China
Africa
Explore chapters and articles related to this topic
Stability Theory
Published in Mayer Humi, Introduction to Mathematical Modeling, 2017
In many important practical applications, one has to consider dynamical systems, Equation (4.71), with periodic motions. That is, some of the trajectories “allowed” by the system satisfy either exactly or approximately x(t + T) = x(t), for some T > 0. In these cases the steady state of the system is represented by a one dimensional curve in Rn (usually referred to as a limit cycle) rather than a point. Perhaps one of the most important examples of such a system relates to the motion of Earth and the other planets around the Sun. This motion is subject to perturbations due to the gravitational field of the other planets and other celestial bodies in the system. It follows then that the answer to the question of stability of this periodic motion is of paramount importance to the ve + y existence of life on Earth. However, periodic motion is important also in many mundane applications, e.g. waves of various types (water waves, electromagnetic waves, etc). In this section we present some elementary techniques, to determine the existence and stability of these limit cycles. However, in general the existence and stability of limit cycles for a given dynamical system is a rather difficult mathematical problem.
Study of fringe effects of a two-rod capacitor embedded in a medium in order to deduce its permittivity
Published in European Journal of Environmental and Civil Engineering, 2022
Xavier Chavanne, Alain Bruère, Jean-Pierre Frangi
Moreover, additional effects modifying the factor g can arise due to practical constraints. Thus, because electrodes are inserted in the medium at a desired depth, whereas the measurement unit is placed at medium surface, a pair of leads - along protective tubes around leads - is required between them. Leads act as a capacitor in parallel with the electrode one. Tubes also may introduce some electric perturbations. An alternative set-up to leads would be to connect directly the electronic unit to electrodes. However, the configuration would require to dig a trench in the medium, to assure through it power supply and data transfer, and would create stronger perturbations of medium and its inner flows. By comparison use of tubes with same geometry as electrodes - except height - reduces the perturbations. Moreover, the alternative set-up would still produce the main fringe effect, with a pattern more complex than in the case of two-rod geometry.
Centre of pressure velocity shows impairments in NCAA Division I athletes six months post-concussion during standing balance
Published in Journal of Sports Sciences, 2020
Moira K. Pryhoda, Kevin B. Shelburne, Kim Gorgens, Aurélie Ledreux, Ann-Charlotte Granholm, Bradley S. Davidson
Our results show differences in the COP ML component of velocity in the double-leg stance and the COP AP component of velocity in the tandem stance between non-concussed and concussed athletes, indicating that concussed athletes shift to the use of a hip control strategy. Winter et al. (1996) explain these mechanisms based on foot position where, in the double-leg stance, the ML component of COP is under hip control, while the AP component is under ankle control. In the tandem stance, this relationship is opposite, with the ML component under ankle control and the AP component under hip control (Winter et al., 1996). In general, the use of an ankle control strategy in quiet stance is dominant in healthy reference populations. The ankle control strategy corrects for small perturbations in the centre of gravity by using ankle plantar and dorsiflexors to keep the centre of gravity within the base of support. When ankle control is ineffective, and larger perturbations of the centre of mass occur, hip control using abductors and adductors is employed. Since the feet are narrow in both of these stances as prescribed by the BESS test, athletes may be even more inclined to use a hip control strategy rather than an ankle control strategy.
A multiscaling-based semi-analytic orbit propagation method for the catalogue maintenance of space debris
Published in Journal of Spatial Science, 2020
Bin Li, Jizhang Sang, Jinsheng Ning
Generally, approaches to orbit propagation can be broadly categorized as analytic methods, semi-analytic methods and numerical methods. The analytic methods solve the equations of motion directly, in which simplification of the perturbation forces is done in order to obtain an analytic solution to the equations of motion. Therefore, the analytic methods can compute the orbital elements at any epoch without consuming too much time, but they are not accurate enough due to the simplifications made in the dynamic forces acting on space objects (Brouwer 1959, Kozai 1959). So there are limitations when using them. For example, the analytic SGP4 (Simplified General Perturbations Version 4) model, which provides the basis to propagate the TLEs (two-line elements) in the NORAD catalogue, only considers simplified gravity and drag (Hoots and Roehrich 1980). The 7-day maximum prediction error could reach several kilometres for a space object with an orbit altitude of 700 km (Sang and Bennett 2014). The numerical methods, considering high-fidelity force models to solve the equations of motion, are highly accurate, but this advantage is shadowed by the high computational burden of the small-step numerical integration of the osculating equations of motion. Taking the 11th-order Cowell numerical propagator for example, the 7-day orbit propagation could consume about 10 s on a normal laptop (Dell Computer with i5 processor and 4 GB RAM) for a single target. Thus, the numerical methods are not suitable for the orbit computations when huge amounts of debris need batch processing.