Arc-Surfaced Frictional Damper for Vibration Control in Container Crane

In this paper, a new arc-surfaced frictional damper (AFD) is proposed and its hysteretic behavior is experimentally studied. Then the device is applied to container crane based on a seesawmechanism.Themajor advantage of the seesaw damping system is that the long tension cables can be utilized as bracing between the seesawmember and the portal legs to avoid compression and buckling of the cables. A simplified trilinear force-displacementmodel on the basis of experimental results is adopted to represent the hysteretic behavior of AFD. After that, seismic responses of container crane with and without dampers to four earthquakes are studied using nonlinear dynamic time-history analysis. Besides this system, a diagonal-brace-AFD system is studied for comparison. A method based on the displacement and energy dissipation ratio is proposed to find the optimum slip force for seesaw damping system. Performance of AFD control system is assessed though various parameters including displacement and maximum portal frame drift angle. Results prove a feasible application of AFD control system to absorb large amounts of seismic energy and significantly reduce the structural responses.


Introduction
Frictional damper has been proved to be an efficient approach to improve the seismic performance of the structures since the device can dissipate lager amounts of seismic energy to prevent structural collapse during earthquake ground motions. For the last few decades, several frictional dampers have been developed for structural application.
Rotational friction damper (RFD) was introduced by Mualla and Belev [1]. Experimental and numerical studies were conducted on a single story frame equipped with RFD. Liao et al. [2] carried out a shake table test on a three-story steel frame with RFD. The results proved the effectiveness of RFD in reducing the seismic responses. Kim et al. [3] utilized RFDs to improve progressive collapse-resisting capacity and mitigate the vibration of concrete moment frame. Sanati et al. [4] updated RFD with viscoelastic pads (RFVD). Experimental results conducted on a scaled steel frame with RFVD indicated a better performance of reducing seismic responses compared to the RFD. Mirzabagheri et al. [5] conducted a test on RFDs with different units, observed that more energy was dissipated by increasing number of units, and then proposed an equivalent method to evaluate the performance of RFDs. Cylindrical friction damper (CFD) was proposed by Mirtaheri et al. [6], both experimental and numerical research were carried out to evaluate hysteretic behavior of the damper. Time-history analyses were conducted on a frame with CFDs and results show that the device is capable of improving seismic performance. Monir and Zeynali [7] proposed a modified friction damper and primary experiments were carried out on a SDOF steel frame equipped with the damper. The experimental and numerical analyses revealed that the friction damper can reduce the displacement and story drifts.
Container crane is one of the most important types of equipment in the port. In the recent years, large cranes are needed more than ever. Consequently, modern container is more vulnerable to earthquakes. A few works have been done in the field of seismic control of container cranes. Sagirli et al. [8] studied self-tuning fuzzy logic controllers to suppress structural vibrations of a crane under the earthquake excitation and numerical studies proved that the control strategy is a feasible method. Azeloglu et al. [9,10]   inequality based mixed 2 / ∞ state-feedback controller to reduce seismic response of a crane. They concluded that the proposed controllers have great potential in vibration control for container cranes. Gong-Xian et al. [11] proposed a seismic control method for a container crane based on the principle of energy dissipation. Four viscous dampers supported by the braces were installed in the structure. The equivalent viscous damping and the stiffness were deduced and the optimization method based on displacement and energy dissipation ratio was proposed. It was concluded that the proposed method could effectively dissipate seismic energy and control the seismic responses.
In spite of efforts to improve seismic response of container crane, there is still a lack of study on the use of frictional dampers to reduce the seismic responses and seismic analyses of the crane with dampers. Their implementation is still limited due to the unavailability of reliable technologies and high costs. This paper proposes a novel AFD which can provide variable frictional force. The hysteretic behavior of the device is then experimentally studied. After that, a simplified trilinear force-displacement model for AFD based on the experimental results is suggested to be utilized in seismic response analyses. The vibration control system based on a seesaw mechanism using AFDs is introduced. Besides this, the diagonal-brace-AFD system is studied for comparison. The optimization method for the slip load of AFDs is proposed. Analyses on seismic responses are conducted for the cranes with AFDs and compared to the bare structure.

Arc-Surfaced Frictional Damper (AFD)
The main parts of AFD are arc plate, slider, convex plate, Polyurethane Elastomer (PUE), and T-shaped link. A schematic illustrate of the proposed AFD is presented in Figure 1. The arc plate is designed to fit with the slider, and their contact surfaces are circular and have same radius. Two blocks of PUE are shrink fitted middle two convex plates which contact with the slider through the surface of the cylinder. Consequently, the sliders and the arc plates are closely banded with each other with the normal force provided by PUE. Two sliders and two blocks of PUE are installed onto the T-shaped link which can make them move together. Furthermore, the distance of the arc plates is getting smaller when two sliders are moving close to the end of AFD. During this process, the slider will rotate at a certain degree, which makes the slider and the plate well matched. Since the distance of the two plates is variable, the damping force will change with the moving of sliders during the motion.

Experimental Setup.
The damper consists of three kinds of materials, which are defined by Chinese Industrial Standards. The arc plate with the width of 65 mm and radius of 1000 mm was made from 45 steel with a 354 MPa yield stress and 598 MPa ultimate strength. The sliders with the radius of 1000 mm were made from brass, as shown in Figure 2(a). To obtain energy dissipation capacity and hysteric behavior of AFD, tests were carried out on the universal machine with a capacity of 50 kN. The specifications are presented in Table 1. Figure 2(b) shows the test set up in which both the ends of AFD were installed in the load cells of the testing machine and the bottom load cell applied cyclic loading to the AFD with predetermined amplitudes. The data acquisition system was used to record the experimental data.

Experimental
Results. The hysteretic behaviors of AFDs under cyclic loading with 0.05 Hz frequency are presented in Figure 3. As one could expect, the hysteretic curves with stable hysteresis loops are saddle-shaped. There is a sharp slope on the upper and bottom hysteretic curves. This was induced by the inertia force of the sliders. In the curves, a step appears with the slope crossing -axis, the reason for this is that there is a gap between the load cell and fixture. As can be seen in Figure 3, the frictional force increases with the increasing  displacement. This is caused by the distance between the two arc surfaces getting smaller so that the compression of PUE increases. So, the normal forces acting on the sliders increase, which results in the increasing frictional force. It is noted that the slip load crossing -axis for case C is 2.395 kN, and the maximal one reaches 11.07 kN. This indicates that the device is capable of providing large damping force. For the limitation of test machine, it may be expected that AFD can demonstrate larger damping force with larger configuration.

Simplified Force-Displacement Model for AFD
To investigate the effectiveness of AFD in reducing seismic response of container crane, a simplified force-displacement model is very helpful and can be used to model AFD in some analysis software. For this purpose, the hysteretic behavior for AFD can be idealized by a trilinear model, as shown in Figure 4. The model is able to consider variable frictional force of AFD. In this figure, , , and are, respectively, the initial lateral stiffness, the nonlinear stiffness, and the large deformation stiffness of AFD. The displacement ( , ) at which the stiffness changes to the nonlinear stiffness and the large deformation stiffness can be calculated, based on the results of experimental results. Figure 3 shows the simplified trilinear force-displacement for AFD. It can be seen that the simplified model can work well as the essential characteristics of AFD.

Behavior of Container Crane with AFDs
The total system is composed of the original system and AFD vibration control system. The idealized force-displacement relationship of container crane with AFDs is illustrated in Figure 5. The overall stiffness of the bare structure includes the effect of stiffness reduction. and are, respectively, the initial and the secondary stiffness of the bare structure. is the yield displacement of the crane, at which the stiffness changes to the secondary stiffness.

Seismic Responses of Container
Crane with AFDs 6.1. Bare Structure. Container crane used for analyses is shown in Figure 6. The properties of the crane that is considered for analysis are shown in Table 2. Under working condition, the wheel mechanism is in the braking stage, the crane cannot move on the rail, and the wheel-rail connection can be negligible, so the portal legs are hinged on the ground. In this model, trolley, counterweight, and payload are considered as point masses lumped at the center of mass of the corresponding element.

Description of Analysis Models.
In order to examine the effectiveness of AFD vibration control system, the program SAP2000 was used to analyze the responses of container crane equipped with AFDs. The crane is modeled using the beam element with plastic hinges at the member ends. Table 3 lists the material properties of A36 used in analysis.  A damping ratio of 2% is considered for all mode shapes in the analysis. Figure 7(a) depicts the details of vibration control system [12,13] including a couple of AFDs, tension cables, and a pin-supported seesaw member. The analysis model is shown in Figure 7(b), in which the cables are modeled by the elastic spring element and some rigid truss members from both the seesaw member and the pin-support. The AFDs are modeled using Multilinear Plastic elements ( ) with simplified hysteretic behavior depicted in Section 4. Figure 8 presents the vibration control system installed in the crane. A couple of AFDs are connected to the beam and the seesaw member with the hinges. The long cables with cross turnbuckle are used in the mechanism to connect the seesaw member to the portal legs, on which the pretension force is applied to keep them from compression and buckling. Figure 9 illustrates the mechanism of the damping system. When a lateral load excites the crane, the portal legs tend to sway dramatically. The beam is inclined to move horizontally, which will resist the deformation of portal legs by the combination of the structure and fictional forces.
Three analysis models of container crane are established. Figure 10(a) presents the bare structure without damping system (Model BS), as depicted in Section 6.1. Others are structures equipped with AFD vibration control system, as presented in Figure 10(b and c). Figure 10(b) shows one case concerns the structure with the diagonal-brace-AFD system (Model DB), connecting the portal leg with the beam at the mid-span. The rods are modeled as rigid body. ℎ represents the distance of the bottom point of each brace from the ground level, as shown in Table 4. By contrast, the structure with seesaw system (Model SS) has two damping devices on the beam, as shown in Figure 10(c).  the 1995 Kobe earthquake, as presented in Table 5. These records are scaled to produce the peak ground acceleration of 0.4 g and 0.62 g. According to the Chinese standards, the two values correspond to 7-and 8-degree earthquakes, respectively.

Optimum Slip Load.
In order to dissipate the input energy as much as possible, some slip forces must be examined to find the optimum slip force. Subsequently, a parametric study based on the displacement and energy dissipation ratio is conducted until the minimum displacement at the top of container crane (point 3, see Figure 6) and the maximum energy dissipation ratio are reached. Figure 11 presents the maximum displacement versus slip load for Model SS under the El Centro earthquake. As can be seen in Figure 11(a), the values of the slip load range from 40 kN to 100 kN, and results show that the maximum displacement at the top of the crane varies according to the slip load. In case where the slip load is equal to 80 kN for PGA = 0.4 g, the minimum displacement at the top of the crane is reached.
The normalized / ratio, in which is the energy dissipated by AFDs and is the input energy, is calculated  and shown in Figure 12. As shown in Figure 12(a), when the slip load ranges from 40 kN to 100 kN, the / ratio increases from 0.53 to 0.73. This indicates that the vibration control system using AFDs is capable of consuming more than half of the input energy. For cases when the slip load is larger than 80 kN, the slip load has little effect on the / ratio. From the results presented in Figures 11 and 12, the optimum slip loads are 80 kN and 90 kN for PGA = 0.4 g and PGA = 0.62 g, respectively. They are used for the time-history response analyses in the following section.

Seismic Responses of Container Crane
6.5.1. Nonlinear Pushover Curves. Static pushover analyses are conducted on the bare structure and models with the optimum slip load whose value is 80 kN. The curves of base shear force versus the displacement at the top of the crane are plotted in Figure 13. As can be seen in this figure, introducing AFDs to container crane alters the force-displacement relationship to a shape which is similar to that plotted in Figure 5. Both the stiffness and the strength of the entire system are changed by adding the vibration control systems. Seesaw damping system, as expected, has a better performance of enhancing strength than that of the diagonal brace control system.  Table 6. The ratios of the attenuation with dampers to that without dampers are also presented, in which the maximum damping ratios for both seismic intensities can reach 61%. More than that, it demonstrates that the time-history of displacement reduction in Model SS is larger than the other. Figures 16 and 17, respectively, show the maximum portal frame drift angle distributions of different points (see Figure 6) for PGA = 0.4 g and 0.62 g. In all cases, the portal frame drift angle distributions were reduced by the vibration control system with AFDs. By contrast, the vibration control system with a seesaw mechanism can achieve a better result. So the seesaw vibration control system with AFDs is a feasible way to improve seismic response of container crane.

Conclusion
A new arc-surfaced frictional damper for seismic reduction of container crane was proposed and assessed experimentally. The variable frictional force and stable hysteretic loop are secured by the compression of PUE. A vibration control system using seesaw mechanism equipped with AFDs was introduced. First, a simplified trilinear force-displacement model of AFD was suggested based on the experimental results.
To investigate the effectiveness of this vibration control system, seismic response analyses were then carried out for a container crane with the system. The diagonal-brace-AFD system was also studied for comparison. Time-history responses of displacement, maximum portal frame drift angle were examined. Comparisons between seismic responses of bare structure and that equipped with damping system were conducted. The results show that the displacement is reduced by up to 61% both for PGA = 0.4 g and PGA = 0.62 g. The maximum drift angle is remarkably less compared to that of diagonal-brace-AFD system. The optimization method for the slip force on the basis of the displacement and the energy dissipation ratio is proposed.
The major advantage of the seesaw damping system is that the long tension cables can be utilized as bracing between the seesaw member and the portal legs to avoid compression and buckling of the cables. Seesaw vibration control system with AFDs is a feasible way to improve seismic response of container crane.