Closed-form optimal calibration of a tuned liquid column damper (TLCD) for ﬂexible structures

The conventional optimal calibration of a TLCD for suppressing structural vibration is based on the classic 2-degrees-of-freedom (2-DOF) model in terms of one speciﬁc structural mode (usually the resonant mode). For ﬂexible multi-degrees-of-freedom (MDOF) structures, this implies that background ﬂexibility contribution from other non-resonant modes is omitted, resulting in an unbalance in the frequency response of the ﬂexible structure-TLCD coupled system. Furthermore, numerical search techniques are usually used for optimizing TLCDs, and extensive computational eﬀorts are required. Therefore, in this paper a novel closed-form optimal calibration procedure accounting for the background ﬂexibility contribution from the ﬂexible structure is developed based on the pole-placement method. The background ﬂexibility is represented by an equivalent stiﬀness that is derived from modal analysis. An analogue 2-DOF model accounting for background ﬂexibility is developed, which turns out to be a generalization of the classic 2-DOF model. The root locus analysis is performed to derive the optimal frequency ratio of the TLCD by the equal modal damping ratio criterion. A straightforward approach for determining the optimal head loss coeﬃcient is proposed based on the bifurcation point present in the root locus. It is seen from the closed-form formulas that the optimal parameters of the TLCD depend not only on the constructive parameters (such as the mass ratio and horizontal length ratio of a TLCD) but also on the structural inherent characteristics. Application to MDOF ﬂexible structures are illustrated by a 10-story shear frame. Results demonstrates that the proposed calibration procedure leads to a balanced frequency response curve and thus improves the performance of the TLCD comparing with the calibration procedure ignoring background ﬂexibility.


Introduction
Flexible structures (such as high-rise buildings, chimneys, wind turbine towers etc.) are lightly damped owing to low material damping and lack of aerodynamic damping in the cross-wind direction, making vibration inevitable [1] .Therefore, it is of crucial need to damp vibrations of these flexible structures for improving the fatigue life and the operational performance.Different types of vibration absorbers have been used for damping vibrations of flexible structures such as the tuned mass damper (TMD) [2][3][4][5] , the pendulum absorber [6][7][8][9] , and the tuned liquid damper (TLD) [10][11][12][13] .Among them, the tuned liquid column damper (TLCD) has several advantages over other devices including low cost, easy installation and maintenance, easy adjustment of damper geometry to the target frequency, and controllable damping property by the orifice opening.It can even serve as a water storage facility in case of fire emergency.It was shown to successfully suppress vibrations of high-rise buildings [14][15][16] , towers [17][18][19] , onshore or offshore wind turbines [20][21][22][23] .
TLCD was first proposed by Sakai et al. [24] for suppressing horizontal vibration of building structures, and since then it has been extensively investigated [25][26][27][28][29][30][31][32][33][34][35][36] .A typical TLCD consists of a U-shaped tube with liquid inside, and an orifice in the middle of the horizontal tube acting as a damping element.The property of the TLCD is specified by a few parameters such as the total length of the liquid inside the tube, the horizontal length ratio, the head loss coefficient (mainly depends on the orifice area), the cross-sectional area of both vertical and horizontal tubes and the liquid characteristics [25] .Many investigations were performed for revealing the influences of the TLCD parameters on its vibration damping efficacy.Colwell and Basu [25] experimentally assessed the effects of three types of liquids, i.e. water, magnetorheological fluid and glycol, on the damping performance of the TLCD for a single-degree-of-freedom (SDOF) system.Gao et al. [26] investigated the influence of several TLCD parameters through a numerical procedure.Balendra et al. [27] evaluated the effectiveness of a TLCD for reducing wind-induced vibration of towers, with emphasis on the role of opening ratio of the orifice.Matteo et al. [28] showed how the TLCD mass ratio and the structure damping ratio influence the damper performance, when the SDOF structure is subjected to earthquake excitations.Hitchcock et al. [29,30] investigated the characteristics of a TLCD having different cross-sectional areas in the horizontal and vertical tubes.Furthermore, the inherent nonlinearities of TLCD, such as the nonlinear amplitude-dependent damping mechanism [28,31] and the nonlinear air spring of a sealed TLCD [32] were also studied.
These parametric studies indicate that a proper determination of TLCD parameters is crucial to the performance of TLCD in damping structural vibrations.To facilitate calibration of the damper, optimal TLCD parameters, especially the optimal frequency ratio and the optimal head loss coefficient, were investigated by many researchers.Most of the studies were based on the classic 2-degree-of-freedom (2-DOF) structure-TLCD model, where one DOF is used for the structure representing one structural resonant mode (usually the 1st mode) and one DOF is used for the TLCD.Gao et al. [26] compared the performance of two types of TLCD (the classic U-shaped and improved V-shaped) under a harmonic excitation.Based on the classic 2-DOF model, the optimal frequency ratio and head loss coefficient were obtained by the numerical search technique, using the criterion of minimizing the maximum dynamic magnification factor of the structural displacement.Wu et al. [37] investigated the TLCD performance for suppressing a SDOF system under a white noise type of wind excitation.Due to the amplitude-related nonlinearity of the TLCD damping element, an iterative procedure together with numerical search were adopted to obtain the optimal parameters.Based on the same 2-DOF model, Debbarma et al. [38] evaluated the effect of structural parameters' uncertainties on the optimal parameters and performance of TLCD.For stiff structure subjected to earthquake excitations, Ghosh and Basu [39] proposed an alternative way of attaching the TLCD, i.e. it is connected to the 1-DOF structure model through an extra spring and a dashpot.Although the optimal parameters of the connecting spring and dashpot were analytically determined from the 'fixed-point' theory by Den Hartog [40] , the optimal TLCD parameters were still obtained by numerical search.For performance enhancement of the TLCD, Al-Saif et al. [41] proposed the use of the so-called tuned liquid column ball damper (TLCBD), where a steel ball was added inside the horizontal portion of the tube.A 2-DOF model extended from the classic 2-DOF structure-TLCD model was developed without performing parametric optimization.Later, Gur et al. [42] evaluated the performance of TLCBD based on a similar 2-DOF model subjected to random earthquake excitations, and the optimal damper parameters were obtained numerically.Recently, the spring-connected TLCD model was proposed to be further enhanced by incorporating an inerter between the TLCD and the ground [43] , which might be practically infeasible.
The classic 2-DOF model can be extended to accommodate the case where multiple TLCDs (MTLCDs) are used for damping one structural resonant mode (the 1st mode).The SDOF representing the structural resonant mode is unchanged as in the last paragraph.However, the frequency of each TLCD is different and distributed around the natural frequency of that mode, in order to enhance the robustness of vibration control in practical applications.By appropriately calibrating parameters of each TLCD, a lower (and wider) frequency response curve of the structure response can be obtained.Further, the MTLCDs provide more flexible installation options than a single TLCD (with the same total water mass), since each TLCD can be installed at a different location on a real structure.Gao et al. [44] assessed the damping performance of MTLCDs on a SDOF system under harmonic excitations.Optimal parameters, such as the number of TLCDs, the frequency ratio and head loss coefficient of each TLCD, were numerically optimized.Similar works were performed on the performance of MTLCDs for a SDOF system that is subjected to wind excitation [45] and earthquake excitation [46] .In the study conducted by Shum and Xu [47] , the capability of MTLCDs in suppressing coupled lateral and torsional vibration of a structure were demonstrated.
All the above-mentioned studies resorted to numerical search techniques for parametric optimization of the damper, which require heavy computational efforts.The iteration needed due to the inherent nonlinear liquid damping further decreases the computational efficiency.To facilitate efficient calibration of the TLCD, Shum [48] derived explicit formulas for the optimal frequency ratio by the 'fixed-point' theory (equalized dynamic amplification at the two invariant points) [40] , where the classic 2-DOF model was used as the basis.An approximate solution for the optimum head loss coefficient of TLCD was also developed by extrapolating Brock's perturbation approach [49] , so that iteration was avoided.Ghosh et al. [50] presented an alternative closedform solution for the optimal frequency ratio of the TLCD attached to a damped SDOF structure subjected to earthquake excitations.The 'fixedpoint' theory was again used with good justification even though the two invariant points do not actually exist for a damped structure.Moreover, Yalla et al. [45] derived explicit formulas for the optimal frequency ratio and head loss coefficient of the TLCD attached to an undamped SDOF system subjected to white noise excitation.Unlike in [48] and [50] , the derivations were based on the gradient method together with random vibration theory, with the criterion of minimizing the variance of the structural response [45] .These studies provided useful calibration formulas for a fast and simple design of the structure-mounted TLCD.However, same as the studies based on numerical optimization techniques, the classic 2-DOF model (with one DOF representing the 1st mode) was consistently employed in these works.This implies that the non-resonant modes (higher modes) of the flexibly structure were completely ignored in these studies.As a result, this might lead to non-optimal performance of the TLCD when installed to a flexible structure whose vibration involves multiple modes.
One solution is to apply semi-active TLCD [50][51][52][53][54] to the flexible structure, where the TLCD parameters can in principle be continuously adjusted by a proper control algorithm.In such cases, an optimal calibration of the TLCD is not necessary in advance, and the influence from all modes of the structure on the TLCD parameters is automatically accounted for during the semi-active control.For example, the head loss coefficient was continuously adjusted by an actively controlled orifice device [51] .Two different control strategies were considered, and the TLCD performance was shown to be modestly improved comparing with the passive counterpart [51] .A semi-active device similar to the one proposed by [50] was also investigated [52] , where the TLCD was connected to the primary structure through an adaptive spring with adjustable stiffness.Two different control strategies (feedforward and feedback) were discussed, and the semi-active TLCD was shown to have better performance against the change of structural parameters [52] .However, the semi-active control might still be too complex and demanding for practical applications.Another alternative is to install MTLCDs into the flexible structure [45,55,56] targeting on several vibrational modes.MTLCDs were numerically investigated in suppressing the 1st and 2nd modal responses of the flexible structure [45,55] , and each TLCD was independently tuned to one mode, either through numerical optimization [45] or simply setting the frequency ratio to be one [55] .Full-scale MTLCDs (two TLCDs) attached to a flexible structure were also experimentally investigated by real-time hybrid simulation (RTHS) technique [56] .One TLCD was tuned to the 1st structural mode and the other was tuned to the 2nd mode, where the classic 2-DOF model was used for parametric optimization.MTLCDs are able to suppress several modal responses of the flexible structure including the dominant resonant mode (the 1st mode) and the higher modes, but at the cost of additional amount of liquid needed.Moreover, the use of the damper targeting on higher modes is not cost effective (its vibration damping effect is not fully exploited) since these mode are not significantly excited.Therefore, a single TLCD targeting at the resonant mode (the 1st mode), with its parameters calibrated accounting for the influence from higher modes, might be preferred in many situations.Developing an efficient optimal calibration procedure for it (a single TLCD on flexible structures) becomes essential.
Therefore, the present paper focuses on the development of closedform formulas for the optimal TLCD parameters taking into account the influence (background flexibility) from the higher modes of a flexible structure.First, an analogue 2-DOF model for the flexible structure-TLCD system is established, where the background flexibility is represented by an equivalent stiffness derived from modal analysis of the flexible structure.It is shown that this analogue 2-DOF model is an generalization of the classic 2-DOF structure-TLCD model that was extensively used in previous studies.Next, based on the characteristic equation of the analogue 2-DOF model, explicit formulas for the optimal TLCD parameters are developed by means of the pole-placement calibration approach [57] .Equal modal damping ratio (of two free vibration modes) criterion is employed for deriving the optimal frequency ratio.This is achieved by guaranteeing the two of the three roots of the characteristic equation being inverse points with respect to a reference frequency [57] .Then, the existence of a bifurcation point in the root locus is used to obtain a reference equivalent damping ratio, whereby the optimal head loss coefficient can subsequently be obtained using linearization technique for the nonlinear damping term.A calibration flow chart is presented as a ready-to-use practical tool.Finally, the proposed explicit calibration procedure is evaluated by a 10-story shear frame model.Results indicate that the proposed calibration formulas lead to a balanced frequency response curve of the flexible structure, and thus enhance the performance of the TLCD comparing with the calibration procedure ignoring background flexibility from the higher modes.

Mathematical modelling
When the structure is flexible, the TLCD will experience its support motion not only from the resonant mode (normally the fundamental mode), but also from non-resonant background modes (higherfrequency modes).This is represented in Fig. 1 (a), where the TLCD support (attachment location) is experiencing a total displacement   that consists of the resonant part   and a displacement   from the nonresonant background modes.This additional displacement   may also be understood as being due to an additional flexibility, i.e. the background flexibility.It turns out that this is equivalent to the so-called quasi-static correction term related to truncated modal expansion in structural dynamics [58] .

Reduced-order model of the flexible structure installed with a TLCD
As shown in Fig. 1 (b), the TLCD is a U-Shaped tube with an orifice installed at the center of the horizontal tube.The horizontal length, the vertical length and the total length of the liquid inside the TLCD are denoted by ,  and , respectively, where  = 2  + .Denoting the liquid density  and the cross sectional area  for both the vertical and horizontal tubes (herein, only the TLCD with uniform cross section is considered), the total mass of the liquid inside the TLCD can be calculated as: The effect of an additional flexibility between the primary structure and the TLCD is shown in the modal analogue in Fig. 1 (b).This effect is modelled by connecting the TLCD to the structural mass (modal mass   ) through a spring with the stiffness   related to the background flexibility.The modal mass   and the modal stiffness   corresponding to the resonant mode, and the spring stiffness   ( 1∕   being the background flexibility) are obtained from the following modal analysis of the flexible structure.
The flexible structure (such as the one in Fig. 1 (a)) is normally modelled by multi-degree-of-freedom (MDOF) model using for example the finite element method (FEM).The inherent structural damping is neglected (  =  ) in the subsequent damper calibration procedure to facilitate deriving closed-form formulas, which is also common practice [2,57] .It is considered reasonable because structural damping is insignificant comparing with the additional damping introduced by the damping device.Moreover, it was shown that the influence of structural damping on the optimal TLCD parameters is minor [48] .Therefore, the equation of motion of the uncontrolled flexible structure can be written in a compact form as: where  and  are the mass and stiffness matrices of the flexible structure. (  ) and  (  ) represent the displacement vector and load vector, respectively.The double dots over  (  ) denotes the second derivative with respect to time.From the generalized eigenvalue problem, the eigenvectors  1 , ⋯ ,   and the corresponding eigenfrequencies  1 , ⋯ ,   can be obtained.Following the standard procedure of modal analysis, Eq. ( 2) can be converted into  decoupled modal equation of motion: where      =  2       .  (  ) is the  th modal coordinate.The displacement vector can be expressed as the following modal expansion: In the context of vibration control using TLCD, the structural response is normally dominated by the 1st mode, i.e. the resonant mode, to which the TLCD is targeted.Quasi-static correction in truncated modal expansion [58] indicates that for the modal coordinates of the higherfrequency modes (non-resonant modes), the influence of inertial forces can be ignored, leaving only the quasi-static part.It should be noted that If a higher mode is targeted on rather than the 1st mode, the inertial forces from other non-resonant modes indeed will play a role.In the present study targeting on the 1st mode, Eq. ( 4) can be approximated as: where the well-known expression has been used to obtain the last line of Eq. ( 5) .
Next, the task is to determine the displacement   (  ) at the location where the damper is attached (the damper support, as shown in Fig. 1 (a) the TLCD is attached to the   th floor), when a force   (  ) is acting at this same location.The connectivity vector  = [0 , ⋯ , 1 , ⋯ , 0]  identifies the fixture point of the damper, with a single unit entry placed at the DOF to which the damper is attached.Therefore,   (  ) and   (  ) can be related to  (  ) and  (  ) as: Using Eq. ( 6) and Eq. ( 5) , the following relationship between   (  ) and   (  ) is obtained: The loads and responses are assumed to be harmonic varying, i.e.   (  ) =  , 0   ,   (  ) =  , 0   ,  1 (  ) =  1 , 0   , where  , 0 ,  , 0 and  1 , 0 are the amplitudes.By use of Eq. ( 3) and Eq. ( 7) , the input-output relationship between  , 0 and  , 0 becomes: where the first term inside the parentheses is related to the resonant mode (the 1  mode), with the corresponding modal stiffness   and modal mass   being defined as: where the normalized mode shape (unit modal displacement at the damper) has been used.
The remaining part inside the parentheses defines the equivalent stiffness   due to the background flexibility [2] :

Equations of motion
As shown in Fig. 1 , the total displacement of the structure consist of two components: the displacement   due to the resonant mode and the relative displacement   due to the non-resonant background modes. is the vertical displacement of liquid in the TLCD.The kinetic energy of the analogue 2-DOF model depicted in Fig. 1 (b) is written as: The potential energy of the system is written as: Inserting Eq. ( 11) and Eq. ( 12) into Lagrange's equation in analytical dynamics [59] , one would obtain the following equations of motion (EOMs) of the system: where  (  ) is the external force,   , = −1∕2 | v | v is the inherent nonlinear damping force of the TLCD;  is the head loss coefficient, which can be controlled by varying the orifice area [60] ;  is the gravitational acceleration.
Although there are three variables in Eq. ( 13) , only   and  are independent variables.Comparison of Eq. (13a) and Eq.(13b) leads to an explicit expression of   in terms of   ,  and their 2nd time derivatives.Inserting this expression of   into any two of the three equations in Eq. ( 13) , the following nonlinear EOMs for the analogue 2-DOF model are obtained: As shown from both previous studies [25,28,48] and our preliminary time-domain simulations using Eq. ( 14), the nonlinearity in the damping term for the TLCD is weak and has insignificant influence on the system responses.By replacing this nonlinear damping force,   , = −1∕2 | v | v , with the equivalent linear damping force,   , = −   v (   being the equivalent damping coefficient of the TLCD, will be discussed in Section 3.4 ), the linearized EOMs becomes: For conciseness of analysis and presentation, a few dimensionless parameters are introduced for Eq. ( 15).The linearized EOMs can thus be recast as: in which the frequency ratio of the TLCD to the structure , the ratio of the modal background stiffness to the modal dynamic stiffness , the ratio of the horizontal length to overall length of the liquid column , the ratio of the liquid mass to the modal mass of the structure  and the equivalent damping ratio of the TLCD   are defined as: where   = 2  is the equivalent stiffness of the TLCD;   =  is the total mass of the liquid, cf.Eq. ( 1) ; Eq. (16b) turns out to be a forth-order ordinary differential equation, due to the existence of the equivalent spring   .As   ⇒ ∞ corresponding to ignoring the background flexibility, Eq. (16a) and Eq.(16b) are reduced to the following second-order differential equations: which are exactly the EOMs of the classic SDOF-TLCD system (classic 2-DOF system with background flexibility ignored).Therefore Eq. ( 16) is the generalization of Eq. ( 18) .Assume a harmonic external force,  (  ) =  0   =      (  is a dimensionless force intensity factor), acting on the structure with an circular frequency  .The normalized (by the static response  0 ∕   ) complex amplitudes of forced modal response,   , and of the liquid response, , can be obtained from Eq. ( 16) as: where  = √ −1 is the imaginary unit.The non-dimensional frequency ratio, , employed in Eq. ( 19) is defined as: For Eq. ( 18) , the normalized complex amplitudes of   and  become:

Optimal calibration of the TLCD
Fig. 2 (a) presents the dynamic amplification (the absolute value of the complex amplitude) of structural response of the analogue 2-DOF system (with background flexibility   ≠ ∞ considered as shown in Fig. (1) ), with the mass ratio  = 0 .02 , the liquid length ratio  = 0 .6 and the frequency ratio  = 1 under various values of the equivalent damping ratio   .In addition, results corresponding to the classic 2-DOF system ignoring the background flexibility (   ⇒ ∞) are also shown in Fig. 2 (b).As seen, two fixed points (neutral frequencies, denoted by  1 and  2 ) around the resonant frequency   exist in both subfigures, with their amplitudes being independent of   .The amplitude of point  1 (0.9660, 20.805) in Fig. 2 (a) is a bit lower than that in Fig. 2 (b) (  1 (0.9664, 21.073)), while the amplitude of point  2 (1.0261, 13.510) in Fig. 2 (a) is around 1% larger than that in Fig. 2 (b) (  2 (1.0264, 13.403)).These results indicate that the fixed-points concept [40] hold for both the analogue 2-DOF systemand the classic 2-DOF system, although the specific amplitudes of  1 and  2 are slightly different.Therefore, the optimal frequency ratio,   , can in principle be derived by equalizing the dynamic amplifications at these two points [40] .For the classic 2-DOF system, the derivation of   is presented in Appendix C.1 .However, a direct application of this procedure for the analogue 2-DOF system will lead to a ninth-degree equation in the independent-of-damping-ratio step, cf.Eq. (B.3) in Appendix B .Hence, this conventional calibration procedure is no longer applicable.
Instead, the pole-placement (root locus) calibration method [57,61] is applied herein.Such calibration procedure outlined in the following sections is concerned with the proper placement of the complex system poles obtained from the characteristic equation of the analogue 2-DOF system.A set of closed-form formulas for optimal tuning of the TLCD in flexible structures can then be derived.Since the current study focuses on deriving the closed-form formulas for optimal calibration of TLCDs accounting for the background flexibility, only the analogue 2-DOF system is presented here.Optimal calibration of the classic 2-DOF system using both the equalized dynamic amplification approach and the pole-placement approach are elaborated respectively in Appendix C .

Characteristic equation
The characteristic equation for the analogue 2-DOF system shown in Fig. 1 (b) is obtained from the denominator of Eq. (19a) : As seen, for   = 0 the roots (denoted by  0 , 1 ,  0 , 2 and  0 , 3 , as shown in Fig. 3 ) of the characteristic equation are determined by the cubic equation (the even-power terms and constant term in Eq. ( 22) ).This limit is reached for vanishing damping of the TLCD.In the opposite limit   ⇒ ∞, the roots (denoted by  ∞, 1 and  ∞, 2 , see Fig. 3 ) are determined by the quadratic equation (the quadratic equation inside the bracket of the last term in Eq. ( 22) ).This limit implies the orifice area inside the TLCD is totally blocked.In this case the TLCD can be treated as a concentrated mass,   , connect to the structural modal mass,   , through the stiffness   , whereby the two roots represent the vibration modes for the coupled 2-DOF system.
When   ⇒ ∞, Eq. ( 22) is reduced to Eq. ( 23) , corresponding to the characteristic equation of the classic SDOF-TLCD system:  The arrows denote the evolution directions of roots of Eq. ( 22) with increasing value of   .The three circles ( ) on the horizontal axis denote the roots  0 , 1 ,  0 , 2 and  0 , 3 of Eq. ( 22) obtained in the limit   = 0 , and the two square markers ( ) on the horizontal axis denote the roots  ∞, 1 and  ∞, 2 obtained in the limit   ⇒ ∞.For the limit case   = 0 , the roots are determined by the quadratic equation (the oven-order terms in Eq. ( 23) ).The opposite limit   ⇒ ∞ leads to only one root  ∞ = 1∕ √ 1 + , associated with an extra mass   added to the modal mass   .Further, as can be seen from Eq. ( 21) , the classic SDOF-TLCD system is reduced to the classic SDOF-TMD system as  trends to 1.

Equal modal damping ratio
Once the values of , , ,  and   are provided, six complex roots of the sextic characteristic equation Eq. ( 22) can be obtained.Let three roots be denoted by  1 ,  2 and  3 , which lied in the first quadrant, and the symmetrically located roots − ς 1 , − ς 2 and − ς 3 are located in the second quadrant, where the bar symbol ( ̄) denotes the complex conjugate.By varying the value of the equivalent damping ratio   from 0 to ∞, the so-called root locus diagrams for these six roots can be plotted in the complex plane.Due to symmetry characteristic of roots, only three of six roots in the first quadrant are investigated.The basic dynamic characteristics of the system can be revealed by these roots [61] .
It was demonstrated in [57] that for the classic SDOF-TMD system, equalized dynamic amplification at the two invariant points leads to equal modal damping ratio of the two free vibration modes.For the present analogue 2-DOF system, the idea is that two of the three complex roots (from the sextic characteristic equation) should have equal modal damping ratio [61,62] .Similar to the pole-placement analysis of the classic SDOF-TMD system [57] , the property of equal modal damping ratio for the current analogue 2-DOF system can be achieved by the two complex poles being inverse points with respect to a reference frequency [61,62] .Fig. 3 depicts three possible types of root locus diagrams for the analogue 2-DOF system, depending how the calibration is performed.The three circles ( ) on the horizontal axis denote the roots  0 , 1 ,  0 , 2 and  0 , 3 of Eq. ( 22) obtained in the limit   = 0 , while the two square markers ( ) on the horizontal axis denote the roots  ∞, 1 and  ∞, 2 obtained in the limit   ⇒ ∞.The specific data of Fig. 3 are obtained from the numerical example in Section 5 Fig. 3 (a), 3 (b) show the case of frequency ratio being slightly larger than the optimal values ( ∕   = 1 .2 ), where   denotes the optimal frequency ratio and will be derived below (as shown in Eq. ( 31) ).In this case, one branch of the root locus forms a local closed curve starting from  0 , 1 and ending at  ∞, 1 with the increasing value of the equivalent damping ratio   .Likewise, another branch (as shown in Fig. 3 (b)) also forms a closed curve starting from  0 , 3 and ending at  ∞, 2 with a coordinate value in the real axis far more larger than the first branch of root locus.In contrast, a non-local branch starts from  0 , 2 and extends to the imaginary axis.Fig. 3 (c), 3 (d) present the case of frequency ratio being smaller than the optimal frequency ratio ( ∕   = 0 .8 ).Comparing Figs.3(a) and 3(c) the configurations of the two branches are interchanged.On the other hand, the third branch shown in Fig. 3 (d) remains the same as in Fig. 3 (b).
Therefor it is anticipated that only the two local branches of the root locus will intersect at the so-called bifurcation point,  * , under the condition  =   , as shown in Fig. 3 (e).The root locus diagrams corresponding to the optimal frequency ratio condition are sketched in Fig. 3 (e), 3 (f).As seen, in one limit   = 0 three roots (  0 , 1 ,  0 , 2 and  0 , 3 ) appear on the positive real axis.When increasing   from 0 to a critical value  * , these three roots move into the complex plane.It follows from the inverse point characteristics [61,62] that the first two roots (  1 and  2 ) lying on the same line containing the origin of the complex plane (as seen in Fig. 3 (e)), have equal argument, corresponding to equal modal damping ratio.When   =  * , the two local branches intersect at the bifurcation point  * .Further increasing   leads to branch off in two directions along a circle with radius of  centered at the origin point of the complex plane.As for the third root, the corresponding root locus diagram (as shown in Fig. 3 (f)) seems independent of the frequency ratio, which is identical to Figs. 3(b) and 3(d) .The references [61,62] present three types of possible root locus diagrams for the similar sixtic charac-teristic equation, cf.Fig. 2 in [61] .However, the systems investigated in [61,62] involves three DOFs, which are different with the current 2-DOF case (   and  ).Therefore, the triple-bifurcation point (employed in references [61,62] ) which involves a intersection between those three branches does not exist in the current case.
In the following derivation of the optimal frequency ratio and damping ratio of the TLCD, the root locus shown in Fig. 3 (e) is aimed for, where the a bifurcation point exists and the two complex roots are inverse points (as long as 0 ≤   ≤  * ).
In the limit   ⇒ ∞, the characteristic equation Eq. ( 22) becomes: from which two roots can be obtained,  2 ∞, 1 and ∞, 1 , the specific expression for  can be obtained from Eq. ( 24) as: In the opposite limit   = 0 , the characteristic equation Eq. ( 22) becomes: Based on the inverse points condition of  1 and  2 for 0 ≤   ≤  * , the following relation can thus be formulated as: in which,  1 and  2 are two of three roots (  1 ,  2 and  3 ) of Eq. ( 22) , as seen the red dots in Fig. 3 (e), 3 (f).In the limit   = 0 , these three roots are denoted by  0 , (i.e. 1 =  0 , 1 ,  2 =  0 , 2 and  3 =  0 , 3 when   = 0 , as seen the red circles in Fig. 3 (e), 3 (f)).A sixtic equation under the limit   = 0 can thus be constructed as: where Re [ ⋅] is the real part operator.By equalizing coefficients in Eq. ( 26) and Eq. ( 28) , one would obtain the following relation: where: The optimal frequency ratio,   , which enables an equal modal damping ratio of the analogue 2-DOF system can thus be obtained from Eq. ( 29) : It is noted that, unlike the optimal frequency ratio for the classic SDOF-TMD system cf.[40,57] , the optimal frequency ratio   for the analogue 2-DOF system is not only a function of mass ratio, , and liquid length ratio, , but also depends on the structural inherent characteristics,  =   ∕   , cf.Eq. ( 30) , (31) .

Bifurcation point
Next, the existence of the bifurcation point (see  * in Fig. 3 (e) marked by ) implies that the two complex roots,  1 and  2 , are equal when   =  * (i.e. 1 =  2 =  * ), whereby the sixtic polynomial equation can be formulated as: where  * , 3 is the third root,  3 , of Eq. ( 22) for   =  * and  =   , as seen the red asterisk in Fig. 3

(f).
For convenience of expression in the following analysis, the coefficients for  in Eq. ( 22) are replaced by six variables, thus Eq. ( 22) can be manipulated into the simplified form: in which: ,  2 = ,  0 = By equalizing Eq. (32) with Eq. ( 33) , the coordinates of the bifurcation point,  * , and the corresponding equivalent damping ratio,  * , can be obtained: Fig. 4 shows the dynamic amplification of the modal response | , 0   ∕  0 | of the 2-DOF structure-TLCD system at the optimal frequency ratio (with fixed  = 0 .02 ,  = 0 .6 and  = 5 .58 ) for two different values of the equivalent damping ratio,   .Fig. 4 (a) presents the results for the analogue 2-DOF system with background flexibility considered.The blue line corresponds to   =  * , while the red line corresponds to   = √ 2 ∕2  * , which is the optimal damping ratio suggested by Krenk [57] .As seen, the amplitudes of these two lines at the invariant points (neutral frequencies),  1 and  2 , are equal under the optimal frequency ratio, which verifies the derivation results obtained based on the poleplacement method.Further, the value suggested by Krenk,   = √ 2 ∕2  * , leads to a fairly level behavior of the curve between the neutral frequency points.On the other hand, the value  * leads to a peak between points  1 and  2 .In the following optimal procedure, this equivalent damping ratio (   = √ 2 ∕2  * ) will be regarded as the optimal damping ratio.Similar conclusion can be drawn in Fig. 4 (b) for the classic 2-DOF system.The optimal frequency ratio  =   employed in Fig. 4 (b) is obtained based on the formulation derived in Appendix C.2 .As seen, one additional black line is added in Fig. 4 (b) corresponding to   =   , where   denotes the theoretical optimal damping ratio obtained from the conventional dynamic amplification-based calibration method, cf.Eq. (C.9) in Appendix C .The red line (   = √ 2 ∕2  * ) almost merges with the black line, implying that the equivalent damping ratio value   = √ 2 ∕2  * can be considered optimal in practical calibration of the TLCD.

Optimal head loss coefficient of the TLCD
In Eqs. ( 15) , ( 16) and ( 18) , the nonlinear damping force of the TLCD,   , = −1∕2 | v | v , is replaced by a linearized equivalent damping force,   , = −   v , under a harmonic excitation.  is the equivalent damping coefficient, which can be represented as [26] : The corresponding equivalent damping ratio,   , can thus be obtained: where | 0 | is the absolute amplitude value of liquid in the vertical tube.The optimal head loss coefficient for the TLCD,   , can be obtained either by the theoretical approximations such as the perturbation-based method [48] or the numerical optimizations such as the particle-swarm optimization method [28] .In this study, a rather straightforward way to determine the head loss coefficient is adopted.As shown in Section 3.3 , the optimal equivalent damping ratio is determined as  , = √ 2 ∕2  * under  =   , where  * can be calculated through Eq. ( 36) and   can be obtained from Eq. (31) .By inserting  , and   into Eq.( 38) , the optimal head loss coefficient is: As seen in Eq. ( 39) ,   depends on the external excitation frequency  .In consideration of practical use, the head loss coefficient should be chosen based on the severest condition [28] , i.e.  =   .Hence, the optimal head loss coefficient is given as: where | 0 |  =   denotes the absolute amplitude value of liquid in the vertical tube, which can be obtained by inserting  =   ,  =   and where the augmented stiffness matrix K , damping matrix Ĉ and mass matrix M are given in Appendix D , with elements   = √ 2    *     and   =  2     being inserted into Ĉ and K .The value of | 0 |  =   corresponds to the last element of û 0 (i.e.| 0 |  =   = û 0 , +1 , 1 ).
Once the optimal head loss coefficient,   , and the optimal frequency ratio,   , are determined, the dynamic amplification of structural response can be obtained through the iterative procedure in the frequency domain [48] .To justify the theoretically derived   and   , Fig. 5 compares the dynamic amplifications of the 2-DOF structure-TLCD system using   and   , and the optimal values obtained by numerical search (  , ,  , ), respectively.The range of  , ,  , during the numerical search procedure are limited within 1 to 8, 0.9 to 1.0, respectively.The specific data of Fig. 5 are obtained from the numerical example in Section 5 Eq.(5) .Fig. 5 (a), 5 (b) show the comparison for the analogue 2-DOF system with background flexibility considered.The red line denotes the results corresponding to   = 2 .24 and   = 0 .987 obtained from Eq. ( 31) , (40) , while the blue line denotes the result corresponding to  , = 2 .12 and  , = 0 .988 obtained from the numerical search.It is seen that the current method leads to a slightly flatter behavior for the structural response curve (which is beneficial), although the numerical search results in a lower trough as expected.As for the liquid response  0 , the numerical search leads to a slightly larger amplitudes.The optimal frequency ratios from the two methods,  , = 0 .988 and   = 0 .987 , are very close to each other.As for the optimal head loss coefficient, the result obtained from current method   = 2 .24 is only around 5% higher than the numerical search result  , = 2 .12 .Hence, the current closed-form method is considered comparable with the numerical search method in terms of the accuracy.Furthermore, it is worth noticing that the computational cost for the numerical search method is much higher than the current method.For the classic 2-DOF system, similar conclusion can be drawn from Fig. 5 (c), 5 (d).

Calibration procedure
For better clarity, the optimal calibration procedure for a TLCD installed on a flexible structure is summarized in Fig. 6 .The equations shown in this flow chat are obtained based on the pole-placement approach, cf.Sections 3.2 and 3.4 .The mass ratio,  =   ∕   , and the liquid length ratio,  = ∕ , are assigned by the designers according to practical constraints.The procedure should then determine the location of the TLCD with respect to the structure, i.e. to determine the location vector  : Once  is determined, the corresponding modal mass,   , modal stiffness,   , and background stiffness,   , can be calculated subsequently, based on Eq. ( 9) , (10) .The effect of the background flexibility on the TLCD calibration is accounted for via the non-dimensional stiffness ratio  =   ∕   .n The next step is to determine the optimal frequency ratio,   , which is related to the radius of the circular inversion  and a few constants   ,   and   . corresponds to the smaller root of Eq. ( 24) when   ⇒ ∞, and can be obtained through Eq. (25) .  ,   and   are functions of , ,  and .Hence the optimal frequency ratio,   , can be calculated using Eq. ( 31) .40) that the optimal head loss coefficient   is a function of the equivalent damping ratio   and the response amplitude of the liquid in the vertical tube | 0 |.The optimal damping ratio employed in Eq. ( 40) is taken as  , = √ 2 ∕2  * , where  * can be obtained from Eq. ( 36) , and the response amplitude of liquid | 0 |  =   corresponding to the severest working condition can be obtained from Eq. (41) .

Numerical example
The calibration procedure for a TLCD attached to a flexible structure introduced in the previous sections is illustrated by a simple 10-story shear frame the data of which have already been used for generating Fig. 3,4,5 .Fig. 1 (a) presents a 10-story shear frame with floor mass  (   = 10 ×  ) and inter-story stiffness  .The horizontal displacements of the individual floors,   (  ) , represent the 10 DOFs, and the vertical liquid displacement of the TLCD denotes the 11 ℎ DOF (  (  ) ).These 11 DOFs are assembled in an augmented displacement vector, û , as shown in Eq. (D.5) .
As shown in Fig. 1 (a), the TLCD is located at   th floor.The influence of the location of the TLCD on the optimal device parameters is investigated here, when the objective is to introduce damping into the first mode (resonant mode).The optimal parameters in terms of different values of   are shown in Table 1 ).The modal mass,   , corresponding to the first mode and the total mass of the TLCD,   , are listed in the second and third columns, which increase with the decreasing value of   .Column four presents the ratio of the modal background stiffness to the modal dynamic stiffness,  =   ∕   .As seen, a local maximum value  = 12 .0880 is obtained near the 8th floor.It is so because the location of 8th floor is close to the nodes of 2nd and 3rd modal shapes of the structure, which means the contributions from higher modes (nonresonant modes) are small [2] .The optimal frequency ratio,   , the optimal equivalent damping ratio,  , , and the optimal head loss coefficient,   , corresponding to the calibration procedure with background flexibility included (   ≠ ∞) or excluded (   ⇒ ∞) are present in the last six columns, respectively.According to Eq. (C.12) ,   and  , are merely dependent on  and  when   ⇒ ∞, see columns six and eight.It is seen that the values of   and  , are nearly increasing when the TLCD moves down, but the variations are not significant when the background flexibility is included, see columns five and seven.On the other hand, the optimal head loss coefficient,   , decreases as the TLCD moves down, for both cases, see the last two columns.Furthermore, the background contribution leads to larger values of   and   .
The effectiveness of the TLCD is evaluated by performing frequency analysis of the structure with a TLCD attained at   th floor.The external load is assumed to be uniformly distributed along the height of structure and has harmonic time variation with angular frequency , i.e. f (  ) = f 0   (where ).The top floor response,   , is selected as reference and the corresponding static displacement  , (  , = û 0 ,, 1 , û 0 = K −1 f 0 ) is used for normalization.Based on the calibration results of the optimal frequency ratio,   , and the optimal head lose coefficient,   , given in Table 1 , the dynamic amplification of the Structure-TLCD coupled system can be calculated through the conventional iterative procedure.Fig. 7 presents the dynamic amplification of the structural top floor displacements and the liquid responses for a TLCD attached at the sixth, forth and second floors, respectively.The red solid line corresponds to the results accounting for the background flexibility (   ≠ ∞), and the blue dashed line corresponds to the results excluding the background flexibility (   ⇒ ∞).It is shown that the background flexibility has modest influence on the dynamic response of the structure  , 0 ∕  , for a TLCD located at  ) represent the results when the background flexibility is ignored.the sixth and forth floors, as shown in Fig. 7 (a), 7 (c).However, when a TLCD is located at the second floor, the dynamic amplification of the structural response becomes severely unbalanced, if neglecting the background flexibility, as shown in Eq.Fig. 7 (e).Similar characteristics can be observed in the liquid response of the TLCD,  0 , as can be observed in Fig. 7 (b), 7 (d), 7 (f).
Due to the implementation of a TLCD, the 1st mode (the resonant mode) of the 10-story shear frame is split into two coupled modes.Fig. 8 presents modal damping ratios corresponding to these two modes versus the external excitation frequency.The main object of the calibration procedure is to achieve equal modal damping ratio of these two modes.It can be observed in Fig. 8 (a), 8 (c), 8 (e) that the object is attained when the background flexibility is included.As seen, these two curves are almost identical.On the other hand, when the background flexibility is omitted in the calibration procedure, the difference between these two modal damping ratios becomes increasingly significant as the TLCD is placed lower, as shown in Fig. 8 (b), 8 (d), 8 (f).Especially for the 2  floor, the maximum relative difference reaches about 25% , as shown in Fig. 8 (f).
Fig. 9 shows the root locus diagram of these two coupled modes.As elaborated earlier, the equal modal damping ratio is equivalent to the existence of a local bifurcation point, which is the intersection of two branches of the root locus.Fig. 9 (a), 9 (c), 9 (e) show the results when the background flexibility is considered.For   = 6 , these two branches almost intersect, and the bifurcation point seems to exist (although actually not), as shown in Fig. 9 (a).As   decreases, these two branches exhibit a very slight branch off.On the other hand, when the background flexibility is ignored in the calibration procedure, the non-intersecting property of the two branches becomes much more significant, as shown in Fig. 9 (b), 9 (d), Fig. 9 (f).Especially for   = 2 (as seen in Fig. 9 (f)), the two branches turn out to be quite far away from each other, indicating unbalanced calibration of the TLCD.Fig. 10 shows the variation of the optimal frequency ratio   with respect to the mass ratio  and the horizontal length ratio .The orange, purple and blue lines represent the results obtained from Eq. ( 31) with   equal to 2, 4, 6, respectively.The black line corresponds to the results obtained from Eq. (C.12b) , in which   is independent of   .Similar to the results in [38,45] ,   decreases almost linearly with increasing  for all cases, as shown in Fig. 10 (a).By including the background flexibility, the value of   will be enlarged especially for   = 2 .It is because the contribution from non-resonant modes (higher-frequency modes) make considerable contributions when the TLCD is placed lower.For the 6 ℎ floor, the influence from the background flexibility on   is minor, as the black and blue lines are close to each other.Similar trend can be observed in Fig. 10 (b), except for the orange line corresponding to the results of   = 2 , which shows a reverse trend in comparison with others.It is obvious from the plots that  exhibits modest influence on   in contrast with .
Fig. 11 shows the influence of the mass ratio , the horizontal length ratio  and the excitation intensity factor  on the optimal head loss coefficient   .The solid lines represent the results including the background flexibility (   ≠ ∞), and the dashed lines represent the results ignoring the background flexibility (   ⇒ ∞).It can be observed from Fig. 11 (a)   increases as  increases.Further, the background flexibility leads to slight enlargement of   , particularly when the TLCD is located at the 2  floor.Similar conclusion can be drawn for , see Fig. 11 (b).The variation of   with  is depicted in Fig. 11 (c).It is seen that   decreases with increasing , although the change becomes insignificant as  increases to a certain value.Similar trend has been observed for the optimal head loss coefficient versus  and  of a TLCD given by Shum and Yalla [45,48] .It is also found that for all the relevant parameters discussed above,  plays the most crucial role in determining   .
The variations of the maximum dynamic amplifications of the structural response | , 0 ∕  , |  and the liquid response | 0 ∕  , |  are depicted in Fig. 12 .The subscript  denotes the peak value within the external excitation frequency interval around the resonance frequency (here  ranges from 0 .5   to 1 .8   ).Fig. 12 (a) shows the evolution of | , 0 ∕  , |  with respect to .It is seen that as  increases, the maximum frequency response of the structure     By use of the amplification being independent of   at the two neutral frequencies,  1 and  2 , as shown in Fig. 2 (a), the following relations can be constructed: The plus sign leads to the root  = 0 , while the minus sign leads to the ninth-degree equation in  1 and  2 : By use of the amplification being independent of   at those two neutral frequencies,  1 and  2 , as shown in Fig. 2 (b), the following relations can be obtained: The plus sign leads to the root  = 0 , while the minus sign leads to the quartic equation in  1 and  2 : (2 + 2  −  2 )  4  By inserting the roots obtained from Eq. (C.3) into Eq.(21a) and assuming   ⇒ ∞, the normalized amplitude of the structural displacement at these two invariant points can be calculated: Based on Eq. (C.5) , the equal dynamic amplification condition at  1 and  2 will further leads to: By equalizing Eq. (C.4) with Eq. (C.6) , the optimal frequency ratio can be obtained, given as: When   ⇒ ∞, the natural frequency ratio of the classic 2-DOF system is  ∞ = 1∕ √ 1 + .By inserting  ∞ and   into Eq.(21a) , the normalized amplitude of  , 0 at  ∞ is: According to the optimal criterion proposed by Krenk [57] , the optimal damping ratio can be obtained by equalizing Eq. (C.8) with Eq. (C.5) , given as: 2 (1 +  −  2 ) (1 + )(1 +  −  2 ) (C.9)

C2. Pole-placement (root locus) calibration approach
For the classic 2-DOF system ignoring the background flexibility (i.e.  ⇒ ∞), the calibration procedure is similar to the classic SDOF-TMD system, which is fully elaborated in [57] .For brevity, detailed derivation is skipped in the present paper.
According to Krenk [57] , by appropriate calibrating the frequency ratio , the two roots,  1 and  2 , of Eq. ( 23) are inverse points with respect to a reference frequency,  ∞ = 1∕ √ 1 + , obtained from the limit case   ⇒ ∞.Hence, the following relation must be hold: where Im [ ⋅] is the imaginary part operator.Comparing Eq. ( 23) and Eq.(C.11) , the following results can be obtained: ) where  ,  , and  , are the structural mass matrix, damping matrix and stiffness matrix, respectively;  (  ) and  (  ) represent the displacement vector and load vector of the structure, respectively, see Eq. ( 2) for detailed definition.  is the floor location where the TLCD is attached to, see Fig. 1 (a).  is the equivalent damping coefficient, cf.Eq. (37) ;   =  2   is the equivalent stiffness of the TLCD.

Fig. 1 .
Fig. 1.Coupled TLCD-structure system under external excitation .(a) The primary flexible structure instrumented with a TLCD on the   th floor.The total displacement of the structure   consists of the displacement   due to the resonant mode and the relative displacement   due to the non-resonant background modes.(b) Modal analogue (analogue 2-DOF model) of the flexible structure equipped with a TLCD under the external force  .The analogue 2-DOF model consists of the modal mass   and modal stiffness   of the resonant mode, as well as an equivalent stiffness   related to the nonresonant modes. and  denote the horizontal and vertical length of liquid inside the TLCD, respectively. is the vertical displacement of liquid in the TLCD.

Fig. 6 .
Fig. 6.Optimal calibration procedure for a TLCD with given mass ratio  and horizontal length ratio

Fig. 10 .
Fig. 10.Variation of the optimal frequency ratio   with (a) the mass ratio  (  = 0 .6 and  = 0 .003 ) and (b) the horizontal length ratio  (  = 0 .02 and  = 0 .003 ), for different values of   .The black lines (   ⇒ ∞) correspond to the results when the background flexibility is ignored; the colored lines (   ≠ ∞) correspond to the results when the background flexibility is included.

Fig. 11 .
Fig. 11.Variation of the optimal head loss coefficient   with (a) the mass ratio  (  = 0 .6 and  = 0 .003 ), (b) the horizontal length ratio  (  = 0 .02 and  = 0 .003 ) and (c) the excitation intensity factor  (  = 0 .02 and  = 0 .6 ) for different values of   .The dashed lines (   ⇒ ∞) correspond to the results when the background flexibility is ignored; the solid lines (   ≠ ∞) correspond to the results when the background flexibility is included.

Fig. 12 .
Fig. 12. Variations of the maximum dynamic amplifications for the flexible structure-TLCD system (in the external excitation frequency range from 0 .5   to 1 .8   ) with (a)(c) the mass ratio  (  = 0 .6 and  = 0 .003 ) and (b)(d) the horizontal length ratio  (  = 0 .02 and  = 0 .003 ), for different values of   .(a)(b) Maximum dynamic amplifications of the structural top displacement, (c)(d) Maximum dynamic amplifications of the liquid response.The dashed lines (   ⇒ ∞) correspond to the results when the background flexibility is ignored; the solid lines (   ≠ ∞) correspond to the results when the background flexibility is included.