Damping of Cables by a Transverse Force

A general asymptotic format is presented for the effect on the modal vibrations of a transverse damper close to the end of a cable. Complete locking of the damper leads to an increase of the natural frequencies, and it is demonstrated that the maximum attainable damping is a certain fraction of the relative frequency increase, depending on the type of damping device. The asymptotic format only includes a real and a complex nondimensional parameter, and it is demonstrated how these parameters can be determined from the frequency increase by locking and from an energy balance on the undamped natural vibration modes. It is shown how the asymptotic format can incorporate sag of the cable, and specific results are presented for viscous damping, the effect of stiffness and mass, fractional viscous damping, and a nonlinear viscous damper. The relation of the stiffness component to active and semiactive damping is discussed.     

Semiactive Damping of Stay Cables

Stay cables, such as are used in cable-stayed bridges, are prone to vibration due to their low inherent damping characteristics. Transversely attached passive viscous dampers have been implemented in many bridges to dampen such vibration. Several studies have investigated optimal passive linear viscous dampers; however, even the optimal passive device can only add a small amount of damping to the cable when attached a reasonable distance from the cable/deck anchorage. This paper investigates the potential for improved damping using semiactive devices. The equations of motion of the cable/damper system are derived using an assumed modes approach and a control-oriented model is developed. The control-oriented model is shown to be more accurate than other models and facilitates low-order control designs. The effectiveness of passive linear viscous dampers is reviewed. The response of a cable with passive, active, and semiactive dampers is studied. The response with a semiactive damper is found to be dramatically reduced compared to the optimal passive linear viscous damper for typical damper configurations, thus demonstrating the potential benefits using a semiactive damper for absorbing cable vibratory energy.     

Experimental Verification of Smart Cable Damping

Stay cables, such as are used in cable-stayed bridges, are prone to vibration due to their low inherent damping characteristics. Transversely attached passive viscous dampers have been implemented in many bridges to dampen such vibration. However, only minimal damping can be added if the attachment point is close to the bridge deck. For longer bridge cables, the relative attachment point becomes increasingly smaller, and passive damping may become insufficient. A recent analytical study by the authors demonstrated that “smart” semiactive damping can provide increased supplemental damping. This paper experimentally verifies a smart damping control strategy employing H2/linear quadratic Gaussian (LQG) clipped optimal control using only force and displacement measurements at the damper for an inclined flat-sag cable. A shear mode magnetorheological fluid damper is attached to a 12.65?m inclined flat-sag steel cable to reduce cable vibration. Cable response is seen to be substantially reduced by the smart damper.     

Vibration Control of Stay Cables of the Shandong Binzhou Yellow River Highway Bridge Using Magnetorheological Fluid Dampers

The Shandong Binzhou Yellow River Highway Bridge is a three-tower, cable-stayed bridge in Shandong Province, China. Because the stay cables are prone to vibration, 40 magnetorheological (MR) fluid dampers were attached to the 20 longest cables of this bridge to suppress possible vibration. An innovative control algorithm for active and semiactive control of mass-distributed dynamic systems, e.g., stay cables, was proposed. The frequencies and modal damping ratios of the unimpeded tested cable were identified through an ambient vibration test and free vibration tests, respectively. Subsequently, a series of field tests were carried out to investigate the control efficacy of the free cable vibrations achieved by semiactive MR dampers, “Passive-off” MR dampers and “Passive-on” MR dampers. The first three modal damping ratios of the cable incorporated with the MR dampers were also identified from the in situ experiments. The field experiment results indicated that the semiactive MR dampers can provide significantly greater supplemental damping for the cable than either the Passive-off or the Passive-on MR dampers because of the pseudonegative stiffness generated by the semiactive MR dampers.     

Free Vibrations of Taut Cable with Attached Damper. I: Linear Viscous Damper

Free vibrations of a taut cable with an attached linear viscous damper are investigated in detail using an analytical formulation of the complex eigenvalue problem. This problem is of considerable practical interest in the context of stay-cable vibration suppression in bridges. An expression for the eigenvalues is derived that is independent of the damper coefficient, giving the range of attainable modal damping ratios and corresponding oscillation frequencies in every mode for a given damper location without approximation. This formulation reveals the importance of damper-induced frequency shifts in characterizing the response of the system. New regimes of behavior are observed when these frequency shifts are large, as is the case in higher modes and for damper locations further from the end of the cable. For a damper located sufficiently near the antinode in a given mode, a regime of solutions is identified for which the damping approaches critical as the damper coefficient approaches a critical value. A regime diagram is developed to indicate the type of behavior in each mode for any given damper location.     

Isolation System for Vibration Control of Stay Cables

Rain-wind induced cable vibration can cause serious problems in cable-stayed bridges. Externally attached dampers have been used to provide an effective means to suppress the vibration of relatively short stay cables. For very long stay cables, however, such damper systems are rendered ineffective, as the dampers need be attached near the end of the cables for aesthetic reasons. This paper investigates a new stay-cable isolation system to mitigate the cable vibration. The proposed isolation system, which consists of a laminated rubber bearing and an internal damper, may be installed inside of the cable anchorage. A simple analytical model of the cable-damper system is developed first based on the taut string representation of the cable. The response of a cable with the proposed isolation system is obtained and then compared to those of the cable with and without an external passive damper. The proposed stay-cable isolation system is shown to perform better than the optimal passive viscous damper, thereby demonstrating its applicability in large cable-stayed bridges.     

Dynamic Behavior of Stay Cables with Rotational Dampers

Vibration reduction in stay cables by means of viscous dampers is of great interest in cable damage prevention and serviceability of structural system supported by such cables. The paper presents a study on the effectiveness, as well as the limits, of rotational viscous dampers and springs inserted at the two ends of a bending-stiff taut cable; influence of rotational stiffness of the springs is also investigated. After a nondimensional expression of the equation of motion has been obtained, as in other cases of nonproportionally damped continuous structures, complex modal analysis is pursued, obtaining complex eigenvalues and eigenfunctions. Comparison with intermediate dampers, widely used in bridge engineering, is performed showing the range of nondimensional parameters for which the proposed approach is of interest. Finally, a numerical technique based on complex mode superposition is presented in order to evaluate time domain responses for transversal distributed excitation. As an example, the procedure is applied to a wind-exposed cable.     

Free Vibrations of Taut Cable with Attached Damper. II: Nonlinear Damper

Free vibrations of a taut cable with a nonlinear power-law damper attached near the end are considered. An approximate analytical solution for the amplitude-dependent effective damping ratios in each mode is developed by assuming the same form of solution as for the linear damper and minimizing the mean-square error in the force equilibrium at the damper. An asymptotic approximate solution for small frequency shifts reveals a nondimensional grouping of parameters allowing the development of an amplitude-dependent “universal estimation curve” for the power-law damper. The shape of the universal curve is slightly different for each value of the damper exponent, but for a given exponent the curve is nearly invariant over the same range of parameters as the universal curve for the linear damper. This formulation yields insights into the dependence of nonlinear damper performance on mode number and amplitude of oscillation, suggesting potential advantages that may be offered by a nonlinear damper over a traditional linear damper.     

Vibration of Tensioned Beams with Intermediate Damper. II: Damper near a Support

Analytical solutions are used to investigate the free vibrations of tensioned beams with a viscous damper attached transversely near a support. This problem is of particular relevance for stay-cable vibration suppression, but no restrictions on the level of axial load are introduced, and the results are quite broadly applicable. Characteristic equations for both clamped and pinned supports are rearranged into forms suitable for numerical solution by fixed-point iteration, whereby the complex eigenfrequencies and corresponding damping ratios can be accurately computed within a few iterations. Explicit asymptotic approximations for the complex eigenfrequencies are also obtained, subject to restrictions on the closeness of the eigenfrequencies to their undamped values. These asymptotic approximations are expressed in the same “universal” form identified in previous studies. It is observed that the maximum attainable modal damping ratios and the corresponding optimal values of the damper coefficient can be significantly affected by bending stiffness and by the nature of the support conditions, and a nondimensional parameter grouping is identified that enables an assessment of when bending stiffness should be considered.     

Damping of Taut-Cable Systems: Two Dampers on a Single Stay

The mitigation of in-plane stay oscillation in cable-stayed bridges is commonly addressed by placing an external mechanical damper, linear or nonlinear, on each stay or by introducing transverse cross-ties among cables. Although the problem of a cable with a single external damper has found significant attention in the past and different techniques have been proposed for the solution of the free-vibration problem, limitations are related to the fact that the location of the damper is usually very close to the cable end (on the bridge deck side) due to geometric constraints, leading to inherently low modal damping in the fundamental modes. In this paper the installation of more than one damper on an individual stay is considered to overcome such limitations and to increase the overall performance of the system. An existing procedure, based on the linearized taut-string theory, was modified to allow for the presence of multiple external discrete viscous dampers. The case of two devices with arbitrary location has been solved, identifying advantages and disadvantages of the proposed solution. In addition, extensions of the practical “universal curve” and the interpretation thereof are presented.     

Combined Damping Effect of Two Dampers on a Stay Cable

The combined effect of two dampers, either on the same end or opposite ends of a stay cable, is analytically studied in this paper. By considering small distances of the dampers from the anchorages, an asymptotic formula for the modal damping ratio of the cable is derived from which the total damping effect is presented in an explicit form. It is shown that when two dampers are installed at opposite ends of the cable, the total damping effect is asymptotically the sum of the contributions from single dampers. On the contrast, if two dampers are at the same end, there is no advantage of increasing the maximum modal damping in the cable over the use of a single damper.     

Analytical Study on Bending Effects in a Stay Cable with a Damper

The effects of bending on the modal properties of a stay cable with a transverse damper are analytically studied. Considering that the value of the flexural rigidity in the stay cable is small in practice, an explicit asymptotic formula for the modal damping of a cable with a general type of damper is derived. For a viscous damper, the asymptotic formula obtained is compact, accurate, and thus is very suitable for practical design. Furthermore, for the first few vibration modes of interest, the asymptotic solution is independent of the modal index. It is shown that flexure in the cable reduces the maximum attainable modal damping, possibly up to 20%, while it significantly increases the optimal damping coefficient of the damper.     

Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.     

Evaluation of Viscous Dampers for Stay-Cable Vibration Mitigation

Many cable-stayed bridges around the world have displayed excessive and unanticipated vibrations of the main stays, often associated with the simultaneous occurrence of wind and rain, and mitigation of these vibrations has become a significant concern in cable-stayed bridge design and retrofit. Much of the previous research on this problem has been conducted using wind tunnels, and there have been relatively few opportunities to measure the vibrations at full-scale. This paper presents results from long-term field measurements of cable vibrations on a cable-stayed bridge in the United States. Characteristics of different types of measured vibrations are summarized, and the effectiveness of passive linear dampers in vibration suppression is evaluated by comparing response statistics from two stays before and after installation of dampers and by investigating in detail the damper performance in a few selected records corresponding to different types of excitation. The dampers are observed to be quite effective, but a fundamental limitation of mode-dependence in linear damper performance is emphasized, and some potential advantages offered by a nonlinear damper are discussed.     

Performance and Repair Design of Stay Cables with Rain/Wind-Induced Vibrations

The results of a multifaceted investigation into large-amplitude vibrations of bridge stay cables are presented. The vibration of the stay cables occurred under light rain and wind conditions, and resulted in low-frequency, large-amplitude oscillations having recorded displacements of over 24?in. (0.6?m). Due to concerns about the implications of the vibrations on the integrity and durability of the stay cables and observed damage to stay-cable anchorage components, an investigation and repair program was developed. The final repair program arose from an investigation that combined the talents of practicing engineers, the Texas Department of Transportation, and representatives from four universities. Ultimately, a combination of supplemental hydraulic dampers and cable restrainers were used to mitigate the vibrations. This paper describes the results of the investigation and presents the practice-based methodology used to develop the implemented repair program.     

Optimal Vibration Absorber with Nonlinear Viscous Power Law Damping and White Noise Excitation

A tuned mass damper with a nonlinear power law viscous damper excited by white noise is considered. The system is analyzed by statistical linearization and stochastic simulation with the objective of minimizing the standard deviation of the response. It is shown that the optimal parameters for the tuned mass damper are unaffected by the magnitude of the structural damping in the linear case. However, in the nonlinear case the structural damping influences the equivalent parameters obtained by statistical linearization and thereby indirectly the optimal values for the damper parameters. Results from stochastic simulation show good agreement with results from statistical linearization in terms of the standard deviation of the response. It is shown that the optimal damping, which can be obtained by the passive device, is the same for the linear and nonlinear damper. However, for the nonlinear tuned mass damper the optimal parameters will depend on both structural damping and excitation intensity (or vibration amplitude). The results are presented in such a way that they can be used directly for the design of a tuned mass damper with damping governed by a nonlinear viscous power law.     

Damping of Taut-Cable Systems: Effects of Linear Elastic Spring Support

The influence of linear elastic support on the damper effectiveness of a cable-damper system was investigated by modeling the system as a taut string, an intermediate damper, and a spring in series. Two types of damper were analyzed in this study: (1)?the linear elastic damper; and (2)?the friction threshold. An exact formulation for the free vibration of the system was developed for the linear viscous damping system, and a complex eigenfrequencies equation was worked to obtain the explicit solution for the frequency shift. A damping ratio equation for different modes, which depicts the effect of the spring, was developed from the frequency shift. An effective flexibility coefficient was introduced to investigate the effect of different values of support stiffness on the effectiveness of the linear viscous damper. A universal curve family diagram was constructed, which indicated that linear elastic support reduces the effectiveness of the linear viscous damper. The universal curve obtained previously by Main and Jones was a special case of this universal curve family for the case in which the stiffness of the support approached infinity. The equation of maximum force introduced to the spring was also derived and was shown to be positively related to the cable tension force and the cable vibration amplitude at the damper attachment location. The influence of the linear elastic support on a cable-damper system with a friction threshold was also investigated by using the result of the linear viscous damper and the equivalent energy method. The result showed that the linear elastic support also reduces the effectiveness of the friction threshold. An equation showing how to select an optimal friction threshold for a stay cable was also proposed.     

Performance of Tuned Liquid Dampers

This paper investigates the performance of unidirectional and bidirectional tuned liquid dampers (TLDs) under random excitation. The performance of the tuned liquid dampers is measured in terms of efficiency and robustness. A series of experimental tests are conducted on model scale structure-tuned liquid damper systems to evaluate their performance, which is then compared to that of the well known tuned mass damper. The effective damping is calculated for each test conducted and the efficiency and robustness are subsequently examined. The performance of a mistuned TLD is experimentally investigated to highlight the robustness of these passive dynamic vibration absorbers. A nonlinear numerical model is used to conduct an extensive parametric study on the performance of a tuned liquid damper. This study has resulted in the development of performance charts for a tuned liquid damper. These charts allow the efficiency of a tuned liquid damper to be examined for a number of varying parameters, which include the excitation amplitude, water depth, and building frequency. These charts are particularly useful for the initial design of a tuned liquid damper when the precise frequency of the structure is not known.     

Dynamics of SDOF Systems with Nonlinear Viscous Damping

The effects of nonlinear viscous damping on the dynamic response of single-degree-of-freedom (SDOF) structural systems are analyzed. This kind of damping characterizes a special class of fluid viscous dampers recently utilized in the field of vibration control as base-isolation devices or viscoelastic elements included in steel braces of framed structures. The analytical relationship adopted to reproduce the mechanical behavior of the fluid viscous dampers is a fractional power-law of the velocity, the exponent of which ranges between 0.1 and 0.2. This function had been previously calibrated on the results of a special experimental survey carried out at the University of Florence. The dynamics of the classical linear-viscous SDOF oscillator is herein reformulated on the basis of the above-mentioned fractional viscous damping (FrVD) relationship. In particular, the transient and steady-state responses are examined in both free and forced vibration conditions. The magnification and transmissibility factors are analytically determined for different damping levels. Moreover, the relation between the viscous damping coefficient and the frequency ratio (i.e., the ratio of the dynamic load to the oscillator frequencies) is defined. The diagrams describing these functions provide direct correlations between the damping as well as the elastic properties of the system and the frequency content of the dynamic action.     

Semiactive Control of the 20-Story Benchmark Building with Piezoelectric Friction Dampers

A new control algorithm is proposed in this paper to control the responses of a seismically excited, nonlinear 20-story building with piezoelectric friction dampers. The passive friction damping mechanism is used for low-amplitude vibration while the active counterpart takes over for high-amplitude vibration. Both the stick and sliding phases of dampers are taken into account. To effectively mitigate the peak story drift and floor acceleration of the 20-story building, multiple dampers are placed on the 20-story building based on a sequential procedure developed for optimal performance of the dampers. Extensive simulations indicate that the proposed semiactive dampers can effectively reduce the seismic responses of the uncontrolled building with substantially less external power than its associated active dampers, for instance, 67% less under the 1940 El Centro earthquake when the passive friction force is equal to 10% of the damper capacity.