The advantages of eddy current dampers over conve ntional fluid dampers include the ability to produce resistive forces with no contact between the components, wherein damping forces are generated, resulting in a less degradable mechanism. Thus far, the apparent mass effect, referred to as inertance, in an eddy current damp er employing rotational motion has not been intentionally applied for vibrati on control. Therefore, this pa per proposes utilizing the inertance effect in an inerter eddy current damper (IECD ) to construct a dynamic vibration absorber for the seismic protection of civil structures, termed as a tuned inerter eddy current damper (TIECD). The IECD consists of a ball screw, a conductor, permanent magnets, and a back iron. The gravitational mass of the back iron disk is converted into inertance via the ball-screw mechanism, which amplifies th e mass and damping effects of the rotational disk and eddy currents, respectively, by converting low-speed translational motion into high-speed rotational motion. An analytical method based on the separa tion of variables is applied to estimate the eddy current damping force of the IECD at di fferent velocities. Concurrently, a succinct damping model is developed to capture the mech anical behavior of the nonlinear eddy current damping. A series of dynamic tests using a sma ll-scale prototype IECD is then conducted to confirm the feasibility and accuracy of the pr oposed numerical damping model. Next, an equivalent damping coefficient for eddy current damping is derived under harmonic excitation. Based on a combination of the fixed-point method, the equivalent damping coefficient of the IECD, and complex eigenanalysis, an optimal design method for the TIECD is developed. Finally, the effectiveness of the proposed TIECD is validated through a shake table test and a numerical simulation. Keywords: Eddy current effect; separation of vari ables; dynamic vibration absorber; tuned inerter eddy current damper (TIECD); optimal design. 1 Introduction Over the past several decades, the practice of adding supplemental energy dissipators to structures to control their responses to earthqua kes and wind loads has b ecome widespread. The development of inerters [1] as vibration contro l devices has further expanded the possibilities of seismic protective systems for civil structures. In Japan, in 1975, before the term “inerter” was defined, the so-called ball-screw inerter was u sed as a mechanical snubber to replace a hydraulic snubber for the vibration-proof nuclear power plant piping at the Tokai No. 2 Power Station [2]. Inerters were initially investigated for use in automobile suspension [3] and railroad [4] 4 / 29 applications, before their scope was gradua lly expanded to include building and civil engineering structures [5–15]. Kawamata proposed a mass pump [16] that could be classified as a fluid inerter [17–19]. Various other types of inerter devices using different mechanisms such as racks and pinions [8,20,21] or electromagnets [22] have also been proposed. Viscous fluids used in conventional viscous mass dampers are mostly employed to provide damping; however, drawbacks such as the cost of changing the damping coefficient and oil leakage [23] have been identified by research ers. Potential damping mechanics has been developed to address these shortcomings and o ffer alternatives. Among them, the eddy current effect [24–30] has attracted significant interest among researchers owing to its various benefits, including the absence of mechanical contact friction, stability at high temperatures, and simple form. Furthermore, the intensity of the eddy curre nt effect can be easily adjusted by changing the gap between the magnetic source and conductor [31–34]. Certain devices that use electromotive forces can even serve as energy harvesters [35–38]. The eddy current effect is generated by the relative motion between a non-magnetic conductive material [24,26,28–30] and a magnetic field or by the time-varying intensity of a stationary magnetic field [27,39,40]. Owing to the superior ity of the eddy current effect, as mentioned above, it has been broadly implemented in the industry [31,32,41–45], including in vehicle brakes [46–48]. The application of eddy curre nt dampers and eddy-current-based tuned mass dampers to engineering structures has been pr oposed to exploit their advantages, such as adjustability, stability, and simple topology [49–57] . However, the large apparent mass effect (inertance) resulting from the small physical mass in a rotary eddy current damper has not been exploited thus far. Therefore, this work propo ses using the inertance generated by a rotary eddy current damper to construct an efficient structur al dynamic vibration absorber without a weight penalty. Alternatively, attempts to develop a hybrid strategy that incorporates the eddy current effect with the use of shape memory alloys [ 58], magnetorheological fluids [59], and friction [60,61] have been undertaken to improve damper control performance. It should be noted that traditional eddy current damping employs the tr anslational motion of the magnetic source and conductor, leading to a low damping density, as co mpared with the cost and size of the damper [49,50,57]. In addition to replacing the conduc tor material with a higher conductivity and reducing the air gap between permanent magnets and conductors [33], magnification of the relative velocity between the permanent magnets and conductors represents a more effective approach. Indeed, using ball screw mechanics [62–64] or a rack–gear system [65–67] can magnify the relative velocity between the tw o terminals by up to 10 times, resulting in considerable amplifica tion of the damping. In addition to the development of new techniqu es to enhance the energy dissipation of inerter- based dampers, challenges have been identif ied in developing effective inerter system 5 / 29 configurations and determining the appropriate design paramete rs. Tuned viscous mass dampers (TVMDs) [5,6,14], tuned inerter dampers (TIDs) [9,15,68], and tuned mass damper inerters (TMDIs) [12,18,69–71] have been developed by altering the topology of the inerter, damper, and spring. Owing to its cost-effective amplif ication of energy dissipation, the TVMD has attracted considerable attention for its cont rol mechanism and the development of optimal design methods [5,6,72,73]. In the proposed tuned inerter eddy current damper (TIECD), the inertance and damping amplification effects of a rotating eddy current damper [34,62] are fully utilized as a component of a TVMD to construc t a compact dynamic vibration absorber without a weight penalty [5,72]. Unlike previous studies on eddy current damping devices without tuning springs [62,67,74,75], the proposed TIECD offers excellent energy dissipation capacity based on the damping enhancement principle [5,72] , with the optimal combination of a tuning spring and the IECD. Herein, we present a systematic investigation of the proposed TIECD. We provide the results of theoretical analyses and the experimental valid ation of the IECD, describe an optimal design with an equivalent damping model, and present th e results of a shake table test for the TIECD. The electromagnetic field in moving eddy current problems can be solved using the magnetic equivalent circuit (MEC) model [76,77], fiel d analysis method [31,32,43,46,62], and finite element method [34,67]. The finite element met hod can deal with complex configurations and offer precise results, albeit at the expense of the computational time. Accurate simulation results for the MEC model are strongly dependent on the known flux path and the moderate eddy current effect. In comparison with the two afor ementioned methods, the field analytical method with a clear physical meaning and fewer comput ational resources was adopted in this study to approximate the electromagnetic torque of the inerter eddy current damper (IECD) [31,32,78]. To investigate the influence of the skin eff ect of the conductor and the nonlinearity in the permeability of the back iron on eddy current damping [32], a finite element analysis considering the saturation of the back iron [32,48] was conducted using the finite element method magnetics (FEMM) software [79]. A su ccinct equation for eddy current damping was then fitted to represent the mechanical behavior of the eddy current damping force according to the input velocity over a significantly large range [25,30,63]. In addition, a series of dynamic tests was conducted for the IECD under harmonic excitation at various frequencies, and the influence of the air gap between the perman ent magnets and conductor on the eddy current damping was investigated. Furthermore, an e quivalent relationship between the eddy current damping and linear viscous damping was derived to associate the behavior of the IECD with that of a viscous mass damper (VMD). Subse quently, an optimal design procedure for the TIECD was established by combining the comple x eigenanalysis, fixed-point method, and equivalent damping model. Nume rical simulations and shake table tests for a single-degree-of- 6 / 29 freedom (SDOF) system were also conducted by employing natu ral ground motions, to verify the seismic control effect of the designed TIECD. The remainder of this paper is organized as follows. Section 2 presents the outline of the developed IECD. Section 3 describes the analytical formulation of the eddy current damping force employed in the proposed IECD. Detailed info rmation related to the realization of a small- scale IECD and the performance test observations ar e provided in Section 4, in addition to the derivation of an equivalent damping coefficient for the eddy current damping. In Section 5, the novel TIECD is described, its optimal design method is explained, and the results of the shake table test to verify its control effect are presen ted. Lastly, conclusions and remarks are presented in Section 6. 2 Outline of inerter eddy current damper As illustrated in Fig. 1, a rotary IECD comp rises a screw shaft, a ball nut, a back iron, a conductor, and permanent magnets . Considerable inertial resistive and damping forces are generated by the mass moment inertia of the rota ry disk and the electromagnetic damping forces induced by the relative circumferential moti on between the conductor and permanent magnets. Therefore, the total axial resistive force 𝐹ୢ of the ball screw is expressed as the sum of the inertial force 𝐹୍, eddy current damping force 𝐹ୣ, and mechanical friction force 𝐹୤, as follows: 𝐹ୢൌ𝐹୍൅𝐹ୣ൅𝐹୤. (1) The inertial force 𝐹୍ is expressed as 𝐹୍ൌ𝑚ୢ𝑥ሷୢ, (2) where 𝑥ሷୢ is the relative acceleration response of the two terminals in the IECD; 𝑚ୢൌ𝐽୲𝑎ୢ denotes the inertance, in which 𝐽୲ is the mass moment of inertia of the disk, 𝑎ୢൌሺ2𝜋𝐿ୢ⁄ሻଶ, and 𝐿ୢ is the lead length of the ball screw. The eddy current damping force 𝐹ୣ is given by 𝐹ୣൌቀଶగ ௅ౚቁ𝑇ୣሺ𝑥ሶୢሻ, (3) where 𝑥ሶୢ and 𝑇ୣሺ𝑥ሶୢሻ denote the relative velocity responses of the two terminals in the IECD and electromagnetic torque induced by the e ddy current effect, respectively. A detailed approximation method for 𝑇ୣ is presented in Section 3. Mechanical friction exists at the contact surf ace between the ball bearings and screw threads [80]. Because the ball bearings are preloaded, thei r frictional force is negligibly affected by the external force acting on the ball screw or the ve locity in the axial direction. The Coulomb friction model [81] with a smaller friction coe fficient was adopted to approximate the rolling contact friction, in order to express the friction force 𝐹୤: 𝐹୤ൌ𝐹෨୤sgnሺ𝑥ሶୢሻ, (4) 7 / 29 where 𝐹෨୤ denotes the maximum friction force identif ied using a very low frequency test, and sgnሺ∙ሻ denotes the signum function. Fig. 1 (a) Schematic of IECD and (b) eddy curre nt effect in electromagnetic field. 3 Determination of el ectromagnetic torque As mentioned in Section 2, the electromagnetic to rque of the IECD is generated by the relative motion between the conductor and permanent ma gnets in the circumferential direction. According to the electromagnetic field theory [82], Maxwell’s equations can describe this electromagnetic behavior in an IECD. Owing to th e small ratio of the permanent magnet radius 𝑟୫୥ to its distance from the central location 𝑅୫୥, the velocity of the permanent magnets at its center, 𝑣୫୥ൌ𝜔 𝑅୫୥, can be used to calculate the eddy current damping forces. Using the general equivalence principle proposed by Wa ng et al. [32], permanent magnet shapes were transformed into rectangles with the same area. Th e original 3-D model can then be transformed into a 2-D analytical model in the Cartesian coordinate system, as shown in Fig. 2. Fig. 2 Analytical model of electromagnetic field. In Fig. 2, 𝜏୫୥ and 𝜏୮୭୪ୣൌ𝜋 𝑅୫୥𝑛୮⁄ denote the edge and pole pitc h lengths of the equivalent permanent magnets, respectively, where 𝑛୮ denotes the number of permanent magnet pole pairs. As shown in Fig. 2, the electromagnetic field in the IECD is composed of five regions: (I) the back iron of the magnetic disk, (II) permanent magn ets and intermediate air, (III) air gap, (IV) conductor, and (V) back iron of the conductor. As documented in [83], the curvature ratio defined by the radial edge length of the e quivalent magnets to the pole pitch length 𝜏୮୭୪ୣ is not greater than unity, and the influe nce of the curvature effect is less than 5%, which can be 8 / 29 neglected in this calculation. Additionally, three assumptions are adopted: 1) the relative permeability and conductivity of the permanent magnets are 𝜇୫୥ൌ1 and 𝜎୫୥ൌ0, respectively, and the relative conductivity of the air is 𝜎ୟൌ0; 2) the thickness of the back iron is sufficient to avoid magnetic saturation, a nd its relative permeability is constant; 3) the conductor in region IV and back iron in region V remain stationary, and the permanent magnets in region II and back iron in region I move with velocity 𝑣୫୥. Considering 𝑨ൌሾ0; 0;𝐴௭ሺ𝑥,𝑦ሻሿ as a magnetic vector potential, the flux density 𝑩ൌ ൣ𝐵௫ሺ𝑥,𝑦ሻ,𝐵௬ሺ𝑥,𝑦ሻ,0൧ can be expressed as 𝑩ൌ∇ൈ𝑨 . (5) Using the Coulomb gauge condition ∇∙𝑨ൌ 0 [82], the following equation holds: ∇ൈ∇ൈ𝑨ൌ∇∙ ሺ∇∙𝑨ሻെ∇ଶ𝑨ൌെ ∇ଶ𝑨ൌ∇ൈ𝑩 . (6) The constitutive relation of the electromagnetic field can then be given as 𝑩ൌ𝜇଴൫𝜇୰𝑯൅𝜇୫୥𝑴൯, (7) where 𝜇୰ and 𝜇୫୥ are the relative permeabilities of the non-magnetic material and permanent magnets, respectively; 𝑯ൌൣ 𝐻 ௫ሺ𝑥,𝑦ሻ,𝐻௬ሺ𝑥,𝑦ሻ,0൧ is the magnetic field intensity; and 𝑴ൌ ൣ0;𝑀௬ሺ𝑥,𝑦ሻ;0൧ is the magnetization of the permanent magnet. According to Ampere’s law, ∇ൈ𝑯ൌ𝑱 , (8) where 𝑱ൌሾ0; 0;𝐽୸ሺ𝑥,𝑦ሻሿ is the induced current density, expressed as the product of the electrical conductivity of the material and the electric field intensity, that is, 𝑱ൌ𝜎 𝑬 . Thus, according to Farah’s law, ∇ൈ𝑬ൌെ𝜕𝑩 𝜕𝑡 ⁄, (9) Substituting Eqs. (7)–(9) into Eq. (6), the governin g equation for the electromagnetic field in the IECD can be obtained as follows: ∇ ଶ𝑨ൌ𝜇଴ቀ𝜇୰𝜎డ𝑨 డ௧െ𝜇୫୥∇ൈ𝑴ቁ. (10) 3.1 Governing equations for regions I and III As there is no relative motion between the perm anent magnets and the related back iron, the magnetic vector potential remains constant in re gion I. Therefore, the governing equation for region I is expressed as డమ஺೥౅ డ௫మ൅డమ஺೥౅ డ௬మൌ0, (11) where 𝐴௭୍ is the modulus of the magnetic vector potential in region I. The insulating property of air implies that the gove rning equation for region III is similar to that for region I; thus, 9 / 29 డమ஺೥౅౅౅ డ௫మ൅డమ஺೥౅౅౅ డ௬మൌ0, (12) where 𝐴௭୍୍୍ is the modulus of the magnetic vector potential in region III. 3.2 Governing equation for region II Fig. 3 Definition of harmonic magnetization along with x-axis. As illustrated in Fig. 3, the magnetizatio n distribution of the permanent magnets 𝑀௬ሺ𝑥,𝑦ሻ can be modeled as a rectangular wave along the x-axis [31,32]. Thus, the Fourier series 𝑀෩௬ሺ𝑥,𝑦ሻ is used to approximate 𝑀௬ሺ𝑥,𝑦ሻ, as follows: 𝑀෩௬ሺ𝑥,𝑦ሻൌ∑ 𝑀௬,௡expሺെ𝑖𝛽𝑥ሻே ௡ୀଵ ,ଷ,ହ,⋯ , (13) 𝑀௬,௡ൌଵ ఛ౦౥ౢ౛׬𝑀௬cos௡గ ఛ౦౥ౢ౛𝑥d𝑥ఛ౦౥ౢ౛ ିఛ౦౥ౢ౛ൌସெ೤ ௡గsin௡గఛౣౝ ଶఛ౦౥ౢ౛, (14) where 𝛽ൌ𝑛 𝜋 𝜏 ୮୭୪ୣ⁄ , 𝑖ൌ√െ1, expሺ∙ሻ is the exponential function, 𝑀௬ൌ𝐵௥൫𝜇୫୥𝜇଴൯ ⁄ , 𝐵୰ is the residual flux density, and 𝜇୫୥ denotes the relative permeability of the permanent magnets. Substituting the assumptions that 𝜇୫୥ൌ1 and 𝜎୫୥ൌ0 into Eq. (10), the governing equation for region II with the first-order component of the Fourier series is given by డమ஺೥౅౅ డ௫మ൅డమ஺೥౅౅ డ௬మൌ𝑖𝜇୫୥𝜇଴𝛽𝑀௬,ଵexpሺെ𝑖𝛽𝑥ሻ, (15) where 𝐴௭୍୍ is the modulus of the magnetic vector potential in region II. 3.3 Governing equations for regions IV and V In regions IV and V, the currents induced in th e conductor and back iron were generated by the time-dependent variations in the electromagne tic field intensity. Therefore, the governing equations for regions IV and V can be derived from Eqs. (10), as follows: డమ஺೥౅౒ డ௫మ൅డమ஺೥౅౒ డ௬మൌ𝜇଴𝜇୰୍୚𝜎୍୚డ஺೥౅౒ డ௧, (16) డమ஺೥౒ డ௫మ൅డమ஺೥౒ డ௬మൌ𝜇଴𝜇୰୚𝜎୚డ஺೥౒ డ௧, (17) where 𝜇୰୍୚ and 𝜇୰୚ denote the relative permeabilities, and 𝜎୍୚ and 𝜎୚denote the electrical conductivities of the conductor with the related b ack iron in regions IV and V, respectively. 3.4 Solution of governing equations Using the separation of variables method (SVM), a general solution of the governing equations for the five regions can be defined as th e product of two independent components: 10 / 29 𝐴௭௝ሺ𝑥,𝑦ሻൌ𝐴௭,௬௝ሺ𝑦ሻexpሺെ𝑖𝛽𝑥ሻ,𝑗ൌ I, II, III, IV, V. (18) The solutions of the magnetic vector potential can then be constructed for the five regions, as follows: 𝐴௭୍ሺ𝑥,𝑦ሻൌ ቀ𝐶୍expሺ𝛽𝑦ሻ൅𝐷୍expሺെ𝛽𝑦ሻቁexpሺെ𝑖𝛽𝑥ሻ 𝐴௭୍୍ሺ𝑥,𝑦ሻൌ ቀ𝐶୍୍expሺ𝛽𝑦ሻ൅𝐷୍୍expሺെ𝛽𝑦ሻቁexpሺെ𝑖𝛽𝑥ሻെ𝑖 𝛽𝜇଴𝑀௬,ଵሺ𝑥ሻexpሺെ𝑖𝛽𝑥ሻ 𝐴௭୍୍୍ሺ𝑥,𝑦ሻൌ ቀ𝐶୍୍୍expሺ𝛽𝑦ሻ൅𝐷୍୍୍expሺെ𝛽𝑦ሻቁexpሺെ𝑖𝛽𝑥ሻ (19) 𝐴௭୍୚ሺ𝑥,𝑦ሻൌ ቀ𝐶୍୚exp൫𝛽୍୚𝑦൯൅𝐷୍୚exp൫െ𝛽୍୚𝑦൯ቁexpሺെ𝑖𝛽𝑥ሻ 𝐴௭୚ሺ𝑥,𝑦ሻൌ ቀ𝐶୚exp൫𝛽୚𝑦൯൅𝐷୍exp൫െ𝛽୚𝑦൯ቁexpሺെ𝑖𝛽𝑥ሻ, where 𝛽୍୚ൌට𝛽ଶെ𝑖𝛽𝜇୰୍୚𝜇଴𝜎୍୚𝑣୫୥, 𝛽୚ൌට𝛽ଶെ𝑖𝛽𝜇୰୚𝜇଴𝜎୚𝑣୫୥, and 𝑣୫୥ൌ𝜔 𝑅୫୥. The boundary conditions (BCs) at 𝑦ଵ and 𝑦଺ and the continuous conditions (CCs) at the interfaces of two adjacent regions are given as follows: BC(I):𝜕𝐴௭୍ 𝜕𝑥ൌ0|௬ୀ௬భ . (20)CC(I–II):𝜕𝐴௭୍ 𝜕𝑥ൌ𝜕𝐴௭୍୍ 𝜕𝑥|௬ୀ௬మ 1 𝜇ୱ𝜕𝐴௭୍ 𝜕𝑦ൌ1 𝜇଴𝜕𝐴௭୍୍ 𝜕𝑦|௬ୀ௬మ CC(II–III):𝜕𝐴௭୍୍ 𝜕𝑥ൌ𝜕𝐴௭୍୍୍ 𝜕𝑥|௬ୀ௬య 1 𝜇଴𝜕𝐴௭୍୍ 𝜕𝑦ൌ1 𝜇଴𝜕𝐴௭୍୍୍ 𝜕𝑦|௬ୀ௬య CC(III–IV):𝜕𝐴௭୍୍୍ 𝜕𝑥ൌ𝜕𝐴௭୍୚ 𝜕𝑥|௬ୀ௬రୀ଴ 𝜕𝐴௭୍୍୍ 𝜕𝑦ൌ1 𝜇୰୍୚𝜕𝐴௭୍୚ 𝜕𝑦|௬ୀ௬రୀ଴ CC(IV–V):𝜕𝐴௭୍୚ 𝜕𝑥ൌ𝜕𝐴௭୚ 𝜕𝑥|௬ୀ௬ఱ 1 𝜇୰୍୚𝜕𝐴௭୍୚ 𝜕𝑦ൌ1 𝜇୰୚𝜕𝐴௭୚ 𝜕𝑦|௬ୀ௬ఱ BC(V):𝜕𝐴௭୚ 𝜕𝑥ൌ0|௬ୀ௬ల Substituting Eq. (19) into Eq. (20), the coefficients related to the solutions for the five regions can be estimated via symbolic calculations. Th e symbolic computational tools provided in Sagemath [84], an open-source software, were em ployed to derive the closed-form solution. The flux density in region IV was estimated as follows: 𝐵௬୍୚ൌడ஺೥౅౒ డ௫ൌെ 𝑖 𝛽ቀ𝐶୍୚exp൫𝛽୍୚𝑦൯൅𝐷୍୚exp൫െ𝛽୍୚𝑦൯ቁexpሺെ𝑖𝛽𝑥ሻ. (21) Furthermore, the induced current density in region IV was determined as 𝐽௭୍୚ൌെ 𝜎୍୚డ஺೥౅౒ డ௫ൌെ 𝑖𝜎୍୚𝜔𝑅୫୥𝛽ቀ𝐶୍୚exp൫𝛽୍୚𝑦൯൅𝐷୍୚exp൫െ𝛽୍୚𝑦൯ቁexpሺെ𝑖𝛽𝑥ሻ. (22) Although the back iron in region V generates e ddy currents, its contribution can be ignored owing to the rapid decay of the flux density with an increase in the distance from the surface of the permanent magnet and the low conductivity of the material [46]. Therefore, the total 11 / 29 electromagnetic torque generated by 𝑛୮ pairs of magnets can be expressed as 𝑇ୣ଴ൌ௉ ఠൌ௡౦ ଶఠఙ౅౒׬ሼ𝐽௭୍୚ሽ∗𝐽௭୍୚d𝑉୴୭୪୳୫ୣ ൌ௡౦ఙ౅౒ఠఉమோౣౝమ௪ౣౝఛ౦౥ౢ౛ ଶ׬ห𝐶୍୚exp൫𝛽୍୚𝑦൯൅𝐷୍୚exp൫െ𝛽୍୚𝑦൯หଶd𝑦଴ ି୼ర, (23) where 𝑤୫୥ is the width of the equivalent permanent magnet. To consider the distribution and actual path of the induced current in the overh ang region [32,47,76,77], the modification factor 𝑘ୖ୳ୱୱୣ୪ [78] is used to obtain the electromagnetic to rque in the 3-D analytical model; correcting the torque in the 2-D analytical results yields the following: 𝑇ୣൌ𝑘ୖ୳ୱୱୣ୪𝑇ୣ଴, (24) where 𝑘ୖ୳ୱୱୣ୪ൌ1െ୲ୟ୬୦ఒౣ ఒౣሺଵା୲ୟ୬୦ఒ ౣ୲ୟ୬୦ఒౙౣሻ. (25) Here, 𝜆୫ൌ𝜋 𝜏୫୥൫2𝜏୮୭୪ୣ൯ ⁄ , 𝜆ୡ୫ൌ𝜋 ൫ 𝑤 ୡെ𝑤୫୥൯൫2𝜏୮୭୪ୣ൯ൗ , and 𝑤ୡ is the width of the conductor. It is worth noting that Eq. (25) holds for a variable velocity between the permanent magnets and conductor in the IECD, beca use the eddy current induced by a moving electromagnetic source in a stationary conductor is equivalent to that induced by a time- dependent electromagnetic field intens ity in a stationary conductor [78]. 4 Performance tests for IECD In this section, the dynamic test conducted usi ng a small-scale IECD is discussed to verify the proposed mechanical model of the IECD and th e effectiveness of the eddy current damping. 4.1 Outline of IECD A small-scale IECD was designed, as shown in Fig. 4 (a). A set of ball screws and ball nuts with lead length 𝐿ୢൌ32 mm and nominal diameter 𝑑థൌ32 mm was employed to transfer the translational motion between the terminals into the rotational motions of the permanent magnets and back iron. As illustrated in Fig. 4 (a), the inertance (apparent mass) of the IECD was 𝑚ୢൌ 451 kg , provided by a rotational mass consisting of the back iron (371 kg), permanent magnets (17 kg), and an aluminum template serving as a positioning guide for the magnets (63 kg). The back iron was manufactured using SS400 steel, and its permeability ratio (relative to that of air) was assumed to be 1000. It was 12 mm thick and sufficient to avoid magnetic saturation. Copper with high electrical conductivity was sel ected as the conductive material. Given a ball screw lead length of 𝐿ୢൌ32 mm , the air gap was designed to be 𝛥ଷଵൌ10 mm to realize the optimal combination of the inertance and eddy cu rrent damping coefficient in the TIECD. The other design parameters related to the eddy cu rrent damping force are listed in Table 1. 12 / 29 Fig. 4 (a) Arrangement of the IECD test, (b) pe rmanent magnets, (c) back iron and aluminum fixture of magnets, and (d) conductor. Table 1 Design parameters related to eddy current damping force. Parameters Value [Unit] 𝐵௥ 1.2 [T] 𝜎ୡୡ୭୮୮ୣ୰ 59.6 [MS/m] 𝜎ୡୱ୲ୣୣ୪ 6.99 [MS/m] 𝜇୰ୡ୭୮୮ୣ୰ 1 𝜇୰ୱ୲ୣୣ୪ 1000 𝑛୮ 7 𝑟୫୥ 10 [mm] 𝑅୫୥ 70 [mm] 𝑤ୡ 40 [mm] 𝛥ଵ 12 [mm] 𝛥ଶ 10 [mm] 𝛥ଷଵ 10 [mm] 𝛥ଷଶ 5 [mm] 𝛥ସ 4 [mm] 𝛥ହ 12 [mm] 4.2 Approximation of the designed damping force Assuming that the equivalent shape of the perm anent magnets was a square, the length of each edge was 𝑤୫୥ൌ𝜏୫୥ൌ17.7 mm . The pole pitch length of the permanent magnets was estimated as 𝜏୮୭୪ୣൌ𝜋 𝑅୫୥𝑛୮⁄ൌ31.4 mm . A reasonable copper width in the z-direction 𝑤ୡ should be less than twice the equivalent width of the permanent magnets. As listed in Table 1, the width of copper, 𝑤ୡൌ40 mm , was more than twice the equivalent width of the permanent magnets; therefore, the width of the conductor was selected as 2𝑤୫୥ൌ35 mm . Given that the translational relative velocity between the termin als varies from 0 to 0.7 m/s, the circumferential velocity of the permanent magnets increases from 0 to 9.62 m/s. The air gap of the IECD was 13 / 29 set as 𝛥ଷଵൌ10 mm and 𝛥ଷଶൌ5 mm to investigate its influence on the eddy current damping force. The estimated eddy current damping fo rces obtained using the methods provided in Sections 2 and 3 are represented by the blue circ les in Figs. 5 (a) and 5 (b), respectively. Fig. 5 Comparison of damping forces for (a) Δଷଵൌ10 mm and (b) Δଷଶൌ5 mm . The finite element method was applied to solve the electromagnetic field equations, as discussed in Section 3, to validate the accuracy of the SVM and explore the influences of the skin effect [85] on the conductor and magnetic saturation of the back iron in region V. The 2-D open-source FEMM software [79] developed by Dr. David Meeker was employed to simulate the electromagnetic field, as shown in Fig. 2. Acco rding to the equivalent current model in [82], the constant rotational velocity of the perm anent magnet was modeled as an electromagnet subjected to an alternating current with a d esignated frequency in FEMM. As in the case of the SVM, the computed results in FEMM are corrected with the modification factor 𝑘 ୖ୳ୱୱୣ୪ to account for the distribution and actual path of the induced current in the overhang region. The simulated results using FEMM based on the assumpti on of a linear back iron should be equal to the results obtained from the SVM. To investig ate the influence of the nonlinear back iron, finite element analyses were also conducted us ing FEMM with the nonlinear B–H relationship, as shown in Fig. 6. The simulated results b ased on the linear and nonlinear assumptions were designated as FEMଵ and FEMଶ, respectively, as shown in Fig. 5. 14 / 29 Fig. 6 Nonlinear B–H relationship of back iron. Figure 5 shows that the simulated SVM results nearly overlap the finite element simulation results FEMଵ based on the linear assumption, when the input velocity is no greater than the critical velocity corresponding to the peak eddy current damping force. As the input velocity exceeds the critical velocity, the divergence between the SVM and FEMଵ progressively increases. This is attributed to the aggravati on of the skin effect induced by the eddy current with an increase in the input velocity [85]. As illu strated in Figs. 5 (a) and 5 (b), the simulation results FEMଶ coincide with FEMଵ when the input velocity is less than 0.2 m s ⁄. The growing discrepancies between them are attributed to the nonlinear permeability and even saturation of the back iron in region V. On comparing Figs. 5 (a) and 5(b), the damping force can be observed to increase by a factor of three as the air gap decreases from 10 mm to 5 mm, with peak damping forces of 818 N and 2201 N, resp ectively, under the linear assumption, and 773 N and 2050 N, respectively, under the nonlinear assumption. The discrepancies between the estimated results of the SVM and nonlinear FEM were 5.5% and 6.9% for the air gaps of 10 mm and 5 mm, respectively. These small errors indicate that the proposed SVM method can be confidently used to approximate the eddy current damping force in the preliminary design of the IECD. Although an analytical formulation of the equiva lent damping force can be derived, exhaustive expressions for the coefficients related to th e electromagnetic field equation may hinder their practical application. To describe the mechani cal behavior of the eddy current damping force more concisely, the fitting mode l proposed by Wouterse [25] and Tonoli [30] can be applied, as follows: 𝐹ୣൌ𝑐଴௩బమ ௫ሶౚమା௩బమ𝑥ሶୢ, (26) where the coefficients describing the critical damping 𝑐଴ and critical velocity 𝑣଴ remain constant and are obtained from the computati onal results, as shown in Fig. 5. The fitting parameters of the eddy current damping are summ arized in Table 2. Figure 5 shows that the derived fitting model using the pa rameters listed in Table 2 perf ectly captures the mechanical characteristics of the eddy current damping, regardless of the linearity assumption applied. Thus far, SVM-based simulation results have been used to approximate the eddy current damping force in the preliminary design of the IECD. T hus, based on Sections 3 and 4.1, by substituting Eq. (26) into Eq. (1), the tota l axial force can be expressed as 𝐹ୢൌ𝑚ୢ𝑥ሷୢ൅𝑐଴௩బమ ௫ሶౚమା௩బమ𝑥ሶୢ൅𝐹෨୤sgnሺ𝑥ሶୢሻ. (27) To validate the accuracy of the proposed da mping model and determine the values of the parameters collected in the mechanical model of the IECD, a dynamic test was conducted on the designed IECD under harmonic excita tions with various frequencies. 15 / 29 Table 2 Fitting parameters related to the eddy current damping model. 𝛥ଷ Parameters SVM FEMଵ FEMଶ 10 mm 𝑐଴ 5286 5219 5218 𝑣଴ 0.309 0.307 0.296 5 mm 𝑐଴ 14763 14597 14322 𝑣଴ 0.298 0.295 0.287 4.3 Dynamic test on IECD with harmonic inputs In the dynamic test, the amplitude of the harm onic excitation was set to 30 mm, and sinusoidal excitation with a very low frequency of 0.05 Hz was employed to identify the maximum friction force 𝐹෨୤ of the pre-loading ball nut in the designed IECD [80,81], as illustrated in Fig. 7. Fig. 7 (a) Time history and (b) hysteretic loop of the resistance force under sinusoidal excitation of 0.05 Hz (amplitude = 30 mm, Δଷଵൌ10 mm , and sampling rate = 100 Hz). As shown in Fig. 7 (a), the fitting value for a rectangular wave that includes the contributions from friction and eddy current damping was identified as 𝐹෨୤୧୲୲୧୬୥ൌ115.7 N . However, the red line in Fig. 7 (b) suggests that the e ddy current force with the maximum value 𝐹ୣ,୫ୟ୶ൌ49.8 N needs to be eliminated. To remove the infl uence of the eddy current damping force, an equivalent force 𝐹෨ୣ୤ with a rectangular shape was defined using the equivalent energy dissipation of the eddy current damping force 𝐹ୣ. As indicated in Fig.7 (b), the value of 𝐹෨ୣ୤ was identified as 35.2 N . Therefore, a corrected approximation of the friction force was obtained with the maximum amplitude 𝐹෨୤ൌ115.7െ35.2ൎ81 N in a direction opposite to that of the velocity. 16 / 29 Fig. 8 Hysteretic loop of IECD under sinusoidal ex citations of (a) 0.2 Hz and (b) 0.5 Hz (amplitude = 30 mm, Δଷଵൌ10 mm , and sampling rate = 100 Hz). The hysteretic loops of the designed IECD under si nusoidal excitations of 0.2 Hz and 0.5 Hz are compared in Fig. 8. This figure shows that the proposed model provides excellent approximations of the hysteretic behaviors under both sinusoida l excitations. Notably, the contribution of the inertial force becomes more evident as the input frequency increases. The dashed blue line in Fig. 8 (b) indicates that the inertance of the modeled IECD was 510 kg. In addition, a dynamic test on the designed IECD was conducted using an air gap Δଷଶൌ5 mm under sinusoidal excitation with an amplitude of 30 mm. Figure 9 shows a comparison of the test results obtained under this condition with sinusoidal excitations of 0.2 Hz and 0.5 Hz. Fig. 9 Hysteretic loops of IECD under sinusoidal excitations of (a) 0.2 Hz and (b) 0.5 Hz (amplitude = 30 mm, Δଷଶൌ5 mm , and sampling rate = 1000 Hz). Figure 9 shows that the approximate results correspond well with the smoothed test data, demonstrating that the proposed mechanical model can effectively estimate the mechanical behavior of the designed IECD. Furthermore, the identified inertance of the IECD was approximately 516 kg, which is nearly equal to the result identified for an air gap of 10 mm. The error between the experimentally observed inertance and the approximated inertance of 451 17 / 29 kg may be attributed to the existence of inerta nce in the ball screw and ball. As the inertance and finite stiffness of the fixture constitute an undesirable vibratory system, the local vibrations of the IECD and fixture gradually increase with the excitation frequency, resulting in the high- frequency noise observed in the hysteresis loops. 4.4 Equivalent formulation of eddy current damping The relationship between the eddy current dampi ng and equivalent linear viscous damping must be established to investigate the mechanical beha vior of the IECD in a generalized manner. In this study, we assume that the electromagnetic field is quasistatic. Thus, given a sinusoidal excitation 𝑢ሺ𝑡ሻൌ𝑢෤sinሺ𝜔𝑡ሻ and 𝑢ሶሺ𝑡ሻൌ𝜔 𝑢෤cosሺ𝜔𝑡ሻ, the energy dissipation 𝐸ଵ owing to the eddy current damping with the critical damping coefficients 𝑐଴ and critical velocity 𝑣଴ in a cycle can be expressed as 𝐸ଵൌ4׬𝑐଴௩బమ ௩బమା௨ሶమ𝑢ሶd𝑢௨෥ ଴ൌ2𝑐଴𝑣଴ଶగ ఠቀ1െ𝑣଴ඥ𝑢෤ଶ𝜔ଶ൅𝑣଴ଶ⁄ ቁ. (28) Meanwhile, the energy dissipation of the lin ear viscous damping can be computed as 𝐸ଶൌ4׬𝑐ୣ𝑢ሶd𝑢௨෥ ଴ൌ𝜋 𝜔 𝑐 ୣ𝑢෤ଶ. (29) By equating Eqs. (28) and (29), the equivalent damping coefficient of eddy current damping is given by 𝑐ୣൌଶ௖బ ఘమ൬ඥఘమାଵିଵ ඥఘమାଵ൰,𝜌ൌ௨෥ఠ ௩బ. (30) Fig. 10 Comparison of the mechanical model for the IECD and equivalent VMD under harmonic excitation with 𝑢෤ൌ 60 mm and frequencies of (a) 0.5 Hz and (b) 1.5 Hz. To illustrate Eqs. (28)–(30), Fig. 10 show s a comparison between the equivalent VMD and designed IECD under harmonic excitation with an amplitude of 60 mm at excitation frequencies of 0.5 Hz and 1.5 Hz. The equivalent viscous damping and total axial forces are represented by 18 / 29 𝐹୴୧ୱ and 𝐹୲୭୲ି୴୧ୱ , respectively. In Figs. 10 (a) and 10 (b), the solid and dash-dotted lines in orange denote the eddy current damping force and equivalent viscous damping force under harmonic excitation, respectively. Figure 10 (a) shows that the eddy current damping force approaches its peak when the velocity reaches its maximum under a harmonic excitation of 0.5 Hz. However, Fig. 10 (b) indicates that the peak eddy current damping force shifts away from zero displacement, implying that the eddy current damping is degraded when the input velocity exceeds 𝑣଴. This special characteristic of the eddy current damping ma y provide a fail-safe mechanism to prevent damage to the connecting part and the damper itself caused by excessive reaction forces generated by the device. Thus, the robustness of the structure equipped with the IECD device can be ensured. Furthermore, it was found that the amplitude of the eddy current damping force was larger than the equivalent viscous dampi ng in the range between the peak displacements and the intersection points of the two damping m odels; this implies that the IECD is more effective than the equivalent VMD in dissipa ting earthquake energies at smaller velocities. 5 Optimal design of TIECD In this section, the optimal design of the propo sed TIECD is described. The resonance of the supplemental vibratory system used for tuning, comprising the spring connected in series to the IECD and the inerter element in the IECD conn ected to the primary structure, enhanced the energy dissipation afforded by the eddy current damper. Figure 11 presents a schematic of an SDOF structure equipped with the proposed TIECD. Fig. 11 SDOF structure equipped with TIECD. 5.1 Determination of TIECD parameters The governing equations of an SDOF syst em equipped with a TIECD are given as 𝑚ୱ𝑥ሷୱ൅𝑐ୱ𝑥ሶୱ൅𝑘ୱ𝑥ୱ൅𝐹ୢൌെ 𝑚ୱ𝑥ሷ଴, (31) 𝐹ୢൌ𝑚ୢ𝑥ሷୢ൅𝐹ୣൌ𝑘ୢሺ𝑥ୱെ𝑥ୢሻ, 𝐹ୣൌ𝑐଴𝑥ሶୢ𝑣଴ଶ൫𝑣଴ଶ൅𝑥ሶୢଶ൯ ൗ , where 𝑚ୱ, 𝑐ୱ, and 𝑘ୱ denote the mass, damping coefficient, and stiffness of the SDOF system, respectively; 𝑚ୢ and 𝑘ୢ denote the inertance and supporting spring stiffness of the TIECD, 19 / 29 respectively. By employing the equivalence be tween the linear viscous damping force and eddy current damping force derived in Section 4.4, a dimensionless formulation can be expressed as 𝑥ሷୱ൅2𝜁ୱ𝜔ୱ𝑥ሶୱ൅𝜔ୱଶ𝑥ୱ൅𝜇ୢ𝛾ୢଶ𝜔ୱଶሺ𝑥ୱെ𝑥ୢሻൌെ 𝑥ሷ଴, (32) 𝜇ୢ𝑥ሷୢ൅2𝜁ୢ𝜔ୱ𝑥ሶୢ൅𝜇ୢ𝛾ୢଶ𝜔ୱଶሺ𝑥ୢെ𝑥ୱሻൌ0, where 𝛾ୢ is the dimensionless frequency ratio, and 𝜁ୢ is the dimensionless damping ratio, as defined in the Nomenclature table. Using the fixed-point-based optimization proce dure [5] and the equivalent damping model defined in Section 4.4, an optimization pro cedure was developed to determine the TIECD parameters. According to Ikago, Saito, a nd Inoue [5], the optimal frequency ratio 𝛾ୢ୭୮୲ and optimal damping ratio 𝜁ୢ୭୮୲ of the equivalent TVMD can be obtained as functions of the mass ratio 𝜇ୢ, as follows: 𝛾ୢ୭୮୲ൌ1ඥ1െ𝜇ୢ ⁄ , (33) 𝜁ୢ୭୮୲ൌఓౚ ଶටଷఓౚ ሺଵିఓౚሻሺଶିఓౚሻ. (34) Subsequently, the TIECD stiffness is obtained as 𝑘ୢൌ𝜇ୢሺ𝛾ୢ୭୮୲ሻଶ𝑘ୱ [5]. The optimal 𝑐ୣ୭୮୲of TIECD can then be computed using the equivale nt relationship expressed in Eq. (30), using a given 𝑣଴. The complex eigenvector was obtained from a complex eigenanalysis of the structure equipped with the equivalent TVMD. Given the ta rget inter-story drift of the SDOF system 𝑥ୱ୫ୟ୶ and the corresponding deformation of the IECD 𝑥ୢ୫ୟ୶, the threshold velocity responses 𝑥ሶୱ୫ୟ୶ and 𝑥ሶୢ୫ୟ୶ can be approximated as 𝑥ሶୱ୫ୟ୶ൌ𝜔ଵ𝑥ୱ୫ୟ୶𝑥ሶୢ୫ୟ୶ൌ𝜔ଵ𝑥ୢ୫ୟ୶, (35) where 𝜔ଵ denotes the fundamental natural frequency ob tained from the complex eigenanalysis. The threshold velocity of the IECD 𝑥ሶୢ୫ୟ୶ is proportional to 𝑣଴, that is, 𝑥ሶୢ୫ୟ୶ൌ𝜅 𝑣଴. When 𝜅൏ 1, the maximum velocity response is smaller than the designed 𝑣଴, indicating that the damping force is always in the ascending interval of th e backbone curve of the eddy current damping. Otherwise, the maximum velocity is larger than 𝑣଴, and the ultimate damping force is smaller than the maximum damping force. In this case, the proportional relationship among the components in the complex eigenvector 𝝓ൌሾ𝜙ୱ,𝜙ୢሿ୘ can be adopted to determine the value of 𝑣଴ for the IECD, as follows: |థ౩| |థౚ|ൌ௫౩ౣ౗౮ ௫ౚౣ౗౮ൌ௫ሶ౩ౣ౗౮ ௫ሶౚౣ౗౮⇒𝑣଴ൌఠభ௫౩ౣ౗౮|థౚ| ఑|థ౩|ൌఠభ௫ౚౣ౗౮ ఑. (36) By letting 𝜌ൌ𝜅 and substituting Eq. (36) into Eq. (30), the designed 𝑐଴ of the TIECD can be determined by 𝑐଴ൌ௖౛ ଶ൫𝜅ଶ൅1൅√𝜅ଶ൅1൯. (37) A design flow chart for the TIECD is shown in Fig. 12. 20 / 29 Fig. 12 Flowchart of design procedure for TIECD. 5.2 Shake table validation of TIECD The SDOF steel structure specimen, shown in Fig. 13, was used to experimentally examine the control effect of the designed TIECD. To a void damage to the leaf spring columns, the maximum drift of the specimens was limited to 20 mm. Basic information related to the SDOF steel structure specimens can be found in [5]. The fundamental natural period of the specimen was 0.56 s, and the time step interval of the i nput ground motion was compressed to half of the original data to simulate a structure whose fundamental natural frequency was close to 1 s. Table 3 lists the optimal parameters of the TI ECD, obtained for the given apparent mass of 451 kg, target inter-story drift 𝑥ୱ୫ୟ୶ of 13 mm, and ratio of modification factor 𝜅 of unity. 21 / 29 Fig. 13 (a) Schematic and (b) photograph of the TIECD test. Table 3 Optimized parameters of designed TIECD Parameters Optimized value 𝛾ୢ୭୮୲ 1.12 𝜁ୢ୭୮୲ 0.067 𝑘ୣ୭୮୲ 72.4 ሾkN m⁄ሿ 𝑐ୣ୭୮୲ 3344 ሾN∙sm⁄ሿ 𝑐଴୭୮୲ 5708 ሾN∙sm⁄ሿ 𝑣଴୭୮୲ 0.320 ሾms⁄ሿ Unlike the TVMD systems applied to real-world bu ildings, in which natural rubber is used as a supporting spring device [86,87], we used a set of soft-leaf springs as the spring element to tune the frequency of the designed TIECD. As illustrated in Fig. 13, the tested TIECD was constructed by connecting the proposed IECD to four soft-leaf springs with a rigid beam. For the TIECD specimen used in the test, the stiffness of the tuning leaf spring 𝑘 ୣൌ95 kN m ⁄, 𝑣଴ൌ 0.309 ms⁄, 𝑐଴ൌ5286 N∙sm⁄, and 𝐹෨୤ൌ81 N . The equivalent damping coefficient was estimated from the sum of the eddy current damp ing and friction forces. The equivalent linear damping coefficient of the friction force was estimated as follows: 𝑐ୣ,୤ൌସி෨౜ గ௫ሶౚౣ౗౮ൌ323 N∙sm⁄. (38) According to Eq. (30), the ratio 𝜌 was estimated as 𝜌ൌ𝜔ଵ𝑥ୢ୫ୟ୶𝑣0⁄ൌ0.32 0.309⁄ൌ1.035, (39) and the contribution of eddy current damping force was computed as 𝑐ୣൌଶ௖బ ఘమ൬ඥఘమାଵିଵ ඥఘమାଵ൰ൌ3011 N∙sm⁄. (40) Therefore, the total equivalent damping coefficient of 3334 N∙sm⁄ was nearly equal to 𝑐eopt. 22 / 29 The ground motion record of the 1940 Imperial Vall ey Earthquake, as recorded at the El Centro station, was then used to evaluate the control ef fect of the TIECD, which was reduced to 80% of its original magnitude to ensure that the maxi mum response displacement of the steel structure specimen remained within the allowable limits. A comparison of the numerical simulation and experimental results for the story drift and defo rmation of the IECD is presented in Fig. 14. Additionally, the numerical simulation results of a similar system equipped with an equivalent TVMD using a tuning spring identical to that of the proposed TIECD are plotted for further comparison. Fig. 14 Comparison of simulated results and experi mental data for (a) drift of the primary structure and (b) deformation of TIECD and TVMD. Figure 14 shows that the simulated displacement response of the primary structure nearly overlaps with the experimental observations, t hus demonstrating the accuracy of the proposed TIECD simulation model. Furthermore, the we ll-matched displacement responses obtained from the TIECD and equivalent TVMD systems indicat e that the proposed equivalent linearization model of the IECD effectively captures the rela tionship between the eddy current damping force and viscous damping force. In addition to the comparisons of the displacement responses and relationships between the damping force, the “damping enhancement” effect [72] can be observed on comparing Figs. 14 (a) and 14 (b). This indicates that th e maximum displacement response of the IECD was considerably larger than that of the primary structure. To clarify the “damping enhancement” effect of the designed TIECD, a numerical simulation of the SDOF system equipped with an IECD was also conduc ted, and detailed comparisons of the resulting hysteretic damping force loops and energy dissipa tions are shown in Figs. 15 (a) and 15 (b), respectively. 23 / 29 Fig. 15 Comparison of (a) hysteretic loops of damp ing force and (b) energy dissipations of IECD and TIECD. As shown in Fig. 15, the damping forces of th e TIECD and IECD consis t of friction and eddy current damping forces. Fig. 15 (a) shows that the energy dissipation is enhanced on attaching a tuned stiffness spring component, rendering the hysteretic loop of the TIECD considerably larger than that of the IECD. The additional en ergy dissipation in Fig. 15 (b) demonstrates that the increase in energy dissipation afforded by the TIECD was half that of the original IECD. Fig. 16 (a) Hysteretic loop of damping force and (b) relationship between eddy current damping force and input velocity of TIECD. Figure 16 (a) compares the hysteretic loops of th e damping parts of the TIECD and equivalent TVMD. The simulated TIECD can be observed to provide a hysteretic loop comparable to that for the equivalent viscous damping. It is notew orthy that eddy current damping provides a larger damping force than equivalent viscous damping at velocities less than 0.236 m/s, as this implies that the eddy current damping mechanism is more efficient across a smaller velocity range than the classical linear viscous damping model. Add itionally, the degradation of the damping force 24 / 29 for input velocities greater than 𝑣଴ may serve as a fail-safe mechanism to protect the device connecting the primary structure components. 6 Conclusions and remarks This study investigated a mechanical model a nd the optimal design of a dynamic vibration absorber for use in civil structures, namely the TIECD. The proposed TIECD employs the inertance and electromagnetic damping produced by rotating back iron disks attached to the permanent magnets and eddy currents, respectivel y. The SVM method was first used to estimate the equivalent eddy current damping force of the IECD, by employing a succinct two-parameter fitting model that described its nonlinear behavior . Subsequently, an experimental evaluation of a small-scale IECD was conducted to validate the effectiveness and accu racy of the proposed electromagnetic equations and fitting model. As expected, the well-matched experimental results demonstrated the effectiveness of th e proposed model. An equivalent damping coefficient for eddy current damping was then derived to explore the energy dissipation ability in a generalized manner. Furthermore, an op timal design procedure for TIECD was established using a combination of the equivalent dampi ng model, fixed-point optimization method, and complex eigenanalysis. Finally, a shake table test was conducted to evaluate the control effect of the TIECD on an SDOF benchmark model. Th e observed agreement between the simulated results and experimental observations confirme d that the designed TIECD could control the system response with a high efficiency. Additi onally, the “damping enhancement” effect of the TIECD was observed. This indicated that th e displacement responses of the IECD were considerably larger than those of the primary structure, resulting in enhanced energy dissipation by the eddy currents. Thus, the characteristics of the eddy current damping force, namely the high damping coefficient in the low-speed range and the low damping coefficient in the high- speed range, may help ensure the structural control effect by protecting secondary components from possible damage caused by extremely strong earthquake excitations. Acknowledgments We would like to thank Editage (http://www.ed itage.com) for editing and reviewing this manuscript for the English language. Funding: This study was supported by a Grant-in-Aid from the Japan Society for the Promotion of Science, Grant/Award Number: JSPS KAKENHI [grant number 18F18761]. 25 / 29