Effect of Finite Vessel Stiffness on Transition from Two-Dimensional Liquid Sloshing to Swirling: Reduced-Order Modeling

Abstract

Liquid sloshing in partially filled tanks is rather complex. Thus, reduced-order dynamical models are often used in attempt to describe the dynamics of the contained liquid. One of the most important sloshing phenomena is the transition from two-dimensional to three-dimensional motion, including swirling. This paper addresses a reduced-order model that describes this transition, with one substantial addition—it considers finite stiffness of the vessel itself. Most classical models were obtained under the assumption of infinite stiffness of the vessel and therefore neglected the interaction between the sloshing liquid and the tank structural modes. However, this interaction was proven to be extremely significant. This paper suggests a reduced-order model of the sloshing liquid in a tank with finite stiffness and analyzes the model in conditions of simple horizontal harmonic forcing. The effect of vessel stiffness on the transition from two-dimensional to three-dimensional motion is studied.

Appendix

$$\begin{aligned} A_{11} & = - \frac{{c_{2} \Omega^{2} \left[ {4\left( {\mu + 1} \right)\Omega^{2} + c_{1}^{2} } \right]}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{12} & = - \frac{{\Omega \left\{ {8\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 6\mu c_{1} c_{2} + 2c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} } \right\}}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{13} & = - \frac{{\Omega^{2} \left\{ { - 4\Omega^{2} c_{1} + \left[ {c_{1}^{2} - 4\mu \kappa \left( {1 + \mu } \right)} \right]c_{2} - 4\mu \kappa c_{1} } \right\}}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{14} & = \frac{{2\Omega \left\{ {\left[ {c_{1} c_{2} \left( {1 + \mu } \right) + c_{1}^{2} - 4\mu \kappa } \right]\Omega^{2} + \mu \kappa c_{1} c_{2} } \right\}}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{21} & = - \frac{{\mu c_{2} \Omega^{2} \left[ { - 4\Omega^{2} + c_{1} c_{2} } \right]}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{22} & = \frac{{2\Omega^{3} \mu c_{2} \left[ {\left( {1 + \mu } \right)c_{2} + c_{1} } \right]}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{23} & = - \frac{{4\Omega^{4} c_{1} + \left[ {c_{1} \left( {1 + \mu } \right)c_{2}^{2} + 4c_{2} \mu^{2} \kappa + 4\mu \kappa c_{1} } \right]\Omega^{2} + \mu \kappa c_{1} c_{2}^{2} }}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ A_{24} & = - \frac{{\left\{ {8\Omega^{4} + \left[ {2\left( {1 + \mu } \right)^{2} c_{2}^{2} + 6\mu c_{1} c_{2} + 4c_{1}^{2} - 8\mu \kappa } \right]\Omega^{2} - c_{2}^{2} \left( {2\kappa \mu^{2} - c_{1}^{2} + 2\mu \kappa } \right)} \right\}\Omega }}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \end{aligned}$$

$$\begin{aligned} B_{11} & = - \frac{{\left[ {4\left( {1 + \mu } \right)^{2} c_{2} + 4\mu c_{1} } \right]\Omega^{2} + c_{1}^{2} c_{2} }}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ B_{12} & = - \frac{{2A_{11} }}{{\Omega c_{2} }} \\ \\ B_{21} & = \frac{{2A_{22} }}{{\Omega c_{2} }} \\ \\ B_{22} & = - \frac{{2A_{21} }}{{\Omega c_{2} }} \\ \\ C_{11} & = - \frac{{Fc_{1} A_{22} }}{{\Omega^{2} c_{2} \mu }} \\ \\ C_{21} & = - \frac{{Fc_{1} A_{21} }}{{\Omega^{2} c_{2} \mu }} \\ \\ C_{31} & = - \frac{{2\left[ {\left( {1 + \mu } \right)^{2} c_{2}^{2} + \mu c_{1} c_{2} + 4\Omega^{2} } \right]\Omega F}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \\ \\ C_{41} & = \frac{{\left[ {\left( {1 + \mu } \right)c_{2}^{2} + 4\Omega^{2} } \right]c_{1} F}}{{16\Omega^{4} + \left[ {4\left( {\mu + 1} \right)^{2} c_{2}^{2} + 8\mu c_{1} c_{2} + 4c_{1}^{2} } \right]\Omega^{2} + c_{1}^{2} c_{2}^{2} }} \\ \end{aligned}$$