Shock and Vibration

Volume 2017 (2017), Article ID 6209205, 17 pages

https://doi.org/10.1155/2017/6209205

## Experimental Study on Variation Rules of Damping with Influential Factors of Tuned Liquid Column Damper

^{1}State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300350, China^{2}Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China^{3}State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China^{4}Beijing Spacecrafts, Beijing 100094, China

Correspondence should be addressed to Lixin Xu

Received 2 May 2017; Revised 16 August 2017; Accepted 26 September 2017; Published 26 October 2017

Academic Editor: Felice Ponzo

Copyright © 2017 Yang Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A tuned liquid column damper (TLCD) is a more effective form of passive control for structural vibration suppression and may be promising for floating platform applications. To achieve good damping effects for a TLCD under actual working conditions, factors that influence the damping characteristics need to be identified. In this study, the relationships between head loss coefficients and other factors such as the total length of the liquid column, opening ratio, Reynolds number, Kc number, and horizontal length of the liquid column were experimentally investigated. By using a hydraulic vibration table, a vibration test system with large-amplitude motion simulation, low-frequency performance, and large stroke force (displacement) control is devised with a simple operation and at low cost. Based on the experimental method of uniform design, a series of experimental studies were conducted to determine the quantitative relationships between the head loss coefficient and other factors. In addition, regression analyses indicated the importance of each factor affecting the head loss coefficient. A rapid design strategy of TLCD head loss coefficient is proposed. This strategy can help people conveniently and efficiently adjust the head loss coefficient to a specified value to effectively suppress vibration.

#### 1. Introduction

The dynamic response of tall structures such as high-rise buildings and ocean platforms is highly apparent under external loads [1–3]. The vibration of such structures can be suppressed by strengthening the structure and increasing its size; however, this leads to a significant increase in its cost. To overcome this problem, in recent years, additional dampers have been widely applied to suppress vibrations. Dampers are used in high-rise buildings to reduce the impact of wind and earthquake loads and in offshore platforms, to reduce the effect of waves.

Thus far, few studies have focused on the motion suppression of floating platforms in complex operating environments. Structural vibration control methods can be categorized into three types: active, semiactive, and passive. Passive control is usually used to suppress the vibration of structures by dissipating and absorbing energy. Passive control dampers are of three main types: tuned mass damper (TMD), tuned liquid damper (TLD), and tuned liquid column damper (TLCD). Of these, a TLCD, which was developed based on the TLD, is the most effective.

The main body of the TLCD is a U-shaped water tunnel that consists of two vertical columns connected by a horizontal column filled with liquid, usually water. An orifice plate is installed at the center of the horizontal column; it changes the damping effect by adjusting the discharge passing through. When the main structure vibrates, a part of the energy is transferred to the liquid column, which absorbs the same energy through its movement. At the same time, the pressure difference between the two vertical liquid columns produces a damping force that suppresses the vibration of the main structure. Although few studies have investigated the application of TLCDs to floating platforms, many existing structures on such platforms, such as oil containers, fresh water containers, and even floating tanks, could be modified to incorporate TLCDs. In addition, given that TLCDs have low manufacturing and installation costs as well as relatively low maintenance requirements, they might actually be greatly advantageous for applications to floating platforms.

Sakai et al. [4] first proposed TLCDs in 1989. Hochrainer [5] presented a detailed derivation of their working principle and applied a bang-bang control scheme based on linear optimal control to reduce transient vibrations. Hitchcock et al. [6–8] optimized the original TLCDs by varying their cross section to overcome their narrow frequency band. They also studied the damping effect of several TLCDs and its influence on the vibration frequency range and head loss coefficient. Lee et al. [9–11] built a structural model on a shaking table with acceleration control and experimentally implemented a TLCD to investigate the control of the response of building structures excited by earthquakes. By comparing the liquid column amplitude and energy consumption between a TLD and a TLCD with the same size and damping effect, they found that the natural vibration frequency of the TLCD has a greater influence on the liquid column amplitude, leading to higher energy consumption for the main structure. In addition, by integrating the characteristics of both TLD and TLCD, they designed a bidirectionally tuned liquid column and sloshing damper that suppressed vibrations along the two principal axes of a structure. Chaiviriyawong et al. [12] used a computational fluid dynamics model to study the variation of the flow velocity in the elbow between horizontal and vertical liquid columns of different widths. Chan and Ding [13] conducted structural experiments to compare the damping effects of TLCDs with liquid columns of different lengths and tilt angles. To make TLCDs more effective in reducing structural vibrations, Al-Saif et al. [14] modified a TLCD by placing a coated steel ball as a moving orifice inside the horizontal section of the damper instead of the orifice plate and disturbed the flow so as to improve the absorber’s attenuation performance. In recent years, TLCDs of different forms have been developed as their range of applications has broadened. Huo and Li [15] assessed the application of a TLCD to the control system of a jacket platform and analyzed the control performance of a circular TLCD on suppressing the coupled torsion vibration of an offshore platform under waves and earthquake loads. Lee et al. [16, 17] were the first to apply a TLCD in a floating platform and provide experimental verification. Moreover, they compared the effects of two different installation arrangements: on the floating platform and underwater.

For the TLCD to have good damping effects, the functional relationships that determine the damping characteristics must be identified. The most direct way to do so is to adjust the relevant parameters of the orifice plate. For a given size and conditions such as external excitation frequency and quality, Chen and Chao [18] proposed a method for calculating the optimal damping ratio for a TLCD that has the optimum damping effect on the main structure. Additionally, the influence of the optimal damping ratio on the harmonic response was discussed for different effective length ratios. Shum and Xu [19, 20] developed a closed-form optimal solution scheme for a TLCD-structure system and determined its optimal damping ratio for suppressing harmonic vibrations. Chakraborty et al. [21] discussed the optimization of the damping coefficient and other parameters with a reasonable maximum amplitude of liquid oscillation. Yalla and Kareem [22] varied the inclination angle of the orifice plate to determine the optimal damping ratio for an orifice plate with a narrow slit and confirmed that the damping ratio of the TLCD strongly influences the amplitude of oscillations in the main structure. Lee et al. [11] discussed the correlation between the liquid motion amplitude and different external excitation amplitudes under different frequency ratios and damping coefficients and analyzed the relationships between the excitation amplitude and the parameters related to the TLCD. Furthermore, some researchers performed experimental tests. Wu et al. [23–26] proposed an analytical method for determining the optimal damping ratio with variable cross sections and found that it is inversely proportional to the external excitation amplitude. According to the two-degree-of-freedom (2-DOF) motion equation and transfer relationship between the main structure and the TLCD, they established an experimental measurement method for the head loss coefficient based on the equivalent linear damping item and motion amplitude. The head loss for a TLCD is mainly produced by oscillation flow in the U-shaped tube. When oscillation occurred along the direction of tube flow, considering the Reynolds number, Kc number, and relative roughness of the cylinder surface, Sarpkaya and Isaacson [27–30] plotted the variation in the curves of the drag coefficient and inertial force coefficient for the oscillation flow of a circular cylinder after numerous experiments under different conditions. They also performed extensive analyses based on changes in the cylinder force and energy transmission. Carberry [31] compared and summarized the advantages and disadvantages of existing forced oscillation tests. They investigated the effect of the Reynolds number on the results of the experimental test when the oscillation is perpendicular to the direction of tube flow. Morse and Williamson [32] confirmed that the fluid force of self-excited vibrations is similar to that of forced oscillation under conditions in which the amplitude, frequency, and Reynolds number are equal. Note that these damping characteristic experiments are only applicable to a cylinder. The damping characteristics differ for orifice plates having a shape other than a circle or square. A similar problem is the vortex-induced vibration in marine engineering [33, 34]. Thus far, few studies have focused on the orifice plate characteristics and optimization damping ratio of TLCDs. In addition, the nonlinear damping term is usually simplified to be linear. Hence, several issues could be studied further: the functional relationship between the head loss coefficient and the orifice plate properties; the degree to which the opening rate and other TLCD parameters affect head loss; and guidelines for quickly adjusting the head loss coefficient to the specified value during engineering practice.

Experimental studies on the damping characteristics of a TLCD can be performed using an offshore engineering model basin or a vibration test system on land. The high construction cost of the former makes it impossible to repeat experiments. In addition, such devices cannot directly control the size of the external force applied to the measured model, and the wave force acting on the structure can only be indirectly changed by adjusting the wave height and period. Thus, measuring the head loss coefficient of a TLCD on a floating platform model is extremely difficult. An on-land vibration test system is a good alternative because of its low cost and easy control. On the other hand, such systems are always designed for civil structures or machines, which differ from floating platforms in some respects such as natural vibration period, amplitude, and external environment. Because existing onshore test systems cannot effectively and inexpensively simulate the working conditions on a floating platform, it is desirable to design an alternative device with large stroke force control components for simulating large-amplitude motions at low frequency.

Current onshore vibration test systems mainly include electromagnetic vibration and electrohydraulic servo test systems. An electromagnetic vibration test system, which is costly and shows low total harmonic distortion (THD), is mainly used for high-frequency (tens to thousands of Hertz) simulation. Kim et al. [35] optimized the dynamic performance of electromagnetic exciters by using finite element analysis. By using a numerical simulation to design the control scheme for an electromagnetic actuator with a 10 kN load, Li et al. [36] found that high-precision control of the actuator signal could be achieved by adaptive inverse control theory. Zhu et al. [37] designed a new type of micro electromagnetic vibration exciter whose resonance frequency could be controlled without changing the total damping. Oliver and Priya’s [38] four-bar magnet geometry increased the output power of the electromagnetic vibration exciter and the accuracy of predicting the optimal load resistance.

The other major type of onshore system is an electrohydraulic servo test system, which produces various types of oscillatory waves through a dynamic loading device and simulates the vibration for an experimental subject on a rigid surface. Despite its speed and high power, its precision is slightly less than that of the electromagnetic vibration test system. Moreover, its running and maintenance costs are quite high. Conte and Trombetti [39] found a potentially strong dynamic interaction between the oil column in the actuator and the payload when their frequencies were similar in a uniaxial servohydraulic shaking table system. Stehman and Nakata [40] used force feedback to ensure stable motion of the shaking table in a perfectly balanced position to optimize acceleration control. Jianjun et al. [41] discussed the use of the least mean square (LMS) adaptive filtering algorithm to control the amplitude and phase of the acceleration signal so as to suppress higher harmonics and reduce distortion [41]. To eliminate adverse effects on the acceleration of vibration control, Dozono et al. [42] introduced adaptive filtering compensation (AFC) to the control theory of shaking tables. To reduce the maintenance and operation cost, Ye et al. [43] invented a pneumatic shaking table that used compressed air instead of electricity as the transmission medium and changed the loading mode from hydraulic to pneumatic actuation. This type of shaking table provides more stable low-frequency harmonic frequencies (above 3 Hz).

An investigation of a shaking table test system reveals that an electromagnetic vibration system is usually used to study the strength and elastoplastic properties of the sample. Conducting experiments with large displacements is difficult because the maximum amplitude of the electromagnetic vibration system is only 5–35 mm. However, electrohydraulic servo test systems suffer from some drawbacks. The input signal is always the acceleration or displacement, which is seldom loaded directly. Nonlinear factors such as the delay of the servo valve actuator, compression of liquid in the actuator, and closeness of the actuator greatly influence precision, especially for a small electric hydraulic servo vibration system. The distortion of the response curve is a serious impediment to processing experimental data at low frequencies (0.2–1 Hz) because the high-order harmonic generation cannot be eliminated. In this study, the frequency of the shaking table must be 0.5–1.5 Hz to simulate the horizontal movement accurately. Moreover, the maximum displacement amplitude is set above 100 mm to simulate the resonance phenomenon between the shaking table and the TLCD. Hence, conducting experiments with an electrohydraulic servo test system is extremely difficult.

As discussed above, it is challenging to find an available test system simultaneously equipped with large-amplitude motion simulation, low-frequency performance, and large stroke force control. In this study, a vibration test system that satisfies the above requirements is independently designed to achieve harmonic wave force loading based on a small hydraulic shaking table with a custom-made loading system and a device for changing stiffness. In the low-frequency range of 0.5–1.5 Hz, the maximum displacement amplitude is above 100 mm without any high-order harmonic generated during the loading process. The THD is less than 0.5%. According to the experimental data, the head loss coefficient of the nonlinear damping term of the TLCD is calculated based on the energy transmission between the TLCD and the experimental vibration system. The functional relationship between the head loss coefficient and the orifice plate properties or the TLCD’s relevant parameters is also investigated. This paper is divided into two parts: the first part describes the rationale for independently designing an available test system with large-amplitude motion simulation, low-frequency performance, and large stroke force control, and the second part discusses the application of a uniform design to determine the functional relationship between the head loss coefficient and the natural frequency of the TLCD, orifice plate parameters, Reynolds number or Kc number in the liquid column of the TLCD, or ratio of the horizontal column to the vertical column. Finally, a type of rapid design strategy is proposed for the TLCD head loss coefficient under working conditions.

#### 2. Design of Experimental System

As noted earlier, a TLCD, a type of passive vibration control device, is widely used in structures such as high-rise buildings. A TLCD suppresses motion in a structure by transferring part of the kinetic energy from the main structure to the liquid column, which causes the latter to undergo severe sloshing and absorb energy. In addition, part of the energy is consumed by damping effects in the TLCD. As shown in Figure 1, is the total length of the liquid column, which can be expressed as ; vertical columns length is denoted by ; horizontal column length is denoted by ; and head loss coefficient is denoted by . By using the Lagrange equation and energy method, the 2-DOF motion equations of the TLCD fluid column and the main structure are established as follows: