As a well-known and reliable control device, tuned liquid column dampers (TLCDs) have been investigated numerically and experimentally and implemented in a number of structures over the last three decades. However, TLCDs basically su er from the lack of multidirectionality, which is the critical need for real structures, in particular under random vibrations such as wind and earthquake excitations. This aspect has garnered the attention of the structural control community to modify this promising damper to achieve more eciency and to extend its application range to multidirectional vibrations. This paper proposes a mathematical modeling and optimization approach for omnidirectional tuned liquid column dampers (O-TLCDs). As an improved and reformed TLCD, O-TLCDs are formed by circularly distributed of n(integern3) L-arms about a common joint point at the center, through which all L-arms are connected to each other. Thanks to this layout, O-TLCDs can control structures with full counteracting force capacity in all transversal directions regardless of the excitation angle of incidence. This paper, in the rst step, proposes the governing equation of motion of O-TLCDs, for which Lagrange's principle is employed, and the equation of motion of the coupled O-TLCD-structure system. In doing so, a formal solution to determine the degree of freedom (DoF) of the O-TLCD is introduced, which proves independence of the O-TLCD response from the number of L-arms as well as from the angle of excitations. Second, for designing O-TLCDs, a set of design criteria and a general optimization scheme, which accommodate the online simulation of coupled O-TLCD-structure system under arbitrary excitations, are proposed. Consequently, without adding extra complication coming from extra DoFs to the motion equation of the damper, the O-TLCD functions as an enhanced liquid damper for multidirectional vibration attenuation. Next, using the O-TLCDs designed with di erent mass ratios, numerical simulations of O-TLCD-structure systems are conducted under seismic loads, free vibration, harmonic excitation and white noise and the controlled and uncontrolled responses of the systems are assessed in the time and the frequency domain. Here, the role of important parameters such as the mass ratio, the head loss coecient, the liquid de ection and the excitation amplitude are evaluated and the in uence of varying conditions on the eciency of the O-TLCD are discussed. Results demonstrate that the proposed O-TLCD can be well tuned to the structure and markedly control the peak and the RMS of responses of the structure. In the 1 end, an experimental study on a prototype O-TLCD is performed using a shaking table, which veri es the proposed mathematical modeling approach. Keywords: TLCD, O-TLCD, structural control, passive vibration control, omnidirectional 1. Introduction Vibratory systems such as structures built by conventional methodologies, which may everyday end up with more massive and sti er structural elements, su er from inherent low damping to face with stochastic and dynamic excitations. Structural control eld, on the other hand, has shed light on dealing with dynamic vibrations compatibly, which was founded by the pioneering work of Yao [1] and profoundly reviewed (e.g. see [2, 3]). Structural control applications can be classi ed as passive, semi-active, active, and hybrid control systems [4{7]. Due to their simple concept and reliability, the widespread applications of tuned dampers, such as tuned mass dampers (TMDs) and tuned liquid dampers (TLDs), which can be accommodated in one or more of the mentioned classes have been noticeable. As a more matured tuned damper, which eliminates the need for problematic installations of TMDs and recti ed the substantial inactive liquid of TLDs, the tuned liquid column damper (TLCD) [8, 9] is invented. A number of notable studies on TLCD may include the mathematical description of TLCDs and their optimum parameters using sinusoidal and/or random vibrations, such as [10{19], passive TLCDs, such as [20{31], and semi-active and active TLCDs, such as [32{41]. In parallel, the attention of the structural control community has been garnered to study the mathematical description of TLCDs operating in other degrees of freedom (DoFs) than the unidirectional translational movement. Hochrainer et al. [42] presented a torsional TLCD with ring-shaped geometry for controlling torsional vibrations of structures; in addition, the paper employed two other TLCDs for the horizontal translational DoFs and presented also experimental results. Similar ring-shaped geometric layout was numerically investigated for plan-asymmetric high-rise buildings exposed to an earthquake with prescribed angle of excitation [43], for simulation of a 27-story building with eccentricity under wind load [44] and for attenuation of the edgewise vibrations of wind turbine blades [45]. These studies showed that annular TLCDs can be e ective when exposed to torsional response. To control multiple degrees of freedom, Heo et al. [46] proposed a tuned liquid mass damper (TLMD), which was basically a TLCD mounted on natural rubbers, to reduce the translational responses in the weak and strong directions of a building. The TLMD worked as a TLCD in one direction where the liquid movement was employed and as a TMD in the orthogonal direction where the total weight of the TLCD was merely the tuned mass. Experimental results showed the TLMD could be tuned to two frequencies for bidirectional response reduction. Min et al. [47] proposed a two-way liquid damper, which was a TLCD with wider dimension perpendicular to the TLCD's plane, so that the damper behaved like a TLCD in one direction and a TLD in the orthogonal one; the eciency of the damper was veri ed for a 64-story building under wind. Similar two-way TLCD was applied by Lee et al. [48] where the inclination of the excitation angle ranging from 0 to 90was also studied. Email addresses: mehrkian@lbb.rwth-aachen.de (Behnam Mehrkian), altay@lbb.rwth-aachen.de (Okyay Altay) Further multidirectional TLCDs were introduced using plus-shaped by Sakai and Takaeda [49], crossed tube-like by Zhang et al. [50], and rectangular-based by Hitchcock et al. [51] layouts. A bidirectional TLCD, which consisted of four vertical columns connected with four horizontal ones, was proposed by Rozas et al. [52]. The damper required less liquid than two independent TLCDs and acted as a TLCD in two orthogonal directions. Shaking-table tests with 45-degree orientation identi ed two distinct natural frequencies of the damper. Numerical investigations of a 3-story scaled model under seismic loads showed that the proposed bidirectional TLCD can control the roof displacements more than the roof accelerations in two rst vibrational modes. Adopting this bidirectional TLCD, Tong et al. [53] studied controlling the pitch and the roll motions of wind turbines in the fore-aft and side-to-side directions, which brought reduction in damage loads of the tower and the uctuations of the rotor speed and generator power. Based on the concept of conventional TLCDs, Coudurier et al. [54] introduced a tuned liquid multi-column damper in three di erent con gurations to control simultaneously the pitch and roll motions of oating wind turbine platforms. The liquid dampers were composed of Nvertical columns, which were connected either simply by horizontal columns over each other or by interconnected horizontal columns. Depending on the arrangement of the horizontal columns, at least N1 DoFs were used to formulate the equations of motion of the dampers. They concluded that, in comparison with crosswise layout of multiple TLCDs over each other, the interconnected-column arrangement can result in a better performance due to its robustness against wave incidence and less parasitic-oscillation creation. Due to changing excitation direction, translational vibrations, such as the one induced by winds and earthquakes on high-rise structures, arise a challenge for structural control. Based on previously proposed TLCD con gurations, a star shaped TLCD with N vertical columns seems to be the best con guration to control translational vibrations. However, to the best of the authors' knowledge, for the control of translational vibrations, the de nition of such an omnidirectional TLCD is missing. This study introduces the mathematical description and optimization scheme for omnidirec- tional tuned liquid column dampers (O-TLCDs) in the class of passive structural control, which can cover every translational vibration in all directions with full capacity and regardless of the excitation angle of incidence. In Section 2, the governing equation of motion is derived based on the Lagrange's principle, which is independent of the number of liquid columns and employs only a single degree of freedom. The independence of the O-TLCD from the excitation direction is ana- lytically proven where the resulted damping of the O-TLCD is formulated by the non-conservative forces. In Section 3, to design an O-TLCD, an optimization scheme is introduced, which also accommodates a simulation of coupled O-TLCD-structure system under an arbitrary excitation. In Section 4, four O-TLCDs with di erent mass ratios are designed and investigated numerically under earthquake records, free vibration, harmonic excitation, and white noise. The performance of the O-TLCD, in general, and the role of critical parameters such as the mass ratio, the head loss coecient, the liquid de ection and the excitations, in particular, are numerically investigated. In Section 5, experimental investigations are performed to study the validity of the proposed math- ematical modeling approach, particularly focusing on the prediction of the natural frequency and the liquid de ection response. Finally, this paper is concluded in Section 6. 2. Mathematical model This section introduces the equations of motion of the O-TLCD and the O-TLCD-structure system. In the following, the geometric layout and the DoF of the O-TLCD will be identi ed. 3 Accordingly, the equations of motion will be derived based on Lagrange's principle and equilibrium of forces. 2.1. Geometry of O-TLCD A TLCD with a horizontal liquid length Hand vertical liquid length Vis assumed, which is located in the direction of a translational excitation wwith arbitrary angle relative to a coordinate system. It is also considered that the TLCD has the same cross sectional area Ain horizontal and vertical tubes. When this TLCD divides into two halves, which are called here as L-shape parts (L-arms), and nnumber of L-arms are arranged symmetrically around an origin, an omnidirectional TLCD (O-TLCD) is created. Accordingly, the horizontal column of each L-arm is H=2 and the angle between L-arms is = 2=n. With this de nition where n2Z, it is highlighted that TLCDs correspond to n= 2 while O-TLCDs to n3. Fig. 1 represents the basic layout of an O-TLCD with three columns. 21 3y xwu 4 �2 𝜓 𝜓 𝜓 α H/2 (a) Plan view z 43 1 2 A x5AVu3 u1 u2 at rest6 (b) Front view Figure 1: An omnidirectional tuned liquid column damper (O-TLCD) with three columns, in the plan (a) and front view (b), where 1to3are the three liquid columns, 4is the ori ce, 5is the liquid level at rest and u1tou3(6) show the positive direction of liquid de ection in each column (the de ections are drawn considering the shown excitation w). 2.2. Degree of freedom of O-TLCD To develop both the motion equation of an O-TLCD and the interaction equation of an O- TLCD-structure system, the liquid movement of each individual L-arm is required. For this pur- pose, the excitation angle is considered with respect to the rst tube. We consider a pseudo (imaginary) TLCD, which is representing the instantaneous damping e ect of an O-TLCD with two L-arms and so a horizontal liquid length of 2 H=2 =Hin the excitation direction. Then, we denote the liquid movement of the assumed TLCD by u(Fig. 2a and 2b). Here, it is considered that the positive vertical liquid movement points to positive zdirection. It is also noted that the vertical and horizontal liquid movements (i.e. displacements, velocities and accelerations) are the same because of the assumption of the same cross sectional areas. Projecting the liquid movement u, which is a pseudo de ection, onto each tube of the O-TLCD, one can nd the liquid movement of ith tubeuias ui=ucos( i); (1) 4 𝜃𝑖 𝑥𝑦 1 𝑖 𝑛2𝜋/𝑛𝑢1𝑢 𝑢𝑖𝑢𝑛𝛼 𝑥𝑧 1 𝑖𝑢𝑖 𝑛 𝑉at rest 𝑢𝑛 𝑢1𝑤 𝐻/2(a) plan and front view 𝑤 𝜃𝑖 𝑥𝑦 1 𝑖𝑛𝑢1 𝑢 𝑢𝑖𝑧 𝑢𝑖𝑢𝑛 𝑢𝑛𝑢1 Pseudo TLCD𝑢 (b) 3D view 060120180240300360 Excitation direction [°]-1-0.500.51ui/uTube 1 Tube 2 Tube 3 (c) Tubes' contribution Figure 2: Graphical representation of the DoF of an O-TLCD and its TLCD representative with the equivalent damping e ect in the excitation direction with horizontal liquid length of H(a,b); the contribution of each tube to the liquid movement udepending on in a typical 3-Larm O-TLCD (c). whereirepresents the location of the ith L-arm with respect to the rst L-arm and is de ned as i= 2(i1)=n: (2) when is in terms of radian. It is noted that, by using the pseudo TLCD, Eq. 1 represents O-TLCDs, regardless of number of L-arms, as a single degree of freedom system (SDoF). Eq. 1 is the key relation in mathematical description of O-TLCDs since it determines both the ow share and ow direction in each L-arm and that only based on u. It is evident that positive and negative values for uiin Eq. 1 demonstrate upward and downward liquid motions, respectively. As an example, Fig. 2c illustrate the liquid movements in the columns of an O-TLCD with three L-arms. It can be proven that a sum over allnL-arms in Eq. 1, the total liquid movements, is zero which was already expected physically as the amount of liquid is constant: nX i=1ui=unX i=1cos( i) = 0: (3) 5 2.3. Equation of motion of O-TLCD To develop the equation of motion of the n-L-arm O-TLCD, the Lagrange's principle is em- ployed: d dt@Ekin @_u @Ekin @u+@Epot @u=Qu; (4) whereEkinis the kinetic energy, Epotis the potential energy and Quis the non-conservative force. Unless otherwise mentioned, it is assumed that n2 in the following. Furthermore, it is considered the geometric parameters, such as Aare constant and do not vary with time. 2.3.1. Kinetic energy The kinetic energy of an O-TLCD with nL-arms can be written as follows: Ekin=1 2AnX i=1H 2j_ uH;ij2+Vj_ uV;ij2 ; (5) whereis the liquid density; _ uH;iand_ uV;iare the total liquid velocity vectors in the horizontal and the vertical parts of the ith L-arm, respectively, which, for the case is a constant, can be de ned considering the magnitude of the relative liquid velocity _ ui= _ucos( i) and the magnitude of excitation velocity _ w(see Eq. 1 and Fig. 2) as _ uT V;i= _wsin _wcos _ucos( i) ; (6) _ uT H;i= _ucos( i) sini+ _wsin _ucos( i) cosi+ _wcos 0 ; (7) which bring, Ekin=1 2A H 2 _u2+ 2 _u_w +V_u2nX i=1cos2( i) +n_w2H 2+V! ; (8) and d dt@Ekin @_u =1 2A(H(u+ w) + 2Vu)nX i=1cos2( i): (9) It is noted that for the case of the time-variant , _ui= _ucos( i)_ usin( i) should be considered for calculating the kinetic energy in Eq. 5. However, for the sake of simplicity in the following, we consider that is constant during an excitation event. 2.3.2. Potential energy The potential energy comes from the in uence of the gravity on the liquid movement in vertical tubes: Epot=nX i=11 2gA(V+ucos( i))2: (10) 6 Expanding the squared term and using Eq. 3, one can end up with Epot=1 2gA nV2+u2nX i=1cos2( i)! ; (11) @Epot @u=gAunX i=1cos2( i): (12) 2.3.3. Non-conservative force, Qu In uid dynamics, the non-conservative force of a pipe portion can be computed using the pressure loss  pLand pipe cross section Aat the ow region under consideration: Qu=pLA; (13) The pLdepends on the stagnation pressure v2=2 and loss factor (Re): pL=v2 2(Re); (14) where the term v2=2 is also known as dynamic pressure [55]. The pressure loss  pLreduces to the following form for turbulent ow ( Re&4000 in circular pipes) pL=v2 2; (15) wherevis the liquid ow velocity and is the loss factor which is here independent of Reynolds numberRe. The dynamic pressure v2=2 represents the pressure rise when the uid in motion is brought to a stop isentropically, which occurs at the so called stagnation point [55]. The O-TLCD presents n ow regions where the liquid in each horizontal tube can reach a stagnation point. Accordingly, the total non-conservative force of O-TLCD can be represented as the sum of non-conservative forces in all L-arms. Therefore, the sum of non-conservative forces in all horizontal tubes can be considered as Qu=nX i=1Qu;i=nX i=1A_u2 i 2=1 2A_u2nX i=1cos2( i)n3 (O-TLCD) : (16) For TLCD, on the other hand, the non-conservative force is de ned in literature as (e.g. see Hochrainer [56]) Qu=1 2A_u2cos2( i)n= 2 (TLCD) ; (17) which, here, is extended to the case where the excitation is not aligned with the TLCD. Introducing Eq. 9, Eq. 12, Eq. 16, and Eq. 17 into Eq. 4, one can end up with the equation of motion of TLCDs ( n= 2) andn-L-arm O-TLCDs ( n3) with identical pipe cross sections as follows: u+j_uj_u+!2 du= w; (18) 7 where the e ective length L, the natural frequency !d, and the geometric factor are de ned as L= 2V+H !d=r 2g L =H L; (19) and the head loss coecient as =( 1 2Lifn= 2 (TLCD) 1 Lifn3 (O-TLCD) :(20) The solution of Eq. 18 is the liquid de ection of the pseudo TLCD which is introduced in Fig. 2. Subsequently, the liquid de ection in the ith L-arm of the O-TLCD is simply obtained using Eq. 1. 2.4. Equation of motion of O-TLCD-structure systems This section presents the equation of motion of a structural system equipped with an O-TLCD (Fig. 3). A symmetric 3-dimentional SDoF structure, with the mass ms, the sti ness coecient ks and the damping coecient cs, is considered. The structure is excited by an external excitation F, which is applied at the mass center and in arbitrary constant direction . Accordingly, the oor moves with single degree of freedom win the direction . y xwu �2 𝜓 𝜓 𝜓 α H/2F (a) Plan view cs ks/4 ks/4F sin α w sin αz A xAVu3 u1 u2 at rest (b) Front view Figure 3: Coupled O-TLCD-structure system in the plan (a) and front view (b), excited by an external excitation F). To derive the interaction equations, rst, the active liquid mass mact;iof theith L-arm is identi ed: mact;i=H=2 n(V+H=2)md=1 n md; (21) which is the horizontal liquid portion of the total liquid mass md. To calculate the resultant inertial forces of the O-TLCD, Eq. 1 is used to form the liquid acceleration vector of all nL-arms: uT= [ucos( 1) ucos( 2):::ucos( n)]: (22) 8 𝑥𝑦 1 2 𝑖3 𝛼𝑤𝑔 𝑤restoring force of liquid in 𝑛𝑡ℎL-arm 𝐹𝑛 resultant restoring force of O-TLCD𝜃𝑖 2𝜋 𝑛(a) Restoring force distribution 0306090120150180 Excitation angle [°]00.20.40.60.81Normalized spectral amplitudeTLCD O-TLCD (b) O-TLCD vs. TLCD Figure 4: Distribution of the restoring force exerted by a typical O-TLCD to a structure (a); comparison between the performance of the O-TLCD and the TLCD when the excitation angle varies (b). Furthermore, since the resultant inertial force of the O-TLCD is in the excitation direction, a vector pis de ned to project the liquid motion of each tube onto this direction: pT= [cos( 1) cos( 2):::cos( n)]: (23) Then, the equation of motion of the O-TLCD-structure system can be obtained based on equilibrium of forces in the excitation direction and using the elements of vectors uandp(Fig. 4a): (ms+md) w+cs_w+ksw+nX i=1 mact;iucos2( i) =F: (24) To nalize Eq. 24, it is necessary to di erentiate TLCDs with n= 2 from O-TLCDs with n3 in calculating the summation. For TLCDs, we have nX i=1cos2( i) = 2 cos2 n = 2 (TLCD) : (25) For O-TLCDs, on the other hand, it can be proven that nX i=1cos2( i) =n 2n3 (O-TLCD) : (26) Applying the Eq. 25, Eq. 26 and Eq. 21 to Eq. 24, the interaction equation of motion can be written as msw+cs_w+ksw=Fmdwq mdu; (27) or equivalently w+ 2Ds!s_w+!2 sw=f( w+q u); (28) 9 for q=( cos2 ifn= 2 1 2ifn3;(29) wheref=F=msmass-normalized external force, Dsis the damping coecient and !sis the natural frequency of the main structure; is the mass ratio between the liquid mass and the mass of the structure. It is noted that, in Eq. 28, the movement of the structure in the xand they direction can be simply obtained by wsin andwcos , respectively; the liquid acceleration  uin the path of the excitation comes from Eq. 18. Furthermore, if the external excitation is a ground motion applied at the projected mass center of the structure on the ground in direction, the equations of motion in Eq. 18 and in Eq. 28 can be presented as u+j_uj_u+!2 du= ( w+ wg); (30) w+ 2Ds!s_w+!2 sw=wg( w+ wg+q u); (31) where wgis the ground acceleration. Eq. 28 (or Eq. 31) states that, in contrast to TLCDs whose counteracting force is -dependent as expected, by employing a symmetric O-TLCD with 3 or greater number of L-arms, the interaction equation of motion is independent of the excitation angle when is not changing during the excitation event. This is signi cant since O-TLCDs can exert identical counteracting forces in all directions to the structure regardless of the excitation angle. For the sake of simplicity, as mentioned before, we present here the equations for the excitation direction, which is not changing during the excitation event. If is changing, the Lagrange formulation with its components can be expanded as well. Concerning the mass ratio in Eq. 28 (or Eq. 31) which is generally de ned as =nA(V+H=2) ms; (32) it is important to note that increasing the number of L-arms nincreases the active mass ratio , which is de ned in analogy with TMDs by introducing u=u= [56] into the coupled equations of motion in Eq. 18 and Eq. 28 (or Eq. 30 and Eq. 31): = 2 1 +(1 2); (33) which in turn decreases the optimal frequency fd;opt(proposed by Warburton [57] for TMDs without superscript *), which can be used for the O-TLCD as fd;opt=f sp 1=2 1 +; (34) where based on the increased mass of the structure m sdue to the dead mass of the liquid: m s=ms(1 +(1 2)); (35) f scan be obtained out of the natural frequency of the main structure fs: f s=fsrms ms: (36) 10 Eq. 34 shows that, according to Eq. 19, while changing the number of L-arms ndoes not a ect the natural frequency of the O-TLCD, the change in nmay demand retuning the O-TLCD, which results in a di erent geometry. Therefore, it is generally not simple to directly nd the in uence of number of L-arms on the control force in Eq. 28 and, for example, nd a clear comparison between the magnitude of control forces produced by O-TLCDs and TLCDs. However, to provide a relative comparison, it is considered here that O-TLCDs and TLCDs are designed individually and optimally to work with full capacity for a single system. Accordingly, the performance of each system can be normalized to their maximum control force and recorded for di erent incremental excitation angle from 0 to 180, which is illustrated in Fig. 4b. 3. Design criteria and optimization scheme This paper proposes a constrained optimization scheme (Fig. 5) to design the O-TLCD. The tuning criteria proposed by Warburton [57], which is basically the extension of the criteria of Den Hartog [58] for harmonic vibrations to random excitations, is employed in this approach. Using Warburton's formulas for TMDs and the de nition of active mass ratio in Eq. 33, the optimal frequency and the optimal damping ratio for O-TLCDs are calculated by Eq. 34 and Eq. 37 for undamped structures, respectively. Dd;opt=s (1=4) 4(1 +)(1=2): (37) The tuning constrains of the optimization scheme are de ned as fdfd;opt fd;optf; Dd;optDd;min; (38) wherefd=!d=2is the natural frequency of the O-TLCD;  fis the tolerance value, for which 0:01, andDd;min is the minimum desired optimal damping, for which 0 :05 is recommended. For comparison with optimal damping ratio in Eq. 37, the damping ratio of the O-TLCD is approxi- mated by a linearized viscose damping ratio used for harmonic vibrations [25]: Dd;lin=4U0 3; (39) whereU0is the liquid vibration amplitude. The mass ratio , de ned by Eq. 32, is constrained by a plausible upper limit ( max) with respect to the total mass of the primary structure that is either known or can be conservatively estimated based on the modal mass. Concerning the geometry, an O-TLCD with rectangular cross section, which is easier to be manufactured than the circular one and brings the advantage of using di erent dimensions for providing the cross section A, is considered. To nd an optimum and feasible geometry a set of linear and nonlinear geometric constraints are de ned, for which Fig. 6 introduces the geometric parameters that facilitates the description of the constraints. In this gure, parameter s, which is equal to ( HitV 2), indicates the net horizontal column of each L-arm and together with the half of the vertical column width tVdetermines the outer horizontal length of the L-arm in O-TLCDs, while the Hidetermines the centered one. It can be found from Fig. 6 that the parameter sdepends on the width of each L-arm b, the angle between 11 Given for O-TLCD &structureInitial condition & objective functionExcitation & boundary condition Constraints are in rangeρ,n,Cb,Dd,min,fs,ms,Ds,Δf μ,μ∗,fd,fd,Opt, Dd,Opt,ballwd,uallwd Sample time, simulation time, solver Simulation: O-TLCD -structure under chosen excitation •Uncontrolled &controlled :RMSoff&RMSwith •Peakliquiddeflection: 𝑢𝑚𝑎𝑥 O-TLCD design is finalized for CRMS,jballwd,uallwd,μmax,Dd,min,Δf,CRMS,jResponse of the structure & O-TLCD No Yes CRMS,j =CRMS,j−CRMS,incrmntIncreasing response reduction capacityLinear & nonlinear constraints Loop j j=+1Optimization algorithm ( fmincon , GA, ... )Excitation, μmax, x0={x0,l,x0,h,x0,r}, lB, uB, CRMS,j,CRMS,incrmnt , i=1 x0=x0i,ObjFunc i=+1j=1 Derived for O- TLCDx=[bVHAδ] i≤3No Yes O-TLCD geometry is not feasibleVariables Loop iFigure 5: The proposed optimization scheme for designing O-TLCDs under an arbitrary excitation. L-arms (i.e. also the number of L-arms n),tVandHi. Accordingly, in order to achieve a desirable space for the horizontal column length of each L-arm, the following constraint can be de ned bCb2(1cos ) sin (HitV 2); (40) whereCbis a user-de ned coecient in [0 ;1] interval; in practice, Cb0:50 results in a reasonable geometry. The right hand side of Eq. 40 is introduced as ballwd which de nes an upper design limit forband will be used for the optimization scheme. The other important constrain comes from the responses of the primary structure and the O- TLCD. In the optimization scheme in each loop ( j;i), simulation of the resulted coupled system using desirable sample and simulation time and a solver type as numerical integration method is performed online. The RMS of the response of the controlled structure (the structure with O- TLCD)RMSwithis constrained to be smaller than the RMS of the response of the uncontrolled structure (the structure with o O-TLCD whose liquid movement is suppressed despite existence) 12 btV sV Hi tHxy z x𝜓 Figure 6: Geometric parameters of the O-TLCD used in the optimization scheme. RMSoffunder arbitrarily chosen excitation. More precisely, RMSwith RMSoffCRMS; (41) whereCRMS is a user-de ned upper limit in the [0 ;1] interval. Although the value for CRMS depends on the intensity of the excitation and the desire of the control designer, CRMS = 0:50 is recommended for reasonable O-TLCD geometries. Incrementally decreasing the CRMS by a prede ned value CRMS;incrmnt injloops: CRMS;j =CRMS;jCRMS;incrmnt ; (42) the algorithm nds the optimum O-TLCD geometry as long as the prescribed RMS upper limit at thejth loop,CRMS;j , can be achieved. Furthermore, the considered equation of motion of O-TLCD would no longer be valid if the liquid de ection uis larger than the theoretical liquid height Vin the vertical column [17]. There- fore, in the optimization scheme, the maximum liquid de ection umaxunder arbitrarily chosen excitation is constrained|to be in the safe side|not only to Vbut rather to the net liquid height located immediately above the horizontal column uallwd =VtH=2. To implement the proposed optimization scheme, the minimum of constrained nonlinear multi- variable (fmincon) function in Matlab is employed; it is worth mentioning that other optimization algorithms such as genetic algorithm (GA) is applicable in the proposed scheme as well. Five parameters b,V,Hi,A(see Fig. 1) and (see Eq. 20) are selected as the independent unknown variables and the , de ned in Eq. 19, is considered as the objective function to be maximized. Finally, it should be highlighted that, in order to fairly compensate the role of the initial point (x0) in the minimization process which may lead to local minima, the developed scheme can initialize the problem at least from three di erent origins. Two of these points are clearly higher (x0;h) and lower ( x0;l) estimated initial guesses and the third one is a random initial point ( x0;r) 13 generated based on either the rst or the second chosen initial point. The initial points are checked successively in iloops. The outcome of the described scheme is a unique geometry for the O-TLCD optimized when the coupled O-TLCD-structure system is exposed to an arbitrary excitation, such as an earthquake, with a particular intensity. In this system, the same O-TLCD can be still utilized under other excitations with higher intensities and di erent content as long as the liquid de ection does not violateuallwd, which can be possibly achieved by increasing the head loss coecient in Eq. 20. On the other hand, it can be also the case under weaker excitations for which lower might be required to employ the advantage of higher liquid de ection for more vibration mitigation during the excitation. However, it should be ascertained that the manipulation of the head loss coecient requires a semi-active control of the O-TLCD, which is beyond the scope of the current study. 4. Numerical Investigations A seismically excited SDoF structure with an O-TLCD is investigated. To evaluate the perfor- mance of O-TLCDs, a symmetric high-rise structure (Fig. 7) with natural frequency fs= 0:280 Hz is taken into account. Furthermore, it is assumed that the rst modal mass is ms= 400 t and the rst modal damping ratio is Ds= 1 % O-TLCD md = 155.74 t (µ = 0.39) fd = 0.201 Hz δreq =1.310 m-1 (δ =0.838 m-1) γ = 0.841 Structure ms = 400 t fs = 0.280 Hz Ds = 1 %xy w wgu3u1u2 𝛼 (a) 8.041.05 2.360.97 5.17 1.05 [m]xy 5.17 z x𝜓 (b) Figure 7: The example structure controlled by the O-TLCD (a) and the plan and the side view of one of the designed O-TLCDs ( = 0:39) (b). In the following, rst, the optimized O-TLCDs using di erent mass ratios are designed, which function with optimum head loss coecient for the described structure under a particular earth- quake. Next, the coupled O-TLCD-structure systems are evaluated under an ensemble of earth- quakes, for which the general need for a higher head loss coecient reqis presented. The O-TLCDs equipped with reqare further investigated under a free vibration, harmonic excitation and white noise, subsequently. 4.1. Optimized O-TLCD To design the O-TLCD, four mass ratios ranging from = 10 to 40 % with respect to the rst modal mass of the structure are chosen as the mass-ratio upper-limits in the optimiza- 14 tion scheme. Estimating a total mass of such structure to be, for example, three times the rst modal mass, the investigated mass ratios cover a quite reasonable range with respect to the total mass of the structure. Other constrains include  f= 0:01, D= 0:1,Cb= 0:50, CRMS; 1= 0:50,CRMS;incrmnt = 0:01, and lower bound lB= [1;0:1;1;0:1;0:01] and upper bound uB= [100;10;100;10;10] for the ve unknown geometric variables [ b;V;Hi;A; ] with the corre- sponding units as [m] for b;V;Hi, [m2] forA, and [m-1] for. To compute the RMS value in Eq. 41, the displacement response of the structure is utilized. The earthquake record Kobe (1995) with the intensity factor of one is utilized for the design of each of four O-TLCDs using the optimization scheme described in the last section. Therefore, as the Kobe record is used in the optimization, the El Centro (1940), Hachinohe (1968) and Northridge (1994) earthquakes are opted for the validation phase. Fig. 8 shows the earthquake records. The simulations are conducted using the ode8 (Dormand-Prince) time integration method in Simulink/Matlab. The simulation time of each calculation is 150 s with a time increment of 0:01 s for Hachinohe and 0 :02 s for other records. 0102030405060 Time [s]-4-2024Acceleration [m/s2] (a) El Centro (1940) 0102030405060 Time [s]-8-6-4-202468Acceleration [m/s2] (b) Kobe (1995) 0102030405060 Time [s]-202Acceleration [m/s2] (c) Hachinohe (1968) 0102030405060 Time [s]-8-6-4-202468Acceleration [m/s2] (d) Northridge (1994) Figure 8: Time-history plots of earthquake records used in this study. Table 1 and 2 summarize the outcomes of the optimization scheme for the aforementioned four mass-ratio upper-limits in two parts and Fig. 9 depicts the obtained geometries. First of all, search- ing for the most optimized geometry, it is noted in the table and the gure that the optimization scheme exploited the mass-ratio upper-limits fully for all cases except for the 40 % limit. This implies that no larger tuned geometry for O-TLCD, which can attenuate uncontrolled responses more than CRMS = 0:25, exists to accomodate liquid mass more than 0 :389ms. Therefore, it is con rmed that the developed optimization scheme is able to converge to the most plausible and 15 ecient O-TLCD for the structure under consideration, which, according to the indices ,Dd;opt, Dd;lin, , andCRMS in Table 1 and 2, is the O-TLCD =0:39. 8.04 1.05 𝛿𝛿=0.838 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟=1.310𝜇𝜇=0.396.34 1.18 1.86 𝛿𝛿=0.639 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟=0.954𝜇𝜇=0.30𝛿𝛿=0.407 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟=0.5793.65 0.98 1.09 𝜇𝜇=0.104.46 1.33 1.30 𝛿𝛿=0.640 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟=0.705𝜇𝜇=0.20 2.36 Figure 9: Characteristics of the investigated O-TLCDs with di erent mass ratios (dimensions are in [m]; the optimal head loss coecient and the required one reqare in [m-1]). Table 1: Parameters (Part 1) of the designed O-TLCDs with di erent mass ratios . b[m]V[m]H[m]A[m2][m-1]req[m-1][-]m s[-] = 0:10 3:65 1:21 5:20 3:58 0:407 0:579 0:05 421:92 = 0:20 4:46 1:25 6:48 5:93 0:640 0:705 0:09 438:38 = 0:30 6:34 1:11 8:50 7:46 0:640 0:950 0:17 444:63 = 0:39 8:04 0:98 10:34 8:45 0:838 1:310 0:25 445:52 Table 2: Parameters (Part 2) of the designed O-TLCDs with di erent mass ratios . f s[Hz]fd;opt [Hz]fd[Hz]Dd;opt [-]Dd;lin[-]uallwd [m] [-]CRMS [-] = 0:10 0:273 0:258 0:255 0:10 0:18 0 :72 0:68 0:46 = 0:20 0:267 0:238 0:235 0:15 0:18 0 :59 0:72 0:34 = 0:30 0:266 0:217 0:215 0:19 0:21 0 :52 0:79 0:27 = 0:39 0:265 0:199 0:201 0:23 0:25 0 :45 0:84 0:25 Beside the geometries, Table 1 and Fig. 9 show the head loss coecient coming from the optimization scheme under the selected earthquake Kobe for all mass ratios. However, since the occurrence of intenser excitations than Kobe is probable, these optimized head loss coecients ( ) might not be sucient to prevent the liquid motion from exceeding the provided liquid de ection 16 uallwd which is also reported in Table 2 for each O-TLCD. This can be investigated when we simulate the performance of the presented O-TLCDs in controlling the structure in Fig. 7 under di erent excitations. 4.2. Earthquake excitation, with  Fig. 10 and Fig. 11 show the response of the structure equipped with the O-TLCD =0:10and the O-TLCD =0:39, which are the lightest and the heaviest O-TLCDs and present the lowest and the highest active mass ratios (Table 1), under all four earthquakes, respectively. In spite of notable uncontrolled-response mitigation, the peak liquid de ections of the O-TLCD =0:10and the O-TLCD =0:39under the Northridge record in Table 3 indicate that the induced peak liquid de ections ( umax= 0:836 m and umax= 0:533 m) indeed exceed the provided liquid de ections for O-TLCD =0:10and O-TLCD =0:39(uallwd = 0:720 m,uallwd = 0:450 m). Similar results are observed when we study the peak liquid de ections of other O-TLCDs with 20 and 30 % mass ratios in Table 3 compared with the provided ones uallwd. Therefore, this observation dictates the need for a higher head loss coecient for the O-TLCDs if they are to be also e ective under intenser excitations such as Northridge. 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (a) El Centro (1940) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (b) Kobe (1995) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD (d) Northridge (1994) Figure 10: Displacement response of the uncontrolled structure versus the controlled structure equipped with O- TLCD=0:10when= 0:407 m-1coming from the optimization process under the selected earthquake Kobe. 17 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD(a) El Centro (1940) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (b) Kobe (1995) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD (d) Northridge (1994) Figure 11: Displacement response of the uncontrolled structure versus the controlled structure equipped with O- TLCD=0:39when= 0:838 m-1coming from the optimization process under the selected earthquake Kobe. 4.3. Earthquake excitation, with req The higher required head loss coecient is denoted here by req. To obtain req, for each mass ratio, the O-TLCD-structure system is simulated under the Northridge earthquake in a loop where the head loss coecient of the O-TLCD increases incrementally as long as the induced liquid de ection is greater than the provided one, uallwd. Consequently, the obtained value for thereqout of each loop for each O-TLCD is reported in Fig. 9. Next, one important aspect is if these values as the head loss coecient are realistic, for which Eq. 19 and Eq. 20 can be used. For example, for O-TLCD =0:39, the loss factor is obtained through the e ective liquid length L= 20:97 + 10:34 = 12:29 m andreq= 1:310 m-1: =L= 1:31012:29 = 16:10; (43) which is plausible, for instance, by the means of butter y valves [59] in the midway of the liquid passage in the horizontal columns of the O-TLCD. Since these required head loss coecients are to be passively assigned to the corresponding O-TLCDs, all the following assessments are conducted considering that the four O-TLCDs are 18 equipped not with the Kobe-optimized head loss coecient but with the Northridge-required head loss coecient req. To do this, in Eq. 18 (or Eq. 30) is replaced with req. It is noted that assigning req, which is generally greater than and therefore limits liquid movements more, in uences the O-TLCD performance|can be either improvement or deterioration depending on the vibration content|under less demanding excitations, but important is it can preserve the functionality of the O-TLCD exposed to severer excitations. The peak liquid motion of all four O-TLCDs functioning with their corresponding andrequnder Northridge record are brought in Table 3 where now all the peak liquid de ections using reqare in the range of uallwd. Subsequently, to evaluate the approximated damping ratio of the O-TLCDs, Dd;linin Table 2, U0 andin Eq. 39 are replaced with uallwd andreq, respectively. For assessment, applying the four earthquake records, Fig. 12 to Fig. 17 present the overall performance of the O-TLCD =0:10and the O-TLCD =0:39withreqby plots of the displacement response of the primary structure in the time and the frequency domain as well as the correspond- ing liquid de ection ufrom Eq. 30. To achieve a ner resolution in the frequency domain, the simulations are conducted for 2000 sec; corresponding sampling rate according to the sample time of each utilized earthquake is employed. For the sake of comparison, the performance of O-TLCDs for all four mass ratios ranging from 10 to 39 % are summarized based on the evaluation of the RMS and the peak of controlled and uncontrolled displacement response of the structure in Fig. 18 with absolute values and in Fig. 19 with relative values with respect to the uncontrolled responses. Here, the uncontrolled responses refer to the structure without O-TLCD (bare structure). Table 3: Peak amplitude of the liquid de ection u(Eq. 30) of all four O-TLCDs mounted on the structure vibrated under Northridge earthquake with head loss coecient orreqshown in Fig. 9 versus the allowed de ection uallwd. = 0:10= 0:20= 0:30= 0:39 Allowed liquid de ection uallwd [m] 0.720 0.590 0.530 0.450 Peak liquid de ection with [m] 0.836 0.610 0.601 0.533 Peak liquid de ection with req[m] 0.722 0.588 0.525 0.450 It is seen in these two gures that the O-TLCD =0:39presents the most e ective performance where, according to Fig. 18a, the RMS of the displacement drops from around 10 cm to almost 5 cm under the Kobe and at most from 19 cm to 7 cm under the Northridge earthquake; 4 cm under the Kobe and 20 cm under the Northridge are the least and the most absolute amounts of peak displacement attenuation, respectively. The gures report that, even with the lowest mass ratio, the O-TLCD signi cantly improves the responses of the structure. According to Fig. 18a, the RMS of the displacement of the structure with the smallest damper O-TLCD =0:10is reduced at least by 31 % under the Kobe and at most by 46 % under the Northridge earthquake. The peak displacement, except a slight increase under the Kobe record, is decreased at least by 11 % under the Northridge and at most by 24 % under the Hachinohe earthquake with this O-TLCD. These results indicate that the structure can experience further vibration mitigation when the attached O-TLCD is designed with a higher mass ratio although this improvement from one mass ratio to the other may not be uniform. 4.4. Free vibration and harmonic excitation, with req For further evaluation, the O-TLCD-structure system is investigated under a harmonic excita- tion for the O-TLCD =0:10and the O-TLCD =0:39. For the uncontrolled case ( without O-TLCD ), 19 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD(a) El Centro (1940) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (b) Kobe (1995) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD (d) Northridge (1994) Figure 12: Displacement response of the uncontrolled structure versus the controlled structure equipped with O- TLCD=0:10whenreq= 0:579 m-1(Northridge-required head loss coecient) under earthquakes. it is obvious that the excitation frequency should equal to !s. To evaluate the resonant frequency of the controlled case ( with O-TLCD ), we conduct a free vibration test with initial displacement of 0:15 m for 2000 s and process the displacement response using fast Fourier transform (FFT) with sampling rate of 100 Hz, which is brought in Fig. 20 and Fig. 21 in time interval of 300 s and frequency band of 0 :5 Hz for the O-TLCD =0:10and the O-TLCD =0:39, respectively. In these gures, as mentioned earlier, o O-TLCD refers to the O-TLCD-structure system where the O-TLCD exists but is deactivated. The response in the frequency domain in Fig. 21b shows that the system with o O-TLCD =0:39 presents fs;d=fsrms ms+md= 0:280r 400 400 + 155:74= 0:238Hz; (44) which is the natural frequency of the structure carrying the whole mass of the O-TLCD =0:39with stationary liquid; fs;d= 0:267 for the O-TLCD =0:10. On the other hand, when the O-TLCD is activated (with O-TLCD), fs;dis split into a lower and a higher frequency, which according 20 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD(a) El Centro (1940) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (b) Kobe (1995) 0 50 100 150 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD (d) Northridge (1994) Figure 13: Displacement response of the uncontrolled structure versus the controlled structure equipped with O- TLCD=0:39whenreq= 1:310 m-1(Northridge-required head loss coecient) under earthquakes. to Fig. 21b, are fs;d;l= 0:187 Hz and fs;d;r = 0:270 Hz, respectively, for the O-TLCD =0:39; fs;d;l= 0:243 Hz and fs;d;r= 0:283 Hz for the O-TLCD =0:10. Therefore, the harmonic excitation is de ned as wg=Asin(!rest); (45) where, for the O-TLCD =0:39, by choosing the right frequency of the coupled system fs;d;r, !res=( !s= 0:282= 1:759 rad s-1for the uncontrolled structure (without O-TLCD) !s;d;r= 0:2702= 1:697 rad s-1for the controlled structure (with O-TLCD) ; (46) andAis an appropriate amplitude (here A= 0:05 m s-2);!s;d;r = 1:778 rad s-1for the O- TLCD=0:10. The displacement response of the coupled system and the corresponding liquid de ection under the harmonic excitation in Eq. 45 are depicted for the O-TLCD =0:10with req= 0:579 m-1and the O-TLCD =0:39withreq= 1:310 m-1in Fig. 22 and Fig. 23, respec- tively. It is noted, since the employed head loss coecients reqare generally higher than the 21 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD(a) El Centro (1940) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (b) Kobe (1995) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (c) Hachinohe (1964) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (d) Northridge (1994) Figure 14: Spectral amplitude (normalized to the peak uncontrolled amplitude) of the response of the uncontrolled structure versus the controlled structure equipped with O-TLCD =0:10whenreq= 0:579 m-1(Northridge-required head loss coecient) under earthquakes. obtained optimal one (Table 1), the peak response of the coupled system may occur not at the right but at a slightly lower frequency, which is the result of excessive damping on the optimal tuning (see e.g. [58]) and is neglected here. It is also worth mentioning that the two points of intersection of with-O-TLCD and o -O- TLCD cases in the middle of the spectrum graph in Fig. 21b are not in the same level, which may be interpreted as not properly optimized tuning of the dampers, for which the reason can be twofold. First, the utilized tuning formulas (Warburton) have been proposed for undamped structures while the structure of the current study presents slight damping. Second, the free vibration causes harmonic oscillations, for which Den Hartog's tuning criterion, fd=fd;opt=f s 1 +=0:265 1 + 0:247= 0:213Hz; (47) for a theoretic O-TLCD =0:39(here we neglect that the layout of a real O-TLCD a ects f sand , too) with the same req= 1:310 m-1results in the performance shown in Fig. 24d, which brings the two points of intersection on the yaxis closer to each other. Similar in uence is seen for the 22 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD(a) El Centro (1940) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (b) Kobe (1995) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (c) Hachinohe (1964) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (d) Northridge (1994) Figure 15: Spectral amplitude (normalized to the peak uncontrolled amplitude) of the response of the uncontrolled structure versus the controlled structure equipped with O-TLCD =0:39whenreq= 1:310 m-1(Northridge-required head loss coecient) under earthquakes. other mass ratios in this gure. This observation, however, cannot necessarily lead to a better performance for a random vibra- tion, because of which the Warburton criteria were chosen for this study although the investigations of the authors (not reported here) on di erent types of vibrations using di erent damper frequen- cies highlight that the Warburton's criteria cannot be taken as impeccable tuning principles for a whole vibration scenario either. This issue, like the discussion of the required head loss coecient mentioned earlier, remarks that there would be a need for a continuous tunability in liquid dampers, in which the O-TLCD is no exception, to realize their full potential. Moreover, Fig. 24 reports that the more the mass ratio increases the wider frequency band is used to split the frequency of the o -O-TLCD-structure system. 4.5. White-noise excitation, with req The last contribution of this study in the numerical investigation part is the application of a random-generated signal as the excitation. In doing so, the performance of the lightest and the heaviest O-TLCDs have been investigated under band-limited white noise, with sample time of 23 0 50 100 150 Time [s]-0.4-0.200.20.4Liquid deflection [m](a) El Centro (1940) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Liquid deflection [m] (b) Kobe (1995) 0 50 100 150 Time [s]-0.4-0.200.20.4Liquid deflection [m] (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Liquid deflection [m] (d) Northridge (1994) Figure 16: Time-history plots of liquid de ection u(from Eq. 30) of O-TLCD =0:10withreq= 0:579 m-1on the structure excited by earthquakes. 0:01 s and noise power of 0 :005, applied to the structure as a base acceleration lasting for 300 s in the Simulink/Matlab. Results in the time and the frequency domain brought in Fig. 25 once again con rm here that even the O-TLCD =0:10has been able to reduce excessive vibrations of the structure acceptably. On the other hand, and similar to the case under the real earthquake records, O-TLCD =0:39performs the highest response reductions with the least liquid movement. Chief among these observations is that this e ective performance of the O-TLCD is not excitation- direction dependent, but omnidirectional. In other words, regardless of the excitation angle of incidence, the O-TLCD provides the structure with the same control force all along a vibration scenario. This omnidirectionality, without adding extra complication coming from extra DoFs to the motion equation of the damper, is accounted as a remarkable advantage compared to other counterparts such as TLCDs whose control forces fade when the excitation diverges from the damper's alignment. 24 0 50 100 150 Time [s]-0.200.2Liquid deflection [m](a) El Centro (1940) 0 50 100 150 Time [s]-0.4-0.200.20.4Liquid deflection [m] (b) Kobe (1995) 0 50 100 150 Time [s]-0.200.2Liquid deflection [m] (c) Hachinohe (1964) 0 50 100 150 Time [s]-0.4-0.200.20.4Liquid deflection [m] (d) Northridge (1994) Figure 17: Time-history plots of liquid de ection u(from Eq. 30) of O-TLCD =0:39withreq= 1:310 m-1on the structure excited by earthquakes. 5. Experimental Investigations This section investigates the validity of the proposed mathematical representation of the O- TLCDs experimentally. For this purpose, an O-TLCD prototype is fabricated and experiments on it are conducted at the Control Engineering Laboratory of the RWTH Aachen University. The focus of the experiments is two governing equations: (i) calculation of the natural frequency of the O-TLCDs (Eq. 19) and (ii) calculation of the response of the O-TLCDs subjected to di erent excitation angle of incidence (Eq. 18). 5.1. Description of the experimental setup The O-TLCD prototype is fabricated using transparent PVC-U pipes (inner diameter = 55 mm) connected to each other by PVC-U angles 90and a PVC-U crosspiece with adhesive sleeves (inner diameter = 63 mm) as shown in Fig. 26. This O-TLCD has four L-arms (see numbers 1to4) with equal vertical and horizontal cross sections distributed symmetrically around the center, where the liquid level in the vertical columns is V= 121 mm and the horizontal length of each L-arm isH=2 = 270 mm. Furthermore, in order to provide a reference device, a classical TLCD with 25 0.000.050.100.150.20 El Centro Kobe Hachinohe NorthridgeRMS of displacement [m] Uncontrolled µ=0.10 µ=0.20 µ=0.30 µ=0.39(a) 0.000.150.300.450.600.75 El Centro Kobe Hachinohe NorthridgePeak displacement [m]Uncontrolled µ=0.10 µ=0.20 µ=0.30 µ=0.39 (b) Figure 18: Displacement absolute values: the RMS (a) and the peak (b) response evaluation of the O-TLCD-structure system (controlled) with di erent mass ratios and with the corresponding req(Fig. 9) versus the response of the bare (uncontrolled) structure under four earthquakes. 0.0015.0030.0045.0060.00 El Centro Kobe Hachinohe NorthridgeRMS displacement [%]µ=0.10 µ=0.20 µ=0.30 µ=0.39 (a) 0.0015.0030.00 El Centro Kobe Hachinohe NorthridgePeak displacement [%]µ=0.10 µ=0.20 µ=0.30 µ=0.39 (b) Figure 19: Displacement reduction percentage: the RMS (a) and the peak (b) response evaluation of the O-TLCD- structure system (controlled) with di erent mass ratios and with the corresponding req(Fig. 9) relative to the response of bare (uncontrolled) structure under four earthquakes. two vertical columns is also fabricated using the same material with the same dimensions of inner diameter,VandH. As shown in Fig. 26, the O-TLCD (as well as the TLCD) prototype is tested on a uniaxial shaking table ( 5). The shaking table is built in a way that a threaded rod, which is operated by a step motor with the frequency range of 0 :10{50 Hz and50 mm stroke, forms the horizontal motion of the O-TLCD. The O-TLCD is mounted on the shaking table using two rectangular wooden platforms ( 6and 7). The lower platform ( 6) is directly xed to the shaking table; the upper one ( 7) is xed to the O-TLCD and can be screwed to the lower platform at two diagonal points ( 8). At these two points, two steel threaded inserts are placed on the lower platform, which guarantee no relative movement between the platforms. Accordingly, on the diagonal quarters of the upper platform, seven holes (orienting holes) are equally spaced at 15in a circular pattern 26 050100150200250300 Time [s]-0.15-0.1-0.0500.050.10.15Displacement [m]off O-TLCD with O-TLCD(a) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCD (b) Figure 20: Free vibration: Displacement response of the O-TLCD =0:10-structure system for two cases of the deacti- vated (o O-TLCD) and the activated (with O-TLCD) O-TLCD with req= 0:579 m-1in the time domain (a) and the frequency domain (b) when 0 :15 m initial displacement is applied to the structure. so that the upper platform xed to the O-TLCD can be oriented in seven di erent ways ranging from 0 to 90with respect to the motion direction of the shaking table (Fig. 26a). With this setup multidirectional vibration tests are conducted on the O-TLCD. The liquid motion is dynamically measured using an ultrasonic distance measuring sensor ( 9), UNDK 30U6112/S14, which is installed over the vertical column of the L-arm 1. Furthermore, the motion of the shaking table is independently monitored by a laser sensor ( 10), by which the accurate application of the requested excitation is validated. The data acquisition is conducted by a micro-controller with an analog-to-digital (A/D) converter and processed in Simulink/Matlab. The Simulink model is compiled in a real-time PC and the control command is sent to the step motor. In this setup, the measurements are recorded with the sample rate of 100 Hz. For the sake of statistical accuracy, each test is conducted at least three times and the average values are reported. 5.2. Natural frequency of the O-TLCD The natural frequencies of the O-TLCD prototype are determined experimentally by frequency sweeps. The O-TLCD is positioned at = 0(L-arm 1 is aligned with the motion of the shaking table as shown in Fig. 26) and a series of tests are conducted under harmonic excitation of the form, w=Asin (2ft); (48) where the excitation amplitude Ais a constant equal to 4 :5 mm and the excitation frequency franges from 0 :65 Hz to 1:05 Hz with an increment of 0 :02 Hz. For the sake of accuracy, the frequency interval is divided by a ner increment of 0 :005 Hz about the expected resonance region. The frequency sweep test is performed for the TLCD prototype as well. From the measured liquid de ection u1(the liquid de ection in L-arm 1) the response curves of both prototypes are determined and illustrated in Fig. 27. The frequencies of the peak values of the response curves correspond to the natural frequencies of the tested prototypes. Using Eq. 19, the analytical natural frequencies of both devices are calculated by substituting V= 121 mm and H=2 = 270 mm. The experimental and analytical results are gathered and compared in Table 4. 27 050100150200250300 Frequency [Hz]-0.15-0.1-0.0500.050.10.15Displacement [m]off O-TLCD with O-TLCD(a) 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCD (b) Figure 21: Free vibration: Displacement response of the O-TLCD =0:39-structure system for two cases of the deacti- vated (o O-TLCD) and the activated (with O-TLCD) O-TLCD with req= 1:310 m-1in the time domain (a) and the frequency domain (b) when 0 :15 m initial displacement is applied to the structure. 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD (a) 0 50 100 150 Time [s]-0.4-0.200.20.4Liquid deflection [m] (b) Figure 22: Harmonic excitation: Displacement response of the uncontrolled (bare) structure versus the controlled structure equipped with O-TLCD =0:10withreq= 0:579 m-1under the base acceleration 0 :05 sin(!rest) where 1:759 rad s-1for the uncontrolled and 1 :778 rad s-1for the controlled structure are considered as !res(a) and the corresponding liquid de ection ufrom Eq. 30 (b). Table 4: Natural frequencies of the O-TLCD and TLCD prototypes determined experimentally from the frequency sweep tests and calculated by Eq. 19. Experiment Calculation Di erence fexp[Hz]fcalc[Hz] f[Hz] O-TLCD 0.830 0.797 0.033 TLCD 0.825 0.797 0.025 The obtained experimental natural frequencies are both for the O-TLCD and TLCD prototypes very close to the calculation results. The frequency di erence is acceptable and can be justi ed 28 0 50 100 150 Time [s]-0.8-0.6-0.4-0.200.20.40.60.8Displacement [m]without O-TLCD with O-TLCD(a) 0 50 100 150 Time [s]-0.200.2Liquid deflection [m] (b) Figure 23: Harmonic excitation: Displacement response of the uncontrolled (bare) structure versus the controlled structure equipped with O-TLCD =0:39withreq= 1:310 m-1under the base acceleration 0 :05 sin(!rest) where 1:759 rad s-1for the uncontrolled and 1 :696 rad s-1for the controlled structure are considered as !res(a) and the corresponding liquid de ection ufrom Eq. 30 (b). with the uncertainties regarding the shaking table response and the frequency resolution of the measured data. Furthermore, the calculations are based on the simpli ed dimensions VandH=2, which do not consider the exact geometries of the prototypes and the liquid ow. 5.3. Response of the O-TLCD under di erent excitation angle To evaluate the response calculation of O-TLCDs, the aforementioned experimental tests are conducted numerically. The main purpose of this investigation is to verify the equation of motion of the O-TLCD (Eq. 18) considering particularly the representation of the e ects of the excitation direction. The core of the presented equation of motion in Eq. 18 is the proposed de nition of the DoF of the O-TLCDs in Eq. 1, which assumes a pseudo TLCD in the excitation direction and relates the liquid motion of this TLCD, u, to the liquid motions in the tubes of the O-TLCD, ui. In the rst part of the investigation, the frequency sweep tests are simulated for both O- TLCD and TLCD prototypes under the harmonic excitation given in Eq. 48 in Simulink/Matlab. The resulted two spectra are added to Fig. 27. Here, the natural frequencies and the head loss coecients of both devices are tuned to the resonance case. Accordingly, we simulate the peak value u1;max = 42:2 mm for O-TLCD with fd= 0:830 Hz and = 2:19103mm-1. For TLCD, we use fd= 0:825 Hz and = 2:26103mm-1. Wider frequency response spectra of the simulation shows that the nonlinear damping representation used in the mathematical models basically predicts more inherent damping than the one the prototypes presented physically in the laboratory. In the second part of the investigation, the free vibration response of the O-TLCD is studied by applying on the damper an impulse excitation by the shaking table. At each test, the orientation of the O-TLCD, , is varied from 0 to 45with the increment of 15with respect to the motion direction of the shaking table. The liquid motions for each excitation direction are recorded in the L-arm 1 for about 80 s and after zero padding till 1000 s are transformed into the frequency domain by the FFT. The results are shown in Fig. 28 which con rms that the natural frequency of the O-TLCD remains invariable by changing the excitation direction. 29 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCDW with O-TLCDD(a)= 0:10 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCDW with O-TLCDD (b)= 0:20 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCDW with O-TLCDD (c)= 0:30 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudeoff O-TLCD with O-TLCDW with O-TLCDD (d)= 0:40 Figure 24: Free vibration: Spectral amplitude (normalized to the peak uncontrolled amplitude) of response of the uncontrolled structure versus the controlled structure equipped with O-TLCDs with Warburton's optimal frequency (O-TLCD W) and with theoretic O-TLCDs with Den Hartog's optimal frequency (O-TLCD D) whenreq= 1:310 m-1 for all mass ratios and 0 :15 m initial displacement are applied to the structure. In the third part of the investigation, the O-TLCD is excited harmonically by the shaking table according to Eq. 48 with A= 8:4 mm and the obtained resonance frequency of the O-TLCD f= 0:830 Hz. Also, these tests are repeated for seven excitation angles , ranging from 0 to 90 with the increment of 15with respect to the motion direction of the shaking table. To compare with the experimental results, the O-TLCD response is simulated numerically under the same harmonic excitation for the considered values. As in the rst part of the investigation, the natural frequency fdand the head loss coecient are tuned to the peak response ( u1;max = 68:2 mm) at = 0. Accordingly, the natural frequency used in the simulations is fd= 0:830 Hz and the head loss coecient is = 1:63103mm-1. Introducing fdandinto Eq. 18, the response of the pseudo TLCD uis obtained. According to this equation, the response uis independent from the excitation direction , since the assumed TLCD is de ned to be always aligned with . On the other hand, as proposed by Eq. 1, the response of the O-TLCD is expected to change depending on . The liquid motion u1of the measured L-arm 1 is calculated by introducing the corresponding excitation direction , L-arm orientation = 0and the calculated liquid de ection 30 050100150200250300 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD(a)= 0:10 050100150200250300 Time [s]-0.4-0.200.20.4Displacement [m]without O-TLCD with O-TLCD (b)= 0:39 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (c)= 0:10 00.1 0.2 0.3 0.4 0.5 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudewithout O-TLCD with O-TLCD (d)= 0:39 050100150200250300 Time [s]-0.4-0.200.20.4Liquid deflection [m] (e)= 0:10 050100150200250300 Time [s]-0.4-0.200.20.4Liquid deflection [m] (f)= 0:39 Figure 25: White noise excitation: Displacement response of the uncontrolled structure versus the controlled structure equipped with O-TLCD =0:10withreq= 0:579 m-1and O-TLCD =0:39withreq= 1:310 m-1in the time (a,b) and the frequency domain (c,d); the corresponding liquid de ections u(from Eq. 30) in (e) and (f), respectively, under white noise with sample time of 0 :01 s and noise power of 0 :005. 31 14 3 26 7 9105 88 𝛼=30°(a) 143 2 679 10 58 (b) Figure 26: The fabricated prototype of the O-TLCD with four L-arms ( 1-4) attached on the uniaxial shaking table (5) in the plan view (a) and the front view (b). A lower platform ( 6) is connected with a rotatable upper platform (7) at two diagonal points ( 8) allowing multidirectional vibration tests with the excitation direction = 0 to 90 with 15increment. Here the O-TLCD is shown at = 30. The liquid de ection is measured by an ultrasonic sensor ( 9). The motion of the shaking table is measured by a laser sensor ( 10). uof the pseudo TLCD into Eq. 1. Fig. 29 compares the time histories of the numerically calculated liquid motion u1with the experimental one at =f0;30;60;90g. It is seen that the proposed mathematical model of the O-TLCD predicts the experimental data accurately. The liquid de ection amplitude changes depending on the excitation direction and matches the experimental data. As previously observed in the frequency spectra in Fig. 27, it is again noted here that the simulation slightly overestimates the inherent damping. Particularly in the response time history of = 60, the simulated response (u1;max= 33:6 mm) is slightly lower than the experimental result ( u1;max= 36:2 mm). During experiments, by rotating the O-TLCD from = 0 to 90, we observed on the liquid surface a gradually increasing wave formation in the excitation direction. At = 90, the mea- sured L-arm is perpendicular to the excitation direction and the liquid de ection at this excitation is theoretically expected to be 0. However, from the recorded time history shown in Fig. 29 the wave-induced liquid motion can be clearly seen. This response can be modeled by mathemati- cal approaches proposed for TLDs and therefore does not belong to the scope of the O-TLCD mathematical model. Physically the wave formation can be prevented in O-TLCD by dividing the column vertically in smaller cells. Fig. 30 gathers the experimental and numerical peak liquid motions for all considered excitation angles. With increasing , the predicted peak liquid de ection ( u1;max)u1reduces corresponding to the experimental results. The curve belonging to the experimental results includes three measured maximum de ection values and a mean value for each excitation angle. From the comparison of the results we see particularly at = 90a marginal di erence in the predicted liquid de ection due to the previously mentioned numerically overestimated inherent damping and the wave e ects. Therefore, it is concluded that the proposed mathematical approach is able to imitate the behavior of the O-TLCDs markedly where the liquid motions in any liquid column can be determined for any excitation direction. 32 0.65 0.75 0.85 0.95 1.05 Frequency [Hz]010203040Peak liquid deflection [mm]O-TLCDexp TLCDexp O-TLCDsim TLCDsim fexp,O=0.830 Hz fexp,T=0.825 HzFigure 27: Experimental (O-TLCD exp, TLCD exp) and numerical (O-TLCD sim, TLCD sim) frequency response curves of the liquid de ection. The numerical calculations are performed for O-TLCD with the natural frequency fd= 0:830 Hz and the head loss coecient = 2:19103mm-1. For TLCD the natural frequency is fd= 0:825 Hz and the head loss coecient is = 2:26103mm-1. Both dampers are subjected to a harmonic excitation w= 4:5 sin(2ft) mm and are aligned with the excitation direction ( = 0). 6. Conclusion This paper proposed a mathematical modeling and optimization scheme for omnidirectional tuned liquid column dampers (O-TLCDs), which are formed by circularly distributed of n(integer n3) L-arms about a common joint region at the center, through which the liquid moves between horizontal tubes. Accordingly, introducing the concept of a pseudo TLCD with the liquid de ection uin the direction of an excitation , this study introduced a formal solution to determine the DoF of the O-TLCDs, by which it was proven that the O-TLCDs behave as an SDoF system; the liquid motion in each L-arm can be then obtained using uand . Next, The paper derived the equations of motion of the O-TLCDs using Lagrange's principle and the coupled system using equilibrium of forces and showed that, owing to its con guration, O-TLCDs can reinforce structures with full counteracting force capacity in all transversal directions regardless of the excitation angle of incidence. For designing O-TLCDs, a set of design criteria and a general optimization scheme, which includes the online simulation of the system equipped with the O-TLCD exposed to an arbitrary excitation, were also proposed. Numerical parametric studies on a SDoF structure with an O-TLCD were performed for varying mass ratios and head loss coecients. Seismic, harmonic and white noise excitations were investigated in time and frequency domains showing the high vibration reduction capability of the damper. Aiming to verify the proposed mathematical approach, experimental investigations were per- 33 00.20.40.60.811.21.4 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudefexp,O = 0.830 Hz(a) = 0 00.20.40.60.811.21.4 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudefexp,O = 0.829 Hz(b) = 15 00.20.40.60.811.21.4 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudefexp,O = 0.830 Hz (c) = 30 00.20.40.60.811.21.4 Frequency [Hz]00.20.40.60.81Normalized spectral amplitudefexp,O = 0.833 Hz(d) = 45 Figure 28: Free vibration: Spectral amplitude of the liquid de ection (normalized to the peak de ection) of the prototype O-TLCD with four orientation angles = 0 to = 45with respect to the axis of the shaking table. formed on an O-TLCD prototype. The prototype was attached to the shaking table and tested under harmonic vibrations, where the liquid motion was measured using an ultrasonic sensor. The natural frequency of the O-TLCD was derived from frequency sweep test. The response of the O-TLCD was measured for di erent excitation angles ranging from 0 to 90. Observations showed that the proposed mathematical model can predict the response of O-TLCDs with circularly dis- tributednL-arms successfully. Results of this study introduces a mathematical description for the O-TLCDs, which requires no extra DoFs and is valid for any lateral excitation. The both numerical and experimental observations regarding the actual optimal frequency and the required head loss coecient for a vibration scenario highlights also the need for a semi-active tuning capacity for the introduced O-TLCDs, which necessitates further investigations. Acknowledgment The authors would like to express their sincere appreciation for the nancial support from the German Federal Ministry of Education and Research (Bundesministerium f ur Bildung und Forschung, BMBF) with the grant number: FKZ 03VP04680. The authors would also like to thank the Control Engineering Institute of RWTH Aachen University, and in particular Mr. Markus Zimmer, for their support during the preparation of the experimental setup. 34 0 10 20 30 40 Time [s]-60-45-30-15015304560Liquid deflection [mm]Experiment Simulation(a) = 0 0 10 20 30 40 Time [s]-60-45-30-15015304560Liquid deflection [mm]Experiment Simulation(b) = 30 0 10 20 30 40 Time [s]-60-45-30-15015304560Liquid deflection [mm]Experiment Simulation (c) = 60 0 10 20 30 40 Time [s]-60-45-30-15015304560Liquid deflection [mm]Experiment Simulation(d) = 90 Figure 29: Comparison of the experimental and numerical liquid de ection u1of the O-TLCD subjected to a harmonic excitationw= 8:4 sin(20:83t) mm at the excitation direction =f0;30;60;90g. At = 0, the observed L- arm 1 of the O-TLCD is aligned with the excitation direction. At = 90, the observed L-arm of the O-TLCD is perpendicular to the excitation direction. The natural frequency is fd= 0:830 Hz and the head loss coecient is = 1:63103mm-1. 35 0 15 30 45 60 75 90 Excitation angle [ °]015304560Liquid deflection [mm]Experiment SimulationFigure 30: Comparison of the experimental and numerical peak liquid de ection ( u1;max)u1of the O-TLCD subjected to a harmonic excitation w= 8:4 sin(20:83t) mm at the excitation direction = 0 to 90. At = 0, the observed L-arm 1 of the O-TLCD is aligned with the excitation direction. At = 90, the observed L-arm 1 of the O-TLCD is perpendicular to the excitation direction. The natural frequency is fd= 0:830 Hz and the head loss coecient is = 1:63103mm-1. 36