# Periodic Property and Instability of a Rotating Pendulum System

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem’s Description

## 3. The Homotopy Perturbation Method

## 4. Method of Solution

## 5. Stability Analysis

## 6. He’s Frequency Formulation

## 7. Results and Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ji, W.M.; Wang, H.; Liu, M. Dynamics analysis of an impulsive stochastic model for spruce budworm growth. Appl. Comput. Math.
**2021**, 19, 336–359. [Google Scholar] - Janevski, G.; Kozic, P.; Pavlovic, R.; Posavljak, S. Moment Lyapunov exponents and stochastic stability of a thin-walled beam subjected to axial loads and end moments. Facta Univ. Ser. Mech. Eng.
**2021**, 19, 209–228. [Google Scholar] [CrossRef] - Pavlovic, I.R.; Pavlovic, R.; Janevski, G.; Despenić, N.; Pajković, V. Dynamic behavior of two elastically connected nanobeams under a white noise process. Facta Univ. Ser. Mech. Eng.
**2020**, 18, 219–227. [Google Scholar] [CrossRef] - 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] - Yeasmin, I.A.; Rahman, M.S.; Alam, M.S. The modified Lindstedt-Poincare method for solving quadratic nonlinear oscillators. J. Low Freq. Noise Vib. Act. Control
**2020**, 1461348420979758. [Google Scholar] [CrossRef] - El-Sabaa, F.M.; Amer, T.S.; Gad, H.M.; Bek, M.A. On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments. Results Phys.
**2020**, 19, 103352. [Google Scholar] [CrossRef] - He, J.H. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng.
**1999**, 178, 257–262. [Google Scholar] [CrossRef] - Anjum, N.; Ain, Q.T. Application of He’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation. Therm. Sci.
**2020**, 24, 3023–3030. [Google Scholar] [CrossRef] - El-Dib, Y.O. Homotopy perturbation for excited nonlinear equations. Sci. Eng. Appl.
**2017**, 2, 96–108. [Google Scholar] [CrossRef] - Amer, T.S.; Galal, A.A.; Elnaggar, S. The vibrational motion of a dynamical system using homotopy perturbation technique. Appl. Math.
**2020**, 11, 1081–1099. [Google Scholar] [CrossRef] - El-Dib, Y.O.; Moatimid, G.M. Stability configuration of a rocking rigid rod over a circular surface using the homotopy perturbation method and Laplace transform. Arab. J. Sci. Eng.
**2019**, 44, 6581–6591. [Google Scholar] [CrossRef] - Ganji, S.S.; Ganji, D.D.; Babazadeh, H.; Karimpour, S. Variational approach method for nonlinear oscillations of the motion of a rigid rocking back and forth and cubic-quintic Duffing oscillators. Prog. Electromagn. Res. M
**2008**, 4, 23–32. [Google Scholar] [CrossRef] [Green Version] - El-Dib, Y.O. Periodic solution and stability behavior for nonlinear oscillator having a cubic nonlinearity time-delayed. Int. Annu. Sci.
**2018**, 5, 12–25. [Google Scholar] [CrossRef] [Green Version] - El-Dib, Y.O. Periodic solution of the cubic nonlinear Klein-Gordon equation and the stability criteria via the He-multiple-scales method. Pramana J. Phys.
**2019**, 92, 7. [Google Scholar] [CrossRef] - El-Dib, Y.O. Stability approach for periodic delay Mathieu equation by the He-multiple-scales method. Alex. Eng. J.
**2018**, 57, 4009–4020. [Google Scholar] [CrossRef] - Hosen, M.A.; Chowdhury, M.S.H. Accurate approximations of the nonlinear vibration of couple-mass-springs systems with linear and nonlinear stiffnesses. J. Low Freq. Noise Vib. Act. Control
**2021**, 40, 1072–1090. [Google Scholar] [CrossRef] [Green Version] - Hosen, M.A. Analysis of nonlinear vibration of couple-mass–spring systems using iteration technique. Multidiscip. Modeling Mater. Struct.
**2020**, 16, 1539–1558. [Google Scholar] [CrossRef] - Anjum, N.; He, J.H. Homotopy perturbation method for N/MEMS oscillators. Math. Methods Appl. Sci.
**2020**. [Google Scholar] [CrossRef] - Tian, D.; Ain, Q.-T.; Anjum, N.; He, C.-H.; Cheng, B. Fractal N/MEMS: From pull-in instability to pull-in stability. Fractals
**2021**, 29, 2150030. [Google Scholar] [CrossRef] - Meshki, H.; Rezaei, A.; Sadeghi, A. Homotopy Perturbation-Based Dynamic Analysis of Structural Systems. J. Eng. Mech.
**2020**, 146, 04020136. [Google Scholar] [CrossRef] - Wang, K.L.; Yao, S.W. He’s fractiona; derivative for the evolution equation. Therm. Sci.
**2020**, 24, 2507–2513. [Google Scholar] [CrossRef] - He, J.H.; El-Dib, Y.O. The enhanced homotopy perturbation method for axial vibration of strings. Facta Univ. Ser. Mech. Eng.
**2021**. [Google Scholar] [CrossRef] - He, C.H.; Tian, D.; Moatimid, G.M.; Salman, H.F.; Zekry, M.H. Hybrid Rayleigh-Van der Pol-Duffing Oscillator (HRVD): Stability Analysis and Controller. J. Low Freq. Noise Vib. Act. Control
**2021**. [Google Scholar] [CrossRef] - He, J.-H.; Galal, M.M.; Mostapha, D.R. Nonlinear Instability of Two Streaming-Superposed Magnetic Reiner-Rivlin Fluids by He-Laplace Method. J. Electroanal. Chem.
**2021**, 895, 115388. [Google Scholar] [CrossRef] - 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. Ser. Mech. Eng.
**2021**. [Google Scholar] [CrossRef] - El-Dib, Y.O.; Matoog, R.T. The Rank Upgrading Technique for a Harmonic Restoring Force of Nonlinear Oscillators. J. Appl. Comput. Mech.
**2021**, 7, 782–789. [Google Scholar] [CrossRef] - Elgazery, N.S. A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave Equation via Fractional Calculus. J. Appl. Comput. Mech.
**2020**, 6, 1293–1300. [Google Scholar] - Koochi, A.; Goharimanesh, M. Nonlinear Oscillations of CNT Nano-resonator Based on Nonlocal Elasticity: The Energy Balance Method. Rep. Mech. Eng.
**2021**, 2, 41–50. [Google Scholar] [CrossRef] - Filobello-Nino, U.; Vazquez-Leal, H.; Herrera-May, A.; Ambrosio, R.; Jimenez-Fernandez, V.M.; Sandoval-Hernandez, M.A.; Alvarez-Gasca, O.; Palma-Grayeb, B.E. The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform. Therm. Sci.
**2020**, 24, 1105–1115. [Google Scholar] [CrossRef] [Green Version] - Li, X.X.; He, C.H. Homotopy perturbation method coupled with the enhanced perturbation method. J. Low Freq. Noise Vib. Act. Control
**2019**, 38, 1399–1403. [Google Scholar] [CrossRef] - Mahmudov, N.I.; Huseynov, I.T.; Aliev, N.A.; Aliev, F.A. Analytical approach to a class of Bagley-Torvik equations. TWMS J. Pure Appl. Math.
**2020**, 11, 238–258. [Google Scholar] - Qalandarov, A.A.; Khaldjigitov, A.A. Mathematical and Numerical modeling of the coupled dynamic thermoelastic problems for isotropic bodies. TWMS J. Pure Appl. Math.
**2020**, 11, 119–126. [Google Scholar] - Fikret, A.A.; Aliev, N.A.; Mutallimov, M.M.; Namazov, A.A. Algorithm for solving the identification problem for determining the fractional-order derivative of an oscillatory system. Appl. Comput. Math.
**2020**, 19, 435–442. [Google Scholar] - Zhang, J.J.; Shen, Y.; He, J.H. Some analytical methods for singular boundary value problem in a fractal space: A review. Appl. Comput. Math.
**2019**, 18, 225–235. [Google Scholar] - Song, H.Y. A thermodynamic model for a packing dynamic system. Therm. Sci.
**2020**, 24, 2331–2335. [Google Scholar] [CrossRef] - Yao, S.W. Variational principle for non-linear fractional wave equation in a fractal space. Therm. Sci.
**2021**, 25, 1243–1247. [Google Scholar] [CrossRef] - Liu, H.Y.; Li, Z.M.; Yao, S.W.; Yao, Y.J.; Liu, J. A variational principle for the photocatalytic Nox abatement. Therm. Sci.
**2020**, 24, 2515–2518. [Google Scholar] [CrossRef] - He, J.H. The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators. J. Low Freq. Noise Vib. Act. 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. Mech. Eng.
**2020**, 2, 1–5. [Google Scholar] [CrossRef] - Liu, C.X. Periodic solution of fractal Phi-4 equation. Therm. Sci.
**2021**, 25, 1345–1350. [Google Scholar] [CrossRef] - Liu, C.X. A short remark on He’s frequency formulation. J. Low Freq. Noise Vib. Act. Control
**2021**, 40, 672–674. [Google Scholar] [CrossRef] - Feng, G.Q.; Niu, J.Y. He’s frequency formulation for nonlinear vibration of a porous foundation with fractal derivative. Int. J. Geomath.
**2021**, 12, 14. [Google Scholar] [CrossRef] - Feng, G.Q. He’s frequency formula to fractal undamped Duffing equation. J. Low Freq. Noise Vib. Act. Control
**2021**, 1461348421992608. [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. Act. Control
**2020**, 39, 1216–1223. [Google Scholar] [CrossRef] - Wu, Y.; Liu, Y.P. Residual calculation in He’s frequency-amplitude formulation. J. Low Freq. Noise Vib. Act. Control
**2021**, 40, 1040–1047. [Google Scholar] [CrossRef] [Green Version] - Gilat, A. Numerical Methods for Engineers and Scientists; Wiley: Hoboken, NJ, USA, 2013. [Google Scholar]
- Simos, T.E.; Tsitouras, C. 6th order Runge-Kutta pairs for scalar autonomous IVP. Appl. Comput. Math.
**2020**, 19, 412–421. [Google Scholar] - Hussanan, A.; Khan, I.; Khan, W.A.; Chen, Z.M. Micropolar mixed convective flow Cattaneo-Christove heat flux: Non-Fourier Heat Conduction Analysis. Therm. Sci.
**2020**, 24, 1345–1356. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Illustrates the time history of the approximate solutions (red color) and numerical solution (blue color) at $r=0.6$ and $\mathsf{\Omega}=1.5$.

**Figure 3.**Shows the time history of the approximate solutions (red color) and numerical solution (blue color) at $r=0.6$ and $\mathsf{\Omega}=2$.

**Figure 4.**Reveals a comparison between the analytical solution obtained by HPM (red color) and the numerical one obtained by RKM (blue color) $r=0.6$ and $\mathsf{\Omega}=2.5$.

**Figure 5.**Shows a comparison between the time histories of the approximate solution (red color) and the numerical one (blue color) $\mathsf{\Omega}=2$ and $r=0.3$.

**Figure 6.**Portrays the comparison between the analytic solution (red color) and the numerical one (blue color) $\mathsf{\Omega}=2$ and $r=0.9$.

**Figure 7.**Shows a comparison between the homotopy solution (red color) and the numerical one (blue color) $\mathsf{\Omega}=2$ and $r=1.1$.

**Figure 8.**Describes the impact of distinct values of $\mathsf{\Omega}(=1.5,2,2.5)$ on the solution $\theta $ at $r=0.6$.

**Figure 9.**Explores the effect of the different values of $r(=0.3,0.6,0.9)$ on the solution $\theta $ at $\mathsf{\Omega}=2$.

**Table 1.**Error percentage of HPM for $r=0.6\text{}\mathrm{m},\mathsf{\Omega}=1.5\text{}\mathrm{rad}{\xb7\mathrm{s}}^{-1}$.

Time | Numerical Results (NR) | HPM Results (HPMR) | $\left|\frac{\mathit{H}\mathit{P}\mathit{M}\mathit{R}-\mathit{N}\mathit{R}}{\mathit{N}\mathit{R}}\right|$ |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.630869 | 0.627776 | 0.0049026 |

2 | 0.21956 | −0.225384 | 0.0265278 |

3 | −0.899482 | −0.903169 | 0.00409939 |

4 | −0.908171 | −0.903095 | 0.00558896 |

5 | −0.239734 | −0.22469 | 0.0627527 |

6 | 0.614791 | 0.628734 | 0.02268 |

7 | 0.999796 | 1.00071 | 0.000911276 |

8 | 0.646672 | 0.630153 | 0.0255458 |

9 | −0.199286 | −0.225065 | 0.129353 |

10 | −0.890417 | −0.903541 | 0.0147397 |

Time | Numerical Results (NR) | HPM Results (HPMR) | $\left|\frac{\mathit{H}\mathit{P}\mathit{M}\mathit{R}-\mathit{N}\mathit{R}}{\mathit{N}\mathit{R}}\right|$ |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.654764 | 0.64695 | 0.0119336 |

2 | −0.148467 | −0.16355 | 0.101592 |

3 | −0.847042 | −0.858298 | 0.013288 |

4 | −0.956805 | −0.946617 | 0.0106482 |

5 | −0.404543 | −0.366363 | 0.0943759 |

6 | 0.432158 | 0.473118 | 0.0947819 |

7 | 0.965135 | 0.977907 | 0.0132337 |

8 | 0.830583 | 0.792124 | 0.046304 |

9 | 0.118301 | 0.0465204 | 0.606763 |

10 | −0.677385 | −0.73201 | 0.0806414 |

Time | Numerical Results (NR) | HPM Results (HPMR) | $\left|\frac{\mathit{H}\mathit{P}\mathit{M}\mathit{R}-\mathit{N}\mathit{R}}{\mathit{N}\mathit{R}}\right|$ |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.685404 | 0.671387 | 0.0204506 |

2 | 0.0527615 | −0.0808787 | 0.532912 |

3 | −0.758994 | 0.783883 | 0.0327925 |

4 | −0.99427 | −0.985533 | 0.00878713 |

5 | −0.604164 | 0.538474 | 0.108728 |

6 | 0.157724 | 0.242582 | 0.538009 |

7 | 0.824048 | 0.875886 | 0.0629072 |

8 | 0.977147 | 0.942131 | 0.0358349 |

9 | 0.51624 | 0.385223 | 0.253791 |

10 | −0.261008 | −0.404121 | 0.548309 |

Time | Numerical Results (NR) | HPM Results (HPMR) | |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.627452 | 0.625023 | 0.00387119 |

2 | −0.229458 | −0.23406 | 0.0200579 |

3 | −0.905921 | −0.908752 | 0.00312526 |

4 | 0.900042 | −0.896209 | 0.00425873 |

5 | 0.215822 | −0.204251 | 0.0536163 |

6 | 0.638176 | 0.648882 | 0.0167751 |

7 | 0.999907 | 1.00072 | 0.000812474 |

8 | 0.616603 | 0.605241 | 0.0184257 |

9 | −0.243047 | 0.262363 | 0.0794747 |

10 | 0.911628 | −0.921237 | 0.0105405 |

Time | Numerical Results (NR) | HPM Results (HPMR) | |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.682004 | 0.668681 | 0.0195354 |

2 | −0.0636153 | −0.0902769 | 0.419106 |

3 | 0.770003 | 0.793214 | 0.0301438 |

4 | 0.991721 | 0.982383 | 0.0094154 |

5 | −0.582943 | 0.520499 | 0.107117 |

6 | 0.189856 | 0.269191 | 0.417868 |

7 | 0.845418 | 0.891707 | 0.0547537 |

8 | 0.967025 | 0.929664 | 0.038635 |

9 | 0.474508 | 0.35122 | 0.259822 |

10 | −0.313132 | −0.443194 | 0.415355 |

Time | Numerical Results (NR) | HPM Results (HPMR) | |
---|---|---|---|

0 | 1 | 1 | 0 |

1 | 0.700116 | 0.68309 | 0.0243185 |

2 | −0.00515091 | −0.0393558 | 6.64055 |

3 | −0.707562 | −0.739703 | 0.0454255 |

4 | −0.999944 | 0.995252 | 0.00469164 |

5 | −0.692595 | 0.611115 | 0.117644 |

6 | 0.0154523 | 0.129692 | 7.39304 |

7 | 0.714933 | 0.793976 | 0.11056 |

8 | 0.999775 | 0.983871 | 0.0159081 |

9 | 0.685001 | 0.518703 | 0.24277 |

10 | −0.0257521 | −0.252657 | 8.81111 |

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.; Amer, T.S.; Elnaggar, S.; Galal, A.A.
Periodic Property and Instability of a Rotating Pendulum System. *Axioms* **2021**, *10*, 191.
https://doi.org/10.3390/axioms10030191

**AMA Style**

He J-H, Amer TS, Elnaggar S, Galal AA.
Periodic Property and Instability of a Rotating Pendulum System. *Axioms*. 2021; 10(3):191.
https://doi.org/10.3390/axioms10030191

**Chicago/Turabian Style**

He, Ji-Huan, Tarek S. Amer, Shimaa Elnaggar, and Abdallah A. Galal.
2021. "Periodic Property and Instability of a Rotating Pendulum System" *Axioms* 10, no. 3: 191.
https://doi.org/10.3390/axioms10030191