56,078 research outputs found
Application of bifurcation methods for the prediction of low-speed aircraft ground performance
The design of aircraft for ground maneuvers is an essential part in satisfying the demanding requirements of the aircraft operators. Extensive analysis is done to ensure that a new civil aircraft type will adhere to these requirements, for which the nonlinear nature of the problem generally adds to the complexity of such calculations. Small perturbations in velocity, steering angle, or brake application may lead to significant differences in the final turn widths that can be achieved. Here, the U-turn maneuver is analyzed in detail, with a comparison between the two ways in which this maneuver is conducted. A comparison is also made between existing turn-width prediction methods that consist mainly of geometric methods and simulations and a proposed new method that uses dynamical systems theory. Some assumptions are made with regard to the transient behavior, for which it is shown that these assumptions are conservative when an upper bound is chosen for the transient distance. Furthermore, we demonstrate that the results from the dynamical systems analysis are sufficiently close to the results from simulations to be used as a valuable design tool. Overall, dynamical systems methods provide an order-of-magnitude increase in analysis speed and capability for the prediction of turn widths on the ground when compared with simulations. Nomenclature co = oleo damping coefficient, N s2 =m2 cz = tire vertical damping coefficient Fco = damping force in oleo due to the orifice,
Zero modes in magnetic systems: general theory and an efficient computational scheme
The presence of topological defects in magnetic media often leads to normal
modes with zero frequency (zero modes). Such modes are crucial for long-time
behavior, describing, for example, the motion of a domain wall as a whole.
Conventional numerical methods to calculate the spin-wave spectrum in magnetic
media are either inefficient or they fail for systems with zero modes. We
present a new efficient computational scheme that reduces the magnetic
normal-mode problem to a generalized Hermitian eigenvalue problem also in the
presence of zero modes. We apply our scheme to several examples, including
two-dimensional domain walls and Skyrmions, and show how the effective masses
that determine the dynamics can be calculated directly. These systems highlight
the fundamental distinction between the two types of zero modes that can occur
in spin systems, which we call special and inertial zero modes. Our method is
suitable for both conservative and dissipative systems. For the latter case, we
present a perturbative scheme to take into account damping, which can also be
used to calculate dynamical susceptibilities.Comment: 64 pages, 15 figure
Lyapunov exponents for conservative twisting dynamics: a survey
Finding special orbits (as periodic orbits) of dynamical systems by
variational methods and especially by minimization methods is an old method
(just think to the geodesic flow). More recently, new results concerning the
existence of minimizing sets and minimizing measures were proved in the setting
of conservative twisting dynamics. These twisting dynamics include geodesic
flows as well as the dynamics close to a completely elliptic periodic point of
a symplectic diffeomorphism where the torsion is positive definite . Two
aspects of this theory are called the Aubry-Mather theory and the weak KAM
theory. They were built by Aubry \& Mather in the '80s in the 2-dimensional
case and by Mather, Ma{\~n}{\'e} and Fathi in the '90s in higher dimension. We
will explain what are the conservative twisting dynamics and summarize the
existence results of minimizing measures. Then we will explain more recent
results concerning the link between different notions for minimizing measures
for twisting dynamics: their Lyapunov exponents; their Oseledet's splitting;
the shape of their support. The main question in which we are interested is:
given some minimizing measure of a conservative twisting dynamics, is there a
link between the geometric shape of its support and its Lyapunov exponents? Or
: can we deduce the Lyapunov exponents of the measure from the shape of the
support of this measure? Some proofs but not all of them will be provided. Some
questions are raised in the last section.Comment: 28 page
Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs
Operator splitting methods allow to split the operator describing a complex
dynamical system into a sequence of simpler subsystems and treat each part
independently. In the modeling of dynamical problems, systems of (possibly
coupled) differential-algebraic equations (DAEs) arise. This motivates the
application of operator splittings which are aware of the various structural
forms of DAEs. Here, we present an approach for the splitting of coupled
index-1 DAE as well as for the splitting of port-Hamiltonian DAEs, taking
advantage of the energy-conservative and energy-dissipative parts. We provide
numerical examples illustrating our second-order convergence results
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Constraining the Solution to the Last Parsec Problem with Pulsar Timing
The detection of a stochastic gravitational-wave signal from the
superposition of many inspiraling supermassive black holes with pulsar timing
arrays (PTAs) is likely to occur within the next decade. With this detection
will come the opportunity to learn about the processes that drive
black-hole-binary systems toward merger through their effects on the
gravitational-wave spectrum. We use Bayesian methods to investigate the extent
to which effects other than gravitational-wave emission can be distinguished
using PTA observations. We show that, even in the absence of a detection, it is
possible to place interesting constraints on these dynamical effects for
conservative predictions of the population of tightly bound supermassive
black-hole binaries. For instance, if we assume a relatively weak signal
consistent with a low number of bound binaries and a low black-hole-mass to
galaxy-mass correlation, we still find that a non-detection by a simulated
array, with a sensitivity that should be reached in practice within a few
years, disfavors gravitational-wave-dominated evolution with an odds ratio of
30:1. Such a finding would suggest either that all existing astrophysical
models for the population of tightly bound binaries are overly optimistic, or
else that some dynamical effect other than gravitational-wave emission is
actually dominating binary evolution even at the relatively high
frequencies/small orbital separations probed by PTAs.Comment: 14 pages, 8 figure
How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models
Invariant manifolds are important constructs for the quantitative and
qualitative understanding of nonlinear phenomena in dynamical systems. In
nonlinear damped mechanical systems, for instance, spectral submanifolds have
emerged as useful tools for the computation of forced response curves, backbone
curves, detached resonance curves (isolas) via exact reduced-order models. For
conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and
their reduced dynamics provide a way to identify nonlinear amplitude-frequency
relationships in the form of conservative backbone curves. Despite these
powerful predictions offered by invariant manifolds, their use has largely been
limited to low-dimensional academic examples. This is because several
challenges render their computation unfeasible for realistic engineering
structures described by finite-element models. In this work, we address these
computational challenges and develop methods for computing invariant manifolds
and their reduced dynamics in very high-dimensional nonlinear systems arising
from spatial discretization of the governing partial differential equations. We
illustrate our computational algorithms on finite-element models of mechanical
structures that range from a simple beam containing tens of degrees of freedom
to an aircraft wing containing more than a hundred-thousand degrees of freedom
The Maintenance of Conservative Physical Laws Within Data Assimilation Systems
In many data assimilation applications, adding an error to represent forcing to certain dynamical equations may be physically unrealistic. Four-dimensional variational methods assume either an error in the dynamical equations of motion (weak constraint) or no error (strong constraint). The weak-constraint methodology proposes the errors to represent uncertainties in either forcing of the dynamical equations or parameterizations of dynamics. Dynamical equations that represent conservation of quantities (mass, entropy, momentum, etc.) may be cast in an analytical or control volume flux form containing minimal errors. The largest errors arise in determining the fluxes through control volume surfaces. Application of forcing errors to conservation formulas produces non-physical results (generation or destruction of mass or other properties), whereas application of corrections to the fluxes that contribute to the conservation formulas maintains the physically realistic conservation property while providing an ability to account for uncertainties in flux parameterizations. The results suggest that advanced assimilation systems must not be liberal in applying errors to conservative equations. Rather systems must carefully consider the points at which the errors exist and account for them correctly. Though careful accounting of error sources is certainly not an entirely new idea, this paper provides a focused examination of the problem and examines one possible solution within the 4D variational framework
Basic Types of Coarse-Graining
We consider two basic types of coarse-graining: the Ehrenfests'
coarse-graining and its extension to a general principle of non-equilibrium
thermodynamics, and the coarse-graining based on uncertainty of dynamical
models and Epsilon-motions (orbits). Non-technical discussion of basic notions
and main coarse-graining theorems are presented: the theorem about entropy
overproduction for the Ehrenfests' coarse-graining and its generalizations,
both for conservative and for dissipative systems, and the theorems about
stable properties and the Smale order for Epsilon-motions of general dynamical
systems including structurally unstable systems. Computational kinetic models
of macroscopic dynamics are considered. We construct a theoretical basis for
these kinetic models using generalizations of the Ehrenfests' coarse-graining.
General theory of reversible regularization and filtering semigroups in
kinetics is presented, both for linear and non-linear filters. We obtain
explicit expressions and entropic stability conditions for filtered equations.
A brief discussion of coarse-graining by rounding and by small noise is also
presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by
filtering. A talk given at the research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena," University of
Leicester, UK, August 24-26, 200
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