Frequency analysis and control of sloshing coupled by elastic walls and foundation with smoothed particle hydrodynamics method

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Abstract

Broadband stochastic excitations from environment such as seismic activities, sea waves, wind gusts, and tremors can lead to the terrible damages of tall structures and buildings which usually have a low amount of vibration damping. Tuned liquid dampers (TLD) have been considered for around thirty years regarding their ability for passive suppression and absorb the vibration energy in such structures. In this paper, the fluid-structure interaction of a two-dimensional rectangular tank with elastic side walls and foundation is investigated by the smoothed particle hydrodynamics (SPH) method. First, the implementation of SPH for the frequency analysis of a rectangular container with flexible boundaries, partially filled with liquid, is described. The governing equations of the sloshing of an inviscid liquid in a two-dimensional Cartesian coordinate system coupled with the thin plate vibration on sides are linearized and solved by the SPH method. The sloshing and bulging modes of the system are derived and discussed in this paper. The results of the frequency spectrum of the sloshing tank coupled with the spring-damper system are analyzed and compared with the mechanical equivalent system. Results are validated by analytical benchmark and show the applicability as well as the simplicity of the SPH method for analytical vibration analysis of a fluid-structure coupled system. Then various control method including: Active mass damping (AMD) control designed based on proportional feedback control, the tuned liquid damper (TLD) control based on the analytical model, and a hybrid method of active tuned liquid damper (ATLD) are modeled and compared in a one-story structure. Results demonstrated that although the AMD is more efficient and stable in a specific range of feedback coefficient, the proposed ATLD is reliable in a wider range of control parameters.

Introduction

The shallow fluid motion has been effectively used to control and damp the seismic load effects on building and structures by suppressing the amplitudes of vibration. The idea of utilizing liquid material to settle the roll movements of tanks was presented by Froude in 1861 [1]. From 1950s liquid dampers are used in ships to control the rolling motions [2] and from the 1960s for satellites [3]. In the 1980s Bauer [4] was investigated the suppression of the structural vibration which that novel idea followed by other researchers. The researchers work on damping of wind-induced oscillations through liquid sloshing [5], and practical applications of tuned sloshing damper for wind-induced vibration of the tower [6]. As well, reduction of wind-induced motion utilizing a tuned sloshing damper [7], suppression of wind-induced vibration of a tall building [8], modeling of tuned liquid damper [9], and study of size and shape effects on tuned liquid damper [10].

In the previous years, numerous papers have studied the TLD experimentally and theoretically [[11], [12], [13]]. Sun et al. showed that the nonlinearities of TLD become stronger as the excitation amplitude increases [9]. Since, applying the TLD damping, for the damping value higher than the optimal damping value is not useful. As well they experimentally studied the effects of TLD geometry on suppression performance with various shapes of the circular, ring and rectangular tanks. It was found that for the same characteristic length, the shape of the tank did not affect the performance of TLD [10]. Farid and Gendelman [14] presented an impact model for both the regime of linear and strongly non-linear sloshing regimes with a collision kernel. Tait et al. [15] proposed the nonlinear term to the shallow water equations to simulate tuned liquid dampers (TLD) with slat screens using the method of virtual work. In the weakly damped free-surface model of Dias et al. [16] a viscous correction is added to the ir-rotational pressure (Bernoulli's equation), and the kinematic boundary condition. The multimodal method which comes from Bateman-Luke variational principle is firstly used by Faltinsen et al. [11] in liquid sloshing dynamics analysis. In the multimodal method, the mechanical system is considered as a conservative system with infinite degrees of freedom where the generalized Lagrange coordinates are used to model the nonlinear velocity field. Instead of considering the full Navier-Stokes formulation on the sloshing movement inside a rectangular tank or the frozen fluid method, they consider inviscid incompressible ir-rotational fluid (where tank velocity is much lower compared to the speed of sound in water) as a rigid body with a modified inertia tensor in a formalism provided by the Lukovsky formulas [12].

Moiseev [17] investigated the case of periodic sloshing caused by excitation of the lowest frequency in the steady-state. In limited liquid depth (liquid depth divided by tank length greater than 0.2) especially for the value higher than the critical depth ratio (h/l = 0.3368), Moiseev's asymptotic is not true and secondary resonances can happen. Faltinsen et al. [18] classified the three-dimensional nonlinear sloshing in a square-base tank with finite depth. Their experiments show that the oscillations can be classified into various types of chaos (weak and strong near the resonance mode), square-like (or diagonal at low amplitude excitations), planar and swirling types (in high frequencies which causes secondary resonances). For the rectangular shape, a diagonal-type (squares) wave (no angle to the excitation plane) is produced.

Active structural control has been used for more than three decades in the suppression of vibrations in tall buildings to increase the comfort of building residents [[19], [20], [21]]. As well through the passive control methods, the first tuned liquid damper used in a hybrid way in Tokyo in 1992 [22]. The simplest modeling sloshing phenomena coupled with structure motion shows that liquid sloshing dynamics could be simplified as the spring-mass model [23] especially in the first sloshing mode [[24], [25], [26]]. Based on the linear wave theory, the analytical solution in closed form could be obtained for the phenomena [[26], [27], [28]]. Increasing the water depth [29], immerged mass [30], wave amplitude [31], surface tension [32,33], wave breaking [34,35], and baffle [36] increases sloshing nonlinearity that make the system more complex.

The effectiveness of TLD in dissipating the energy of the system could be measured by its damping ratio. By tuned mass damper (TMD) analogy, the properties of tuned liquid dampers are derived by Sun et al. [37]. The damping factor is typically 0.5% for a liquid [38]. Analytical modeling [[39], [40], [41]] and experimental [[42], [43], [44], [45], [46], [47], [48], [49], [50]] efforts are done to model the damping ratio of TLD. As shown the damping ratio in TLDs increases by wave breaking [42,51], the amplitude of excitation [[45], [46], [47], [48], [49]], and length to height ratio [50]. Some studies are investigating fluid-solid interaction [51,52]. The finite element method shows its efficiency in the solution of such problems [51]. By assumption of inviscid flow, the analytical solution of sloshing is possible [52]. The force reaction examination of the movement of a tank and the fluid contained in it to seismic excitation has gotten expanding consideration in evaluating the safety of huge storing tanks. In various examinations, excitations got from seismic tremor records. Recently, many methods are developed and the coupled sloshing problem has been investigated by many researchers. An elastic wall considered tank can equip with piezoelectric actuators and piezoelectric sensors to respond to external disturbances and internal changes. Piezoelectric actuators wafers have the advantages of compactness, robustness, resistance to humidity and high-temperature environments.

In fluid-structure interaction sometimes just an approximation of fluid effect on the structure is used to solve the coupled problem [53]. Different methods are recommended by researchers to calculate the sloshing phenomena effects. Some of these approaches are based on the analytical solution of fluid and structural modes amplitudes, approximate solutions, and discretization of governing equations. In some of the applications, the fluid forces on solid are neglected [54]. Because of the complex nature of fluid-structure coupling, usually the high number of discretization is needed to solve the vibration attenuation effects of elastic walls [55]. Thus, experts have introduced another approach for fluid-solid introduction problems based on meshless Lagrangian methods. Sloshing mode is a specific fluid potential distribution of a fluid interacting structure, in which the fluid motion rules the system motion [56]. As the well-bulging mode is a specific fluid potential distribution coupled by structure motion, in which the surrounding structure motion dominates the system motion [57,58]. There is some importance for sloshing modes compared to bulging modes. First, they are fundamental modes of a usual system which is important to the response of seismic load. Second, they can absorb seismic load energy in the tuned liquid dampers. Third, sloshing modes are independent of the movement of solid boundary conditions [59,60]. Although, the SPH method was used to simulate the fluid flow motion [[61], [62], [63], [64]], it is not used for frequency analysis of a fluid-structure coupled system yet.

Previous studies are all about the sloshing and bulging modes of fluid motion coupled with beams and plates. However, no works have been reported for rectangular fluid motion modes based on fluid and structural mode calculation with SPH. In this paper, the natural modes of a rectangular tank are derived and the fluid motion is suppressed by a wall harmonic modes. As well, an examination of the hydro elastically coupled movement of a rectangular compartment with versatile dividers mostly loaded up with fluid, and due to a non-stationary irregular excitation is performed. The examination has been completed as an initial stage in the advancement of various control methods. In this study, the idea of ATLD for the structure is developed and used for a one-story structure case. By the aid of existing theories active mass damping AMD control designed based on proportional feedback control, the tuned liquid damper TLD model based on the simplified model considering dominant modes in the linear case, and a hybrid method are presented. In the second step fluid-solid coupling of a rectangular tank with a one-story structure is studied. Finally, the hybrid method (ATLD) is applied and compared with AMD and TLD. The remainder of this paper is organized as follows. Vibration motion equations of the fluid-structure motion are derived in section 2. The structural mode shapes and natural frequencies are presented in this section. In section 3, the coupled motion of a rigid tank with external spring-damper is proposed. Finally, section 4 concludes this paper.

Section snippets

Governing equation

Consider a rectangular tank with the length L, height h and wall thickness of thickness hp. A Cartesian coordinate system is attached to the elastic tank as shown in Fig. 1. Where x is the axis along its length, y is the axis along with its height, and u and v are the displacements in the x and directions, respectively. Laplacian of flow potential over the domain is zeroΔφ=0inΩ,

The top kinetic boundary conditionφt|z=η=gηxX¨

The top kinematic boundary conditionηt=φz|z=η

The bottom

TLD modeling

In this section, the mathematical modeling of the structure and controlling systems are presented and discussed. TLD device required no power and control forces to work properly. The designed TLD here is used to suppress the first natural frequency of structure. Through the horizontal seismic load is considered in one direction. As well the linearized motion is assumed for liquid through the simulation and nonlinear effects are ignored. Fig. 5 presents a schematic of a vessel partially filled

SPH results for natural frequency

Finding a technique for numerically demonstrating sloshing has progressed toward becoming a key role in figuring out what impact the slosh will have on a certain rocket, particularly the right common . To find the natural frequency a system with parameters presented in Table 1 is considered. The relative error of natural frequency is plotted in Fig. 11. By increase of node number, the error is decreased. A comparison of natural sloshing frequency is presented in Table 2.

Structure system identification

In this section, the

Discussion and conclusion

This paper was studied the passive and active control of a sloshing tank, based on coupled sloshing modes. For the first time, the SPH method was developed using SPH kernels then was used to control the sloshing in low frequencies and linear fluid motion. To this end, vibration equations of motion were obtained by combining the energies belonging to the elastic wall and the fluid motion. Natural frequencies and structural mode shapes were computed and presented by solving the eigenvalue

Author contributions section

The work is done by MYAJ and corresponding author is responsible for ensuring that the descriptions are accurate. Since all: Conceptualization Ideas; ethodology; Software; Validation; Formal analysis; Investigation; Resources; Data curation; Writing - original draft; Writing - review & editing; Visualization; Supervision; and Project administration is done by himself solely and the research receive no Fund.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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