Nonlinear Responses of Sloshing in Square Tanks Subjected to Horizontal Random Ground Excitation

Abstract. Nonlinear responses of the predominant two sloshing modes in a square tank have been investigated when the tank is subjected to horizontal, narrow-band random ground excitation. Galerkin’s method is applied to derive the modal equations of motion for nonlinear sloshing. Then the Monte Carlo simulation is used to calculate the mean square responses of these two modes. These to modes are nonlinearly coupled with each other, known as ‘autoparametric interaction’. The responses differ significantly from those of the corresponding linear model, depending on the characteristics of the narrow-band ground excitation such as the bandwidth, center frequency and the intensity. In addition, it is found that the direction of the excitation is a significant factor in predicting the mean square responses.


Introduction
Sloshing dynamics is one of the most important issues in mechanical, civil, marine and aeronautical engineering. Housener [1] presented a linear sloshing model to investigate the responses of free surfaces in partiallyfilled liquid tanks subjected to horizontal, seismic excitation. It is well known that sloshing at large amplitudes exhibits nonlinear behaviour, thus nonlinear models have been developed to obtain more accurate results. A comprehensive book on nonlinear sloshing dynamics was compiled by Ibrahim [2]. Ground-breaking studies on nonlinear sloshing behaviour in cylindrical tanks subjected to horizontal, and/or vertical harmonic excitation were theoretically and experimentally examined by Hutton [3] and Abramson et al. [4]. Threedimensional sloshing in square tanks subjected to horizontal, harmonic excitation was theoretically investigated and their results were compared with experimental data [5,6]. Few examples of nonlinear sloshing behaviour under random excitation exist. However, Sakata et al. investigated a cylindrical tank under random base excitation using modal equations of motion for sloshing [7]. Responses of sloshing in a rectangular tank were also investigated using the finite element method [8]. Furthermore, nonlinear responses of elastic structures with cylindrical tanks subjected to vertical random excitation were also investigated [9,10].
The present paper investigates the system in which the square tank is subjected to horizontal random excitation using the model of reference [6]. Galerkin's method is employed to derive the nonlinear modal equations of motion for sloshing. These modal equations are solved using the Monte Carlo simulation and the mean square responses of the predominant two sloshing modes are calculated when the liquid tank is subjected to horizontal, narrow-band ground random excitation. The influences of the bandwidth and center frequency of the random excitation and the deviation angle of the tank are examined.    Figure 1 shows a model for the theoretical analysis. A nearly square tank with length l and breadth w is partially filled with liquid to the level h. The Cartesian coordinate system O-xyz is fixed to the tank where the xy-plane coincides with the undisturbed liquid surface. The tank is horizontally subjected to random ground excitation x g (t) and the direction of excitation deviates from the tank length by angle . In the theoretical analysis, the liquid is assumed to be a perfect fluid; hence the velocity potential (x, y, z, t) can be introduced. P(x, y, z, t) is the fluid pressure,  is the fluid density, and(x, y, t) is the liquid elevation at position (x, y) in the tank. The following dimensionless quantities are introduced:

Equations of motion
The ground excitation x g (t) is assumed to be generated from the linear shaping filter as follows: ( Here g is the acceleration of gravity and p ij represents the natural frequency of (i, j) sloshing mode. Figures 2(a) and 2(b) show the shapes of (1, 0) and (0, 1) sloshing modes, respectively. Their nodal lines coincide with the y-and xaxes, respectively. All primes " ′ " in Eq. (1) will hereafter be omitted for simplicity, although the quantities are still dimensionless in the theoretical analysis and results. Laplace's equation and Euler's energy equation for the fluid motion are expressed in the dimensionless form, respectively: , sin cos . The boundary conditions for the fluid velocity at the tank walls and bottom are: In addition, the kinematic boundary condition at the liquid free surface is: Because at the liquid free surface, the boundary condition for Eq. (5) is:

Modal equations of motion for sloshing
Galerkin's method is used to derive modal equations of motion for sloshing.  and  are assumed in terms of the eigenfunctions which can be obtained from the corresponding linear system, as follows: in which represent eigenfunctions: where m and n are integers. Note that ij  in Eq. (10) represent dimensionless quantities given by Eqs. (1) and (2). a ij (t) and b ij (t) in Eqs. (9a,b) are unknown functions of time. The coordinates x and y in Eqs. (4) and (7) are expanded in terms of the eigenfunctions of Eq. (10): where the coefficients r i0 and r 0j are determined by the method used in the previos paper [11].  is introduced as a bookkeeping parameter to determine the approximate solutions when two sloshing modes (1, 0) and (0, 1) predominantly appear. Therefore, the orders of a ij (t), b ij (t), x g , and the system parameters are assumed: Equations (6) and (7) Because (1, 0) mode is directly excited, b 10 and  x oscillate violently. Furthermore (0, 1) mode is indirectly excited because the two modes are nonlinear coupled and b 01 and  y intermittently oscillate. This is known as "autoparametric interaction." . Note that linear viscous damping terms    modes, it appears at slighlty higher values than .

Influence of Bandwidth
where the symbols S n (n=1, 2, …, 36) are constants defined by the system parameters and their perfect expressions are omitted here. Because the nonlinear terms of b 10 Figure 7 shows the time histories at =0.98 in figure   6. It can be seen that b 01 oscillates at higher amplitudes than b 10 . When the value of  increases from 0 to 45, even though (1, 0) mode receives more energy from the ground excitation, its mean square value is not usually larger than that of (0, 1) mode. When the bandwidth is narrow, this result is similar to that of a system under harmonic excitation, where multi-valued response curves are observed [6].

Conclusions
The mean square responses of the predominant sloshing modes in a square tank have been investigated when subjected to horizontal, narrow-band random ground excitation. The results are summarized as follows: 1. When the bandwidth is narrow and =0 autoparametric interaction occurs and the mean square responses of (1, 0) mode under direct excitation are decreased by the occurrence of (0, 1) mode depending on the center frequency.
2. Increasing the bandwidth results in less autoparametric interaction and flatter simulation results of the mean square responses.
3. When 0<<45, although both modes are directly excitied, (1, 0) mode receives more energy from the ground excitation. However, its mean square values are not usually larger than those of (0, 1) mode.
For further work, the influence of the intensity of random ground excitation and liquid level should be investigated as well as the risk of overspill, most likely at two opposing corners of the tank.