# Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies.

## 1. Introduction

_{2}symmetry [40]. Taking the HB as an example, to obtain an explicit solution with high accuracy, one needs to solve a set of complicated nonlinear algebraic equations.

## 2. The Basic Idea of the Dynamic Frequency Method

- order 1:$$\begin{array}{l}\frac{1}{2}{({a}_{0}\mathrm{sin}{\omega}_{1,0}t)}^{2}({\omega}_{1,0}^{2}+2{\omega}_{1,0}p{\omega}_{1,1}(t))={E}^{\ast}-\frac{1}{2}{\omega}_{0}^{2}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{2}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{i=2}^{M}\frac{{\alpha}_{i,0}}{i+1}}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i+1}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \int {\displaystyle \sum _{i=0}^{K}{\displaystyle \sum _{j=1}^{K-i}{\beta}_{i,j}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i}{(-{a}_{0}{\omega}_{1,0}\mathrm{sin}{\omega}_{1,0}t)}^{j+1}}}}dt+\mathrm{O}({p}^{2}),\end{array}$$
- order 2:$$\begin{array}{l}\frac{1}{2}{({a}_{0}\mathrm{sin}{\omega}_{1,0}t)}^{2}({\displaystyle {\displaystyle \sum}_{i=0}^{2}}{\displaystyle {\displaystyle \sum}_{j=0}^{2-i}}{p}^{i+j}{\omega}_{1,i}(t){\omega}_{1,j}(t))={E}^{\ast}-\frac{1}{2}{\omega}_{0}^{2}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{2}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{i=2}^{M}\frac{{\alpha}_{i,0}}{i+1}}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i+1}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \int {\displaystyle \sum _{i=0}^{K}{\displaystyle \sum _{j=1}^{K-i}{\beta}_{i,j}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i}{(-{a}_{0}{\omega}_{1,0}\mathrm{sin}{\omega}_{1,0}t)}^{j}[-{a}_{0}({\omega}_{1,0}+p{\omega}_{1,1}(t))\mathrm{sin}{\omega}_{1,0}t]}}}dt+\mathrm{O}({p}^{3}),\end{array}$$
- order $k$:$$\begin{array}{l}\frac{1}{2}{({a}_{0}\mathrm{sin}{\omega}_{1,0}t)}^{2}({\displaystyle {\displaystyle \sum}_{i=0}^{k}}{\displaystyle {\displaystyle \sum}_{j=0}^{k-i}}{p}^{i+j}{\omega}_{1,i}(t){\omega}_{1,j}(t))={E}^{\ast}-\frac{1}{2}{\omega}_{0}^{2}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{2}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \sum _{i=2}^{M}\frac{{\alpha}_{i,0}}{i+1}}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i+1}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \int {\displaystyle \sum _{i=0}^{K}{\displaystyle \sum _{j=1}^{K-i}{\beta}_{i,j}{({a}_{0}\mathrm{cos}{\omega}_{1,0}t+{b}_{0})}^{i}{(-{a}_{0}{\omega}_{1,0}\mathrm{sin}{\omega}_{1,0}t)}^{j}[-{a}_{0}({\omega}_{1,0}+{\displaystyle \sum _{i=1}^{k-1}{p}^{i}{\omega}_{1,i}(t)})\mathrm{sin}{\omega}_{1,0}t]}}}dt+\mathrm{O}({p}^{k+1}).\end{array}$$

## 3. Examples

#### 3.1. Duffing Oscillators

#### 3.2. Oscillator with Coordinate-Dependent Mass

## 4. Strategy to Improve the Accuracy of Computation

#### 4.1. Modified Energy Balance Method

_{1}to G

_{4}between the first order dynamic frequency and the modified EBM which transfers the analysis from an isolated linear collocation point to globally effect of nonlinearity.

#### 4.2. The Nonlinear Strength Index

_{1}to G

_{4}, the enlargement of $\delta $ synchronizes the intensification of the nonlinearity, as well as the loosing accuracy of the first order analytical approximate solutions. Then, two major nonlinear intervals are logically classified, where the boundary values $\delta =0.3$ are a coarse estimate, only for the choice of the following analytical strategies. The nonlinear strength indexes and belonging intervals from G

_{1}to G

_{4}are presented in Table 4.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgement

## Conflicts of Interest

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**Figure 1.**Comparison of analytical approximations and numerical solution for $\gamma =-1,A=\sqrt{0.8}$ in Equation (14).

**Figure 2.**Absolute errors of analytical approximations for different orders (

**a**) the first order approximate solution and (

**b**) the second order approximate solution.

**Figure 3.**Comparison of approximate solutions to Equation (33) for (

**a**) $\alpha =0.1$; (

**b**) $\alpha =0.3$; (

**c**) $\alpha =0.5$ and (

**d**) $\alpha =0.8$: Runge-Kutta method (red solid line), Homotopy Perturbation method (orange dashed line), the present method (black dotted line).

**Figure 4.**Comparison of approximate solutions of Equation (45) under different sets of parameters: (

**a**) G

_{1}; (

**b**) G

_{2}; (

**c**) G

_{3}; (

**d**) G

_{4}; Runge-Kutta method (red solid line), the first order dynamic frequency method (black dotted line), modified energy balance method with a

_{1}(blue dashed line).

$\mathit{\gamma}{\mathit{A}}^{\mathbf{2}}$ | ${\mathit{T}}_{\mathbf{1}}/{\mathit{T}}_{\mathbf{ex}}$ | ${\mathit{T}}_{\mathbf{2}}/{\mathit{T}}_{\mathbf{ex}}$ |
---|---|---|

−0.8 | 0.8939 | 1.0216 |

−0.7 | 0.9481 | 0.9973 |

−0.5 | 0.9862 | 0.9966 |

−0.1 | 0.9997 | 0.9999 |

0.1 | 0.9998 | 0.9999 |

0.5 | 0.9975 | 0.9988 |

0.7 | 0.9960 | 0.9981 |

0.8 | 0.9953 | 0.9977 |

1 | 0.9939 | 0.9970 |

10 | 0.9751 | 0.9859 |

$\mathit{\alpha}$ | T_{1} | T |
---|---|---|

0.1 | 0.7885 | 0.7861 |

0.2 | 0.7300 | 0.7491 |

0.3 | 0.6841 | 0.7169 |

0.4 | 0.6468 | 0.6886 |

0.5 | 0.6160 | 0.6633 |

0.6 | 0.5898 | 0.6406 |

0.7 | 0.5674 | 0.6201 |

0.8 | 0.5479 | 0.6015 |

Groups | ${\mathit{\omega}}_{\mathbf{0}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{\alpha}}_{\mathbf{3}}$ | ${\mathit{\alpha}}_{\mathbf{4}}$ | ${\mathit{\alpha}}_{\mathbf{5}}$ | ${\mathit{\beta}}_{\mathbf{0}\mathbf{,}\mathbf{1}}$ | ${\mathit{\beta}}_{\mathbf{2}\mathbf{,}\mathbf{1}}$ | ${\mathit{\theta}}_{\mathbf{1}}$ |
---|---|---|---|---|---|---|---|---|

G_{1} | 2 | −2 | −4 | 2 | −2 | 1 | −3 | $\pi /4.22$ |

G_{2} | 2 | −2 | −4 | 2 | −2 | 2 | −3 | $\pi /4.28$ |

G_{3} | 2 | −2 | −6 | 2 | −2 | 2 | −3 | $\pi /4.43$ |

G_{4} | 2 | −2 | −6 | 2 | −2 | 4 | −3 | $\pi /4.62$ |

Groups | G_{1} | G_{2} | G_{3} | G_{4} |
---|---|---|---|---|

Index δ | 0.22 | 0.28 | 0.43 | 0.62 |

Interval | weakly nonlinear | strongly nonlinear |

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**MDPI and ACS Style**

Zhang, Z.; Wang, Y.; Wang, W.; Tian, R.
Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method. *Symmetry* **2019**, *11*, 676.
https://doi.org/10.3390/sym11050676

**AMA Style**

Zhang Z, Wang Y, Wang W, Tian R.
Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method. *Symmetry*. 2019; 11(5):676.
https://doi.org/10.3390/sym11050676

**Chicago/Turabian Style**

Zhang, Zhiwei, Yingjie Wang, Wei Wang, and Ruilan Tian.
2019. "Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method" *Symmetry* 11, no. 5: 676.
https://doi.org/10.3390/sym11050676