Vibration Mitigation of Sagged Cables Using a Viscous Inertial Mass Damper: An Experimental Investigation

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Introduction
Cables are widely used in cable-supported bridges and longspan structures due to their economy, aesthetic efect, and light weight.However, cables are prone to severe vibration because of their low inherent damping and their high fexibility, and this vibration can be detrimental to the serviceability and safety of the pertinent structures.
Recently, the superiority, concerning vibration mitigation, of passive devices possessing properties of negative stifness, such as negative stifness damper (NSD) and inerter damper (ID), has been widely studied by many researchers [1][2][3].Te damper characteristics and control performance have been investigated under random excitations [4,5], pulse-like excitations [6], and earthquake excitations [7][8][9][10].Furthermore, the efectiveness of the negative stifness devices has been systematically analyzed for high buildings [11][12][13][14][15][16], bridges [17][18][19], cables [20,21], and ocean structures [22,23].Among them, the NSD and ID for cables have attracted strong attention because of their high efciency.Shi et al. [24] systematically discussed the dynamic behavior of a cable-NSD system including stability, mode shapes, and modal damping ratios.Javanbakht et al. [25] proposed a refned design formula for a cable-NSD system, and the control performances of negative, zero, and positive stifness dampers were investigated.A novel multimode control approach considering the efect of NSD stifness was further developed [26].Sun et al. [27] studied a system of two cables interconnected with a tuned inerter damper (TID), in which the dynamic characteristics of the system were obtained by complex modal analysis, and an approximate method was proposed to optimize the TID parameters.Chen et al. [28] discussed diferent types of inerter-based vibration absorbers (IVAs) and made comparisons between NSDs and IVAs.Furthermore, Chen proposed a control strategy for multimode cable vibration using NSDs and IVAs, wherein the damper parameters were obtained by maximizing the lowest value of the modal damping ratio of the target modes.It was reported that a viscous inertial mass damper (VIMD) may achieve a modal damping ratio of a cable upto 10 times the values reached by a traditional viscous damper [29,30].Many experimental studies have been conducted to validate the high efciency on the structural vibration mitigation by NSD and ID [31][32][33].An experimental study on NSD for a taut scaled cable was reported by Shi et al. [34].Te achieved maximum modal damping ratio was 10.31% when the damper was installed at a location of 5% of the cable length away from a cable end.Li et al. [35] showed the results of a full-scaled cable experiment using an electromagnetic shunt damper-inerter damper (EMSD-ID).It was reported that a modal damping ratio of 6% was observed with the damper at the 5% position of the cable, which was one of the highest modal damping ratios obtained in full-scale experiments.Moreover, Li et al. [36] carried out another fullscale experiment using an electromagnetic inertial mass damper (EIMD), wherein the obtained modal damping ratio of the cable-EIMD system was enhanced almost two times the values using a viscous damper (VD).Overall, the high control efciency of the NSD and ID was predicted from theoretical study in comparison with those of the traditional viscous damper.However, the control performance from the experiments was not as high as the value predicted by the theory, and it still requires validation.
As a traditional design procedure, the fxed-point method (FPM) was recently applied for optimizing inerter damper and negative stifness devices, which was originally proposed by Den Hartog for a TMD system [37].Ikago et al. [38] proposed a new tuned viscous mass damper (TVMD) with a ball screw as a mass amplifer, which was used and optimized for an MDOF system with complex modal analysis.Te mode vectors of the controlled and uncontrolled system were compared, and a mode-split phenomenon was found.Lazar et al. proposed a control strategy for multistory buildings [39] and cables [40] based on the FPM with an inerter-based device.Jin et al. [41] studied a system of a beam with two IDs and proposed a vibration suppression design using the FPM.It was shown that the target structural mode was split into two modes due to introducing additional inertial mass, which was also found in the cable-ID system [29,30].Marian and Giaralis [42] optimized the parameters of the tuned mass-damper inerter (TMDI) by adopting the FPM and showed that the TMDI is more efective than the TMD with the same mass.Gonzalez-Buelga et al. [43] analytically and experimentally assessed a ball-screw-type TID.Te dry friction of the TID was considered as a nonlinearity, and the TID parameters were optimized by the FPM.Wang et al. [6] optimized a negative stifness amplifying damper (NSAD) for an SDOF system, and the performance was simulated and discussed under numerous types of excitations.Aiming at reducing the cable response directly, the present authors extended the FPM for optimizing VIMD parameters for a taut cable and analyzed the cable mode-split phenomenon in detail [44].Nevertheless, the experimental observations of the mode-split phenomenon and the fxed points on FRF curves are rarely reported, particularly the concerning cables.
Te eddy current damper (ECD) was developed and investigated in the past half century because of its advantages of insensitivity to temperature and its noncontact nature.Its feasibility and control performances were studied numerically and experimentally for cantilever beams [45][46][47], vehicle suspension [48], an SDOF structure [49], and a robotic milling machine [50].Recently, many novel ECDs combining with the ID have been proposed, because they are efective at emulating viscous dampers and have only small friction.Nakamura et al. [51] studied and proposed an inerter-based ECD named electromagnetic inertial mass damper (EIMD), where its mechanism is similar to a VIMD.Li et al. [36] and Zhu et al. [52] designed an EIMD prototype and investigated its control performance on a 135 m long cable.Gao et al. [53] proposed a novel negative stifness inerter damper (NSID) consisting of a magnetic negative stifness device and an inerter-combined ECD.Wang et al. [54] developed an eddy current inertial mass damper (ECIMD) with a rotational ECD and a ball screw, and an experimental investigation for cable vibration mitigation using two ECIMDs was carried out.
In this paper, we propose a VIMD prototype with adjustable damper parameters (inertial mass and damping).In it, an ECD is used to emulate the mechanical behavior of the viscous damper.Te characteristics of the VIMD prototype were analyzed by a series of performance tests.An experimental study was carried out on a sagged scaled cable for various cases with the proposed VIMD for vibration mitigation.A series of vibration tests were conducted with diferent damper parameters.Numerical simulation analysis was also conducted for a cable-VIMD system, and the results were compared with the experimental results.We analyzed the characteristics of the cable-VIMD system, such as the cable responses and modal damping ratios, and we investigated the infuence of sag on the control performance.Te frequency-response functions (FRFs) obtained experimentally from the responses of the cable-VIMD system were analyzed in depth, and they revealed a mode-split phenomenon and two fxed points as in the theory of the FPM for the optimum design of a TMD.Te performance of the optimum VIMD by the FPM was also investigated in comparison with that by the maximum modal damping method [29].
Tis paper is organized as follows.In Section 2, the VIMD prototype and the performance test are introduced.Section 3 establishes the model of the cable-VIMD system and briefy analyzes its modal characteristics.Performance of the two optimum VIMD designs is briefy discussed.Te scaled-cable experiment is presented in Section 4, where the experimental results are analyzed with respect to the modal damping ratio, sag efect, and frequency-response function curve in detail.Section 5 summarizes the main conclusions of the work.

Confguration of VIMD.
In this study, a parameteradjustable VIMD was designed and manufactured.Te confguration of the damper prototype is shown in Figure 1.Te VIMD consists of a viscous damping part and an inertial mass part.Te damping part is essentially a planar eddy current damper that is comprised of a copper plate and two electromagnets with opposite magnetic poles, and the inertial mass part comprises a ball screw and a fywheel with four weights.Te VIMD has two terminals as shown in the fgure.Te connecting terminal is used to connect with cables, and the damper frame, as a fxed terminal, is usually connected to a support frame that is fxed to the bridge deck or ground.Te cable vibration leads a linear reciprocating movement of the conductor plate in the magnetic feld, which generates an electromagnetic viscous damping force.Meanwhile, the linear movement of the screw rod is converted to a rotational movement of the fywheel through the ball nut, which can provide a large inertial mass.
According to Bae et al. [45], the electromagnetic viscous damping force f ecd can be expressed as the following equation: where v is the moving velocity of the copper plate, c e is the electromagnetic viscous damping coefcient, α is a geometric factor, S is the projection area of the copper plate in the magnetic feld, t is the thickness of the copper plate, σ is the conductivity of copper, and B is the magnetic fux density.Te magnetic fux density can be adjusted by giving the electromagnets diferent current inputs I, which are approximately obtained according to Sodano and Inman with the Biot-Savart Law [55].In this study, we simulated and analyzed the planar eddy current damper by an FE model using the ANSYS-EMAG module.Te electromagnetic viscous damping coefcient was experimentally obtained and verifed using a performance test.
Based on the mechanism of a ball screw, the amplifed inertial mass m e can be calculated by the following equation: where m i is a constant value comprised of the initial mass of the sensors, screw rod, and copper plate and the inertial mass of the ball nuts, drum, and spokes; r is a converting factor determined by the screw pitch; I f is the moment of inertia of the fywheel; and r 1 , r 2 , h, and m are the inner radius, outer radius, height, and mass of the weight, respectively.R is the drum radius and d determines the weight position.Te inertial mass m e can be adjusted by changing the weight position d.
A support spring with a constant stifness k n is designed to avoid an initial cable deformation due to m i as shown in Figure 1.Hence, the theoretical damper force of the prototype can be expressed as follows: where x, _ x, and € x are, respectively, the damper displacement, velocity, and acceleration.

Performance
Tests on the VIMD.Initially, a performance test is conducted on the VIMD as shown in Figure 2(a).One terminal of the VIMD was connected to an actuator and the other to the support frame.One force sensor and one position sensor were employed to measure the damper force f d and damper displacement x .A sinusoidal excitation was given to the prototype by the actuator, and thus the velocity and acceleration can be obtained numerically by taking the derivatives of x.Based on the measured x and the calculated _ x and € x, the unknown parameters in equation ( 3) (m e and c e ) can be identifed using a curve ftting method, to make the damper force calculated by the equation (3) consistent with the measured experimental f d .However, the friction between the ball nut and the screw rod will afect the accuracy on the identifcation of parameters due to the small size of the prototype.Terefore, a modifed Bingham model [56] with considering the friction was used to identify the damper parameters from the measured results as follows: where f fr (t) is the friction term; f c means the friction; _ x 0 is a regularization parameter that has a velocity dimension and controls the exponential growth of the damping force.k n is zero because no support springs were used in the performance test.A least-square method was utilized, and the unknown parameters (m e , c e , f c , and _ x 0 ) can be calculated by ftting the damper force calculated by equation (4) to the measured results of f d (t), wherein a nonlinear least-squares function of lsqcurveft in MATLAB was used.Ten, the friction term f fr (t) in equation (4a) can be linearized as follows: where c f is the linearized friction damping coefcient and e(t) is the linearized error.One case of the identifed results is shown in Figures 2(b)-2(e), wherein no weights were used and the ECD was not activated.Te identifed inertial mass is m e � 16.07 kg, which is close to the true value: m i � 15.38 kg.Te identifed c e , f c , and corresponding c f are 0.01 Ns/m, 2.42 N, and 56.8 Ns/m.Strong agreement can be observed between the experimental results (blue curves) and the ftted damping forces (red curves).Te force-velocity relationship shows good applicability of the modifed Bingham model to this VIMD, and the negative stifness property can be observed clearly in Figure 2(c).

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where Y i is the identifed m e and  Y i is the theoretical value.Te test results using the smaller weights are shown in Figure 3. Te left and right vertical axes denote the inertial mass value and nRMSE value, respectively, and the horizontal axis is for weight position d.We have found that the test results strongly match the theoretical value in equation ( 2), which verifes the feasibility of the VIMD for adjusting m e .In most cases, the nRMSE values are smaller than 5%, which means that the damper provides an accurate m e .

Viscous Damping Coefcient.
Te parameters of the planar ECD are shown in Table 2. Te FE model for a quarter of the VIMD is shown in Figure 4(a).3D elements with 20 nodes (Solid236) in ANSYS were employed for simulating the copper plate.Te total number of FE elements was 47,240.Both edge-fux degree of freedom (AZ) and electric degree of freedom (VOLT) were considered.By giving a moving velocity to the conductor and diferent inputcurrent values to the electromagnets, the electromagnetic c e was evaluated based on the Lorentz force (using command EMFT).Te simulation results of the fux density are shown in Figure 4(b), and the relationship between the equivalent c e and the current is obtained by a regression analysis as shown in Figure 4(c), c e � 3.08•I 2 .
Four groups of tests were carried out with two inertial mass values (4.88 and 45.98 kg) and two excitation frequencies (1.31 and 1.94 Hz).We note that the ball nut and fywheels were not used in the present cases of m e � 4.88 kg, which contain only the mass of the copper plate, screw rod, and force sensor.Te test results are ftted to a parabolic equation as shown in Figure 4(c).We can observe that the test results show good accuracy for small current but are larger than the ANSYS results for larger current.Tis is mainly due to the two electromagnets attracting each other as the magnetic force increases.Te damper frame had a small deformation, and the two electromagnets moved closer to the copper plate.Tis increases the magnetic fux between the magnets, thus making the test results larger than the simulation results.Two regression relationships between the damping and input current, the blue curve and black curve as shown in Figure 4(c), showed good agreement with diference of 10%, whereas the diference was less than 4% in the low current range.Te regression curve, the blue curve, based on the test results is used in the present study with c e � 3.408•I 2 .Hence, a theoretical maximum damping coefcient is 218.1 Ns/m with considering the safe

Sagged Cable-VIMD System
3.1.Model of Cable-VIMD System.Referring Figure 5, the current sagged cable-damper system can be modeled by a nondimensional equation of motion as follows [60][61][62]: where w(x, t) is the nondimensional displacement of the cable perpendicular to the chord in the vertical plane from the initial static sagged state; x is the nondimensional position along the chord (0 ≤ x ≤ 1); c is the cable inherent damping; p(x, t) is the external vertical force on the cable; and F d (t) is the force from the VIMD at x � x d .Te nondimensional quantities are related to their dimensional counterparts, shown with overbars, according to the following relations: in which ρ, L, and T are the cable mass per unit length, cable chord length, and cable tension in the chord direction; is the frst natural frequency of the cable without a VIMD, computed without considering the sag efect.
Te nondimensional parameter λ 2 for the sag efect is defned as follows [63]: in which θ and L e are the cable inclination angle and the stretched cable length and E, A, and g are Young's modulus of the cable, area of the cable cross section, and gravitational acceleration, respectively.By introducing a section-wise linear defection shape of the cable on both sides of the VIMD and a series of sinusoidal shapes on the whole cable section, the initially assumed shape functions of the cable are taken as follows [60]: where H(x − x d ) is a Heaviside function.In this study, 20 sinusoidal shape functions were used.By using the shape functions in equation ( 8), the cable response w(x, t) can be approximately expressed as follows: where q i (t) is the generalized coordinate.Ten, the equation of motion, equation (6), can be discretized into a matrix form as follows: where M, C, and K are the generalized mass, damping, and stifness matrices, which are related to the assumed shape functions ϕ i (x) in equation (8).In reference [60], f is the generalized external load vector:

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where the external force on the cable is assumed as p(x, t) � p 0 (x)f t (t).
Te damper force of the VIMD at location x � x d can be expressed as follows: where m e , c d , and k n are the nondimensional damper parameters defned as follows: in which c d � c e + c f .Using equations ( 10) and ( 12), it can be rewritten as follows: Ten, the state-space expression can be obtained as follows [29,60]: where z is the state vector, A s is the state matrix, and B s is the input matrix.

Analysis of Modal Damping
Ratio.Conjugate pairs of the complex eigenvalues, Λ j and Λ * j , can be obtained from the state matrix A s and is expressed as follows: where ω j and ξ j are the jth natural frequency and modal damping ratio and i is the imaginary unit.Tus, the modal damping ratio ξ j can be obtained as follows: Figure 6 shows two 3D-plots for ξ 1 and ξ 2 with respect to the damper parameters (m e and c d ; k n � 0 and x d � 0.021) in two diferent sag conditions (λ 2 � 3.9 and 6.6).Te fgures show that, for a specifed damper parameter (m e or c d ), ξ j always increases frst and then decreases with the other increasing parameter (c d or m e ).Tus, there exists a theoretical maximum modal damping ratio at a set of damper parameters.Te maximum ξ 1 is obtained as 8.29% and 7.25% for two λ 2 values, as shown in Table 3, which are 12 and 14 times than those obtained by optimal viscous dampers (VD) [64].Moreover, we can see that the maximum ξ 1 is signifcantly reduced with the growing sag parameter.ξ 1 decreases from 8.29% to 7.25% as λ 2 increases from 3.9 to 6.6, respectively.However, the maximum ξ 2 is only slightly afected by the sag efect: 9.36% and 9.35% for two λ 2 .Two 3D graphs of ξ 2 are almost the same for two λ 2 as shown in Figure 6(b).

Characteristics of a
Cable with a VIMD.Te present authors proposed an optimal design procedure for a VIMD on a taut cable using the fxed-point method (FPM: Reference [44]).Te FPM was originally proposed for optimizing a tuned mass damper (TMD) for ordinary structures by Den Hartog [37], where the mode-split phenomenon and two fxed points on the frequency response function (FRF) curves were analyzed.In this study, the FPM is extended for a sagged cable, and the typical characteristics are briefy discussed.Based on equations ( 9), ( 11), (14), and (15), the FRF of the cable response at x under a harmonic excitation, p(x, t) � p 0 (x)f t (t) with f t (t) � e iωt , can be expressed using the state-space formulations as follows: where f 0 is the modal force coefcient vector associated with the shape functions ϕ(x) and the load distribution p 0 (x) as in equation ( 11 where ψ and ψ * are the corresponding conjugate pairs of eigenvectors, which represent the enhanced complex modes of cable vibration from the assumed shape functions ϕ i (x) in equation (8).Assuming λ 2 � 3.9 and x d � 0.021 with a concentrated harmonic actuation p(x, t) � δ(x − x a )e iωt at x a � 0.99, Figure 7(a) shows the FRF curves of the cable with diferent VIMDs shown in Section 4, at the cable midpoint obtained for various values of m e and c d using equation (18).In the fgure, the blue curves are with zero damping and the red curve is with infnite damping.Te blue dashed curve is for the cable without the VIMD, which shows the original frst cable frequency (ω 1 �1.148 ω 0 ).It is interesting to observe that the blue solid curve (m e � 3.528) has two resonant frequency peaks, ω 1−1 � 1.074 ω 0 and ω 1−2 � 1.263 ω 0 .Te blue dot curve and the red curve are the same although the damper parameters are diferent.Tis shows that the original frst cable mode splits into two modes by introducing a proper inertial mass, and that the cable is clamped at the damper location when c d or m e approaches infnity.Figure 7(b) shows four FRF curves at the cable midpoint for various c d values with m e � 3.528.It clearly shows that all the four curves pass through two points, P and Q, as in the theory of the FPM, which shows that the locations of P and Q are independent of the VIMD damping when m e is fxed.It can be also seen that the responses around the frst resonant frequency (ω 1 ) can be signifcantly suppressed with a properly selected damping, e.g., c d � 0.88, so that the optimum VIMD design may be obtained using the FPM.Te main idea for the optimum design in the FPM is to fnd the VIMD parameters (m e and c d ) so that the FRF curves may have maxima with the same amplitude at the two fxed points (P and Q).Te detailed optimization procedures can be found in reference [44].

Comparison of Two Optimum VIMD Design Methods.
Two optimum VIMD design methods were introduced in the previous sections.One is by the maximum modal damping ratio as in Figure 6, and the other is by the FPM as illustrated by VIMD-1 in Figure 7. Figure 8 shows the comparisons of the cable responses computed for diferent types of excitations, wherein uniformly distributed sinusoidal excitations are used in Figures 8(a) and 8(b), and distributed random excitations are in Figures 8(c) and 8(d).Two kinds of VIMDs with diferent parameters were considered, where VIMD-1 is the optimum design to the frst mode obtained by the FPM (m e � 3.528, c d � 0.88, and ξ 1 � 5.32%) and VIMD-2 is the one by the maximum modal damping ratio (m e � 3.528, c d � 1.426, and ξ 1 � 8.29%) as shown also in Figure 7(b).Tree diferent excitation frequencies were used for the sinusoidal excitation, which were Ω � 1.164, 1.232, and 1.35.For Ω � 1.164, the maximum nondimensional response amplitude, w in equation ( 6), using VIMD-1 is 0.048 in Figure 8(a), whereas the result using VIMD-2 is 0.075.However, the amplitudes using two

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VIMDs become nearly the same for Ω � 1.232 as expected in Figure 7(b).Te amplitude using VIMD-1 is slightly larger than the result using VIMD-2 when Ω � 1.35, which are 0.0218 and 0.0206, respectively.Figure 8(b) shows the root mean-square (RMS) values of the nondimensional displacement along the cable.It is obvious that the cable responses using VIMD-2 is considerably larger than those using VIMD-1 when Ω � 1.164, although VIMD-2 provides higher modal damping ratio (ξ 1 � 8.29%) than the one (ξ 1 � 5.32%) by VIMD-1.Te vibration amplitudes then become similar with the increasing Ω as expected by Figure 7(b).Te diferences of the cable responses at the midpoint are 5.63% and 5.78% for Ω � 1.232 and 1.35, respectively.Figure 8(c) shows the nondimensional cable responses subjected to distributed random excitations of white noises.Figure 8(d) shows the mean RMS values of the nondimensional cable responses considering 20 sets of diferent distributed random excitations, which shows that the control performances of two VIMDs are very similar under random excitations, where the reduction of cable responses using VIMD-1 is slightly better around the cable midpoint.Experimental validations of the abovementioned comparison on the two methods are provided in Section 4. It is to be noted that the optimum design method by the maximum modal damping method can be used more conventionally in practice, because it provides a VIMD that is independent of the response location, whereas the optimum VIMD by the FPM depends on the response location.

Scaled-Cable Experiments and Performance Evaluation
Experiments are carried out on a scaled cable with the parameters shown in Table 4. Figure 9 shows the experimental setups.Te actuator and the damper were installed at 1% and 2.1% of cable length L near the two ends.A series of experiments was carried out for two sag conditions, i.e., λ 2 � 3.9 and 6.6.Te frst two cable modes were obtained for each sag condition with various damper parameters.Te measured natural frequencies (ω 1 and ω 2 ) for the frst two cable modes without a VIMD in two sag conditions are also shown in Table 4, which are found to be very close to the theoretical values.Te natural frequencies (ω 0 ) computed considering the cable tension only without the sag efect are smaller than the frequencies including the sag efect.Te maximum modal damping ratios and the corresponding damper parameters theoretically obtained by the state-space formulation [29,60] are given in Table 5, which indicates that the modal damping ratios can be increased to a range of 6.1-8.5% by introduction of a VIMD.A sinusoidal excitation with ω e close to ω 1 or ω 2 was applied to the cable.Te electromagnets were then activated after the vibration reached a steady state, and the excitation was removed after reaching another steady state.Te cable displacements at the midpoint (for ω e near ω 1 ) and the quarter point near the bottom anchorage (for ω e near ω 2 ) were measured using noncontact measurement with a digital camera.Te modal damping ratios were identifed from the free vibration part.Te damper force and the damper displacement were recorded by a force sensor and a position sensor, respectively.

Damper Parameter Identifcation.
Figure 10(a) shows the experimental time histories of the damper force and the damper displacement (for λ 2 � 6.6).It is to be noted that the results shown in Figure 10 were measured from a cable-VIMD system, whereas the data of Figures  Teoretically, the friction should be identical for all experimental cases.However, it was changed to a small degree for diferent inertial mass values because the weight positions were manually adjusted.Figures 10(c) and 10(d) show comparisons between the experimental results and the ftted results using the identifed parameters.Te ftted damper force strongly matches the measured counterpart.

Analysis of Modal Damping Ratio.
In this section, we discuss mainly the experimental results of λ 2 � 3.9, while the results for a diferent sag (λ 2 � 6.6) are shown in the next  Structural Control and Health Monitoring section.Figure 11(a) shows the free vibration response at the midpoint without a VIMD after being excited with ω e � ω 1 � 1.333 Hz.Te envelope curve gives the cable inherent modal damping ratio: ξ � 0.24%.It is to be noted that this inherent modal damping ratio includes the infuence of the actuator, which is very small compared with that provided by the damper.Tis inherent modal damping ratio is simulated as a mass proportional damping matrix C in equations ( 10) and ( 14) and included the damping parameters shown in Table 5. Figure 11(b) shows the Fourieramplitude-spectrum (FAS) results with and without a damper.We can see that the response amplitude is signifcantly reduced by the VIMD, and the cable frequency is slightly enlarged.5. Te envelope curve of cable responses H(t) was obtained using the Hilbert-transform, and the modal damping ratio ξ was obtained by taking the slope of the logarithm of H(t) in the free decaying part.Figure 12(b) shows the response at L/4 (after being excited at ω e � 2.29 Hz) with the damper parameters optimized to the second mode (by maximizing ξ 2 ), where the identifed values are m e � 21.4 kg and c d � 113.4 Ns/m, which are also close to the target values (21.4 kg, 122.2 Ns/m).Te identifed modal damping ratios are 6.98% and 8.15% for the two modes, which are very close to the theoretical maximum values (7.23% and 8.52%).Te identifed damping ratios are found to be 10 and 7.5 times of those using a traditional VD shown in Table 3. Figures 12(c) and 12(d) give comparisons between the theoretical damping ratios (by the state-space method) and the experimental values.Strong agreement can be found between the two results.Te results show that there exists a pair of m e and c d that provides a maximum modal damping ratio for each mode.

Analysis of Sag Efect.
To study the sag efect on the control performance, experiments were also conducted on a cable with λ 2 � 6.6.Te cable tension T has been changed from 1497.5 N to 1266.7 N, as shown in Table 4.Note that the load cell gives the tension at the bottom anchorage.Figure 13(a) shows an example time history of the cable response at L/4 with excitation frequency ω e � 2.33 Hz, where the electromagnets were turned on at 3 s and the excitation was removed at 40 s.Te identifed parameters were m e � 22.6 kg, c f � 47.4 Ns/m (0-3 s), and c d � 112.5 Ns/ m (3-50 s), respectively.Te vibration was suppressed in 2 s after the excitation was removed, and the identifed ξ 2 was 7.34%, where the theoretical maximum value was 8.34% as in Table 5. Figures 13(b) and 13(c) show comparisons between the theoretical modal damping ratios and the experimental results for the various VIMD parameters (m e and c d ), wherein good agreements can be generally found.Compared with the case of λ 2 � 3.9 (Figures 12(c) and 12(d)), it can be observed that both m e and c d , corresponding to the maximum point of the frst modal damping ratio ξ 1 , reduced considerably, whereas those of the second mode remained almost the same, as expected by Figure 6. Figure 13(d) shows the theoretical maximum damping ratios (ξ 1 and ξ 2 ) and the experimental maximum results for two sag conditions.Te solid curves represent the theoretical value and the dashed curves with the markers "○" represent the experimental values.Te experimental maximum value of ξ 1 is reduced from 6.98% (λ 2 � 3.9) to 5.27% (λ 2 � 6.6), whereas ξ 2 decreases from 8.15% to 7.34%.It can be observed that the vibration-reduction performance of the frst cable mode is signifcantly afected by the sag, whereas the second mode is   Structural Control and Health Monitoring not.Larger diferences are observed between the experimental and theoretical results for the maximum modal damping ratios of the second modes in the low range of c d , which might be caused by the identifcation error in the data process.However, the changes in the cable responses are not always the same for all the cases.Comparing the cable displacement responses at L/4 in Figures 13(a) and 14(a), diferent trends in the response amplitude changes (before/ after the activation of the electromagnets) can be found in two sag conditions, although the excitation is almost the same.Te amplitude increases from 20.1 mm to 42.3 mm for λ 2 � 3.9 after activating the electromagnets but decreases from 44.6 mm to 22.8 mm for λ 2 � 6.6.Te identifed ξ 2 is 2.7%, 8.15%, and 3.47% for c d � 62.2, 113.4,and 189.2 Ns/m with λ 2 � 3.9 in Figure 14(b).Te blue curve with the smallest ξ 2 gives the smallest cable response amplitude at ω e � 2.3 Hz, whereas the green curve with the largest c d gives the largest cable response.

Analysis of Maximum
Te abovementioned phenomena can be explained through the analysis of the FRF curves.Figure 15 shows FRF curves at L/4 obtained for various parameters using equation (18).Te solid curve is without a VIMD, the rest of the curves are with a VIMD using diferent values of m e and c d considered in the experimental cases in Figure 14.It can be seen that the peak frequencies without a VIMD are almost the same as those from the experiments on the scaled cable shown in      Structural Control and Health Monitoring 2.153 Hz, which is slightly higher than original frequency.It can be also observed that all the experimental FAS curves pass through two fxed points (P and Q), regardless of the c d value, as expected.
Figure 18 shows the FAS curves of the responses at L/2 under the same sine sweep excitation for the case of λ 2 � 3.9 and m e � 66.1 kg.Te same phenomenon of mode splitting and two fxed points can be clearly observed.

Conclusions
In this study, a VIMD prototype with adjustable damper parameters was developed.Performance tests were carried out on the VIMD and a scaled cable with the damper for the vibration-mitigation efect.Te experimental results were systematically analyzed along with the simulated results by FEM, and the conclusions are summarized as follows: (1) Te damper parameters of the proposed VIMD prototype are continuously and easily adjustable.Te inertial mass can be adjusted by moving the weights, and the electromagnetic viscous damping coefcient is adjusted by changing the input current to the electromagnets.Te results of the performance tests show strong agreement with those from the theory and the simulation.Good feasibility and accuracy were found on the VIMD parameter adjustment.
(2) Te modal damping ratios of the frst two cable modes are remarkably enhanced with the installing of the VIMD.Te identifed maximum modal damping ratios of the frst two modes are 6.89% and 8.15%, which are almost 10 and 8 times the maximum values of those using a viscous damper.Tose results are one of the highest damping ratios obtained in experiments with a damper location at 2.1% so far.Besides, we fnd that there is one maximum modal damping ratio for each mode with a set of the damper parameters (m e and c d ), as in the theory.(3) Te frst modal damping ratio is signifcantly reduced due to the larger sag efect.For example, the identifed maximum frst modal damping ratio decreases by 24.5% when λ 2 increases from 3.9 to 6.6.However, the maximum second modal damping ratio is reduced by only 9.9%.Te VIMD parameters corresponding to the maximum frst modal damping ratio also reduce, but those for the second mode do not.(4) Te cable mode is split into two modes by introducing a VIMD with a proper inertial mass.For   Structural Control and Health Monitoring a selected inertial mass value, two split frequencies move closer to each other with the increasing damping coefcient.Te two frequencies converge to one frequency, as the damping approaches infnite.Besides, two fxed points are observed on the FAS results of the cable responses subjected to sinusoidal sweep excitations.For a selected inertial mass value, all the FRF curves pass through the two fxed points regardless of damping, as in the fxedpoint method (FPM).Tis can be used to obtain the optimum VIMD design.(5) Te optimum VIMD by the FPM may provide considerably smaller modal damping ratio than the one by the maximum modal damping method.But the FPM may give signifcantly smaller cable responses particularly for harmonic excitations near the resonant frequency, though it is expected to give very similar RMS responses for wide-banded random excitations.
Further studies are suggested for the full-scale experimental validation with considering higher modes and the multimode cable vibration control using multiple VIMDs.

Figure 3 :
Figure 3: Test and simulation results of inertial mass with the small weights at various positions (d).

Figure 4 :
Figure 4: Test and simulation results of electromagnetic viscous damping coefcient.(a) FE model of planar ECD: a quarter model.(b) Simulation results of magnetic fux density.(c) Comparison of test and simulation results.

Figure 6 :
Figure 6: Variation of the modal damping ratio with respect to VIMD parameters under λ 2 � 3.9 and 6.6.(a) First cable mode.(b) Second cable mode.

Ω = 1 Figure 8 :
Figure 8: Comparison of simulated cable responses by two optimum VIMDs (red: VIMD-1 with m e � 3.528 and c d � 0.88; blue: VIMD-2 with m e � 3.528 and c d � 1.426).(a) Displacements w under uniformly distributed sinusoidal excitations at L/2.(b) RMS values of displacements w along the cable under uniformly distributed sinusoidal excitations.(c) Displacements w at L/2 under distributed random excitations.(d) RMS values of displacements w under distributed random excitation.

Figure 12 (
a) shows the vibration response at L/2 (after being excited at ω e � 1.414 Hz) with the damper parameters optimized to the frst mode (by maximizing ξ 1 ), where the identifed values are m e � 66.1 kg and c d � 164.2 Ns/m, which are very close to the target values (66.4 kg, 168.1 Ns/m) shown in Table
Responses and FRF Curves 4.4.1.Responses to Harmonic Excitations.Figure 14(a) shows the typical time-history responses of the damper force, the damper displacement, and the cable displacement at L/4 in the case: ω e � 2.3 Hz, λ 2 � 3.9, m e � 21.4 kg, and c d � 189.2 Ns/m.Te exciting frequency ω e was taken to be close to ω 2 � 2.314 Hz.Te external excitation was introduced at 7.5 s, and the electromagnets were activated at 40 s.Tis means that c d � c f equals 40.3Ns/m (c e � 0) during the period of 0-40 s, and it becomes 189.2 Ns/m (c e � 148.9 Ns/m) after 40 s, whereas the inertial mass remains constant as m e � 21.4 kg.Ten, the external excitation was removed at 70 s.It can be seen in Figure 14(a) that both the damper force and displacement responses decrease after the activation of the electromagnets, whereas the cable displacement at L/4 increases.Comparisons of the time-history responses with the same ω e and m e but for three diferent c d Logarithm of |H (t

Figure 16 (
a) shows three experimental displacement time histories at 0.5 L with m e � 66.1 kg, ω e � 1.414 Hz, and λ 2 � 3.9.Te total damping coefcient (c d � c e + c f ) is 82.7, 121.1, and 164.2 Ns/m, for those cases, which are the frst, second, and fourth yellow points in Figure 12(c).Te corresponding modal damping ratio ξ 1 increases with the increasing damping as in Figure 12(c).However, the steady state amplitudes of those three cases are almost the same as in Figure 16(a).

Figure 16 (Figure 14 :
Figure 16(b) shows three other experimental results at 0.25 L with the same m e � 22.6 kg, ω e � 2.33 Hz, and λ 2 � 6.6, but diferent c d � 47.4, 112.5, and 152.9 Ns/m, which are the frst, highest, and last yellow points in Figure 13(c).Te

Table 1 :
Parameters of single weight.
[57][58][59]ontrol and Health Monitoring current-carrying capacity of the coil (max.I � 8 A).In this performance test, the supply current to the electromagnets is amplifed by the AE Techron 7224 and set in the range of [0-6] A to avoid an overheat of the coil, and the corresponding adjustment range of damping coefcient is thus [0-122.7]Ns/m.Te test results also demonstrate that the varying excitation frequencies have little efect on the damper parameters in low frequency and velocity ranges as indicated by references[57][58][59].

Table 4 :
Parameters for two sagged scaled cables without a VIMD.

Table 5 :
Maximum ξ and corresponding theoretical damper parameters by the state-space formulation.
Te VIMD parameters in Table5are diferent from those in Table3due to k n of the additional spring in the VIMD prototype.

Table 4 .
Te original second mode (the solid curve with ω 2 � 2.236 Hz) has split into two, as in the dashed curve with ω 2−1,2 � 2.194 and 2.549 Hz shown in Figure 15, after introducing a VIMD with m e � 21.4 kg and (18)2.5 Ns/m with the highest modal damping ratio, ξ 2 � 7.34%, is close to the response for c d � 152.9 Ns/m with ξ 2 � 3.63%, where it is smaller than the response for c d � 47.7 Ns/m with ξ 2 � 2.15%.Tese diferent trends of the cable responses can be explained by the FRF curves obtained from equation(18)shown in Figures16(c) and 16(d).Te excitation frequency (ω e � 1.414 Hz) close to the one of the fxed points Q (ω Q �1.418 Hz as shown in Figure 16(c)) leads the similar vibration amplitudes in Figure 16(a).
Figure 16(d)indicates that the cable responses for a small c d � 47.4 Ns/m with the smallest ξ 2 (2.15%) at ω e � 2.33 Hz is to be higher than the other two cases with larger ξ 2 (7.34% and 3.63%), wherein the latter two cases have very similar FRF amplitudes.Te abovementioned results reveal that a higher modal damping ratio does not guarantee a lower cable vibration amplitude.2 � 2.13 Hz) has split into two (1.963 and 2.307 Hz) in the case of c d � 25.3 Ns/m.As c d increases, the amplitudes of the two split modes are frst reduced, and the two peaks move closer to each other.Ten, the two split modes become one mode, and the amplitude keeps growing with the increasing c d .Te peak frequency for c d � 144.1 Ns/m is obtained as