the periodic solution of the time-space fractional Sasa-Satsuma equation arising in the monomode optical fibers. EPL
**2021**. [Google Scholar] [CrossRef] - He, J.H. The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators. J. Low Freq. Noise Vib. Active Control
**2019**, 38, 1252–1260. [Google Scholar] [CrossRef] [Green Version] - Qie, N.; Hou, W.F.; He, J.H. The fastest insight into the large amplitude vibration of a string. Rep. Mechan. Eng.
**2020**, 2, 1–5. [Google Scholar] [CrossRef] - Feng, G.Q. He’s frequency formula to fractal undamped Duffing equation. J. Low Freq. Noise Vib. Active Control
**2021**. [Google Scholar] [CrossRef] - Liu, C.X. A short remark on He’s frequency formulation. J. Low Freq. Noise Vib. Active Control
**2021**, 40, 672–674. [Google Scholar] [CrossRef] - Liu, C.X. Periodic solution of fractal Phi-4 equation. Therm. Sci.
**2021**, 25, 1345–1350. [Google Scholar] [CrossRef] - Elías-Zúñiga, A.; Palacios-Pineda, L.M.; Jiménez-Cedeño, I.H.; Martínez-Romero, O.; Trejo, D.O. He’s frequency-amplitude formulation for nonlinear oscillators using Jacobi elliptic functions. J. Low Freq. Noise Vib. Active Control
**2020**, 39, 1216–1223. [Google Scholar] [CrossRef] - Elías-Zúñiga, A.; Palacios-Pineda, L.M.; Jiménez-Cedeño, I.H.; Martínez-Romero, O.; Olvera-Trejo, D. Enhanced He’s frequency-amplitude formulation for nonlinear oscillators. Results Phys.
**2020**, 19, 103626. [Google Scholar] [CrossRef] - Wu, Y.; Liu, Y.P. Residual calculation in He’s frequency-amplitude formulation. J. Low Freq. Noise Vib. Active Control
**2021**, 40, 1040–1047. [Google Scholar] [CrossRef] [Green Version] - Zuo, Y.T. A gecko-like fractal receptor of a three-dimensional printing technology: A fractal oscillator. J. Math. Chem.
**2021**, 59, 735–744. [Google Scholar] [CrossRef] - He, C.H. An introduction to an ancient Chinese algorithm and its modification. Int. J. Numer. Methods Heat Fluid Flow
**2016**, 26, 2486–2491. [Google Scholar] [CrossRef] - Khan, W.A. Numerical simulation of Chun-Hui He’s iteration method with applications in engineering. Int. J. Numer. Methods Heat Fluid Flow
**2021**. [Google Scholar] [CrossRef] - Sedighi, H.M.; Shirazi, K.H.; Attarzadeh, M.A. A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches. Acta Astronaut.
**2013**, 91, 245–250. [Google Scholar] [CrossRef] - Sedighi, H.M. Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut.
**2014**, 95, 111–123. [Google Scholar] [CrossRef] - Sedighi, H.M.; Keivani, M.; Abadyan, M. Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity, surface energy and nonlocal effect. Comp. Part B-Eng.
**2015**, 83, 117–133. [Google Scholar] [CrossRef] - Zhang, Y.N.; Tian, D.; Pang, J. A fast estimation of the frequency property of the microelectromechanical system oscillator. J. Low Freq. Noise Vib. Active Control
**2021**. [Google Scholar] [CrossRef] - Wang, K.L.; Wei, C.F. A powerful and simple frequency formula to nonlinear fractal oscillators. J. Low Freq. Noise Vib. Active Control.
**2021**, 40, 1373–1379. [Google Scholar] [CrossRef] - El-Dib, Y.O. The frequency estimation for non-conservative nonlinear oscillation. ZAMM-Z. Angew. Math. Mech.
**2021**. [Google Scholar] [CrossRef] - Wang, K.J. A fast insight into the nonlinear oscillation of nano-electro mechanical resonators considering the size effect and the van der Waals force. EPL Lett. J. Explor. Front. Phys.
**2021**. [Google Scholar] [CrossRef] - Popov, M. Friction under Large-Amplitude Normal Oscillations. Facta Univ. Ser. Mech. Eng.
**2021**, 19, 105–113. [Google Scholar] [CrossRef]

**Figure 1.**The approximate solution of Equation (12) vs. the exact one of Equation (5). (

**a**) (a,b) = (0, 0.2); (

**b**) (a,b) = (0.3, 0); (

**c**) (a,b) = (0.5, 0.3); and (

**d**) (a,b) = (0.8, 0.3).

**Figure 2.**The approximate solution ($w=A\mathrm{cos}(\omega t+\phi )$) vs. the exact one of Equation (27). (

**a**) (a,b) = (0, 0.2); (

**b**) (a,b) = (0.5, 0); (

**c**) (a,b) = (0.6, 0.2); and (

**d**) (a,b) = (0.8, 0.3).

**Figure 3.**The comparison of approximate solution $w=A\mathrm{cos}(\omega t+\phi )$ with exact one of Equation (34). (

**a**) (k,a,b) = (0.8, 0.4, 0.1); (

**b**) (k,a,b) = (0.6, 0.8, 0.1); (

**c**) (k,a,b) = (0.3, 0.5, 0.2); (

**d**) (k,a,b) = (0.6, 0.7, 0.2); and (

**e**) (k,a,b) = (1, 0.5, 0).

**Figure 4.**The MEMS oscillator with different values of b. (

**a**) b = 0.05; (

**b**) b = 0.10; and (

**c**) b = 0.15.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

He, J.-H.; Yang, Q.; He, C.-H.; Khan, Y.
A Simple Frequency Formulation for the Tangent Oscillator. *Axioms* **2021**, *10*, 320.
https://doi.org/10.3390/axioms10040320

**AMA Style**

He J-H, Yang Q, He C-H, Khan Y.
A Simple Frequency Formulation for the Tangent Oscillator. *Axioms*. 2021; 10(4):320.
https://doi.org/10.3390/axioms10040320

**Chicago/Turabian Style**

He, Ji-Huan, Qian Yang, Chun-Hui He, and Yasir Khan.
2021. "A Simple Frequency Formulation for the Tangent Oscillator" *Axioms* 10, no. 4: 320.
https://doi.org/10.3390/axioms10040320