# Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Modeling of Nose Landing Gear Shimmy

#### 2.1. Coordinate System and Degrees of Freedom

- (1)
- The landing gear strut can rotate around the directional axis with an angle of $\psi $.
- (2)
- In the X-axis direction, it is considered that the stiffness of the landing gear strut is sufficient, and the deformation has a small impact on the shimmy.
- (3)
- The lateral displacement $\lambda $ of the left and right tires is used to describe the deformation of the tires.
- (4)
- The degree of freedom $\phi $ of the axle torsion deformation indicates that the two wheels do not rotate together.

#### 2.2. Tire Dynamics Model

#### 2.3. Nonlinear Analysis of Pendulum Reducers

#### 2.4. Dynamic Equations of the System

## 3. Hopf Bifurcation Theory of the Oscillation Model

#### 3.1. Linear Stability Theory

#### 3.2. Hopf Bifurcation Theory

#### 3.3. Dimension Reduction in the Central Manifold Theorem

#### 3.4. Analysis of the Time-Delay Effect under Linear Damping

## 4. Stability Analysis of Oscillation System

#### 4.1. Numerical Verification of the Theoretical Results

^{7}, ${b}_{1}$ = −1.0603 × 10

^{9}, ${b}_{2}$ = −2.6470 × 10

^{6}, ${b}_{3}$ = 4.8429 × 10

^{10}, and ${b}_{4}$ = 2.5335 × 10

^{11}. By simplifying the equation, it can be concluded that:

#### 4.2. The Influence of the Stability Distance on the Stability of Shimmy

^{2}, and we have:

#### 4.3. Analysis of Shimmy Stability

#### 4.4. Hopf Bifurcation Analysis

^{3}, and the initial swing angle to 0.001 rad. The following curve can be obtained using numerical simulation:

#### 4.5. The Influence of Parameters on Amplitude

## 5. Conclusions

- Using theoretical analysis and numerical simulation, the accuracy of the derived formula has been validated for linear systems. When the stability distance exceeds the combined length of half of the tire in contact with the ground and the relaxation length, the stable shimmy performance is maintained by the aircraft while sliding at a specific speed. However, when the stability distance is small, only a limited portion of the speed range exhibits stability. Therefore, adjusting either the stability distance or the tire size can alter the stability performance of aircraft shimmy.
- The critical speed of the landing gear system’s linear oscillation was determined based on the stability criterion. Furthermore, a theoretical derivation was conducted to obtain the expression for the first Lyapunov coefficient of the landing gear system. By analyzing its sign, the first Lyapunov coefficient can determine the type of Hopf bifurcation that will occur in the landing gear system without requiring a solution for its bifurcation diagram. It has been demonstrated that higher-order nonlinear damping leads to supercritical Hopf bifurcation in the landing gear system.
- Using quantitative analysis of the parameters’ influence on the limit cycle, the research findings indicate that an increase in velocity induces shimmy motion in the landing gear system through Hopf bifurcation from its equilibrium point. Initially, the amplitude of the limit cycle increases and subsequently decreases, ultimately converging to the equilibrium point via Hopf bifurcation, resulting in cessation of shimmy motion. To mitigate undesired oscillations, it is recommended to enhance the linear damping coefficient within a stability margin range. Similarly, for reducing the limit cycle amplitude, designing the nonlinear damping coefficient within a reasonable range is advisable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Arreaza, C.; Behdinan, K.; Zu, J.W. Linear stability analysis and dynamic response of shimmy dampers for main landing gears. J. Appl. Mech.
**2016**, 83, 081002. [Google Scholar] [CrossRef] - Norman, S. Currey. Aircraft Landing Gear Design: Principles and Practices; American Institute of Aeronautics and Astronautics: Washington, DC, USA, 1988. [Google Scholar]
- Shengli, L.; Xiaochuan, L.; Rongyao, C. The influence of local stiffness of body joint on the shimmy stability of landing gear of light aircraft. J. Vib. Eng.
**2017**, 30, 249–254. [Google Scholar] - Somieski, G. Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods. Aerosp. Sci. Technol.
**1997**, 1, 545–555. [Google Scholar] [CrossRef] - Tartaruga, I.; Lowenberg, M.H.; Cooper, J.E.; Sartor, P.; Lemmens, Y. Bifurcation analysis of a nose landing gear system. In Proceedings of the 15th Dynamics Specialists Conference, San Diego, CA, USA, 4–8 January 2016; p. 1572. [Google Scholar]
- Thota, P.; Krauskopf, B.; Lowenberg, M. Bifurcation analysis of nose-landing-gear shimmy with lateral and longitudinal bending. J. Aircr.
**2010**, 47, 87–95. [Google Scholar] [CrossRef] - Pacejka, H.B. Tire and Vehicle Dynamics; Elsevier Ltd.: Waltham, MA, USA, 2012. [Google Scholar]
- Ruan, S.; Zhang, M.; Hong, Y.; Nie, H. Influence of clearance and structural coupling parameters on shimmy stability of landing gear. Aeronaut. J.
**2023**, 127, 1591–1622. [Google Scholar] [CrossRef] - Rahmani, M.; Behdinan, K. Investigation on the effect of coulomb friction on nose landing gear shimmy. J. Vib. Control.
**2018**, 25, 255–272. [Google Scholar] [CrossRef] - Nayfeh, A.H.; Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods; Wiley-Vch Verlag: New York, NY, USA, 2008. [Google Scholar]
- Thota, P.; Krauskopf, B.; Lowenberg, M. Interaction of torsion and lateral bending in aircraft nose landing gear shimmy. Nonlinear Dyn.
**2009**, 57, 455–467. [Google Scholar] [CrossRef] - Ghadami, A.; Epureanu, B.I. Forecasting critical points and post-critical limit cycles in nonlinear oscillatory systems using pre-critical transient responses. Int. J. Nonlinear Mech.
**2018**, 101, 146–156. [Google Scholar] [CrossRef] - Thota, P.; Krauskopf, B.; Lowenberg, M. Multi-parameter bifurcation study of shimmy oscillations in a dual-wheel aircraft nose landing gear. Nonlinear Dyn.
**2012**, 70, 1675–1688. [Google Scholar] [CrossRef] - Cheng, L.; Cao, H.; Zhang, L. Two-parameter bifurcation analysis of an aircraft nose landing gear model. Nonlinear Dyn.
**2021**, 103, 367–381. [Google Scholar] [CrossRef] - Zhang, T.; Dai, H. Bifurcation analysis of high-speed railway wheel-set. Nonlinear Dyn.
**2016**, 83, 1511–1528. [Google Scholar] [CrossRef] - Dong, H.; Zeng, J. Normal form method for large/small amplitude instability criterion with application to wheelset lateral stability. Int. J. Struct. Stab. Dyn.
**2014**, 14, 1350073. [Google Scholar] [CrossRef] - Beregi, S.; Takacs, D.; Stepan, G. Bifurcation analysis of wheel shimmy with non-smooth effects and time delay in the tyre–ground contact. Nonlinear Dyn.
**2019**, 98, 841–858. [Google Scholar] [CrossRef] - Wang, N.; Mao, Z.; Ren, C.; Gao, D. Hopf Bifurcation and Sensitivity Analysis of Vehicle Shimmy System. Mech. Sci. Technol. Aerosp. Eng.
**2023**, 42, 559–565. [Google Scholar] - Knothe, K.; Bohm, F. History of stability of railway and road vehicles. Veh. Syst. Dyn.
**1999**, 31, 283–323. [Google Scholar] [CrossRef] - Nath, Y.; Jayadev, K. Influence of yaw stiffness on the nonlinear dynamics of railway wheelset. Commun. Nonlinear Sci. Numer. Simul.
**2005**, 10, 179–190. [Google Scholar] [CrossRef] - Beckers, C.J.J.; Öngüt, A.E.; Verbeek, G.; Fey, R.H.B.; Lemmens, Y.; van de Wouw, N. Bifurcation-based shimmy analysis of landing gears using flexible multibody models. In Nonlinear Structural Dynamics and Damping; Springer: Lisboa, Portugal, 2018. [Google Scholar]
- Yan, Y.; Zeng, J.; Huang, C.; Zhang, T. Bifurcation analysis of railway bogie with yaw damper. Arch. Appl. Mech.
**2019**, 89, 1185–1199. [Google Scholar] [CrossRef] - Yan, Y.; Zeng, J.; Mu, J. Complex vibration analysis of railway vehicle with tread conicity variation. Nonlinear Dyn.
**2020**, 100, 173–183. [Google Scholar] [CrossRef] - Xu, Q. Hopf bifurcation study of wheelset system. Chin. J. Theor. Appl. Mech.
**2021**, 53, 2569–2581. [Google Scholar]

Symbolic | Description | Value | Unit |
---|---|---|---|

$e$ | Stability distance | 0.08 | [m] |

$I$ | Moment of inertia around the axis of the pillar | 0.25 | [kg∙m^{2}] |

$\sigma $ | Tire slack length | 0.1376 | [m] |

$h$ | Half length of tire touching the ground | 0.0517 | [m] |

${K}_{\lambda}$ | Tire lateral stiffness | 2 × 2 × 10^{5} | [N/m] |

${K}_{\phi}$ | Tire torsional stiffness | 2 × 1.2 × 10^{3} | [Nm/rad] |

$C$ | Linear damping coefficient | 10 | [Nms] |

${C}_{3}$ | Nonlinear damping coefficient | 1 | [Nms^{3}] |

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**MDPI and ACS Style**

Ruan, S.; Zhang, M.; Yang, S.; Hu, X.; Nie, H.
Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System. *Aerospace* **2024**, *11*, 104.
https://doi.org/10.3390/aerospace11020104

**AMA Style**

Ruan S, Zhang M, Yang S, Hu X, Nie H.
Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System. *Aerospace*. 2024; 11(2):104.
https://doi.org/10.3390/aerospace11020104

**Chicago/Turabian Style**

Ruan, Shuang, Ming Zhang, Shaofei Yang, Xiaohang Hu, and Hong Nie.
2024. "Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System" *Aerospace* 11, no. 2: 104.
https://doi.org/10.3390/aerospace11020104