The use of Classical Rolling Pendulum Bearings (CRPB) for vibration control of stay-cables

Cables are efficient structural elements that are used in cable-stayed bridges, suspension bridges and other cable structures. A significant problem which arose from the praxis is the cables’ rain-wind induced vibrations as these cables are subjected to environmental excitations. Rain-wind induced stay-cable vibrations may occur at different cable eigenfrequencies. Large amplitude Rain-Wind-Induced-Vibrations (RWIV) of stay cables are a challenging problem in the design of cable-stayed bridges. Several methods, including aerodynamic or structural means, have been investigated in order to control the vibrations of bridge’s stay-cables. The present research focuses on the effectiveness of a movable anchorage system with a Classical Rolling Pendulum Bearing (CRPB) device. An analytical model of cable-damper system is developed based on the taut string representation of the cable. The gathered integral-differential equations are solved through the use of the Lagrange transformation. . Finally, a case study with realistic geometrical parameters is also presented to establish the validity of the proposed system.


Introduction
Cable-stayed (C-S) bridges have been known since the beginning of the 18th century, but there were difficulties in their static and dynamic analysis.A significant problem, which arose from the practice, is the cables rain-wind induced vibrations.Large amplitude Rain-Wind-Induced-Vibrations (RWIV) of stay-cables constitutes a challenging problem in the design of C-S bridges.Several methods, have been investigated in order to control the vibrations of bridge's stay cables.
This paper investigates the effectiveness of a movable anchorage system with a Classical Rolling Pendulum Bearing (CRPB) device.An analytical model of the cable-damper system is developed herein based on the taut string representation of the cable.The gathered integral-differential equations are solved through the use of the Lagrange transformation.Finally, a case study with realistic geometrical parameters is also presented to establish the validity of the proposed system, while the required device for the studied case is designed (see Fig. 1).

Basic assumptions
The deformed shape of the cables under dead and live loads is a catenary curve, with displacements wo and tensile forces To (see Fig. 2), which because of its very shallow form can be approximated by a second-order parabola.
z ,w

Fig. 2. Cable and reference axes
Under the action of the dynamic loads py(x,t) and pz(x,t), the cable takes the form of Fig. 3, with additional displacements ud , υd, wd and tensile forces Td.

Equilibrium of horizontal forces
Projecting on xoz-plane and taking the equilibrium of horizontal forces, we obtain:

Equilibrium of vertical forces
Projecting on xoz-plane and taking the equilibrium of vertical forces, we obtain:

Catenary and the parabola approach
It is usual to use the parabola as a curve that is very close to the catenary one, especially for shallow forms of cables.For a cable's shallow form the equation of a parabola passing from the points (0,0), (L,0) and having is given by the following formula:

The cable
The stay-cable model with the considered anchoragebearing system is shown in Fig. 6.Fig. 6.The stay-cable model

The rolling pendulum bearing system
Let us consider a C.R.P.B. device with one concave rolling, like the one of Fig. 7.The C.R.P.B. system is made from material like the one of the classical ballbearings having surfaces elaborated wery diligently, with coefficient of rolling friction ranging from 0.002 to 0.005.Therefore, the developed friction forces can be neglected in this preliminary study.On the other hand, the angle of the friction cone amounts up to 0.34o, which corresponds to a very small static friction.

The equation of motion
We obtain the following equation of motion:

The free vibrating cable
The equation of motion of a free vibrating cable with movable anchorage is: We are searching for a solution of separated variables of the form: One can determine the following form of the shape functions:

The forced vibrating cable
We are searching for a solution of the form: and we conclude finally to: 5 Numerical results and discussion

The cables
Let us consider a C-S bridge with dense distribution of cables from which we study a cable having tension To=300000 dN/cable, cross-sectional area F=7.5•10-3m2, diameter D=0.13 m, weight G=70 dN/m, mass per unit length m=7kg/m, and variable length L=150, 250, and 350m.

The rain-wind combination
It has been observed that the rain-wind-induced vibration in bridge cables usually occurs in a frequency range from 0.5 to 4 sec-1.For the study of the vibration of a cable under the action of a rain-wind combination we choose the following loading: , where ω=1, 2, 3, 4 sec-1 (for the study of the above cables without a damping system) and ω=3 sec-1 (for the study of the above cables with the proposed C.R.P.B. devise).In this case, big deformations appear, because the eigenfrequency of the external load is near to the first one of the cable (ω1 =2.82 sec-1).In this case, big deformations appear, because the eigenfrequency of the external load is near to the first one of the cable (ω1 =2.15 sec-1).

The damping system
In the followings, we will use a C.R.P.B. device, based on the operation principle of the simple system of Fig. 7.
We will consider devices with concave radii R=1, 2, and 3 meters.
In the plots of Figs 12, 13, 14, we see the oscillations of the middle and of the anchor head of a cable of length 150m, tension 300000dN, and for different values of R. From the above plots of Figs 11 to 13, we ascertain that smaller radii are more effective than the greater ones.
The above results are valid for both the cables' deformations and the anchorages' motion.
In the plots of Figs 14, 15, and 16, we see the oscillations of the mid-length and of the anchor head of a cable of length 250m, tension 300000dN, and for different values of R.  From the above plots of Figs 14 to 16, we observe that although the oscillations' amplitude is remarkable large the effectiveness of the system is obvious.We ascertain, again, that smaller radii are more effective than the greater ones.The above results are valid for both the cables' deformations and the anchorages' motion.

Conclusions
In this paper, a movable anchorage system with Classical Rolling Pendulum Bearing (C.R.P.B.) for vibration control of stay cables has been proposed and investigated.A model for the control system has been formulated, based on the taut string representation in which the proposed device has been incorporated.From the studied cases, one can conclude to the followings: a) The constant of the equivalent spring of the CRPB system has been assessed.b) The results of cable response show that the proposed CRPB device can effectively reduce the oscillation magnitude of the cable, proving the efficiency of the system.c) The observed decrease of the cable's oscillations amounts from 15% to 50%, while the motion of the anchor-head of the selected and designed device amounts from 1.5 to 6 cm.One must note that the CRPB device is very effective even for external loads acting with frequencies equal or near to one of the eigenfrequencies of the strained cable.d) The design parameters of the CRPB system for the selected cables are identified and the proper device has been designed.e) The proposed anchorage system is shown to perform more efficiently than the conventional passive external dampers, presenting a better solution from aesthetics point of view.

Fig. 8 .
Fig. 8. Oscillations of the mid-length of a cable of lengthL=150m, without any damping system

Fig. 9 .
Fig. 9. Oscillations of the mid-length of a cable of lengthL=250m, without any damping system

Fig. 10 .
Fig. 10.Oscillations of the mid-length of a cable of lengthL=350m, without any damping system

Fig. 14 .
Fig. 14.Oscillations of the mid-length and of its anchor head of a cable of L=250m, R=3m ___ with, ---without C.R.P.B

Table 1 .
Eigenfrequencies of the cables In Figs 8 to 10, one can see the oscillations of the midlength of the studied cables with length L=150, L=250 and L=350m, subjected to loadings acting with frequencies ω=1, 2, 3, 4 sec-1.