Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.     

Number of Cable Effects on Buckling Analysis of Cable-Stayed Bridges

Cables instead of interval piers support cable-stayed bridges, and the bridge deck is subjected to strong axial forces due to the horizontal components of cable reactions. The structural behavior of a bridge deck becomes nonlinear because of the axial forces, large deflection, and nonlinear behavior of the cables and the large deformation of the pylons as well as their interactions. The locations and amplitude of axial forces acting on the bridge deck may depend on the number of cables. Agrawal indicated that the maximum cable tension decreases rapidly with the increase in the number of cables. This paper investigates the stability analysis of cable-stayed bridges and considers cable-stayed bridges with geometry similar to those proposed in Agrawal's paper. A digital computer and numerical analysis are used to examine 2D finite element models of these bridges. The eigen buckling analysis has been applied to find the minimum critical loads of the cable-stayed bridges. The numerical results indicate that the total cumulative axial forces acting on the bridge girder increase as the number of cables increases, yet because the bridge deck is subjected to strong axial forces, the critical load of the bridges decreases. Increasing the number of cables may not increase the critical load on buckling analysis of this type of bridge. The fundamental critical loads increase if the ratio of Ip∕Ib increases until the ratio reaches the optimum ratio. If the ratio of Ip∕Ib is greater than the optimum ratio, depending on the geometry of an individual bridge, the fundamental critical load decreases for all the types of bridges considered in this paper. In order to make the results useful, they have been normalized and represented in graphical form.     

Stability Analysis of Box-Girder Cable-Stayed Bridges

The objective of this study is to investigate the stability characteristics of box-girder cable-stayed bridges by three-dimensional finite-element methods. Cable-stayed bridges have many design parameters, because they have a lot of redundancies, especially for long-span bridges. Cable-stayed bridges exhibit several nonlinear behaviors concurrently under normal design loads because of large displacements; the interaction among the pylons, the stayed cables, and the bridge deck; the strong axial and lateral forces acting on the bridge deck and pylons; and cable nonlinearity. A typical two-lane, three-span, steel box-girder cable-stayed bridge superstructure was selected for this paper. The numerical results indicate that, if the ratio of the main span length with respect to the total span length, L1∕L, is small, the structure usually has a higher critical load. If the ratio Ip∕Ib increases, the critical load of the bridge decreases, in which Ip is the moment of inertia of the pylon and Ib is the moment of inertia of the bridge deck. When the ratio Ip∕Ib is greater than 10.0, the decrement becomes insignificant. For cable arrangements, bridges supported by a harp-type cable arrangement are the better design than bridges supported by a fan-type cable arrangement on buckling analysis. The numerical results also indicate that use of either A-type or H-type pylons does not significantly affect the critical load of this type of structure. In order to make the numerical results useful, the buckling loads have been nondimensionalized and presented in both tabular and graphical forms.     

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.     

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.     

Effect of Hanger Flexibility on Dynamic Response of Suspension Bridges

Linearized continuum models of a suspended span with unloaded backstays and of a symmetric three-span suspension bridge are used to study the effects of the flexibility of the hangers on the vertical vibrations of suspension bridges. The models include elastic parabolic cables, flexible distributed hangers with variable length, and a stiffening girder represented by an elastic beam. It is shown that the free vibrations of a suspended span with unloaded backstays are controlled by five dimensionless parameters, while six dimensionless parameters control the response of a symmetric three-span suspension bridge. The results indicate that the flexibility of the hangers has a significant effect on the natural frequencies of the higher modes only when the relative stiffness of the girder is very high. The effects of hanger flexibility on the response of a suspension bridge to localized impulsive loads are also found to be small. These findings confirm the traditional, albeit previously untested, assumption of inextensible hangers. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.     

Deterministic Control of Column under Horizontal-Vertical Excitation

A control methodology is presented for a column, modeled as a single-degree-of-freedom system subjected to simultaneous horizontal and vertical support motion. It is assumed that the support motions are uncertain, time varying, and norm bounded, and that the column itself has no uncertainties associated with it. The control signal is based on Lyapunov theory and noise-free state feedback measurements. The states of the resulting closed-loop system (namely displacement and velocity) are uniformly and ultimately bounded within a neighborhood of the zero state. To illustrate the features of the proposed controller, examples of nonhysteretic and hysteretic columns subjected to combined horizontal and vertical nonstationary seismic excitation are considered. Numerical results for the uncontrolled and controlled responses are obtained and analyzed. Several issues involved in the controller design are examined, and results illustrating the control performance and effectiveness are presented and discussed.     

Scheduled Controllers for Buildings under Seismic Excitation with Limited Actuator Capacity

A new class of scheduled controllers is presented for improving the response of seismic-excited buildings under actuators with limited capacity. The controller is scheduled based on the response of the system on-line, to take full advantage of the inevitably limited actuation. While the controller is nonlinear, the overall approach relies on a variety of techniques from linear designs, which makes it possible to generalize the approach to a variety of systems and control objectives. For example, sufficient conditions for feasibility of a feedback controller are expressed in terms of linear matrix inequalities, thus solvable with standard software. Both state feedback and observer-based controllers are discussed. Performance of the proposed technique is illustrated through simulations of a six-story building subject to earthquake ground motion.     

Design of Mechanical Viscous Dampers for Stay Cables

External dampers have been utilized in a number of cable-stayed bridges to suppress transverse cable vibrations. However, simple and accurate damper design recommendations that concurrently consider all important cable parameters are lacking. Previous efforts have been based on the idealization of cables as taut strings. In this paper, the governing differential equation for vibration of cables containing a viscous damper was first converted to a complex eigenvalue problem containing nondimensional cable parameters. Then, a parametric study was conducted involving repeated solutions of the eigenvalue problem for a wide range of nondimensional parameters. Based on the results of the parametric study, the effects of dampers on first mode vibration frequencies and first mode cable damping ratios were presented in nondimensional format. It is shown that for the range of parameters involved in most stay cables, the influence of cable sag is insignificant, whereas the cable bending stiffness can have a significant influence on the resulting cable damping ratios. Simplified nondimensional relationships are proposed for calculating damper-induced changes in the first mode cable damping ratios. Results of laboratory tests on a scaled model cable are compared with the estimated values using the formulation presented. Finally, example problems are presented for comparison with other relationships, and for the design of mechanical viscous dampers for suppression of cable vibrations including rain-wind induced vibrations.     

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.     

Automatic Downstream Water-Level Feedback Control of Branching Canal Networks: Theory

Most of the research on the design of feedback controllers for irrigation canals has been concentrated on single, in-line canals with no branches. Because the branches in a network are hydraulically coupled with each other, it may be difficult to automatically control a branching canal network by designing separate feedback controllers for each branch and then letting them run simultaneously. Thus feedback control of an entire branching canal system may be more efficient if the branching flow dynamics are explicitly taken into account during the feedback controller design process. This paper develops two different feedback controllers for branching canal networks. The first feedback controller was developed using linear quadratic regulator theory and the second using model predictive control. Both algorithms were able to effectively control a simple branching canal network example with relatively small flow changes.     

Multiobjective Optimal Fuzzy Logic Controller Driven Active and Hybrid Control Systems for Seismically Excited Nonlinear Buildings

The third generation benchmark control problem for seismically excited nonlinear buildings is an effort to evaluate the developed control strategies in order to apply them in field applications. As the fuzzy logic control systems have been applied effectively in various fields, including vibration control of structures, a multiobjective optimal fuzzy logic control system has been proposed in this paper. Two types of control devices, namely, active and hybrid, driven by a fuzzy logic controller (FLC) have been considered in the present study. Nondimensionalized peak interstory drift ratio and peak floor acceleration have been used as the two objective functions for the multiobjective optimal design problem. A two-branch tournament genetic algorithm has been used to find a set of Pareto-optimal solutions, as the optimization problem is not necessarily continuous or convex. Performance of the FLC driven active and hybrid control systems have been evaluated for all three third generation benchmark problems for seismically excited nonlinear buildings (3-, 9- and 20-story). Acceleration and velocity information of different floors have been used as feedback to the FLC. This approach provides a set of Pareto optimal designs, from which a controller design can be selected for the required performance. The FLC driven active control system performs better than the sample controller given in the benchmark problem. Though the number of sensors and control devices are far less, the performance of the hybrid is close to the active control system.     

Study of Passive Deck-Flaps Flutter Control System on Full Bridge Model. I: Theory

A passive aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Control flap rotations are governed by prestressed springs and additional cables spanned between the control flaps and an auxiliary transverse beam supported by the main cables of the bridge. The rotational movement of the flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited forces for the whole three-dimensional suspension bridge model is obtained through a rational function approximation of the generalized Theodorsen function and implemented in the FEM formulation. This paper lays the theoretical groundwork for the one that follows.     

轧机液压压下系统非线性振动诱因分析

 具有液压压下系统的轧机被广泛应用在板带生产过程中,为解决液压压下系统引起的轧机非线性振动问题,在闭环控制情况下分别研究了不同阻尼系数、泄漏系数以及控制器比例系数3种系统参数对压下系统垂振的影响。考虑压下缸非线性液压弹簧力建立了轧机液压压下系统垂振模型,通过振动位移时域图和位移-速度相图分析了不同系统参数与系统垂振的相关性。仿真结果表明,由于轧机液压压下系统采用传统PID控制器,液压系统本身阻尼系数、泄漏系数等具有慢时变特性,会使得原有PID控制可能出现功能失效情况,造成系统垂振的发生,该研究为后续设计控制器消除液压系统参数变动引起的垂振有一定的理论和工程实际意义。     

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.     

Suppression of Bridge Flutter by Active Deck-Flaps Control System

A theoretical study on an aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Their rotational movement, commanded via feedback control law, is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time domain formulation of self-excited and buffeting forces is obtained through the rational function approximation of the generalized Theodorsen function. The optimal configuration of the deck-flaps system is found with respect to the performance index based on stability robustness of the system. A control system with the rotational center of the flaps that is located on the edges of the deck was found to be the most effective. It is also shown that this control system can provide sufficient aerodynamic damping and satisfactory stability robustness of the system with a relatively small flap size for the considered range of wind speed.     

Three-Dimensional Elastic Catenary Cable Element Considering Sliding Effect

The nonlinear behavior of cable-supported bridges is governed by the geometric nonlinearity of cables, which is attributable to sag and sliding effects at the saddle. In a cable-stayed bridge with a midspan saddle, and in all suspension bridges, cable sliding can occur at the saddle under extreme forces, such as those caused by an earthquake or typhoon. However, the conventional method of analysis of cable-supported bridges does not consider the effect of cable sliding at the saddle; instead it regards those cables as fixed. This assumption might lead to a misunderstanding of the global structure system. The goal of this study is to develop a three-dimensional (3D) elastic cable finite element that considers the sliding effect and uses a geometric nonlinear cable finite element based on elastic catenary theory. In this study, two types of sliding were considered: the roller sliding condition without friction and the frictional sliding condition. These were formulated to derive the nodal force vectors and tangential stiffness matrices. To validate the proposed 3D cable sliding element, experiments were conducted for both sliding conditions, and results were compared with calculations of the amount of sliding and displacement at the loading point. In addition, a cable-supported structural system was analyzed to investigate the characteristics of a realistic structure with cable sliding. Overall calculations using the 3D cable sliding model were in very good agreement with the measured values.     

Evaluation, Rehabilitation Planning, and Stay-Cable Replacement Design for the Hale Boggs Bridge in Luling, Louisiana

The Hale Boggs Bridge opened to traffic on October 5, 1983. At the time, it was the first U.S. cable-stayed crossing over the Mississippi River. The PE (polyethylene) protective sheathing was damaged in many of the cables before and during installation, and after the opening of the bridge to traffic. Repairs were attempted to correct the defects in cable sheathing. Many of the repairs performed poorly and failed to protect the main tension element. The condition of 39 out of 72 cables indicated a critical need for repair and timely action was recommended. To address these damages, and to assure the structural integrity of the bridge structure, several strategies involving a range of repair and replacement options were evaluated using life cycle cost analysis. It was concluded that the strategy to replace all cables presents the best value among evaluated alternatives. The design of the complete 72 cable array replacement is the first occasion on which this process is attempted in North America. The final design of the replacement cables is heavily influenced by the geometric restrictions of the existing anchorage locations. The replacement cables are being designed for a 75-year design life and incorporated with the latest advancements in corrosion protection and vibration control. Maintenance of traffic design is an essential part of the project. The bridge is a critical regional link and constitutes a hurricane evacuation route. Traffic maintenance during cable replacement was designed to be as unobtrusive to the public and commerce as practical. This paper describes efforts associated with cable condition assessment, rehabilitation strategy, and design considerations and concepts, undertaken by the writers since 2002 to improve the condition of this major river crossing.     

Automatic Downstream Water-Level Feedback Control of Branching Canal Networks: Simulation Results

Previous research on canal automation has dealt with the control of single, in-line canals, while canal operators typically have to control an entire network of canals. Because the branches in a network are hydraulically coupled with each other, control of a branching canal network based on separate controllers for each branch may not be the most effective control strategy. A methodology by which existing automatic control systems could be modified to control branching canal networks is provided in a companion paper. This paper presents results of hydraulic simulations of the new methodology to estimate the controllability of a large portion of the branching canal network operated by the Salt River Project (SRP). Two types of controllers were used for this study: (1) linear quadratic regulator (LQR) and (2) model predictive control (MPC). Both controllers used the same underlying process model [integrator-delay (ID) model], and both controllers were capable of feedback and feedforward control. Under feedback control alone, both controllers gave similar performance, but were unable to effectively control the overall system because of the long delay times. When feedforward control was added to the feedback controller, both of these control systems were able to effectively control the branching canal network operated by SRP. For the LQR controller, the volume compensation method for routing known demand change was used as the feedforward controller. For the MPC controller, the ID model was used as the feedforward controller. Slight differences were noted between the performance of the two feedforward controllers.     

Nonlinear Analytical Solution for Cable Truss

In this technical note the nonlinear closed-form static solution of the suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span is presented. Irvine’s linearized forms of the deflection and the cable equations are modified because the effects of the nonlinear truss behavior needed to be incorporated in them. The concrete form of the system of two nonlinear cubic cable equations is derived and presented. From a solution of a nonlinear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The transformation analytical model serves to determine the response, i.e., horizontal components of cable forces and deflection of the geometrically nonlinear truss, due to the applied loading, considering effects of elastic deformations, temperature changes, and elastic supports. The deflection of asymmetric prestressed cable trusses has been compared with Irvine’s linear solution as well as the nonlinear finite element model results.