# Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation

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*J*in 2022)

## Abstract

**:**

## 1. Introduction

## 2. The Duffing Equation

## 3. The Helmholtz Nonlinear Oscillator

## 4. The Simple Pendulum

## 5. Vertical Oscillations under the Influence of Nonlinear Elastic Forces

## 6. Discussion

## 7. Conclusions

## 8. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The $\mathrm{U}=\mathrm{f}\left(\mathrm{x}\right)$ graph (

**a**) for $-2\mathrm{m}\le \mathrm{x}\le 2\mathrm{m}$ and (

**b**) for $-0.5\mathrm{m}\le \mathrm{x}\le 0.5\mathrm{m}$. The graphs were plotted using Equation (11).

**Figure 2.**(

**a**) The $T=f\left(n\right)$ functions using the accurate solution (Equation (21)) and the approximate Equations (26), (28) and (30). (

**b**) The $T=f\left(n\right)$ functions using Equation (21) (black curve) and (30) (red curve). Equation (30) is the more accurate approach in this case.

**Figure 4.**The error in period’s calculation if using Equation (30) with respect to the exponent n of the power law elastic force.

**Table 1.**Accurate and approximate solutions for Duffing equation for $\mathrm{m}=1\mathrm{kg}$, ${\mathrm{c}}_{1}=1{\mathrm{kgs}}^{-2}$ and ${\mathrm{c}}_{2}=1{\mathrm{kgm}}^{-2}{\mathrm{s}}^{-2}$.

A (m) | T(s) (Using Equation (7)) | T (s) (Using Equation (9)) | Error (%) |
---|---|---|---|

0.1 | 6.2598 | 6.2597 | 1.5975 $\times {10}^{-3}$ |

0.3 | 6.0818 | 6.0813 | 8.2213 $\times {10}^{-3}$ |

0.5 | 5.7689 | 5.7658 | 5.3736 $\times {10}^{-2}$ |

1 | 4.7680 | 4.7496 | 3.8591 $\times {10}^{-1}$ |

2 | 3.1797 | 3.1416 | 1.1982 |

5 | 1.4419 | 1.4138 | 1.9488 |

10 | 0.7362 | 0.72073 | 2.1013 |

100 | 0.07416 | 0.07254 | 2.1845 |

1000 | 0.007416 | 0.007255 | 2.1710 |

**Table 2.**Accurate and approximate solutions for oscillators with a quadratic restoring force for $\mathrm{m}=1\mathrm{kg}$, ${\mathrm{c}}_{1}=1{\mathrm{kgs}}^{-2}$ and ${\mathrm{c}}_{2}=1{\mathrm{kgm}}^{-1}{\mathrm{s}}^{-2}$.

A (m) | T(s) (Using Equation (13)) | T (s) (Using Equation (14)) | Error (%) |
---|---|---|---|

0.05 | 6.1540 | 6.1318 | 0.3607 |

0.10 | 6.0326 | 5.9908 | 0.6929 |

0.15 | 5.9183 | 5.8591 | 1.0003 |

0.20 | 5.8102 | 5.7357 | 1.2822 |

0.25 | 5.7080 | 5.6199 | 1.5434 |

0.30 | 5.6111 | 5.5107 | 1.7893 |

0.35 | 5.5189 | 5.4077 | 2.0149 |

0.40 | 5.4312 | 5.3103 | 2.2260 |

0.45 | 5.3476 | 5.2179 | 2.4254 |

0.50 | 5.2678 | 5.1302 | 2.6121 |

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**MDPI and ACS Style**

Kontomaris, S.V.; Chliveros, G.; Malamou, A. Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation. *J* **2023**, *6*, 140-151.
https://doi.org/10.3390/j6010010

**AMA Style**

Kontomaris SV, Chliveros G, Malamou A. Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation. *J*. 2023; 6(1):140-151.
https://doi.org/10.3390/j6010010

**Chicago/Turabian Style**

Kontomaris, Stylianos Vasileios, Georgios Chliveros, and Anna Malamou. 2023. "Approximate Solutions for Undamped Nonlinear Oscillations Using He’s Formulation" *J* 6, no. 1: 140-151.
https://doi.org/10.3390/j6010010