Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation

Abstract

Nonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.

Nonlinear energy transfer between fluid sloshing and vessel motion

“…In that case both the pure modes have the exact same stability properties. Secondly the problem of energy transfer is not considered in the analysis in Feng & Sethna (1989), Faltinsen et al (2003) or Ikeda et al (2012). The analysis of energy transfer is made apparent in the present paper by showing that the orbits of the normal form (1.2) can be projected onto a spheroid in three dimensions.…”

“…The most well-known context is sloshing in three dimensions when the horizontal cross-section of the vessel is square or almost square. This configuration has been studied by Feng & Sethna (1989);Faltinsen et al (2003); Ikeda et al (2012); see also §9.2.3 of Faltinsen & Timokha (2009). There are two significant differences from the analysis here.…”

Response regimes in equivalent mechanical model of moderately nonlinear liquid sloshing

“…This analytical method describes both regular and irregular liquid sloshing motion, and managed to obtain irregular solutions that were described by Chester and Bones [24], [25] and numerically by Frandsen [3]. In the work done by Ikeda et al [26], [27] on nonlinear sloshing responses in square tank under obliquely horizontal excitation, super-position of two modes sloshing motion is combined to yield a swirling motion.…”

“…Equivalent mechanical models for linear small-amplitude sloshing were based on pendulums and mass-spring systems were studied for different excitations and tank shapes [1], [38], [39]. However, since nonlinear sloshing regimes cannot be described by the linear models, major attempts were made to formulate nonlinear reduced order equivalent mechanical models [7], [8], [10], [26], [40], [41] that may predict and describe nonlinear responses observed experimentally. For instance, nonlinear rotary sloshing in a scale model of Centaur propellant tank at low fill level was modeled by combination of ordinary linear pendulum and a spherical pendulum (i.e.…”

An efficient computational procedure for the dynamic analysis of liquid storage tanks

“…An important aspect of this formulation is its suitability for both time-history and frequency analysis of fluids with free surface. More recently, an increased number of even more complex procedures have been proposed, for example taking into account nonlinear sloshing due to large free surface motions [21][22][23][24] and including identification of damping effects introduced at the tank walls due to viscosity effects in the thin interface layer [25]. However, implementation of these procedures is almost exclusively relegated to scientific and research professional environments due to the complexity of the formulations and the high level of expertise required for their implementation.…”