Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections R.O. Ruiz a,b, A.A. Taflanidis b,n, D. Lopez-Garcia a,c a Department of Structural & Geotechnical Engineering, Pontificia Universidad Catolica de Chile, Av. Vicuna Mackenna 4860 Macul, Santiago, RM 782-0436, Chile b Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN 46556, USA c National Research Center for Integrated Natural Disaster Management CONICYT/FONDAP/15110017, Santiago, Chile

a r t i c l e i n f o

abstract

Article history: Received 31 March 2015 Received in revised form 29 November 2015 Accepted 9 January 2016 Handling editor: M.P. Cartmell

A recently proposed new type of liquid mass damper, called Tuned Liquid Damper with Floating Roof (TLD-FR), is the focus of this paper. The TLD-FR consists of a traditional TLD (tank filled with liquid) with the addition of a floating roof. The sloshing of the liquid within the tank counteracts the motion of the primary structure it is placed on, offering the desired energy dissipation in the vibration of the latter, while the roof prevents wave breaking phenomena and introduces an essentially linear response. This creates a dynamic behavior that resembles other types of linear Tuned Mass dampers (TMDs). This investigation extends previous work of the authors to consider TLDs-FR with arbitrary tank cross-sections, whereas it additionally offers new insights on a variety of topics. In particular, the relationship between the tank geometry and the resultant vibratory characteristics is examined in detail, including the impact of the roof on these characteristics. An efficient mapping between these two is also developed, utilizing Kriging metamodeling concepts, to support the TLD-FR design. It is demonstrated that the overall behavior can be modeled through introduction of only four variables: the liquid mass, the frequency and damping ratio of the fundamental sloshing mode of the TLD-FR, and the efficiency index, which is related to the portion of the total mass that participates in this mode. Comparisons between TLDs-FR and other types of mass dampers are established through the use of the latter index. A design example is presented considering the dynamic response of a structure under stationary excitation. It is illustrated in this example that for complex tank cross-sectional geometries there exists a manifold of tank configurations leading to the same primary vibratory characteristics and therefore same efficiency of the TLD-FR. Considerations about excessive displacements of the roof can be then incorporated to indicate preference towards some of these candidate configurations. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Tuned Liquid Dampers Floating roof Mass dampers Sloshing

1. Introduction Mass dampers, with main representative the Tuned Mass Damper (TMD), are widely acknowledged as a highly effective device for suppressing structural vibrations [1–7]. They consist of a secondary mass attached to the primary structure n

Corresponding author. E-mail address: a.tafl[email protected] (A.A. Taflanidis).

http://dx.doi.org/10.1016/j.jsv.2016.01.014 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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(whose vibration is to be controlled) through a spring and a dashpot. Through proper selection (tuning) of the spring/ dashpot parameters the vibration of the secondary mass counteracts the motion of the primary structure in some chosen mode (typically fundamental mode), facilitating the desired energy dissipation for this motion. The main requirement in this tuning is that the frequency of oscillation of the mass damper is close to the fundamental frequency of the primary structure. Additionally, it involves a proper selection of the dashpot characteristics, providing damping directly for the vibration of the secondary mass, with dashpot values lower than the optimal one significantly reducing the implementation effectiveness [5]. Many variations of such mass dampers have been proposed in the literature, mainly distinguished by the approach for creating the secondary mass of the damper and facilitating the equivalent spring/dashpot connection [5,8–11]. Liquid dampers, creating this mass through water within some type of tube or tank, are one of the most popular variants due to their lower cost, easy tuning, and potential alternative use of the added mass [8,9,11]. Within the class of liquid mass dampers one can further distinguish between Liquid Column Dampers (LCDs), such as the Tuned Liquid Column Damper (TLCD) and the Liquid Column Vibration Absorber (LCVA), and Tuned Liquid Dampers (TLDs). LCDs consist of a U-shaped tube filled with water [10–12]. In this case the water movement within the horizontal part of the tube is responsible for counteracting the motion of the structure, resulting in an effectiveness of the TLCD/LCVA that is directly related to the liquid mass within that part, and not to the overall liquid mass [5]. The frequency of oscillation of the liquid is related to the length of the liquid column, which is the only parameter that can be adjusted for tuning, whereas the equivalent dashpot effect can be created through the placement of an element (typically an orifice plate [11] or even through the recently proposed inclusion of a spherical ball [13]) that facilitates energy dissipation (damping). LCD behavior can be predicted through a single degree of freedom model which facilitates a simple design process, like for many other types of mass dampers, though they are typically restricted to one-directional applications, and this behavior is still nonlinear because the aforementioned damping is amplitude-dependent [14]. Nevertheless, they have been proven efficient in reducing both wind-induced [10,15] and earthquake-induced [14,16,17] vibrations. TLDs on the other hand, consist of a tank filled with liquid, the sloshing of which facilitates the desired energy dissipation while providing seamlessly capabilities for bi-directional applications [8,9,18]. The frequency of oscillation is related to the dimensions of the tank and the depth of the liquid, whereas establishing the desired optimal level of damping requires addition of submerged obstacles [19,20] whose behavior is in general difficult to reliably predict. Additionally, their dynamic behavior is typically nonlinear due to wave breaking phenomena [18,21]. Like LCDs their implementation has been examined for different type of excitations [7,22,23]. Motivated by these challenges for liquid dampers a variation of the traditional TLD was recently presented [24] by introducing a floating roof as shown in Fig. 1. The new device is called TLD with floating roof, TLD-FR. The roof, being stiffer than water, prevents wave breaking, hence facilitating a linear response even at large amplitudes while it also accommodates the addition of supplemental devices (such as viscous dampers) with which the level of damping in the liquid vibration can be substantially augmented to reach the desired optimal value. The new device maintains desired attributes of both LCDs and TLDs [25]: its dynamics can be ultimately characterized by considering a single degree of freedom, linear model; modification of the natural frequency is highly versatile; independent operation in both directions can be supported; damping in the liquid vibration can be easily enhanced. The studies for TLDs-FR [24,25] have been restricted, though, to rectangular tank geometries and placed focus on the experimental validation of the proposed numerical models for characterizing their dynamic behavior. Here, this analysis is extended to tanks with arbitrary cross-sections and emphasis is placed on the theoretical assessment/design framework, aiming to offer new insights on a variety of topics; examine what is the relationship between the tank geometry and the resultant vibratory characteristics, develop an efficient mapping between these two, examine how the inclusion of the floating roof impacts the vibratory characteristics. After the theoretical discussions all these concepts are illustrated within a design example that examines the dynamic response of a structure under stationary seismic excitation. It is demonstrated in this example that for complex tank geometries there exist a manifold of configurations leading to the same primary vibratory characteristics, and therefore efficiency for the

Damper Floating Roof

η

Liquid

H

L Fig. 1. Tuned Liquid Damper with Floating Roof (TLD-FR) concept. (a) Schematic with different components and (b) Photo from experimental configuration at Pontificia Universidad Catolica de Chile.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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Fig. 2. Concept for numerical modeling of TLD-FR. (a) Complete SSM-FR model, and decomposition to the two different components, (b) Beam FE model and (c) SSM model.

TLD-FR. Considerations about excessive displacements of the roof can be then incorporated to indicate preference towards some of these candidate configurations.

2. Numerical model of TLDs-FR The numerical model to describe the dynamic behavior of a TLD-FR is first reviewed in this section. This modeling, presented in detail in [25], considers first two independent components, the liquid vibration within an arbitrary tank and the vibration of the floating roof, and then combines them through the pressures generated at the liquid-roof interface to derive the final equation of motion, including the forces transferred at the supporting base, used to couple the TLD-FR with the primary structure. A schematic of this concept is illustrated in Fig. 2. Although the TLD-FR has the potential to work in a bidirectional mode here we focus on the case that the liquid vibration characteristics are tuned to the vibration characteristics of the structure for only a specific translational mode and that the tank “length” is aligned with that direction. This leads ultimately to a unidirectional implementation since in the transverse direction the liquid vibration is detuned from the structure that supports it. Additionally, the transversal geometry of the liquid tank is considered to be constant. With respect to the bi-directional potential of TLD-FR we should point out that this requires proper configuration of the roof to facilitate displacements in both directions, an implementation that still requires further investigation (to prove how practical it can be). 2.1. Liquid vibration within an arbitrary tank The vibration of the liquid within a tank with arbitrary geometry is modeled through the Simplified Sloshing Model (SSM) presented recently by the authors [26]. The approach assumes an ideal behavior (irrotational, incompressible and inviscid fluid), which allows for a Finite Element Method (FEM) formulation based on potential variables with only one degree of freedom considered per node. Fig. 2 presents a scheme of the liquid tank where the free surface elevation η is measured from the equilibrium position defined by Γo. Additionally, tank walls and bottom (Γp) are assumed rigid, which simplifies the introduction of the boundary conditions, while the free surface presents a non-zero pressure that comes from the interaction with the floating roof. The procedure relies on implementation of Galerkin’s Weighted Residual Methods to force a condensation and express the problem only in terms of free surface variables, an idea discussed first in [27–29]. This leads [26] to a system of nm (number of nodes located at the free surface) equations given by 1 € s þ Ks η s ¼  Rs u€ b  Ps Ms η

ρ

(1)

where ρ is the fluid density, ηs is a vector that contains the nodal elevation of the free surface, u€ b corresponds to the acceleration input at the base of the tank, Ps is the vector that contains the nodal pressure of each node located at the free surface (due to the presence of the floating roof), and matrices Ms, Ks and vector Rs are obtained using the SSM and can be interpreted as equivalent mass and stiffness matrices, and as an influence coefficients vector. Note that Eq. (1) might appear dimensionally inconsistent but neither Ms has mass units nor Ks has stiffness units as discussed in [26]. The calculation of these quantities involves an appropriate iterative meshing process for the liquid and the implementation of a FEM analysis. More details on this formulation, including definition of the different matrices as well as the derivation process for Eq. (1) may be found in [26]. It should be stressed that these matrices depend on the tank geometry, meaning that any different geometry involves a new meshing process. Note that the inherent damping of the fluid, for example related to drag between liquid and tank or floating roof (for the TLD-FR application), can be incorporated into this formulation by adding the term _ s in the left hand side of Eq. (1) where Cs is the damping matrix which can be calculated based on any appropriate Cs η assumption (for example, assuming modal damping). This issue will be further discussed in Section 6.2. Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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Another important component of this modeling is the total transmitted force to the base of the tank, which is given by € s  ρ d B u€ b F ¼ ρ d A η

(2)

where d corresponds to the tank width (perpendicular direction to the ground motion) while A (a row vector) and B (scalar variable) come also from the SSM implementation [26], and similar to Ms, Ks and Rs have a strong dependence on the tank geometry. 2.2. Floating roof model Modeling of the floating roof requires consideration of its own vibratory behavior as well as of the effect of the viscous dampers that are connected to it. The former is established assuming it behaves like a flexible elastic beam (FEM modeling dependent upon the modulus of elasticity E and the moment of inertia I of the beam) and considering a coincident mesh between the floating roof and the fluid surface as shown in Fig. 2. Proper condensation of the beam FEM model [25] and additional assumption that the fluid free surface and the vertical displacements of the floating roof are the same, leads to the following equation € s þ Cf η _ s þ Kf η s ¼ F s Mf η

(3)

where Fs is the vector of nodal forces, Mf and Kf correspond to the condensed mass and stiffness matrices, respectively, of the floating roof, and Cf is the damping matrix that takes into account the damping introduced by the external dampers (non-intrinsic damping is assumed in the floating roof). If the system has Nd dampers and the ith damper has a damping coefficient Cdi then the damping matrix is given by Cf ¼ RTd Cd R d where Cd is the diagonal matrix consisting of all these coefficients 2 3 0 C d1 6 7 ⋱ (4) Cd ¼ 4 5 0 C dNd and the collocation matrix Rd is defined such that the ith row represents the relationship between damper displacements and ηs. If nodal positions and damper locations coincide then each row is simply a row vector of zeros with a single unit element at the node corresponding to the damper location. Note that the nodal forces in Eq. (3) are related to the nodal pressures Ps by [25] Z Fs ¼ G Ps with G ¼ Nη T Nη dΓ s (5) Γs

where Nη is the interpolation function within the FEM scheme used in the SSM, representing the pressure variation between nodes [26]. 2.3. TLD-FR numerical model The two models for the fluid and the floating roof can be finally combined through the nodal pressures to yield the differential equation of motion € s þ Ca η _ s þKa η s ¼  Ra u€ b Ma η

(6)

where the augmented matrices involved are defined as

h i Ma ¼ ρðMs Þ þ G1 Mf h i Ca ¼ G1 Cf h i Ka ¼ ρðKs Þ þG1 Kf Ra ¼ ρðR s Þ

(7)

If inherent damping is assumed for the fluid motion, then matrix Ca should include component þρCs as well. Eq. (6) represents a second order linear system of differential equations where the dependent variable is the vertical displacement of the floating roof and the input corresponds to the acceleration imposed at the base of the TLD-FR. An important characteristic of the TLD-FR is the force transmitted to its base, facilitating the coupling with the structure that supports it, which is given by Eq. (2), and depends on the tank geometry, the sloshing amplitude and the horizontal acceleration of the tank. The sloshing amplitude is computed by Eq. (1) if the floating roof is not installed (traditional TLD) or by Eq. (6) if the configuration corresponds to a TLD-FR. The behavior of a structure equipped with a TLD-FR can be then characterized by combining its equation of motion with Eq. (6), with u€ b corresponding to the absolute acceleration of the floor in which the damper is installed, and by utilizing Eq. (2) to estimate the force transmitted by the TLD-FR to the structure. The dynamic behavior of a traditional TLD can be ultimately described by Eq. (1) and Eq. (2) (which corresponds to Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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the SSM procedure) while the dynamics of the TLD-FR is defined by Eq. (6) and Eq. (2), denoted herein as Simplified Sloshing Model with Floating Roof (SSM-FR). Experimental validations [25] have demonstrated the accuracy of both the SSM-FR and SSM numerical models. These validated models are utilized here to investigate in detail the impact of the floating roof on the behavior of the liquid damper and establish a formal design process for it.

3. Influence of the floating roof on the fundamental vibratory behavior

Normalized modal shape

The first question investigated is how does the addition of the floating roof impact the fundamental vibration behavior of the TLD? First symmetric, rectangular, tanks are investigated (same patterns hold for non-rectangular symmetric tanks as well). The geometry of the tanks is described by the length L and water depth H as shown earlier in Fig. 1, or equivalently L and R¼H/L which uniquely define the fundamental period of the TLD [26]. The three different configurations are: (i) one with a larger TLD fundamental period (L¼8 m and R ¼0.5), (ii) one with a medium period (L ¼4 m and R ¼0.5) and (iii) one with a lower TLD fundamental period (L¼1.5 m and R¼0.5). The floating roof is represented by an elastic beam with modulus of elasticity E ¼3 GPa (corresponding to polystyrene) whereas two different values are examined for its thickness (2 cm and 18 cm), leading to values of EI ¼1400 Nm2 and EI ¼1400 kNm2, respectively. The first two modal shapes for the tank with R¼0.5, L¼8 m for the TLD (no floating roof) and the TLD-FR with EI ¼1400 kNm2 for the roof-beam are presented in Fig. 3 (same patterns hold for all other cases). Then Fig. 4 presents the frequency response functions for all cases, obtained 0.8 1st Mode 0.4 0.0 −0.4 −0.8 0.8 2nd Mode 0.4 0.0 −0.4 EI =1400 kNm2 −0.8 -L/2

TLD−FR TLD

L/2

0

Location along length of tank Fig. 3. Modal shapes for first and second mode of vibration for rectangular tank with characteristics R ¼0.5, L ¼8 m, for a TLD and a TLD-FR with EI ¼1400 kNm2. Both mode shapes are normalized to max displacement of 1.

10 2

10 2 10 0

[s2]

H (iω)

10

[s2]

−2

H (iω)

10 0

TLD−FR EI=1400 Nm2 TLD

TLD−FR EI=1400 kNm2 TLD

R=0.5 and L=8m

R=0.5 and L=8m

R=0.5 and L=4m

R=0.5 and L=4m

R=0.5 and L=1.5m

R=0.5 and L=1.5m

10−2

[s2]

100

H (iω)

10 2

10−2 0

5

10

ω [rad/s]

15 0

5

10

15

ω [rad/s]

Fig. 4. Frequency response function for TLD and TLD-FR with (plots a, c, e) EI¼ 1400 Nm2 [left column] or (plots b, d, f) EI¼ 1400 kNm2 [right column] for three different rectangular tank configurations, corresponding to (plots a, b) R¼ 0.5 and L ¼ 8 m [first row], (plots c, d) R¼ 0.5 and L ¼ 4 m, [second row] (plots e, f) R¼ 0.5 and L ¼1.5 m [third row].

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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0.8

1st Mode

Normalized modal shape

0.4

H h L h h= H

H R= L

h = 0.7

0.0 TLD−FR TLD

−0.4 −0.8 0.8

2nd Mode

0.4 0.0 −0.4 −0.8

L = 1.5m

-L/2

R = 0.5

L/2

0 Location along length of tank

H (iω) [s2]

10 2 TLD−FR TLD 10 0

10−2 0

5

ω [rad/s]

10

15

Fig. 5. Results for asymmetric tank for TLD and TLD-FR (a) Schematic of tank and geometrical details, (b) modal shapes for first and second mode (both normalized to max displacement of 1) and (c) frequency response function. Value of EI is 1400 kNm2 for the TLD-FR.

for the TLD and the TLD-FR, respectively, as

     1  Rs  TLD: HðiωÞ ¼   ω2 Ms þiωCs þKs

(8)

     1  TLD  FR: HðiωÞ ¼   ω2 Ma þ iωCa þ Ka Ra 

(9)

For all cases the tank width is taken as 1 m while a 0.5% of damping ratio (inherent damping) is assumed for all modes to define Ca or Cs. It is observed that the behavior around the first sloshing period and the first modal shape are practically the same for both the TLD and the TLD-FR whereas the floating roof moves higher sloshing modes to higher frequencies (lower periods), allowing a larger separation between these modes. Additionally, higher modes are suppressed (amplitude reduced), whereas for floating roofs with higher stiffness no higher modes are observed. This behavior is easy to explain, especially when coupled with the information in Fig. 3 about the mode shapes of different modes for the TLD: for the fundamental mode the roof is allowed to participate as a rigid body so it has limited impact on the vibratory characteristics. Neither the mode shape nor, more importantly, the fundamental period change. Coupling in higher modes requires the roof to vibrate according to the coupled dynamics, which is difficult due to its higher stiffness (note that this is the case even for the smaller EI considered). This ultimately leads to the observed suppression of the higher modes. This further supports the intended tuning of the TLD-FR to only a specific mode, since its own modes are better separated (compared to the TLD), with higher modes significantly suppressed. Fig. 5 shows similar results for a non-symmetric tank. The geometry, mode shapes for the first two modes as well as frequency response function are all shown in this figure. It is again evident that the floating-roof suppresses higher modes. In this case, since the mode shape in the fundamental mode is not completely symmetric the inclusion of the floating roof has a minor impact on it as well. These trends related to the impact of the floating roof agree with the ones reported in the experimental validation in [25]. It was further shown there that the inclusion of the floating roof induces a practically linear behavior on the sloshing (no amplitude dependence of vibratory characteristics and no wave-breaking), and that even under random excitations (earthquake excitation was utilized) the TLD-FR practically responds only in its first sloshing mode. Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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These observations mean that the TLD-FR demonstrates ultimately the same characteristics as other type of linear mass dampers and can be described by a second order system tuned to a single frequency. Within this context the force transmitted by the TLD-FR to the structure can be considered equivalent to the force transmitted by mass dampers (through the spring/dashpot connection) to the primary structure [5]. An immediate question is then how can the behavior of the TLD-FR, dependent on specific characteristics of the tank geometry, be simplified so that easier comparisons to linear mass dampers can be drawn. A parametric formulation is introduced next to support this goal. This formulation has two components: (a) simplification of equations of motion through only four variables and (b) derivation of simplified relationships to relate these variables to tank geometry and external dampers, to support the design of TLDs-FR. These two components are independently discussed. Additionally, insight is provided into the effect of the transverse tank geometry on the TLD-FR fundamental vibratory characteristics.

4. Simplification of equations of motion through parametric formulation The parametric formulation is based on a modal reduction retaining only the first mode of the TLD-FR. Let Φ represent the eigenvector associated with the first sloshing frequency and y the corresponding modal coordinate, so that ηs ¼ Φ y. The mode shape Φ is normalized so that the modal value at the wall (which is expected to be the maximum for the fundamental mode) is equal to one. This is chosen so that the modal displacement corresponds to the displacement of the floating roof at the wall. Both these terminologies, modal displacement or floating roof displacement, will be used herein to describe y. The proposed modal reduction leads then to the scalar differential equation mm y€ þ cm y_ þ km y ¼ Rm u€ b

(10)

where mm ¼ Φ Ma Φ T

cm ¼ Φ Ca Φ T

km ¼ Φ Ka Φ T

Rm ¼ Φ R a T

(11)

Further introduction of the natural frequency and the damping ratio, given, respectively, by sffiffiffiffiffiffiffiffi km ωm ¼ mm

ξm ¼

cm 2mm ωm

(12)

and of the normalized modal displacement (having the same units as y) yn ¼ ymm =Rm

(13)

y€ n þ 2ξm ωm y_ n þ ω2m yn ¼  u€ b

(14)

lead to

which is the standard equation describing the behavior of any linear mass damper [5]. Note that for liquid column dampers derivation of Eq. (14) requires definition of a similar normalized damper displacement [14]. The transmitted force can be expressed using the previous transformations as   Rm (15) y€  ρdBu€ b F ¼  ρd A Φ Mm n The first term in this equation is related to the liquid sloshing and the second to the acceleration imposed at the base of the tank. When the floating roof has small accelerations the first component can be neglected and the total transmitted force to the base corresponds to the total mass of liquid multiplied by the acceleration of the tank base (fluid-tank behaves as a rigid body). This shows that term ρdB may be considered to correspond to the liquid mass, denoted from now on as mliq, while B may be interpreted as the transversal area of the tank. Of course, for general excitation conditions the effect of the sloshing on the transmitted force needs to be taken into account and for better characterizing this effect the following non-dimensional parameter, termed efficiency index, is defined  R (16) γ ¼ AΦ m Bmm Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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The introduction of the efficiency index allows to rewrite Eq. (15) as F ¼  mliq ½γ y€ n mliq u€ b

 ¼ mliq ½γ2ξm ωm y_ n þ mliq γω2m yn  mliq ½1 γ u€ b

(17)

where for deriving the second equality Eq. (14) was utilized. The efficiency index represents the amount of liquid that participates in the sloshing for the fundamental mode; an efficiency index equal to zero indicates a non-sloshing condition (case described previously) in which the liquid acts like a rigid body. On the other extreme, an efficiency index equal to 1 indicates that the whole liquid mass is participating in the sloshing and contributing dynamically to the transmitted force, condition which corresponds to a traditional TMD. In general, only part of the liquid is expected to contribute dynamically in TLDs [30], which indicates that the values of the efficiency index range between zero and one. The efficiency index discussed here is equivalent to the similar parameter utilized to describe the behavior of liquid column dampers (TLCDs/LCVAs) [14]. In that particular case, the efficiency index is related to the percentage of the total mass that participates in the horizontal liquid vibration and its calculation is straightforward, as explicit relationships exist the relate the efficiency index to the geometry of the liquid columns. For the TLD-FR it needs to be ultimately calculated through the FEM approach discussed in Section 2. It leads, though, to an identical representation of the transmitted force to the supporting structure, with only one part of the total liquid mass contributing through inertia effects to that force,

 represented by term mliq γ y€ n in Eq. (17). Should be also pointed out that different definitions of the efficiency index have been proposed for mass dampers (see, for example, [5]), but the definition adopted here is preferable (which is also the same presented in [14]) since it gives direct information about the amount of liquid that participates in the fundamental vibration mode. Finally, the parametric formulation leads to a description of a TLD-FR operating in its fundamental mode through only 4 variables when considering response in terms of normalized model displacements: its mass mliq , its efficiency index γ , its natural frequency ωm , and its damping ratio ξm . The first three variables are related to the tank geometry, with the latter two completely dependent upon the tank transversal area and the first one independently defined through proper selection of the tank width, whereas the fourth variable is ultimately related to the external dampers and (as will be discussed later) any internal damping existent in the vibration of the liquid. If the modal displacement is required then knowledge of ratio mm =Rm is also needed to invert the relationship of Eq. (14). But if only the response of the primary structure is of interest then the latter is not needed. This formulation also leads to identical formulation of the equations of motion for the liquid and the transmitted force to its base as established for liquid column dampers (for any γ) and TMDs (for γ ¼1). This means that the behavior of TLDs-FR and the behavior of liquid column dampers can be directly related, and results from the analysis of the latter dampers [5,14] can directly extrapolate to the TLD-FR. For example, higher values for γ or of mliq lead to better efficiency (greater suppression of the vibration of the primary structure) whereas they also affect the optimal tuning values for ωm and ξm . In particular, the optimal values of both ωm and ξm of the TLD-FR decrease if mliq or γ increase whereas the efficiency (suppression of primary structure vibration) is sensitive to the frequency of the TLD-FR but not to damping ratios lower than the optimal damping. The question that then arises is how does the transversal area affect the efficiency index and the sloshing frequency and what values of the efficiency index are possible for typical tank configurations? This question is examined next by investigating the sensitivity of these two properties to the tank geometry characteristics.

5. Relationship of transversal geometry of the tank to vibratory characteristics The values of the efficiency index γ and sloshing period Tm (defined as 2π/ωm) for different geometries of TLDs-FR are examined in this section. In all instances the identification of γ and Tm are obtained through the SSM-FR described in Section 2

Fig. 6. Schematic of different tank geometries used in the investigation: (a) rectangular, (b) V-type and (c) T-type.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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for moderately high values of the floating roof stiffness (though these values ultimately do not impact the fundamental mode behavior as demonstrated earlier). The different tank geometries examined are shown in Fig. 6. 5.1. Case 1: Rectangular tanks The simplest configuration is a rectangular tank [plot(a) of Fig. 6], completely defined as discussed earlier by the length L and the water depth H or the aspect ratio R ¼H/L. Results for this tank are presented in Fig. 7. It is observed that for aspects ratios greater than certain value (approximately R 40.8) the sloshing period only depends on L. On the other hand, the

Fig. 7. Iso-curves as variation of R and L for rectangular TLD-FR for (a) fundamental period and (b) efficiency index γ.

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Fig. 8. Iso-curves for V-tank TLDs-FR. Variation with respect to h and a is examined for fundamental period in top row (plots a, b, c) and efficiency index in bottom row (plots d, e, f). The cases in the three columns correspond to (plot a, d) R¼ 0.2 and L ¼ 4 m, (plot b, e) R ¼0.2 and L ¼ 6 m, and (plot c, f) R¼ 0.4 and L ¼ 4 m.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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efficiency index is practically independent of L and is completely defined by the aspect ratio of the tank R. Values of R 40.8 lead to rather small efficiency indexes. The combination of these two trends demonstrates that such values for R (shaded area in Fig. 7) should be avoided; not only do they lead to lower values for the efficiency index but also they prohibit the modification of the natural period by changing the water depth H (sloshing period is not sensitive to changes of H for these configurations). 5.2. Case 2: Non-rectangular tanks Further manipulation of the efficiency index and sloshing period can be established by considering more complicated, non-rectangular tanks. Here two different tank geometries are examined, also shown in Fig. 6, the V-Tank and the T-Tank. In order to keep same representation as for rectangular tanks, the aspect ratio is also utilized here to characterize both tanks. Additionally, two new ratios are introduced for each type of tank, h and a for V-Tank and r and e for T-Tank. Details for these ratios are included in Fig. 6 [parts(b) and (c), respectively]. In both types of tanks, the section shape is defined by three dimensionless ratios: h, a and R for V-Tanks and r, e and R for T-Tanks. Then, the incorporation of the length L allows to scale the shape to a particular size (similar to rectangular tanks where R defines the shape while L defines the size of the section), such that it is feasible to have tanks with the same shape (i.e., same dimensionless ratios) but different sizes (i.e., different lengths). The sensitivity of the efficiency index and sloshing period to changes in the geometry of V-Tanks is studied first and the results are presented in Fig. 8. Set of iso-curves for sloshing period and efficiency index are shown for three different configurations: (i) R¼ 0.2 and L¼4 m, (ii) R¼0.2 and L ¼6 m and (iii) R¼0.4 and L ¼4 m. In that sense, configurations (i) and (ii) correspond to tanks with the same shape but different sizes, while configurations (i) and (iii) correspond to tanks with the same length L but different shapes. The first interesting result is that the sensitivity of the sloshing period increases for lower values of a and higher values of h. These configurations correspond to tanks with transversal section close to conical. Moreover, these configurations present also a higher efficiency index indicating that the largest part of the liquid [more than 88% for configuration (i) and (ii)] contributes to the sloshing. It is also observed that configurations (i) and (ii) present the same efficiency index, indicating that this index is sensitive only to the shape and not to the size of the tank. On the other hand, comparing configurations (i) and (iii), it is evident that for higher values of R the efficiency index shows an important

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Fig. 9. Iso-curves for T-tank TLDs-FR. Variation with respect to r and e is examined for fundamental period in top row (plots a, b, c) and efficiency index in bottom row (plots d, e, f). The cases in the three columns correspond to (plot a, d) R ¼0.2 and L ¼ 4 m, (plot b, e) R¼ 0.2 and L ¼6 m, and (plot c, f) R ¼0.4 and L ¼ 4 m.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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reduction. These trends for variation with respect to R and L follow the ones identified for rectangular tanks, but additionally demonstrate an important sensitivity to the other transversal area characteristics. A similar analysis (Fig. 9) is performed for the T-Tank examining the same combination for R and L. Large sensitivity of the sloshing period for lower eccentricities e and higher radius values r is again reported, whereas the efficiency index is sensitive to the shape but not to the size of the tank, with lower aspect ratios contributing to larger efficiency indexes. The overall discussion shows the following trends: (1) tanks with lower aspect ratios present higher efficiency indexes (it is easier for shallow tanks to incorporate greater amount of liquid in the sloshing vibration), (2) the efficiency index depends entirely on the shape and not on the size of the tank (tanks can be scaled to different lengths without affecting the efficiency index), and (3) the sloshing period sensitivity due to changes in the geometry is higher for configurations that also correspond to higher efficiency indexes. The results show that the potential values of the efficiency index established are high, especially for non-rectangular tanks, definitely higher than the values that can be established for liquid column dampers, for which it is impractical to accomplish γ 40.6–0.7 (even such cases require significant portion of the mass to be concentrated in the horizontal part of the tube). TLDs-FR can therefore accomplish higher efficiency that liquid column dampers while maintaining some of their desirable characteristics (behavior that is linear and concentrated within a single vibration mode). Furthermore, tanks with non-rectangular cross sections present significant advantages over rectangular tanks as they can help accommodate larger values of the efficiency index while facilitating a more versatile tuning process. Should be stressed, though, that for such tanks bi-directional control might be impractical, since the cross-sectional tank geometry in the transverse direction is nonuniform.

6. Design of TLDs-FR and relationships between parametric description and tank configuration As established in the previous two sections the behavior of the TLD-FR can be described through only four variables, representing characteristics of the fundamental sloshing mode, whereas the tank geometry has a big impact on these variables. The design of the TLD-FR can be performed in this four-dimensional parametric space, adopting any appropriate criteria and design approach that has been proposed for mass dampers [5,31–34], for example, suppression of harmonic responses [1,35] or control of stationary response [2,36,37] or even optimization based on first-passage reliability-based criteria [14,38]. The selection of the damper configuration and tank geometries can be then subsequently established based on the identified optimal parametric configuration. This two-stage approach is preferable over the alternative, i.e. the optimization directly for the tank/damper configurations since the latter (1) can be a non-convex problem since it is evident from the discussion in the previous section that multiple geometries may lead to exactly same vibratory characteristics, and (2) has a large associated cost since it requires implementation of the SSM-FR at each iteration of the optimization to derive the equations of motion for the TLD-FR. Of course the proposed two-stage approach still requires a match of the parametric formulation to the physical characteristics of the TLF-FR. For mliq this is easily established by selecting an appropriate transverse width d, since the latter has no effect on the other vibratory characteristics. If architectural constraints do not allow for the use of such a large d, or if constraints for d exist so that TLD-FR has a desired fundamental frequency also in the transverse direction, then multiple TLDs-FR can be considered with sum of widths equal to the targeted width. The tank geometry can be then chosen to match the intended efficiency index and frequency while also considering the allowable displacement of the roof, whereas the dampers can be chosen to match the targeted damping ratio. These two are separately discussed next, with greater emphasis placed on the first topic. 6.1. Transverse tank geometry The tank geometry can be chosen based on the targeted efficiency index and frequency. For rectangular tanks this is a one-to-one mapping as shown in Fig. 7 and the two vibratory characteristics lead to a unique definition of the two tank dimensions (H–L). This feature does not extend to more complex tank geometries that are characterized by a larger number of geometrical parameters (defining the transverse tank area), such as the V and T tanks discussed previously (also shown in Fig. 6). In this case different tank geometries can lead to the same characteristics of vibration. An additional challenge in this identification is the fact that this mapping does not have an analytic closed-form solution, relying on FEM analysis based on the SSM-FR approach. To circumvent the computational burden of repeating the FEM analysis for every geometrical configuration examined, derivation of an approximate relationship based on Kriging metamodeling is adopted here. Kriging [39,40] in this case offers a surrogate model between the input parameters (tank geometrical characteristics) and the output of interest (vibratory characteristics used to provide the simplified parametric formulation) and is built using an extensive database of high-fidelity (established through the FEM analysis) evaluations of the input–output pair, also known as support points. Once generated, the metamodel can be then used to represent the input–output mapping and find all configurations that lead to the targeted vibration properties, for example through an exhaustive blind search [41]. Kriging is chosen here for this purpose (over other available metamodels) because it has been proven accurate in approximating complex functions and is highly efficient in blind search settings since it can be easily parallelized [42] as it relies simply on matrix manipulations. Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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The intended selection of tank geometries is established finally through the following process: Step 1: First a Kriging metamodel is developed to approximate the metamodel output of interest z, corresponding to the natural frequency (period Tm is used here), the efficiency index γ and the parameters needed to describe the actual liquid modal displacement (ratio Rm/mm) so that z ¼ ½ T m γ mm =Rm T , as a function of the tank geometry, corresponding to the metamodel input x. The latter depends on the tank characteristics, for example, for V-Tanks the input is defined by x ¼ ½ L R a h T . For developing this metamodel an initial database of nsup (¼8000 in the case study discussed later) support points is established based on the FEM approach and some choice for the EI value (this selection typically does not affect the dynamic of the system as demonstrated earlier Section 3). These support points can be selected using Latin Hypercube Sampling based on the range X that the input x is expected to take values in, supplemented, if necessary, with any adaptive approach to improve accuracy in specific regions of interest [40,43]. Note that this entire process is independent of the structure and application of interest, i.e. the metamodel is developed once and used then as needed. Step 2: Once the metamodel is available, the input parameters x are identified that lead to the intended targeted values for Tm and γ. This can be established in a blind-search approach, especially for complex geometries for which a manifold is expected to exist in the x space that provide same z values (more on this in the case study discussed later). Such manifolds cannot be identified through optimization algorithms yielding only countable solutions. This blind-search entails a derivation of another, very extensive in this case, database (10,000,000 components in the case study discussed later), something that is performed by utilizing the already established surrogate model. Note that this latter database is also independent of the application considered and can be used multiple times. Step 3: Finally, the value mm/Rm can be used to examine the implications of the liquid vibration, i.e. whether the floating roof might encounter obstacles in its vibration and hits side walls or bottom of the tank. This can be accommodated by calculating the free space available as will be illustrated in the example considered later. This latter consideration leads then to the final selection of the tank configuration. 6.2. Damper configuration The external dampers can be chosen to facilitate the desired damping ratio ξm . The relationship between this damping ratio and the external dampers can be established by utilizing Eqs. (7) and (11), which leads to 2 3 C d1 0 7 T 1 T6 ⋱ Φ G Rd 4 (18) 5R d Φ ¼ 2ξm ωm mm 0 Cd N Based on known ξm (and ωm ) the damper coefficients can be then chosen so that the above equality is satisfied. Using the same coefficients for all dampers will lead to a unique solution, though the approach allows the adoption of dampers with different coefficients. Note that Eq. (18) can also facilitate a direct calculation of the damping ratio ξm if damper characteristics and tank geometry are known (in other words, can be used to solve the forward problem). It was experimentally demonstrated [25], though, that TLDs-FR present some level of inherent damping (i.e., damping that is not related to the external dampers), probably associated to drag between tank, liquid and the floating roof. This inherent damping also contributes to the total damping ratio ξm and this can be taken into account in the design process by expressing the total damping ratio as the sum of the inherent (ξinh) and the external (ξext) damping ratios, such that ξm ¼ ξinh þ ξext. The external dampers are then designed based on ξext, i.e. the value for ξext is used in Eq. (18) instead of ξm. Further experimental investigations are required, though, to obtain a deeper understanding of ξinh, including suggesting appropriate values for it, since the results reported in [25] demonstrate significantly higher damping values than the ones reported for TLDs [19] (in other words ξinh is related to the floating roof itself). A conservative approach is to use a small value for ξinh; as discussed earlier high sensitivity is expected for values of ξm lower than the optimal one (common characteristic of linear mass dampers [5]), yielding a significant deterioration of the level of protection offered by the TLDFR. The suggestion above will guarantee a damping coefficient equal or greater than the optimal damping ratio, avoiding such detrimental effects (stemming from having a damping ratio smaller than the optimal one). Of course if ξinh can be actually accurately quantified, then its value should be directly adopted. 6.3. Design process summary Combining the previous concepts the following design process is established for the TLD-FR. Step 1: Select values for the efficiency index γ and the mass mliq of the TLD-FR based on any constraints (higher values for these parameters will always lead to better performance). Different values for each of them can be examined. Step 2: Based on the desired optimization approach select the natural period Tm and damping ratio ξm of the TLD-FR. This entails proper modeling of the excitation and adoption of some performance objective (an example will be discussed in the next section). Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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Step 3: Based on the γ and Tm select the tank geometry. For tanks with non-uniform cross sections a manifold of different tank configurations can be identified through the process discussed in Section 6.1. For each tank configuration also obtain information for the roof displacement. Step 4: For each identified tank geometry calculate the width d to facilitate the desired mliq. Consider using multiple TLDs-FR if achieving the desired d with a single one is impractical. Step 5: If multiple tank geometries have been identified, either because different γ values have been considered in Step 1 or non-uniform cross section is examined in Step 3, incorporate any additional architectural constraints as well as constraints related to the roof displacement to select final one. Step 6: Assume an inherent damping ratio ξinh and an appropriate location for the external dampers, and then calculate the characteristics of these dampers through the process discussed in Section 6.2 [goal is to establish an external damping ratio equal to ξext ¼ ξm  ξinh].

7. Case study for design of TLDs-FR The TLD-FR design procedure discussed above is demonstrated in this Section in a practical case study. The focus is here on the selection of the characteristics of the TLD-FR so a relatively simple, but very popular for mass dampers [37,44,45], performance quantification is adopted, considering the variance reduction under stationary earthquake excitation. The TLDFR is applied to a single-degree of freedom (SDOF) oscillator, representing the fundamental mode for the structure that is controlled. Additionally, considerations about the maximum floating-roof displacement are incorporated into the design process. Though design procedures for mass dampers mainly focus on the suppression of the vibration of the primary structure, the displacement of the secondary mass is also of concern [46,47]. This is particular important for TLDs-FR or even for TLCD and LCVA applications, so that undesired behaviors that compromise its effectiveness can be avoided, e.g., collision of the floating roof at the bottom part of the tank for the case of TLD-FR or displacement of parts of the liquid outside of a column in TLCDs or LCVAs. For TLD-FR applications with complex geometries such concerns can be indirectly incorporated in the analysis by selecting a configuration that allows greater movement of the floating roof among the manifold of solutions yielding the same vibratory characteristics. 7.1. Equation of motion and response under stationary excitation The equations of motion of a structure with a TLD-FR are established through the parametric formulation by utilizing Eq. (17) to describe the transmitted force and Eq.(14) to characterize the TLD-FR vibration. The mass ratio, r, defined as the ratio of the liquid mass mliq to the total mass of the structure ms (i.e. mliq ¼rms), and the tuning ratio φ, defined as the ratio of the fundamental sloshing frequency ωm to the SDOF frequency ωs (i.e. ωm ¼ φωs), are further introduced to simplify the parametric description. The dynamic behavior of the coupled system is described by four non-dimensional parameters: the mass ratio r, the damping ratio ξm, the tuning ratio φ and the efficiency index γ. If ωs, ξs and ms are, respectively, the natural frequency, the damping ratio and the mass of the SDOF, u denotes its the displacement relative to the ground and u€ g the base acceleration, then the coupled system of equations of motion is " #" # " #" #    " € # u u_ 1 0 u 2ξs 0 1 þ r γr 1þr 2 þ ωs þ ¼  ω (19) u€ g 2 s y_ n yn 0 φ y€ n 0 2ξm φ 1 1 1 Seismic excitation is modeled as a stationary process though the well known Kanai–Tajimi filter [48] whereas the response to this excitation is calculated through a state space approach [49,50], briefly summarized in Appendix A. This approach ultimately yields the structure response variance σ 2u and the normalized floating roof response variance σ 2yn , which are the performance objectives that will be needed in the design process next. 7.2. Design procedure The optimal design is formulated by identifying the characteristic of the TLD-FR that minimize σ 2u :   q ¼ arg min σ 2u ðqÞ

(20)

qAQ

where vector q contains the TLD-FR variables that are going to be optimized and Q represents the admissible design space. The optimization is typically conducted assuming specific values for the efficiency index and the mass ratio (since larger values for either of them lead to higher efficiency in terms of reduction of the displacement of the primary structure), meaning that q corresponds to the damping ratio ξm and the frequency ratio φ. This process will lead to a desired configuration in the parametric space, characterized by ropt [or equivalently mliq,opt ¼ roptms], γopt, φopt [or equivalently Topt ¼2π/(ωsφopt)] and ξm,opt. It is necessary then to select the tank geometry and the external damper configuration that lead to this configuration, following the process discussed in Section 6. As discussed earlier, an additional preference can be then incorporated in the final selection considering the amplitude of vibration of the floating roof among the configurations leading to the same efficiency. This is established through the Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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following approach: a maximum admissible amplitude is adopted yadm as a percentage of the water depth H, leading to a constraint for the standard deviation of the response, i.e. we desire that the actual response of the floating roof be bounded by yadm. This entails multiplication of σ yn by ratio Rm =mm so that the standard deviation of the normalized displacement is converted to the standard deviation of the actual displacement of the roof by the wall. Therefore, the identification of the transversal geometry of the tank is established so that Tm is equal to 2π/(ωsφopt), γ is equal to γopt and σ 2yn Rm =mm is less than a fraction (say 1/3 or 1/5) of yadm. The latter fraction is utilized to convert statistical quantities to expected maximum displacements; in the previous example (reported within the parenthesis) the maximum displacement is considered, respectively, 3 or 5 standard deviations away from the mean. An alternative approach would have been to consider the firstpassage problem to identify the probability that the displacement of the roof exceeds the acceptable threshold [14]. 7.3. Illustrative implementation For the illustrative implementation the structure is considered to have natural frequency ωs ¼ π rad/s (corresponding to 2 s of natural period) and damping ratio ξs ¼2% while the TLD-FR corresponds to the V-Tank shown in Fig. 6 earlier [plot(b)]. For the excitation common recommendations for the Kanai–Tajimi filter are adopted: ωg ¼2π ¼4ωs, ξg ¼50% and σg ¼0.11 g. Note that the values of ωs and σg do not affect the optimization in the parametric space, but only the selection of the TLD-FR geometry, with the former influencing the exact period of the TLD-FR and the latter impacting the constraint for the response of the floating roof. For the proposed blind-search identification a Kriging metamodel is established as discussed in Section 6.1. The input to this metamodel are the characteristics of the V-Tank geometry x ¼ ½ L R a h T whereas the output is z ¼ ½ T m γ mm =Rm T . The number of support points is selected as 8000 utilizing Latin-hypercube sampling within the following domain, X, for x: L A ½2m; 4m; R A ½0:2; 0:8; a A ½0:05; 0:95; h A ½0:05; 0:95. The accuracy is evaluated through a cross-validation approach [51] and the maximum error reported is lower than 2%, demonstrating a high accuracy (average error is of the order of 0.1%). For the blind-search 10.000.000 different tank configurations are generated within X. Then, the optimization of Eq. (20) is performed for different combinations of mass ratios r and efficiency indexes γ. The optimal damping and frequency ratios are reported in Fig. 10 whereas Fig. 11 shows the variation as function of r and γ of the percentage reduction of the response though the introduction of the TLD-FR, and the standard deviation of the normalized amplitude of the floating roof σ yn . The results follow anticipated trends: higher values of r and γ lead to higher values for the optimal damping ratio, to lower values for the optimal frequency, and to higher efficiency (higher reduction of the response). Simultaneously they lead to lower values for the normalized roof displacement. Ultimately the design is performed through the process presented in Section 6.3. First (step 1 in Section 6.3), the designer has to choose the values for the mass ratio and the efficiency index based on particular constrains for the specific implementation under consideration. For the illustration here we will assume that a mass ratio equal to 1% (typical value for TMDs [4]) and an efficiency index equal to 0.5 (typical value for Tuned Liquid Column Dampers [14]) are utilized. Then (step 2 in Section 6.3) the optimal frequency and damping ratios are identified. Based on the results in Fig. 10 the optimal configuration (for γ ¼0.5, r ¼1%,) of the TLD-FR correspond to: φopt ¼ 0.9875 [leading to Tm,opt ¼2.03 s ( ¼Ts/φopt)] and ξopt ¼3.53%. The use of a TLD-FR with these characteristics leads to a response reduction of 22.47% and a normalized floating roof displacement of σ yn ¼ 0:15m (Fig. 11). All these details are specified in Figs. 10 and 11. At this point (Step 3 in Section 6.3) a blind-search is performed to identify which geometries match with γ ¼ 0.5 and Tm,opt ¼2.03 s. As additional constraint, related to floating roof displacement, the constraint σ yn Rm =mm r f o yadm is utilized, and three different

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0.99

3 0.6

0.7 0.8 Efficiency Index γ

0.9

1.0 0.5

0.6

0.7 0.8 0.9 Efficiency Index γ

0.99 1.0

Fig. 10. (a) Optimal damping ratio and (b) optimal frequency ratio for different efficiency indexes and mass ratios. The implementation corresponding to mass ratio 1% and efficiency index 0.5 (discussed in detail in the case study) is also specified in the plot.

Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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0.1

35

1.8

Mass Ratio r [%]

RMS Normalized Response of the Floating Roof [m]

RMS Displacement Reduction for SDOF [%] 30

2.0

1.6

15

0.0

8

30

1.4

0.1 30

1.0 22.47 % 0.8

2

30

25

1.2

2

20 0.7

0.8

0.12

0.2 0.9

1.0

0.5

0.6

Efficiency Index γ

0.12

0.14 0.16

0.16

0.18

25

20 0.6

0.1

0.15 m 0.14

25

0.6 0.4 0.5

0.1

25

20

0.08

0.1

0.1

0.18 0.7

0.8

0.9

0.14 0.16 1.0

Efficiency Index γ

Fig. 11. (a) Reduction of the RMS displacement of the SDOF and (b) RMS normalized displacement of the floating roof. The implementation corresponding to mass ratio 1% and efficiency index 0.5 (discussed in detail in the case study) is also specified in the plot.

1.0

1.0

0.8

0.8

0.6

h

0.6

a 0.4

0.4

0.2

0.2

0.0 0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.8

0.2

0.3

0.4

1.5

3.0

0.5

Floating roof displacement 1/4 yadm ≤ σy ≤ 1/3 yadm 1/5 yadm ≤ σy ≤ 1/4 yadm σy ≤ 1/5 yadm 0.2

0.3

0.4

0.5

R

0.6

0.7

0.8

0.6

0.7

0.8

L [m]

3.5

Rm/mm

2.0

1.0

0.5

R

R

0.6

2.5

0.7

0.8

2.0 0.2

0.3

0.4

0.5

R

Fig. 12. Characteristics of the different tank geometries that match the optimal values of the TLD-FR and satisfy the various chosen constraints for the roof displacement (different symbols utilized to distinguish between the different cases for the constraints). In all plots relationship to R (horizontal axis for all plots) is shown for (a) h, (b) a, (c) Rm/mm, and (d) L. Dashed lines indicate the configuration discussed in detail in the manuscript. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

fractions fo are considered, taking the following values fo ¼1/3, 1/4 and 1/5 whereas yadm is defined as H–h (total available space for roof dispalcement). The respective geometries that match with these characteristics are presented in Fig. 12. The different symbols categorize the standard deviation of the amplitude of the floating roof in three groups: (i) amplitudes lower than 1/5 of the available Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

16

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space [identified with black squares], (ii) amplitudes between 1/5 and 1/4 of the available space [identified with blue plus signs] and (ii) amplitudes between 1/4 and 1/3 of the available space [identified with red circles]. An immediate observation is the fact that a large number of geometries exist that satisfy the desired vibratory characteristics while having different associated amplitudes for the floating roof. The ability to identify all these configurations demonstrates the efficiency of the proposed blind-search approach. Instead of finding a single configuration (if a traditional optimization approach was adopted) all these different configurations are successfully identified. Recall that our interest in the floating roof amplitude stems from the desire to avoid potential impacts with the bottom part of the tank. In that sense, geometries identified with black squares correspond to the most conservative case. For this case, lower values of h are obtained, which is expected since a lower h is associated with larger space in the vertical section of the wall. On the other hand, practically any a can be chosen for the three cases presented, indicating that its impact on the displacement of the floating roof is not as relevant as h. Additionally, the amplitude of the floating roof presents a low sensitivity with respect to the tank length L and the ratio Rm =mm , while it has an important sensitivity with respect to R when h and a have higher and lower values, respectively. The latter condition can be explained if one considers that lower values of h and higher values of a represent configurations close to a rectangular tank for which, as explained earlier, the mapping between the vibratory characteristic and the geometry is one-to-one. Therefore only few aspect ratios R match the desired vibratory characteristics (with limiting case a single match). Once the transversal area of the TLD-FR is defined (based on the chosen geometry for the cross-sectional area), the identification of d (tank width) is performed (Step 4 in Section 6.3) in order to match the desired mass (1% of the mass of the primary system). For example if the mass of the SDOF is 2.000.000 kg and a configuration with the following characteristics is chosen (among the ones available in Fig. 12), L¼2.95 m, R ¼ 0:5, h ¼ 0:10 and a ¼ 0:43, then d is established as 4.7313 m [this configuration is explicitly specified in Fig. 12]. If this width is excessive for a single tank then as discussed earlier multiple, identical tanks can be used to provide the total required width. The design choice for the tank geometry for a particular application is ultimately made (Step 5 in Section 6.3) by combining the blind-search information (Fig. 12) and the associated required width, taking into account the architectural/space constraints (which will be unique to the application examined). As a last step (Step 6 in Section 6.3), the characteristics of the external dampers can be obtained directly by the solution of Eq. (18), assuming a known inherent damping ξinh and defining the external damping ratio as ξext ¼ ξopt  ξinh. Since damping ratios lower than the optimal one deteriorate the protection offered by the TLD-FR, a conservative scenario, as discussed earlier, is to assume small values of inherent damping. For the case discussed above (mass of SDOF equal to 2.000.000 kg and TLD-FR configuration equal to L ¼2.95 m, R ¼ 0:5, h ¼ 0:10, a ¼ 0:43 and d¼4.7313 m) and assuming a pair of dampers symmetrically placed at distance 0.7375 m from the middle of the tank, the damper coefficient is established as 2.195 kN s/m assuming ξinh ¼0.

8. Conclusions A liquid mass damper consisting of a traditional Tuned Liquid Damper (TLD) with the addition of a floating roof, termed TLD-FR, was discussed in this paper. The floating roof prevents wave-breaking phenomena (common for TLDs) and provides an essentially linear behavior, while it also accommodates the placement of damping devices to facilitate the desired optimal damping level for the vibration of the liquid. Focus was placed in this manuscript on describing the behavior and establishing design processes for TLDs-FR with arbitrary cross-section geometries for the tank containing the liquid, as well as on providing a deeper understanding of the impact of the floating roof on the TLD. The already experimentally validated numerical model for describing the TLD-FR was first reviewed and subsequently this model was utilized to demonstrate that the addition of the floating roof suppresses higher sloshing modes of vibration without affecting the period or mode shape of the fundamental mode, even for asymmetric tanks. These results validate previous experimental observations showing that the TLD-FR behaves linearly and in a single sloshing mode. This linear vibratory behavior was then exploited to establish a simplification of the equation of motion, leveraging modal analysis/truncation. Additionally, through a proper parameterization and response-normalization and, in particular, through the introduction of a variable, termed efficiency index, that for TLD-FR represents the portion of the liquid mass contributing to the sloshing vibration in its fundamental mode, complete equivalence to the behavior of liquid column mass dampers was established. In the parameterized format the behavior of any structure with a TLD-FR can be described by only four parameters that are common to the parameters used for other types of mass dampers: the mass, frequency and damping ratios and the efficiency index. Due to this established equivalence all results for liquid column mass dampers and TMDs (representing the limiting case of efficiency index equal to 1) are directly extendable to the behavior of TLDs-FR. Additionally, it was shown through parametric investigation that higher values of the efficiency index are easy to obtain for the TLD-FR and that the efficiency index is related to the shape of the tank while the frequency is sensitive to both its size and shape. For complex cross-sectional geometries (i.e. non-rectangular tanks), different tank configurations can lead to the same efficiency index or fundamental frequency, creating a mapping that is not one-to-one. Motivated by the latter observation a design procedure was established to relate the four aforementioned parameters to geometrical properties of the TLD-FR. The targeted mass ratio (or more generally mass) can be established through proper selection of the tank width whereas the optimal damping ratio through proper dimensioning of the external Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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17

dampers. In the latter case a conservative estimate of the inherent damping was suggested to avoid damping values lower than the optimal one. The efficiency index and optimal frequency can be then chosen based on the cross-sectional tank geometry through a blind-search approach, so that all possible solutions can be identified. These solutions correspond, though, to different roof displacement amplitudes, and considerations for the maximum allowable displacement to avoid collisions of the floating roof with the walls or the bottom of the tank can provide an additional preference between these solutions. To support the aforementioned blind-search a surrogate modeling approach was adopted. The overall design procedure was demonstrated in the design of a TLD-FR considering stationary earthquake excitation. It was shown in this example that the proposed approach facilitates an efficient identification of different geometries corresponding to the same vibratory behavior. Additional criteria related to the maximum allowable floating roof displacement can be then incorporated to select the most appropriate design configuration between these candidates.

Acknowledgments Financial support was provided by the Pontificia Universidad Catolica de Chile, by the National Research Center for Integrated Natural Disaster Management CONICYT/FONDAP/15110017 (Chile) and by the Ministerio de Educacion Superior de Chile. This financial support is gratefully acknowledged.

Appendix A In this Appendix the estimation of the stationary response of a SDOF with a TLD-FR is reviewed. For this purpose Eq. (19) is re-arranged in state space format as x_ o ¼ Ao xo þ Bo u€ g y o ¼ Co x o

2

022  " γr  1 1 0 1

6  1þr Ao ¼ 6 4 ω2s 1 2 6 Bo ¼ 4



#

0

φ2

ωs



1þr

γr

1

1

I2   1"

2ξs

0 2ξm φ

0

3

021  1þr

  1

γr

1

1

 1  1 þr 7 5C o ¼ 0 1

0

0

0

1

0

0

3 #7 7 5

 (21)

where the state vector is defined as xo ¼ ½ u yn u_ y_ n T , the output vector is composed of the displacement for the SDOF and the normalized roof displacement of the TLD-FR, yo ¼ ½ u yn T , 0jxk is a matrix of zeros with dimension of jxk and Ij is the identify matrix of dimension j. The state-space form for the Kanai–Tajimi filter is given as x_ g ¼ Ag xg þ Bg w u€ g ¼ Cg xg

" Ag ¼

0 ω

pffiffiffiffiffiffiffiffiffiffih 2 Cg ¼ 2π so ωg

2 g

1

#

 2ξg ωg i

; Bg ¼

2ξg ωg ; so ¼

  0 1

2ξg σ g

πωg 4ξ2g þ 1

(22)

where the input w represents zero mean Gaussian white noise, ωg corresponds to the dominant frequency of the ground motion, ξg controls its bandwidth while σg defines the RMS value of the acceleration u€ g . Augmentation of the two previous state space representations ultimately leads to the final desired state-space formulation: x_ s ¼ As xs þ Bs w

h

T xs ¼ x o

xTg

iT

" ; As ¼

Ao 024

y o ¼ Cs x s # " # Bo C g 041

; Bs ¼ ; Cs ¼ Co Ag Bg

022



(23)

that has as input the zero-mean white noise w and provides output yo composed of the displacements of the SDOF system and the floating roof. Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i

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18

Under these assumptions (i.e. w being white noise sequence), the output has zero mean and a covariance matrix defined by Kyy ¼ Cs PCTs

(24)

where the state covariance matrix P is determined by the solution of the Lyapunov equation [49]: As Pþ PATs þBTs Bs ¼ 0

(25)

The structure response variance σ 2u corresponds to the first diagonal element of the covariance matrix Kyy, and the normalized floating roof response variance σ 2yn to the second diagonal element.

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Please cite this article as: R.O. Ruiz, et al., Characterization and design of tuned liquid dampers with floating roof considering arbitrary tank cross-sections, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.014i