# Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

## 3. Classification of Periodic Solution

## 4. Periodic Solution

**Theorem**

**1.**

**Note 1**Conditions (20) and (32) are sufficient and unnecessary conditions for system (6) with the first kind of collision periodic solution.

**Note 2**The collision recovery matrices ${R}_{2},{R}_{3},{R}_{4}$ of the second type, the third type, and the fourth type of periodic solutions can be shown as

## 5. Numerical Simulation

_{j}, b

_{j}(j = 1, 2) can be obtained. Hence, we can detect the third kind of collision periodic solution. The phase diagram and Poincare section of the periodic 2 motion are shown in Figure 7a and Figure 8a. As can be seen in Figure 7a, there is a big gap between the phase diagram curve and the blue solid line, indicating that the upper pendulum does not collide with the right wall and the velocity diagram further confirms the fact in Figure 7c. In Figure 8a,c, the phase diagram and the velocity at the collision point have a momentary jump, showing that lower pendulum and the rigid baffle collided with the right wall.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The first kind of collision periodic solutions: (

**a**) Phase diagram and Poincare section of periodic 3 motion of upper pendulum; (

**b**) Time-displacement diagram of the upper pendulum; (

**c**) Time-velocity diagram of upper pendulum.

**Figure 4.**The first kind of collision periodic solution: (

**a**) Phase diagram and Poincare section of periodic 3 motion of lower pendulum; (

**b**) The time-displacement diagram of the lower pendulum; (

**c**) The time-velocity diagram of the lower pendulum.

**Figure 5.**The second kind of collision periodic solutions: (

**a**) Phase diagram and Poincare section of periodic 2 motion of upper pendulum; (

**b**) Time-displacement diagram of the upper pendulum; (

**c**) Time-velocity diagram of upper pendulum.

**Figure 6.**The second kind of collision periodic solution: (

**a**) phase diagram and Poincare section of periodic 2 motion of lower pendulum; (

**b**) the time-displacement diagram of the lower pendulum; (

**c**) the time-velocity diagram of the lower pendulum.

**Figure 7.**The third kind of collision periodic solutions: (

**a**) Phase diagram and Poincare section of periodic 2 motion of upper pendulum; (

**b**) Time-displacement diagram of the upper pendulum; (

**c**) Time-velocity diagram of upper pendulum.

**Figure 8.**The third kind of collision periodic solution: (

**a**) Phase diagram and Poincare section of periodic 2 motion of lower pendulum; (

**b**) The time-displacement diagram of the lower pendulum; (

**c**) The time-velocity diagram of the lower pendulum.

Type of Solution | Constraint Condition |
---|---|

The first kind of periodic solution | $({s}_{1},{x}_{1})$ |

The second kind of periodic solution | $({s}_{1},{x}_{0})$ |

The third kind of periodic solution | $({s}_{0},{x}_{1})$ |

The forth kind of periodic solution | $({s}_{0},{x}_{0})$ |

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**MDPI and ACS Style**

Guo, X.; Zhang, G.; Tian, R.
Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain. *Symmetry* **2019**, *11*, 886.
https://doi.org/10.3390/sym11070886

**AMA Style**

Guo X, Zhang G, Tian R.
Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain. *Symmetry*. 2019; 11(7):886.
https://doi.org/10.3390/sym11070886

**Chicago/Turabian Style**

Guo, Xiuying, Gang Zhang, and Ruilan Tian.
2019. "Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain" *Symmetry* 11, no. 7: 886.
https://doi.org/10.3390/sym11070886