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Meeting	ACGS Committee Meeting 111 - Reno, Nevada - March 2013
Agenda Location	6 SUBCOMMITTEE D – DYNAMICS, COMPUTATIONS, AND ANALYSIS 6.2 Bifurcation Analysis for Subscale Aircraft
Title	Bifurcation Analysis for Subscale Aircraft
Presenter	Mark Lowenberg
Affiliation	Bristol University
Available Downloads*	presentation
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Abstract	The paper highlights recent activities using bifurcation methods to gain insight into nonlinear dynamics of a subscale aircraft model, and also preliminary work on ways in which the approach may help support control law design. The project described is being conducted by PhD candidate Stephen Gill, and is focused on the upset characteristics of the NASA Generic Transport Model (GTM). Bifurcation analysis and the numerical continuation algorithms used to generate bifurcation diagrams are briefly described, as is the upset problem and the GTM model. Bifurcation diagrams for the open-loop model are then used to categorise the types of behaviour exhibited by the GTM over its usable elevator range, in particular those relating to upset. It is shown that there are multiple system attractors and that upset may be associated with a pair of stable steep spiral equilibrium solutions ¡V which arise as a result of a strong asymmetry just below the stall incidence ¡V and two forms of steep spin oscillatory (periodic orbit) solutions at higher angle of attack. Whilst the steep spirals are strong stable attractors that will dominate behaviour as the stick is pulled back, the steep spins are more difficult to acquire. Nevertheless, time histories show how a pilot trying to recover from a steep spiral upset condition using rudder can easily end up in a spin; a bifurcation diagram in rudder deflection explains why this occurs. On the other hand, pushing the stick forward can effect recovery from both the spiral and the spin upset scenarios. Further results depict changes in the underlying dynamical systems structure as centre of gravity is moved fore or aft, and show how this changes the spin attractors in particular and the mechanism for entering them. The technique is then applied to a closed-loop system, using a fixed-gain PI controller designed for the GTM at NASA Langley. This preliminary study focuses on the manner in which the open-loop equilibria are modified when the loop is closed. Given that the spiral mode is marginally unstable but acceptable, the desired symmetric trim solutions are achieved up to an incidence of approx. 21„a, apart from a small-amplitude limit cycle around stall. Furthermore, the steep spiral upset solution branch is transformed by the controller into a gentle helical turning solution. Thus the behaviour is largely improved at low-to-moderate angles of attack. However, with the loop closed, a high frequency large amplitude coupled longitudinal/lateral limit cycle is introduced for incidences above 21„a, whereas the open-loop spiral mode was stable up to around 27„a. These investigations continue and the paper concludes with suggestions for further studies of the closed-loop system.