Theoretical and Experimental Analysis of a Winding Rope Fluid Viscous Damper

Zhang, Cheng; Zhang, Wenxue; Su, Lijun; Chen, Ying; Du, Xiuli

doi:https://doi.org/10.1155/2023/7307224

Structural Control and Health Monitoring

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Abstract Introduction Conclusions and Limitations Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 7307224 | https://doi.org/10.1155/2023/7307224

Theoretical and Experimental Analysis of a Winding Rope Fluid Viscous Damper

Cheng Zhang,¹Wenxue Zhang,¹Lijun Su,^1,2Ying Chen,¹and Xiuli Du¹

Academic Editor: Lucia Faravelli

Received23 Nov 2022

Revised17 Feb 2023

Accepted25 Mar 2023

Published12 Apr 2023

Abstract

This paper investigated a winding rope fluid viscous damper (WRFVD) that uses the frictional amplification mechanism of a winding rope to obtain the operating characteristics of a viscous damper-like device. The WRFVD combined the advantages of fluid viscous dampers and friction dampers. It not only retained the mechanical characteristics of the fluid viscous damper but also reduced manufacturing costs. First, the construction and working principles of the WRFVD were introduced. A theoretical model that could accurately simulate the hysteretic characteristics of the damper was derived. Then, a series of dynamic tests were performed on six prototypes of the WRFVD. The dynamic performance under different displacement amplitudes and device parameters was analyzed based on the test results. In addition, the theoretical model was validated by experimental results. Finally, a series of parametric analyses of the WRFVD were performed. The experimental results showed that the WRFVD is a type of velocity-dependent damper with smooth and plump hysteretic curves under sinusoidal displacement excitation. Fatigue loading tests showed that the WRFVD had excellent fatigue resistance capacity and stable operational performance. The established theoretical model was reasonable and satisfactorily reproduced the hysteretic properties of the WRFVD. The parametric analyses showed that it was not recommended to improve the performance of the WRFVD by the method of adjusting the damping coefficient and pretightening load. It was reasonable to adjust the performance of the WRFVD by setting a smaller velocity exponent, an appropriate winding turn of winding ropes, and a suitable friction coefficient.

1. Introduction

Structural control is an effective method to reduce the dynamic response of a structure. This concept was proposed by Yao in the 1970s [1]. Structural control can be divided into active control, passive control, semiactive control, and hybrid control [2–5]. Passive control does not require an external power source, which includes vibration isolation, vibration absorption, energy dissipation, and vibration damping control measures [6, 7]. Over the past decades, a wide variety of energy dissipation and damping devices have been developed, including metal dampers [8], friction dampers [9, 10], lead dampers [11, 12], viscoelastic dampers [13], and fluid viscous dampers [14].

The fluid viscous damper was a velocity-dependent damping device. It can be divided into three categories according to its mechanical properties: linear fluid viscous dampers, nonlinear fluid viscous dampers, and superlinear fluid viscous dampers, among which nonlinear fluid viscous dampers are more commonly used in general engineering [15–17]. In the 1990s, the viscous fluid properties and damping force characteristics of fluid viscous dampers were studied in detail by Makris and Constantinou et al. [18, 19], and they found that the damping force of such dampers was a power function of the velocity of the piston rod; the damping coefficient of the dampers was constant when the piston rod of the dampers moved at low frequency relative to the cylinder.

To improve the damping effect of fluid viscous dampers under small displacement inputs, some scholars have combined displacement amplification devices with fluid viscous dampers. The toggle-brace-damper (TBD) device was proposed by Constantinou et al. based on the lever principle, which amplifies the displacement of a fluid viscous damper by several times [20]. The TBD allows fluid viscous dampers to function at small displacements of the structure, and the effectiveness of such devices has been verified by experimental and theoretical studies. Several studies on the application of TBDs and the optimization of their form have shown that the application of TBDs can be effective in reducing the vibration response of structures [21–23]. Huang [24] summarized various types of displacement amplification mechanisms of elbow-joint supports. The amplification principle and the derivation of the amplification coefficients of the displacement amplification mechanism of the elbow-joint support were proposed. The results indicated that, with a properly designed amplification system, the application of elbow-joint supports on fluid viscous dampers is an effective solution to reduce the structural response.

The fluid viscous damper has good performance, and its range of application can be greatly extended by the use of some mechanical means. However, the fluid viscous damper is expensive and has high replacement and maintenance costs. The above amplification methods only amplify the structural displacement response to give the fluid viscous damper a suitable input displacement, but do not solve the problem of the size and cost of the fluid viscous damper, and may even increase its maintenance and replacement costs.

Friction dampers are favored by the engineering community due to the low cost and high performance. In 1928, Pall and Marsh [25] introduced the idea of improving the seismic and damage control potential of structures by installing sliding friction devices in frame structures. This device consists of sliders and bolts. The friction of the damper is obtained by applying a certain load to sliders through bolts. This friction damper has shown reliability in multicycle loading, and its energy dissipation effect is widely recognized by the engineering community. Many scholars have proposed various forms of friction dampers to suit different engineering conditions based on the principle of plate friction dampers. Wei and Chen [26] improved on the plate friction damper and proposed a piston constant friction damper, which can be more easily installed in various structural conditions and can utilize spring prestress to adjust the magnitude of the frictional force. Experimental studies showed that the hysteretic curve of the damper is approximately rectangular, which is consistent with the theoretical analysis. To increase the self-restoring capacity of the structure, a rotational friction damper was proposed, which provides additional stiffness by combining a rotational friction damper with a torsional spring. Numerical simulation studies proved that it can effectively improve the seismic performance and cost-effectiveness of the structure compared to conventional friction dampers [27]. Most conventional friction dampers are unidirectional. To solve this problem, Suk and Altintaș proposed a multidirectional friction damper consisting of a friction ball and four friction blocks surrounding the friction ball. Prestress was applied by adjusting the bolts on friction blocks. Experimental and numerical studies showed that the hysteretic behavior of this device was stable. It could positively affect the performance of the structure by reducing its period and displacement [28]. Inspired by the displacement amplification mechanisms of levers and ball screws, some scholars have also used these mechanical principles to amplify the response of friction dampers to achieve greater damping forces. Studies have shown that these classes of amplified dampers have better vibration control capabilities than conventional friction dampers [29, 30]. However, due to the uncertainty of seismic action, the activation force of friction dampers is difficult to determine, so conventional friction dampers cannot accommodate both large and small earthquakes. Because conventional friction dampers require the friction surface to remain in a high-pressure state, their performance may change due to bolt loosening and friction surface wear [31, 32].

To overcome the problem of the constant friction force and limited application range of traditional friction dampers, this paper proposes a winding rope fluid viscous damper (WRFVD). Previous studies have shown that the WRFVD can amplify the damping force of a fluid viscous damper by using a winding rope, which not only preserves the mechanical properties of the fluid viscous damper but also reduces the cost [33]. Due to these novel configurations and characteristics, it is necessary to further understand the hysteretic behavior and energy consumption capacity of the WRFVD.

The contents of this paper are organized as follows: First, the configuration and the working mechanism of the WRFVD are explained in Section 2. The theoretical model of the WRFVD is established in Section 3. Then, a laboratory test program on six prototypes of the WRFVD is introduced in Section 4. The test observations and results are presented, where the mechanical performance of the WRFVD is elaborately investigated. The established theoretical model is validated, followed by a parametric analysis of the WRFVD parameters in Section 5. Finally, the conclusions are summarized in Section 6.

2. Design and Working Principle of the WRFVD

2.1. Basic Construction

Figure 1 shows the construction of the WRFVD. It mainly consists of winding rope friction dampers (FDs) and a viscous damper (FVD). The FD part consists of winding ropes and a friction shaft. The winding rope is wrapped around the friction shaft for a number of turns, and its two ends are bolted to the piston rod of the FVD and the girder connection element, respectively. These bolts are called pretightening load-adjusting bolt; the pretightening load in the winding rope can be adjusted by the pretightening load-adjusting bolt. The friction shaft is fixed to the friction shaft support and the friction shaft. The FD works with the friction force generated by the reciprocating movement of the winding rope around the friction shaft. The FVD is fixed to the fluid viscous damper support. The damping force of the FVD is obtained by the pressure generated by the reciprocating motion of the viscous fluid on both sides of the piston [34].

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2.2. Working Mechanism

The WRFVD can be applied between the sliding piers and the girder of a continuous girder bridge. The fluid viscous damper supports and the friction shaft supports are fixed to the top of the sliding piers, and the girder connection elements are fixed to the girder, as shown in Figure 1(a). Note that Figure 1 shows only the installation locations rather than the actual sizes of each component. The girder of the bridge is supported by sliding bearings on the pier, which are not shown in the figure.

The relative displacement between the sliding pier and the girder of the continuous girder bridge is generated under the seismic load. Meanwhile, the WRFVD is activated, and the winding ropes in its two FDs make reciprocating motion around the friction shafts, pulling the piston rod of the FVD to move reciprocally. This makes each sliding pier share part of the seismic force originally applied to the fixed pier. In addition, the WRFVD consumes seismic energy by its damping, thus improving the overall seismic capacity of the structures.

The WRFVD has superior mechanical performance, adjustable energy consumption capacity, and simple construction. Its energy consumption capacity and stroke can be adjusted according to the actual demands of the structures. The specific adjustment methods are as follows:(1)Replace the FVD with different design strokes and different lengths of the winding rope to change the maximum stroke of the WRFVD(2)Adjust the number of winding turns and the pretightening load of the winding rope to adjust the maximum damping force and energy consumption capacity of the WRFVD

3. Theoretical Model of the WRFVD

3.1. Mechanical Model of the Fluid Viscous Damper

According to previous research, the recommended methods for calculating the damping force of fluid viscous dampers are the linear model, Kelvin model, Maxwell model, and fractional derivative model [35–37]. In this paper, the Maxwell model, which does not consider stiffness, is used to fit the parameters of FVDs [38, 39]. The fitting formula is shown in equation (1), and Table 1 shows the fitting results:where F_FVD is the damping force of the FVD, c is the damping coefficient of the FVD, α is the velocity exponent of the FVD, is the relative velocity between the FVD piston rod and the cylinder, and is a signed function of the piston rod velocity.

Figure 2 shows the comparison between the theoretical and experimental hysteretic curves of the FVDs under f = 0.4 Hz and u₀ = 100 mm for the third cycle, where Exp. and The. represent the experimental results and theoretical calculations, respectively. The experimental hysteretic response of the FVD can be satisfactorily simulated by the nonlinear Maxwell model. The parameters of the FVDs were accurate and could be used for subsequent theoretical analysis and experimental verification.

3.2. Mechanical Model of the Winding Rope Friction Damper

We take the ideal state of the rope wrapped around the friction axis to analyze and derive the formula of FD damping force. It is based on the following assumptions: (1) the mass of the rope is not considered, (2) the bending stiffness of the rope is ignored, and (3) neither the rope nor the friction shaft produces elastic deformation.

As shown in Figure 3, an infinitesimal arc segment was taken at any contact point P between the friction shaft and the winding rope [40]. The total radian of the rope wrapped around the winding shaft is θ. F_a is the tension at the active end of the winding rope. is the tension at the passive end of the winding rope. and are the tensions at each end of the rope. is the degree of the corresponding circular angle of the tiny arc segment. μ_s is the friction coefficient of the contact surface. is the normal force exerted by the rope to the point P. df is the friction force between the contact surfaces. The following equations can be obtained:

Because the arc segment is infinitesimal, its corresponding circular angle dθ tends to zero; then, and , and in equilibrium, the system of equilibrium equations is as follows:

The following equation can be obtained by performing a Taylor expansion of at θ and neglecting the squared terms of , higher order infinitesimal terms and the Peyano remainder.

Combining equations (2) to (5) and neglecting the squared terms of dθ, that is,

The relationship between the tension at both ends of the winding rope can be obtained as follows:

In the WRFVD, the winding of the rope on the friction shaft is in a radian of an integer multiple of 2π, when the number of winding turns is n, that is,

The damping force of FD is as follows:

Equation (9) shows that the damping force of FD is determined by , , and n. “” can be defined as the “rope equivalent friction factor” μ_e, so equation (9) can be regarded as the product of F_p and μ_e.

3.3. Model Establishment of the WRFVD

As shown in Figure 4, the WRFVD connects the FD and FVD using winding ropes. Assuming that the upper loading beam moves to the right with a velocity of , the winding rope at the right end is considered the active end and the winding rope at the left end is considered the passive end. The damping force generated by FVD is , the damping force generated by FD at the right end is , the damping force produced by FD at the left end is , and the damping force generated by the WRFVD is . The following equations can be obtained:

The isolators taken for WRFVD are shown in Figure 5. In the initial state, a symmetric preload of winding ropes is required to create a pretightening load of in winding ropes. In the operation condition, the following equations can be obtained:where and are the tensions in the winding ropes at the active end of right and left FDs, respectively, and and are the tensions in the winding ropes at the passive end of right and left FDs, respectively.

The following equations can be obtained from equation (8):

Combining equations (12) to (15), the following equation can be obtained:

Combining equations (1), (10), (11), and (16), the following equation can be obtained:

Let and , and the damping force of the WRFVD can be regarded as the linear sum of the equivalent frictional force () and the equivalent viscous damping force (). Figure 6 shows the restoring force model of the WRFVD under sinusoidal excitation, where u denotes the displacement of the damper. The hysteretic curve of the equivalent friction force is rectangular, and the hysteretic curve of the equivalent viscous damping force is elliptical. These two are superimposed to form the hysteretic curve of the WRFVD. Figure 7 shows the components in the mechanical model of the WRFVD, including the equivalent damping () and the equivalent frictional force . This model could be used for experimental validation and parametric analysis.

4. Mechanical Property Test of WRFVD

4.1. Experimental Setup and Program

To investigate the mechanical performance of the proposed WRFVD, a series of laboratory tests were conducted on six different prototypes of the WRFVD. Table 2 summarizes the list of specimens. The rules for specimen numbering in the table are as follows: the number after the letter “D” represents the number of the FVD, the number after the letter “N” represents the number of winding turns, and the number after the letter “R” represents the diameter of the winding rope in centimeters.

Figure 8(a) shows the physical dimensions of the FVD. The dimensions of the three FVDs used in the test are consistent. The maximum displacement of the FVD is ±120 mm. The friction shaft configuration used in the test is shown in Figure 8(b). The material of the friction shaft is steel, and the interior is solid. The winding ropes used in the test are 18 × 7-WSC wire ropes with diameters of 8 mm and 14 mm, respectively, as shown in Figure 8(c) [41]. Each FD was applied with two winding ropes on the winding shaft. The surface of the ropes was evenly coated with lubricating grease before the test.

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The test setup of the WRFVD is shown in Figure 9. The tests were completed at the Beijing Key Laboratory of Engineering Seismic and Structural Diagnosis, Beijing University of Technology. The test bench used for the test was a 3000 kN damper tester [42]. The upper loading beam was bolted to the actuator of the tester, which is used to simulate the girder in a continuous girder bridge. The lower support beam was bolted to the fixing rod of the tester, which is used to simulate the top of the sliding pier in a continuous girder bridge. The specimens of the WRFVD were installed between the lower support beam and the upper loading beam.

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Force sensors 1 and 2 were installed at the connection between the winding ropes and the upper loading beam to measure the tension force in winding ropes. Force sensors 3∼6 were installed at the connection between the FVD piston rod and the winding rope to measure the damping force of the FVD. Linear variable differential transformer No. 1 (LVDT1) was installed at the end of the upper loading beam to measure the displacement of the winding rope. To measure the displacement of the FVD piston rod, linear variable differential transformer No. 2 (LVDT2) was installed at the end of the FVD piston rod. The force and displacement in each part of the WRFVD can be calculated from the measured values of the sensors. As shown in Figure 9(a), rightward loading was positive and leftward loading was negative.

All tests were conducted under a sinusoidal displacement loading protocol. In order to ensure that the winding rope has a certain pretensioning force, the adjusting bolts should be adjusted in advance. Table 3 shows the pretightening load and loading protocol. Before loading, a pretightening load of 2 kN was applied to the winding ropes of D1N1R14, D2N1R14, and D3N1R14. The specimens were tested under different u₀ and fixed f to explore the mechanical properties of the WRFVD at different displacement amplitudes. D2N1R14, D2N2R14, D3N1R08, and D3N2R08 were subjected to different pretightening loads before loading. The specimens were tested under the same u₀ and f to investigate the mechanical properties of WRFVD under different pretightening loads. The winding ropes of D3N1R08 were subjected to a pretightening load of 6 kN. The fatigue performance of the WRFVD was tested by loading for 30 cycles at a loading frequency of 0.2 Hz and a maximum displacement of 100 mm.

4.2. Selection of Evaluation Indices

In order to compare the mechanical properties of the specimens under different loading conditions, the following evaluation indices were selected. As recommended by the Chinese standard (JG/T209-2012) [43] and the international standard (ISO 22762-1:2010) [44], the third cycle hysteretic curve was chosen as the evaluation criterion for the mechanical properties of the WRFVD:(1)The maximum damping force is calculated as follows: where F_max⁺ and F_max⁻ are the maximum and minimum damping forces for one load cycle, respectively.(2)W is the energy consumed by the damper for each hysteretic cycle, and it represents the energy dissipation capacity of the damper. It is obtained by calculating the area of a single force-displacement hysteretic loop.

4.3. Analysis of Loading Condition Effects

D1N1R14, D2N1R14, and D3N1R14 were tested from u₀ = 10 mm to u₀ = 70 mm with f = 0.4 Hz and F₀ = 2 kN for five cycles. The hysteretic curves of the WRFVD under different u₀ are shown in Figure 10. The load-velocity curves of the WRFVD under different u₀ are shown in Figure 11. The variation curve of the maximum damping force of the WRFVD with u₀ and is shown in Figure 12.

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As shown in Figure 10, the hysteretic curves of each specimen showed a rounded rectangular shape and were very plump. The hysteretic curve of each specimen showed a drop phenomenon at the maximum displacement, which was caused by the relaxation of the winding ropes during loading. A certain tilt phenomenon was observed in the hysteretic curves of each specimen. This is caused by three reasons. First, the stiffness of the upper loading beam is relatively low, showing a “bow” shape in loading. Second, the winding rope is elastic, which will be slightly stretched during loading. The third is the small amount of stiffness contribution from FVDs.

Figure 11 shows that the maximum damping force of the WRFVD varies with the loading velocity. The force-velocity curves do not overlap in one cycle, indicating a certain hysteretic between the damping force and the loading velocity, which also indicate the existence of stiffness in the WRFVD. However, the stiffness contribution of the upper loading beam in this test is the highest among the WRFVD specimens. This contribution is not actually part of the WRFVD because the loaded upper beam is only used to simulate the girder of the bridge, and the stiffness of the girder can be considered infinite. Therefore, it will not be the subject of further discussion in this paper. Figure 12 shows that the WRFVD is a velocity-dependent damping device, since the maximum damping force of the WRFVD increases linearly with an increase of u₀ and under the same f.

Figure 13 shows the energy dissipation of the WRFVD. It can be seen that the energy dissipation of WRFVD, FVD, and FD increases exponentially with an increase of u₀ at the same frequency. Table 4 shows the energy dissipation and maximum damping force of the WRFVD. The energy consumption ratio of FDs and FVDs of the same specimen does not vary significantly with u₀, indicating that the WRFVD could work stably under different displacement amplitudes. The energy consumption ratio of the FD is higher than that of the FVD in each specimen, although it varies between specimens. Therefore, it can be concluded that the energy consumption ratio of each part of the WRFVD is effectively influenced by the structure and parameters of the device but is less influenced by the amplitude of the load displacement.

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4.4. Analysis of the Device Parameter Effects

Figure 14 shows the hysteretic curves of D2N1R14, D2N2R14, D3N1R08, and D3N2R08 under u₀ = 50 mm, f = 0.2 Hz, with different F₀. Figure 15 shows the variation curve of F_WRFVD,max with F₀.

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It can be seen from Figure 14 that the hysteretic curves of D2N1R14, D2N2R14, and D3N2R08 basically overlap as F₀ increases. Furthermore, comparison with Figure 15 shows that the maximum damping force of the specimen almost does not change and that the maximum damping force does not fluctuate by more than 5% relative to the average value. This indicates that changing the pretightening load did not have much effect on the maximum damping force of the WRFVD.

A comparison of the specimens with different winding turns (D2N1R14 and D2N2R14 and D3N1R08 and D3N2R08) shows that the maximum damping force of D2N2R14 is about 2.2 times that of D2N1R14 and that the maximum damping force of D3N2R08 is also about 2.3 times that of D3N1R08 under the same loading conditions. As a result, the maximum damping force of the WRFVD can be significantly increased by increasing the number of windings.

The hysteretic curves of D2N1R14 in Figure 14 at F₀ = 6 kN and F₀ = 8 kN show a sharp angle in the third quadrant. This is because one end of the winding rope was caught in the groove created by wear on the friction shaft at the beginning of the test. As loading proceeded, the winding rope slipped out of the groove, resulting in a slight increase in the winding diameter of the winding rope and an increase in the tension in the winding rope. The hysteretic curves of D3N1R08 almost overlapped when F₀ < 6 kN, and its maximum damping force did not change with a change of F₀. Its hysteretic curve expanded when F₀ ≥ 6 kN, and its maximum damping force showed a linear increasing law with an increase of F₀. In contrast, the hysteretic curves of the other specimens remained almost unchanged as F₀ increased. The hysteretic curves of D3N1R08 showed a different pattern from the other specimens. The reasons for these differences are as follows.

4.4.1. Elasticity of the Winding Rope

We assume that the upper loading beam is rigid; i.e., L_r (shown in Figure 16(a)) remains constant during the reciprocal loading. During the loading process, there was a tensile force T in the winding rope, which will cause the winding rope to produce a certain amount of elongation Δl, as shown in equation (19). Assuming that the load direction is to the right, the winding rope in the right FD remains taut, where the tension causes the winding rope to elongate Δl, while the end of the winding rope of the left FD moves Δl to the right. The pretightening load also stretches the rope, causing the rope in the left FD to elongate Δl_p. If Δl > Δl_p, the effect of the pretightening load disappears and the rope slackens, leaving this end of the FD with almost no frictional force. If Δl ≤ Δl_p, the pretightening load still keeps the winding rope taut and the FD at this end provides frictional force, so the damping force of the WRFVD was greater. According to equation (19), as the number of winding turns increases, the winding rope becomes longer and Δl increases. Therefore, a larger pretightening load is required to obtain larger Δl_p to ensure that the winding rope is tight. This is the reason why the hysteretic curves of D3N1R08 and D3N2R08 are different. The winding rope of D3N1R08 has fewer winding turns and shorter rope length, so the pretightening load required to keep the rope taut is low. However, the winding rope of D3N2R08 has more winding turns and longer rope length, so the pretightening load required to keep the rope taut is high:where T is the tension in the winding rope, l is the length of the winding rope, is the elongation of the winding rope under the tension of T, E is the modulus of elasticity of the winding rope, and A is the cross-sectional area of the winding rope.

4.4.2. Bending Stiffness of the Winding Rope

The winding rope used in the test is made of the steel wire, and its bending stiffness is not negligible. The wrapping of the wire rope around the friction shaft must overcome its bending stiffness. Part or all of the pretightening loads applied in the test are used to overcome the bending stiffness of the wire rope, and the remaining part is used to elongate the rope. As the diameter of the rope increases, so does its bending stiffness. Therefore, as the diameter of the rope increases, the component of the pretightening load to overcome the bending stiffness of the rope increases and the remaining component to overcome the elastic effect decreases. This results in differences in the damping force of the WRFVD with different diameters of winding ropes. This is the reason why the hysteretic curves of D3N1R08 and D2N1R14 are different. The winding rope of D3N1R08 has a small diameter and low bending stiffness, which requires fewer pretightening load components to overcome its bending stiffness. The pretightening load has more residual components to overcome the elastic effect. While the winding rope of D2N1R14 has a large diameter and high bending stiffness, which requires more pretightening load components to overcome its bending stiffness. The pretightening load has fewer residual components to overcome the elastic effect.

The phenomenon of the slackness of the winding ropes was indeed found in the test, as shown in Figure 16. The drop phenomenon at the maximum displacement of the hysteretic curves in Figure 14 is caused by the slackness of the winding ropes. The slack phenomenon was more obvious in the specimen with a larger diameter of the winding rope and more winding turns. However, this phenomenon decreased or even disappeared with an increase of F₀, and the drop phenomenon in the hysteretic curves was also reduced or even disappeared.

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Table 5 shows the energy dissipation of D2N1R14, D2N2R14, D3N1R08, and D3N2R08. The energy dissipation of D2N1R14, D2N2R14, and D3N2R08 slightly increased with an increase of F₀. There was no significant change in the energy consumption ratio of each part in the specimen. This was because the relaxation of the winding ropes of D2N1R14, D2N2R14, and D3N2R08 diminished with an increase of F₀, and the hysteretic curve was plumper. However, only one FD worked to consume energy, so the energy consumption ratio of the FD remained the same. The energy dissipation of D3N1R08 increased with an increase of F₀. The energy consumption ratio of the FD in the specimen was also increased. This was because the relaxation of the winding ropes of D3N1R08 diminished until it disappeared as F₀ increased. The two FDs worked simultaneously to consume energy, so total energy consumption increased, and the energy consumption ratio of FDs increased.

Comparing D2N1R14 and D2N2R14 and D3N1R08 and D3N2R08, it could be found that the energy dissipation of the WRFVD increased significantly with an increase of n. The energy consumption ratio of FDs was also substantially increased. It is indicated that the increasing number of winding turns leads to an increase in the frictional energy consumption capacity of the WRFVD.

In summary, the adjustment of the pretightening load can be used to prevent the relaxation phenomenon of the winding ropes. Using this method to increase the energy consumption capacity and maximum damping force of the WRFVD is not stable enough. However, increasing the winding turns of the winding ropes can effectively improve the energy consumption capacity and maximum damping force of the WRFVD.

4.5. Fatigue Property

The hysteretic curves of D3N1R08 under 30 cycles of the low-cycle fatigue test are shown in Figure 17. The hysteretic curves of the WRFVD, FVD, and FD remained consistent and plump after 30 cycles of loading. There was no breaking and slipping of the winding rope and no leaking and cylinder bursting of the FVD during loading, proving that the WRFVD was safe and stable with excellent energy absorption capability.

Table 6 shows the 3rd and 30th cycle performance index values and attenuations. After 30 cycles of loading, the attenuation of its main performance parameters was maintained in the range of 1.08% to 2.69%, which fully complies with the requirements of the Chinese standard (GB 50011-2010) [36], which stipulates that the error and attenuation of the main design indices of dampers should not exceed 15%.

5. Parameter Analysis of the WRFVD

5.1. Comparison between Theoretical and Experimental Results

It was obtained by analyzing the tension data of the winding rope with 8 mm diameter that F₀ decreased about 4.0 kN due to bending stiffness and decayed about 80% due to elasticity when n = 1 and that F₀ decreased about 8.4 kN due to bending stiffness and decayed about 97.5% due to elasticity when n = 2. The proposed model in Section 3.3 is implemented by using MATLAB software, and the friction coefficient of μ_s was determined by the experiment. The parameters of the WRFVD are shown in Table 7, and the hysteretic curves under different F₀ with f = 0.2 Hz and u₀ = 50 mm were simulated by the model using sinusoidal displacement excitation.

The hysteretic curves of the WRFVD predicted by the theoretical model and the corresponding experiment results under f = 0.2, u₀ = 50 mm, and different F₀ are shown in Figure 18. As can be seen in the figure, most of the curves are coincident, and the theoretical results agree with the experimental curves. Because the effect of damper stiffness was not considered in the theoretical calculation, the theoretical hysteretic curves were not inclined, which differed from the experimental ones. The phenomenon that the winding rope embedded in the worn groove of the friction shaft was found in the test, resulting in a right angle at the corner of the experimental hysteretic curve and a lower degree of agreement with the theoretical one.

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Table 8 shows the calculated results of the evaluation indexes to verify the validity of the theoretical model. F_WRFVD,max is calculated accurately, and the maximum error is not more than 4.66%. As F₀ increases, the calculation error of the energy dissipation for each loop W decreases. When F₀ is small, the proportion of test errors caused by the slack of the winding rope is larger. However, with an increase of F₀, the proportion of this error becomes smaller. The maximum calculation error of W does not exceed 9.80%. Therefore, the theoretical model is reasonable and provides a satisfactory reproduction of the hysteretic properties of the WRFVD.

5.2. Parameter Analysis

To illustrate the effects of each device parameter on the mechanical properties of the WRFVD in more detail, a numerical simulation was conducted based on the proposed model. The parameters affecting the performance of the WRFVD are as follows: velocity exponent α, damping coefficient c, pretightening load F₀, friction coefficient μ_s, and winding turns n. Figure 19 plots the hysteretic curves of the WRFVD under sinusoidal displacement loading under u₀ = 100 mm and f = 0.4 Hz, when only one parameter was changed and the other parameters were the same. Figure 20 shows the variation of the energy dissipation and maximum damping force of the WRFVD with a certain parameter.

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(d)

(e)

Figure 19(a) shows the hysteretic curves for different values of α for F_FVD,max = 20 kN, F₀ = 1 kN, μ_s = 0.15, and n = 2. Figure 20(a) shows the variation curves of the maximum damping force and energy consumption with α. As can be seen in the figures, the hysteretic curve of the WRFVD gradually changed from a shuttle shape to a rounded rectangle with a decrease of α and also became plumper. The energy consumption capacity of the WRFVD decreased with an increase of α. Its maximum damping force did not change at certain F_FVD,max. Therefore, smaller α could make the device easier to be excited and obtain a better energy consumption capacity.

Figure 19(b) shows the hysteretic curves for different values of c for α = 0.3, F₀ = 1 kN, μ_s = 0.15, and n = 2. Figure 20(b) shows the variation curves of the maximum damping force and energy consumption with c. As can be seen in the figures, the hysteretic curves of the WRFVD with different c were similar in shape, and the hysteretic curve of larger c was on the periphery of the hysteretic curve of smaller c. With an increase of c, both the energy dissipation and the maximum damping force of the WRFVD increased in a linear trend. However, when α is fixed, increasing c is equivalent to increasing the maximum damping force of the FVD. This will increase the size and cost of the FVD, which does not meet the original design intention of the WRFVD. Therefore, it is unsuitable to improve the performance of the WRFVD by increasing c.

Figure 19(c) shows the hysteretic curves for different values of F₀ for F_FVD,max = 20 kN, α = 0.3, μ_s = 0.15, and n = 2. Figure 20(c) shows the variation curves of the maximum damping force and energy consumption with F₀. Note that F₀ here is the effective pretightening load without considering the effect of the slack in winding ropes. As can be seen in the figures, the hysteretic curves of the WRFVD with different F₀ were similar in shape, and the hysteretic curve of larger F₀ was on the periphery of the hysteretic curve of smaller F₀. With an increase of F₀, both the energy dissipation and the maximum damping force of the WRFVD increased in a linear trend. Although the adjustment of F₀ is easy to operate and does not change the cost of the WRFVD, it is obvious that enhancing the performance of the WRFVD by increasing F₀ is an unreliable way considering the relaxation effect of the winding ropes. Therefore, the pretightening load should be used only as a means to ensure the tightness of winding ropes and reduce slackness.

Figure 19(d) shows the hysteretic curves for different values of μ_s for F_FVD,max = 20 kN, F₀ = 1 kN, α = 0.3, and n = 2. Figure 20(d) shows the variation curves of the maximum damping force and energy consumption with μ_s. As can be seen in the figures, the hysteretic curves of WRFVD with different μ_s were similar in shape, and the hysteretic curve of larger μ_s was on the periphery of the hysteretic curve of smaller μ_s. With an increase of μ_s, both the energy dissipation and the maximum damping force of the WRFVD increased in an exponential trend. The adjustment of μ_s is demanding on the production process of the WRFVD. Therefore, the performance of the WRFVD can be adjusted by handling the friction coefficient between the winding rope and the friction shaft in a reasonable way. However, due to the nature of the material, this paper does not recommend using the method of adjusting the friction coefficient to optimize the mechanical properties of the WRFVD.

Figure 19(e) shows the hysteretic curves for different values of n for F_FVD,max = 20 kN, F₀ = 1 kN, α = 0.3, and μ_s = 0.15 Figure 20(e) shows the variation curves of the maximum damping force and energy consumption with n. As can be seen in the figures, the hysteretic curves of WRFVD with different n were similar in shape, and the hysteretic curve of larger n was on the periphery of the hysteretic curve of smaller n. With an increase of n, both the energy dissipation and the maximum damping force of the WRFVD increased in an exponential trend. Increasing n can substantially improve the performance of the WRFVD and adds little to the cost of the WRFVD. Although increasing n increases the effect of slack in winding ropes, this effect can be eliminated by properly adjusting the pretightening load. So it is a reasonable way to improve the performance of the WRFVD by increasing the number of winding turns.

6. Conclusions and Limitations

This paper proposes a winding rope fluid viscous damper (WRFVD) that uses FDs to amplify the damping force of FVDs. A theoretical model capable of describing the hysteretic responses of the WRFVD was established and verified by experiments. Subsequently, sinusoidal loading protocols were used to test six different WRFVD prototypes. Mechanical properties of WRFVDs have been extensively studied with respect to displacement amplitude effects, pretightening load effects, winding turn effects, and fatigue performance. Finally, a series of parametric analyses of the performance parameters of the WRFVD were performed. A summary of the main results and conclusions of this study are presented as follows:(1)The WRFVD has a plump hysteretic curve with rounded rectangles and is a velocity-dependent damping device. The maximum damping force and energy consumption capacity increase with an increase in loading displacement under a certain loading frequency.(2)The established theoretical model is reasonable and provides a satisfactory reproduction of the hysteretic properties of the WRFVD. It accurately predicts the energy dissipation capability of the WRFVD.(3)The smaller velocity exponent improves the energy consumption capacity of the WRFVD. The larger damping coefficient also improves the energy consumption capacity and maximum damping force of the WRFVD, but it is not recommended to improve the performance of the WRFVD by increasing the damping coefficient.(4)The pretightening load can effectively reduce the slack phenomenon of winding ropes. In the case that the winding ropes are not slack, the maximum damping force and energy dissipation of the WRFVD increase with an increase in the pretightening load. However, the method of improving the performance of the WRFVD by increasing the pretightening load is not recommended.(5)The number of winding turns and the friction coefficient can significantly affect the performance of the WRFVD. As the winding turns and the friction coefficient increase, the maximum damping force and energy consumption of the WRFVD increase. However, due to the nature of the material, it is more recommended to optimize the mechanical properties of the WRFVD by adjusting the number of winding turns.(6)The WRFVD has good fatigue resistance capacity. The hysteretic curves almost coincided before and after the fatigue test. After 30 times of cyclic loading, the attenuation of the main performance parameters did not exceed 15%.

The potential limitation in this study is that the stiffness of winding ropes is not considered in the theoretical model of the WRFVD. A more accurate model is required for future research. Moreover, the durability of the WRFVD under long-term slow loading was not investigated in this study. The issue of the durability and maintainability of each component in the WRFVD could be investigated in the future.

Data Availability

(1) The test result data for FVDs in Section 3.1 used to support the findings of this study are available from the corresponding author upon request. (2) The test result data for WRFVDs in Sections 4.3 to 4.5 used to support the findings of this study are available from the corresponding author upon request. (3) The theoretical calculation result data for WRFVDs in Sections 5.1 to 5.2 used to support the findings of this study are available from the corresponding author upon request. (4) The energy dissipation and maximum damping force data for WRFVDs in Sections 4.3 to 4.4 used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (grant no. 51778022).

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