This paper presents a new type of semi-active tuned liquid column damper (S-TLCD) for the lateral vibration control of high-rise civil engineering structures. Analogous to the passive tuned liquid column damper (TLCD), the S-TLCD comprises a U-shaped tank consisting of two vertical columns, which are arranged at a distance from each other and communicating through a horizontal passage. The tank is partially lled with a Newtonian uid until the liquid reaches a certain level in the columns. In contrast to the passive TLCD, the S-TLCD provides also mechanisms for a continuous adaptation of both its natural frequency and damping behaviour in real-time. In the rst part of the paper, the governing equations of the S-TLCD are derived based on the Bernoulli equation of a non-stationary incompressible liquid ow. The natural frequency of the S-TLCD is revealed to depend on the scaled length of the liquid. The scaling amount of the liquid length is formulated in dependence of the cross-sectional area ratios of the tank segments. The mathematical description of the S-TLCD is concluded by providing the state-space representation of a multi-degree-of-freedom structure with several S-TLCDs. In the second part of the paper, the derived natural frequency equation is veri ed and the proof of concept of the S-TLCD is shown by experimental investigations, which are performed on an S-TLCD model utilizing a test structure and shaking table tests. Keywords: Semi-active tuned liquid column dampers; adaptive sti ness; adaptive damping; natural frequency tuning; high-rise structures; lateral vibrations 1. Introduction Semi-active damper systems can adapt themselves to the changing structural conditions and loading situations based on the data captured by their sensors. This autonomous tuning capability allows the semi-active structural control devices to outperform passive vibration mitigation methods. Furthermore, semi-active systems can alternate their parameters also in real-time ensuring a superior, stable and low- energy control performance compared to active systems. In the last decades, many semi-active systems have been developed. However, most of these devices can adapt only their damping parameters. Besides the active variable sti ness (AVS) system developed by Kobori et al. [1], only a few semi-active systems have been developed with sti ness adaptation capabilities. Some rare examples include the adaptive length pendulum tuned mass dampers (APTMD) developed by Nagarajaiah, which are also applied in the USA, Japan and China [2{4]. Semi-active systems with tunable natural frequency capabilities are especially necessary for slender high-rise structures, such as wind turbines. During the service time, these structures change their natural frequencies due to degradation and soil e ects. Thereby passive systems lose their eciency by frequency o -tuning. In this context, Sun developed for wind turbines subjected to multi- hazards a sophisticated simulation environment, where degradation and soil e ects can be simulated [5, 6]. He proposed and investigated for monopile o shore wind turbines a semi-active tuned mass damper (STMD), which can adapt its damping and frequency parameters. The results show a signi cant improvement of the control system due to semi-active adaptation capability. Besides these mechanical tuned mass dampers, the so far proposed liquid based tuned mass dampers, which possess both natural frequency and damping adaptation capabilities, are still in the very early devel- opment stage. Due to their geometric versatility, low prime costs and stable dynamic behaviour, the tuned liquid column dampers (TLCD) are regarded as one of the most promising solutions. However, an applicable semi-active TLCD with continuously variable natural frequency does not exist yet. First invented by Frahm [7] already in 1909 for ships and then proposed by Sakai [8] in 1991 for civil engineering structures, the TLCD is one of the rst structural control devices. The TLCD consists of a U-shaped tube with two vertical columns, which are open at one end and connected at the other end by a horizontal segment. The tube is attached on a structure, the vibrations of which are to be controlled. For high-rise structures the TLCDs are usually attached at the top of the structure, where the maximum lateral motion is expected. The tube is partially lled with a Newtonian uid, which is prescribed to ow with phase shift with respect to the structure. The phase shift must be tuned by the natural frequency and the damping parameters of the liquid motion. The oscillating liquid mass evokes restoring forces. During this oscillation local friction and turbulence e ects in the tube dissipate vibration energy, allowing a control of the structural vibrations. To tune the damping of the TLCDs, Sakai installed an ori ce in the horizontal segment. This ori ce can be closed manually to introduce additional damping by increasing the local friction [8]. Frahm and Matsuo closed the open ends of the vertical columns with ori ces, which in uence the air ow and reduce the speed of liquid level motion [7, 9]. As this method in uences also the natural frequency, the approach proposed by Sakai counts as an established method for the damping tuning of the TLCDs. Several studies have been conducted considering the calculation and control of the damping e ects introduced by the ori ce, such as by Yalla and La et al. In their study, Yalla et al. equipped a TLCD with an electro-pneumatic valve to tune its inherent damping in a semi-active manner [10]. They developed a control strategy and conducted experimental investigations. The results show that the semi-active tuning of the inherent damping provides the TLCD an up-to 25% performance improvement. Furthermore, La et al. derived in their study a general on-o controller for the damping tuning of the TLCDs using the ori ce [11]. They documented the robust performance of the controller numerically using a ve-degree-of-freedom shear frame, which is subjected to earthquake. Both of these investigations use TLCDs without frequency adaptation capabilities. On the other hand, to tune the natural frequency of the TLCDs, Csupor divided the horizontal segment into two channels, each with a di erent cross-sectional area [12]. As the uid ow velocity in the tube depends on the cross-sectional area, this approach allowed the TLCD to switch between two natural frequencies. However, for the mitigation of lateral vibrations the volume of the horizontal segment is decisive. Therefore, the TLCD proposed by Csupor alternates unfavorably also its restoring force level depending on the volume of the chosen channel. Another approach for the natural frequency tuning was proposed by Nomichi et al. [13]. By sealing the open ends of the vertical columns they introduced an air spring, which increased the natural frequency of the liquid oscillation. Yoshimura et al. enhanced this approach by sensors and a controller to tune the natural frequency in a semi-active manner [14]. Kagawa et al. divided the vertical column above the liquid level into several chambers, which are connected by ori ces [15, 16]. By closing the ori ces, the air volume is modulated causing also an adjustment of the sti ness coecient of the air spring. This allowed the TLCD to switch its natural frequency incrementally. Hochrainer et al. investigated the application of this sealed tuned liquid column gas damper (TLCGD) on structures [17{19]. Reiterer et al. implemented TLCGD in vertical direction [20]. The rst application of TLCGD showed that the approach using overpressure is eligible primarily for small-scale distributed TLCD systems [21]. Moreover, as the supplementary air spring increases the natural frequency of the TLCD, it impedes the application of the damper on slender structures, which usually oscillate with low frequencies. A further method for frequency tuning is proposed by Yoshimura et al. [22]. They integrated in TLCDs 2 mechanical spring elements in uencing the ow velocity of the liquid. They attached the liquid column damper with period adjustment (LCD-PA) on a hotel building in Tokyo [23]. Also Ghosh and Sonmez et al. changed the natural frequency of the TLCD by interposing spring elements between TLCD and structure [24{26]. Both of these hybrid solutions control the natural frequency mechanically using spring elements while the TLCD continues its operation in a passive manner. Kim et al. divided the vertical columns in upright cells allowing a frequency tuning of the TLCD [27]. They applied this TLCD on four skyscrapers in South Korea to tune the natural frequency after installation of the damper in a passive adaptive manner. By closing the chambers, the cross-sectional area of the columns can be adjusted, which in uences the liquid ow speed and enables an incremental frequency tuning. This paper introduces a new type semi-active tuned liquid column damper (S-TLCD), which can change both its natural frequency and damping continuously. Section 2 presents the general properties of the S- TLCD. Section 3 derives the governing equations of the S-TLCD including its equation of motion, natural frequency and state-space representation. Section 4 veri es the natural frequency equation and shows the proof of concept of the S-TLCD experimentally by using a small-scale model of the S-TLCD and performing shaking table tests. Section 5 concludes the mathematical de nition and the results of the experimental studies. 2. Semi-active Tuned Liquid Column Damper Inspired by the passive TLCD, the S-TLCD is a tank lled with a Newtonian liquid, such as a mixture of water and antifreeze. The tank has a U-shaped geometry consisting of two rectangular liquid columns, which are arranged at a distance from each other and communicating by a horizontal passage as shown in Figure 1. In the tank, mechanisms for tuning both the natural frequency and the damping are provided, which can adjust the parameters of the damper in real-time. For this purpose, perpendicular to the ow direction, movable vertical panels are attached to the column walls, which allow to change the cross-sectional area of the columns without changing the horizontal distance between the liquid columns. By changing the cross-sectional area, the velocity of the liquid ow can be controlled enabling the natural frequency tuning capability of the S-TLCD. Each movable panel consists of a transition and an upright segment. The liquid level oscillates in the upright segment range allowing the S-TLCD to keep its natural frequency constant during its operation. The gaps between movable panels and column walls are sealed by waterproof elastic membranes. For the tuning of the damping, several butter y valves are installed in the connection passage of the liquid columns. By closing these valves, local friction e ects can be introduced allowing an increase in damping. The motion of the panels and the butter y valves is controlled by actuators. The liquid motion is measured by a capacitance transducer. A demonstrator of S-TLCD with such actuators and sensors will be presented in Section 4.1. 3. Mathematical Description In uid dynamics, the absolute acceleration aof a single liquid particle with in nitesimal mass d mand volume dVis determined by the surrounding force eld f, such as gravity, and the pressure gradient grad p, as de ned by the Euler impulse equation for homogeneous frictionless incompressible uids in Eq. 1, [28]. In Eq. 2 the terms are divided by the liquid density . Besides gravity, magnetism and electricity cause also a force eld, which can be used to control magnetorheological or electrorheological liquids. dma= dmfdVgradp (1) ,a=f1 gradp (2) Figure 2 shows the liquid particle moving through a tube between the 1○and 2○with the cross-sectional areasA1andA2accordingly. The starting and ending velocities of the particle are v1andv2. The direction 3 Figure 1: The proposed semi-active tuned liquid column damper (S-TLCD) with natural frequency and damping tuning capabilities. dm Figure 2: Homogeneous frictionless incompressible uid particle moving through a tube from 1○to2○. 4 vector tangent to the tube streamline is labeled as et. The force eld facting on the particle and the absolute acceleration aof the liquid particle are also shown in Figure 2. Based on Eq. 2 Sakai et al. [29] derived the equation of motion of passive TLCDs attached to horizontally vibrating high-rise structures. Xue et al. [30] enhanced this equation for pitching structures. Hochrainer et al. [17] derived the equation of motion for TLCGDs attached to horizontally vibrating high-rise structures according to the approach developed by Sakai. Reiterer et al. [31] enhanced this equation for TLCGDs at- tached to vertically and rotationally oscillating structures, such as footbridges. In this section, the governing equations of the S-TLCD, which is attached to a horizontally vibrating high-rise structure, will be derived in a similar manner based on the Bernoulli equation of the non-stationary incompressible ow of the liquid. 3.1. Equation of Motion The total acceleration of a liquid column in a tank can be calculated in a general form by integrating the Eq. 2 along the streamline for the force eld of gravity gleading to Z aetds=Z getds1 Z gradpetds: (3) The integral of the pressure gradient along the streamline of the S-TLCD equals to the pressure loss  p, which is caused by turbulance and local friction e ects in the tank. Analytical methods for the calculation of pressure loss can be found in the literature, such as in Eq. 4 according to [30] p=j_uj_u; (4) where _uis the velocity of the liquid ow and the head loss coecient, which is a function representing the turbulence e ects depending on Reynolds number of the liquid stream together with the local friction e ects. The absolute liquid acceleration ain Eq. 3 is de ned as the sum of the acceleration aMof the main structure, on which the S-TLCD is attached, and the relative acceleration a0of the liquid column yielding aet=aMet+a0et: (5) The relative acceleration a0of the uid is separated into local and convective accelerations as shown below in Eq. 6. The local acceleration describes the force eld induced time-dependent change in the velocity _uof the liquid particle. The convective acceleration describes the position-dependent change in the velocity of the liquid stream. a0et=@_u @t+ _u@_u @s(6) From Eq. 3, by integrating the scalar product getand using the pressure loss  pas described before in Eq. 4, a modi ed version of the Bernoulli equation can be determined for the S-TLCD as shown in Eq. 7, whereuis the displacement of the liquid in the tank. Z aMetds+Z@_u @tds+Z@ @s_u2 2 ds=2gu1 p (7) As shown in Figure 3, the main parameters describing the geometry of the S-TLCD tank are in vertical direction the height Vand in horizontal direction the length Hof the streamline. Further parameter is di, wherei= [1;4] giving the cross-sectional areas of the vertical columns and the horizontal passage. As seen in the side view, the cross-sectional area of the vertical columns can change depending on the inclination angle of the movable panels. The vertical columns are hereby divided into three sections: beginning section, transition section and end section, V1,V2andV3respectively. The height V2is a function of and furthermore, for S-TLCDs with constant liquid amount also V3changes depending on . The cross-sectional areas of the tank are calculated using Eq. 8 and 10. Hereby, AV1is the cross-sectional area of the column-top, AV2is the cross-sectional area of the column-beginning and AHis the cross-sectional 5 uV Figure 3: Mechanical model showing the governing parameters of the S-TLCD. Left: Front view of the S-TLCD. Right: Side view of the S-TLCD. area of the connecting passage of the vertical columns. Because d3is a function of the panel position , AV1can be modi ed by changing . AV1=d1d3 (8) AV2=d1d4 (9) AH=d2d4 (10) The calculation of the acceleration e ects, which are induced by the oscillations of the main structure, are performed in Eq. 11, along the streamline through connection points 1-8 of the sections of the damper tank as shown in Figure 3. Here, the acceleration of the main structure is assumed to occur only in horizontal x-direction, which is usually the relevant vibration direction for high-rise structures subjected to wind and earthquake loads. Therefore, for the acceleration, it is assumed that aM= xex. The scalar product of the horizontal direction vector exwithetvector, which is parallel to the streamline, is conducted in Eq. 12-14. The result equals to the length of the horizontal passage of the damper Hmultiplied with the acceleration xof the main structure. Z aMetds= xZ2 1exetds+Z3 2exetds++Z8 7exetds =Hx (11) 1!4 :exet= 0 (12) 4!5 :exet= 1 (13) 5!8 :exet= 0 (14) Figure 4 shows the velocity pro le of the liquid stream in the S-TLCD. The velocity changes in dependence with the cross-sectional area of each segment. This is applied in Eq. 15-23 to determine the local acceleration of the liquid ow. The result corresponds to the acceleration  uof the liquid scaled by L1, which we call rst e ective length of the S-TLCD. Eq. 16 determines L1by solving the integral in Eq. 15. For a TLCD with equal horizontal and vertical cross-sections AH=AV, the e ective length corresponds to the total length of the liquid streamline. 6 u V HEquilibrium levelu 1 2 3 48 7 6 5V2 V1V3 u. u. u..AV1 AV2 u..AV2 AHu..AV1 AV2x yFigure 4: Velocity pro le of the liquid stream in the damper tank. Z@_u @tds=7X i=1Zi+1 i@_u @tds=L1u (15) L1= 2V3+V2+ (2V1+V2)AV1 AV2+HAV1 AH(16) 1!2 :@_u @t= u (17) 2!3 :@_u @t=u 2 1 +AV1 AV2 (18) 3!4 :@_u @t= uAV1 AV2(19) 4!5 :@_u @t= uAV1 AH(20) 5!6 :@_u @t= uAV1 AV2(21) 6!7 :@_u @t=u 2 1 +AV1 AV2 (22) 7!8 :@_u @t= u (23) As shown in Eq. 24, due to the symmetry of the damper the integral of the convective acceleration becomes zero: Z8 1@ @s_u2 2 ds=1 2( _u2 8_u2 1) = 0: (24) The sum of Eq. 11, 15 and 24 gives according to Eq. 7 the expanded nonlinear equation of motion of the S-TLCD for a liquid motion along the streamline as shown in Eq. 25 and modi ed in Eq. 26, where !Dis the adaptable natural circular frequency of the semi-active damper, which is calculated according to Eq. 27. In Eq. 26, we call the parameter 1as the rst geometric factor of the S-TLCD, which is given by Eq. 28. Both!Dand 1depend on the rst e ective length of the streamline L1. By changing the cross-sectional area of the vertical liquid columns, the e ective length and following that the circular frequency !Dand the geometric factor 1change. 7 As a result, the equation of motion of the S-TLCD reads L1u+ 2gu+1 p=Hx; (25) ,u+!2 Du+j_uj_u= 1x; (26) where the natural frequency of the S-TLCD is given by !D=r 2g L1(27) and the rst geometric factor of the S-TLCD is expressed as 1=H L1: (28) As shown in Eq. 27, the natural frequency !Dof the S-TLCD depends only on the rst e ective length L1 . This equation and the calculation of the rst e ective length L1accordingly will be veri ed experimentally in Section 4.2. In the equation of motion of the S-TLCD, the nonlinear term j_uj_ucan be linearized according to [31] into 2DD!D_u, whereDDis the equivalent viscous damping ratio of the S-TLCD. The DDcan be calculated by demanding for both linear and nonlinear cases that the energy dissipation is for each cycle equal. This yields the relation shown in Eq. 29, where Tis the vibration period. ZT 0j j_uj_u+!2 Du _ujdt=ZT 0j 2DD!D_u+!2 Du _ujdt (29) As described in [31], from Eq. 29 by substituting the time harmonic function u=U0cos!Dt, the equivalent viscous damping ratio DDis determined proportional to the vibration amplitude U0as shown in Eq. 30. DD=4U0 3(30) Using this approach the equation of motion of the S-TLCD can be linearized as u+!2 Du+ 2DD!D_u= 1x: (31) This approach simpli es, at least for the harmonic excitations, the tuning of the damping. However, especially in case of random excitations with changing amplitudes, for an accurate tuning of the damping, the nonlinear term j_uj_ugiven in Eq. 26 should be used. The accurate tuning of this parameter can be accomplished by using soft-computing methods such as Fuzzy control, neural networks and evolutionary algorithms [32], which is the subject of future research. 3.2. Restoring Force The restoring force of the S-TLCD on a horizontally oscillating high-rise structure is derived from the momentum equation. The momentum Iinduced by the liquid motion is computed in Eq. 32 using the integral of the resulting liquid velocity over the liquid mass mf. The resulting velocity is assembled by the velocity of the structure vAat the attachment point Aof the S-TLCD to the structure and the liquid velocity v. I=Z (vA+v)dmf=Z8 1 _xet+ _uAV1 A(s)et A(s)ds (32) 8 The momentum projected in lateral x-direction is computed in Eq. 33. The restoring force is determined in Eq. 34 from the time derivative of the momentum equation. In Eq. 35, the liquid mass mfis calculated. The second geometric factor 2is calculated in Eq. 36 and the second ecient length L2is given in Eq. 37. Ix= AV12V3_x+ (AV1+AV2)V2_x+AV22V1_x+AHH _x+ _uAV1 AH (33) Fx=dIx dt=mf(x+ 2u) (34) mf=AV1L2 (35) 2=H L2(36) L2= 2V3+V2+ (2V1+V2)AV2 AV1+HAH AV1(37) The equation of motion of a single-degree-of-freedom (SDoF) system with an S-TLCD subjected to a time-dependent load F(t) =f(t)mSis shown in Eq. 38, where =mf=mSis the ratio of the liquid mass to the modal mass of the structure. In the same equation, the damper interaction forces are represented by x+ 2u, which is derived previously in Eq. 34. In Eq. 38, !Sis the natural frequency of the structure andDSis the damping ratio. x+ 2DS!S_x+!2 Sx=x 2u+f (38) If the structure is subjected to a ground acceleration  xg, the equation of motion of the SDoF system with the S-TLCD is rewritten as x+ 2DS!S_x+!2 Sx=xg(x+ xg) 2u+f: (39) Based on this equation, the equation of motion of a multi-degree-of-freedom (MDoF) system with several S-TLCDs will be derived in Section 3.4 by using the state-space representation. 3.3. Geometric Factors The geometric factors 1and 2determine the ratio of the horizontal length Hof the S-TLCD to the ecient lengths L1andL2. For an S-TLCD with a constant liquid amount, the cross-sectional area AV1, the height V2and the height V3depend on . As a result, L1andL2, as well as 1and 2accordingly, become also a function of . For the sake of convenience, the equations of the ecient lengths and geometric factors representing this general case are enlisted below: L1( ) = 2V3( ) +V2( ) + (2V1+V2( ))AV1( ) AV2+HAV1( ) AH; (40) L2( ) = 2V3( ) +V2( ) + (2V1+V2( ))AV2 AV1( )+HAH AV1( ); (41) 1( ) =H L1( ); (42) 2( ) =H L2( ): (43) For a TLCD with uniform cross-sectional area, the total length of the liquid column Lequals to both L1andL2. For a TLCD with a constant cross-sectional area of the vertical columns AVand a di erent cross-sectional area of the horizontal passage AH, as proposed by Hitchcock et al. [33, 34], both ecient lengthsL1andL2are given by the sum of the total liquid length in vertical direction and the horizontal length, which is scaled by the ratio of the cross-sectional areas AVandAH. For this type of TLCD, if the 9 liquid amount is constant, a change of AVwould in uence the rst ecient length L1linearly and the second ecient length L2in a nonlinear manner. In case of the S-TLCD with a constant liquid amount, a change of the panel position in uences both L1andL2nonlinearly. As the natural frequency !Ddepends also onL1, a change of the panel position will also cause !Dand consequently the geometric factors 1and 2to change in a nonlinear manner. This e ect can be seen in Sections 4.2 and 4.3 from the experimental results. The interaction forces between S-TLCD and structure are scaled by 1and 2. Therefore, the geometric factors are expected to e ect the eciency of the damper depending on . The rst geometric factor 1scales the acceleration of the structure  x, as given in the equation of motion of the S-TLCD in Eq. 31. Accordingly, together with the acceleration  xof the structure, an increase in the geometric factor 1ampli es the liquid motion amplitude u. In the equation of motion of the structure, given in Eq. 39, the second geometric factor 2scales the acceleration of the liquid motion with the term  2u, where the minus sign represents the restoring character of the interaction force. Consequently, an increase in the geometric factor 2mitigates the lateral motionxof the structure. From the arguments above, it can be assumed that 2will dominate the eciency of the S-TLCD. However, 1correlates directly with L1and with the natural frequency !Daccordingly, which is vital for the eciency of the damper. This e ect can also be observed from the experimental results presented in Section 4.2. Furthermore, to increase the active mass fraction of the S-TLCD, both 1and 2must be maximized, which will be shown below. Analogous to the TLCD, the active mass fraction for the S-TLCD de nes the amount of liquid mass participating to the structural control by inducing the restoring force. The rest of the liquid mass is necessary for the tuning of the natural frequency of the damper. The amount of active mass fraction can be determined according to Hochrainer et al. [17] by introducing the scaled liquid motion  u=u= 1 in the equations of motion of the structure and the S-TLCD, and by multiplying these equations with diag (1=(1 +(1 1 2));1= 1), which yields 1 +  1 1x u +2DS!S 0 0 2DD!D_x _u +!2 S0 0!2 Dx u =1 +  1 xg+1=mS 0 f; (44) where the ratio between the active mass fraction  mDof the S-TLCD and the sum of the modal mass of the structure mSand the rest liquid mass of the damper is denoted by  , which is given by =mD mS=mD 1 2 mS(1 +(1 1 2))= 1 2 1 +(1 1 2): (45) From the numerator of Eq. 45 it can be clearly seen, that the active mass fraction  mDequals to the total liquid mass of the damper scaled by 1and 2: mD=mD 1 2: (46) In Eq. 44,  !Sand DSare given by dividing the natural frequency !Sand the damping ratio DSbyp 1 +(1 1 2). 3.4. State-Space Representation In this section, the state-space representation of a n-DoF structure subjected to a horizontal ground acceleration  xgand horizontal excitation forces F(t) withkS-TLCDs will be derived. The equation of motion of the system is given in Eq. 47. In this equation, MMis a hypermatrix consisting of the mass of the structure and the S-TLCDs. The lateral displacement vector of the system and the liquid motion vector of the S-TLCDs are denoted by xandurespectively. CSandCDare the damping matrices of the structure and the S-TLCDs. KSandKDare the sti ness matrices. MSis the modal mass matrix of the structure andMDis the mass matrix of the S-TLCDs. The vector rdistributes the ground excitation  xgto the DoFs 10 of the structure. As shown in Eq. 48, ris an identity vector for a horizontally excited structure with DoFs only in horizontal direction. MM x  u +CS0 0 C D _ x _ u +KS0 0 K D x u =MSr+LMDi 1i xg+f 0 (47) rT= (1;1;:::;1)1n (48) The S-TLCDs are distributed to the DoFs of the structure by using an incidence matrix Lwith the dimensionnk. Eq. 49 below shows an example of an incidence matrix for a structure with three DoF with two S-TLCDs attached to the second and third DoF of the structure. L=2 40 0 1 0 0 13 5 (49) For a horizontally excited system with all S-TLCDs also in the horizontal direction, the vector iis an identity vector as shown in Eq. 50. iT= (1;1;:::;1)1k (50) Furthermore, in Eq. 47, 1is a matrix containing the geometric factors of the S-TLCDs. The excitation forces acting on the structure are represented by f. The hypermatrix MMis assembled in Eq. 51. It contains besides the mass matrices MSandMD, and the incidence matrix Lalso the matrices 1and2with the geometric factors, as given in Eq. 52. The hypermatrix includes furthermore an identity matrix I, as shown in Eq. 53. MM=MS+LMDLTLMD2 1LTI (n+k)(n+k)(51) i= diag ( i;1; i;2;::: i;k) (52) IT= (1;1;:::;1)kk (53) In the state-space representation, the displacements x,uand the velocities _ x,_ uof the system are composed in the state vector zand its rst derivative _ zas zT= (x;u;_ x;_ u)2(n+k)1; (54) _ zT= (_ x;_ u; x; u)2(n+k)1: (55) Using the state vector zand its rst derivative _ z, a general representation of a horizontally excited MDoF system with several S-TLCDs is derived. The distribution vector egand the distribution matrix E are provided in Eq. 57 and 58: _ z=Arzegxg+Ef; (56) eg=2 6640 0 M1 M MSr+LMDi 1i3 775 2(n+k)n; (57) 11 E=2 6640 0 M1 MI 03 775 2(n+k)n: (58) The hypermatrix Arde nes dynamic properties of the system and consists of two further components AandBRas given below in Eq. 60-62: Ar=A+BR; (59) A=2 40 I M1 MKS0 0 0 M1 MCS0 0 03 5 2(n+k)2(n+k); (60) B=2 40 I M1 M I 0 0 I M1 M I 0 0 I3 5 2(n+k)2(n+k); (61) R=2 6640 0 0 0 0 K D 0 0 0 0 0 0 0 0 0 C D3 775 2(n+k)2(n+k): (62) 4. Experimental Investigations To verify the natural frequency equation derived in Section 3 and to show the proof of concept of the S-TLCD, experimental investigations are conducted at the control engineering laboratory of RWTH Aachen University. The study focuses on the natural frequency adaptation capability, e ects on the inherent damping and the vibration control performance of the S-TLCD considering frequency tuning e ects and in uences of the geometric factors. 4.1. Experimental Setup The experimental setup consists of a small-scale model of the S-TLCD with full adaptation capability and a model structure, on which the S-TLCD is attached and excited by an uniaxial shaking table. The model of the S-TLCD, as shown in Figure 5, has the intended adaptation features and can change its natural frequency and damping behaviour. For the U-shaped tank, acrylic glass is chosen to allow visual observation of the liquid motion and position change of the movable components. The tank is built by rectangular glass elements, which are glued with each other, compressed by screws and sealed with silicone. The tank is 430 mm length, 570 mm height and 130 mm width. The material thickness of the acrylic glass is 15 mm. The cross-sectional area of the vertical columns of the tank is 50 mm 100 mm. The horizontal passage has a cross-sectional area of 100 mm 100 mm. The S-TLCD model has in each vertical column two movable panels, which can change the cross-sectional area of the vertical columns and allow the tuning of the natural frequency, see 1○in Figure 5 and Figure 6. Each panel consists of two segments, which are connected by joints. The rst transition segment is 60 mm long and connects the second segment with the vertical column of the tank. The second segment is vertical and has a length of 385 mm. In each vertical column, the panels are connected together with a threaded rod, which allows the panels to change their position simultaneously perpendicular to the liquid ow and accordingly to the operation direction of the S-TLCD. Each threaded rod is connected to an actuator, which 12 1 32Figure 5: S-TLCD model. 1○: Movable panels for natural frequency tuning. 2○: Actuators adjusting the panels. 3○: Butter y valves for damping tuning. allows an electronic adjustment of the panel positions, see 2○in Figure 5 and Figure 6. The gaps between the panels and the tank walls are sealed with water-resistant and highly-expandable plastic membranes. The construction allows the panels to change their position within a range of = [45;72]. For the tuning of the inherent damping, two butter y valves are installed in the middle of the horizontal passage, see 3○in Figure 5 and 4○in Figure 6. Each valve consists of a rectangular plate with the dimensions 98 mm45 mm. Each plate is rotatable around a vertical axis perpendicular to the liquid ow direction. By opening the valves (rotation angle of the plate '= 90) and closing them ( '= 0), the inherent damping can be changed due to modulation of the local friction e ects. As shown in Figure 6 by 5○, both valves are connected by gear wheels to an actuator, which allows a simultaneous electronic adjustment of the valves. For the measurement of the liquid motion, a capacitance transducer is developed and installed in one of the vertical columns of the S-TLCD model, which can dynamically measure the liquid level change, see 3○in Figure 6. The liquid motion is recorded at a sample rate of 100 Hz allowing the identi cation of relevant dynamic parameters, such as maximum liquid amplitude and natural frequency of the S-TLCD. The measured liquid motion is also utilized to investigate the inherent damping e ects of the S-TLCD. For the experimental investigations, a uniaxial horizontal shaking table is built, which can generate harmonic vibrations, see 3○in Figure 7. The shaking table is operated by a linear actuator in a frequency range of 0:1 - 50 Hz with 100 mm ( 50 mm) stroke. The motion of the shaking table is registered by a microelectromechanical system (MEMS) accelerometer. During the experimental investigations, the S- TLCD is directly attached on the shaking table to study its natural frequency and inherent damping. During further tests regarding the performance of the damper, the S-TLCD is attached to a model structure, which is excited by the shaking table as shown in Figure 7. The model structure consists of a pendulum attached to a rigid frame with four steel wires, see 2○ in Figure 7. The rigid frame is built by aluminium elements and has its natural frequency at 9 :20 Hz. The pendulum consists of a platform, where the S-TLCD can be attached and has its natural frequency at 13 45 123Figure 6: The actuator of the butter y valves to control the inherent damping of S-TLCD. S-TLCD attached on the pendulum of the model structure (Left). S-TLCD lled with liquid (water and colorant) (Right) 1○: Movable panels for frequency tuning. 2○: Actuators adjusting the movable panels. 3○: Sensor measuring the liquid motion. 4○: Buttel y valves for damping tuning. 5○: Actuator adjusting the valves. 1 32 Figure 7: Model structure on shaking table with the S-TLCD. 1○: S-TLCD model. 2○: Pendulum connected to a rigid frame structure, which is excited by the shaking table in horizontal direction. 3○: Uniaxial shaking table. 14 05101520253035 0.50 0.75 1.00 1.25 1.50 1.75 2.00Liquid motion u[mm] Excitation frequency [Hz]Damper Liquid: 4.69 L=45°=58°=72° 05101520253035 0.50 0.75 1.00 1.25 1.50 1.75 2.00Liquid motion u[mm] Excitation frequency [Hz]Damper Liquid: 4.77 L=45°=58°=72° 05101520253035 0.50 0.75 1.00 1.25 1.50 1.75 2.00Liquid motion u[mm] Excitation frequency [Hz]Damper Liquid: 4.83 L=45°=58°=72° 05101520253035 0.50 0.75 1.00 1.25 1.50 1.75 2.00Liquid motion u[mm] Excitation frequency [Hz]Damper Liquid: 4.95 L=45°=58°=72°Figure 8: Frequency response spectra of the four damper con gurations with the liquid volumes 4 :69, 4:77, 4:83 and 4:95 L at the panel positions 45, 58and 72. The butter y valve position is '= 0(closed). fS= 1:15 Hz, which can be changed by modifying the length of the wires. The damping ratio of the pendulum is aboutD= 1:0%. The mass of the pendulum together with the mass of the S-TLCD are documented in Section 4.4.1 and Table 4. The motion of the pendulum is registered also by a MEMS accelerometer. The model structure with the pendulum simulates a horizontally oscillating SDoF system with a low natural frequency, which allows the design of a very compact S-TLCD utilizing reasonable actuators adequate for the laboratory environment, where the tests are conducted. 4.2. Investigations on the Natural Frequency For the veri cation of the natural frequency equation of the S-TLCD, four damper con gurations with the liquid volumes 4 :69, 4:77, 4:83 and 4:95 L. with variable panel positions are investigated. During these investigations, the S-TLCD model is directly attached on the shaking table and excited harmonically by excitation frequencies 0 :50-2:00 Hz with an increment of 0 :10 Hz. For each damper con guration the experiments are repeated for panel positions between 45and 72with an increment of 0 :5. The damper response is determined by measuring the liquid motion. From the maximum values of the liquid motion, the frequency response curves for the conducted frequency sweeps are determined. Figure 8 shows the measured frequency response curves for the tested four damper con gurations and selected panel positions 45, 58 and 72. The peak values of the frequency curves correspond to the resonant response of the damper, in which the excitation frequency matches with the natural frequency. From the corresponding frequency the natural frequency of the damper is determined. In Figure 8, the change of the natural frequency is clearly observed from the horizontal shift of the peak of the frequency response curve. The shape of the frequency response curve shows that the panel position has also a certain e ect on the damping ratio of the damper. The wide frequency curve at the panel position 45indicates more damping compared to the panel positions 58and 72. Further discussion regarding this damping e ect can be found in Section 4.3. 15 Table 1: Parameters of the S-TLCD con guration with 4 :69 and 4:95 L liquid volume and comparison of the calculated natural frequencyfD;cwith the measured natural frequency fD;m. V 1V2V3H A V1AV2AHL1fD;cfD;m [] [m] [m] [m] [m] [m2] [m2] [m2] [m] [Hz] [Hz] 4:69 L 45 0:050 0:042 0:115 0:350 0:181020:501021:001020:383 1:14 1:15 58 0:050 0:051 0:052 0:350 0:281020:501021:001020:328 1:21 1:21 72 0:050 0:057 0:020 0:350 0:421020:501021:001020:373 1:15 1:16 4:95 L 45 0:050 0:042 0:189 0:350 0:181020:501021:001020:531 0:97 0:99 58 0:050 0:051 0:098 0:350 0:281020:501021:001020:430 1:07 1:07 72 0:050 0:057 0:052 0:350 0:421020:501021:001020:436 1:07 1:06 For each damper con guration and panel position, the mathematical parameters are determined, as introduced in Section 3.1. From these parameters the natural frequencies are calculated by using equations of the natural frequency !Dand rst ecient length L1, which are repeated below in Eq. 63 and Eq. 64. Table 1 shows the damper parameters for two damper con gurations with 4 :69 and 4:95 L, and compares the calculated and measured natural frequencies. In Figure 9, the calculated natural frequencies fD;care compared with the measured natural frequencies fD;mfor the tested four damper con gurations within the operation range of the panels between 45and 72. The results correspond to each other very well proving the applicability of the mathematical equation of the natural frequency of S-TLCD. fD;c=1 2r 2g L1(63) L1= 2V3+V2+ (2V1+V2)AV1 AV2+HAV1 AH(64) 4.3. Investigations on the Inherent Damping 4.3.1. E ects of the Butter y Valves on the Inherent Damping For the investigation of the e ects of the butter y valves on the inherent damping of the S-TLCD four di erent valve positions are tested: '=f0;30;60;90g, where'= 0corresponds to closed position and 90to open. Each test is repeated for four times to increase the statistical accuracy of the results. The tests are performed on the S-TLCD con guration with 4 :69 L at a panel position of = 45. The S-TLCD is attached directly on the shaking table and a free vibration of the liquid is induced by a sudden start and stop of the table after a certain initial velocity. For this purpose, the motion of the shaking table is tuned in such a way that the rst peak of the liquid motion is approximately umax15 mm. As shown in Figure 10, the damping ratios are determined from the recorded decay curves of the liquid motion beginning with this peak. In Figure 10, the left diagram shows the comparison of the time histories for the open and closed valve positions. The measured initial liquid motion peaks of the shown data sets are u= 14:97 mm for the valve in closed position and u= 15:24 mm for the valve in open position. The decay of the liquid de ection increases by closing the valve, which shows the increased damping behaviour. The right diagram shows the determined damping ratios, which vary between 4 :6% and 7:4% for the tested valve positions. The test results prove that the damping ratio of the S-TLCD can be modi ed eciently by changing the valve position. For the investigation of the e ects coming from the nonlinearity of the S-TLCD damping and its coherence with the de ection amplitude, further tests are conducted with higher liquid de ection amplitudes. For this 16 0.70.80.91.01.11.2 30 40 50 60 70 80 90Natural frequency f[Hz] Panel position [°]Damper Liquid: 4.69 L Calculation Experiment 0.70.80.91.01.11.2 30 40 50 60 70 80 90Natural frequency f[Hz] Panel position [°]Damper Liquid: 4.77 L Calculation Experiment 0.70.80.91.01.11.2 30 40 50 60 70 80 90Natural frequency f[Hz] Panel position [°]Damper Liquid: 4.83 L Calculation Experiment 0.70.80.91.01.11.2 30 40 50 60 70 80 90Natural frequency f[Hz] Panel position [°]Damper Liquid: 4.95 L Calculation ExperimentFigure 9: Comparison of analytical and experimental results showing the relationship between the natural frequency and the panel position for four damper con gurations with liquid volumes: 4 :69, 4:77, 4:83 and 4:95 L. The non-grey areas are showing the operation range of the panels. The butter y valve position is '= 0(closed). -20-1001020 0 5 10 15 20Liquid motion u[mm] Time t[s]Valve closed (=0°) Valve open ( =90°) 3456789 0 30 60 90Damping ratio D[%] Valve position [°] Figure 10: Time histories of the liquid motion for the valve positions '= 0and'= 90(Left). Relationship between the damping ratio Dand the valve position 'determined from time series with umax15 mm and 5 oscillations (Right). The panel position is = 45. 17 -90-60-300306090 0 5 10 15 20u1,1=75.12 mm u2,1=15.12 mm u1,6=9.64 mm u2,6=3.60 mmLiquid motion u[mm] Time t[s]Valve open (=90°)Figure 11: Investigation of the nonlinearity depending on the damping ratio and the liquid motion amplitude. The panel position is = 45. Table 2: Parameters of the experimental investigations on the nonlinear inherent damping e ects of the S-TLCD. uiui+nn D [mm] [mm] [-] [-] [%] Data set 1 75 :12 9 :64 5 0 :41 6 :5 Data set 2 15 :12 3 :60 5 0 :29 4 :6 purpose, the motion of the shaking table is increased, which caused an initial liquid motion peak of u= 75:12 mm, as shown in Figure 11. During these tests, the valve position is set as '= 90(open). The damping ratios are determined from several decay curves using di erent amplitude combinations. For the amplitude pairs 75:12-9:64 mm and 15 :12-3:60 mm the determined damping ratios are 6 :5% and 4:6% as evaluated in Table 2, where uiandui+nare the amplitude pairs, nis the number of oscillations between the amplitude pairs,  is the logarithmic decrement and Dis the damping ratio of the S-TLCD. It is evident from these results, that for larger amplitudes the damping ratio increases slightly in a nonlinear manner. Due to the nonlinearity of the inherent damping, for real applications, the tuning of the damping requires a mapping of the relationship between the liquid stream velocity amplitude and the damping term j_uj_u. Sophisticated control algorithms, such as using soft-computing methods can conduct this [32]. This approach can consider the nonlinearity of the damper and allow an accurate tuning of the damping depending on the liquid velocity. On the other hand, es explained in Section 3, the nonlinear damping term of the equation of motion can also be linearized by introducing the equivalent damping ratio DD. This approximation approach, which was proposed in [31], has sucient accuracy at least for harmonic excitations with low liquid amplitudes. At higher amplitudes or random excitations, the linearization underestimates the inherent damping and cause mistuning e ects, which will sink the performance of the damper. 4.3.2. E ects of the Movable Panels on the Inherent Damping For the investigation of the e ects of the movable panels on the inherent damping of the S-TLCD further tests were conducted for the three di erent panel positions at 45, 58and 72using the damper con guration with 4:69 L and a constant valve position at 90(open). Further parameters are shown in Table 3. Similar to the previous investigations on the valve e ects, the damping ratios are determined from the de ection decay curves of the liquid motion. Figure 12 (Left) shows the time histories of the liquid de ection for the tested three panel positions and the amplitude pairs, which are used for the determination of the damping ratios. The highest damping ratio with 6 :5% was determined for 45. The lowest damping ratio is 4 :8% at panel position 72. By closing the panels the damping ratio increases. This result matches with the previous results from frequency sweep tests, which are presented in Section 4.2. During frequency sweeps, the measured frequency response curve of the panel position 45was wider than of the panel positions 58 18 Table 3: Parameters of the experimental investigations on the damping e ects caused by the panels. u i ui+n n  D [] [mm] [mm] [-] [-] [%] 45 75 :12 9 :64 5 0 :41 6 :5 58 75 :10 14 :98 5 0 :32 5 :1 72 75 :14 16 :44 5 0 :30 4 :8 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 45 58 72 β[ °]9060300φ[ ] -90-60-300306090 0 5 10 15 20Liquid motion u[mm] Time t[s]β=72° β=58° β=45° Figure 12: Investigation of the e ect of the panel position on the damping behaviour. Valve position is '= 90(open) (Left). Damping ratio [%] of the S-TLCD for di erent valve/panel ( '= ) positions determined from time series with umax15 mm using 5 oscillations (Right). and 72showing that the damping behaviour is a ected not only by the valves but also by the panel position, cf. Figure 8. As mentioned previously in Section 2, the inherent damping of the S-TLCD is caused by local friction e ects in the tank. Therefore, every tank component, which is introducing additional friction, leads to a complementary increase in the inherent damping. To investigate this e ect, we rstly changed in Section 4.3.1 only the position of the butter y valves. As shown in Section 4.3.1, at the panel position = 45, the inherent damping decreases by just opening the butter y valves from '= 0to 90. This e ect is known also from the conventional TLCDs. Secondly, we changed in Section 4.3.2 also the panel position. As shown in this section, the inherent damping decreases further also by opening the movable panels from = 45to 72. This e ect is unique for the S-TLCD. To investigate the interaction between the valve and panel positions 'and in more detail, we conducted further tests. During these tests, we determined the damping ratio for di erent 'and from decay curves withumax15 mm using 5 oscillations. The results are shown in Figure 12 (Right). As determined also previously, the S-TLCD reaches the maximum damping ratio at a valve position of '= 0(closed) with a panel position of = 45. By opening the valves to '= 90and the panels to = 72the S-TLCD reaches its minimum damping ratio. The determined maximum and minimum damping ratios are 7 :4% and 3:4% respectively. Other variations, as also shown in Figure 12, lead to damping ratios, which are between these two limit values. It is worth noting that the inherent damping of the S-TLCD, as previously discussed in Section 4.3.1 and shown in Figure 11, also depends on the amplitude of the liquid motion. 4.4. Investigations on the Vibration Control Performance The following sections will present the results of the investigations on the vibration control performance of the S-TLCD. As mentioned in Section 1, the damping adaptation using an ori ce, as proposed by [8], counts as an established method, which is also utilized in the S-TLCD in a similar manner by the butter y valves. Studies on the performance e ects of this approach can be found in the literature such as in [10, 11]. 19 Table 4: Parameters of the performance study. f D( )fSmDmPemDemS=mPe+mDe [] [Hz] [Hz] [kg] [kg] [kg] [kg] 4572 0:971:07 1:15 4:950 1:500 9:360 10 :860 The major novelty of the proposed S-TLCD is its frequency adaptation capability. Therefore, and for the sake of brevity, this paper focuses on the e ects of the frequency adaptation on the performance of the S-TLCD. In this regard, Section 4.4.1 discusses the e ects of the natural frequency tuning on the damper performance. Section 4.4.2 documents the side e ects of the frequency adaptation, which are initiated by the change of geometric factors. 4.4.1. E ects of the Natural Frequency Tuning on the Damper Performance The natural frequency tuning is decisive for the eciency of structural control systems utilizing auxiliary masses. As a semi-active system, the S-TLCD is able to change its natural frequency and is therefore expected to reach a higher performance compared to a passive detuned damper. This e ect is investigated during an experimental campaign with the S-TLCD. During these tests, the S-TLCD with 4 :95 L liquid amount is attached on the pendulum. As experi- mentally determined in Section 4.2, this S-TLCD con guration can change its natural frequency within the range of 0:97 Hz to 1:07 Hz. For a 1 :15 Hz pendulum natural frequency the optimum natural frequency is determined using the system parameters documented in Table 4 according to the Den Hartog criteria [35] given in Eq. 65 - 68 for the S-TLCD. In Table 4, fDis the natural frequency of the damper depending on the panel position andfSis the natural frequency of the pendulum. Further parameters introduced in Table 4 are the liquid mass of the damper mD, the mass of the pendulum mPeand the mass of the damper prototypemDe. AsmPeandmDeare "dead masses", the vibration of which is aimed to be controlled, these masses are compiled in mSand called as the main mass, whereas mDis the secondary mass. As derived in Section 3.3 and shown in Eq. 65, the amount of liquid mass, which is actively participating to the vibration control, is calculated by scaling the total liquid mass mDwith the geometric factors 1 und 2. The rest liquid mass is also a "dead mass", which is necessary only for the tuning of the natural frequency of the damper. Therefore, this mass is added in Eq. 66 to the main mass mS. In Eq. 66, the mass ratio , which is necessary for the application of the Den Hartog criteria, is calculated from the actively participating liquid mass  mDand the total amount of the dead mass  mS. This mass ratio together with the natural frequency of the system fSis substituted in Eq. 68 to determine the optimum natural frequency of the damper. The pendulum, on which the S-TLCD model is attached, can be assumed to be a mathematical pendulum, the natural frequency of which is independent from the mass. Therefore, in Eq. 68, the natural frequencyfSis not changed by the additional mass. mD=mD 1 2 (65) =mD mS=mD 1 2 mS+mD(1 1 2)= 1 2 1 +(1 1 2)(66) =mD mS(67) fD;opt =fS 1 + (68) The optimum inherent damping of the S-TLCD can be approximated in a similar manner to the frequency tuning according to the Den Hartog criteria. The result will be a linearized damping ratio as given in Eq. 69, where is again the mass ratio of actively participating liquid mass  mDand the total amount of the 20 0306090 0110 220 330 440 550Panel position β[ °] Time t[s]β[] 0.900.951.001.051.101.15 0110 220 330 440 550Frequency f[Hz] Time t[s]fD[Hz] fD,opt [Hz] 0.00.10.20.30.40.50.60.70.80.9 0110 220 330 440 550Geometric factors i[-] Time t[s] 1[-]2[-]1⋅ 2[-]Figure 13: Time history of the panel position (Left) in uencing the natural frequency fDand the geometric factors i. Change offDcompared with the optimum natural frequency fD;opt (Centre). Change of 1and 2(Right). dead mass mM. DD;opt =s 3 8(1 + )3(69) As stated before,  depends on the geometric factors 1and 2. As shown in Figure 13, a position change of the movable panels initiates a change of the geometric factors, which will be discussed in Section 4.4.2 in detail. This change in the geometric factors has according to Eq. 69 also an in uence on the optimum damping ratio. In this study, the in uence of geometric factors on the optimum damping ratio is neglected and the damping ratio kept constant by adjusting the butter y valves to the closed position with '= 0. For the investigation of the frequency tuning e ect, the pendulum and S-TLCD system is excited har- monically by the shaking table with the natural frequency of the pendulum inducing resonant vibrations. During the excitation, the panel position is changed gradually in its operation range of 45-72. Figure 13 (left) shows the time history of for a test sequence of 550 seconds. Figure 13 (centre) shows for the same test sequence the time history of the measured natural frequency fDand the calculated optimum natural frequencyfD;opt of the damper according to Den Hartog. As illustrated in this diagram, fDcatchesfD;opt exactly with 1 :083 Hz at = 62. At = 72,fDcomes also very close to fD;opt by 1:068 Hz compared to 1:065 Hz. Figure 13 (right) shows for the same test sequence the time history of the calculated geometric factors 1, 2and their multiplication 1 2. As discussed in Section 3.3, with changing panel position the natural frequency and the geometric factors change in a nonlinear manner. Figure 14 (left) shows the time history of the measured acceleration of the pendulum. The maximum acceleration is observed at the beginning of the test sequence due to the detuned natural frequency of the damper. At = 45, the natural frequency of the damper is 0 :97 Hz and the optimum frequency is 1 :12 Hz. As the natural frequency comes closer to the optimum frequency, the acceleration reduces. However, the minimum acceleration 0 :032m=s2is reached at the panel position = 72, rather than at = 62, where the natural frequency is exactly equal the optimum frequency. At = 62, the acceleration is measured as 0:037m=s2. This e ect is caused by the geometric parameters 1and 2due to increased amount of active mass fraction  mD, which will be discussed in the next Section 4.4.2. A sample of the liquid motion time history, which is measured during the investigations, is shown in Figure 14 (right). The liquid oscillation decreases as the pendulum vibration decreases. Also this e ect will be discussed in the next Section 4.4.2. The e ects of the frequency tuning is determined by comparing the measured maximum and minimum pendulum accelerations shown in Figure 14 (left). As a result, during the experimental investigations, the frequency adaptation capability of S-TLCD enabled a performance improvement of over 77%. 21 -0.20-0.15-0.10-0.050.000.050.100.150.20 0 110 220 330 440 550max. Acceleration: 0.145 m/s2 min. Acceleration: 0.032 m/s2Pendulum acceleration [m/s2] Time t[s]-100-50050100 0 110 220 330 440 550umax=84.72 mm umin=28.23 mmLiquid motion u[mm] Time t[s]Figure 14: Time history of the pendulum acceleration induced by resonant harmonic excitation (Left). Time history of the liquid motion of the damper (Right). 4.4.2. E ects of the Geometric Factors on the Damper Performance As derived in Section 3, the geometric factors, 1and 2, determine the amount of active liquid mass  mD participating to the vibration control. Therefore, besides frequency tuning e ects, they are both expected to have an in uence on the performance of the damper. Furthermore, 1is directly connected with the e ective length L1, which determines the natural frequency !D. Therefore, it is expected to have a close correlation with the damper performance. 2scales the restoring force, as shown in Eq. 39. Therefore, it is also expected to be relevant for the performance of the damper. As mentioned in Section 4.4.1, in this study the damping ratio is tuned by the butter y valves with '= 0. Figure 15 (left) shows the change of the geometric factors depending on the panel position . The diagram in Figure 15 (centre) shows the frequency detuning determined from the frequency di erence  f between the optimum damper frequency fD;opt and the natural frequency of the damper fD. In Figure 15 (right) the S-TLCD induced reduction of the pendulum acceleration is illustrated. From the shape of the three curves, 1, fandR, the direct correlation between these parameters can be observed. Compared to these three curves, the two other curves, 2and 1 2, are showing a di erent progress as the panel position changes. To investigate this e ect, the relevant parameters are compiled in Table 5 for discrete panel positions . The shown parameters are the natural frequency of the damper fDand the optimum frequency fD;opt, the maximum liquid de ection umaxand the maximum pendulum acceleration  xmax. From these parameters the correlation coecients are calculated to investigate the relationship between the parameters. The correlation coecient of the frequency detuning  f=fD;optfDwith the acceleration reductionRis0:992, which shows an almost full correlation. The correlation between the rst geometric factor 1andRis about 0:971. The correlation between 2andRis 0:893. Comparing these three correlation values, it can be seen, that the frequency detuning has the highest e ect on the damper performance. The standard deviations of the compared data sets of  f, 1and 2are 0:052, 0:056 and 0:059 respectively, which are almost equal showing a comparable data distribution for the investigated parameters. These results match with the mathematical background derived in Section 3 and with the interpretations at the beginning of this section. 5. Conclusions A new-type semi-active tuned liquid column damper (S-TLCD) is proposed to mitigate the lateral vi- brations of high-rise civil engineering structures. The S-TLCD comprises a U-shape tank, which is partially lled with a Newtonian uid. The S-TLCD provides mechanisms for the real-time continuous tuning of both natural frequency and damping parameters. In the rst part of the paper the governing equations of the S-TLCD are derived based on the non- stationary incompressible liquid ow. The equation of motion is compiled rst for a SDOF system with 22 0.00.20.40.60.81.0 455055606570Geometric factors γ i[-] Panel position β[°]γ1[-] γ2[-] γ1⋅γ2[-] 0.000.040.080.120.160.20 455055606570Frequency difference Δf[Hz] Panel position β[°]Δf[Hz] 020406080100 455055606570Acceleration reduction R[%] Panel position β[°]R[%]Figure 15: Change of the geometric parameters 1and 2depending on the panel position (left). Frequency detuning f=fD;optfDin relevance to the panel position (centre). Comparison of the panel position induced natural frequency tuning e ect and reduction of pendulum acceleration (right). Table 5: Change of damper performance depending on frequency tuning fD;optfDand geometric factors i. f D fD;opt umax xmax [] [Hz] [Hz] [mm] [ m=s2] 45 1 :12 0 :97 143 0 :145 50 1 :11 1 :02 75 0 :110 55 1 :10 1 :06 57 0 :059 60 1 :09 1 :08 36 0 :040 65 1 :08 1 :08 34 0 :037 70 1 :07 1 :07 33 0 :033 72 1 :07 1 :07 30 0 :032 23 a single S-TLCD. The natural frequency is formulated using a scaled liquid length. The scaling factors are derived from the tank geometry. Further geometric parameters, which in uence the eciency of the S-TLCD, are formulated. The interaction force between S-TLCD and structure is derived from the equation of momentum, which is induced by the liquid motion. For multi-degree-of-freedom (MDoF) systems with several S-TLCDs, the state-space representation is formulated allowing the simulation of S-TLCDs with MDoF systems including sensors and control algorithm. In the second part of the paper the natural frequency equation is veri ed by experimental studies using a small-scale S-TLCD model attached to a shaking table. Further experiments are conducted to show proof of concept. The results show that the natural frequency of the S-TLCD can be controlled in the intended manner. E ects on the inherent damping of the S-TLCD are investigated. The results show that the inherent damping of the S-TLCD can also be controlled in the foreseen manner. Further experimental investigations are performed to study the e ects in uencing the control performance of the S-TLCD. For this investigation, the S-TLCD is attached to a model structure, which is excited by the shaking table. The results show that the tank geometry in uences directly the eciency of the damper. Hereby the most relevant aspect is revealed as the natural frequency tuning. During the experimental tests, the natural frequency adaptation capability enabled the S-TLCD a performance improvement of over 77%. 6. Future Work The presented S-TLCD enables new possibilities for the application of new control algorithms, which can adapt both frequency and damping parameters. Furthermore, due to natural frequency adaptation capability, system identi cation algorithms are necessary, which can determine the relevant structural and loading parameter in real-time. Moreover, an optimum design of the S-TLCD requires multi-objective optimization algorithms, which can calculate the physical geometry parameters of the S-TLCD and its mechanisms. Acknowledgements This research work is funded by the Excellence Initiative of the German federal and state governments. The authors would like to express their sincere appreciation for the support (Grant number: OPSF226). The authors would like to thank Control Engineering Institute of RWTH Aachen University for their support during the preparation of the experimental investigations. The authors also wish to thank Margarita Chasapi for her kind comments.