Nonlinearity of Finite-Amplitude Sloshing in Rectangular Containers

2016 ◽  
Vol 84 (3) ◽  
Author(s):  
Mohammed F. Daqaq ◽  
Yawen Xu ◽  
Walter Lacarbonara
Keyword(s):  
Surface Tension ◽  
High Frequency ◽  
Finite Amplitude ◽  
Internal Resonances ◽  
Small Surface ◽  
High Frequency Mode ◽  
Nonlinear Modes ◽  
Coupled Mode ◽  
Nonlinear Normal ◽  
Mode Response

This paper investigates the influence exerted by small surface tension on the nonlinear normal sloshing modes of a two-dimensional irrotational, incompressible fluid in a rectangular container. To this end, the influence of surface tension on the modal frequencies is investigated by assuming pure slipping at the contact line and a 90 deg contact angle between the fluid surface and the walls. The regions of possible nonlinear internal resonances up to the fifth mode are highlighted. Away from the highlighted regions, the influence of surface tension on the effective nonlinearity of the lowest four modes is studied and used to shed light onto its effect on the softening/hardening behavior of the uncoupled nonlinear modes. Subsequently, the response of the sloshing waves near two-to-one internal resonances is studied. It is shown that, in the vicinity of such internal resonance, the steady-state sloshing response can either contain a contribution from the two interacting modes (coupled-mode response) or only the high-frequency mode (high-frequency uncoupled mode response). The regions where the coupled mode uniquely exists are shown to depend on the surface tension. Moreover, it is demonstrated that such regions may be underestimated considerably when neglecting the influence of the cubic nonlinearities.

Nonlinearity of Finite-Amplitude Waves in Rectangular Containers

2016 ◽  
Author(s):  
Mohammed F. Daqaq ◽  
Yawen Xu ◽  
Walter Lacarbonara
Keyword(s):  
Surface Tension ◽  
Finite Amplitude ◽  
Frequency Mode ◽  
Internal Resonances ◽  
High Frequency Mode ◽  
Nonlinear Modes ◽  
Coupled Mode ◽  
Sloshing Dynamics ◽  
Nonlinear Normal ◽  
Mode Response

This paper investigates the two-dimensional nonlinear finite-amplitude sloshing dynamics of an irrotational, incompressible fluid in a rectangular container. In specific, the paper addresses the influence of surface tension represented by a coefficient, β, and the ratio between the fluid height and the container’s width, represented by h/L, on the nonlinear normal sloshing modes. To achieve this objective, we first study the influence of, β, and, h/L, on the modal frequencies and generate a map in the (h/L, β) parameters’ space to highlight regions of possible nonlinear internal resonances up to the fifth mode. The map is used to characterize the regions where a single uncoupled nonlinear mode is sufficient to capture the response of surface waves. For these regions, we study the influence of surface tension on the effective nonlinearity of the first four modes and illustrate its considerable influence on the softening/hardening behavior of the uncoupled nonlinear modes. Subsequently, we investigate the response of the sloshing waves near internal resonance of the two-to-one type. We show that, in the vicinity of these internal resonances, the steady-state sloshing response can either contain a contribution from the two interacting modes (coupled-mode response) or only the higher frequency mode (un-coupled high-frequency mode response). We show that the regions where the coupled-mode uniquely exists has a clear dependence on the surface tension.

Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

2003 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Richard H. Rand
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Resonance Capture ◽  
Periodic Motions ◽  
Internal Resonances ◽  
Dynamical Mechanism ◽  
Elliptic Orbits ◽  
Weakly Coupled ◽  
Resonant Dynamics ◽  
Nonlinear Normal

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes — NNMs), as well as, asynchronous periodic motions (elliptic orbits — EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Discrete breathers modeling from first principles in graphene and in classical approximation in fcc Ni: Comparison

2019 ◽  
Vol 04 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ivan P. Lobzenko
Keyword(s):  
Molecular Dynamics ◽  
First Principles ◽  
High Frequency ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
First Principles Calculations ◽  
Discrete Breathers ◽  
Points Of View ◽  
Nonlinear Normal ◽  
Dynamics Simulations

Properties of discrete breathers are discussed from two points of view: (I) the ab initio modeling in graphene and (II) classical molecular dynamics simulations in the ace-centered cubic (fcc) Ni. In the first (I) approach, the possibility of exciting breathers depends on the strain applied to the graphene sheet. The uniaxial strain leads to opening the gap in the phonon band and, therefore, the existence of breathers with frequencies within the gap. In the second (II) approach, the structure of fcc Ni supports breathers of another kind, which possess a hard nonlinearity type. It is shown that particular high frequency normal mode can be used to construct the breather by means of overlaying a spherically symmetrical function, the maximum of which coincides with the breather core. The approach of breathers excitation based on nonlinear normal modes is independent of the level of approximation. Even though breathers could be obtained both in classical and first-principles calculations, each case has advantages and shortcomings, that are compared in the present work.

High Frequency Mode Cascades in the ASDEX Upgrade Tokamak

1998 ◽  
Vol 80 (2) ◽  
pp. 293-296 ◽  
Author(s):  
K. Hallatschek ◽  
A. Gude ◽  
D. Biskamp ◽  
S. Günter ◽  
the ASDEX Upgrade Team
Keyword(s):  
High Frequency ◽  
Frequency Mode ◽  
High Frequency Mode

scholarly journals Level measurement for saline with a small surface area using high frequency electromagnetic sensing technique

Measurement ◽  
2017 ◽  
Vol 101 ◽  
pp. 118-125 ◽  
Author(s):  
Tiemei Yang ◽  
Qian Zhao ◽  
Kin Yau How ◽  
Kai Xu ◽  
Mingyang Lu ◽  
...  
Keyword(s):  
Surface Area ◽  
High Frequency ◽  
Small Surface ◽  
Level Measurement ◽  
Electromagnetic Sensing ◽  
Small Surface Area

scholarly journals On the deflection of a liquid jet by an air-cushioning layer

2018 ◽  
Vol 846 ◽  
pp. 711-751 ◽  
Author(s):  
M. R. Moore ◽  
J. P. Whiteley ◽  
J. M. Oliver
Keyword(s):  
Surface Tension ◽  
Equations Of Motion ◽  
Constant Thickness ◽  
Liquid Jet ◽  
Pressure Jump ◽  
Length Scales ◽  
Small Surface ◽  
Numerical Studies ◽  
Impact Problems ◽  
Systematic Derivation

A hierarchy of models is formulated for the deflection of a thin two-dimensional liquid jet as it passes over a thin air-cushioning layer above a rigid flat impermeable substrate. We perform a systematic derivation of the leading-order equations of motion for the jet in the distinguished limit in which the air pressure jump, surface tension and gravity affect the displacement of the centreline of the jet, but not its thickness or velocity. We identify thereby the axial length scales for centreline deflection in regimes in which the air layer is dominated by viscous or inertial effects. The derived length scales and reduced equations aim to expand the suite of tools available for future analyses of the evolution of lamellae and ejecta in impact problems. Assuming that the jet is sufficiently long that tip and entry effects can be neglected, we demonstrate that the centreline of a constant-thickness jet moving with constant axial speed is destabilised by the air layer for sufficiently small surface tension. Expressions for the fastest-growing modes are obtained in both the viscous-dominated air and inertia-dominated air regimes. For a finite-length jet emanating from a nozzle, we show that, in one particular asymptotic limit, the evolution of the jet centreline is akin to the flapping of an unfurling flag above a thin air layer. We discuss the distinguished limit in which tip retraction can be neglected and perform numerical investigations into the resulting model. We show that the cushioning layer causes the jet centreline to bend, leading to rupture of the air layer. We discuss how our toolbox of models can be adapted and utilised in the context of recent experimental and numerical studies of splash dynamics.

scholarly journals Solution selection of axisymmetric Taylor bubbles

2018 ◽  
Vol 843 ◽  
pp. 518-535 ◽  
Author(s):  
A. Doak ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Constant Velocity ◽  
Numerical Scheme ◽  
Solution Space ◽  
Selection Mechanism ◽  
Small Surface ◽  
Taylor Bubble ◽  
Solution Selection ◽  
Selection Of ◽  
Taylor Bubbles

A finite difference scheme is proposed to solve the problem of axisymmetric Taylor bubbles rising at a constant velocity in a tube. A method to remove singularities from the numerical scheme is presented, allowing accurate computation of the bubbles with the inclusion of both gravity and surface tension. This paper confirms the long-held belief that the solution space of the axisymmetric Taylor bubble for small surface tension is qualitatively similar to that of the plane Taylor bubble. Furthermore, evidence suggesting that the solution selection mechanism associated with plane bubbles also occurs in the axisymmetric case is presented.

scholarly journals High frequency mode generation by toroidal Alfvén eigenmodes

2019 ◽  
Vol 26 (7) ◽  
pp. 074501
Author(s):  
Shizhao Wei ◽  
Peiwan Shi ◽  
Liming Yu ◽  
Wei Chen ◽  
Ningfei Chen ◽  
...  
Keyword(s):  
High Frequency ◽  
Frequency Mode ◽  
High Frequency Mode ◽  
Alfven Eigenmodes ◽  
Mode Generation
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