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Research Papers

Operational Parameter Study of Aircraft Dynamics on the Ground

[+] Author and Article Information
James Rankin1

Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UKj.rankin@bris.ac.uk

Bernd Krauskopf, Mark Lowenberg

Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK

Etienne Coetzee

Landing Gear Systems, Airbus, Bristol BS99 7AR, UK

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(2), 021007 (Feb 18, 2010) (11 pages) doi:10.1115/1.4000797 History: Received February 24, 2009; Revised July 09, 2009; Published February 18, 2010

The dynamics of passenger aircraft on the ground are influenced by the nonlinear characteristics of several components, including geometric nonlinearities, aerodynamics, and interactions at the tire-ground interface. We present a fully parameterized mathematical model of a typical passenger aircraft that includes all relevant nonlinear effects. The full equations of motion are derived from first principles in terms of forces and moments acting on a rigid airframe, and they include implementations of the local models of individual components. The overall model has been developed from and validated against an existing industry-tested SIMMECHANICS model. The key advantage of the mathematical model is that it allows for comprehensive studies of solutions and their stability with methods from dynamical systems theory, particularly, the powerful tool of numerical continuation. As a concrete example, we present a bifurcation study of how fixed-radius turning solutions depend on the aircraft’s steering angle and center of gravity position. These results are represented in a compact form as surfaces of solutions, on which we identify regions of stable turning and regions of laterally unstable solutions. The boundaries between these regions are computed directly, and they allow us to determine ranges of parameter values for safe operation. The robustness of these results under the variation in additional parameters, specifically, the engine thrust and aircraft mass, are investigated. Qualitative changes in the structure of the solutions are identified and explained in detail. Overall our results give a complete description of the possible turning dynamics of the aircraft in dependence on four parameters of operational relevance.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 7

The bifurcation curves L and H for a light aircraft and for thrust levels of (a) T=14%, (b) T=15.4%, and (c) T=16% are shown in projection onto the (δ,Vx)-plane (first column), onto the (δ,CG)-plane (second column), and onto the (Vx,CG)-plane (third column). Note that the Vx-axis has been reversed in the third column to remain consistent with the surfaces, as plotted in Fig. 6. The black dots in panels (b) represent degenerate Bogdanov–Takens points and in panels (c) two non-degenerate Bogdanov–Takens points. Compare panels (a) and (c) with Fig.  66, respectively.

Grahic Jump Location
Figure 1

Schematic diagram showing relative positions of force elements F∗ acting on the airframe with dimensions defined by l∗ in Table 1. Three projections are shown in the aircraft’s body coordinate system: (a) the (x,z)-plane in panel, (b) the (x,y)-plane in panel, and (c) the (y,z)-plane in panel. The center of gravity position is represented by a checkered circle, the aerodynamic center by a white circle, and the thrust center of each engine by a white square.

Grahic Jump Location
Figure 2

Lateral force Fy plotted against slip angle α as calculated from Eq. 16. The maximum point Fy max that can be generated by the tire occurs at the optimal slip angle αopt.

Grahic Jump Location
Figure 3

Comparison between the mathematical model (Eqs. 1,2,3,4,5,6) (gray curves) and the SIMMECHANICS model (black curves). Panel (a) shows one-parameter bifurcation diagrams for varying steering angle δ and fixed CG=14% and T=19%. There is a single branch of turning solutions; stable parts are solid and unstable parts are dashed. Changes in stability occur at the bifurcation points L1, L3, L4 and H2. The maximum and minimum forward velocity of a branch of periodic solutions originating at H2 are also shown. Panel (b) shows the branch of periodic solutions plotted in the (δ,Vy,Vx)-projection; the (gray) surface was computed from the mathematical model and the individual orbits (black closed curves) on the surface were computed with the SIMMECHANICS model. Panel (c) shows a comparison of the individual periodic orbits at δ=10 deg in the (Wz,Vx)-projection. The corresponding CG-trace of the aircraft in the (X,Y) ground plane is shown in panel (d) with markers indicating the orientation of the aircraft at regular time intervals. In all figures velocities Vx and Vy are measured in m/s, rotational velocity Wz in deg/s, and distances X and Y in m.

Grahic Jump Location
Figure 4

Surfaces of turning solutions in (δ,Vx,CG)-space for a heavy aircraft (as specified in Table 1) and for three fixed values of the thrust: (a) T=16%, (b) T=18%, and (c) T=20%. Stable solutions are black and unstable solutions are gray; limit point bifurcations occur along the thick black curve L and Hopf bifurcations occur along the thick gray curve H; the black dots in panel (c) are Bogdanov–Takens bifurcation points labeled BT. The specific cases plotted in Fig. 5 are shown as dashed curves and labeled CG30 in each panel.

Grahic Jump Location
Figure 5

One-parameter bifurcation curves at fixed CG=30% for varying δ∊(0,10) plotted against Vx for: (a) T=16%, (b) T=18%, and (c) T=20%. Stable solutions are black and unstable solutions are gray. Limit point bifurcations L1 and L2 are marked with solid dots and Hopf bifurcations H1 and H2 are marked with stars.

Grahic Jump Location
Figure 6

Surfaces of turning solutions in (δ,Vx,CG)-space for a light aircraft (as specified in Table 1) and for three fixed values of the thrust: (a) T=12%, (b) T=14%, and (c) T=16%. Stable solutions are black and unstable solutions are gray. Limit point bifurcations occur along the thick black curve L and Hopf bifurcations occur along the thick gray curve H; the black dots in panel (c) are Bogdanov–Takens bifurcation points labeled BT.

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