Research Papers

Shimmy of an Aircraft Main Landing Gear With Geometric Coupling and Mechanical Freeplay

[+] Author and Article Information
C. Howcroft

Department of Engineering Mathematics,
University of Bristol,
Bristol BS8 1TR, UK
e-mail: c.howcroft@bristol.ac.uk

M. Lowenberg

Department of Aerospace Engineering,
University of Bristol,
Bristol BS8 1TR, UK
e-mail: m.lowenberg@bristol.ac.uk

S. Neild

Department of Mechanical Engineering,
University of Bristol,
Bristol BS8 1TR, UK
e-mail: simon.neild@bristol.ac.uk

B. Krauskopf

Department of Mathematics,
University of Auckland,
Private Bag 92019,
Auckland 1142, New Zealand
e-mail: b.krauskopf@auckland.ac.nz

E. Coetzee

Future Projects,
Filton BS99 7AR, UK
e-mail: etienne.coetzee@airbus.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 9, 2014; final manuscript received October 16, 2014; published online April 6, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 10(5), 051011 (Sep 01, 2015) (14 pages) Paper No: CND-14-1070; doi: 10.1115/1.4028852 History: Received March 09, 2014; Revised October 16, 2014; Online April 06, 2015

The self-sustained oscillation of aircraft landing gear is an inherently nonlinear and dynamically complex phenomenon. Although such oscillations are ultimately driven from the interaction between the tires and the ground, other effects, such as mechanical freeplay, and geometric nonlinearity, may influence stability and add to the complexity of observed behavior. This paper presents a bifurcation study of an aircraft main landing gear (MLG), which includes both mechanical freeplay and significant geometric coupling, the latter achieved via consideration of a typical side-stay orientation. These aspects combine to produce complex oscillatory behavior within the operating regime of the landing gear, including longitudinal and quasi-periodic shimmy. Moreover, asymmetric forces arising from the geometric orientation produce bifurcation results that are extremely sensitive to the properties at the freeplay/contact boundary. However, this sensitivity is confined to the small amplitude dynamics of the system. This affects the interpretation of the bifurcation results; in particular bifurcations from high amplitude behavior are found to form boundaries of greater confidence between the regions of different behavior given uncertainty in the freeplay characteristics.

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Grahic Jump Location
Fig. 1

The six degrees of freedom used in characterizing the deflected state of the dual-wheel MLG system. ψ, δ, and β denote angular deflections of the gear structure, and η, λL, and λR linear deflections of the main strut and tire contact patches, respectively. Global (X, Y, Z) coordinates are defined relative to the (X, Y) ground-plane (X forward, Z up); (x, y, z) are defined locally with z aligned with the main strut, x perpendicular to the main strut and side-stay, and y chosen to complete the right-handed coordinate system. For the zero rake angle case shown here z = Z.

Grahic Jump Location
Fig. 2

Depiction of the side-stay angles m, p, and the MLG rake angle φ; for the landing gear of this study (m,p,φ)=(40,0,-7) deg

Grahic Jump Location
Fig. 3

Panel (a) is a depiction of the landing gear torque link geometry. Here an apex freeplay of yfp mm translates into a torsional freeplay of ψfp°. Panel (b) shows an example moment, deflection profile of the ψ-DOF for Fz*=1.0 (giving rtlk = 0.477 m) and ε = 0.1 mm.

Grahic Jump Location
Fig. 4

One-parameter bifurcation diagram in V for the MLG system defined by Eqs. (1)–(7) with zero freeplay and fixed Fz*=1

Grahic Jump Location
Fig. 5

Two-parameter bifurcation diagram of the MLG system (1)(7) with zero torque link freeplay. The region enclosed by the dashed line indicates a typical operating envelope of the MLG.

Grahic Jump Location
Fig. 6

Two-parameter bifurcation diagram of the MLG system for ±1 mm of torque link freeplay

Grahic Jump Location
Fig. 7

One-parameter bifurcation diagram in V with Fz*=1 and yfp = 1 mm. The minimum and maximum amplitude of solutions are expressed in ψ (a) and xground (b), while panel (c) shows their frequency. Points of bifurcation along these curves are as indicated in the key in panel (a).

Grahic Jump Location
Fig. 8

Simulation results in ψ, δ, β, and η for (V,Fz*)=(110,1); corresponding frequency spectra are shown to the right of each response

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Fig. 9

One-parameter bifurcation diagram in V for Fz*=0.2 and yfp = 1 mm. The minimum and maximum amplitude of solutions are expressed in ψ (a), xground (b), and frequency (c).

Grahic Jump Location
Fig. 10

Example multifrequency solution branches. Panel (a) shows the one-parameter bifurcation diagram in V for Fz*=0.2, yfp = 1 mm. Oscillations along these branches are shown for constant V = 70, 85, 103, and 120 m/s in panels (b)–(e), respectively, their maximum amplitudes indicated by the corresponding points B–E in panel (a). Frequency spectra are also shown to the right of each time series, sampled over 100 s of settled oscillation.

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Fig. 11

Additional isolated periodic solution branch for Fz*=0.2, yfp = 1 mm. Panel (a) shows an extended time series over 1200 s for V = 120. Panel (b) indicates 6 s of the eventual settled response as well as the corresponding frequency spectrum sampled over 100 s. Panel (c) shows the isolated solution branch to which these oscillations approach with the point Es marking the amplitude of settled oscillation observed in panels (a) and (b).

Grahic Jump Location
Fig. 12

One-parameter diagram for ε = 0.1 mm and a freeplay of ±1 mm. Fz* is the bifurcation parameter and V is fixed at 50 m/s. The boundary of torsional freeplay is indicated by the dashed gray curve.

Grahic Jump Location
Fig. 13

One-parameter bifurcation diagram in Fz* showing the MLG stationary solution (solid line) for a nonsmooth freeplay of ±1 mm. The boundary of torsional freeplay is shown by the dashed gray curve. The black dashed-dotted line indicates the boundaries of the loading range over which the stationary solution rests up against the boundary of torsional freeplay.

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Fig. 14

The Hopf bifurcation curves in the (V,Fz*)-plane of the MLG for a freeplay of yfp = 1 mm and smoothing of ε = 1 × 10−2mm (a), ε = 1 × 10−3mm (b) and the nonsmooth case ε = 0 mm (c). The Hopf bifurcation curves for ε = 0 mm are also shown as thicker curves in panels (a) and (b). Below each two-parameter bifurcation diagram we show the corresponding freeplay stiffness profile with a zoom of the local smoothing at the freeplay boundary.

Grahic Jump Location
Fig. 15

One-parameter bifurcation diagram showing together, the torsional solution branch for ε = 1 × 10−2 mm (branching from the ψ = 0.13 deg solution at V ≈ 196 m/s) and ε = 1 × 10−3 mm (branching at V ≈ 273 m/s). Note that above a maximum ψ of 0.135 deg there is no discernible difference between the amplitude or stability of the two ε cases.



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