1. Dynamics of Aircraft Main Landing Gears Chris Howcroft Academic Supervisors: Bernd Krauskopf Mark Lowenberg Industrial Supervisor: Simon Coggon Research funded by: University of Bristol

2.  Shimmy oscillations represent a significant problem for a variety of rolling systems  Implications for safety / maintenance of such systems  May also occur for aircraft landing gears  Usually appears in the nose gear dynamics but has also been observed for main landing gears.  Motivation for project stems from the need to design shimmy resistant landing gear configurations. Example of aircraft ground tracks following a shimmy event Motivation Project Motivation – Shimmy Phenomenon

3. Modelling Procedure  Capture landing gear dynamics by considering components of lateral, longitudinal and torsional modes  By considering these modes a mathematical model may be derived expressing the free dynamics  Important to consider tyre effects in full model – resultant coupling of modes Main Gear Modes:

4. Von Schlippe Tyre Model May approximate the tyre as a mass-less stretched string of infinite length. -Constant pre tension -Uniform elasticity in lateral direction -No sliding occurs in contact region h L Stiffness force may be integrated over the contact region to give an effective restoring force and torque acting on the tyre

5. General Landing Gear Model Vertical load Lateral tyre force Tyre moment Torsional Mode Lateral Mode Longitudinal Mode Lateral Tyre Deflection

6. Characteristics of Torsional Shimmy

7. Characteristics of Lateral Shimmy

8. Single Parameter Bifurcation Analysis Fz = 200 KN

9. Single Parameter Bifurcation Analysis Fz = 400 KN

10. Two Parameter Continuation

11. Lateral Shimmy Torsional Shimmy Bi-stability No Shimmy

12. Orientation of attachment points  Attachment points not necessarily horizontal and perpendicular to the direction of travel.  May parameterise their orientation using the angles ρ & μ.  By moving the attachment points, the gear geometry may approach that of a nose gear. V

13. Vector Model  More flexible modelling approach  Easy to incorporate changes in geometry as well as extra external forces acting on the system

14. Variation of μ μ = 0° μ = 23° μ = 65°μ = 90° μ = 25° μ = 10°μ = 21° μ = 45°

15. Eigenvalues Longitudinal Torsional Lateral

16. Eigenvalues Longitudinal Torsional Lateral μ = 0° μ = 23° μ = 65°μ = 90° μ = 25° μ = 10°μ = 21° μ = 45°

17. Two Parameter Continuation Hopf curve Fold curve Torus curve

18. Two Parameter Continuation

19. Fz = 500 kN

20. Matlab Simulation

21. Conclusion  Existence of stable periodic solutions – shimmy  Areas of shimmy and no shimmy identified in the (V, F Z ) plane  Vector model developed to easily accommodate changes in geometry  Side stay orientation parameterised by ρ and μ angles  Longitudinal mode comes into dynamics for μ ≠ 0 case  Two parameter continuation highlights region of tri-stability  Confirmed using numerical simulation in parameter region of interest Future project direction:  Study more realistic set of main gear parameters  Effects of free-play in the system, dynamics of shock absorber  Different tyre model  Incorporate flexibility of the wing into the dynamics