Development and design of a deep tank damper with submerged suspended plate

Abstract

An appendage, in the form of a submerged suspended plate (SSP) in a liquid-containing deep tank, is proposed for converting functional liquid storage tanks into passive tank dampers for structural vibration mitigation. The oscillating frequency of the SSP is designed to be tuned to the structural frequency so that the tank damper is independent of detuning effects that could be caused by liquid level fluctuations. Further, the impulsive, rather than the sloshing, liquid mass is utilized here. This alleviates the disadvantage of low energy dissipation associated with the relatively smaller proportion of sloshing liquid mass of a deep tank. The theoretical model and working mechanism of the deep tank damper with submerged suspended plate (DTD-SSP) are developed and the expression for the tuning frequency derived. A time-domain formulation for the structure-DTD-SSP system is presented and the design and effectiveness of the DTD-SSP for an example structure under base excitation examined. Further, the performance sensitivity to tuning ratio and comparison with the traditional tuned mass damper (TMD) are presented. Results indicate that with the proposed design, liquid storage tanks installed for other functional requirements can be converted into DTD-SSPs that can achieve significant response reductions comparable with that of TMDs.

Keywords

structural vibration control deep tank damper submerged suspended plate impulsive liquid mass base excitation

Introduction

The effectiveness of the dynamic vibration absorber (DVA) to enhance structural performance under environmental loading is well-established. A DVA consists of a secondary mass with resilient and dissipative elements and is designed so that its natural frequency is tuned, that is, maintained close to the frequency of the structural mode to be controlled. Depending on the basic constitution, DVAs are divided into two categories, namely, (a) the tuned mass damper (TMD), and (b) the tuned liquid damper (TLD).

In the TMD, typically, the secondary mass is a solid one that is connected to the primary structure through external spring and damping elements. The TMD has its root in the Fraham’s absorber, developed in the early 20th century (Frahm, 1911). Ormondroyd and Den Hartog (1928) and Den Hartog (1947) are credited with the development of the theoretical background of the TMD. However, the study on the use of the TMD in civil engineering structures commenced in the 1970s (McNamara, 1977). Over the years, the TMD has established itself as an effective vibration control device and there are several real-life implementations of the device (Soto and Adeli, 2013). The potential of TMDs to mitigate structural vibrations induced by strong wind (Elias et al., 2019; Wang et al., 2020; Zhang, 2022), as well as earthquake (Zhao et al., 2018; Shu et al., 2019; Zhang et al., 2022) is well demonstrated. In recent years, to improve performance and applicability, several variations of the conventional TMD have been developed, such as the particle TMD (Lu et al., 2017), the tuned impact damper (Lu et al., 2018), the pendulum-type TMD (Shu et al., 2019), the inerter-assisted TMD (Patsialis et al., 2021), etc.

The term TLD stands for the class of DVAs, in which liquid, partly or fully, constitutes the inertial mass of the damper and also contributes to the damping mechanism. Over the last three decades, the TLD has emerged as one of the most attractive passive devices to suppress structural vibrations, with low-cost involvement, easy operation, and amenability to design modifications. Further, the TLD can be incorporated easily into new or existing structures and requires minimal maintenance. The applicability of the device in controlling wind-induced structural vibrations is well established (Kareem, 1990; Konar and Ghosh, 2021c, 2022b; Modi and Seto, 1997; Shankar and Balendra, 2002; Zhang et al., 2016). The effectiveness of the tank dampers in seismic vibration control of structures has also been reported by many researchers (Banerji et al., 2000; Das et al., 2022; Jin et al., 2007; Kareem and Sun, 1987; Lotfollahi-Yaghin et al., 2016; Pandit and Biswal, 2020).

Despite the development of several different configurations of the TLD, it is conventionally associated with the tank damper, in which liquid resides in a circular or rectangular vessel, though other shapes have also been used. Tank dampers are chiefly of three types; (a) shallow tank damper, (b) intermediate-depth tank damper, and (c) deep tank damper (DTD) (Konar and Ghosh, 2022a). This classification is based on the type of sloshing wave produced by the liquid within the tank when laterally excited. Shallow liquid level in a tank induces travelling sloshing waves, whereas deep liquid level in a tank causes standing sloshing waves in the fundamental mode. In the intermediate-depth tank damper, transitional waves are generated during lateral vibration. For practical purposes, the differentiation between the three types of tank dampers is made based on the ratio of liquid depth to the tank dimension along the direction of wave propagation. This ratio is restricted to within 0.1 for shallow tank dampers and is higher than 0.5 for DTDs. The ratio lies between 0.1 and 0.5 for intermediate-depth tank dampers.

Most of the studies on tank dampers have dealt with shallow tanks as deep tanks inherently have much lower energy dissipation capacity (Konar and Ghosh, 2022d). Some efforts have been directed toward enhancing the energy dissipation capacity of DTDs through the incorporation of additional flow damping devices such as baffles, screens, nets, etc., within the damper liquid (Konar and Ghosh, 2021c). Kaneko and Yoshida (1999) studied a DTD with a submerged net across the length of the tank for suppressing structural vibration. Anderson et al. (2000a, 2000b) investigated a DTD with horizontal baffles, placed close to the free liquid surface, to control transient vibrations of a single-degree-of-freedom (SDOF) system. Nguyen et al. (2018) studied a multi-degree-of-freedom structure under seismic excitation with multiple DTDs mounted with slat screens and reported structural displacement response reductions with the requirement of considerably high mass ratios. Recently, Fu et al. (2018) and Xu et al. (2018) proposed a combination of TMD and sloshing damper, in which the TMD is kept submerged within the sloshing liquid, with which it interacts to dissipate energy during vibration.

Civil engineering structures often support liquid (mostly water) containing secondary components such as overhead water tanks on buildings, elevated water tanks, liquid storage tanks on ground supported or offshore platforms, etc. However, despite the success of the TLD as a damping device, these liquid tanks are seldom utilized in the energy dissipation of the primary structure when subjected to lateral excitation. The primary reasons for not using these liquid-containing tanks as damping devices are that these are deep tanks and subject to liquid level fluctuations. Thus, despite being very common engineering structures, there are only very few instances of work that have been devoted to utilizing these deep tanks as damping devices. Hemalatha and Jaya (2008) studied a flexibly supported overhead reservoir on a building, in which the impulsive frequency of the whole tank, and not the liquid sloshing frequency, was tuned to the fundamental frequency of the building for seismic response reduction. Bandyopadhyay et al. (2018) suggested a specific profile of the overhead water tank for which the sloshing frequency of the water within is depth-independent. It may, however, be difficult to procure such non-standard tank shapes for liquid storage purposes. Another technique to have a constant sloshing frequency in an overhead water tank with fluctuating water level was recently developed by Konar and Ghosh (2021b, 2022c) through the concept of a tank damper with a floating base. Rai et al. (2011) demonstrated a case study of seismic retrofitting of a building by compartmentalization of the existing overhead tank to act as shallow tank dampers, along with the installation of additional tank dampers. However, compartmentalization of existing deep tanks to allow them to perform as shallow tank dampers may not always be practically feasible and would disturb the functional utility of the tanks as well. It is also noted that though the incorporation of baffles can help convert standing waves into sloshing waves, they are usually provided near the top of the liquid level and would not serve the purpose when there would be chances of fluctuation in the liquid level in the tank.

In the DTD, a large portion of the liquid is the impulsive liquid, which behaves as though rigidly attached to the tank container during vibration. For example, if a rectangular tank has a height of liquid equal to the tank length, the impulsive liquid mass would be to the tune of 80% of the total liquid (ACI 350.3-06, 2006). The impulsive mass does not engage in the damping mechanism and hence the overall volumetric efficiency of the DTD is low. With this in mind, Konar and Ghosh (2021a) developed the idea of a DTD with a submerged cylindrical pendulum appendage (DTD-CPA). In this, the impulsive mass is energized by the laterally excited pendulum and the drag exerted by the surrounding liquid on the pendulum leads to energy dissipation. However, both the added mass effect from the impulsive mass and the drag effect from the surrounding liquid are limited in case of the DTD-CPA.

The major contribution of the present work is to utilize the basic concept of the DTD-CPA to extract higher energy dissipation from the deep tank, by replacing the cylindrical pendulum of the DTD-CPA with a submerged suspended plate (SSP). The primary advantage of having functional liquid tanks, such as overhead water reservoirs, to be designed as dampers, with allowance for fluctuations in liquid level is retained. The proposed DTD with SSP is henceforth referred to as the DTD-SSP. The DTD-SSP retains all the advantages of the DTD-CPA and further provides higher drag and enhanced added mass. In what follows, first the proposed damper system and its working mechanism are described. The oscillating frequency of the submerged plate in the tank is then derived through free vibration analysis. The theoretical formulation is validated through a comparison with experimental results available in literature. Next, the equations of motion of a laterally excited SDOF structural system with DTD-SSP are derived. The design and performance of the DTD-SSP system in controlling the vibrations of an example building structure under base excitation, through tuning the oscillating frequency of the SSP to that of the primary structure, are examined through a numerical study. Further, the sensitivity of the damper performance to tuning is investigated and a comparison of the effectiveness of the proposed damper with that of a traditional TMD is presented.

Description and working mechanism of DTD-SSP

Figure 1(a) and (b) depict the model of the proposed DTD-SSP. A rectangular deep tank having internal dimensions $L_{T}$ and $B_{T}$ parallel to the $x$ -, and $z$ -axis respectively is considered. The length, height, and thickness of the SSP are $L_{1}$ , $L_{2}$ , and $t_{p}$ respectively. The plate is considered to be suspended from a hanging arrangement, which is connected to the tank walls parallel to the $z$ -axis. The plate is hinged along its length at the top and can oscillate freely about the x-axis. When the oscillating frequency of the plate in the DTD-SSP is tuned to the frequency of the predominant mode of the structure intended for control, vibrational energy is transferred to the SSP and an oscillatory motion is induced in the plate. The surrounding liquid opposes the oscillation of the plate and this leads to the dissipation of energy. Further, a part of the impulsive liquid mass acts as an added mass to the oscillating plate during its vibration. Thus, in the DTD-SSP, the impulsive liquid mass is utilized as a part of the supplemental damping mechanism.

Figure 1.

(a) Elevation of DTD-SSP; (b) 3-dimensional view of DTD-SSP.

It may be noted that if the plate is placed too close to the bottom of the tank, the added mass would increase due to the blockage effects and this may lead to an overestimation of the frequency of oscillation of the plate. In the case of oscillation of a plate close to a rigid wall, the increase in added mass depends on the ratio $∆ / α$ , where, $∆$ is the width of the gap between the plate and the wall, and $α$ is the dimension of the plate parallel to the wall (Kaneko et al., 2014). As expected, this effect is more when $∆ / α$ is less and reduces with the increase in $∆ / α$ . For $∆ / α = 2.5$ , the added mass increases by only 3% (Kaneko et al., 2014). In the present case of the DTD-SSP, the tank bottom functions as a rigid wall, and the dimension of the plate parallel to the tank bottom is represented by the plate thickness, $t_{p}$ . In the model of the DTD-SSP the influence of the distance from the plate to the bottom of the tank is not considered. Thus, in the design of the DTD-SSP, the ratio $∆ / t_{p}$ is assumed to be not less than 2.5.

Free vibration of plate in DTD-SSP

Theoretical formulation

The following assumptions are made in the free vibration analysis of the SSP.

(i) The thickness of the suspended plate is uniform.

(ii) The mass of the suspended plate per unit volume is constant all over the plate.

(iii) The suspended plate is rigid and is placed perpendicular to the direction of the anticipated lateral load.

(iv) The suspended plate is able to oscillate freely when subjected to external excitations.

(v) The fluid in the tank is incompressible and the tank container is rigid.

(vi) The suspended plate remains fully submerged in the tank liquid during vibration.

The equation of dynamic equilibrium of the oscillating plate is expressed as follows.

I \ddot{θ} = \sum τ

(1)

Here, $θ$ is the angular displacement of the plate measured from the vertical equilibrium position at time $t$ , $I$ is the moment of inertia of the oscillating plate and $\sum τ$ is the net torque acting on the plate.

The effect of the liquid on the oscillation of the submerged plate is taken care of through the consideration of an added inertial mass and drag force.

The effective mass of the oscillating plate may thus be taken to be

M_{e f f} = M_{p} + M_{a d d}

(2)

Here, $M_{p}$ is the mass of the plate and $M_{a d d}$ is the added inertial mass.

Hence,

I = M_{e f f} \frac{{L_{2}}^{2}}{3}

(3)

To determine the net torque acting on the system, an element of the oscillating SSP of infinitesimal height, $d y,$ at a distance, $y,$ from the point of suspension of the plate is considered, as shown in Figure 2(a). The different forces acting on the infinitesimal element are represented in Figure 2(b).

Figure 2.

(a) Enlarged view of the oscillating suspended plate; (b) An infinitesimal element of the oscillating suspended plate.

The gravitational force acting on the element is given by

f_{g} = \frac{M_{p}}{L_{2}} g d y

(4)

where,

g

is the gravitational acceleration.

The buoyant force acting on the element is represented by

f_{b} = \frac{1}{R_{ρ}} \frac{M_{p}}{L_{2}} g d y

(5)

Here, $R_{ρ}$ is the density ratio given by the ratio of the mass density of the plate, $ρ_{p,}$ to the mass density of the liquid, $ρ_{l}$ .

Now, in case of a fixed plate submerged in fluid of mass density, $ρ_{f}$ , flowing with velocity, $v_{f}$ , the drag force acting on the plate along the direction of the flow of fluid is expressed as,

f_{d} = \frac{1}{2} C_{d} (A_{p}) . (ρ_{f}) . {(v_{f})}^{2}

(6)

where,

C_{d}

and

A_{p}

, denote the drag coefficient and projected area of the plate perpendicular to the direction of flow respectively.

In the present case, the SSP is oscillating within the impulsive or dead liquid mass of the tank. The oscillation of the plate is opposed by the liquid through drag. Hence, equation (6) for the plate element in Figure 2(b) may be written as (Nakayama and Boucher, 1999),

f_{d} = \frac{1}{2} C_{d} L_{1} ρ_{l} (y | \dot{θ} |) y \dot{θ} d y

(7)

Here $\dot{θ}$ is the angular velocity of the plate element.

The net torque acting on the plate may be expressed as

\sum τ = \int_{y = 0}^{y = L_{2}} (- f_{d} - f_{g} \sin θ + f_{b} \sin θ) y

(8)

Substitution of equations (4), (5), and (7) in equation (8) and the assumption of a constant drag coefficient, $C_{d},$ leads to

\sum τ = - \frac{1}{2} C_{d} L_{1} ρ_{l} \frac{{L_{2}}^{4}}{4} | \dot{θ} | \dot{θ} - M_{p} g \frac{L_{2}}{2} \sin θ + \frac{1}{R_{ρ}} M_{p} g \frac{L_{2}}{2} \sin θ

(9)

For a rectangular plate the added inertial mass is given by (Dong, 1978)

M_{a d d} = \frac{π}{4} k ρ_{l} L_{1} {L_{2}}^{2}

(10)

Here, $k$ is the added mass coefficient.

Now, on using equations (3), (9), and (10), equation (1) can be written as,

(R_{ρ} + \frac{π}{4} k \frac{L_{2}}{t_{p}}) \ddot{θ} + \frac{3}{8} C_{d} \frac{L_{2}}{t_{p}} | \dot{θ} | \dot{θ} + \frac{3 g}{2 L_{2}} (R_{ρ} - 1) \sin θ = 0

(11)

Let consider

λ = (1 + \frac{π k}{4 R_{ρ}} \frac{L_{2}}{t_{p}})

(12)

Now, equation (11) may be recast as

\ddot{θ} + \frac{3 L_{2} C_{d}}{8 t_{p} R_{ρ} λ} | \dot{θ} | \dot{θ} + {ω_{p}}^{2} \sin θ = 0

(13)

Here, $ω_{p}$ is the linearized oscillating frequency of the SSP given by equation (14). The corresponding linearized natural period is denoted by $T_{p} [= 2 π / ω_{p}]$ .

ω_{p} = \sqrt{\frac{3 g}{2 λ L_{2}} (1 - \frac{1}{R_{ρ}})}

(14)

For the purpose of design of the damper, this linearized frequency of the SSP given in equation (14) would be used for tuning to the structural frequency. However, during evaluation of the performance of the damper, the nonlinear equation given in equation (13) would be used.

Validation of theoretical formulation

The veracity of the above theoretical formulation is examined through the work of Lindholm et al. (1965) who had experimentally determined the frequencies of steel cantilever plates of different aspect ratios submerged in water and compared them with the theoretical frequencies of the plates in vacuum. In their study, the plates were clamped at one edge, while, in the present study, the plate is hinged at the top edge. However, in both cases, the frequency of the plate is influenced by the surrounding liquid and the frequency reduces in the submerged condition as compared to the frequency of the plate in vacuum. It may be noted that in vacuum, both

λ

and

(1 - 1 / R_{ρ})

in equation (14) become equal to 1. As the end conditions of the plate in the study carried out by Lindholm et al. (1965) are different from the present study, a direct comparison of the frequencies is not possible. Rather, a validation can be made in terms of the reduction in the plate frequency in the submerged condition as compared to the plate frequency in vacuum. A comparison between this reduction in plate frequency as obtained from the current theoretical formulation for the proposed SSP and the experimental results of Lindholm et al. (1965) for different plate aspect ratios is presented in Table 1. It is observed that there is good agreement between the two. The density of water and steel are considered 1000 kg/m³ and 7700 kg/m³ respectively (Lindholm et al., 1965).

Table 1.

Comparison between theoretical and experimental results regarding reduction in plate frequency in submerged condition as compared to that in vacuum for different plate aspect ratios.

$L_{2} / L_{1}$	$t_{p} / L_{1}$	Reduction in plate frequency in submerged condition as compared to the frequency in vacuum (%)
$L_{2} / L_{1}$	$t_{p} / L_{1}$	Theoretical value from present study	Experimental result by Lindholm et al. (1965)
1.0	0.0238	46.3	48.3
0.5	0.0131	54.9	52.5
1.0	0.0131	56.8	58.1
0.5	0.009	61.1	60.1
1.0	0.009	62.9	63.2

Further, the approach adopted in the present study to cater to the effect of the presence of the surrounding liquid on the free vibration characteristics of the submerged plate is similar to that used by Mathai et al. (2019) in their study on the motion of a cylindrical pendulum submerged in liquid. Through extensive studies, they have reported good agreement between the natural frequency of oscillation as determined from free vibration experiments and from theoretical analysis. It is thus expected that the actual behavior of the present system may be well represented by the mathematical model given by equations (13) and (14).

The assessment of the drag coefficient, $C_{d}, is$ difficult, as, in addition to the aspect ratio of the oscillating object, it depends upon the Reynolds Number $(R e)$ . For a pendulum oscillating in liquid, a wide variation in the instantaneous value of $R e$ occurs as $R e$ is directly proportional to the velocity. In their study on the submerged cylindrical pendulum, Mathai et al. (2019) showed that the consideration of a constant value of $C_{d}$ , based on the aspect ratio of the oscillating object, throughout the entire time span of oscillation has an insignificant influence on the frequency of the pendulum. However, it results in a slight under-prediction of the system damping. In the same line, for the present study, a constant value of $C_{d}$ , determined from the aspect ratio of the plate, ( $L_{1} / L_{2})$ , is assumed during the entire time span of oscillation of the plate. With this consideration, the expected under-prediction of the damping would lead to a conservative estimation of the effectiveness of the energy dissipation by the proposed damper, which is acceptable.

Equations of motion of laterally excited SDOF structure with DTD-SSP

The DTD-SSP system described in the foregoing section is considered to be rigidly attached to an SDOF structural system and subjected to lateral excitation (see Figure 3). The mass, stiffness, and damping of the SDOF system are $M_{s}$ , $k_{s}$ , and $c_{s}$ respectively. The time-dependent external force acting laterally on the structure is considered as $P (t)$ . The horizontal displacement of the structure is u $(t)$ . The mass of the tank is assumed to be lumped with the structural mass. It is also assumed that, apart from the added mass of liquid participating in the plate oscillation, the remaining mass of the liquid in the tank is lumped with the structure. This is justifiable as a large proportion of the liquid mass in a deep tank is impulsive mass.

Figure 3.

Model of structure with DTD-SSP.

The kinetic energy of the structure-DTD-SSP system is given by

T = \frac{1}{2} M_{s} {\dot{u}}^{2} + \frac{1}{2} \int_{y = 0}^{y = L_{2}} \frac{M_{e f f}}{L_{2}} {(\dot{u} + y \dot{θ} \cos θ)}^{2} d y + \frac{1}{2} \int_{y = 0}^{y = L_{2}} \frac{M_{e f f}}{L_{2}} {(y \dot{θ} \sin θ)}^{2} d y

(15)

Equation (15) can be simplified as

T = \frac{1}{2} (M_{s} + M_{e f f}) {\dot{u}}^{2} + \frac{1}{2} M_{e f f} (\dot{u} \dot{θ} L_{2} \cos θ + \frac{{L_{2}}^{2}}{3} {\dot{θ}}^{2})

(16)

Now, the potential energy of the system is expressed as

V = \frac{1}{2} k_{s} u^{2} + \int_{y = 0}^{y = L_{2}} \frac{M_{p}}{L_{2}} g y (1 - \cos θ) d y - \int_{y = 0}^{y = L_{2}} \frac{1}{R_{ρ}} \frac{M_{p}}{L_{2}} g y (1 - \cos θ) d y

(17)

On simplification of equation (17), the following is obtained.

V = \frac{1}{2} k_{s} u^{2} + (1 - \frac{1}{R_{ρ}}) M_{p} g \frac{L_{2}}{2} (1 - \cos θ)

(18)

Hence, the Lagrangian of the system may be written as

L = \frac{(M_{s} + M_{e f f})}{2} {\dot{u}}^{2} + \frac{M_{e f f}}{2} (\dot{u} \dot{θ} L_{2} \cos θ + \frac{{L_{2}}^{2}}{3} {\dot{θ}}^{2}) - \frac{k_{s}}{2} u^{2} - (1 - \frac{1}{R_{ρ}}) M_{p} g \frac{L_{2}}{2} (1 - \cos θ)

(19)

For a non-conservative system such as the present one, Lagrange’s equation may be written as follows (Minguzzi, 2015).

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} = Q_{i}

(20)

Here, $q_{i}$ and $Q_{i}$ respectively denote the generalized coordinate and the non-conservative generalized force corresponding to the $i$ ^th degree of freedom.

For the structure-damper system, the generalized coordinates are

q_{1} = u a n d q_{2} = θ

(21)

In this system, the external force and the damping forces are the non-conservative forces. Using the principle of virtual work, the non-conservative generalized force corresponding to the $first$ and $the second$ degrees of freedom may be written as

Q_{1} = P (t) - c_{s} \dot{u}

(22)

Q_{2} = - \int_{y = 0}^{y = L_{2}} \frac{1}{2} C_{d} L_{1} ρ_{l} (y | \dot{θ} |) y \dot{θ} y d y

(23)

o r, Q_{2} = - \frac{1}{2} C_{d} L_{1} ρ_{l} \frac{{L_{2}}^{4}}{4} | \dot{θ} | \dot{θ}

(24)

Now, the equations of dynamic equilibrium of the structure-DTD-SSP system may be obtained from

\frac{d}{d t} (\frac{\partial T}{\partial \dot{u}}) - \frac{\partial L}{\partial u} = Q_{1}

(25)

\frac{d}{d t} (\frac{\partial T}{\partial \dot{θ}}) - \frac{\partial L}{\partial θ} = Q_{2}

(26)

Substitution of equations (16), (19), and (22) in equation (25) leads to

(M_{s} + M_{e f f}) \ddot{u} + M_{e f f} \frac{L_{2}}{2} \ddot{θ} \cos θ - M_{e f f} \frac{L_{2}}{2} {\dot{θ}}^{2} \sin θ + k_{s} u = P (t) - c_{s} \dot{u}

(27)

Let $μ = M_{p} / M_{s}$ , represent the mass ratio, defined by the ratio of the plate mass to the structural mass.

Normalizing equation (27) with respect to $(1 + μ λ) M_{s}$ the following is obtained.

\ddot{u} + \frac{2 ζ_{s} ω_{s}}{(1 + μ λ)} \dot{u} + \frac{{ω_{s}}^{2}}{(1 + μ λ)} u = \frac{P (t)}{(1 + μ λ) M_{s}} - \frac{μ λ}{(1 + μ λ)} \frac{L_{2}}{2} \ddot{θ} \cos θ + \frac{μ λ}{(1 + μ λ)} \frac{L_{2}}{2} {\dot{θ}}^{2} \sin θ

(28)

Here, $ζ_{s} [= \sqrt{c_{s} / M_{s}}]$ , is the damping ratio of the structural system and $ω_{s} [= \sqrt{k_{s} / M_{s}}]$ , is the fundamental structural frequency.

Again, on substitution of equations (16), (19), and (24) in equation (26), the following is obtained.

\frac{1}{2} M_{e f f} L_{2} \ddot{u} \cos θ + \frac{1}{2} M_{e f f} \frac{2 {L_{2}}^{2}}{3} \ddot{θ} + \frac{1}{2} (1 - \frac{1}{R_{ρ}}) M_{p} g L_{2} \sin θ = - \frac{1}{2} C_{d} L_{1} ρ_{l} \frac{{L_{2}}^{4}}{4} | \dot{θ} | \dot{θ}

(29)

Rearranging terms in equation (29) leads to

\ddot{θ} + \frac{3 C_{d} L_{2}}{8 R_{ρ} λ t_{p}} | \dot{θ} | \dot{θ} + {ω_{p}}^{2} \sin θ = - \ddot{u} \frac{3}{2 L_{2}} \cos θ

(30)

The coupled equations, (28) and (30), govern the motion of the structure-DTD-SSP system.

Instead of an external force, $P (t)$ , acting on the structure, for an input base acceleration, ${\ddot{Z}}_{0} (t)$ , the governing equations of the structure-DTD-SSP system may be modified as,

\ddot{u} + \frac{2 ζ_{s} ω_{s}}{(1 + μ λ)} \dot{u} + \frac{{ω_{s}}^{2}}{(1 + μ λ)} u = - {\ddot{Z}}_{0} - \frac{μ λ}{(1 + μ λ)} \frac{L_{2}}{2} \ddot{θ} \cos θ + \frac{μ λ}{(1 + μ λ)} \frac{L_{2}}{2} {\dot{θ}}^{2} \sin θ

(31)

\ddot{θ} + \frac{3 C_{d} L_{2}}{8 R_{ρ} λ t_{p}} | \dot{θ} | \dot{θ} + {ω_{p}}^{2} \sin θ = - (\ddot{u} + {\ddot{Z}}_{0}) \frac{3}{2 L_{2}} \cos θ

(32)

Numerical study

Example structure and design of DTD-SSP

For the numerical study, an example SDOF structure is considered that has fundamental period ( $T_{s}$ ) and mass $(M_{s})$ equal to 1.58s and 598,471 kg respectively (Whittaker et al., 2003). The damping ratio $(ζ_{s})$ of the structure is considered as 5.0%.

For the mass ratio $(μ)$ of the damper, a practically feasible value equal to 1.5% is taken. For an inertia based damper with low mass ratio, the tuning ratio $(γ)$ , defined as the ratio of the damper frequency to the structural frequency, normally remains close to unity. Hence, for the present study, the value of $γ$ is considered 1.0 for simplicity. One of the main parameters of the DTD-SSP is the height of the plate, $L_{2}$ . On substituting equation (12) in equation (14) and taking the real root of the resultant quadratic equation of $L_{2}$ , the following is obtained.

L_{2} = \frac{2 R_{ρ} t_{p}}{π k} [\sqrt{1 + \frac{3 π g k}{2 R_{ρ} t_{p} {ω_{p}}^{2}} (1 - \frac{1}{R_{ρ}})} - 1]

(33)

The value of $L_{2}$ depends upon $t_{p}$ , $R_{ρ}$ and $k,$ in addition to $ω_{p}$ and $g$ , both of which have fixed values for a particular design problem. Depending upon the aspect ratio of the plate, $L_{1} / L_{2}$ , the value of $k$ ranges from 0.47 to 1.0 for $L_{1} / L_{2} < 3.0$ and for $L_{1} / L_{2} \geq 3.0$ , $k$ becomes constant and equal to 1.0 (Dong, 1978). For the DTD-SSP of the example structure, the variation in $L_{2}$ with $t_{p},$ for different values of $R_{ρ}$ and $k,$ is presented in Figure 4. It is observed that $L_{2}$ increases with the increase in $t_{p}$ as well as in $R_{ρ}$ . On the other hand, $L_{2}$ increases when $k$ decreases. It is thus clear from Figure 4 that for a given $ω_{p},$ the value of $L_{2}$ can be selected from a workable range of values by a suitable variation of the plate thickness, plate material and plate aspect ratio. This is significant as it allows design of the plate dimensions that are compatible with an existing tank’s dimensions and thereby constitutes an advantage of the DTD-SSP.

Figure 4.

Height of suspended plate $(L_{2})$ versus its thickness $(t_{p})$ for different values of $k$ and $R_{ρ}$ .

For the present design, the liquid within the DTD-SSP is assumed to be water, with mass density, $ρ_{l},$ equal to 1000 kg/m³. The plate is assumed to be made of copper, of mass density, $ρ_{p}$ , equal to 8960 kg/m³. Thus, $R_{ρ}$ = 8.96. It is assumed that there are five numbers of identical DTD-SSPs placed on the structure. The SSPs of DTD-SSPs are assumed to be of identical size and mass. Hence to provide the total considered $μ$ of 1.5%, the mass of each plate should be 0.3% of the mass of the example structure.

A suitable thickness of the plate,

t_{p}

, equal to 125 mm, is assumed. Next,

L_{2}

is calculated using equation (33). The length of the plate,

L_{1}

, is determined using the values of

M_{s}

μ

ρ_{p}

t_{p}

and

L_{2}

. Based on the ratio

(L_{1} / L_{2}

), the value of the drag coefficient,

C_{d}

(Nakayama and Boucher, 1999) and added mass coefficient,

k

(Dong, 1978), are determined. The values of

L_{2}

and

k

are obtained through an iterative solution of equation (33), with an initial value of

k = 1

. The values of all the parameters of the designed SSP appendage are provided in Table 2.

Table 2.

Parameters of the suspended plates of the designed DTD-SSPs

Parameters	values
$t_{p}$	125 mm
$L_{2}$	0.586 m
$L_{1}$	2.736 m
$L_{1} / L_{2}$	4.670
$k$ (Dong, 1978)	1.000
$C_{d}$ (Nakayama and Boucher, 1999)	1.201

In the present problem, it is assumed that the tank dimensions are such that the internal length ( $L_{T}$ ) and internal width ( $B_{T}$ ) of each tank are equal to 3.5 m and 1.35 m respectively, and the depth of liquid in the tanks in the tank-full condition is 2.7 m. As mentioned earlier, for existing tanks on a building, the plates may be designed suitably so that, while fulfilling tuning requirements, each plate is accommodated comfortably within the impulsive zone of liquid in a tank. Further, it is assumed that in each tank, the plate is suspended such that the point of suspension is at a height of 0.9 m from the bottom of the tank. Thus, the gap between the bottom of the tank and the plate, $∆$ , is equal to 0.314 m, and the ratio $∆ / t_{p}$ is equal to 2.51. This satisfies the condition mentioned earlier as regards neglecting the influence of the tank bottom on the modelling of the DTD-SSP.

In a rectangular tank, the mass of the impulsive liquid, $M_{0}$ , is given by (Housner, 1963),

M_{0} = M_{T} \frac{\tanh (0.85 B_{T} / h)}{0.85 B_{T} / h}

(34)

Here, $h$ , and $M_{T}$ respectively denote the depth of liquid, and the total mass of the liquid in the tank.

Equation (34) dictates that in order to have the SSP fully submerged within the impulsive liquid mass, the minimum water depth to be maintained in each tank is 1.171 m, which is 43% of the total depth of water in the tank in the tank-full condition. Thus, 57% of the total volume of the water tanks is available for other functional usages.

In the design of the DTD-SSP, the dimensions of the tank should be so chosen that the fundamental sloshing frequency of the liquid in the tank is well away from the natural frequency of the structure, so as to minimize the effect of liquid sloshing on the structural response and on the damper performance. In the present case, the fundamental liquid sloshing frequency is always more than 20% of the natural frequency of the structure, for the permissible range of fluctuation in liquid depth. The fundamental liquid sloshing frequency in rectangular tanks is derived from (Graham and Rodriguez, 1952),

ω_{s l o s h} = {[\frac{π g}{B_{T}} \tanh \frac{π h}{B_{T}}]}^{1 / 2}

(35)

Performance of DTD-SSP under base excitation

Here, the performance of the designed damper system is evaluated in terms of the ratio of the difference in the displacement responses of the uncontrolled and controlled structure to the displacement response of the uncontrolled structure, expressed as a percentage. The performance of the DTD-SSP is first examined by applying harmonic acceleration to the base of the structure. Three different amplitudes (

A_{0})

of the harmonic excitation, namely, 0.10 m/s², 0.15 m/s² and 0.20 m/s², are considered to gauge the amplitude dependency of the control achieved by the proposed damper system. In each case, the excitation frequency is taken equal to that of the example SDOF structural system. The governing equations of the structure-DTD-SSP system, given in equations (31) and (32), are solved with the help of the explicit Runge–Kutta formula of order 4 and 5. The amplitude of steady-state displacement response of the structure

(U_{s s}),

for both the controlled and uncontrolled cases are presented in Table 3, along with the reductions in

U_{s s}

that are achieved by the damper system. Sample displacement time histories of the structure, with and without DTD-SSP, for the case of harmonic acceleration with 0.15 m/s² amplitude are presented in Figure 5(a). The time history of plate rotation under harmonic base acceleration of amplitude 0.15 m/s² is given in Figure 5(b). The effectiveness of the proposed DTD-SSP system in controlling the structural response is clearly visible with control efficiency in the range of 31%–55% (Table 3).

Table 3.

Reductions achieved in steady-state displacements response of structure $(U_{s s})$ by DTD-SSP system under harmonic acceleration at base of structure.

Amplitude of harmonic base acceleration ( $A_{0})$	Amplitude of steady-state displacement $(U_{s s})$			Maximum plate rotation at steady-state condition
Amplitude of harmonic base acceleration ( $A_{0})$	Without damper (m)	With damper (m)	Reduction by damper (%)	Maximum plate rotation at steady-state condition
0.10 m/s²	0.063	0.028	55.4	39.5^o
0.15 m/s²	0.095	0.055	41.5	50.0^o
0.20 m/s²	0.126	0.087	31.4	57.8^o

Figure 5.

Response of structure-DTD-SSP system under harmonic base acceleration of amplitude 0.15 m/s² (a) time histories of displacement of example structure, with and without DTD-SSP system, (b) time history of plate rotation.

It is clear from Table 3 that the performance of the DTD-SSP is dependent on the excitation amplitude and that with the increase in excitation amplitude the response reduction reduces. This is analyzed as follows. In the present system, the control is achieved by the drag force that acts perpendicular to the surface of the oscillation plate. For very high angular displacement of the plate, the horizontal component of the drag force is small. Hence, when the structure-damper system is subjected to large amplitude excitation, the angular displacements of the plate are high, resulting in lower control effectiveness of the damper.

Next, the damper performance is examined under seismic excitation. Five recorded earthquake ground motions, details of which are given in Table 4, are considered. The input ground motions are so chosen so as to have the natural frequency of the example structure in the dominant frequency range of the excitations.

Table 4.

Details of considered earthquake motions.

Sl. No.	Earthquake	Year	Station	Magnitude ( $M_{w}$ )	Component	Peak ground acceleration ( $g$ )
1.	Western Washington	1949	Olympia	6.7	86^o	0.280
2.	Alkion	1981	Korinthos-OTE Building	6.6	N–S	0.230
3.	Northridge	1994	LA-Saturn St	6.7	20^o	0.468
4.	Chile	2010	Santiago Maipu	8.8	E-W	0.488
5.	New Zealand	2010	Darfield high School	7.1	S17E	0.458

The results of the time history analysis of the structure-DTD-SSP system under the chosen ground motions are summarized in Table 5. Reductions achieved in the peak displacement and in the root-mean-square (rms) displacement of the structure by the DTD-SSP system respectively range from 13%–18% and 20%–31%, which are reasonable for a passive damper. The reduction in peak response indicates reduction in transmission of maximum force due to seismic excitation, whereas, reduction in rms response represents the dissipation of energy transmitted to the structure due to seismic excitation (Pandey et al., 2019). As expected for a passive damper, the DTD-SSP system produces more reduction in rms response than in peak reduction. Here, it may be noted that the reduction of response over the duration of the earthquake is given by the rms response reduction, hence it gives a more consistent measure of the effectiveness of the damper (Pandey et al., 2019).

Table 5.

Reductions achieved in displacement response of structure by DTD-SSP system under earthquake excitation.

Earthquake data	rms displacement		Peak displacement		Maximum plate rotation
Earthquake data	Without damper (m)	Reduction by damper (%)	Without damper (m)	Reduction by damper (%)	Maximum plate rotation
Western Washington	0.020	21.6	0.085	14.5	47.8^o
Alkion	0.039	24.7	0.118	12.7	66.1^o
Northridge	0.034	31.0	0.106	14.3	52.8^o
Chile	0.035	20.0	0.187	18.4	72.4^o
New Zealand	0.034	19.9	0.159	17.0	68.5^o

The displacement time histories of the uncontrolled and controlled structure subjected to the Northridge earthquake excitation are shown in Figure 6(a). It is observed that during the initial phase of vibration, the effect of the DTD-SSP system is not perceptible, possibly due to the time taken by the SSPs to start to oscillate and mobilize the control force. However, after the initial phase, the effectiveness of the damper is clearly visible.

Figure 6.

Response of structure-DTD-SSP system under Northridge earthquake excitation (a) time histories of displacement of example structure, with and without DTD-SSP system, (b) time history of plate rotation.

The oscillations of the SSP of the designed dampers are also studied and it is found that under the considered ground motions, the maximum plate rotations are obtained in the range of 47.8^o to 72.4^o, which are quite significant as the damper dissipates energy through the drag resistance provided by the surrounding liquid on the oscillating plate. It also indicates that a small-angle linear approximation of the plate oscillation would be inappropriate. Representative time history of plate rotation of the proposed dampers for the example structure for the Northridge earthquake is presented in Figure 6(b).

Sensitivity of DTD-SSP to tuning and comparison with traditional TMD

The sensitivity of the DTD-SSP system to tuning is analyzed under harmonic base excitation to the structure. Three different amplitudes ( $A_{0})$ of the input acceleration are considered, namely, 0.10 m/s², 0.15 m/s² and 0.20 m/s², and in each case the excitation frequency is considered equal to the frequency of the SDOF structure. To investigate the effect of mass ratio ( $μ$ ) on the optimum tuning ratio $(γ_{o p t})$ , three different mass ratios, 1.0%, 1.5% and 2.0%, are considered. The design of DTD-SSP system for mass ratio of 1.5% has been presented at the beginning of the numerical study. A similar procedure is followed for the design of the DTD-SSP with the other two mass ratios. For 1.5% mass ratio, the thickness of the SSP was considered as 125 mm. While designing the DTD-SSP for mass ratios 1.0% and 2.0%, the thickness of the plate is changed proportionately and obtained as 83.3 mm and 166.7 mm respectively. The parameters of the primary structure, mass density of plate material and damper liquid are taken to be the same as those considered in the section ‘Example structure and design of DTD-SSP’. Now, the reduction in the amplitude of steady-state displacement of the structure by the DTD-SSP system over a range of $γ$ from 0.8 to 1.2, for different combinations of mass ratio and input amplitude, are presented in Figure 7(a)-(c). As the system is nonlinear, the curves exhibit a slight tilt to the right. The $γ_{o p t}$ and the maximum reduction in $U_{s s}$ in all nine cases, are indicated in Table 6.

Figure 7.

Percentage reduction in steady-state structural displacement $(U_{s s})$ versus tuning ratio $(γ)$ for different mass ratios ( $μ$ ) and harmonic base accelerations of different amplitude ( $A_{o}$ ) (a) $μ$ = 1.0%, (b) $μ$ = 1.5%, (c) $μ$ = 2.0%.

Table 6.

Optimal tuning ( $γ_{o p t}$ ) ratio and corresponding reduction in amplitude of steady-state structural displacement $(U_{s s})$ for different amplitudes of harmonic base acceleration.

Mass ratio ( $μ$ ) (%)	1.0			1.5			2.0
Amplitude of harmonic base acceleration ( $A_{0})$ (m/s²)	0.10	0.15	0.20	0.10	0.15	0.20	0.10	0.15	0.20
$γ_{o p t}$ of DTD-SSP	1.031	1.058	1.090	1.024	1.052	1.088	1.016	1.042	1.078
Reduction in $U_{s s}$ for $γ = γ_{o p t}$ by DTD-SSP	42.3%	33.3%	26.8%	58.7%	48.4%	39.9%	71.0%	61.3%	52.4%
Reduction in $U_{s s}$ for $γ = γ_{o p t}$ by TMD	41.6%	41.6%	41.6%	51.6%	51.6%	51.6%	58.4%	58.4%	58.4%

As it would be of interest to compare the performance of the proposed DTD-SSP with that of the well-established TMD, the reductions in $U_{s s}$ by a conventional linear TMD for the said range of tuning ratio are also plotted in Figure 7(a)–(c). The mass ratio of the TMD is considered same as the corresponding DTD-SSP. The damping ratios of TMD $(ζ_{T M D})$ for the different mass ratios are calculated from equation (36) (Leung and Zhang, 2009), which provides the optimum value of $ζ_{T M D}$ for maximum reduction in $U_{s s}$ by a TMD for a damped SDOF structure under harmonic base excitation.

ζ_{T M D} = \sqrt{\frac{3 μ}{8 (1 + μ) (1 - μ / 2)}} + 0.1277 ζ_{s} + (0.5277 ζ_{s} + 2.3657 {ζ_{s}}^{2}) μ

(36)

It is observed that the performance of the DTD-SSP, which is inherently nonlinear, is expectedly dependent on the excitation amplitude and with the increase in excitation amplitude, the maximum reduction in $U_{s s}$ reduces. This is in contrast to the behavior of the linear TMD. Further, with the increase in excitation amplitude, the performance of the DTD-SSP is less sensitive to tuning. It is also observed that with the increase in mass ratio, the increase in the effectiveness of the DTD-SSP is greater as compared to the TMD. Moreover, the results indicate that the values of optimal tuning ratios for DTD-SSP are slightly more than unity, which, again, is different from the established results for inertia-based dampers, wherein the value of the optimal tuning ratio normally falls on the lower side of unity. In the DTD-SSP, for a given mass ratio and plate thickness, with higher tuning ratio, the required value of the height of the plate reduces and the length of the plate increases. This results in an increase in the ratio $L_{1} / L_{2}$ that leads to a greater value of the drag coefficient and thereby to enhanced damping. Due to this, the optimal tuning of this proposed damper is slightly shifted towards the higher side of unity. The shift is greater when the amplitude of excitation is higher. Thus, the response reduction capacity of the DTD-SSP falls more sharply when mistuned towards the higher side of $γ_{o p t}$ . For example, in the current example case, for a mistuning of +10%, the drop in the response reduction by the DTD-SSP from that achieved under the optimally tuned case is in the range of 17.0%–42.6%, depending on the excitation amplitude, whereas, −10% mistuning leads to 5.4%–34.0% drop in the response reduction by the DTD-SSP. As already mentioned, the DTD-SSP is less sensitive to tuning at higher excitation amplitudes, thus, the lower values in the drop in the response reduction by the DTD-SSP due to mistuning are for the higher excitation amplitude. In the case of TMD, the response reduction by the damper falls by 21.2%–25.3% and 19.2%–20.4% respectively, for mistuning of +10% and −10%.

Overall, the performance of DTD-SSP is found to be comparable and even superior to that of the TMD for low to medium-intensity excitation. Though the effectiveness of the DTD-SSP reduces for higher excitation amplitude, it is still significant. However, with higher mass ratio, the structural response reduction achieved by the DTD-SSP for higher excitation amplitude is close to that achieved by the TMD.

To further highlight the effectiveness of the DTD-SSP, the performance of the damper is compared with that of a TMD, both having the same mass ratio,

μ,

of 1.5%, under the seismic base excitations given in Table 4. The example structure and the DTD-SSP considered are the same as described in the section ‘Example structure and design of DTD-SSP’. The maximum reduction in rms displacement response of the structure by the DTD-SSP and the TMD for the considered earthquakes are given in Table 7. Table 8 provides the maximum reduction in peak structural displacement by the DTD-SSP and by the TMD. The

γ_{o p t}

for the DTD-SSP for the considered earthquakes and for rms and peak structural displacement reduction are also given in Tables 7 and 8 respectively. It may be noted that the effectiveness of the DTD-SSP is again comparable to and in some instances better than that of the traditional TMD.

Table 7.

Optimal tuning ( $γ_{o p t}$ ) ratio and corresponding reduction in rms structural displacement for different earthquake base excitation.

Earthquake	Western Washington	Alkion	Northridge	Chile	New Zealand
$γ_{o p t}$ of DTD-SSP for rms displacement reduction	0.96	1.00	1.02	1.06	1.06
Reduction in rms displacement for $γ = γ_{o p t}$ by DTD-SSP	23.6%	24.6%	31.9%	22.5%	22.4%
Reduction in rms displacement for $γ = γ_{o p t}$ by TMD	21.1%	24.3%	27.9%	26.6%	23.8%

Table 8.

Optimal tuning ( $γ_{o p t}$ ) ratio and corresponding reduction in peak structural displacement for different earthquake base excitation.

Earthquake	Western Washington	Alkion	Northridge	Chile	New Zealand
$γ_{o p t}$ of DTD-SSP for peak displacement reduction	0.99	1.06	1.00	1.00	1.05
Reduction in peak displacement for $γ = γ_{o p t}$ by DTD-SSP	17.3%	13.6%	14.3%	18.4%	20.1%
Reduction in peak displacement for $γ = γ_{o p t}$ by TMD	14.3%	11.2%	10.6%	17.1%	17.7%

Conclusions

In the proposed device (DTD-SSP), the impulsive liquid mass in the deep tank is energized by the SSP and participates in the energy dissipation mechanism. The oscillating frequency of the SSP is tuned to the fundamental structural frequency. The tank is thereby converted into a viable damping device without disturbing its functionality. Unlike the conventional TMD, the DTD-SSP does not require any spring or damping elements that are costly and require frequent maintenance. Further, it can be incorporated easily into an existing tank with little modification of the original tank structure. Thus, the proposed design is suitable for new construction as well for retrofitting work. The only additional requirement of the proposed damper is to maintain a minimum liquid depth in the tank. This minimum liquid storage can meet the requirement for reserve storage and for firefighting. The major findings of the study are summarized below.

(1) A good agreement of the theoretically obtained reduction in the plate frequency in a submerged condition with respect to the plate frequency in vacuum with experimental results available in literature (Lindholm et al., 1965) provides a validation of the proposed formulation.

(2) The performance of the DTD-SSP is dependent both on the amplitude of the excitation as well as on the mass ratio. For harmonic base acceleration input of different amplitudes, an optimally tuned DTD-SSP can achieve significant response reductions in the range 26%–71.0% with mass ratio as low as 1%–2%. The efficiency of the DTD-SSP is higher for lower amplitude excitations.

(3) The optimal tuning ratio is close to unity, and depends both on the mass ratio as well as on the input excitation amplitude.

(4) For a given mass ratio, the required tuning frequency of the DTD-SSP can be achieved for a range of plate dimensions and density of plate material. This brings flexibility to the design of the proposed damper.

(5) Under the considered seismic base excitations, the DTD-SSP is able to achieve 13%–18% and 20%–31% reductions in the peak and rms displacement of the structure respectively.

(6) Under both harmonic and seismic base excitations, the performances of the proposed damper are found to be comparable and, in some instances, even superior to that of the traditional TMD.

Overall, the proposed damper design is believed to have significant potential in the utilization of functional liquid tanks in the mitigation of structural response to lateral excitation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Aparna (Dey) Ghosh

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