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This book suggests a new common approach to the study of resonance
energy transport based on the recently developed concept of
Limiting Phase Trajectories (LPTs), presenting applications of the
approach to significant nonlinear problems from different fields of
physics and mechanics. In order to highlight the novelty and
perspectives of the developed approach, it places the LPT concept
in the context of dynamical phenomena related to the energy
transfer problems and applies the theory to numerous problems of
practical importance. This approach leads to the conclusion that
strongly nonstationary resonance processes in nonlinear oscillator
arrays and nanostructures are characterized either by maximum
possible energy exchange between the clusters of oscillators
(coherence domains) or by maximum energy transfer from an external
source of energy to the chain. The trajectories corresponding to
these processes are referred to as LPTs. The development and the
use of the LPTs concept a re motivated by the fact that
non-stationary processes in a broad variety of finite-dimensional
physical models are beyond the well-known paradigm of nonlinear
normal modes (NNMs), which is fully justified either for stationary
processes or for nonstationary non-resonance processes described
exactly or approximately by the combinations of the non-resonant
normal modes. Thus, the role of LPTs in understanding and analyzing
of intense resonance energy transfer is similar to the role of NNMs
for the stationary processes. The book is a valuable resource for
engineers needing to deal effectively with the problems arising in
the fields of mechanical and physical applications, when the
natural physical model is quite complicated. At the same time, the
mathematical analysis means that it is of interest to researchers
working on the theory and numerical investigation of nonlinear
oscillations.
This book suggests a new common approach to the study of resonance
energy transport based on the recently developed concept of
Limiting Phase Trajectories (LPTs), presenting applications of the
approach to significant nonlinear problems from different fields of
physics and mechanics. In order to highlight the novelty and
perspectives of the developed approach, it places the LPT concept
in the context of dynamical phenomena related to the energy
transfer problems and applies the theory to numerous problems of
practical importance. This approach leads to the conclusion that
strongly nonstationary resonance processes in nonlinear oscillator
arrays and nanostructures are characterized either by maximum
possible energy exchange between the clusters of oscillators
(coherence domains) or by maximum energy transfer from an external
source of energy to the chain. The trajectories corresponding to
these processes are referred to as LPTs. The development and the
use of the LPTs concept a re motivated by the fact that
non-stationary processes in a broad variety of finite-dimensional
physical models are beyond the well-known paradigm of nonlinear
normal modes (NNMs), which is fully justified either for stationary
processes or for nonstationary non-resonance processes described
exactly or approximately by the combinations of the non-resonant
normal modes. Thus, the role of LPTs in understanding and analyzing
of intense resonance energy transfer is similar to the role of NNMs
for the stationary processes. The book is a valuable resource for
engineers needing to deal effectively with the problems arising in
the fields of mechanical and physical applications, when the
natural physical model is quite complicated. At the same time, the
mathematical analysis means that it is of interest to researchers
working on the theory and numerical investigation of nonlinear
oscillations.
Asymptotic methods belong to the, perhaps, most romantic area of
modern mathematics. They are widely known and have been used in me
chanics, physics and other exact sciences for many, many decades.
But more than this, asymptotic ideas are found in all branches of
human knowledge, indeed in all areas of life. In this broader
context they have not and perhaps cannot be fully formalized.
However, they are mar velous, they leave room for fantasy, guesses
and intuition; they bring us very near to the border of the realm
of art. Many books have been written and published about asymptotic
meth ods. Most of them presume a mathematically sophisticated
reader. The authors here attempt to describe asymptotic methods on
a more accessi ble level, hoping to address a wider range of
readers. They have avoided the extreme of banishing formulae
entirely, as done in some popular science books that attempt to
describe mathematical methods with no mathematics. This is
impossible (and not wise). Rather, the authors have tried to keep
the mathematics at a moderate level. At the same time, using simple
examples, they think they have been able to illustrate all the key
ideas of asymptotic methods and approaches, to depict in de tail
the results of their application to various branches of knowledg-
from astronomy, mechanics, and physics to biology, psychology and
art. The book is supplemented by several appendices, one of which
con tains the profound ideas of R. G.
This book describes significant tractable models used in solid
mechanics - classical models used in modern mechanics as well as
new ones. The models are selected to illustrate the main ideas
which allow scientists to describe complicated effects in a simple
manner and to clarify basic notations of solid mechanics. A model
is considered to be tractable if it is based on clear physical
assumptions which allow the selection of significant effects and
relatively simple mathematical formulations. The first part of the
book briefly reviews classical tractable models for a simple
description of complex effects developed from the 18th to the 20th
century and widely used in modern mechanics. The second part
describes systematically the new tractable models used today for
the treatment of increasingly complex mechanical objects - from
systems with two degrees of freedom to three-dimensional continuous
objects.
This book covers developments in the theory of oscillations from
diverse viewpoints, reflecting the fields multidisciplinary nature.
It introduces the state-of-the-art in the theory and various
applications of nonlinear dynamics. It also offers the first
treatment of the asymptotic and homogenization methods in the
theory of oscillations in combination with Pad approximations. With
its wealth of interesting examples, this book will prove useful as
an introduction to the field for novices and as a reference for
specialists.
This book describes significant tractable models used in solid
mechanics - classical models used in modern mechanics as well as
new ones. The models are selected to illustrate the main ideas
which allow scientists to describe complicated effects in a simple
manner and to clarify basic notations of solid mechanics. A model
is considered to be tractable if it is based on clear physical
assumptions which allow the selection of significant effects and
relatively simple mathematical formulations. The first part of the
book briefly reviews classical tractable models for a simple
description of complex effects developed from the 18th to the 20th
century and widely used in modern mechanics. The second part
describes systematically the new tractable models used today for
the treatment of increasingly complex mechanical objects - from
systems with two degrees of freedom to three-dimensional continuous
objects.
In this book a detailed and systematic treatment of asymptotic
methods in the theory of plates and shells is presented. The main
features of the book are the basic principles of asymptotics and
their applications, traditional approaches such as regular and
singular perturbations, as well as new approaches such as the
composite equations approach. The book introduces the reader to the
field of asymptotic simplification of the problems of the theory of
plates and shells and will be useful as a handbook of methods of
asymptotic integration. Providing a state-of-the-art review of
asymptotic applications, this book will be useful as an
introduction to the field for novices as well as a reference book
for specialists.
In this book a detailed and systematic treatment of asymptotic
methods in the theory of plates and shells is presented. The main
features of the book are the basic principles of asymptotics and
their applications, traditional approaches such as regular and
singular perturbations, as well as new approaches such as the
composite equations approach. The book introduces the reader to the
field of asymptotic simplification of the problems of the theory of
plates and shells and will be useful as a handbook of methods of
asymptotic integration. Providing a state-of-the-art review of
asymptotic applications, this book will be useful as an
introduction to the field for novices as well as a reference book
for specialists.
Asymptotic methods belong to the, perhaps, most romantic area of
modern mathematics. They are widely known and have been used in me
chanics, physics and other exact sciences for many, many decades.
But more than this, asymptotic ideas are found in all branches of
human knowledge, indeed in all areas of life. In this broader
context they have not and perhaps cannot be fully formalized.
However, they are mar velous, they leave room for fantasy, guesses
and intuition; they bring us very near to the border of the realm
of art. Many books have been written and published about asymptotic
meth ods. Most of them presume a mathematically sophisticated
reader. The authors here attempt to describe asymptotic methods on
a more accessi ble level, hoping to address a wider range of
readers. They have avoided the extreme of banishing formulae
entirely, as done in some popular science books that attempt to
describe mathematical methods with no mathematics. This is
impossible (and not wise). Rather, the authors have tried to keep
the mathematics at a moderate level. At the same time, using simple
examples, they think they have been able to illustrate all the key
ideas of asymptotic methods and approaches, to depict in de tail
the results of their application to various branches of knowledg-
from astronomy, mechanics, and physics to biology, psychology and
art. The book is supplemented by several appendices, one of which
con tains the profound ideas of R. G."
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