It was well established that rate-independent linear damping (RILD) can bene t base-isolated building structures by reducing the isolator displacement without increasing the oor response acceleration. In the present study, we further extended the application of RILD for seismic vibration mitigation of multi-story building structures. For this purpose, a gradient-based optimization method was presented to determine the locations and required quantities of RILD elements. A ten-story benchmark building structure was employed as an analytical example, and the e ectiveness of the proposed method was veri ed by comparing the performance of the RILD controlled structures using di erent damper placement methods. Moreover, parametric studies were conducted to investigate the e ect of RILD on the dynamic behavior of the optimally controlled structure. The performance curves of the benchmark structure enhanced with optimally placed RILD and viscous dampers were further compared in terms of the control cost and structural inter-story drift. It is suggested that RILD can be advantageous over the conventional viscous damper by more e ectively mitigating the seismic vibration response of a multi-story building structure without increasing the control cost. Keywords: rate-independent linear damping, vibration mitigation, optimal damper placement, multi- story building structure, gradient-based optimization method. 1. Introduction In the past decades, passive control technolo- gies have been extensively applied for mitigating harmful vibration responses of engineering struc- tures subject to extreme excitations ( i.e., strong 5 earthquake and wind) [1, 2]. By installing addi- tional passive energy dissipation systems into en- gineering structures, the energy dissipation capac- ity can be increased without requiring a external power source, and the structural vibration ampli- 10 tude can be reduced. For this purpose, various pas- sive energy dissipation systems have alreadly been developed, e.g., metallic yield dampers [3], friction dampers [4], uid viscous dampers [5, 6, 7], tuned mass dampers [8, 9], tuned liquid dampers [10, 11], 15 inerter-based dampers [12, 13, 14, 15, 16], negative sti ness dampers [17, 18, 19, 20, 21], and so on. Correspondence to: T.Y. Zhu, School of Civil and Hy- draulic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. Email address: tyzhu@163.com (Tianyi Zhu)Di erent from the viscous damping or nonlin- ear hysteretic damping in traditional passive energy dissipation systems, rate-independent linear damp- 20 ing (RILD) is an idealized damping model which provides a control force proportional to the defor- mation whereas in phase with the velocity. It was also known as complex damping, linear hysteretic damping, or structural damping in earlier litera- 25 tures [22, 23, 24]. Recently, the application of RILD for vibration mitigation of engineering structures have attracted increasing attentions. Inaudi and Kelly [24] obtained the optimal coecient of RILD for minimizing the mean-square responses of a base- 30 isolation system under stationary random excita- tions. Ikago and Inoue [25] indicated that RILD is capable of reducing the isolator displacement of a base-isolated structure without increasing oor response accelerations when subjected to strong 35 high-frequency ground motions. Because RILD is non-causal in the time domain [26, 27], many re- searchers [28, 29, 30, 31, 32, 33, 34] attempted to approximate the behavior of RILD by developing Preprint submitted to Elsevier July 16, 2024 causal models. 40 More recently, in terms of RILD used for the performance improvement of seismically laoisolated structures, Keivan et al. [35, 36] proposed a semi- active method of approximating RILD based on a rst-order all-pass lter, and veri ed its e ective- 45 ness by by conducting real-time hybrid simulation tests on seismically isolated structures. Later, in order to passively realize RILD, Luo et al. [37] pro- posed a mechanical model combining a Maxwell el- ement coupled in parallel with a negative sti ness 50 element (hereafter referred to as Maxwell-negative- sti ness (MNS) model). Furthermore, Luo and Ik- ago [27] proposed a unifying causal model that can include existing causal models of RILD in the litera- ture, and explored its bene ts in improving the seis- 55 mic performance of high-rise building structures. Liu and Ikago [38, 39] implemented the MNS model by using a physical Maxwell-type damper and veri- ed its e ectiveness by conducting real-time hybrid simulation tests on a base-isolated structure. Luo 60 et al. [40] proposed a novel physical device to real- ize RILD for performance enhancement of seismi- cally isolated structures. Wu et al. [41] presented an inerter-based device to mimic the behavior of RILD for seismic protection of base-isolated struc- 65 tures. Zhu et al. [42] examined the feasibility of RILD for simultaneously reducing the isolator and sloshing displacements of base-isolated exible liq- uid storage tanks. Luo et al. [43] proposed the use of base-isolation systems with RILD for seismic pro- 70 tection of high-rise building structures subject to both near- and far-fault ground motions. It is well established that RILD can bene t a base-isolated structure by e ectively reducing the isolator dis- placement without increasing the structural oor 75 response acceleration, whereas the application of RILD for seismic vibration mitigation of base- xed building structures is still limited. In the present study, we further explored the bene t of RILD for seismic vibration mitigation of a multi-story shear- 80 ing building. To fully utilize the capacities of RILD elements, an optimal design procedure for simultaneously de- termining their locations and quantities at di er- ent stories is to be developed. Indeed, many re- 85 searchers have worked on the optimal methods of placing damping devices in multi-story shear buildings for seismic performance enhancement. Zhang and Soong [44] presented a sequential pro- cedure for nding the optimal locations of vis- 90 coelastic dampers in a frame structure. Later,Wuet al. [45] further extended this sequential pro- cedure to 3-D unsymmetrical structures by mini- mizing translation-torsion coupling e ects. Take- waki [46] proposed a gradient-based procedure for 95 optimally placing viscous dampers in a shear build- ing. Later, Takewaki et al. [47] further presented a augmented gradient-based procedure for nding the optimal locations and required capacities of added viscous dampers in a 3-D shear building model. 100 Park et al. [48] proposed a gradient-based optimiza- tion procedure for designing viscoelastic dampers and their supporting braces installed in a struc- ture. Lavan and Levy [49] conducted the fully stressed design method of added viscous dampers 105 installed in a frame structure subject to realistic ground motions. Kanno [50] developed a mixed- integer programming method of optimally placing supplemental viscous dampers used in shear build- ing structures. Mart nez et al. [51] proposed a op- 110 timization procedure for designing nonlinear hys- teretic dampers installed in planar structures un- der seismic excitations. Shen et al. [52] presented an elastic{plastic design method of a structure with metallic yield dampers based on the elastic{plastic 115 response reduction curve. Pollini et al. [53] pur- sued the minimum-cost optimization design for seis- mic retro tting of structures using nonlinear uid viscous dampers. Sanati and Karamodin [54] pre- sented an optimum design procedure of frame struc- 120 tures with metallic yielding dampers considering life-cycle cost. Hao et al. [55] presented a design method of designing viscoelastic dampers to retro t reinforced concrete structures based on the struc- tural design capacity redundancy ratio. However, 125 to the best of the authors' knowledge, few research work on the optimization method of designing the locations and required quantities of RILD elements in a multi-story shear building structure have been reported. 130 In the present study, a gradient-based method was developed to optimally place RILD elements for seismic vibration mitigation of a multi-story shear- ing building structure, and a ten-story benchmark building structure was employed to verify the e ec- 135 tiveness of the developed method and identify the advantages of RILD over conventional dampers for structural seismic vibration mitigation. 2. Single-degree-of-freedom system In structural dynamics, damping is an idealized 140 concept to represent a process by which vibra- 2 tion steadily diminishes in amplitude [56]. Sev- eral damping models are available to represent the damping e ect for engineering applications, and two types of linear damping model are considered 145 for comparison. 2.1. Two types of linear damping models 2.1.1. Linear viscous damping The linear viscous damping (LVD) model has been often used to represent damping since its use by Lord Rayleigh [57]. With respect to this model, it is assumed that the resisting force is proportional to relative vocility, i.e., f1(t) =cd_x(t) (1) whichf1andxare the damping force and relative displacement of the LVD model, respectively; the 150 overdot denotes the derivative with respect to time; cdis the viscous damping coecient. By conducting the Fourier transform to Eq.(1), one can reformulate the LVD force in the frequency domain as follows: F1(i!) =i!cdX(i!) (2) whereidenotes the imaginary unit, and !is the excitation frequency. From the above equation, the transfer function of an LVD model from the de- formation to the resisting force, also known as the dynamic sti ness, is obtained as follows: K1(i!) =F1(i!) X(i!)=i!cd (3) The LVD model is one of the most widely used linear damping model, which is partly becuase it provides great convenience to the mathematical analyses of structure in both the frequency and time domains. It should be noticed that, in one cycle of harmonic vibration, the energy dissipated by the LVD model can be readily derived as follows: E1=!c dX2(4) which suggests that the energy dissipated by the LVD model is proportional to the vibration fre- quency. 155 2.1.2. Rate-independent linear damping The RILD model is another type of damping model considered here, and it was rst used for the simulation of internal friction in engineering materi- als. In 1927, Kimball and Lovell [58] reported that, 160from tests on eighteen solids with di erent physical properties, the energy loss per sinusoidal strain cy- cle is proportional to the square of strain amplitude, but independent on the strain rate over a consider- able frequency range. This led to the introduction 165 of an idealized concept of RILD. The resisting force of the RILD model is usually expressed in terms of dynamic sti ness in the fre- quency domain as follows: K2(i!) =F2(i!) X(i!)=ikdsgn (!) (5) wherekddenote the complex sti ness coecient; sgn() denotes the signum function. It is suggested that the force of RILD is proportional to the defor- mation with a phase lead of =2 rad. The energy dissipated by the RILD model in one cycle of har- monic vibration is expressed as follows: E2=kdX2(6) It is worthy noting that E2is independent on the excitation frequency !. Thus, it can be used to simulate the rate-independent dissipation behavior of some solid materials. 170 Fig. 1 depicts the dynamic sti nesses of the LVD and RILD models, where the magnitude of dynamic sti ness is normalized by using the complex sti ness coecientkd, and!0=kd=cddenotes a parame- ter with a unit of circular frequency. It is shown 175 that with! < ! 0, RILD generates higher control force compared with LVD. This implies that RILD can more e ectively control the structural response displacement under low-frequency excitations. As ! > ! 0, RILD generates lower control force than 180 LVD. This implies that RILD can avoid generating excessive control force under high-frequency excita- tions. /s49/s48/s45/s50 /s49/s48/s45/s49 /s49/s48/s48 /s49/s48/s49 /s49/s48/s50/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s109/s97/s103/s110/s105/s116/s117/s100/s101 /s47/s32/s76/s86/s68 /s32/s82/s73/s76/s68 /s49/s48/s45/s50 /s49/s48/s45/s49 /s49/s48/s48 /s49/s48/s49 /s49/s48/s50/s47/s50/s80/s104/s97/s115/s101/s32/s97/s110/s103/s108/s101 /s47/s32/s76/s86/s68 /s32/s82/s73/s76/s68 /s48 /s40/s97/s41 /s40/s98/s41 Figure 1: Normalized dynamic sti ness 3 2.2. Response ampli cation factors Let's further consider a generalized linear damp- ing model comprising LVD and RILD elements in- corporated into an SDOF structural system, which is governed by the following equation of motion:  m!2+k+Kp(i!) X(i!) =m!2Xg(i!) (7) wheremandkdenote the structural mass and sti ness, respectively; X(i!) andXg(i!) are the frequency-domain response displacement relative to the ground and the excitation acceleration of the ground, respectively; Kp(i!) denotes the dynamic sti ness of the generalized damping model Kp(i!) =i!cd+ikdsgn (!) (8) which reduces to Eqs.(3) and (5) in the cases of kd= 0 andcd= 0, respectively. Letting !1=p k=m, cd= 2m! 1,kd=k, one can rewrite Eq.(8) as follows: Kp(i!) =im! 1[2!+!1sgn (!)] (9) From Eqs.(7) and (9), one can obtain the relative displacement response ampli cation factor (RAF) of the damped structural system as follows: DRAF= 2 q ( 21)2+ [2 +sgn ( )]2(10) where =!=! 1. Similarly, one can also obtain the structural absolute acceleration RAF as follows: ARAF=s 1 + [2 +sgn ( )]2 (1 2)2+ [2 +sgn ( )]2(11) For example, letting = 2, Fig. 2 depicts 185 the relative displacement and absolute acceleration RAFs of the LVD and RILD damped SDOF struc- tural systems. It is well established that the ad- dition of more LVD can decrease the displacement and acceleration RAFs in the resonant zone at the 190 expense of increased acceleration RAFs at higher frequencies beyond !=p 2!0. However, as shown in Fig. 2, the incorporation of more RILD can achieve a similar reduction in displacement and ac- celeration RAFs in the resonant zone as LVD, with- 195 out compromising the high-frequency responses. This indicates that RILD can be more bene cial than LVD in reducing structural response accelera- tion when it is integrated into base-isolated struc- tures subject to strong earthquake-induced ground 200 motion [18]. /s49/s48/s45/s50 /s49/s48/s45/s49 /s49/s48/s48 /s49/s48/s49 /s49/s48/s50/s49/s48/s45/s52/s49/s48/s45/s50/s49/s48/s48/s82/s101/s108/s97/s116/s105/s118/s101/s32/s100/s105/s115/s112/s108/s97/s99/s101/s109/s101/s110/s116 /s47/s76/s86/s68/s40 /s61/s48/s46/s48/s50/s41 /s76/s86/s68/s40 /s61/s48/s46/s50/s48/s41 /s82/s73/s76/s68/s40 /s61/s48/s46/s48/s52/s41 /s82/s73/s76/s68/s40 /s61/s48/s46/s52/s48/s41 /s40/s97/s41 /s40/s98/s41/s49/s48/s45/s50 /s49/s48/s45/s49 /s49/s48/s48 /s49/s48/s49 /s49/s48/s50/s49/s48/s45/s52/s49/s48/s45/s50/s49/s48/s48/s65/s98/s115/s111/s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110 /s47/s76/s86/s68/s40 /s61/s48/s46/s48/s50/s41 /s76/s86/s68/s40 /s61/s48/s46/s50/s48/s41 /s82/s73/s76/s68/s40 /s61/s48/s46/s48/s52/s41 /s82/s73/s76/s68/s40 /s61/s48/s46/s52/s48/s41Figure 2: Response ampli cation factor It is well established that RILD can e ectively reduce the structural response displacement with lower control forces compared with other types of damping devices when subjected to strong ground 205 motions [37]. Several attempts have been made to investigate the e ect of RILD on the seismic responses of high-rise building structures [27, 43], however, there is still limited research work on the application of optimally placed RILD elements for 210 the seismic vibration mitigation of building struc- tures. 3. Optimal damping placement Here, we presented a gradient-based optimization method of placing RILD elements for the seismic 215 vibration mitigation of multi-story building struc- tures. 3.1. Sensitivities of the response ampli cation fac- tor Let's consider supplemental generalized linear damping elements incorporated into a multi-degree- of-freedom building structure model, which is gov- erned by the following equation of motion formu- lated in the frequency domain:  !2M+i!C+Kp(i!) +K X(i!) =!2M1TXg(i!) (12) where M,C, and Kare the structural mass, 220 damping, and sti ness matrices, respectively; X= fX1;X2;:::;XngTis the displacement vector rela- tive to the ground, where ndenotes the story num- ber, and the superscript ' T' denotes the matrix transposition operator; 1=f1;1;:::;1g;Kp(i!) is 225 the dynamic sti ness matrix contributed by supple- mental generalized linear damping elements. 4 By introducing a transformation matrix, i.e., T=2 6666641 0 0 0 1 1 0 0 ............... 0 0 1 0 0 0  1 13 777775(13) one can formulate the dynamic sti ness matrix of generalized linear damping elements as follows: Kp(i!) =qp(i!)TTdiag(kd)T (14) where diag() denotes the operation of transforming a vector into the corresponding diagonal matrix; kd=fkd;1;kd;2;:::;kd;ngdenotes a vector of sti - ness coecients of supplemental damping elements; qp(i!) denotes the frequency-dependent term asso- ciated with the dynamic sti ness of the p-th type of linear damping element, i.e., one has qp(i!) =( i!=! 1; p = 1; isgn (!); p= 2:(15) which are associated with the LVD and RILD ele- ments, respectively. Furthermore, let's introduce the following auxil- iary matrix A=1 !2 1 K+i!1C+Kp(!1)!2 1M (16) By substituting Eq.(14) into Eq.(16) and conduct- ing the partial di erentiations of the resulting ma- trixAwith respect to kd, one can obtain A0 j=@A @kd;j=q(i!1) !2 1TTDjT (17) withj= 1;2;:::;n , where Djdenotes a square ma- 230 trix whose elements are zeros, except for a unity element at the j-th entry in the diagonal. By using Eq.(16), Eq.(12) can be reformulated as follows: A~X=MrT(18) with ~X=X(!1)=Xg(!1). By performing partial di erentiations of both sides of Eq.(18) with respect tokd;j, one has A0 j~X+A~X0 j=0 (19) withj= 1;2;:::;n . It follows the rst-order sensi- tivity of the transfer function of relative displace- ment vector with respect to kd;j ~X0 j=A1A0 j~X; j= 1;2;:::;n: (20)Notice that the inter-story drift vector can be ex- pressed in terms of the relative displacement vector as follows: ^U=T~X (21) Similarly, one can obtain the rst-order sensitivity of^Uwith respect to kd;jas follows: ^U0 j=TA1A0 jT1^U (22) Substituting Eq.(17) into Eq.(22) gives ^U0 j=q(i!1) !2 1TA1TTDj^U (23) withj= 1;2;:::;n . Notice that the complex quan- tity ^Ucan also be formulated in terms of its real and imaginary parts as follows: ^U= Re[ ^U] +iIm[^U] (24) From the above equation, one can obtain ^U0 j= [Re( ^U)]0 j+i[Im(^U)]0 j (25) Therefore, with ^U0 j=f(^U1)0 j;(^U2)0 j;:::;(^Un)0 jgTob- tained from Eq.(23), one can calculate the rst- order sensitivity of the RAF of inter-story drift at thel-th story,j^Ulj, with respect to kd;jby using the following equation j^Ulj0 j=1 j^Ulj Re(^Ul)[Re( ^Ul)]0 j+ Im( ^Ul)[Im( ^Ul)]0 j (26) withl= 1;2;:::;n . Furthermore, by conducting the partial di eren- tiation of Eq.(23) with respect to kd;r, one can ob- tain the following equation ^U00 jr=q(i!1) !2 1TA1h TTDj^U0 rA0 rA1TTDj^Ui (27) where the relation ( A1)0 r=A1A0 rA1is used. On substitution of Eq.(17) into Eq.(27) and use of Eq.(23), the following equation can be obtained ^U00 jr=q(i!1) !2 1TA1TTh Dj^U0 r+Dr^U0 ji (28) From the above equation, the real and imaginary parts of ^U00 jrcan be obtained, respectively, which 235 are useful to calculate the partial di erentiation of Eq.(26) with respect to kd;ras follows: j^Ulj00 jr=1 j^Ulj2fj^Ulj([Re( ^Ul)]0 r[Re(^Ul)]0 j+ Re( ^Ul)[Re( ^Ul)]00 jr+ [Im( ^Ul)]0 r[Im(^Ul)]0 j + Im( ^Ul)[Im( ^Ul)]0 jr)j^Ulj0 r(Re( ^Ul)[Re( ^Ul)]0 j+ Im( ^Ul)[Im( ^Ul)]0 j)g(29) 5 3.2. Formulation of the optimal problem The objective of the optimal damping placement problem in a multi-story building model is to nd optimal kdso that the sum of the structural inter- story drift RAFs at the undamped fundamental fre- quency is minimized, i.e., MinV=1j^Uj (30) subjected to constraints on the sums of sti ness co- ecients of added linear damping elements 1kd=w (31) wherewis a speci ed constant value. The La- grangian for the aforementioned problem can be ex- pressed in terms of Lagrange multiplier as follows: L1(kd;) =1j^Uj+(1kdw) (32) The optimality criteria for the above problem is ob- tained from the stationary conditions of L(kd;) with respect to kdandas follows: ( 1j^Uj0 j+= 0 1kdw= 0(33) withj= 1;2;:::;n . It is assumed that the sti - ness coecients of added linear damping elements should be non-negative. In the cases of kd;j= 0, the corresponding equation of the optimality crite- ria should be modi ed as follows: 1j^Uj0 j+0 (34) By introducing the following auxiliary quantities j=Bj+1 B1; j= 1;2;:::;n1: (35) where Bj=1j^Uj0 j; j= 1;2;:::;n: (36) the optimality criteria, Eq.(33), can be alterna- tively reformulated without the Lagrange multipli- ers as follows: ( j1 = 0 1kdw= 0(37) withj= 1;2;:::;n1. The linear increment  j of jcan be written in terms of increment  kdas follows:  j=1 B1[(Bj+1)0(B1)0 j] kd (38)where (Bj)0=@Bj=@kd. By combining Eq.(38) with1kd=0, one can obtain a set of simultane- ous linear equations with respect to  kdas follows: 2 66641 B1[(B2)0(B1)0 1] ... 1 B1[(Bn)0(B1)0 n1] 13 7775kd=8 >>>< >>>: 1 ...  n1 09 >>>= >>>;(39) where thel-th entry in the vector ( Bj)0is calculated from the following equation (Bj)0 l=1j^Uj00 jl (40) withj;l= 1;2;:::;n . Recall that j^Uj00 jlcan be obtained from Eq.(29), therefore, one can readily evaluate  kdfrom Eq.(39) with  = f 1; 2;:::; n1gTprovided. For this pur- pose, letting  =1 N( F 0) (41) where Fand 0denote the target and initial val- ues of the quantities de ned in Eq.(35), respec- 240 tively;Ndenotes the number of incremental steps. From Eq.(37), one can initially assume that F= f1;1;:::;1gTso that  can be readily evaluated. It should be mentioned that the sti ness coe- cients of supplemental damping elements are non- 245 negative. In the case of kd;j= 0, the expres- sion of the optimality criteria in Eq.(37) with re- spect to j1should be modi ed as follows. For j= 2;3;:::;n , ifkd;16= 0 andkd;j= 0, j11; if kd;1= 0 andkd;j6= 0, j11. It should be no- 250 ticed that a similar formulation was presented by Takewaki [46] to optimally place viscous dampers in shear buildings. 4. Numerical examples In order to verify the e ectiveness of the afore- 255 mentioned method and illustrate the bene t of RILD, a ten-story benchmark shear building struc- ture [59] was used as an analytical example, as shown in Fig. 3, whose structural parameters are listed in Table 1. Modal analyses were conducted 260 for the undamped primary linear structure, and the rst three undamped fundamental natural periods are 2.01 s, 0.76 s, and 0.46 s. The inherent damping matrix of the primary building structure was as- sumed to be proportional to the structural sti ness 265 matrix with a inherent damping ratio of 0= 0:02, i.e.,C0= 20=!1K. 6 Figure 3: A 10-story benchmark shear building structure Table 1: Primary structural properties j m j(ton)kj(kN/m) cj(kNs/m) Height (m) 1 700 279 960 3 586 6 2 682 383 550 4 913 4 3 680 383 020 4 906 4 4 676 328 260 4 204 4 5 670 306 160 3 921 4 6 667 291 890 3 739 4 7 660 244 790 3 135 4 8 656 220 250 2 821 4 9 649 180 110 2 307 4 10 875 158 550 2 031 4 4.1. E ectiveness of the optimal method Additional RILD elements are supplemented to mitigate the vibration of the primary structure un- 270 der strong earthquakes. For comparison, three dif- ferent cases are considered: (i) the complex sti ness coecients of RILD elements are proportional to the structural sti ness coecients, i.e.,Pkd=k (proportional case, = 2= 0:1 is considered in 275 this example); (ii) the coecients of RILD elements are uniformly distributed at each story of the build- ing structure (uniform case); (iii) the coecients of RILD elements are optimally obtained by using the developed method (optimal case, 0is obtained 280 from the uniform case). The sums of the complex sti ness coecients of RILD elements used in the three cases are assumed to be equal. For the last case,N= 100 was used in this example. The com- plex sti ness coecients of RILD elements obtained 285 in the three cases are also listed in Table 2.4.1.1. Frequency response analyses Fig. 4 depicts the inter-story drift and absolute acceleration RAFs of the uncontrolled structure and those of the controlled structure equipped with 290 RILD elements evaluated at the undamped funda- mental frequency. It is noticed that in the opti- mal case, RILD elements are placed in the rst and fourth stories (as shown in Table 2), where the inter-story drifts of the uncontrolled structure are 295 larger than those of other stories (see the solid black line in Fig. 4(a)). From Fig. 4, it is observed that compared with the proportional and uniform cases, the structure equipped with optimally placed RILD elements are more e ective in reducing both inter- 300 story drift and absolute acceleration RAFs evalu- ated at the undamped fundamental frequency. /s48 /s49 /s50 /s51 /s52 /s53/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48/s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s97/s41/s32/s73/s110/s116/s101/s114/s115/s116/s111/s114/s121 /s32/s100/s114/s105/s102/s116/s32/s82/s65/s70/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s79/s112/s116/s105/s109/s97/s108 /s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s98/s41/s32/s65/s98/s115/s111/s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s82/s65/s70/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s79/s112/s116/s105/s109/s97/s108 Figure 4: Structural RAFs at the fundamental frequency Figs. 5 and 6 depict the rst story drift and rooftop acceleration RAFs of the controlled struc- ture equipped with RILD elements, respectively. It 305 is shown that compared with other two cases, the optimally placed RILD elements are more e ective in reducing the the rst story drift RAFs all over the frequency, without compromising the rooftop acceleration RAFs. It is expected that RILD can be 310 used to simultaneous mitigating the response dis- placement and acceleration of a multi-story build- ing structure subject to strong earthquakes. 4.1.2. Response history analyses Time-history response analyses were conducted 315 to investigate the performance of the di er- ently controlled benchmark structures under strong earthquakes. For example, Figs. 7 and 8 depict the peak responses of the uncontrolled and controlled 7 Table 2: Properties of RILD elements in three di erent cases jjkd(kN/m) j Proportional Uniform Optimal Proportional Uniform Optimal 1 27 996 27 765 186 748 0.51 0.51 0.88 2 38 355 27 765 0 0.46 0.47 0.82 3 38 302 27 765 0 0.56 0.56 1.00 4 32 826 27 765 90 906 0.54 0.54 0.95 5 30 616 27 765 0 0.47 0.46 0.83 6 29 189 27 765 0 0.48 0.48 0.86 7 24 479 27 765 0 0.38 0.38 0.68 8 22 025 27 765 0 0.30 0.29 0.54 9 18 011 27 765 0 0.14 0.13 0.24 10 15 855 27 765 0 { { { /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s70/s105/s114/s115/s116/s32/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s82/s65/s70 /s47/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s79/s112/s116/s105/s109/s97/s108 Figure 5: Structural rst story drift RAFs /s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s82/s111/s111/s102/s116/s111/s112/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s82/s65/s70 /s47/s32/s85 /s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108 /s32/s85 /s110/s105/s102/s111/s114/s109 /s32/s79/s112/s116/s105/s109/s97/s108 Figure 6: Rooftop absolute acceleration RAFs structures under the 1940 Imperial Valley Earth- 320 quake (El Centro, N{S component, peak ground velocity (PGV) = 0.5 m/s) and the 1968 Tokachi- oki Earthquake (Hachinohe, N{S component, PGV = 0.5 m/s), respectively. Figs. 9 and 10 depict the corresponding response histories of the controlled 325 structure equipped with RILD elements. It can be observed from Figs. 7-10 that compared with the proportional and distributed RILD elements hav- ing equal sum of coecients, the optimally placed RILD elements are much more e ective in reduc- 330 ing the inter-story drifts without compromising the oor response acceleration. This example veri es /s48 /s49/s47/s49/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48 /s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48/s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s97/s41/s32/s73 /s110/s116/s101/s114/s115/s116/s111 /s114/s121 /s32/s100/s114/s105/s102/s116/s32/s97/s110/s103 /s108/s101/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100/s32 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108/s32 /s32/s79/s112/s116/s105/s109/s97/s108 /s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s98/s41/s32/s65 /s98/s115/s111 /s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111 /s110/s32/s40/s109/s47/s115/s50 /s41/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100/s32 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108/s32 /s32/s79/s112/s116/s105/s109/s97/s108Figure 7: Structural peak oor responses (El Centro) /s48 /s49/s47/s49/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48 /s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48/s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s97/s41/s32/s73 /s110/s116/s101/s114/s115/s116/s111 /s114/s121 /s32/s100/s114/s105/s102/s116/s32/s97/s110/s103 /s108/s101/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100/s32 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108/s32 /s32/s79/s112/s116/s105/s109/s97/s108 /s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s98/s41/s32/s65 /s98/s115/s111 /s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111 /s110/s32/s40/s109/s47/s115/s50 /s41/s32/s85/s110/s99/s111/s110/s116/s114/s111/s108/s108/s101/s100/s32 /s32/s85/s110/s105/s102 /s111/s114/s109 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108/s32 /s32/s79/s112/s116/s105/s109/s97/s108 Figure 8: Structural peak oor responses (Hachinohe) the e ectiveness of the developed optimal RILD placement method for seismic vibration mitigation of a multi-story shear building model. 335 8 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s54/s48/s54 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s53/s48/s53/s73/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s40/s99/s109/s41 /s84/s105/s109/s101/s32/s40/s115/s41/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97 /s108 /s85/s110/s105/s102/s111/s114/s109 /s79 /s112/s116/s105/s109/s97 /s108 /s53/s46/s50 /s45/s51/s46/s56/s65/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s40/s109/s47/s115/s50 /s41 /s84/s105/s109/s101/s32/s40/s115/s41/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97 /s108 /s85/s110/s105/s102/s111/s114/s109 /s79 /s112/s116/s105/s109/s97 /s108/s40/s97/s41/s32/s70/s105/s114/s115/s116/s32/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102 /s116 /s40/s98/s41/s32/s82/s111/s111/s102 /s116/s111/s112/s32/s97/s98/s115/s111/s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32Figure 9: Structural Response histories (El Centro) /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s54/s48/s54 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s53/s48/s53/s73/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s40/s99/s109/s41 /s84/s105/s109/s101/s32/s40/s115/s41/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97 /s108 /s85/s110/s105/s102/s111/s114/s109 /s79 /s112/s116/s105/s109/s97 /s108 /s52/s46/s56 /s45/s51/s46/s52/s65/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s40/s109/s47/s115/s50 /s41 /s84/s105/s109/s101/s32/s40/s115/s41/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97 /s108 /s85/s110/s105/s102/s111/s114/s109 /s79 /s112/s116/s105/s109/s97 /s108/s40/s97/s41/s32/s70/s105/s114/s115/s116/s32/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102 /s116 /s40/s98/s41/s32/s82/s111/s111/s102 /s116/s111/s112/s32/s97/s98/s115/s111/s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32 Figure 10: Structural Response histories (Hachinohe) 4.2. Parametric studies Here, we further increase the sum of complex sti ness coecients of RILD elements and investi- gate the distribution pattern variation of optimally designed RILD elements at di erent damping lev- 340 els. To this end, the sum of the coecients of proportionally designed RILD elements is consid- ered,w=1kd=1k, and the loss factor is in- creased from 0 to 0 :16 with an interval of 0 :002. By applying the developed procedure, the opti- 345 mally designed coecients of RILD elements can be determined. Figs. 11 and 12 depict the corre- sponding objective functions and the designed coef- cients of RILD elements at di erent damping lev- els. For comparison, the sums of the fundamental 350modal RAFs of the inter-story drift of the structure equipped with proportionally and uniformly dis- tributed RILD elements are also plotted. It should be mentioned that the optimally designed coe- cients of RILD elements at the other stories are ze- 355 ros in this case studies, and thus they are not shown in Fig. 12 for brevity. /s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50 /s48/s46/s49/s54/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53 /s32/s80/s114/s111/s112/s111/s114/s116/s105/s111/s110/s97/s108 /s32/s85 /s110/s105/s102/s111/s114/s109 /s32/s79/s112/s116/s105/s109/s97/s108/s79/s98/s106/s101/s99/s116/s105/s118/s101/s32/s102/s117/s110/s99/s116/s105/s111/s110 /s76/s111/s115/s115/s32/s102/s97/s99/s116/s111/s114/s61/s48/s46/s49/s48 Figure 11: Objective functions /s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50 /s48/s46/s49/s54/s48/s49/s50/s51/s52/s53/s65/s100/s100/s101/s100/s32/s99/s111/s109/s112/s108/s101/s120/s32/s115/s116/s105/s102/s102/s110/s101/s115/s115/s32/s40/s78/s47/s109/s41 /s76/s111/s115/s115/s32/s102/s97/s99/s116/s111/s114/s32/s107 /s100/s44/s49/s32 /s32/s107 /s100/s44/s52 /s32/s107 /s100/s44/s53/s32 /s32 /s107 /s100/s44/s106/s49/s48/s56 /s61/s48/s46/s49/s48 Figure 12: Optimally designed complex sti ness coecients It is observed from Fig. 11 that the objective functions decrease with the increase of the sum of complex sti ness coecients of RILD elements 360 (or loss factors), and the lowest objective functions among the three di erent cases at the same damp- ing level are achieved by the optimally designed RILD elements. This further veri es the e ective- ness of the developed optimal design method. 365 It is observed from Fig. 12 that with 2[0;0:16], there exist three di erent patterns for the optimal distribution of RILD elements in the benchmark structure. At low damping levels, the RILD ele- ments are installed at the rst story only. Gradu- 370 ally, as the sum of RILD complex sti ness coe- cients or loss factor increases, part elements start 9 to be distributed at the fourth story, and then they are distributed at the rst, fourth, and fth stories. This example veri es that the developed procedure 375 can well adapt the distribution pattern variation of added RILD elements in a multi-story shearing building structure. 4.3. Comparison between di erent types of dampers 4.3.1. Performance curves 380 In order to further illustrate the bene t of RILD incorporated in multi-story shear buildings, the seismic responses of the benchmark building struc- ture equipped with additional optimally placed LVD and RILD elements are compared. 385 To this end, the maximum value of the sum of peak damping forces of added damping elements used in the structure normalized by the total struc- tural weight W,i.e.,I1=Pmax t(jfd(t)j)=W, is considered as a cost-related index in this study, be- 390 cause the damper force is directly related with the costs of a damper and its supporting members in engineering practice, and the maximum value of the peak inter-story drift angles of the controlled structure, i.e.,I2= max[max t(ju(t)j:=h)], where h 395 denotes the structural height vector, is considered as a performance index. By increasing the sums of LVD and RILD coecients and conducting struc- tural response history analyses, we obtained the performance curves of the optimally designed LVD 400 and RILD controlled structure in terms of the cost- related index I1and performance-related index I2, respectively, as shown in Figs. 13 and 14. It should be noted that the developed procedure can be read- ily used for the optimal design of added LVD ele- 405 ments in the benchmark structure with slight mod- i cation. /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s49/s46/s51/s49/s46/s52/s49/s48/s45/s50/s77/s97/s120/s46/s32/s105/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s97/s110/s103/s108/s101/s44/s32 /s73 /s50 /s83/s117/s109/s32/s111/s102/s32/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s102/s111/s114/s99/s101/s115/s44/s32 /s73 /s49/s32/s76/s86/s68 /s32/s82/s73/s76/s68 Figure 13: Performance curves (El Centro) /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s49/s46/s51/s49/s46/s52/s49/s48/s45/s50/s77/s97/s120/s46/s32/s105/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s97/s110/s103/s108/s101/s44/s32 /s73 /s50 /s83/s117/s109/s32/s111/s102/s32/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s102/s111/s114/s99/s101/s115/s44/s32 /s73 /s49/s32/s76/s86/s68 /s32/s82/s73/s76/s68Figure 14: Performance curves (Hachinohe wave) It is shown in Figs. 13 and 14 that with an equal cost-related index I1, the RILD controlled structure generates smaller response inter-story drift than the 410 LVD controlled structure. This veri es the bene t of RILD used for mitigating the seismic vibration of a multistory shear building structure subject to strong ground motions. 4.3.2. Comparison with equal cost-related indexes 415 To illustrate the performance di erence of dif- ferently controlled structures, we assumed that the cost-related index is targeted at I1= 0:15 when the structure is subjected to the El Centro wave, and optimally determined the distribution of added 420 LVD and RILD elements in the multistory build- ing structure, respectively. The designed damper properties and the corresponding normalized con- trol forces are listed in Table 3. Fig. 15 plots the maximum oor responses of 425 three types of structures subject to the El Centro wave. For example, Fig. 16 plots the oor response histories of those structures in terms of the second oor inter-story drift (which results in the largest inter-story drift angle) and rooftop response accel- 430 eration, when they are subjected to the El Centro wave. Fig. 17 plots the corresponding hysteresis loops of the added damping elements installed at the rst and fourth stories. It is shown in Figs. 15 and 16 that the RILD 435 elements can be more e ective than the LVD el- ements in reducing the structural inter-story drift and response absolute acceleration, when they are optimally designed with the equal cost-related in- dexesI1(i.e., sum of maximum normalized damp- 440 ing forces) for uses in a multistory building struc- ture. 10 Table 3: Designed properties of LVD and RILD elements for comparison (El Centro wave) jLVD RILD cd;j(kNs/m) Normalized control force kd;j(kN/m) Normalized control force 1 57 411 0.114 182 420 0.091 4 22 604 0.036 79 806 0.033 5 0 0 54 300 0.026 other 0 0 0 0 sum 80 015 I1= 0:15 316 526 I1= 0:15 /s48 /s49/s47/s49/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48 /s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57/s49/s48/s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s97/s41/s32/s73 /s110/s116/s101/s114/s115/s116/s111 /s114/s121 /s32/s100/s114/s105/s102/s116/s32/s97/s110/s103 /s108/s101/s85/s110/s99/s111/s110/s116/s114/s111/s108 /s76/s86/s68 /s82/s73/s76/s68 /s83/s116/s111/s114/s121/s32/s110/s117/s109/s98/s101/s114 /s40/s98/s41/s32/s65 /s98/s115/s111 /s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111 /s110/s32/s40/s109/s47/s115/s50 /s41/s85/s110/s99/s111/s110/s116/s114/s111/s108 /s76/s86/s68 /s82/s73/s76/s68 Figure 15: Maximum oor responses (El Centro) /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s52/s48/s52 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s53/s48/s53/s73/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102/s116/s32/s40/s99/s109/s41 /s84/s105/s109/s101/s32/s40/s115/s41/s76/s86/s68/s32 /s82/s73/s76/s68 /s51/s46/s51 /s45 /s51/s46/s54 /s52/s46/s53 /s52/s46/s48/s65/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32/s40/s109/s47/s115/s50 /s41 /s84/s105/s109/s101/s32/s40/s115/s41/s76/s86/s68 /s82/s73/s76/s68/s40/s97/s41/s32/s83/s101/s99/s111/s110/s100/s32/s102 /s108/s111/s111/s114/s32/s105/s110/s116/s101/s114/s115/s116/s111/s114/s121/s32/s100/s114/s105/s102 /s116 /s40/s98/s41/s32/s82/s111/s111/s102 /s116/s111/s112/s32/s97/s98/s115/s111/s108/s117/s116/s101/s32/s97/s99/s99/s101/s108/s101/s114/s97/s116/s105/s111/s110/s32 Figure 16: Structural response acceleration (El Centro) Furthermore, a large number of response his- tory analyses are conducted to investigate the per- formance of the RILD optimally-controlled struc- 445 ture under ground motions containing various phase properties. To this end, three types of recorded excitations are employed for time-history dynamic analyses. According to the design practice in /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s99/s111/s110/s116/s114/s111/s108/s32/s102/s111/s114/s99/s101 /s68/s101/s102/s111 /s114/s109/s97/s116/s105/s111 /s110/s40/s99/s109/s41/s32/s76/s86/s68/s32 /s32/s82/s73/s76/s68 /s40/s97/s41 /s40/s98/s41/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s99/s111/s110/s116/s114/s111/s108/s32/s102/s111/s114/s99/s101 /s68/s101/s102/s111 /s114/s109/s97/s116/s105/s111 /s110/s40/s99/s109/s41/s32/s76/s86/s68/s32 /s32/s82/s73/s76/s68Figure 17: Hysteresis loops (El Centro): (a) rst story, (b) fourth story Japan [37], the three natural records were scaled 450 such that their peak ground velocities (PGVs) are equal to 0.5 m/s, which corresponds to design level earthquakes. Table 4 lists details of these ground motions. Each time-history excitation is identi ed by a number, as shown in Table 4. 455 Table 4: Recorded ground motions No. Earthquake Year Station Component PGV 1 Imperial Valley 1940 El Centro N-S 0.5 2 Kern County 1952 Taft E-W 0.5 3 Tokachi-oki 1968 Hachinohe N-S 0.5 Table 5 summaries the maximum seismic re- sponses of the aforementioned LVD and RILD con- trolled structures under the chosen ground motions. It is shown that with equal cost-related indexes, the optimally designed RILD elements can be more ef- 460 fective than the viscous dampers in reducing the peak inter-story drift and oor response accelera- tion of a multi-story building structure under strong ground motions. This example also veri es that by minimizing the sum of structural inter-story drift 465 RAFs, the developed method can be readily used to e ectively mitigate the peak seismic vibration of a multi-story building structure under earthquake waves containing various phase properties. 11 Table 5: Maximum seismic responses of LVD and RILD controlled structures No.LVD RILD I1 I2 Floor acc.(m/s2)I1 I2 Floor acc.(m/s2) 1 0.15 1/112 4.49 0.15 1/122 4.08 2 0.15 1/118 4.73 0.14 1/122 4.37 3 0.14 1/130 3.92 0.15 1/135 3.83 Max. 0.15 1/112 4.79 0.15 1/122 4.08 5. Conclusions 470 In this paper, we presented a gradient-based op- timization method of placing RILD elements for seismic vibration mitigation of a multi-story build- ing structure. Firstly, the relative displacement and absolute acceleration RAFs of the RILD controlled 475 SDOF system were investigated by considering var- ious damping levels. It is revealed that increasing RILD can e ectively reduce the structural resonant responses without compromising the responses at high-frequency regions. This is one of the main ad- 480 vantages of RILD over the conventional LVD used for structural vibration mitigation. A ten-story benchmark building structure was employed as an analytical example to verify the ef- fectiveness of the proposed method. The conven- 485 tional sti ness proportional and uniform distribu- tion methods were considered for comparison with the proposed method. It was shown that with equal sum of RILD complex sti ness coecients, the pro- posed method contributed to smaller structural vi- 490 bration responses than the conventional methods. Parametric studies were conducted to investigate the e ect of RILD on the performance of the opti- mally controlled structure by gradually increasing the sum of complex sti ness ecients. The perfor- 495 mance curves of the RILD controlled structure were obtained and compared with those of the controlled structure equipped with optimally placed viscous dampers. It is suggested that with equal sum of control forces, the optimally placed RILD elements 500 can be more e ective than viscous dampers in mit- igating the inter-story drift and oor response ac- celeration of a multi-story building structure un- der strong ground motions containing various phase properties. 505 Acknowledgments The authors gratefully acknowledge the sup- port by the National Natural Science Foundation of China (No. 52208314 and No. 51838006), the National Key R&D program of China (No. 5102023YFC3805700), and the China Postdoctoral Science Foundation (No. YJ20210457 and No. 2022M711258).