Nonlinear low-frequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations

Abstract

In the experiments of a water storage cylindrical shell, excited by a horizontal external force of sufficient large amplitude and high frequency, it has been observed that gravity water waves of low frequencies may be generated. This paper intends to investigate this phenomenon in order to reveal its mechanism. Considering nonlinear fluid–structure interactions, we derive the governing equations and the numerical equations describing the dynamics of the system, using a variational principle. Following the developed generalized equations, a four-mode approximation model is proposed with which an experimental case example is studied. Numerical calculation and spectrum analysis demonstrate that an external excitation with sufficient large amplitude and high frequency can produce gravity water waves with lower frequencies. The excitation magnitude and frequencies required for onset of the gravity waves are found based on the model. Transitions between different gravity waves are also revealed through the numerical analysis. The findings developed by this method are validated by available experimental observations.

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part II: Theoretical analysis

“…This can be explained that because the ratio of natural frequencies of two modes is O(e À1 ), differential of the Lagrangian can produce higher coupling order. Results from numerical simulation of the equations in [9] reveal good agreement with experimental observations. The objective of this paper is to further explore analytically the mechanism of the experimental phenomenon through a concise theoretical model.…”

“…A set of nonlinear equations involving four modes interactions is then obtained. Unlikeness Mile's [7] assertion, the resulted mode coupling order in [9] is O(1) in mode interaction equations. This can be explained that because the ratio of natural frequencies of two modes is O(e À1 ), differential of the Lagrangian can produce higher coupling order.…”

“…Ref. [9] gives detailed description about the calculation process. Here we just cite the result equations.…”

“…Noticing that previous researches concentrated only on the water waves in the fluid domain excited by prescribed vessel motions cannot give precise description of this nonlinear water-shell interaction problem, Zhou et al [9] establish a model of nonlinear modal interactions based on variational principle of fluid-structure coupled system to analysis this phenomenon. A set of nonlinear equations involving four modes interactions is then obtained.…”

“…The objective of this paper is to further explore analytically the mechanism of the experimental phenomenon through a concise theoretical model. Based on the theory given in [9], a two mode interaction model is therefore adopted to assist analysis for the major factors affecting nonlinear interactions and the stabilities of the dynamic system. Compared with four-mode approximation used in [9], twomode approximation is more convenient to study mathematically the rules of mode interaction and to make clear the key parameters causing the nonlinear resonance.…”