Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems

We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries characterising the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proven here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported

Analytical determination of eclipse entry and exit points considering a conical shadow and oblate Earth

This paper presents a new analytical procedure to model the umbra generated during an eclipse considering an oblate ellipsoid of rotation as occulting body and a conical shadow. The method is based on purely geometrical considerations and results in the analytical definition of the entry and exit points from the conical shadow starting from the knowledge of the Sun position vector, the occulting body position vector and the orbital elements of the spacecraft orbiting the occulting body. The conical shadow also permits analytical definition of the entry and exit points of the penumbra region, which cannot be defined by using the classic cylindrical approach. Some numerical applications are proposed to test the effectiveness of the analytical formulations and to check the error in the prediction of the time spent in the shadow by the satellite. Finally, a discussion between the new conical shadow model and the classic cylindrical eclipse is carried out to see the improvements introduced by the refined geometry and the effects on space missions focusing on the cumulative error when multiple revolutions are considered

A regularized damage model for structural analyses of concrete dams in the presence of alkali-silica reaction

Alkali-silica reaction is a chemical phenomenon that affects concrete structures built some decades ago and subject to a very wet environment, e.g. dams. The starting point of this work is a bi-phase damage model present in the literature. In general, finite element solutions with damage models for material having a softening behaviuor exhibit a sensitivity to the element size and do not converge to physically meaningful solutions as the mesh is refined. In literature, some regularization techniques have been proposed and the fracture energy one has been implemented in the bi-phase chemo-damage model. The limit of this approach is that the solution remains mesh-dependent, so if the mesh is refined the damage localizes in a band of width fixed by the element size. In this work the nonlocal formulation of this damage model has been developed, validated with simple examples and applied to an existing concrete gravity dam, subject to service loading and affected by the ASR. A comparison between fracture energy regularization approach and nonlocal formulation is performed

Micro-to-macro: astrodynamics at extremes of lengths-scale

This paper investigates astrodynamics at extremes of length-scale, ranging from swarms of future `smart dust' devices to the capture and utilisation of small near Earth asteroids. At the smallest length-scales families of orbits are found which balance the energy gain from solar radiation pressure with energy dissipation due to air drag. This results in long orbit lifetimes for high area-to-mass ratio `smart dust' devices. High area-to-mass hybrid spacecraft, using both solar sail and electric propulsion, are then considered to enable `pole-sitter' orbits providing a polar-stationary vantage point for Earth observation. These spacecraft are also considered to enable displaced geostationary orbits. Finally, the potential material resource available from captured near Earth asteroids is considered which can underpin future large-scale space engineering ventures. The use of such material for geo-engineering is investigated using a cloud of unprocessed dust in the vicinity of the Earth-Sun $L_1$ point to fractionally reduce solar insolation

Orbital dynamics of "smart dust" devices with solar radiation pressure and drag

This paper investigates how perturbations due to asymmetric solar radiation pressure, in the presence of Earth shadow, and atmospheric drag can be balanced to obtain long-lived Earth centred orbits for swarms of micro-scale 'smart dust' devices, without the use of active control. The secular variation of Keplerian elements is expressed analytically through an averaging technique. Families of solutions are then identified where Sun-synchronous apse-line precession is achieved passively to maintain asymmetric solar radiation pressure. The long-term orbit evolution is characterized by librational motion, progressively decaying due to the non-conservative effect of atmospheric drag. Long-lived orbits can then be designed through the interaction of energy gain from asymmetric solar radiation pressure and energy dissipation due to drag. In this way, the usual short drag lifetime of such high area-to-mass spacecraft can be greatly extended (and indeed selected). In addition, the effect of atmospheric drag can be exploited to ensure the rapid end-of-life decay of such devices, thus preventing long-lived orbit debris

Trajectory design and optimisation for lunar transfer

This paper deals with the design and optimization of transfer trajectories from the Earth to the Moon. In particular, the requirements of the ESMO mission have been considered. This mission, currently in its phase A, is completely designed by European students: because of this, the budget must be kept as low as possible. The mission analysis has thus to focus on low-energy transfers, in order to obtain very low cost trajectories. Two different chemical transfers are considered: a trajectory through L1 lagrangian point, considering a restricted three body problem, and a more complex Belbruno transfer, taking into account the presence of the Sun. Some results, from another low-cost lunar mission, are presented