Dry-friction-induced self-excitation of a rectangular liquid-filled tank

Abstract

The self-excitation of spring–mass system subjected to dry friction has been studied for decades because of its interesting and complex nonlinear dynamics behaviors. In this paper, a fluid–structure interaction problem, which describes the dry-friction-induced self-excitation of a rectangular liquid-filled tank, is proposed here about how the liquid sloshing affects the tank’s motion. To study responses of both the tank and liquid in depth, an equivalent pendulum model is developed to consider the coupling effects of structure elasticity, liquid sloshing, and dry friction. Another model based on smoothed particle hydrodynamics (SPH) method is also built for studying the liquid behavior while sloshing. The multidimensional-mode convergence is first analyzed, and then, the parameter analysis on the mode and frequency of the tank–spring–liquid–belt coupled system is analytically and numerically conducted. With the increasing liquid, the first two modes of the system approach the structure frequency and liquid frequency, respectively, at first, and then, the tendency converts for the intersection of the structure frequency and liquid frequency. Based on Lyapunov stability theorem, the periodic motion of the liquid-filled tank, including the stick–slip phase and slip phase, is both caused by Hopf bifurcation of equilibrium point. The sloshing effect results in a comparatively smaller bifurcation point. When there is no liquid or a little liquid in tank, the system experiences a subcritical Hopf bifurcation. If the liquid mass is heavy enough, the subcritical Hopf bifurcation converts to the supercritical Hopf bifurcation. During the self-excitation motion, liquid in tank sometimes sloshes linearly with small-amplitude free-surface deformation based on SPH simulation and sometimes sloshes violently with impacting, turning, breaking, and merging. Results by equivalent model show that the swing range of the equivalent pendulum can even reach ± 25.0° at this time. Hence, the dry-friction-induced self-excitation of the liquid-filled tank with lots of interesting nonlinear dynamics behaviors is tightly parameter-related. Results show that the equivalent pendulum model and SPH model match each other well and have different advantages on handling the nonlinear dynamical behaviors of such systems.

Appendices

Appendix 1

The elements of coefficient matrix A_f of the linearized dynamical equations for the tank–spring–liquid–belt coupled FSI system are:

$$ \begin{aligned} {\varvec{A}}_{\text{f12}} & = 1,\quad {\varvec{A}}_{\text{f21}} = - \frac{k}{m},\quad {\varvec{A}}_{\text{f22}} = \frac{{(M + m)g\left( {\mu_{\alpha } - 3\mu_{\beta } v_{0}^{2} } \right)}}{m} \\ {\varvec{A}}_{\text{f23}} & = \frac{{m_{1} g}}{m},\quad {\varvec{A}}_{\text{f24}} = \frac{{c_{1} }}{{ml_{1} }},\quad {\varvec{A}}_{\text{f34}} = 1 \\ {\varvec{A}}_{\text{f41}} & = \frac{k}{{ml_{1} }},\quad {\varvec{A}}_{\text{f42}} = - \frac{{(M + m)g\left( {\mu_{\alpha } - 3\mu_{\beta } v_{0}^{2} } \right)}}{{ml_{1} }}, \\ {\varvec{A}}_{\text{f43}} & = - \frac{(M + m)g}{{ml_{1} }},\quad {\varvec{A}}_{\text{f44}} = - \frac{{c_{1} (M + m)}}{{mm_{1} l_{1}^{2} }}. \\ \end{aligned} $$

Other elements in matrix A_f are zero.

Appendix 2

The coefficients a in the characteristic equation are:

$$ \begin{aligned} a_{1} & = \frac{{\left[ {c_{1} - m_{1} gl_{1}^{2} \left( {\mu_{\alpha } - 3\mu_{\beta } v_{0}^{2} } \right)} \right] \cdot (M + m)}}{{m_{1} ml_{1}^{2} }} \\ a_{2} & = \frac{{kl_{1}^{2} m_{1} - g\left[ {c_{1} \left( {\mu_{\alpha } - 3\mu_{\beta } v_{0}^{2} } \right) - m_{1} l_{1} } \right] \cdot (M + m)}}{{m_{1} ml_{1}^{2} }} \\ a_{3} & = \frac{{c_{1} k - m_{1} g^{2} l_{1} \left( {\mu_{\alpha } - 3\mu_{\beta } v_{0}^{2} } \right) \cdot (M + m)}}{{m_{1} ml_{1}^{2} }} \\ a_{4} & = \frac{gk}{{ml_{1} }}. \\ \end{aligned} $$

Appendix 3

The coefficients b for the Hurwitz determinant Δ₃ are:

$$ \begin{aligned} b_{1} & = - \frac{{(M + m)^{3} c_{1} g^{4} }}{{m_{1} m^{3} l_{1}^{3} }} \\ b_{2} & = (M + m)g^{2} m_{1} \left[ {\frac{{(M + m)^{2} c_{1}^{2} g}}{{m_{1}^{3} m^{3} l_{1}^{5} }} + (M + m)g\frac{{km_{1} l_{1} + (M + m)m_{1} g}}{{m_{1}^{2} m^{3} l_{1}^{2} }}} \right] \\ & \quad - \frac{{(M + m)^{2} g^{4} }}{{m^{2} l_{1}^{2} }} - \frac{{(M + m)^{2} g^{3} k}}{{m^{3} l_{1} }} + \frac{{(M + m)^{2} c_{1}^{2} g^{2} k}}{{m_{1}^{2} m^{3} l_{1}^{4} }} \\ b_{3} & = \frac{{2(M + m)c_{1} kg^{2} }}{{m_{1} m^{2} l_{1}^{3} }} - c_{1} k\left[ {\frac{{(M + m)^{2} c_{1}^{2} g}}{{m_{1}^{3} m^{3} l_{1}^{6} }} + (M + m)g\frac{{km_{1} l_{1} + (M + m)m_{1} g}}{{m_{1}^{2} m^{3} l_{1}^{3} }}} \right] \\ & \quad - (M + m)^{2} c_{1} g^{2} \frac{{km_{1} l_{1} + (M + m)m_{1} g}}{{m_{1}^{2} m^{3} l_{1}^{4} }} + \frac{{2(M + m)^{2} c_{1} g^{2} k}}{{m_{1} m^{3} l_{1}^{3} }} \\ b_{4} & = (M + m)c_{1}^{2} k\frac{{km_{1} l_{1} + (M + m)m_{1} g}}{{m_{1}^{3} m^{3} l_{1}^{5} }} - \frac{{c_{1}^{2} k^{2} }}{{m_{1}^{2} m^{2} l_{1}^{4} }} - \frac{{(M + m)^{2} c_{1}^{2} gk}}{{m_{1}^{2} m^{3} l_{1}^{5} }}. \\ \end{aligned} $$