scholarly journals NONLINEAR DYNAMICS OF LONG-WAVE PERTURBATIONS OF THE INVISCID KOLMOGOROV FLOW

2019 ◽  
Vol 47 (1) ◽  
pp. 64-65
Author(s):  
M.V. Kalashnik ◽  
M.V. Kurgansky
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Oscillations ◽  
Initial Perturbation ◽  
Analytical Formula ◽  
Basic Research ◽  
Elliptic Functions ◽  
Structural Features ◽  
Kolmogorov Flow ◽  
Long Wave ◽  
The Galerkin Method

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).

Instability of surface quasigeostrophic spatially periodic flows

2020 ◽  
Author(s):  
Maksim Kalashnik ◽  
Michael Kurgansky ◽  
Sergey Kostrykin
Keyword(s):  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Classical Mechanics ◽  
Basic Research ◽  
Growth Increment ◽  
Russian Science ◽  
Periodic Flows ◽  
Galerkin Approximations ◽  
Spatially Periodic ◽  
The Galerkin Method

<p>The surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.</p><p>The work was supported by the Russian Foundation for Basic Research (Project 18-05-00414) and the Russian Science Foundation (Project 19-17-00248).</p>

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

2012 ◽  
Author(s):  
Eulher Chaves Carvalho ◽  
Paulo Batista Gonçalves ◽  
Zenon J. G. N. del Prado
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Torsional Stiffness ◽  
Complex Dynamic ◽  
Runge Kutta Method ◽  
The Galerkin Method

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.

Side-Wall Effect on the Long-Wave Instability in Kolmogorov Flow

1998 ◽  
Vol 67 (5) ◽  
pp. 1597-1602 ◽  
Author(s):  
Hiroaki Fukuta ◽  
Youichi Murakami
Keyword(s):  
Side Wall ◽  
Wall Effect ◽  
Wave Instability ◽  
Kolmogorov Flow ◽  
Long Wave

Nonlinear Oscillations of Hyperelastic Annular Membranes With Varying Density

2017 ◽  
Author(s):  
Renata M. Soares ◽  
Paulo B. Gonçalves
Keyword(s):  
Hypergeometric Functions ◽  
Nonlinear Oscillations ◽  
Outer Boundary ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Element Formulation ◽  
The Galerkin Method ◽  
Stretching Ratio ◽  
Annular Membrane ◽  
Membrane Density

This research presents the mathematical modeling for the nonlinear oscillations analysis of a pre-stretched hyperelastic annular membrane with varying density under finite deformations. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean and the variation of the membrane density in the radial direction is investigated. The membrane is first subjected to a uniform radial traction along its outer circumference and the stretched membrane is fixed along the outer boundary. Then the equations of motion of the pre-stretched membrane are derived. From the linearized equations of motion, the natural frequencies and mode shapes of the membrane are obtained analytically. The vibration modes are described by hypergeometric functions, which are used to approximate the nonlinear deformation field using the Galerkin method. The results are compared with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the influence of the stretching ratio and varying density on the linear and nonlinear oscillations of the membrane.

scholarly journals Intrastrand backbone-nucleobase interactions stabilize unwound right-handed helical structures of heteroduplexes of L-aTNA/RNA and SNA/RNA

2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Yukiko Kamiya ◽  
Tadashi Satoh ◽  
Atsuji Kodama ◽  
Tatsuya Suzuki ◽  
Keiji Murayama ◽  
...  
Keyword(s):  
Nucleic Acid ◽  
Nucleic Acids ◽  
Crystal Structures ◽  
Prebiotic Chemistry ◽  
Basic Research ◽  
Structural Features ◽  
Helical Structures ◽  
Base Pairs ◽  
Triplex Structure ◽  
Rna And Dna

Abstract Xeno nucleic acids, which are synthetic analogues of natural nucleic acids, have potential for use in nucleic acid drugs and as orthogonal genetic biopolymers and prebiotic precursors. Although few acyclic nucleic acids can stably bind to RNA and DNA, serinol nucleic acid (SNA) and L-threoninol nucleic acid (L-aTNA) stably bind to them. Here we disclose crystal structures of RNA hybridizing with SNA and with L-aTNA. The heteroduplexes show unwound right-handed helical structures. Unlike canonical A-type duplexes, the base pairs in the heteroduplexes align perpendicularly to the helical axes, and consequently helical pitches are large. The unwound helical structures originate from interactions between nucleobases and neighbouring backbones of L-aTNA and SNA through CH–O bonds. In addition, SNA and L-aTNA form a triplex structure via C:G*G parallel Hoogsteen interactions with RNA. The unique structural features of the RNA-recognizing mode of L-aTNA and SNA should prove useful in nanotechnology, biotechnology, and basic research into prebiotic chemistry.

Nonlinear Vibrations of Rectangular Mooney-Rivlin Membrane Resting on a Nonlinear Elastic Foundation

2014 ◽  
Author(s):  
Renata M. Soares ◽  
Paulo B. Gonçalves
Keyword(s):  
Elastic Foundation ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Incompressible Material ◽  
Mode Shapes ◽  
Element Formulation ◽  
Nonlinear Elastic Foundation ◽  
The Galerkin Method ◽  
Nonlinear Elastic ◽  
Foundation Parameters

The aim of the present work is to investigate the nonlinear vibration response of a pre-stretched rectangular hyperelastic membrane resting on a nonlinear elastic foundation. The membrane is composed of an isotropic, homogeneous and hyperelastic material, which is modeled as a Mooney-Rivlin incompressible material. The elastic foundation is described by a Winkler type nonlinear model with cubic nonlinearity. First the exact solution of the membrane under a biaxial stretch is obtained. Then the equations of motion of the pre-stretched membrane resting on the nonlinear foundation are derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are obtained analytically. Then the natural modes are used to approximate the nonlinear deformation field using the Galerkin method. The results compare well with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the strong influence of the initial stretching ratio and foundation parameters on the linear and nonlinear oscillations and stability of the membrane.

On the Chaotic Vibrations of Electrostatically Actuated Arch Micro/Nano Resonators: A Parametric Study

2015 ◽  
Vol 25 (08) ◽  
pp. 1550106 ◽  
Author(s):  
Farid Tajaddodianfar ◽  
Mohammad Reza Hairi Yazdi ◽  
Hossein Nejat Pishkenari
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Partial Differential Equation ◽  
Beam Theory ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Chaotic Vibrations ◽  
Numerical Tools ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam ◽  
Insight Into

Motivated by specific applications, electrostatically actuated bistable arch shaped micro-nano resonators have attracted growing attention in the research community in recent years. Nevertheless, some issues relating to their nonlinear dynamics, including the possibility of chaos, are still not well known. In this paper, we investigate the chaotic vibrations of a bistable resonator comprised of a double clamped initially curved microbeam under combined harmonic AC and static DC distributed electrostatic actuation. A reduced order equation obtained by the application of the Galerkin method to the nonlinear partial differential equation of motion, given in the framework of Euler–Bernoulli beam theory, is used for the investigation in this paper. We numerically integrate the obtained equation to study the chaotic vibrations of the proposed system. Moreover, we investigate the effects of various parameters including the arch curvature, the actuation parameters and the quality factor of the resonator, which are effective in the formation of both static and dynamic behaviors of the system. Using appropriate numerical tools, including Poincaré maps, bifurcation diagrams, Fourier spectrum and Lyapunov exponents we scrutinize the effects of various parameters on the formation of chaotic regions in the parametric space of the resonator. Results of this work provide better insight into the problem of nonlinear dynamics of the investigated family of bistable micro/nano resonators, and facilitate the design of arch resonators for applications such as filters.

Nonlinear oscillations in warm plasmas with initial velocity perturbations

1976 ◽  
Vol 16 (2) ◽  
pp. 103-118 ◽  
Author(s):  
S. Cuperman ◽  
M. Mond
Keyword(s):  
Present Model ◽  
Nonlinear Oscillations ◽  
Initial Perturbation ◽  
Electron Plasma ◽  
Particle Simulation ◽  
Nonlinear Frequency ◽  
Boundary Integration ◽  
Discrete Particle Simulation ◽  
Space Boundary ◽  
Number Of Particles

The Vlasov equation is solved by a phase-space boundary integration method in order to investigate the nonlinear frequency of an electron plasma mode. Unlike the familiar discrete particle simulation codes in which a limited number of particles may be considered, the present model represents a more realistic plasma case in which a very large number (practically unlimited) of plasma particles evolving in their self-consistent collective field is investigated. The unperturbed collisionless one-dimensional plasma system consists of warm electrons having a water-bag distribution to simulate equilibrium, and of static ions. The initial perturbation is introduced by changing the boundaries of the electron plasma such that υu = υ0 (1 + α.sin kx) and υl = − υ0(1 − α sin kx), υu and υl representing the upper and the lower boundaries, respectively. This corresponds to a standing wave perturbation. The results obtained for different wavelength perturbations λ ≡ 2π/k, and for several perturbation amplitudes α, are presented and discussed.

Nonlinear Dynamics of Electrically-Actuated Carbon Nanotube Resonator

2008 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis
Keyword(s):  
Nonlinear Dynamics ◽  
Carbon Nanotube ◽  
Natural Frequency ◽  
Primary Resonance ◽  
Beam Model ◽  
Harmonic Load ◽  
Shooting Technique ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

This paper presents an investigation into the nonlinear dynamics of a carbon nanotube (CNT) actuated electrically by a DC force and an AC harmonic load. The CNT is described by an Euler Bernoulli beam model that accounts for the system nonlinearities due to mid-plane stretching and electrostatic forcing. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic response of the CNT. The static deflection of the CNT and its pull-in voltage are calculated and validated by comparing them to published results. It was found that mid-plane stretching has a major impact on the pull-in prediction of CNT. Dynamic analysis is conducted to explore the nonlinear oscillation of the CNT near its fundamental natural frequency (primary resonance) and near one half, twice, and three times its natural frequency (secondary resonances). The nonlinear analysis is carried out using a shooting technique combined with the Floquet theory to capture periodic orbits and analyze their stability. The results show that these resonances can lead to complex nonlinear dynamics phenomena such as hysteresis, dynamic pull-in, hardening and softening behaviors, and frequencies bands with an inevitable escape from a potential well.

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