Lock-in Problem for Parallel Rotor-router Walks

The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al.(2003) proved that a single rotor-router walk on any graph with m edges and diameter $D$ stabilizes to a traversal of an Eulerian circuit on the set of all 2m directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most 2mD steps. The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the conjecture that a system of k \textgreater{} 1 parallel walks also stabilizes with a period of length at most $2m$ steps. In this work we disprove this conjecture, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of $O(m^4 D^2 + mD \log k)$ and $O(m^5 k^2)$ on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of $k = 2$ walks

Dynamical response of the "GGG" rotor to test the Equivalence Principle: theory, simulation and experiment. Part I: the normal modes

Recent theoretical work suggests that violation of the Equivalence Principle might be revealed in a measurement of the fractional differential acceleration $\eta$ between two test bodies -of different composition, falling in the gravitational field of a source mass- if the measurement is made to the level of $\eta\simeq 10^{-13}$ or better. This being within the reach of ground based experiments, gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in low orbit around the Earth is likely to provide a much better accuracy. We report on the progress made with the "Galileo Galilei on the Ground" (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test. In the present and following paper (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation -in particular its normal modes (Part I) and rejection of common mode effects (Part II)- can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining quantitative agreement with the available experimental data on the frequencies of the normal modes, and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus.Comment: Accepted for publication by "Review of Scientific Instruments" on Jan 16, 2006. 16 2-column pages, 9 figure

Demonstration of a coupled floating offshore wind turbine analysis with high-fidelity methods

This paper presents results of numerical computations for floating off-shore wind turbines using, as an example, a machine of 10-MW rated power. The aerodynamic loads on the rotor are computed using the Helicopter Multi-Block flow solver developed at the University of Liverpool. The method solves the Navier–Stokes equations in integral form using the arbitrary Lagrangian–Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform are computed using the Smoothed Particle Hydrodynamics method, which is mesh-free and represents the water and floating structures by a set of discrete elements, referred to as particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the results are presented in this paper. The importance of coupling is assessed and the loosely coupled algorithm used is described in detail alongside the obtained results

Investigation of Anti-Phase Asymmetric Quiet Rotor Technology

The future of urban air mobility has a well-known tall pole challenge in the form of community acceptance which largely comes from the noise. This paper presents a proposed anti-phase rotor technology that could reduce noise sources such as blade vortex interaction noise. The anti-phase rotor technology includes a rotor design with various anti-phase alternating trailing edge patterns and a rotor design with an asymmetric blade tip. Four small-scale anti-phase rotors are fabricated by 3D printing for acoustic measurements conducted in a low-speed open-circuit wind tunnel to assess the effectiveness of the proposed anti-phase rotor technology. Preliminary test results appear to be promising and indicate that the anti-phase rotor designs could be a practical means of reducing blade vortex interactions and noise. The four tested anti-phase rotor designs have peak acoustic performance depending on the RPM and thrust which suggests improved performance through design optimization could be achieved for specific mission requirements

Control of rotordynamic instability in a typical gas turbine's power system

The effect of rotor internal friction on the system's stability was studied when operated above the first critical speed. This internal friction is commonly caused by sliding press fits or sliding splines. Under conditions of high speed and low bearing damping, these systems will occassionally whirl at a frequency less than the shaft's rotational speed. This subsynchronous precession is a self excited phenomenon and stress reversals are created. This phenomenon was observed during engine testing. The reduction of spline friction and/or the inclusion of squeeze film damping have controlled the instability. Case history and the detail design of the squeeze film dampers is discussed

CFD investigation of a complete floating offshore wind turbine

This chapter presents numerical computations for floating offshore wind turbines for a machine of 10-MW rated power. The rotors were computed using the Helicopter Multi-Block flow solver of the University of Glasgow that solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform were computed using the Smoothed Particle Hydrodynamics method. This method is mesh-free, and represents the fluid by a set of discrete particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the loosely coupled algorithm used is described in detail alongside the obtained results

Study of hybrid air vehicles stability using computational fluid dynamics

This paper uses Computational Fluid Dynamics to predict aerodynamic damping of airships or hybrid air vehicles. This class of aircraft is characterised by large lifting bodies combining buoyancy and circulatory lift. Damping is investigated via forced oscillations of the vehicle in pitch and yaw. The employed method is verified using data for lighter than air vehicles. The use of fins and stabilisers was found to be beneficial. The rear part of the body was dominated by separated flow that containedmore frequencies than the forcing frequency imposed on the body. The final design is seen to be dynamically stable across a range of conditions for small pitch angles