Resonantly forced surface waves in a circular cylinder
TL;DR: In this paper, a weakly nonlinear, weakly damped response of the free surface of a liquid in a vertical circular cylinder that is subjected to a simple harmonic, horizontal translation is examined by extending the corresponding analysis for free oscillations.
Abstract: The weakly nonlinear, weakly damped response of the free surface of a liquid in a vertical circular cylinder that is subjected to a simple harmonic, horizontal translation is examined by extending the corresponding analysis for free oscillations. The problem is characterized by three parameters, α, β, and d/a, which measure damping, frequency offset (driving frequency–natural frequency), and depth/radius. The asymptotic (t↑∞) response may be any of: (i) harmonic (at the driving frequency) with a nodal line transverse to the plane of excitation (planar harmonic); (ii) harmonic with a rotating nodal line (non-planar harmonic); (iii) a periodically modulated sinusoid (limit cycle); (iv) a chaotically modulated sinusoid. It appears, from numerical integration of the evolution equations, that only motions of type (i) and (ii) are possible if 0.30 < d/a < 0.50, but that motions of type (iii) and (iv) are possible for all other d/a in some interval (or intervals) of β if α is sufficiently small. Only motion of type (i) is possible if α exceeds a critical value that depends on d/a.
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19 May 2005TL;DR: In this article, the authors present a detailed review of liquid sloshing dynamics in rigid containers, including linear forced and non-linear interaction under external and parametric excitations.
Abstract: Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.
920 citations
TL;DR: In this paper, a multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth is presented, where the modality is modelled as a set of modalities.
Abstract: Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth
351 citations
319 citations
TL;DR: In this paper, the authors review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems, and discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems.
Abstract: The authors review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, they discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.
262 citations
TL;DR: In this paper, a modal theory based on an innite-dimensional system of nonlinear ordinary dierential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes is presented.
Abstract: Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Innite tank roof height and no overturning waves are assumed. The modal theory is based on an innite-dimensional system of nonlinear ordinary dierential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an innite-dimensional system of ordinary dierential equations with fth-order polynomial nonlinearity by assuming suciently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in dierent domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l. Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h=l =0 :173, where the previous model by Faltinsen et al. (2000) failed. The new model agrees well with experiments.
216 citations
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
TL;DR: The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics as discussed by the authors.
Abstract: The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincare's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.
189 citations
TL;DR: In this article, the weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio δ, natural frequency ω 0 ) to the planar displacement e l cos ω t (e ⪡ 1) of its point of suspension is examined in a four-dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion.
128 citations
TL;DR: In this paper, the equations of motion for a lightly damped spherical pendulum that is subjected to harmonic excitation in a plane are approximated in the neighborhood of resonance by discarding terms of higher than the third order in the amplitude of motion.
Abstract: The equations of motion for a lightly damped spherical pendulum that is subjected to harmonic excitation in a plane are approximated in the neighborhood of resonance by discarding terms of higher than the third order in the amplitude of motion. Steady-state solutions are sought in a four-dimensional phase space. It is found that: (a) planar harmonic motion is unstable over a major portion of the resonant peak, (b) non-planar harmonic motion is stable in a spectral neighborhood above resonance that overlaps neighborhoods of both stable and unstable planar motions, and (c) no stable, harmonic motions are possible in a finite neighborhood of the natural frequency. The spectral width of these neighborhoods is proportional to the two-thirds power of the amplitude of excitation. The steady-state motion in the last neighborhood is quasisinusoidal (at the forcing frequency) with slowly varying amplitude and phase. The waveform, as determined by an analog computer, is periodic but quite complex.
96 citations
TL;DR: In this article, the two dominant, linearly independent surface-wave modes in a circular cylinder, which differ only by an azimuthal rotation of ½π and have equal natural frequencies, are nonlinearly coupled, both directly and through secondary modes.
Abstract: The two dominant, linearly independent surface-wave modes in a circular cylinder, which differ only by an azimuthal rotation of ½π and have equal natural frequencies, are nonlinearly coupled, both directly and through secondary modes. The corresponding, weakly nonlinear free oscillations are described by a pair of slowly modulated sinusoids, the amplitudes and phases of which are governed by a four-dimensional Hamiltonian system that is integrable by virtue of conservation of energy and angular momentum. The resulting solutions are harmonic in a particular, slowly rotating reference frame. Harmonic oscillations in the laboratory reference frame are realized for three special sets of initial conditions and correspond to a standing wave with a fixed nodal diameter and to two azimuthally rotating waves with opposite senses of rotation. The finite-amplitude corrections to the natural frequencies of these harmonic oscillations are calculated as functions of the aspect ratio d/a (depth/radius). A small neighbourhood of d/a = 0.1523, in which the natural frequency of the dominant axisymmetric mode approximates twice that of the two dominant antisymmetric modes, is excluded. Weak, linear damping is incorporated through a transformation that renders the evolution equations for the damped system isomorphic to those for the undamped system.
96 citations