Two-dimensional global manifolds of vector fields

We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle

Influence of Variable Side-Stay Geometry on the Shimmy Dynamics of an Aircraft Dual-Wheel Main Landing Gear

Commercial aircraft are designed to fly but also need to operate safely and efficiently as vehicles on the ground. During taxiing, take-off, and landing the landing gear must operate reliably over a wide range of forward velocities and vertical loads. Specifically, it must maintain straight rolling under a wide variety of operating conditions. It is well known, however, that under certain conditions the wheels of the landing gear may display unwanted oscillations, referred to as shimmy oscillations, during ground maneuvers. Such oscillations are highly unwanted from a safety and a ride-comfort perspective. In this paper we conduct a study into the occurrence of shimmy oscillations in a main landing gear (MLG) of a typical midsize passenger aircraft. Such a gear is characterized by a main strut attached to the wing spar with a side-stay that connects the main strut to an attachment point closer to the fuselage center line. Nonlinear equations of motion are developed for the specific case of a two-wheeled MLG configuration and allow for large angle deflections within the geometrical framework of the system. The dynamics of the MLG are expressed in terms of three degrees of freedom: torsional motion, in-plane motion, and out-of-plane motion (with respect to the side-stay plane). These are modeled by oscillators that are coupled directly through the geometric configuration of the system as well as through the tire/ground interface, which is modeled here by the von Schlippe stretched string approximation of the tire dynamics. The mathematical model is fully parameterized and parameters are chosen to represent a generic (rather than a specific) landing gear. In particular, the positions of the attachment points are fully parameterized so that any orientation of the side-stay plane can be considered. The occurrence of shimmy oscillations is studied by means of a two-parameter bifurcation analysis of the system in terms of the forward velocity of the aircraft and the vertical force acting on the gear. The effect of a changing side-stay plane orientation angle on the bifurcation diagram is investigated. We present a consistent picture that captures the transition of the two-parameter bifurcation diagram as a function of this angle, with a considerable complexity of regions of different types of shimmy oscillations for intermediate and realistic side-stay plane orientations. In particular, we find a region of tristability in which stable torsional, in-plane, and out-of-plane shimmy oscillations coexist

Crocheting the Lorenz manifold

You have probably seen a picture of the famous butterfly-shaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s T-shirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the two-dimensional stable manifold of the origin of the Lorenz system, which we call the Lorenz manifold for short. This surface organizes the dynamics in the three-dimensional phase space of the Lorenz system. It is invariant under the flow (meaning that trajectories cannot cross it) and essentially determines how trajectories visit the two wings of the Lorenz attractor. We have been working for quite a while on the development of algorithms to compute global manifolds in vector fields and have computed the Lorenz manifold up to considerable size. Its geometry is very intriguing and we explored different ways of visualizing it on the computer [6, 9]. However, a real model of this surface was still lacking. During the Christmas break 2002/2003 Hinke was relaxing by crocheting hexagonal lace motifs when Bernd suggested: “Why don’t you crochet something useful? ” The algorithm we developed ‘grows ’ a manifold in steps. We start from a small disc in the stable eigenspace of the origin and add at each step a band of a fixed width. In other words, at any time of the calculation the computed part of the Lorenz manifold is a topological disc whose outer rim is (approximately) a level set of the geodesic distance from the origin. What we realized then and there is that the mesh generated by our algorithm can directly be interpreted as chrochet instructions! After some initial experimentation, the first model of the Lorenz manifold was 1 Osinga & Krauskopf Chrocheting the Lorenz manifold

A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant