Using magnetorheological (MR) dampers in multiswitch open-loop control mode has been shown to be cost-effective for cable vibration mitigation. In this paper, a method for analyzing the damping performance of taut cables incorporating MR dampers in open-loop control mode is developed considering the effects of damping coefficient, damper stiffness, damper mass, and stiffness of the damper support. Making use of a three-element model of MR dampers and complex modal analysis, both numerical and asymptotic solutions are obtained. An analytical expression is obtained from the asymptotic solution to evaluate the equivalent damping ratio of the cable-damper system in the open-loop control mode. The individual and combined effects of the damping coefficient, damper stiffness, damper mass and stiffness of damper support on vibration control effectiveness are investigated in detail. The main thrust of the present study is to derive a general formula explicitly relating the normalized system damping ratio and the normalized damper parameters in consideration of all concerned effects, which can be easily used for the design of MR dampers to achieve optimal open-loop vibration control of taut cables.

In this paper, a method for analyzing the damping performance of stay cables incorporating magnetorheological (MR) dampers in the passive control mode is developed taking into account the cable sag and inclination, the damper coefficient, stiffness and mass, and the stiffness of damper support. Both numerical and asymptotic solutions are obtained from complex modal analysis. With the asymptotic solution, analytical formulas that evaluate the equivalent damping ratio of the sagged cable-damper system in consideration of all the above parameters are derived. The main thrust of the present study is to develop an general design formula and a universal curve for the optimal design of MR dampers for adjustable passive control of sagged cables. Two sag-affecting coefficients are derived to reflect the effects of cable sag on the maximum attainable damping ratio and the optimal damper coefficient. For the cable configurations commonly used in cable-stayed bridges, the sag-affecting coefficients are directly expressed in terms of the sag-extensibility parameter to facilitate the control design. A case study on adjustable passive vibration control of the longest cable (536 m) on Stonecutters Bridge is carried out to demonstrate the influence of the sag for the damper design, and to figure out the necessity of adjustability of damper coefficients for achieving maximum damping ratio for different vibration modes.

Modeling of magnetorheological (MR) dampers for cable vibration control to facilitate the design of even more effective and economical systems is still a challenging task. In this study, a parameter-adaptive three-element model is first established for a full-scale MR damper based on laboratory tests. The parameters of the model are represented by a set of empirical formulae in terms of displacement amplitude, voltage input, and excitation frequency. The model is then incorporated into the governing equation of cable-damper system for investigation of open-loop vibration control of stay cables in a cable-stayed bridge. The concept of optimal voltage/current input achieving the maximum damping for the system is put forward and verified. Multi-mode suboptimal and Single-mode optimal open-loop control method is then developed. Important conclusions are drawn on application issues and unique characteristics of open-loop cable vibration control using MR dampers.

One of the most effective countermeasures for mitigating cable vibration is to install mechanical dampers near the anchorage of the cable. Most of the dampers used in the field are so-called passive dampers where their parameters cannot be changed once designed. The parameters of passive dampers are usually determined based on the optimal damper force obtained from the universal design curve for linear dampers, which will provide a maximum additional damping for the cable. As the optimal damper force is chosen based on a predetermined principal vibration mode, passive dampers will be most effective if cable undergoes single-mode vibration where the vibration mode is the same as the principal mode used in the design. However, in the actual engineering practice, multi-mode vibrations are often observed for cables. Therefore, it is desirable to have dampers that can suppress different modes of cable vibrations simultaneously. In this paper, MR dampers are proposed for controlling multi-mode cable vibrations, because of its ability to change parameters and its adaptability of active control without inquiring large power resources. Although the highly nonlinear feature of the MR material leads to a relatively complex representation of its mathematical model, effective control strategies can still be derived for suppressing multi-mode cable vibrations based on nonlinear modelling, as proposed in this paper. Firstly, the nonlinear Bouc-wen model is employed to accurately portray the salient characteristics of the MR damper. Then, the desired optimal damper force is determined from the universal design curve of friction dampers. Finally, the input voltage (current) of MR damper corresponding to the desired optimal damper force is calculated from the nonlinear Bouc-wen model of the damper using a piecewise linear interpolation scheme. Numerical simulations are carried out to validate the effectiveness of the proposed control algorithm for mitigating multi-mode cable vibrations induced by different external excitations.

In this paper, pounding tuned mass dampers (PTMDs) were designed to mitigate the multi-mode vortex-induced vibration (VIV) of stay cable utilizing the viscous-elastic material's energy-dissipated ability. The PTMD device consists of a cantilever metal rod beam, a metal mass block and a specially designed damping element covered with viscous-elastic material layer. Wind-tunnel experiment on VIV of stay cable model was set up to validate the effectiveness of the PTMD on multi-mode VIV mitigation of stay cable. By analyzing and comparing testing results of all testing cases, it could be verified that the PTMD with viscous-elastic pounding boundary can obviously mitigate the VIV amplitude of the stay cable. Moreover, the installed location and the design parameters of the PTMD device based on the controlled modes of the primary stay cable, would have a certain extent suppression on the other modal vibration of the stay cable, which means that the designed PTMDs are effective among a large band of frequency for the multi-mode VIV control of the stay cable.

Cables are prone to vibration due to their low inherent damping characteristics. Recently, negative stiffness dampers have gained attentions, because of their promising energy dissipation ability. The viscous inertial mass damper (termed as VIMD hereinafter) can be viewed as one realization of the inerter. It is formed by paralleling an inertial mass part with a common energy dissipation element (e.g., viscous element) and able to provide pseudo-negative stiffness properties to flexible systems such as cables. A previous study examined the potential of IMD to enhance the damping of stay cables. Because there are already models for common energy dissipation elements, the key to establish a general model for IMD is to propose an analytical model of the rotary mass component. In this paper, the characteristics of the rotary mass and the proposed analytical model have been evaluated by the numerical and experimental tests. First, a series of harmonic tests are conducted to show the performance and properties of the IMD only having the rotary mass. Then, the mechanism of nonlinearities is analyzed, and an analytical model is introduced and validated by comparing with the experimental data. Finally, a real-time hybrid simulation test is conducted with a physical IMD specimen and cable numerical substructure under distributed sinusoidal excitation. The results show that the chosen model of the rotary mass part can provide better estimation on the damper's performance, and it is better to use it to form a general analytical model of IMD. On the other hand, the simplified damper model is accurate for the preliminary simulation of the cable responses.

Inerter-based damping devices (IBBDs), which consist of inerter, spring and viscous damper, have been extensively investigated in vehicle suspension systems and demonstrated to be more effective than the traditional control devices with spring and viscous damper only. In the present study, the control performance on cable vibration reduction was studied for four different inerter-based damping devices, namely the parallel-connected viscous mass damper (PVMD), series-connected viscous mass damper (SVMD), tuned inerter dampers (TID) and tuned viscous mass damper (TVMD). Firstly the mechanism of the ball screw inerter is introduced. Then the state-space formulation of the cable-TID system is derived as an example for the cable-IBBDs system. Based on the complex modal analysis, single-mode cable vibration control analysis is conducted for PVMD, SVMD, TID and TVMD, and their optimal parameters and the maximum attainable damping ratios of the cable/damper system are obtained for several specified damper locations and modes in combination by the Nelder-Mead simplex algorithm. Lastly, optimal design of PVMD is developed for multi-mode vibration control of cable, and the results of damping ratio analysis are validated through the forced vibration analysis in a case study by numerical simulation. The results show that all the four inerter-based damping devices significantly outperform the viscous damper for single-mode vibration control. In the case of multi-mode vibration control, PVMD can provide more damping to the first four modes of cable than the viscous damper does, and their maximum control forces under resonant frequency of harmonic forced vibration are nearly the same. The results of this study clearly demonstrate the effectiveness and advantages of PVMD in cable vibration control.

Passive control may not provide enough damping for a stay cable since the control devices are often restricted to a low location level. In order to enhance control performance of conventional passive dampers, a new type of damper integrated with a rotary electromagnetic damper providing variable damping force and a flywheel serving as an inertial mass, called the rotary electromagnetic inertial mass damper (REIMD), is presented for suppressing the cable vibrations in this paper. The mechanical model of the REIMD is theoretically derived according to generation mechanisms of the damping force and the inertial force, and further validated by performance tests. General dynamic characteristics of an idealized taut cable with a REIMD installed close to the cable end are theoretically investigated, and parametric analysis are then conducted to investigate the effects of inertial mass and damping coefficient on vibration control performance. Finally, vibration control tests on a scaled cable model with a REIMD are performed to further verify mitigation performance through the first two modal additional damping ratios of the cable. Both the theoretical and experimental results show that control performance of the cable with the REIMD are much better than those of conventional passive viscous dampers, which mainly attributes to the increment of the damper displacement due to the inertial mass induced negative stiffness effects of the REIMD. Moreover, it is concluded that both inertial mass and damping coefficient of an optimum REIMD will decrease with the increase of the mode order of the cable, and oversize inertial mass may lead to negative effect on the control performance.

The stochastic stability control of the parameter-excited vibration of an inclined stay cable with multiple modes coupling under random and periodic combined support disturbances is studied by using the direct eigenvalue analysis approach based on the response moment stability, Floquet theorem, Fourier series and matrix eigenvalue analysis. The differential equation with time-varying parameters for the transverse vibration of the inclined cable with control under random and deterministic support disturbances is derived and converted into the randomly and deterministically parameter-excited multi-degree-of-freedom vibration equations. As the stochastic stability of the parameter-excited vibration is mainly determined by the characteristics of perturbation moment, the differential equation with only deterministic parameters for the perturbation second moment is derived based on the $It{\hat{o}}$ stochastic differential rule. The stochastically and deterministically parameter-excited vibration stability is then determined by the deterministic parameter-varying response moment stability. Based on the Floquet theorem, expanding the periodic parameters of the perturbation moment equation and the periodic component of the characteristic perturbation moment expression into the Fourier series yields the eigenvalue equation which determines the perturbation moment behavior. Thus the stochastic stability of the parameter-excited cable vibration under the random and periodic combined support disturbances is determined directly by the matrix eigenvalues. The direct eigenvalue analysis approach is applicable to the stochastic stability of the control cable with multiple modes coupling under various periodic and/or random support disturbances. Numerical results illustrate that the multiple cable modes need to be considered for the stochastic stability of the parameter-excited cable vibration under the random and periodic support disturbances, and the increase of the control damping rather than control stiffness can greatly enhance the stochastic stability of the parameter-excited cable vibration including the frequency width increase of the periodic disturbance and the critical value increase of the random disturbance amplitude.

Using cross-ties to connect cables together when forming a cable network is regarded as an efficient method of mitigating cable vibrations. Cross-ties have been extended and fixed on bridge decks or towers in some engineering applications. However, the dynamics of this kind of system need to be further studied, and the effects of extending cross-links to bridge decks/towers on the modal response of the system should be assessed in detail. In this paper, a system of two cables connected by an inter-supported cross-link with another lower cross-link extended to the ground is proposed and analyzed. The characteristic equation of the system is derived, and some limiting solutions in closed form of the system are derived. Roots of cable system with special configurations are also discussed, attention being given to the case when the two cables are identical. A predictable mode behavior was found when the stiffness of inter-connection cross-link and the cross-link extended to the ground were the same. The vector of mode energy distribution and the degree of mode localization index are proposed so as to distinguish global and local modes. The change of mode behaviors is further discussed in the case when the two cables are not identical. Effects of cross-link stiffness, cross-link location, mass-tension ratio, cable length ratio and frequency ratio on $1^{st}$ mode frequency and mode shape are addressed.

Vibration mitigation of cables or hangers is one of the crucial problems for cable supported bridges. Previous research focused on the behaviors of cable with dampers or crossties, which could help engineering community apply these mitigation devices more efficiently. However, less studies are available for hybrid applied cross-ties and dampers, especially lack of both analytical and experimental verifications. This paper studied damping and frequency of two parallel identical cables with a connection cross-tie and an attached damper. The characteristic equation of system was derived based on transfer matrix method. The complex characteristic equation was numerically solved to find the solutions. Effects of non-dimensional spring stiffness and location on the maximum cable damping, the corresponding optimum damper constant and the corresponding frequency of lower vibration mode were further addressed. System with twin small-scale cables with a cross-link and a viscous damper were tested. The damping and frequency from the test were very close to the analytical ones. The two branches of solutions: in-phase modes and the out-of-phase modes, were identified; and the two branches of solutions were different for damping and frequency behaviors.

In August 2012, during the passage of the typhoon Haikui (1211), large amplitude vibrations were observed on long hangers of the Xihoumen suspension Bridge, which destroyed a few viscoelastic dampers originally installed to connect a pair of hanger ropes transversely. The purpose of this study is to identify the cause of vibration and to develop countermeasures against vibration. Field measurements have been conducted in order to correlate the wind and vibration characteristics of hangers. Furthermore, a replica aeroelastic model of prototype hangers consisting of four parallel ropes was used to study the aeroelastic behavior of hanger ropes and to examine the effect of the rigid spacers on vibration mitigation. It is shown that the downstream hanger rope experiences the most violent elliptical vibration for certain wind direction, and the vibration is mainly attributed to wake interference of parallel hanger ropes. Based on wind tunnel tests and field validation, it is confirmed that four rigid spacers placed vertically at equal intervals are sufficient to suppress the wake-induced vibrations. Since the deployment of spacers on hangers, server hanger vibrations and clash of hanger ropes are never observed.

The number of cable-stayed bridges has been increasing worldwide, causing issues in maintaining the structural safety and integrity of bridges. The stay cable, one of the most critical members in cable-stayed bridges, is vulnerable to wind-induced vibrations owing to its inherent low damping capacity. Thus, vibration mitigation of stay cables has been an important issue both in academia and practice. While a semi-active control scheme shows effective vibration reduction compared to a passive control scheme, real-world applications are quite limited because it requires complicated equipment, including for data acquisition, and power supply. This study aims to develop an Arduino-based integrated cable vibration control system implementing a semi-active control algorithm. The integrated control system is built on the low-cost, low-power Arduino platform, embedding a semi-active control algorithm. A MEMS accelerometer is installed in the platform to conduct a state feedback for the semi-active control. The Linear Quadratic Gaussian control is applied to estimate a cable state and obtain a control gain, and the clipped optimal algorithm is implemented to control the damping device. This study selects the magnetorheological damper as a semi-active damping device, controlled by the proposed control system. The developed integrated system is applied to a laboratory size cable with a series of experimental studies for identifying the effect of the system on cable vibration reduction. The semi-active control embedded in the integrated system is compared with free and passive mode cases and is shown to reduce the vibration of stay-cables effectively.