Nonlinear Resonances

Author: Shanmuganathan Rajasekar
Publisher: Springer
ISBN: 3319248863
Format: PDF
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This introductory text presents the basic aspects and most important features of various types of resonances and anti-resonances in dynamical systems. In particular, for each resonance, it covers the theoretical concepts, illustrates them with case studies, and reviews the available information on mechanisms, characterization, numerical simulations, experimental realizations, possible quantum analogues, applications and significant advances made over the years. Resonances are one of the most fundamental phenomena exhibited by nonlinear systems and refer to specific realizations of maximum response of a system due to the ability of that system to store and transfer energy received from an external forcing source. Resonances are of particular importance in physical, engineering and biological systems - they can prove to be advantageous in many applications, while leading to instability and even disasters in others. The book is self-contained, providing the details of mathematical derivations and techniques involved in numerical simulations. Though primarily intended for graduate students, it can also be considered a reference book for any researcher interested in the dynamics of resonant phenomena.

Nonlinear Resonance Analysis

Author: Elena Kartashova
Publisher: Cambridge University Press
ISBN: 1139493086
Format: PDF, Mobi
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Nonlinear resonance analysis is a unique mathematical tool that can be used to study resonances in relation to, but independently of, any single area of application. This is the first book to present the theory of nonlinear resonances as a new scientific field, with its own theory, computational methods, applications and open questions. The book includes several worked examples, mostly taken from fluid dynamics, to explain the concepts discussed. Each chapter demonstrates how nonlinear resonance analysis can be applied to real systems, including large-scale phenomena in the Earth's atmosphere and novel wave turbulent regimes, and explains a range of laboratory experiments. The book also contains a detailed description of the latest computer software in the field. It is suitable for graduate students and researchers in nonlinear science and wave turbulence, along with fluid mechanics and number theory. Colour versions of a selection of the figures are available at www.cambridge.org/9780521763608.

Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum

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We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). As a result, we rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.

Nonlinear Resonances of an Idealized Saccular Aneurysm

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Abstract: This paper investigates the occurrence of dynamic instabilities in idealized intracranial saccular aneurysms subjected to pulsatile blood flow and surrounded by cerebral spinal fluid. The problem has been approached extending the original 2D model of Shah and Humphrey (1999) to a 3D framework. The justification for using a 3D formulation arises from the works of Suzuki and Ohara (1978), MacDonald et al. (2000) and Costalat et al. (2011) who showed experimental evidences of intracranial aneurysms with a ratio between wall thickness and inner radius larger that 0.1. Two different material models have been used to describe the mechanical behaviour of the aneurysmal wall: Neo-Hookean and Mooney–Rivlin. To the authors' knowledge, for the first time in literature, the dynamic response of the aneurysm has been analysed using complete nonlinear resonance diagrams that have been obtained from a numerical procedure specifically designed for that purpose. Our numerical results show that, for a wide range of wall thicknesses and both constitutive models considered, the saccular aneurysms are dynamically stable within the range of frequencies associated to the normal heart rates, which confirms previous results of Shah and Humphrey (1999). On the other hand, our results also show that the geometric and material nonlinearities of the problem could bring closer than expected the resonance frequencies of the aneurysm to the frequencies of the pulsatile blood flow.

Contemporary Accelerator Physics

Author: Stephan I. Tzenov
Publisher: World Scientific
ISBN: 9789812794734
Format: PDF, Mobi
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This book adopts a non-traditional approach to accelerator theory. The exposition starts with the synchro-betatron formalism and continues with the linear and nonlinear theories of transverse betatron motion. Various methods of studying nonlinear dynamical systems (the canonical theory of perturbations and the methods of multiple scales and formal series) are explained through examples. The renormalization group approach to studying nonlinear (continuous and discrete) dynamical systems as applied to accelerators and storage rings is used throughout the book. The statistical description of charged particle beams (the BalescuOCoLenard and Landau kinetic equations as well as the Vlasov equation) is dealt with in the second part of the book. The processes of pattern formation and formation of coherent structures (solitons) are also described. Contents: Hamiltonian Formulation of Single Particle Dynamics; Linear Betatron Motion; Nonlinear Resonances of Betatron Oscillations; Canonical Perturbation Theory; Special Methods in Accelerator Theory; Transfer Maps; Statistical Description of Charged Particle Beams; Statistical Description of Non-Integrable Hamiltonian Systems; The Vlasov Equation; Nonlinear Waves and Turbulence in Intense Beams. Readership: Graduate students, academics and researchers in accelerator & experimental physics."