Instability of sloshing motion in a vessel undergoing pivoted oscillations

Abstract

-AbstractSuspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearized for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1 : 1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.-October 23, 2014-

Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing

“…A nonlinear normal form is derived close to an internal 1 : 1 resonance, with the energy transfer manifested by a heteroclinic connection, which connects the purely symmetric sloshing modes to the purely antisymmetric sloshing modes. In [43] the dynamic coupling problem is studied for a suspended rectangular container from a single rigid pivoting rod and with interior partially filled with an inviscid fluid. A characteristic equation is derived for the natural frequencies of the coupled system and it is shown that the coupled pendulum sloshing dynamics exhibits unstable solutions if the length of the rigid pole is below some critical value.…”

Liquid sloshing in a horizontally forced vessel with bottom topography

“…In these cases the sloshing fluid produces a force on the vessel walls, causing it to move which hence produces a subsequent force on the fluid. This dynamic coupling effect has been widely studied in homogeneous bottomed vessels (Cooker, 1994;Yu, 2010;Alemi Ardakani and Bridges, 2010;Herczyński and Weidman, 2012;Alemi Ardakani et al, 2012;Turner et al, 2015a) and interesting phenomena such as energy transfer between the sloshing modes has been identified (Turner and Bridges, 2013). The results of this paper suggest that the inclusion of a step or hump topography would decrease the feedback mechanism from the fluid to the vessel as these topographies reduce the size of the fluid amplitudes at the side walls.…”

How to Mitigate Sloshing

“…Some experimental observations of this set-up are reported in [3]. The more realistic, but more complicated, case of a mug in a cradle that can oscillate as a simple pendulum has been modeled in [8]. It involves the added complication of the rotational motion of the mug, but we will see that our model predicts very similar response characteristics.…”

“…These ω n are the frequencies which correspond to the odd sloshing modes in a container without a spring; the even modes would also be included in the solution if waves were generated by a wavemaker at one end of the tank with the other end fixed. These odd modes are also the only ones that occur in the full pendulum model analyzed in [8].…”

“…Note that the function Δ(ω) above would be replaced by D(ω) from equation (3.15) in [8] if we were dealing with the full swinging pendulum problem. Downloaded 04/08/19 to 13.66.222.141.…”