One of the lesser¹ known concerns about commercial aircraft is their stability on the ground during taxiing（滑行）, takeoff, and landing. During these processes, planes must maintain stability under various operating conditions. However, in some situations, the aircraft landing gear displays unwanted oscillations（振动）, which are referred to as shimmy oscillations. In a paper published last month in the SIAM Journal on Applied² Dynamical Systems, authors Chris Howcroft, Bernd Krauskopf, Mark Lowenberg, and Simon Neild study the dynamics³ of aircraft landing gear using nonlinear models. The dynamics of landing gear shimmy and the wheel-ground interaction are fundamentally nonlinear.

 

"Shimmy oscillations of aircraft landing gear have long been a problem, and their prediction and prevention remains⁴ an ongoing⁵ challenge in landing gear design," explains author Chris Howcroft. "The issue is that a landing gear may display the desired behavior during ground take-off/landing manoeuvres over several hundred or so flights, but then suddenly oscillate given just the right -- or rather, the wrong -- conditions."

 

Fortunately, mathematical models provide cost-effective ways to study the dynamics of the main landing gear (MLG) and determine the types of oscillations that may occur under different conditions. "The work we conducted clarifies under which conditions shimmy oscillations can be encountered in the MLG of a representative midsize passenger aircraft. We identified different types of shimmy oscillations and showed where they occur," says Howcroft.

 

"Having the right mathematical model is really the key," he adds. "Actual testing is extremely expensive; however, nonlinear analysis methods are very well suited to identifying these hard-to-find dynamics. They may also be employed to determine the shimmy characteristics before the aircraft has actually been built."

 

The model can provide insights not only into aircraft operation, but also design features, and can aid in adjusting both for optimum stability.

 

Aircraft landing gear supports the weight of the aircraft during landing and ground operations. In addition to their wheels, landing gear also have shock absorbing equipment or "shock struts⁶" as well as brakes, retraction⁸ mechanisms⁹, controls, and structural¹⁰ entities¹¹ that attach the gear to the aircraft.

 

The model in the paper takes into account tire-contact dynamics and the orientation¹² of the side-stay, the part of the aircraft that supports the shock strut⁷. It characterizes the motion of the system in terms of dynamics of the MLG, which are expressed as three degrees of freedom: rotation¹³ about the main strut, and in-plane and out-of-plane motion with respect to the plane of main strut and side-stay. After determining the dynamics for the simplest geometric case, where the side-stay is perpendicular¹⁴ to the direction of travel, the authors use the model to study different side-stay orientations¹⁵.

 

"For the specific case of MLG, we developed a nonlinear and fully¹⁶ parameterised model that allowed us to map out how its dynamics depends on operational parameters¹⁷, such as aircraft velocity¹⁸ and loading, and design parameters describing the geometry of the landing gear," explains Howcroft. "In contrast to the more traditional approach of performing large numbers of simulations, this was achieved by employing advanced tools from dynamical systems that track solutions and stability changes in parameters directly."

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