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

Authors: Shenghao Dong; Yuanfeng Duan; Shilang Xu; Chungbang Yun
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

In this paper, we describe the design and manufacturing of a viscous inertial mass-damper (VIMD) prototype. The damper parameters, the inertial mass, and the viscous damping coefficient can be continuously adjusted. Experimental validations were carried out on the VIMD with various parameters and a scaled sagged cable of 19.47 m long with the VIMD for vibration mitigation, along with a simulation study using the finite element method (FEM). The experimental results show that the modal damping ratio of the scaled cable was remarkably increased by the VIMD. The achieved maximum modal damping ratios for the first two modes were 6.98% and 8.15%, respectively, which are about 10 times the maximum values using a conventional viscous damper. The sag influence on damper-control performance was also studied by changing the cable tension. It showed that the first modal damping ratio provided by the VIMD was significantly reduced by the increased sag, whereas the second modal damping ratio was not. Fourier amplitude spectra of the cable responses subjected to sine sweep excitations show that the cable mode split into two and a pair of fixed points occurred by introducing a VIMD as in the theory of the fixed-point method (FPM) for the optimum design of a tuned mass damper. The performance of the optimum VIMD by the FPM was investigated in comparison with that by the maximum modal damping method.


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## Body

## 1. Introduction

Cables are widely used in cable-supported bridges and long-span structures due to their economy, aesthetic effect, and light weight. However, cables are prone to severe vibration because of their low inherent damping and their high flexibility, 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 stiffness, such as negative stiffness damper (NSD) and inerter damper (ID), has been widely studied by many researchers [1–3]. The damper characteristics and control performance have been investigated under random excitations [4, 5], pulse-like excitations [6], and earthquake excitations [7–10]. Furthermore, the effectiveness of the negative stiffness devices has been systematically analyzed for high buildings [11–16], bridges [17–19], cables [20, 21], and ocean structures [22, 23]. Among them, the NSD and ID for cables have attracted strong attention because of their high efficiency. 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 refined design formula for a cable-NSD system, and the control performances of negative, zero, and positive stiffness dampers were investigated. A novel multimode control approach considering the effect of NSD stiffness 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 different 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 efficiency on the structural vibration mitigation by NSD and ID [31–33]. An experimental study on NSD for a taut scaled cable was reported by Shi et al. [34]. The 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 full-scale 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 efficiency 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 fixed-point method (FPM) was recently applied for optimizing inerter damper and negative stiffness 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 amplifier, which was used and optimized for an MDOF system with complex modal analysis. The 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 effective than the TMD with the same mass. Gonzalez-Buelga et al. [43] analytically and experimentally assessed a ball-screw-type TID. The 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 stiffness 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 fixed points on FRF curves are rarely reported, particularly the concerning cables.The 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 [45–47], 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 effective 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 stiffness inerter damper (NSID) consisting of a magnetic negative stiffness 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. The 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 different 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 influence of sag on the control performance. The 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 fixed points as in the theory of the FPM for the optimum design of a TMD. The performance of the optimum VIMD by the FPM was also investigated in comparison with that by the maximum modal damping method [29].This paper is organized as follows. In Section2, the VIMD prototype and the performance test are introduced. Section 3 establishes the model of the cable-VIMD system and briefly analyzes its modal characteristics. Performance of the two optimum VIMD designs is briefly discussed. The scaled-cable experiment is presented in Section 4, where the experimental results are analyzed with respect to the modal damping ratio, sag effect, and frequency-response function curve in detail. Section 5 summarizes the main conclusions of the work.


## 2. VIMD Prototype and Performance Tests

### 2.1. Configuration of VIMD

In this study, a parameter-adjustable VIMD was designed and manufactured. The configuration of the damper prototype is shown in Figure1. The VIMD consists of a viscous damping part and an inertial mass part. The 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 flywheel with four weights. The VIMD has two terminals as shown in the figure. The connecting terminal is used to connect with cables, and the damper frame, as a fixed terminal, is usually connected to a support frame that is fixed to the bridge deck or ground. The cable vibration leads a linear reciprocating movement of the conductor plate in the magnetic field, which generates an electromagnetic viscous damping force. Meanwhile, the linear movement of the screw rod is converted to a rotational movement of the flywheel through the ball nut, which can provide a large inertial mass.Figure 1
Configuration of the VIMD prototype.According to Bae et al. [45], the electromagnetic viscous damping force fecd can be expressed as the following equation:(1)fecd=cev,ce=αStσB2,where v is the moving velocity of the copper plate, ce is the electromagnetic viscous damping coefficient, α is a geometric factor, S is the projection area of the copper plate in the magnetic field, t is the thickness of the copper plate, σ is the conductivity of copper, and B is the magnetic flux density. The magnetic flux density can be adjusted by giving the electromagnets different 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. The electromagnetic viscous damping coefficient was experimentally obtained and verified using a performance test.Based on the mechanism of a ball screw, the amplified inertial massme can be calculated by the following equation:(2)me=mi+r2If,If=m3r12+r22+h212+mR+d2,where mi 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; If is the moment of inertia of the flywheel; and r1, r2, 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. The inertial mass me can be adjusted by changing the weight position d.A support spring with a constant stiffnesskn is designed to avoid an initial cable deformation due to mi as shown in Figure 1. Hence, the theoretical damper force of the prototype can be expressed as follows:(3)fd=mex¨+cex˙+knx,where x, x˙, and x¨ are, respectively, the damper displacement, velocity, and acceleration.


### 2.2. Performance Tests on the VIMD

Initially, a performance test is conducted on the VIMD as shown in Figure2(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 fd 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) (me and ce) can be identified using a curve fitting method, to make the damper force calculated by the equation (3) consistent with the measured experimental fd. However, the friction between the ball nut and the screw rod will affect the accuracy on the identification of parameters due to the small size of the prototype. Therefore, a modified Bingham model [56] with considering the friction was used to identify the damper parameters from the measured results as follows:(4)fdt=mex¨t+cex˙t+knxt+ffrt,(4a)ffrt=fc1−exp−sgnx˙tx˙tx˙0sgnx˙t,where ffrt is the friction term; fc means the friction; x˙0 is a regularization parameter that has a velocity dimension and controls the exponential growth of the damping force. kn is zero because no support springs were used in the performance test. A least-square method was utilized, and the unknown parameters (me, ce, fc, and x˙0) can be calculated by fitting the damper force calculated by equation (4) to the measured results of fdt, wherein a nonlinear least-squares function of lsqcurvefit in MATLAB was used. Then, the friction term ffrt in equation (4a) can be linearized as follows:(4b)ffrt=cfx˙t+et,where cf is the linearized friction damping coefficient and et is the linearized error.Figure 2
VIMD performance test. (a) Test setup35×35×115cm. (b) Force time history. (c) Force vs. displacement. (d) Force vs. velocity. (e) Force vs. acceleration.
(a)(b)(c)(d)(e)One case of the identified results is shown in Figures2(b)–2(e), wherein no weights were used and the ECD was not activated. The identified inertial mass is me = 16.07 kg, which is close to the true value: mi = 15.38 kg. The identified ce, fc, and corresponding cf 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 fitted damping forces (red curves). The force-velocity relationship shows good applicability of the modified Bingham model to this VIMD, and the negative stiffness property can be observed clearly in Figure 2(c).


#### 2.2.1. Inertial Mass

The parameters of the weights are shown in Table1. Two sizes of weight (small and large) were designed. A total of six different weight positions (d = 2.5, 7.5, 14.5, 17.5, 22.5, and 27.5 (mm)) were tested. Hence, the corresponding theoretical adjustment ranges of the total me are (20.6–45.79) kg and (34.35–107.62) kg. Two sinusoidal excitations with 1.31 Hz and 1.94 Hz, respectively, were used in the tests. The number of the presented tests was 3–6 for each d.Table 1
Parameters of single weight.
ComponentsSmall weightsLarge weightsSize (mm)r1×r2×h2×10×52×20×5Mass (kg)0.0110.032me with dmin=2.5mm (kg)1.3044.742me with dmax=27.5mm (kg)7.60123.06A normalized root mean-square error (nRMSE) value defined as in the following equation was used for quantifying the test results.(5)nRMSE=1/n∑i=1nYi−Y^i21/n∑i=1nY^i2,where Yi is the identified me and Y^i is the theoretical value.The test results using the smaller weights are shown in Figure3. The 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 verifies the feasibility of the VIMD for adjusting me. In most cases, the nRMSE values are smaller than 5%, which means that the damper provides an accurate me.Figure 3
Test and simulation results of inertial mass with the small weights at various positions (d).


#### 2.2.2. Viscous Damping Coefficient

The parameters of the planar ECD are shown in Table2. The 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. The total number of FE elements was 47,240. Both edge-flux degree of freedom (AZ) and electric degree of freedom (VOLT) were considered. By giving a moving velocity to the conductor and different input-current values to the electromagnets, the electromagnetic ce was evaluated based on the Lorentz force (using command EMFT). The simulation results of the flux density are shown in Figure 4(b), and the relationship between the equivalent ce and the current is obtained by a regression analysis as shown in Figure 4(c), ce=3.08∙I2.Table 2
Parameters of ECD.
ParametersValuesCoil size (r1×r2×h) (mm)21× 40 × 80Coil turns1500Plate size (H×W×D) (mm)300× 170 × 10Copper conductivity (σ) (S/m)5.8×107Gap (mm)1Figure 4
Test and simulation results of electromagnetic viscous damping coefficient. (a) FE model of planar ECD: a quarter model. (b) Simulation results of magnetic flux density. (c) Comparison of test and simulation results.
(a)(b)(c)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 flywheels were not used in the present cases ofme=4.88 kg, which contain only the mass of the copper plate, screw rod, and force sensor. The test results are fitted 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. This is mainly due to the two electromagnets attracting each other as the magnetic force increases. The damper frame had a small deformation, and the two electromagnets moved closer to the copper plate. This increases the magnetic flux 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 difference of 10%, whereas the difference was less than 4% in the low current range. The regression curve, the blue curve, based on the test results is used in the present study with ce=3.408∙I2. Hence, a theoretical maximum damping coefficient is 218.1 Ns/m with considering the safe current-carrying capacity of the coil (max. I=8A). In this performance test, the supply current to the electromagnets is amplified 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 coefficient is thus [0–122.7] Ns/m. The test results also demonstrate that the varying excitation frequencies have little effect on the damper parameters in low frequency and velocity ranges as indicated by references [57–59].


## 2.1. Configuration of VIMD

In this study, a parameter-adjustable VIMD was designed and manufactured. The configuration of the damper prototype is shown in Figure1. The VIMD consists of a viscous damping part and an inertial mass part. The 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 flywheel with four weights. The VIMD has two terminals as shown in the figure. The connecting terminal is used to connect with cables, and the damper frame, as a fixed terminal, is usually connected to a support frame that is fixed to the bridge deck or ground. The cable vibration leads a linear reciprocating movement of the conductor plate in the magnetic field, which generates an electromagnetic viscous damping force. Meanwhile, the linear movement of the screw rod is converted to a rotational movement of the flywheel through the ball nut, which can provide a large inertial mass.Figure 1
Configuration of the VIMD prototype.According to Bae et al. [45], the electromagnetic viscous damping force fecd can be expressed as the following equation:(1)fecd=cev,ce=αStσB2,where v is the moving velocity of the copper plate, ce is the electromagnetic viscous damping coefficient, α is a geometric factor, S is the projection area of the copper plate in the magnetic field, t is the thickness of the copper plate, σ is the conductivity of copper, and B is the magnetic flux density. The magnetic flux density can be adjusted by giving the electromagnets different 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. The electromagnetic viscous damping coefficient was experimentally obtained and verified using a performance test.Based on the mechanism of a ball screw, the amplified inertial massme can be calculated by the following equation:(2)me=mi+r2If,If=m3r12+r22+h212+mR+d2,where mi 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; If is the moment of inertia of the flywheel; and r1, r2, 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. The inertial mass me can be adjusted by changing the weight position d.A support spring with a constant stiffnesskn is designed to avoid an initial cable deformation due to mi as shown in Figure 1. Hence, the theoretical damper force of the prototype can be expressed as follows:(3)fd=mex¨+cex˙+knx,where x, x˙, and x¨ are, respectively, the damper displacement, velocity, and acceleration.


## 2.2. Performance Tests on the VIMD

Initially, a performance test is conducted on the VIMD as shown in Figure2(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 fd 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) (me and ce) can be identified using a curve fitting method, to make the damper force calculated by the equation (3) consistent with the measured experimental fd. However, the friction between the ball nut and the screw rod will affect the accuracy on the identification of parameters due to the small size of the prototype. Therefore, a modified Bingham model [56] with considering the friction was used to identify the damper parameters from the measured results as follows:(4)fdt=mex¨t+cex˙t+knxt+ffrt,(4a)ffrt=fc1−exp−sgnx˙tx˙tx˙0sgnx˙t,where ffrt is the friction term; fc means the friction; x˙0 is a regularization parameter that has a velocity dimension and controls the exponential growth of the damping force. kn is zero because no support springs were used in the performance test. A least-square method was utilized, and the unknown parameters (me, ce, fc, and x˙0) can be calculated by fitting the damper force calculated by equation (4) to the measured results of fdt, wherein a nonlinear least-squares function of lsqcurvefit in MATLAB was used. Then, the friction term ffrt in equation (4a) can be linearized as follows:(4b)ffrt=cfx˙t+et,where cf is the linearized friction damping coefficient and et is the linearized error.Figure 2
VIMD performance test. (a) Test setup35×35×115cm. (b) Force time history. (c) Force vs. displacement. (d) Force vs. velocity. (e) Force vs. acceleration.
(a)(b)(c)(d)(e)One case of the identified results is shown in Figures2(b)–2(e), wherein no weights were used and the ECD was not activated. The identified inertial mass is me = 16.07 kg, which is close to the true value: mi = 15.38 kg. The identified ce, fc, and corresponding cf 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 fitted damping forces (red curves). The force-velocity relationship shows good applicability of the modified Bingham model to this VIMD, and the negative stiffness property can be observed clearly in Figure 2(c).


### 2.2.1. Inertial Mass

The parameters of the weights are shown in Table1. Two sizes of weight (small and large) were designed. A total of six different weight positions (d = 2.5, 7.5, 14.5, 17.5, 22.5, and 27.5 (mm)) were tested. Hence, the corresponding theoretical adjustment ranges of the total me are (20.6–45.79) kg and (34.35–107.62) kg. Two sinusoidal excitations with 1.31 Hz and 1.94 Hz, respectively, were used in the tests. The number of the presented tests was 3–6 for each d.Table 1
Parameters of single weight.
ComponentsSmall weightsLarge weightsSize (mm)r1×r2×h2×10×52×20×5Mass (kg)0.0110.032me with dmin=2.5mm (kg)1.3044.742me with dmax=27.5mm (kg)7.60123.06A normalized root mean-square error (nRMSE) value defined as in the following equation was used for quantifying the test results.(5)nRMSE=1/n∑i=1nYi−Y^i21/n∑i=1nY^i2,where Yi is the identified me and Y^i is the theoretical value.The test results using the smaller weights are shown in Figure3. The 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 verifies the feasibility of the VIMD for adjusting me. In most cases, the nRMSE values are smaller than 5%, which means that the damper provides an accurate me.Figure 3
Test and simulation results of inertial mass with the small weights at various positions (d).


### 2.2.2. Viscous Damping Coefficient

The parameters of the planar ECD are shown in Table2. The 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. The total number of FE elements was 47,240. Both edge-flux degree of freedom (AZ) and electric degree of freedom (VOLT) were considered. By giving a moving velocity to the conductor and different input-current values to the electromagnets, the electromagnetic ce was evaluated based on the Lorentz force (using command EMFT). The simulation results of the flux density are shown in Figure 4(b), and the relationship between the equivalent ce and the current is obtained by a regression analysis as shown in Figure 4(c), ce=3.08∙I2.Table 2
Parameters of ECD.
ParametersValuesCoil size (r1×r2×h) (mm)21× 40 × 80Coil turns1500Plate size (H×W×D) (mm)300× 170 × 10Copper conductivity (σ) (S/m)5.8×107Gap (mm)1Figure 4
Test and simulation results of electromagnetic viscous damping coefficient. (a) FE model of planar ECD: a quarter model. (b) Simulation results of magnetic flux density. (c) Comparison of test and simulation results.
(a)(b)(c)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 flywheels were not used in the present cases ofme=4.88 kg, which contain only the mass of the copper plate, screw rod, and force sensor. The test results are fitted 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. This is mainly due to the two electromagnets attracting each other as the magnetic force increases. The damper frame had a small deformation, and the two electromagnets moved closer to the copper plate. This increases the magnetic flux 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 difference of 10%, whereas the difference was less than 4% in the low current range. The regression curve, the blue curve, based on the test results is used in the present study with ce=3.408∙I2. Hence, a theoretical maximum damping coefficient is 218.1 Ns/m with considering the safe current-carrying capacity of the coil (max. I=8A). In this performance test, the supply current to the electromagnets is amplified 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 coefficient is thus [0–122.7] Ns/m. The test results also demonstrate that the varying excitation frequencies have little effect on the damper parameters in low frequency and velocity ranges as indicated by references [57–59].


## 2.2.1. Inertial Mass

The parameters of the weights are shown in Table1. Two sizes of weight (small and large) were designed. A total of six different weight positions (d = 2.5, 7.5, 14.5, 17.5, 22.5, and 27.5 (mm)) were tested. Hence, the corresponding theoretical adjustment ranges of the total me are (20.6–45.79) kg and (34.35–107.62) kg. Two sinusoidal excitations with 1.31 Hz and 1.94 Hz, respectively, were used in the tests. The number of the presented tests was 3–6 for each d.Table 1
Parameters of single weight.
ComponentsSmall weightsLarge weightsSize (mm)r1×r2×h2×10×52×20×5Mass (kg)0.0110.032me with dmin=2.5mm (kg)1.3044.742me with dmax=27.5mm (kg)7.60123.06A normalized root mean-square error (nRMSE) value defined as in the following equation was used for quantifying the test results.(5)nRMSE=1/n∑i=1nYi−Y^i21/n∑i=1nY^i2,where Yi is the identified me and Y^i is the theoretical value.The test results using the smaller weights are shown in Figure3. The 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 verifies the feasibility of the VIMD for adjusting me. In most cases, the nRMSE values are smaller than 5%, which means that the damper provides an accurate me.Figure 3
Test and simulation results of inertial mass with the small weights at various positions (d).


## 2.2.2. Viscous Damping Coefficient

The parameters of the planar ECD are shown in Table2. The 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. The total number of FE elements was 47,240. Both edge-flux degree of freedom (AZ) and electric degree of freedom (VOLT) were considered. By giving a moving velocity to the conductor and different input-current values to the electromagnets, the electromagnetic ce was evaluated based on the Lorentz force (using command EMFT). The simulation results of the flux density are shown in Figure 4(b), and the relationship between the equivalent ce and the current is obtained by a regression analysis as shown in Figure 4(c), ce=3.08∙I2.Table 2
Parameters of ECD.
ParametersValuesCoil size (r1×r2×h) (mm)21× 40 × 80Coil turns1500Plate size (H×W×D) (mm)300× 170 × 10Copper conductivity (σ) (S/m)5.8×107Gap (mm)1Figure 4
Test and simulation results of electromagnetic viscous damping coefficient. (a) FE model of planar ECD: a quarter model. (b) Simulation results of magnetic flux density. (c) Comparison of test and simulation results.
(a)(b)(c)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 flywheels were not used in the present cases ofme=4.88 kg, which contain only the mass of the copper plate, screw rod, and force sensor. The test results are fitted 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. This is mainly due to the two electromagnets attracting each other as the magnetic force increases. The damper frame had a small deformation, and the two electromagnets moved closer to the copper plate. This increases the magnetic flux 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 difference of 10%, whereas the difference was less than 4% in the low current range. The regression curve, the blue curve, based on the test results is used in the present study with ce=3.408∙I2. Hence, a theoretical maximum damping coefficient is 218.1 Ns/m with considering the safe current-carrying capacity of the coil (max. I=8A). In this performance test, the supply current to the electromagnets is amplified 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 coefficient is thus [0–122.7] Ns/m. The test results also demonstrate that the varying excitation frequencies have little effect on the damper parameters in low frequency and velocity ranges as indicated by references [57–59].


## 3. Sagged Cable-VIMD System

### 3.1. Model of Cable-VIMD System

Referring Figure5, the current sagged cable-damper system can be modeled by a nondimensional equation of motion as follows [60–62]:(6)w¨x,t+cw˙x,t−1π2w″x,t+λ2π2∫01wζ,tdζ=px,t+Fdtδx−xd,where wx,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; px,t is the external vertical force on the cable; and Fdt is the force from the VIMD at x=xd. The nondimensional quantities are related to their dimensional counterparts, shown with overbars, according to the following relations:(6a)t=ω0t¯,x=x¯L,c=c¯ρω0,wx,t=w¯x¯,t¯L,δx−xd=Lδ¯x¯−x¯d,px,t=Lp¯x¯,t¯π2T,Fdt=F¯dt¯π2T,in which ρ, L, and T are the cable mass per unit length, cable chord length, and cable tension in the chord direction; ω0=π/LT/ρ is the first natural frequency of the cable without a VIMD, computed without considering the sag effect.Figure 5
A sagged cable with a VIMD.The nondimensional parameterλ2 for the sag effect is defined as follows [63]:(7)λ2=ρgLcosθTEALTLe,Le=L1+18ρgLcosθT2,in which θ and Le 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 deflection 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]:(8)ϕix=12+λ212+λ2−3λ2xd1−xdxxd+1−xxdHx−xd1−xd−3λ2x1−x12+λ2,i=1,sini−1πx,i≥2,where Hx−xd is a Heaviside function. In this study, 20 sinusoidal shape functions were used.By using the shape functions in equation (8), the cable response wx,t can be approximately expressed as follows:(9)wx,t=∑qitϕix=ϕTxqt,where qit is the generalized coordinate.Then, the equation of motion, equation (6), can be discretized into a matrix form as follows:(10)Mq¨+Cq˙+Kq=f+ϕxdFdt,where M, C, and K are the generalized mass, damping, and stiffness matrices, which are related to the assumed shape functions ϕix in equation (8). In reference [60],f is the generalized external load vector:(11)f=∫01ϕxpx,tdx=∫01ϕxp0xdxftt=f0ftt,where the external force on the cable is assumed as px,t=p0xftt.The damper force of the VIMD at locationx=xd can be expressed as follows:(12)Fdt=−m¯eϕTxdq¨+c¯dϕTxdq˙+k¯nϕTxdq,where m¯e, c¯d, and k¯n are the nondimensional damper parameters defined as follows:(13)m¯e=meLω02π2T,c¯d=cdLω0π2T,k¯n=knLπ2T,in which cd=ce+cf.Using equations (10) and (12), it can be rewritten as follows:(14)M+m¯eϕxdϕTxdq¨+C+c¯dϕxdϕTxdq˙+K+k¯nϕxdϕTxdq=Maq¨+Caq˙+Kaq=f.Then, the state-space expression can be obtained as follows [29, 60]:(15)z˙=q˙q¨=0I−Ma−1Ka−Ma−1Caqq˙+0Ma−1f=Asz+Bsf,where z is the state vector, As is the state matrix, and Bs is the input matrix.


### 3.2. Analysis of Modal Damping Ratio

Conjugate pairs of the complex eigenvalues,Λj and Λj∗, can be obtained from the state matrix As and is expressed as follows:(16)Λj,Λj∗=−ξjωj±iωj1−ξj2,j=1,2,3,…n,where ωj and ξj are the jth natural frequency and modal damping ratio and i is the imaginary unit. Thus, the modal damping ratio ξj can be obtained as follows:(17)ξj=−ReΛjAbsΛj,j=1,2,3,…n.Figure6 shows two 3D-plots for ξ1 and ξ2 with respect to the damper parameters (m¯e and c¯d; k¯n = 0 and xd = 0.021) in two different sag conditions (λ2 = 3.9 and 6.6). The figures show that, for a specified damper parameter (m¯e or c¯d), ξj always increases first and then decreases with the other increasing parameter (c¯d or m¯e). Thus, there exists a theoretical maximum modal damping ratio at a set of damper parameters. The maximum ξ1 is obtained as 8.29% and 7.25% for two λ2 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 ξ1 is significantly reduced with the growing sag parameter. ξ1 decreases from 8.29% to 7.25% as λ2 increases from 3.9 to 6.6, respectively. However, the maximum ξ2 is only slightly affected by the sag effect: 9.36% and 9.35% for two λ2. Two 3D graphs of ξ2 are almost the same for two λ2 as shown in Figure 6(b).Figure 6
Variation of the modal damping ratio with respect to VIMD parameters underλ2=3.9 and 6.6. (a) First cable mode. (b) Second cable mode.
(a)(b)Table 3
Theoretical maximum modal damping ratios of VD and VIMD (k¯n = 0).
Mode numbersλ2xdVDVIMDc¯dξ (%)m¯ec¯dξ (%)10 (Reference [26])0.025.051.024.8129.93.90.0214.20.693.5281.4268.296.63.880.523.0451.1617.2520 (Reference [26])0.022.551.021.21.059.053.90.0212.421.081.141.029.366.62.421.081.141.029.35


### 3.3. Characteristics of a Cable with a VIMD

The present authors proposed an optimal design procedure for a VIMD on a taut cable using the fixed-point method (FPM: Reference [44]). The 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 fixed 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 briefly discussed. Based on equations (9), (11), (14), and (15), the FRF of the cable response at x under a harmonic excitation, px,t=p0xftt with ftt=eiωt, can be expressed using the state-space formulations as follows:(18)Hx,ω=ϕTx0ΨiωI−Λ00Λ∗−1ΨTBsf0,where f0 is the modal force coefficient vector associated with the shape functions ϕx and the load distribution p0x as in equation (11); Λ and Λ∗ are the diagonal matrices for conjugate pairs of eigenvalues; and Ψ is the eigen matrix of the state matrix As, normalized as ΨTΨ=I,(19)Ψ=ψψ∗ψΛψ∗Λ∗,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 ϕix in equation (8).Assumingλ2 = 3.9 and xd = 0.021 with a concentrated harmonic actuation px,t=δx−xaeiωt at xa = 0.99, Figure 7(a) shows the FRF curves of the cable with different VIMDs shown in Section 4, at the cable midpoint obtained for various values of m¯e and c¯d using equation (18). In the figure, the blue curves are with zero damping and the red curve is with infinite damping. The blue dashed curve is for the cable without the VIMD, which shows the original first cable frequency (ω1 = 1.148 ω0). It is interesting to observe that the blue solid curve (m¯e = 3.528) has two resonant frequency peaks, ω1−1 = 1.074 ω0 and ω1−2 = 1.263 ω0. The blue dot curve and the red curve are the same although the damper parameters are different. This shows that the original first 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 infinity. 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 fixed. It can be also seen that the responses around the first resonant frequency ω1 can be significantly suppressed with a properly selected damping, e.g., c¯d = 0.88, so that the optimum VIMD design may be obtained using the FPM. The main idea for the optimum design in the FPM is to find the VIMD parameters (m¯e and c¯d) so that the FRF curves may have maxima with the same amplitude at the two fixed points (P and Q). The detailed optimization procedures can be found in reference [44].Figure 7
FRF curves for cable-VIMD system atL/2: λ2 = 3.9 and ω0 = 1.157 Hz. (a) Different m¯e and c¯d. (b) Different c¯d with m¯e = 3.528.
(a)(b)


### 3.4. 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 Figure6, 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 different 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 different parameters were considered, where VIMD-1 is the optimum design to the first mode obtained by the FPM (m¯e = 3.528, c¯d = 0.88, and ξ1 = 5.32%) and VIMD-2 is the one by the maximum modal damping ratio (m¯e = 3.528, c¯d = 1.426, and ξ1 = 8.29%) as shown also in Figure 7(b). Three different 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 VIMDs become nearly the same for Ω = 1.232 as expected in Figure 7(b). The 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 (ξ1 = 8.29%) than the one (ξ1 = 5.32%) by VIMD-1. The vibration amplitudes then become similar with the increasing Ω as expected by Figure 7(b). The differences 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 different 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.Figure 8
Comparison of simulated cable responses by two optimum VIMDs (red: VIMD-1 withm¯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.
(a)(b)(c)(d)


## 3.1. Model of Cable-VIMD System

Referring Figure5, the current sagged cable-damper system can be modeled by a nondimensional equation of motion as follows [60–62]:(6)w¨x,t+cw˙x,t−1π2w″x,t+λ2π2∫01wζ,tdζ=px,t+Fdtδx−xd,where wx,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; px,t is the external vertical force on the cable; and Fdt is the force from the VIMD at x=xd. The nondimensional quantities are related to their dimensional counterparts, shown with overbars, according to the following relations:(6a)t=ω0t¯,x=x¯L,c=c¯ρω0,wx,t=w¯x¯,t¯L,δx−xd=Lδ¯x¯−x¯d,px,t=Lp¯x¯,t¯π2T,Fdt=F¯dt¯π2T,in which ρ, L, and T are the cable mass per unit length, cable chord length, and cable tension in the chord direction; ω0=π/LT/ρ is the first natural frequency of the cable without a VIMD, computed without considering the sag effect.Figure 5
A sagged cable with a VIMD.The nondimensional parameterλ2 for the sag effect is defined as follows [63]:(7)λ2=ρgLcosθTEALTLe,Le=L1+18ρgLcosθT2,in which θ and Le 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 deflection 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]:(8)ϕix=12+λ212+λ2−3λ2xd1−xdxxd+1−xxdHx−xd1−xd−3λ2x1−x12+λ2,i=1,sini−1πx,i≥2,where Hx−xd is a Heaviside function. In this study, 20 sinusoidal shape functions were used.By using the shape functions in equation (8), the cable response wx,t can be approximately expressed as follows:(9)wx,t=∑qitϕix=ϕTxqt,where qit is the generalized coordinate.Then, the equation of motion, equation (6), can be discretized into a matrix form as follows:(10)Mq¨+Cq˙+Kq=f+ϕxdFdt,where M, C, and K are the generalized mass, damping, and stiffness matrices, which are related to the assumed shape functions ϕix in equation (8). In reference [60],f is the generalized external load vector:(11)f=∫01ϕxpx,tdx=∫01ϕxp0xdxftt=f0ftt,where the external force on the cable is assumed as px,t=p0xftt.The damper force of the VIMD at locationx=xd can be expressed as follows:(12)Fdt=−m¯eϕTxdq¨+c¯dϕTxdq˙+k¯nϕTxdq,where m¯e, c¯d, and k¯n are the nondimensional damper parameters defined as follows:(13)m¯e=meLω02π2T,c¯d=cdLω0π2T,k¯n=knLπ2T,in which cd=ce+cf.Using equations (10) and (12), it can be rewritten as follows:(14)M+m¯eϕxdϕTxdq¨+C+c¯dϕxdϕTxdq˙+K+k¯nϕxdϕTxdq=Maq¨+Caq˙+Kaq=f.Then, the state-space expression can be obtained as follows [29, 60]:(15)z˙=q˙q¨=0I−Ma−1Ka−Ma−1Caqq˙+0Ma−1f=Asz+Bsf,where z is the state vector, As is the state matrix, and Bs is the input matrix.


## 3.2. Analysis of Modal Damping Ratio

Conjugate pairs of the complex eigenvalues,Λj and Λj∗, can be obtained from the state matrix As and is expressed as follows:(16)Λj,Λj∗=−ξjωj±iωj1−ξj2,j=1,2,3,…n,where ωj and ξj are the jth natural frequency and modal damping ratio and i is the imaginary unit. Thus, the modal damping ratio ξj can be obtained as follows:(17)ξj=−ReΛjAbsΛj,j=1,2,3,…n.Figure6 shows two 3D-plots for ξ1 and ξ2 with respect to the damper parameters (m¯e and c¯d; k¯n = 0 and xd = 0.021) in two different sag conditions (λ2 = 3.9 and 6.6). The figures show that, for a specified damper parameter (m¯e or c¯d), ξj always increases first and then decreases with the other increasing parameter (c¯d or m¯e). Thus, there exists a theoretical maximum modal damping ratio at a set of damper parameters. The maximum ξ1 is obtained as 8.29% and 7.25% for two λ2 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 ξ1 is significantly reduced with the growing sag parameter. ξ1 decreases from 8.29% to 7.25% as λ2 increases from 3.9 to 6.6, respectively. However, the maximum ξ2 is only slightly affected by the sag effect: 9.36% and 9.35% for two λ2. Two 3D graphs of ξ2 are almost the same for two λ2 as shown in Figure 6(b).Figure 6
Variation of the modal damping ratio with respect to VIMD parameters underλ2=3.9 and 6.6. (a) First cable mode. (b) Second cable mode.
(a)(b)Table 3
Theoretical maximum modal damping ratios of VD and VIMD (k¯n = 0).
Mode numbersλ2xdVDVIMDc¯dξ (%)m¯ec¯dξ (%)10 (Reference [26])0.025.051.024.8129.93.90.0214.20.693.5281.4268.296.63.880.523.0451.1617.2520 (Reference [26])0.022.551.021.21.059.053.90.0212.421.081.141.029.366.62.421.081.141.029.35


## 3.3. Characteristics of a Cable with a VIMD

The present authors proposed an optimal design procedure for a VIMD on a taut cable using the fixed-point method (FPM: Reference [44]). The 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 fixed 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 briefly discussed. Based on equations (9), (11), (14), and (15), the FRF of the cable response at x under a harmonic excitation, px,t=p0xftt with ftt=eiωt, can be expressed using the state-space formulations as follows:(18)Hx,ω=ϕTx0ΨiωI−Λ00Λ∗−1ΨTBsf0,where f0 is the modal force coefficient vector associated with the shape functions ϕx and the load distribution p0x as in equation (11); Λ and Λ∗ are the diagonal matrices for conjugate pairs of eigenvalues; and Ψ is the eigen matrix of the state matrix As, normalized as ΨTΨ=I,(19)Ψ=ψψ∗ψΛψ∗Λ∗,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 ϕix in equation (8).Assumingλ2 = 3.9 and xd = 0.021 with a concentrated harmonic actuation px,t=δx−xaeiωt at xa = 0.99, Figure 7(a) shows the FRF curves of the cable with different VIMDs shown in Section 4, at the cable midpoint obtained for various values of m¯e and c¯d using equation (18). In the figure, the blue curves are with zero damping and the red curve is with infinite damping. The blue dashed curve is for the cable without the VIMD, which shows the original first cable frequency (ω1 = 1.148 ω0). It is interesting to observe that the blue solid curve (m¯e = 3.528) has two resonant frequency peaks, ω1−1 = 1.074 ω0 and ω1−2 = 1.263 ω0. The blue dot curve and the red curve are the same although the damper parameters are different. This shows that the original first 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 infinity. 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 fixed. It can be also seen that the responses around the first resonant frequency ω1 can be significantly suppressed with a properly selected damping, e.g., c¯d = 0.88, so that the optimum VIMD design may be obtained using the FPM. The main idea for the optimum design in the FPM is to find the VIMD parameters (m¯e and c¯d) so that the FRF curves may have maxima with the same amplitude at the two fixed points (P and Q). The detailed optimization procedures can be found in reference [44].Figure 7
FRF curves for cable-VIMD system atL/2: λ2 = 3.9 and ω0 = 1.157 Hz. (a) Different m¯e and c¯d. (b) Different c¯d with m¯e = 3.528.
(a)(b)


## 3.4. 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 Figure6, 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 different 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 different parameters were considered, where VIMD-1 is the optimum design to the first mode obtained by the FPM (m¯e = 3.528, c¯d = 0.88, and ξ1 = 5.32%) and VIMD-2 is the one by the maximum modal damping ratio (m¯e = 3.528, c¯d = 1.426, and ξ1 = 8.29%) as shown also in Figure 7(b). Three different 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 VIMDs become nearly the same for Ω = 1.232 as expected in Figure 7(b). The 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 (ξ1 = 8.29%) than the one (ξ1 = 5.32%) by VIMD-1. The vibration amplitudes then become similar with the increasing Ω as expected by Figure 7(b). The differences 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 different 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.Figure 8
Comparison of simulated cable responses by two optimum VIMDs (red: VIMD-1 withm¯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.
(a)(b)(c)(d)


## 4. Scaled-Cable Experiments and Performance Evaluation

Experiments are carried out on a scaled cable with the parameters shown in Table4. Figure 9 shows the experimental setups. The 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., λ2 = 3.9 and 6.6. The first two cable modes were obtained for each sag condition with various damper parameters. The measured natural frequencies (ω1 and ω2) for the first 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. The natural frequencies ω0 computed considering the cable tension only without the sag effect are smaller than the frequencies including the sag effect. The 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 ωe close to ω1 or ω2 was applied to the cable. The electromagnets were then activated after the vibration reached a steady state, and the excitation was removed after reaching another steady state. The cable displacements at the midpoint (for ωe near ω1) and the quarter point near the bottom anchorage (for ωe near ω2) were measured using noncontact measurement with a digital camera. The modal damping ratios were identified from the free vibration part. The damper force and the damper displacement were recorded by a force sensor and a position sensor, respectively.Table 4
Parameters for two sagged scaled cables without a VIMD.
λ2L (m)θ (°)ρ (kg/m)EA (106 N)T (N)k¯nFrequency (Hz)ω0ω1ω23.919.4711.620.7380.7161497.51.485(1.157)1.333 (1.333)2.271 (2.314)6.61266.71.756(1.065)1.32 (1.319)2.153 (2.129)The frequencies in parentheses are the theoretical values.Figure 9
Setups of the scaled-cable experiment.Table 5
Maximumξ and corresponding theoretical damper parameters by the state-space formulation.
Test modesλ2m¯eme (kg)c¯dcd (Ns/m)Max.ξ(%)1st3.94.6266.41.61168.17.236.64.1659.81.34128.76.122nd3.91.4921.41.17122.28.526.61.5622.41.19114.38.38The VIMD parameters in Table5 are different from those in Table 3 due to kn of the additional spring in the VIMD prototype.


### 4.1. Damper Parameter Identification

Figure10(a) shows the experimental time histories of the damper force and the damper displacement (for λ2 = 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 2(b)–2(e) are for a VIMD without the cable. The cable was excited under a harmonic excitation with ωe = 2.1 Hz, and the electromagnets were not activated in this case. With the least-square method given in Section 2.2, the damper parameters in equation (4) of this case were identified as me = 21.3 kg, ce = 0 Ns/m, and fc = 3.39 N, wherein the design values for me and ce considering the second cable mode are 21 kg and 0 Ns/m, respectively. The identified friction term in equation (4b) is linearized by fitting the force-velocity hysteretic loop as shown in Figure 10(b). The blue curve is the friction term with fc = 3.39 N, and the red curve is the equivalent damping with cf = 27.16 Ns/m. Theoretically, the friction should be identical for all experimental cases. However, it was changed to a small degree for different inertial mass values because the weight positions were manually adjusted. Figures 10(c) and 10(d) show comparisons between the experimental results and the fitted results using the identified parameters. The fitted damper force strongly matches the measured counterpart.Figure 10
Identification of damper force;λ2 = 6.6 and ce = 0. (a) Recorded experimental time histories. (b) Equivalent damping. (c) Comparisons between the experimental results and the fitted results. (d) Comparisons of the hysteretic loops.
(a)(b)(c)(d)


### 4.2. Analysis of Modal Damping Ratio

In this section, we discuss mainly the experimental results ofλ2 = 3.9, while the results for a different sag (λ2 = 6.6) are shown in the next section. Figure 11(a) shows the free vibration response at the midpoint without a VIMD after being excited with ωe=ω1=1.333Hz. The 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 influence of the actuator, which is very small compared with that provided by the damper. This 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 Fourier-amplitude-spectrum (FAS) results with and without a damper. We can see that the response amplitude is significantly reduced by the VIMD, and the cable frequency is slightly enlarged.Figure 11
Dynamic cable characteristics without VIMD:λ2=3.9 and ωe=1.33Hz. (a) Free vibration without VIMD. (b) FAS of displacement at the cable midpoint.
(a)(b)Figure12(a) shows the vibration response at L/2 (after being excited at ωe = 1.414 Hz) with the damper parameters optimized to the first mode (by maximizing ξ1), where the identified values are me = 66.1 kg and cd = 164.2 Ns/m, which are very close to the target values (66.4 kg, 168.1 Ns/m) shown in Table 5. The envelope curve of cable responses Ht was obtained using the Hilbert-transform, and the modal damping ratio ξ was obtained by taking the slope of the logarithm of Ht in the free decaying part. Figure 12(b) shows the response at L/4 (after being excited at ωe = 2.29 Hz) with the damper parameters optimized to the second mode (by maximizing ξ2), where the identified values are me = 21.4 kg and cd = 113.4 Ns/m, which are also close to the target values (21.4 kg, 122.2 Ns/m). The identified 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%). The identified 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. The results show that there exists a pair of me and cd that provides a maximum modal damping ratio for each mode.Figure 12
Experimental results of the modal damping ratio,λ2=3.9. (a) Cable vibration response at L/2: ωe = 1.414 Hz (me = 66.1 kg, cd = 164.2 Ns/m, and ξ = 6.98%). (b) Cable vibration response at L/4: ωe = 2.29 Hz (me = 21.4 kg, cd = 113.4 Ns/m, and ξ = 8.15%). (c) Comparisons of the first modal damping ratios. (d) Comparisons of the second modal damping ratios.
(a)(b)(c)(d)


### 4.3. Analysis of Sag Effect

To study the sag effect on the control performance, experiments were also conducted on a cable withλ2 = 6.6. The 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 ωe = 2.33 Hz, where the electromagnets were turned on at 3 s and the excitation was removed at 40 s. The identified parameters were me = 22.6 kg, cf = 47.4 Ns/m (0–3 s), and cd = 112.5 Ns/m (3–50 s), respectively. The vibration was suppressed in 2 s after the excitation was removed, and the identified ξ2 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 (me and cd), wherein good agreements can be generally found. Compared with the case of λ2 = 3.9 (Figures 12(c) and 12(d)), it can be observed that both me and cd, corresponding to the maximum point of the first modal damping ratio ξ1, 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 (ξ1 and ξ2) and the experimental maximum results for two sag conditions. The solid curves represent the theoretical value and the dashed curves with the markers “○” represent the experimental values. The experimental maximum value of ξ1 is reduced from 6.98% (λ2 = 3.9) to 5.27% (λ2 = 6.6), whereas ξ2 decreases from 8.15% to 7.34%. It can be observed that the vibration-reduction performance of the first cable mode is significantly affected by the sag, whereas the second mode is not. Larger differences are observed between the experimental and theoretical results for the maximum modal damping ratios of the second modes in the low range of cd, which might be caused by the identification error in the data process.Figure 13
Experimental results of the modal damping ratios. (a) Cable vibration response atL/4: ωe = 2.33 Hz (me = 22.6 kg, cd = 112.5 Ns/m, and λ2 = 6.6). (b) Comparisons of the first modal damping ratio, ξ1 for λ2 = 6.6. (c) Comparisons of the second modal damping ratio, ξ2 for λ2 = 6.6. (d) Comparisons of the maximum modal damping ratios for λ2 = 3.9 (me = 66.1 and 21.4 kg) and λ2 = 6.6 (me = 60.3 and 22.6 kg).
(a)(b)(c)(d)


### 4.4. Analysis of Maximum Responses and FRF Curves

#### 4.4.1. Responses to Harmonic Excitations

Figure14(a) shows the typical time-history responses of the damper force, the damper displacement, and the cable displacement at L/4 in the case: ωe = 2.3 Hz, λ2 = 3.9, me = 21.4 kg, and cd = 189.2 Ns/m. The exciting frequency ωe was taken to be close to ω2 = 2.314 Hz. The external excitation was introduced at 7.5 s, and the electromagnets were activated at 40 s. This means that cd=cf equals 40.3Ns/m (ce = 0) during the period of 0–40 s, and it becomes 189.2 Ns/m (ce = 148.9 Ns/m) after 40 s, whereas the inertial mass remains constant as me = 21.4 kg. Then, 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 ωe and me but for three different cd are also given in Figure 14(b). It can be observed that in all the cases the amplitudes of the damper force and damper displacement decreased with the increasing cd, whereas the cable displacement L/4 increased.Figure 14
Vibration responses atL/4 with ωe = 2.3 Hz, λ2 = 3.9, and me = 21.4 kg. (a) Measured time histories with cd = 189.2 Ns/m. (b) Measured cable responses with various cd.
(a)(b)However, the changes in the cable responses are not always the same for all the cases. Comparing the cable displacement responses atL/4 in Figures 13(a) and 14(a), different 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. The amplitude increases from 20.1 mm to 42.3 mm for λ2 = 3.9 after activating the electromagnets but decreases from 44.6 mm to 22.8 mm for λ2 = 6.6. The identified ξ2 is 2.7%, 8.15%, and 3.47% for cd= 62.2, 113.4, and 189.2 Ns/m with λ2 = 3.9 in Figure 14(b). The blue curve with the smallest ξ2 gives the smallest cable response amplitude at ωe = 2.3 Hz, whereas the green curve with the largest cd gives the largest cable response.The abovementioned phenomena can be explained through the analysis of the FRF curves. Figure15 shows FRF curves at L/4 obtained for various parameters using equation (18). The solid curve is without a VIMD, the rest of the curves are with a VIMD using different values of me and cd 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 Table 4. The original second mode (the solid curve with ω2 = 2.236 Hz) has split into two, as in the dashed curve with ω2−1,2 = 2.194 and 2.549 Hz shown in Figure 15, after introducing a VIMD with me = 21.4 kg and cd = 62.2 Ns/m. These two modes move closer to each other by increasing cd as shown by the dotted and dash-dot curves, wherein the two split frequencies are 2.285 and 2.449 Hz for cd = 113.4 Ns/m and 2.364 and 2.37 Hz for cd = 189.2 Ns/m, respectively. Then, they converge to one frequency, i.e., f2−∞ = 2.364 Hz, when cd becomes infinity. From the FRF curves, it can be clearly seen that the response amplitude increases from Point A to Point C at ωe = 2.3 Hz as cd increases from 62.2 to 189.2 Ns/m, which confirms the trends of the experimental results in Figure 14(b). In addition, two fixed points (P and Q) can be observed in Figure 15 denoted with the markers “×.” It is evident that all the three FRF curves with the same me = 21.4 kg pass through the fixed points “×” regardless of the cd values, as expected. It is to be noted that the case with me = 21.4 kg and cd = 113.4 Ns/m represents the optimum VIMD with the maximum modal damping ratio ξ2 of 8.15%. Besides, the red curve is for the optimum VIMD obtained by the FPM, which provides a flat “region” of the FRF curve around the original second frequency.Figure 15
Simulated FRF curves atL/4 near ω2 for experimental cases of Figure 14: λ2 = 3.9.Figure16(a) shows three experimental displacement time histories at 0.5 L with me = 66.1 kg, ωe = 1.414 Hz, and λ2 = 3.9. The total damping coefficient cd=ce+cf is 82.7, 121.1, and 164.2 Ns/m, for those cases, which are the first, second, and fourth yellow points in Figure 12(c). The corresponding modal damping ratio ξ1 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
Experimental responses for various VIMD parameters and corresponding FRF curves by equation (18). (a) Vibration responses at L/2 with ωe = 1.414 Hz, λ2 = 3.9, and me = 66.1 kg. (b) Vibration responses at L/4 with ωe = 2.33 Hz, λ2 = 6.6, and me = 22.6 kg. (c) FRF curves at L/2 with λ2 = 3.9 and me = 66.1 kg. (d) FRF curves at L/4 with λ2 = 6.6 and me = 22.6 kg.
(a)(b)(c)(d)Figure16(b) shows three other experimental results at 0.25 L with the same me = 22.6 kg, ωe = 2.33 Hz, and λ2 = 6.6, but different cd = 47.4, 112.5, and 152.9 Ns/m, which are the first, highest, and last yellow points in Figure 13(c). The corresponding modal damping ratios ξ2 are 2.15%, 7.34%, and 3.63%, respectively. The steady state response amplitude for cd = 112.5 Ns/m with the highest modal damping ratio, ξ2 = 7.34%, is close to the response for cd = 152.9 Ns/m with ξ2 = 3.63%, where it is smaller than the response for cd = 47.7 Ns/m with ξ2 = 2.15%. These different trends of the cable responses can be explained by the FRF curves obtained from equation (18) shown in Figures 16(c) and 16(d). The excitation frequency (ωe = 1.414 Hz) close to the one of the fixed points Q (ωQ = 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 cd = 47.4 Ns/m with the smallest ξ2 (2.15%) at ωe = 2.33 Hz is to be higher than the other two cases with larger ξ2 (7.34% and 3.63%), wherein the latter two cases have very similar FRF amplitudes. The abovementioned results reveal that a higher modal damping ratio does not guarantee a lower cable vibration amplitude.


#### 4.4.2. Responses to Sinusoidal Sweep Excitations

Figure17(a) shows the measured cable responses at L/4 of the cable with me of a VIMD optimized to the second mode (by maximizing ξ2) subjected to a 60 s long sinusoidal sweep excitation, wherein the sweep frequency range is from 0.5 to 3.5 Hz. Four time-history curves are shown with the same sag and inertial mass (λ2 = 6.6, me = 22.6 kg) but with different damping. The corresponding FAS curves are shown in Figure 17(b). It is to be noted that the general shapes of the FAS curves of the responses for the sinusoidal sweeping excitation are similar to the FRF curves in the corresponding frequency range. It can be seen that the original frequency (ω2 = 2.13 Hz) has split into two (1.963 and 2.307 Hz) in the case of cd = 25.3 Ns/m. As cd increases, the amplitudes of the two split modes are first reduced, and the two peaks move closer to each other. Then, the two split modes become one mode, and the amplitude keeps growing with the increasing cd. The peak frequency for cd = 144.1 Ns/m is obtained as 2.153 Hz, which is slightly higher than original frequency. It can be also observed that all the experimental FAS curves pass through two fixed points (P and Q), regardless of the cd value, as expected.Figure 17
Cable responses atL/4 subjected to a sweep excitation: me = 22.6 kg and λ2 = 6.6. (a) Displacement time history. (b) FAS of cable displacement.
(a)(b)Figure18 shows the FAS curves of the responses at L/2 under the same sine sweep excitation for the case of λ2 = 3.9 and me = 66.1 kg. The same phenomenon of mode splitting and two fixed points can be clearly observed.Figure 18
FASs of measured cable responses atL/2 for various cd with the optimized me = 66.1 kg to the 1st mode.


## 4.1. Damper Parameter Identification

Figure10(a) shows the experimental time histories of the damper force and the damper displacement (for λ2 = 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 2(b)–2(e) are for a VIMD without the cable. The cable was excited under a harmonic excitation with ωe = 2.1 Hz, and the electromagnets were not activated in this case. With the least-square method given in Section 2.2, the damper parameters in equation (4) of this case were identified as me = 21.3 kg, ce = 0 Ns/m, and fc = 3.39 N, wherein the design values for me and ce considering the second cable mode are 21 kg and 0 Ns/m, respectively. The identified friction term in equation (4b) is linearized by fitting the force-velocity hysteretic loop as shown in Figure 10(b). The blue curve is the friction term with fc = 3.39 N, and the red curve is the equivalent damping with cf = 27.16 Ns/m. Theoretically, the friction should be identical for all experimental cases. However, it was changed to a small degree for different inertial mass values because the weight positions were manually adjusted. Figures 10(c) and 10(d) show comparisons between the experimental results and the fitted results using the identified parameters. The fitted damper force strongly matches the measured counterpart.Figure 10
Identification of damper force;λ2 = 6.6 and ce = 0. (a) Recorded experimental time histories. (b) Equivalent damping. (c) Comparisons between the experimental results and the fitted results. (d) Comparisons of the hysteretic loops.
(a)(b)(c)(d)


## 4.2. Analysis of Modal Damping Ratio

In this section, we discuss mainly the experimental results ofλ2 = 3.9, while the results for a different sag (λ2 = 6.6) are shown in the next section. Figure 11(a) shows the free vibration response at the midpoint without a VIMD after being excited with ωe=ω1=1.333Hz. The 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 influence of the actuator, which is very small compared with that provided by the damper. This 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 Fourier-amplitude-spectrum (FAS) results with and without a damper. We can see that the response amplitude is significantly reduced by the VIMD, and the cable frequency is slightly enlarged.Figure 11
Dynamic cable characteristics without VIMD:λ2=3.9 and ωe=1.33Hz. (a) Free vibration without VIMD. (b) FAS of displacement at the cable midpoint.
(a)(b)Figure12(a) shows the vibration response at L/2 (after being excited at ωe = 1.414 Hz) with the damper parameters optimized to the first mode (by maximizing ξ1), where the identified values are me = 66.1 kg and cd = 164.2 Ns/m, which are very close to the target values (66.4 kg, 168.1 Ns/m) shown in Table 5. The envelope curve of cable responses Ht was obtained using the Hilbert-transform, and the modal damping ratio ξ was obtained by taking the slope of the logarithm of Ht in the free decaying part. Figure 12(b) shows the response at L/4 (after being excited at ωe = 2.29 Hz) with the damper parameters optimized to the second mode (by maximizing ξ2), where the identified values are me = 21.4 kg and cd = 113.4 Ns/m, which are also close to the target values (21.4 kg, 122.2 Ns/m). The identified 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%). The identified 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. The results show that there exists a pair of me and cd that provides a maximum modal damping ratio for each mode.Figure 12
Experimental results of the modal damping ratio,λ2=3.9. (a) Cable vibration response at L/2: ωe = 1.414 Hz (me = 66.1 kg, cd = 164.2 Ns/m, and ξ = 6.98%). (b) Cable vibration response at L/4: ωe = 2.29 Hz (me = 21.4 kg, cd = 113.4 Ns/m, and ξ = 8.15%). (c) Comparisons of the first modal damping ratios. (d) Comparisons of the second modal damping ratios.
(a)(b)(c)(d)


## 4.3. Analysis of Sag Effect

To study the sag effect on the control performance, experiments were also conducted on a cable withλ2 = 6.6. The 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 ωe = 2.33 Hz, where the electromagnets were turned on at 3 s and the excitation was removed at 40 s. The identified parameters were me = 22.6 kg, cf = 47.4 Ns/m (0–3 s), and cd = 112.5 Ns/m (3–50 s), respectively. The vibration was suppressed in 2 s after the excitation was removed, and the identified ξ2 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 (me and cd), wherein good agreements can be generally found. Compared with the case of λ2 = 3.9 (Figures 12(c) and 12(d)), it can be observed that both me and cd, corresponding to the maximum point of the first modal damping ratio ξ1, 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 (ξ1 and ξ2) and the experimental maximum results for two sag conditions. The solid curves represent the theoretical value and the dashed curves with the markers “○” represent the experimental values. The experimental maximum value of ξ1 is reduced from 6.98% (λ2 = 3.9) to 5.27% (λ2 = 6.6), whereas ξ2 decreases from 8.15% to 7.34%. It can be observed that the vibration-reduction performance of the first cable mode is significantly affected by the sag, whereas the second mode is not. Larger differences are observed between the experimental and theoretical results for the maximum modal damping ratios of the second modes in the low range of cd, which might be caused by the identification error in the data process.Figure 13
Experimental results of the modal damping ratios. (a) Cable vibration response atL/4: ωe = 2.33 Hz (me = 22.6 kg, cd = 112.5 Ns/m, and λ2 = 6.6). (b) Comparisons of the first modal damping ratio, ξ1 for λ2 = 6.6. (c) Comparisons of the second modal damping ratio, ξ2 for λ2 = 6.6. (d) Comparisons of the maximum modal damping ratios for λ2 = 3.9 (me = 66.1 and 21.4 kg) and λ2 = 6.6 (me = 60.3 and 22.6 kg).
(a)(b)(c)(d)


## 4.4. Analysis of Maximum Responses and FRF Curves

### 4.4.1. Responses to Harmonic Excitations

Figure14(a) shows the typical time-history responses of the damper force, the damper displacement, and the cable displacement at L/4 in the case: ωe = 2.3 Hz, λ2 = 3.9, me = 21.4 kg, and cd = 189.2 Ns/m. The exciting frequency ωe was taken to be close to ω2 = 2.314 Hz. The external excitation was introduced at 7.5 s, and the electromagnets were activated at 40 s. This means that cd=cf equals 40.3Ns/m (ce = 0) during the period of 0–40 s, and it becomes 189.2 Ns/m (ce = 148.9 Ns/m) after 40 s, whereas the inertial mass remains constant as me = 21.4 kg. Then, 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 ωe and me but for three different cd are also given in Figure 14(b). It can be observed that in all the cases the amplitudes of the damper force and damper displacement decreased with the increasing cd, whereas the cable displacement L/4 increased.Figure 14
Vibration responses atL/4 with ωe = 2.3 Hz, λ2 = 3.9, and me = 21.4 kg. (a) Measured time histories with cd = 189.2 Ns/m. (b) Measured cable responses with various cd.
(a)(b)However, the changes in the cable responses are not always the same for all the cases. Comparing the cable displacement responses atL/4 in Figures 13(a) and 14(a), different 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. The amplitude increases from 20.1 mm to 42.3 mm for λ2 = 3.9 after activating the electromagnets but decreases from 44.6 mm to 22.8 mm for λ2 = 6.6. The identified ξ2 is 2.7%, 8.15%, and 3.47% for cd= 62.2, 113.4, and 189.2 Ns/m with λ2 = 3.9 in Figure 14(b). The blue curve with the smallest ξ2 gives the smallest cable response amplitude at ωe = 2.3 Hz, whereas the green curve with the largest cd gives the largest cable response.The abovementioned phenomena can be explained through the analysis of the FRF curves. Figure15 shows FRF curves at L/4 obtained for various parameters using equation (18). The solid curve is without a VIMD, the rest of the curves are with a VIMD using different values of me and cd 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 Table 4. The original second mode (the solid curve with ω2 = 2.236 Hz) has split into two, as in the dashed curve with ω2−1,2 = 2.194 and 2.549 Hz shown in Figure 15, after introducing a VIMD with me = 21.4 kg and cd = 62.2 Ns/m. These two modes move closer to each other by increasing cd as shown by the dotted and dash-dot curves, wherein the two split frequencies are 2.285 and 2.449 Hz for cd = 113.4 Ns/m and 2.364 and 2.37 Hz for cd = 189.2 Ns/m, respectively. Then, they converge to one frequency, i.e., f2−∞ = 2.364 Hz, when cd becomes infinity. From the FRF curves, it can be clearly seen that the response amplitude increases from Point A to Point C at ωe = 2.3 Hz as cd increases from 62.2 to 189.2 Ns/m, which confirms the trends of the experimental results in Figure 14(b). In addition, two fixed points (P and Q) can be observed in Figure 15 denoted with the markers “×.” It is evident that all the three FRF curves with the same me = 21.4 kg pass through the fixed points “×” regardless of the cd values, as expected. It is to be noted that the case with me = 21.4 kg and cd = 113.4 Ns/m represents the optimum VIMD with the maximum modal damping ratio ξ2 of 8.15%. Besides, the red curve is for the optimum VIMD obtained by the FPM, which provides a flat “region” of the FRF curve around the original second frequency.Figure 15
Simulated FRF curves atL/4 near ω2 for experimental cases of Figure 14: λ2 = 3.9.Figure16(a) shows three experimental displacement time histories at 0.5 L with me = 66.1 kg, ωe = 1.414 Hz, and λ2 = 3.9. The total damping coefficient cd=ce+cf is 82.7, 121.1, and 164.2 Ns/m, for those cases, which are the first, second, and fourth yellow points in Figure 12(c). The corresponding modal damping ratio ξ1 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
Experimental responses for various VIMD parameters and corresponding FRF curves by equation (18). (a) Vibration responses at L/2 with ωe = 1.414 Hz, λ2 = 3.9, and me = 66.1 kg. (b) Vibration responses at L/4 with ωe = 2.33 Hz, λ2 = 6.6, and me = 22.6 kg. (c) FRF curves at L/2 with λ2 = 3.9 and me = 66.1 kg. (d) FRF curves at L/4 with λ2 = 6.6 and me = 22.6 kg.
(a)(b)(c)(d)Figure16(b) shows three other experimental results at 0.25 L with the same me = 22.6 kg, ωe = 2.33 Hz, and λ2 = 6.6, but different cd = 47.4, 112.5, and 152.9 Ns/m, which are the first, highest, and last yellow points in Figure 13(c). The corresponding modal damping ratios ξ2 are 2.15%, 7.34%, and 3.63%, respectively. The steady state response amplitude for cd = 112.5 Ns/m with the highest modal damping ratio, ξ2 = 7.34%, is close to the response for cd = 152.9 Ns/m with ξ2 = 3.63%, where it is smaller than the response for cd = 47.7 Ns/m with ξ2 = 2.15%. These different trends of the cable responses can be explained by the FRF curves obtained from equation (18) shown in Figures 16(c) and 16(d). The excitation frequency (ωe = 1.414 Hz) close to the one of the fixed points Q (ωQ = 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 cd = 47.4 Ns/m with the smallest ξ2 (2.15%) at ωe = 2.33 Hz is to be higher than the other two cases with larger ξ2 (7.34% and 3.63%), wherein the latter two cases have very similar FRF amplitudes. The abovementioned results reveal that a higher modal damping ratio does not guarantee a lower cable vibration amplitude.


### 4.4.2. Responses to Sinusoidal Sweep Excitations

Figure17(a) shows the measured cable responses at L/4 of the cable with me of a VIMD optimized to the second mode (by maximizing ξ2) subjected to a 60 s long sinusoidal sweep excitation, wherein the sweep frequency range is from 0.5 to 3.5 Hz. Four time-history curves are shown with the same sag and inertial mass (λ2 = 6.6, me = 22.6 kg) but with different damping. The corresponding FAS curves are shown in Figure 17(b). It is to be noted that the general shapes of the FAS curves of the responses for the sinusoidal sweeping excitation are similar to the FRF curves in the corresponding frequency range. It can be seen that the original frequency (ω2 = 2.13 Hz) has split into two (1.963 and 2.307 Hz) in the case of cd = 25.3 Ns/m. As cd increases, the amplitudes of the two split modes are first reduced, and the two peaks move closer to each other. Then, the two split modes become one mode, and the amplitude keeps growing with the increasing cd. The peak frequency for cd = 144.1 Ns/m is obtained as 2.153 Hz, which is slightly higher than original frequency. It can be also observed that all the experimental FAS curves pass through two fixed points (P and Q), regardless of the cd value, as expected.Figure 17
Cable responses atL/4 subjected to a sweep excitation: me = 22.6 kg and λ2 = 6.6. (a) Displacement time history. (b) FAS of cable displacement.
(a)(b)Figure18 shows the FAS curves of the responses at L/2 under the same sine sweep excitation for the case of λ2 = 3.9 and me = 66.1 kg. The same phenomenon of mode splitting and two fixed points can be clearly observed.Figure 18
FASs of measured cable responses atL/2 for various cd with the optimized me = 66.1 kg to the 1st mode.


## 4.4.1. Responses to Harmonic Excitations

Figure14(a) shows the typical time-history responses of the damper force, the damper displacement, and the cable displacement at L/4 in the case: ωe = 2.3 Hz, λ2 = 3.9, me = 21.4 kg, and cd = 189.2 Ns/m. The exciting frequency ωe was taken to be close to ω2 = 2.314 Hz. The external excitation was introduced at 7.5 s, and the electromagnets were activated at 40 s. This means that cd=cf equals 40.3Ns/m (ce = 0) during the period of 0–40 s, and it becomes 189.2 Ns/m (ce = 148.9 Ns/m) after 40 s, whereas the inertial mass remains constant as me = 21.4 kg. Then, 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 ωe and me but for three different cd are also given in Figure 14(b). It can be observed that in all the cases the amplitudes of the damper force and damper displacement decreased with the increasing cd, whereas the cable displacement L/4 increased.Figure 14
Vibration responses atL/4 with ωe = 2.3 Hz, λ2 = 3.9, and me = 21.4 kg. (a) Measured time histories with cd = 189.2 Ns/m. (b) Measured cable responses with various cd.
(a)(b)However, the changes in the cable responses are not always the same for all the cases. Comparing the cable displacement responses atL/4 in Figures 13(a) and 14(a), different 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. The amplitude increases from 20.1 mm to 42.3 mm for λ2 = 3.9 after activating the electromagnets but decreases from 44.6 mm to 22.8 mm for λ2 = 6.6. The identified ξ2 is 2.7%, 8.15%, and 3.47% for cd= 62.2, 113.4, and 189.2 Ns/m with λ2 = 3.9 in Figure 14(b). The blue curve with the smallest ξ2 gives the smallest cable response amplitude at ωe = 2.3 Hz, whereas the green curve with the largest cd gives the largest cable response.The abovementioned phenomena can be explained through the analysis of the FRF curves. Figure15 shows FRF curves at L/4 obtained for various parameters using equation (18). The solid curve is without a VIMD, the rest of the curves are with a VIMD using different values of me and cd 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 Table 4. The original second mode (the solid curve with ω2 = 2.236 Hz) has split into two, as in the dashed curve with ω2−1,2 = 2.194 and 2.549 Hz shown in Figure 15, after introducing a VIMD with me = 21.4 kg and cd = 62.2 Ns/m. These two modes move closer to each other by increasing cd as shown by the dotted and dash-dot curves, wherein the two split frequencies are 2.285 and 2.449 Hz for cd = 113.4 Ns/m and 2.364 and 2.37 Hz for cd = 189.2 Ns/m, respectively. Then, they converge to one frequency, i.e., f2−∞ = 2.364 Hz, when cd becomes infinity. From the FRF curves, it can be clearly seen that the response amplitude increases from Point A to Point C at ωe = 2.3 Hz as cd increases from 62.2 to 189.2 Ns/m, which confirms the trends of the experimental results in Figure 14(b). In addition, two fixed points (P and Q) can be observed in Figure 15 denoted with the markers “×.” It is evident that all the three FRF curves with the same me = 21.4 kg pass through the fixed points “×” regardless of the cd values, as expected. It is to be noted that the case with me = 21.4 kg and cd = 113.4 Ns/m represents the optimum VIMD with the maximum modal damping ratio ξ2 of 8.15%. Besides, the red curve is for the optimum VIMD obtained by the FPM, which provides a flat “region” of the FRF curve around the original second frequency.Figure 15
Simulated FRF curves atL/4 near ω2 for experimental cases of Figure 14: λ2 = 3.9.Figure16(a) shows three experimental displacement time histories at 0.5 L with me = 66.1 kg, ωe = 1.414 Hz, and λ2 = 3.9. The total damping coefficient cd=ce+cf is 82.7, 121.1, and 164.2 Ns/m, for those cases, which are the first, second, and fourth yellow points in Figure 12(c). The corresponding modal damping ratio ξ1 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
Experimental responses for various VIMD parameters and corresponding FRF curves by equation (18). (a) Vibration responses at L/2 with ωe = 1.414 Hz, λ2 = 3.9, and me = 66.1 kg. (b) Vibration responses at L/4 with ωe = 2.33 Hz, λ2 = 6.6, and me = 22.6 kg. (c) FRF curves at L/2 with λ2 = 3.9 and me = 66.1 kg. (d) FRF curves at L/4 with λ2 = 6.6 and me = 22.6 kg.
(a)(b)(c)(d)Figure16(b) shows three other experimental results at 0.25 L with the same me = 22.6 kg, ωe = 2.33 Hz, and λ2 = 6.6, but different cd = 47.4, 112.5, and 152.9 Ns/m, which are the first, highest, and last yellow points in Figure 13(c). The corresponding modal damping ratios ξ2 are 2.15%, 7.34%, and 3.63%, respectively. The steady state response amplitude for cd = 112.5 Ns/m with the highest modal damping ratio, ξ2 = 7.34%, is close to the response for cd = 152.9 Ns/m with ξ2 = 3.63%, where it is smaller than the response for cd = 47.7 Ns/m with ξ2 = 2.15%. These different trends of the cable responses can be explained by the FRF curves obtained from equation (18) shown in Figures 16(c) and 16(d). The excitation frequency (ωe = 1.414 Hz) close to the one of the fixed points Q (ωQ = 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 cd = 47.4 Ns/m with the smallest ξ2 (2.15%) at ωe = 2.33 Hz is to be higher than the other two cases with larger ξ2 (7.34% and 3.63%), wherein the latter two cases have very similar FRF amplitudes. The abovementioned results reveal that a higher modal damping ratio does not guarantee a lower cable vibration amplitude.


## 4.4.2. Responses to Sinusoidal Sweep Excitations

Figure17(a) shows the measured cable responses at L/4 of the cable with me of a VIMD optimized to the second mode (by maximizing ξ2) subjected to a 60 s long sinusoidal sweep excitation, wherein the sweep frequency range is from 0.5 to 3.5 Hz. Four time-history curves are shown with the same sag and inertial mass (λ2 = 6.6, me = 22.6 kg) but with different damping. The corresponding FAS curves are shown in Figure 17(b). It is to be noted that the general shapes of the FAS curves of the responses for the sinusoidal sweeping excitation are similar to the FRF curves in the corresponding frequency range. It can be seen that the original frequency (ω2 = 2.13 Hz) has split into two (1.963 and 2.307 Hz) in the case of cd = 25.3 Ns/m. As cd increases, the amplitudes of the two split modes are first reduced, and the two peaks move closer to each other. Then, the two split modes become one mode, and the amplitude keeps growing with the increasing cd. The peak frequency for cd = 144.1 Ns/m is obtained as 2.153 Hz, which is slightly higher than original frequency. It can be also observed that all the experimental FAS curves pass through two fixed points (P and Q), regardless of the cd value, as expected.Figure 17
Cable responses atL/4 subjected to a sweep excitation: me = 22.6 kg and λ2 = 6.6. (a) Displacement time history. (b) FAS of cable displacement.
(a)(b)Figure18 shows the FAS curves of the responses at L/2 under the same sine sweep excitation for the case of λ2 = 3.9 and me = 66.1 kg. The same phenomenon of mode splitting and two fixed points can be clearly observed.Figure 18
FASs of measured cable responses atL/2 for various cd with the optimized me = 66.1 kg to the 1st mode.


## 5. 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 effect. The experimental results were systematically analyzed along with the simulated results by FEM, and the conclusions are summarized as follows:(1)
The damper parameters of the proposed VIMD prototype are continuously and easily adjustable. The inertial mass can be adjusted by moving the weights, and the electromagnetic viscous damping coefficient is adjusted by changing the input current to the electromagnets. The 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)
The modal damping ratios of the first two cable modes are remarkably enhanced with the installing of the VIMD. The identified maximum modal damping ratios of the first two modes are 6.89% and 8.15%, which are almost 10 and 8 times the maximum values of those using a viscous damper. Those results are one of the highest damping ratios obtained in experiments with a damper location at 2.1% so far. Besides, we find that there is one maximum modal damping ratio for each mode with a set of the damper parameters (me and cd), as in the theory.(3)
The first modal damping ratio is significantly reduced due to the larger sag effect. For example, the identified maximum first modal damping ratio decreases by 24.5% whenλ2 increases from 3.9 to 6.6. However, the maximum second modal damping ratio is reduced by only 9.9%. The VIMD parameters corresponding to the maximum first modal damping ratio also reduce, but those for the second mode do not.(4)
The cable mode is split into two modes by introducing a VIMD with a proper inertial mass. For a selected inertial mass value, two split frequencies move closer to each other with the increasing damping coefficient. The two frequencies converge to one frequency, as the damping approaches infinite. Besides, two fixed 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 fixed points regardless of damping, as in the fixed-point method (FPM). This can be used to obtain the optimum VIMD design.(5)
The 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 significantly 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.



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*Source: 4361740-2023-02-09.xml*