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Abstract

Tuned mass-high damping rubber damper (TM-HDR-D), as a high-efficiency damper, has great practical feasibility and good application prospect for cable vibration mitigation. To investigate the effect of TM-HDR-D on the cable dynamics, a refined model which considers the cable sag effect is established via complex modal method. Then, based on this model, the influence of cable sag and the stiffness, location, loss factor, and mass of the TM-HDR-D on the cable dynamics is systematically investigated. Meanwhile, the multi-mode damping effect of TM-HDR-D is also analyzed. Results show that TM-HDR-D can significantly increase the additional damping of the system. The cable sag only affects the first-order modal damping, and the effect of TM-HDR-D is significantly affected by its frequency, loss factor, and installation position. Improving the mass of the damper can improve the damping effect. Reasonable selection of damper tuning frequency can achieve multi-order modal damping enhancement.

Keywords

Cable vibration control shallow cable tuned mass-high damping rubber damper (TM-HDR-D)parametric analysis damping effect

Introduction

Cable is composed of high-strength steel wire, which can withstand huge axial tension, and it has been widely used in the field of bridge engineering, such as the cable-stayed bridge, suspension bridge, and other types of cable-supported bridges. However, cables are extremely easy to be excited by environmental disturbances due to their light weight, high transverse flexibility, and low intrinsic damping.^1–5 The long-term vibration of the cable may cause local fatigue damage, affecting the bridge safety and service life. Meanwhile, the large amplitude vibration of the cable can also cause a deep anxiety in the public. Therefore, it is necessary to control the vibration of the cables in the practical bridge engineering. Up to now, there are three types of countermeasures are frequently used for cable vibration mitigation in practice, including aerodynamic treatment of cable surfaces, installing dampers, and connecting cables via cross-ties.^6–9 Among them, installing damper between the cable and bridge deck is the most commonly applied and effective measure in cable vibration control.

The purpose of installing the damper on the cable is to suppress the vibration by increasing the cable additional damping. In cable-stayed bridge engineering, many different types of dampers have been successfully used, such as oil damper,^10,11 viscous-shear damper,^7,12 high damping rubber (HDR) damper,^13–15 friction damper,^16,17 and magnetorheological (MR) damper.^18–21 Although the damper performs a good vibration reduction effect, there are also many shortcomings in practical application. First of all, the installation of the damper generally requires a support, for this reason, the damper can only be installed near to the cable end. However, the damping contribution of the damper is proportional to its installation length; thus, the damper location limitation restricts the increase of the cable attainable damping.^22–25 In additions, some damper defects occur after long-term use, for example, the oil damper has the problem of oil leakage,²⁶ and the friction damper causes the wear of friction plate.

Unlike the traditional cable mechanical dampers, the TMD-type dampers without damper support can be installed anywhere along the cable.²⁸ Meanwhile, since the installation position is not limited to the end of the cable, the additional damping provided by this kind damper will be greatly increased. In practice, tuned mass dampers (TMDs) were used in Øresund Bridge for the two longest cables vibration control.²⁷ Wu and Cai conducted a more in-depth study on the cable with a TMD-magnetorheological damper (TMD-MR) system.^28–31 In their research, theoretical and experimental methods were used to explore the influence of TMD-MR on the dynamic characteristics of cable. Their study demonstrated the superiority of TMD for cable vibration control. Theory of a taut cable and a TMD-HDR damper system was explored in detail by Cu et al.³² At the same time, they also carried out theoretical research on a taut cable with viscous damper and tuned mass damper system.³³ Di et al. conducted a full-scale cable test to study the reduction effect of the Stockbridge damper.³⁴ Recently, Di et al. proposed a strategy of using TMD to supplement the modal damping of the cable high-frequency vortex-induced vibration mode for suppressing this kind cable vibration.³⁵

In recent years, the span of cable-stayed bridges has continuously broken records, and the length of the cables is increasing as well, for example, the longest cable of Sutong Bridge is 577 m.⁷ Due to the limitation of the damper installation height and the cable sag effect, the cable damper can no longer provide enough damping.¹⁵ Therefore, in order to supplement the insufficiency of some order modal damping and overcome the limited installation of the damper support, the use of TMD-type damper control is a better choice. Moreover, considering the easy leakage of the actual liquid damper, the TM-HDR-D has a good application prospect. Therefore, on the basis of the previous research, the model of a cable with a TM-HDR-D system is established by considering cable sag via the complex modal method. Then, based on this model, the cable sag and the various parameters of the TM-HDR-D are analyzed in detail.

In this study, the model of the cable equipped with a TM-HDR-D is first established by taking cable sag into account. The influence of the cable sag effect as well as the parameters of TM-HDR-D on the cable modal damping is analyzed based on numerical solutions. Multi-mode damping effect of cable-TM-HDR-D system is also investigated. Finally, the paper concludes with a summary of this study.

Formulation of cable system with a tuned mass-high damping rubber damper (TM-HDR-D)

A shallow cable equipped with a TM-HDR-D is shown in Figure 1. The cable is fixed at two ends. A coordinate system is defined for cable with x starting from the left cable anchorage pointing rightwards along the cable, $y (x)$ represent the static displacement of the cable and v (x, t) is the dynamic displacement relative to the static position of the cable. The TM-HDR-D is installed at location $a$ from the left cable end, and $a^{'} = L - a$ . u(t) denotes the dynamic displacement of the mass of TM-HDR-D. The mass, stiffness and loss factor of TM-HDR-D is denoted as $M, K, φ$ , respectively. The cable is placed horizontally, its chord length is denoted by $L$ , the horizontal tension is $T$ , the mass per unit length is $m$ , the axial stiffness is $E A$ , and the inclination angle is $θ$ . The bending stiffness and inherent damping of cable are here not taken into account.

Figure 1.

A shallow cable with a TM-HDR-D.

For the stay cable, its static state shape can be assumed to be a parabolic function

y (x) = 4 d (1 - \frac{x}{L}) \frac{x}{L}

(1)

where

d

is the cable sag in its middle span.

The governing equation of the cable-TM-HDR-D system motion can be expressed by the following partial differential equation, as³⁶

\frac{\partial^{2} v}{\partial x^{2}} + h \frac{d^{2} y}{d x} - m \frac{\partial^{2} v}{\partial t^{2}} = f_{d} (t) δ (x - a)

(2)

where

δ (\cdot)

denotes Dirac delta function,

f_{d} (t)

is the damping force of the TM-HDR-D.

Under free vibration of cable, the displacement, cable additional horizontal tension and the damping force can be expressed as³⁶

v (x, t) = \tilde{v} (x) e^{i ω t}, h (t) = \tilde{h} e^{i ω t} a n d f_{d} (t) = {\tilde{f}}_{d} e^{i ω t}

(3)

where

i = \sqrt{- 1}

ω

is the cable complex frequency, and

\tilde{v}

is the corresponding complex mode shape.

\tilde{h}

and

{\tilde{f}}_{d}

denote the amplitude of the cable additional horizontal tension and the damping force. Substituting equation (3) into equation (2), one finds

\frac{d^{2} \tilde{v}}{d x^{2}} + β^{2} \tilde{v} = \frac{8 d}{L^{2}} \frac{\tilde{h}}{T}

(4)

where

β

is complex wave number,

β = ω \sqrt{m / T}

The elastic deformation of cable under the action of additional horizontal tension has the following relationship

\frac{\tilde{h} L_{e}}{E A} = \frac{m g}{T} [\int_{0}^{a} \tilde{v} (x) d x + \int_{0}^{a^{'}} \tilde{v} (x^{'}) d x^{'}]

(5)

where

L_{e} = \int_{0}^{L} {(d s / d x)}^{3} d x ≅ [1 + 8 {(d / L)}^{2}] L

, and ds denotes the arch length of cable element.²⁴

With ${\tilde{v}}_{d}$ denotes the cable mode shape amplitude at the damper location, then, the solution of the Equation (4) can be expressed as³⁶

\tilde{v} (x) = {\tilde{v}}_{d} \frac{\sin (β x)}{\sin (β a)} + \frac{8 d \tilde{h} [1 - \cos (β x)] \sin (β a) - [1 - \cos (β a)] \sin (β x)}{{(β L)}^{2} T \sin (β a)}, 0 \leq x \leq a \tilde{v} (x^{'}) = {\tilde{v}}_{d} \frac{\sin (β x^{'})}{\sin (β a^{'})} + \frac{8 d \tilde{h} [1 - \cos (β x^{'})] \sin (β a^{'}) - [1 - \cos (β a^{'})] \sin (β x^{'})}{{(β L)}^{2} T \sin (β a^{'})}, 0 \leq x^{'} \leq a^{'}

(6)

After introducing equation (6) into Equation (5), Equation (5) can be rewritten as

\frac{8 d}{{(β L)}^{2}} \frac{\tilde{h}}{T} [2 \tan \frac{β a}{2} + 2 \tan \frac{β a^{'}}{2} - β L + \frac{1}{λ^{2}} {(β L)}^{3}] = {\tilde{v}}_{d} (\tan \frac{β a}{2} + \tan \frac{β a^{'}}{2})

(7)

where

λ^{2}

is Irvine parameter

λ^{2} = {(\frac{m g L \cos θ}{T})}^{2} \frac{E A L}{T L_{e}}

The internal force equilibrium at the point of TM-HDR-D can be written as

T (\frac{d \tilde{v}}{d x} |_{+ a} - \frac{d \tilde{v}}{d x} |_{- a}) = {\tilde{f}}_{d}

(8)

For the TM-HDR-D, its equilibrium equation is written as

K (1 + i φ) ({\tilde{v}}_{d} - \tilde{u}) + ω^{2} M \tilde{u} = 0

(9)

The displacement amplitude ratio between the mass of TM-HDR-D and the cable at the point of damper is denoted by

α = \tilde{u} / {\tilde{v}}_{d}

, the TM-HDR-D vibration circular frequency is

ω_{d} = \sqrt{K / M}

, and the frequency ratio between the cable fundamental circular frequency and the TM-HDR-D vibration circular frequency is denoted by

ρ = ω_{1}^{0} / ω_{d}

. Let

s = ω / ω_{1}^{0}

. Note that

ω_{1}^{0}

is the cable fundamental circle frequency without considering cable sag. Then, using equation (9), we can derive

α

α = \frac{1 + i φ}{1 + i φ - ρ^{2} s^{2}}

(10)

Meanwhile, the damping force can be rewritten as

{\tilde{f}}_{d} = - {α ω}^{2} M {\tilde{v}}_{d}

(11)

Using Equations (8) and (11), the characteristic equation of a shallow cable equipped with a TM-HDR-D is thus written as

\cot (β a) + \cot (β a^{'}) + \frac{{(\tan \frac{β a}{2} + \tan \frac{β a^{'}}{2})}^{2}}{2 \tan \frac{β a}{2} + 2 \tan \frac{β a^{'}}{2} - β L + \frac{{(β L)}^{3}}{λ^{2}}} = - \frac{{\tilde{f}}_{d}}{T β {\tilde{v}}_{d}}

(12)

The complex wave number

β

can be obtained by solving the characteristic equation (12) via the Newton method, and then, the damping is obtained according to the following formula

ζ_{n} = \frac{I m (β_{n})}{| β_{n} |}, n = 1,2, \dots

(13)

Dynamic analysis of cable-TM-HDR-D system

To understand the effect of the TM-HDR-D on the cable vibration characteristics, the influence of cable sag and the stiffness, installation location, loss factor and mass of the TM-HDR-D on the cable dynamics is systematically investigates in this section.

Impact of cable sag effect

Figure 2 shows the effects of the cable sag on the first two modal damping ratios of the cable, where the curves correspond to the different installation positions of the TM-HDR-D. Note that, in the following analysis, the loss factor $φ$ of TM-HDR-D is set to 0.5, the installation mass ratio $μ = M / (m L)$ is 0.01, and the damper is tuned for the first-order and second-order cable mode, that is, $ρ = 1$ and $ρ = 0.5$ , respectively. In real-world cable-stayed bridges, the cable sag $λ^{2}$ is generally less than 3, and thus the range of sag in the analysis is set to 0∼5. Note that $a / L$ indicates the installation position of the TM-HDR-D.

Figure 2.

Modal damping ratio due to the influence of the cable sag ( $φ = 0.5, μ = 0.01$ ): (a) 1st mode ( $ρ = 1$ ); (b) 2nd mode ( $ρ = 0.5$ ).

It can be seen from Figure 2, for the first mode, the effect of cable sag on the modal damping ratio is obvious, especially when the TM-HDR-D is installed at the middle of the cable span, that is, $a / L = 0.5$ . The cable modal damping ratio first increases and then decreases with the increase of sag. The main reason for the phenomenon is that the cable sag changes the cable fundamental frequency, which changes the frequency ratio between the cable fundamental circular frequency and the TM-HDR-D vibration circular frequency, and then the damping effect of the TM-HDR-D is altered due to the frequency sensitivity of TM-HDR-D. More specifically, for the case of the frequency ratio $ρ = 1$ , when the cable sag parameter is close to 2, cable can be tuned best with the TM-HDR-D, leading to the maximum cable modal damping. The increase and decrease of sag parameters will make the optimal tuning frequency ratio deviate, resulting in the decrease of damping ratio, as depicted in Figure 2(a). Previous studies³⁶ have also shown that the actual cable sag can only change the first modal frequency, and the rest of other modal frequencies are almost unchanged. Therefore, the sag has little effect on the second-order modal damping ratio, as shown in Figure 2(b), owning to the cable sag not altering the second-mode frequency tuning.

Impact of frequency ratio ( $ρ$ ) of TM-HDR-D

In this section, the effect of the frequency ratio ( $ρ$ ) of TM-HDR-D on the modal damping is discussed. Figure 3 shows the dependence of modal damping ratio on the frequency ratio $ρ$ for the different cable sag and different TM-HDR-D mass ratio $μ$ . In this case, the loss factor of TM-HDR-D is $φ = 0.5$ , and the installation location is $a / L = 0.5$ .

Figure 3.

Influence of frequency ratio $ρ$ on the first modal damping ratio ( $φ = 0.5, \frac{a}{L} = 0.5$ ) for: (a) different cable sag $λ^{2}$ ( $μ = 0.01$ ); (b) different TM-HDR-D mass ratio $μ$ .

Figure 3 demonstrates that the TM-HDR-D is an absorber with high-frequency sensitivity, of which the damping effect is significantly reduced when the damper vibration frequency deviates from the optimal tuning frequency. For the first mode, cable sag reduces the optimal tuning frequency ratio, as illustrated in Figure 3(a). At the same time, this can also explain the phenomenon shown in Figure 2(a). Increasing the mass of the damper can significantly enhance the attainable modal damping ratio of the cable, and it is worth noting that the effect mass ratio $μ$ on the optimal tuning frequency ratio is not very obvious, as depicted in Figure 3(b). It can be seen from the above analysis that, as a frequency tuning vibration damping device,^28,29 the TM-HDR-D is more sensitive to frequency. For a TM-HDR-D with a mass ratio $μ = 0.02$ as shown in Figure 3(b), when the frequency ratio $ρ$ changes from 0.98 to 0.8 (18% decrease), the damping ratio is reduced by 53%; when the $ρ$ changes from 0.98 to 1.2 (22% increase) the damping ratio is reduced by 51%. Therefore, in practice, the TM-HDR-D mass need to be increased for ensuring that the additional damping provided by the damper is still satisfactory in the absence of precise tuning.

Impact of location ( $a / L$ ) of TM-HDR-D

This section investigates the impact of the TM-HDR-D installation location on the cable modal damping ratio. Figure 4 shows the relationship between the installation position ( $a / L$ ) of the TM-HDR-D and the cable modal damping ratio for the different mass ratio. Note that the TM-HDR-D loss factor is $φ = 0.5$ , the cable sag $λ^{2} = 2$ , and tuning frequency ratios of damper are $ρ = 1$ for the first mode and $ρ = 0.5$ for the second mode.

Figure 4.

Influence of location $\frac{a}{L}$ on the modal damping ratio ( $φ = 0.5, λ^{2} = 2$ ) for: (a) 1st mode ( $ρ = 1$ ); (b) 2nd mode ( $ρ = 0.5$ ).

For the first mode, the closer the TM-HDR-D is installed to the midpoint of the cable, that is, $\frac{a}{L} = 0.5$ , the greater the modal damping obtained by the cable system, as shown in Figure 4(a). For the second mode, as shown in Figure 4(b), when the TM-HDR-D mass is relatively small, such as $μ \leq 0.03$ , the modal damping ratio reaches the maximum value with the damper installed at the quarter point ( $a / L = 0.25$ ) of the cable, and when the installation mass is large, the optimal installation position will be close to the cable end ( $a / L = 0$ ), for example, the optimal installation position is $a / L = 0.18$ for the mass ratio $μ = 0.07$ . Increasing the mass ratio can decrease the location and significantly improve the damping ratio (this can be seen in Figure 4(b) for cases 3%, 5%, and 7% of mass ratio). In practical engineering, when the TM-HDR-D has to be installed near to the cable end ( $a / L = 0$ ) for the convenience of installation, its mass can be increased to supplement the damping loss caused by the installation position limitation.

Impact of loss factor ( $φ$ ) of TM-HDR-D

The influence of loss factor $φ$ of TM-HDR-D on the cable modal damping ratio is analyzed in this section. Figure 5(a) and (b) show the cable modal damping curves along with the loss factor of TM-HDR-D under different mass ratio $μ$ , for the first and second modes, respectively. Note that, in this case, the cable sag $λ^{2} = 2$ , the frequency ratio $ρ = 1$ and location $a / L = 0.5$ for the first mode, and the frequency ratio $ρ = 0.5$ and location $a / L = 0.25$ for the second mode.

Figure 5.

Influence of loss factor $φ$ on the first modal damping ratio ( $λ^{2} = 2$ ): (a) 1st mode ( $ρ = 1, \frac{a}{L} = 0.5$ ); (b) 2nd mode ( $ρ = 1, \frac{a}{L} = 0.25$ ).

From Figure 5(a), it can be seen that there is a peak point in the damping curve, in other words, an optimal value of the loss factor can make the modal damping ratio reach the maximum value. As the TM-HDR-D mass increases, the optimal loss factor increases to a certain extent, for example, when mass ratio $μ = 0.005$ , the optimal loss factor is $φ = 0.205$ , but when mass ratio $μ = 0.03$ , the optimal loss factor increases to $φ = 0.365$ . It is interesting that the actual high damping rubber loss factor value is very close to the optimal loss factor analyzed in this section. For the second mode, Figure 5(b) shows the cable damping curves with different damper masses; meanwhile, the corresponding optimal loss factor values are marked as well. The increase of the TM-HDR-D mass can significantly improve the modal damping of the system, as well as the optimal loss factor, as seen in Figure 5(b). Comparing optimal loss factors in the first and second mode under the same installation mass, it can be found that the optimal value of the second mode is lower than that of the first mode. In general, the modal damping is more sensitive to the loss factor, especially in the second cable mode, that is, when the loss factor deviates from the optimal value, the modal damping of the system decreases significantly.

Impact of mass ( $μ$ ) of TM-HDR-D

In this section, the influence of TM-HDR-D mass on its modal damping effect is analyzed. Figure 6 plots the modal damping ratio with respect to the mass ratio $μ$ for the first and second cable modes, respectively. Different damping curves in the figure correspond to different loss factors. Note that cable sag is set to $λ^{2} = 2$ . The TM-HDR-D is installed at the midpoint of the cable with the frequency ratio $ρ = 1$ for the first mode, and installed at the quarter point of the cable with the frequency ratio $ρ = 0.5$ for the second mode.

Figure 6.

Influence of mass $μ$ on the first modal damping ratio ( $λ^{2} = 2$ ): (a) 1st mode ( $ρ = 1, \frac{a}{L} = 0.5$ ); (b) 2nd mode ( $ρ = 0.5, \frac{a}{L} = 0.25$ ).

As depicted in Figure 6, the damping curve increases first and then tends to be stable. For the first mode, as shown in Figure 6(a), the damping curve with the smallest loss factor ( $φ = 0.2$ ) has a larger damping value ( $ζ_{1} = 0.02$ ), when the mass is small ( $μ = 0.014$ ). However, with the increase of mass ratio $μ$ , it goes to stability first, and the maximum achievable damping ratio is lower than curves with the loss factor $φ = 0.4,0.6 a n d 0.8$ . For the case with $φ = 0.4$ , the damping curve tends to be stable after the mass ratio $μ$ reaches about 0.06; for the case with $φ = 0.6$ , the damping curve tends to be stable after the mass ratio $μ$ is about 0.08. Similarly, for the second mode, the damping curves also perform the similar law of the first mode. It is interesting that, for the case $φ = 0.2$ , with the increase of installation mass ratio $μ$ , the second modal damping ratio first increases rapidly and then decreases slowly. The reason for the decrement of damping ratio is that the tuning frequency deviates from the optimal value due to the increase of TM-HDR-D mass. Generally speaking, for a TM-HDR-D with a certain loss factor, there is a most suitable installation mass, and when the mass of the TM-HDR-D exceeds the optimal value, it is less efficient to enhance the cable modal damping by continuing to increase the mass. The value of optimal loss factor for the small mass of TM-HDR-D is relatively small, as demonstrated in Figure 5. For this reason, in all damping curves, the smaller the loss factor value, the sooner the stability is reached.

Multi-mode damping effect of cable-TM-HDR-D system

Design of TM-HDR-D

As demonstrated in previous studies, TMD or other tuned-type dampers have a high sensitivity to frequency, and this section thus discusses the damping effectiveness of TM-HDR-D in multiple modes. In the following analysis, the cable sag is $λ^{2} = 2$ , the loss factor of TM-HDR-D is $φ = 0.5$ , and the TM-HDR-D is installed at $a / L = 0.25, 0.3$ and 0.4, respectively. Figure 7 and Figure 8 show the first two modal damping ratios with respect to the frequency ratio $ρ$ for the TM-HDR-D mass ratio $μ = 0.01$ and $μ = 0.03$ , respectively.

Figure 7.

Influence of frequency ratio $ρ$ on the first two modal damping ratios ( $λ^{2} = 2, μ = 0.01, φ = 0.5$ ): (a) $\frac{a}{L} = 0.25$ ; (b) $\frac{a}{L} = 0.3$ ; (c) $\frac{a}{L} = 0.4$ .

Figure 8.

Influence of frequency ratio $ρ$ on the first two modal damping ratios ( $λ^{2} = 2, μ = 0.03, φ = 0.5$ ): (a) $\frac{a}{L} = 0.25$ ; (b) $\frac{a}{L} = 0.3$ ; (c) $\frac{a}{L} = 0.4$ .

It can be found from the Figure 7 that the first modal damping curve reaches its peak when $ρ$ is about 1, and the second modal damping curve reaches its peak when $ρ$ is near to 0.5. However, when one mode achieves the optimal tuning of the TM-HDR-D to obtain the optimal modal damping ratio, the damping of the other order mode is very low. To ensure that the cable modal damping ratios of first two modes reach a good level, the optimal frequency tuning $ρ$ can be selected from the frequency ratio corresponding to the intersection of the two damping curves, marked by black dot as shown in Figure 7. Comparing Figures 7(a)–(c), it can be seen that when the TM-HDR-D is installed at $a / L = 0.3$ , the modal damping ratio of the first two modes is relatively large, by 0.0028, with the frequency tuning $ρ = 0.750$ . Figure 8 shows the damping curves of the system with large TM-HDR-D mass, that is, $μ = 0.03$ . The damping ratios of the first two modes have been significantly improved due to the improved the TM-HDR-D mass, for example, the optimal damping ratio is increased from 0.0028 to 0.0082, when the damper is installed at $a / L = 0.3$ . Meanwhile, it should be worth noting that increasing the TM-HDR-D mass has not a considerable influence on the optimal tuning frequency ratio $ρ$ .

Comparison of effects of different dampers

At present, the commonly used dampers for cable vibration reduction include viscous dampers and HDR dampers.^1,15 In this section, viscous damper, HDR damper and TMD are used to reduce the vibration of a cable in a real-world cable-stayed bridge in China, and cable parameters are listed in Table 1.

Table 1.

Parameters of cable for present study.

Length, L (m)	Tension, T (kN)	Mass, m (kg/m)	Diameter, D (m)	Fundamental frequency (Hz)
255.8	7698	108.5	0.13	0.52

In the design, the first two cable modes are considered for optimizing of the dampers. In the following discussion, assume that the viscous damper or HDR damper is installed at 0.02 L. For the viscous damper, the optimal damping coefficient is 335 kN.s/m. For HDR damper, when the loss factor ( $φ$ ) is 0.5, the optimal stiffness parameter is 1360 kN/m. Besides, a TM-HDR-D is installed at $a / L = 0.3$ with mass ratio $μ = 0.02$ and loss factor $φ = 0.5$ . Considering the increase of the first two cable modal damping ratios, the frequency ratio $ρ$ of TM-HDR-D is taken as 0.776. Comparison of damping effects of different vibration reduction measures is shown in Figure 9.

Figure 9.

Comparison of damping effects of different vibration reduction measures.

From Figure 9, it can be seen that, only from the perspective of lifting cable modal damping, the effect of TM-HDR-D is between the HDR damper and the viscous damper. The main reason is that TM-HDR-D is relatively sensitive to frequency, and the use of multiple TM-HDR-Ds can further improve its damping effect. However, in practical engineering, the installation of viscous dampers requires supports, and there is a problem of damping fluid leakage. Therefore, using TM-HDR-Ds or using TM-HDR-Ds and cable-end damper to control cable vibration in practical engineering has great advantages. In additions, Figure 10 shows the cable with a TM-HDR-D system mode shapes of the first two modes. It can be seen that the mode shapes did not change significantly after adding TM-HDR-D, due to the relatively small installation quality of TM-HDR-D.

Figure 10.

Mode shapes of cable system with ( $μ = 0.02, ρ = 0.776, \frac{a}{L} = 0.3$ ): (a) 1st mode; (b) 2nd mode.

Conclusions

In this paper, a shallow cable with a TM-HDR-D system was studied in detail. Firstly, the refined model by taking the sag effect into account was established using the complex modal method. Then, the critical parameters of the system, including cable sag and the stiffness, installation location, loss factor and mass of the TM-HDR-D, are studied based on numerical solutions. Meanwhile, the damping effect of the TM-HDR-D on the first two modes of the cable is analyzed as well. The following conclusions can be drawn:

(1) The cable sag only has a considerable influence on the first-order modal damping, and changes the damping effect of the TM-HDR-D mainly caused by changing the tuning frequency ratio $ρ$ of the damper.

(2) TM-HDR-D has a high-frequency sensitivity, and the sag will reduce the first-order tuning frequency ratio while the mass has little effect on the optimal frequency ratio; the modal damping is better when the TM-HDR-D is located at the place where the amplitude of the corresponding mode is larger; the modal damping is relatively sensitive to the loss factor of the TM-HDR-D, especially in the second cable mode; for a TM-HDR-D with a certain loss factor, there is the most suitable installation mass.

(3) Due to the sensitivity of the TM-HDR-D to frequency, a single damper cannot achieve optimal damping for all concerned cable modes. However, it is possible for the modal damping of each mode to reach the target level by selecting the appropriate tuning frequency $ρ$ , installation position and mass of the TM-HDR-D.

Footnotes

Author Contributions

Z.Z. carried out the studies, participated in the derivation and drafted the manuscript. P.L. conceived of the study, and participated in its design and coordination and helped to draft the manuscript. C.L. carried out the numerical solution and participated in the design of the study. All authors read and approved the final manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data Availability Statement

All data, models, and codes to support the findings of this study are available from the corresponding author upon request.

ORCID iD

Zheng Zhang

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A shallow cable with a tuned mass-high damping rubber damper

Abstract

Keywords

Introduction

Formulation of cable system with a tuned mass-high damping rubber damper (TM-HDR-D)

Dynamic analysis of cable-TM-HDR-D system

Impact of cable sag effect

Impact of frequency ratio ( ρ ) of TM-HDR-D

Impact of location ( a / L ) of TM-HDR-D

Impact of loss factor ( φ ) of TM-HDR-D

Impact of mass ( μ ) of TM-HDR-D

Multi-mode damping effect of cable-TM-HDR-D system

Design of TM-HDR-D

Comparison of effects of different dampers

Conclusions

Footnotes

Author Contributions

Declaration of Conflicting Interests

Funding

Data Availability Statement

ORCID iD

References

Impact of frequency ratio ( $ρ$ ) of TM-HDR-D

Impact of location ( $a / L$ ) of TM-HDR-D

Impact of loss factor ( $φ$ ) of TM-HDR-D

Impact of mass ( $μ$ ) of TM-HDR-D