Semi-Active Vibration Control of Stay Cables Incorporated with Magneto-Rheological Fluid Damper

Abstract

To explore the addressed issue of closed-loop vibration control using semi-active magneto-rheological (MR) fluid damper, an innovative control algorithm based on feedback of acceleration response at several points along stay cable for active/semi-active control of stay cables is firstly proposed in the present paper. Theoretical analysis, laboratory investigation, and in-situ tests are conducted in this study to verify the proposed control algorithm. Moreover, this paper presents the negative stiffness characteristic of the combined stay cable/semi-active MR damper system. These investigations indicate that the semi-active MR damper can achieve much better mitigation efficacy than the other passive MR dampers and the proposed control algorithm is proved to be practical. The effect of the negative stiffness on response reduction is demonstrated by the improved energy dissipation ability of the damper due to the enhanced displacement of the stay cable at the location attached with the damper. Introduction In the past few decades, many control methods have been proposed and some have been implemented to mitigate the unaccepted stay cable vibration (Johnson, 2003). At present, semi-active dampers have been proven to be effective in many applications and can potentially achieve performance levels that are almost the same as comparable active devices. For stay cables vibration control incorporated with semi-active dampers, most of the existing control strategies are based on full-state feedback of generalized coordinates. However, the generalized displacements and velocities of the stay cable cannot be measured directly. Since accelerometers can readily provide reliable and inexpensive measurements of the stay cable accelerations at strategic points on a stay cable, development of control method based on acceleration feedback at limited locations along stay cable is an ideal solution to solve the addressed practical problem. Moreover, even though the characteristic of negative stiffness of active control of stay cable vibration has been observed by some researchers(Johnson, 2003), the effects of negative stiffness on the stay cable vibration control using active/semi-active dampers 2 has not been fully researched in the present literatures. In this paper, a control algorithm based on feedback of acceleration response at several points for active and semi-active control of mass-distributed dynamic systems, e.g. stay cables is firstly proposed. Using the proposed control algorithm, the vibration control for the stay cable model attached with one MR damper and a series of field tests are carried out to investigate the control efficacy achieved by different control strategies. These investigations address that the semi-active MR damper can achieve much better mitigation efficacy than other passive MR dampers and the proposed control algorithm is proved to be practicable and available. Moreover, the effect of the negative stiffness provided by semi-active MR dampers on response reduction of cables is theoretically and numerically demonstrated by the improved energy dissipation ability of the damper due to the enhanced displacement of the cable at the location attached with the damper. Control Algorithm Based on Acceleration Feedback at Several Locations Governing equation of the combined stay cable/MR damper system shown in Figure 1 is given as (Irvine, 1981) ) ( ) ( ) , ( ) , ( ) , ( ) , ( d s x x t u t x f t x v c t x v T t x v m − + = + ′ ′ − δ & & & (1) with the boundary conditions v(0,t)=v(l,t)=0 for all t .where v(x,t) is the transverse deflection of the cable, l is the length of the cable, T is the cable tension, m and c are the cable mass and viscous damping per unit length, respectively, xd is the distance of the installed MR damper position from the left end, us(t) is the transverse control force provided by the MR damper at location x= xd, f(x,t) is the distributed load along the cable. δ(.) is the Dirac delta function. Figure1. Combined stay cable/MR damper system The transverse deflection could be approximated using a finite series ) ( ) ( ) , ( ∑ = = r j j j x t q t x v 1 φ (2) where the φj(x) is a set of shape modal functions and continuous with piecewise continuous slope to satisfy the geometric boundary conditionsφj(0)= φj(l)=0. Utilizing the approximate series solution and a standard Galerkin approach, the motion equation of the cable-MR damper system in matrix form is written as ) ( ) ( f Kq q C q M q t u x s d φ + = + + & & & (3)