Many of the kinematic and dynamic concepts relating to rotational motion have been generalized for N -dimensional rigid bodies. In this paper a new derivation of the generalized Euler equations is presented. The development herein of the N -dimensional rotational equations of motion requires the introduction of a new symbol, which is a numerical relative tensor, to relate the elements of an N × N skew-symmetric matrix to a vector form. This symbol allows the Hamel coefficients associated with general N -dimensional rotations to be computed. Using these coefficients, Lagrange's equations are written in terms of the angular-velocity components of an N -dimensional rigid body. The new derivation provides a convenient vector form of the equations, allows the study of systems with forcing functions, and allows for the sensitivity of the kinetic energy to the generalized coordinates.
Researchers at the NASA Langley ResearchCenter are developing the Airborne Precision Spacing (APS) concept to increase runway arrival throughput at capacity-constrained airports. Under APS operations, arrival capacity is increased by improving the precision of inter-arrival spacing at the runway threshold. Flight crews achieve this improved precision with the assistance of a new flight-deck system, which allows spacing operations to commence even while the aircraft and its lead are on different arrival routes to the runway. However, the increases in traffic volume that could be enabled by APS operations could also elevate noise concerns at busy airports.Noise-and fuel-efficiency concerns have independently motivated the air traffic management community to investigate Continuous Descent Approaches (CDAs) as alternatives to traditional arrival route profiles. However, uncertainties associated with CDA operations can cause runway capacity to be sacrificed. For this reason, CDA routes have only been implemented during lowdensity traffic operations.In this paper we report on current research into the integration of these two advancements, by employing APS operations to merge and space aircraft that are flying CDA arrival routes. Results obtained to date indicate that, while retaining the benefits of both techniques may be unachievable, APS operations can achieve narrow distributions of inter-aircraft spacing errors at the runway threshold for CDA routes. Analysis of the data also indicates that the spacing error distribution may be sensitive to inaccuracies in modeling the CDA trajectories for the aircraft and its lead.
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