Free vibration of a taut cable with a parallel-connected viscous mass damper and a grounded cross-tie

Abstract

Recent studies have shown that the combination of external dampers and cross-ties can overcome their respective deficiencies while retaining their respective merits. Inspired by the advantages of a viscous damper (VD) and a cross-tie on a single stay cable, the coupled damping effect of a parallel-connected viscous mass damper (PVMD) and a grounded cross-tie on a single stay cable is investigated in this paper. The complex wave number equations of the cable–PVMD–cross-tie system are first formulated through the complex modal analysis. Subsequently, the asymptotic and iterative complex wave numbers are compared to evaluate the applicability of the asymptotic and iterative solutions. Furthermore, parametric studies are carried out to investigate the effects of the stiffness coefficients and the installation positions of the cross-tie on the first supplemental modal damping ratio and frequency of the cable. Finally, the installation positions of the cross-tie are optimized to achieve the optimal vibration control of the cable with the PVMD and the cross-tie.

Keywords

Stay cable vibration control viscous mass damper cross-tie modal damping ratio

Introduction

In recent years, structural vibration control has received extensive attentions, especially the vibration control of stay cables in cable-stayed bridges.^1-3 Stay cables, as the critical load-bearing component of cable-stayed bridges, are highly vulnerable to environmental excitations such as wind, wind-rain, and parametric excitations.^4-6 Frequent vibrations may shorten the cable life and impair public confidence in the safety of cable-stayed bridges. Consequently, several countermeasures have been proposed to suppress the excessive vibrations of stay cables, including changing the surface characteristics of the cable to improve its aerodynamic characteristics,^7,8 connecting cables via the cross-tie,⁹ and attaching external dampers on cables.^10-12 Among these countermeasures, attaching external dampers on cables is the most common and effective measure. However, it is difficult to effectively mitigate vibrations for super-long cables in large-span cable-bridges, such as the longest cable in the Stonecutters Bridge (536 m) and the longest cable in the Sutong Bridge (577 m). Hence, it is necessary to combine available countermeasures to efficiently mitigate these cable vibrations.^13-15

To improve the control performance of passive dampers, a semi-active control strategy using magneto-rheological (MR) damper is proposed.¹⁶ MR dampers, as one of the most representative semi-active control devices, will provide a semi-active control method for stay cables to balance effectiveness and practicality. It is found that the negative stiffness characteristic of the semi-active MR dampers helps to improve the supplemental modal damping ratio of stay cables.^17,18 Inspired by the advantages of the negative stiffness characteristic of the semi-active MR dampers, the passive negative stiffness dampers (NSDs), including pre-spring NSDs^19,20 and magnetic NSDs,^21,22 have received more attention in the field of cable vibration control. Moreover, a series of theoretical and experimental investigations have proved the effectiveness of NSDs in mitigating cable vibrations.^23-25 Shi et al.^23,24 demonstrated that the negative stiffness characteristic of the NSDs was conducive to providing more sufficient supplemental damping than that of passive viscous dampers. Javanbakht et al.²⁵ verified that the NSDs presented superior performance over the conventional positive stiffness dampers (PSDs) in cable vibration control. However, excessive passive negative stiffness may induce the instability problem to the cable.²² Recent studies have shown that passive VD and inertial mass elements are connected in parallel to consist of a parallel-connected viscous mass damper (PVMD), which can improve the vibration control performance of structures.^26-29 Compared with conventional VD, PVMD can provide more supplemental damping for stay cables, which has been verified by both theoretical analyses^30-33 and experimental investigations.^34,35

A combination strategy of using both cross-ties and external dampers is also proposed to suppress the cable vibrations. Connecting cables via cross-ties can effectively enhance the in-plane stiffness and frequencies of stay cables,^36-38 while it is difficult to effectively improve the supplemental damping of stay cables.³⁹ It has been found that the hybrid techniques of using cross-ties and dampers can overcome their respective deficiencies while retaining their respective merits.^40-43 Among them, Zhou et al.⁴³ simplified the cross-ties as a discrete spring to investigate the damping and frequency of a taut cable with a damper and a spring. Di et al.⁴⁴ proposed a simplified analytical method to investigate the effects of cross-ties on the dynamics of a hybrid cable network system. Furthermore, recent studies have shown that when two dampers are installed at opposite cable ends, the combined damping effect is approximately the sum of the contributions from each damper.^45-47 In addition, Yang et al.⁴⁸ explored a novel optimal design of two viscous dampers near to a cable anchorage, which can achieve a potential improvement in damping ratios of cables for lower and higher vibration modes. Thus, the combination strategy of using two dampers, either or a damper and a cross-tie, can achieve superior cable vibration control performance over that of using a single damper.

Inspired by the advantages of a VD and a grounded cross-tie on a single stay cable, the coupled damping effects of a single stay cable with a parallel-connected viscous mass damper (PVMD) and a grounded cross-tie are further investigated in this paper. The paper is organized as follows. Firstly, the complex wave number equations of the cable–PVMD–cross-tie system are formulated by using the complex modal analysis. Then, the asymptotic and iterative solutions for the PVMD and the grounded cross-tie, either at the same cable end or at opposite cable ends, are obtained, respectively. Furthermore, the applicability of the iterative solution is evaluated, and the coupled damping effect of the cable with a PVMD and a cross-tie is further discussed when the cross-tie is connected at different positions of the cable. Finally, the maximum first supplemental modal damping ratio of a cable with a PVMD and a grounded cross-tie is highlighted when the cross-tie is installed at the cable mid-span.

Formulation of the cable–PVMD–cross-tie system

A taut cable with a PVMD and a grounded cross-tie is shown in Figure 1. The length, the tension, and the mass per unit length of the cable are denoted as l, T, and m, respectively. The coordinate system defines that the x-axis and the u-axis are along the cable chord and the transverse direction. The PVMD and the cross-tie are respectively fixed at distances x₁ and x₂ from the left end of the cable (x₂≥x₁). The distance between the cross-tie and the right end of the cable is denoted as $x_{2}^{*} = l - x_{2}$ . The stiffness coefficient of the cross-tie is denoted as k_d. The damping coefficient and inertial mass of the PVMD are denoted as c_d and b_d, respectively. The free vibration of the cable–damper–cross-tie system in the transverse direction can be described by the following equation⁴⁵

T \frac{\partial^{2} u (x, t)}{\partial x^{2}} - m \frac{\partial^{2} u (x, t)}{\partial t^{2}} = = \sum_{j = 1}^{2} f_{j} (t) δ (x - x_{j})

(1)

where u(x,t) is the cable transverse displacement; δ(⋅) is the Dirac delta function; f₁(t) is the damping force of the PVMD located at x=x₁; and f₂(t) is the concentrated force of the cross-tie located at x=x₂.

Figure 1.

A taut cable with a PVMD and a grounded cross-tie.

Equation (1) satisfies the boundary conditions at the cable ends, and there is

u (0, t) = u (l, t) = 0

(2)

At the damper and the cross-tie location, u(x) needs to satisfy the following boundary conditions

\begin{array}{l} u (x_{1}^{-}, t) = u (x_{1}^{+}, t) = u (x_{1}, t) \\ u (x_{2}^{-}, t) = u (x_{2}^{+}, t) = u (x_{2}, t) \\ T ({\frac{\partial u (x, t)}{\partial x} |}_{x_{j}^{+}} - {\frac{\partial u (x, t)}{\partial x} |}_{x_{j}^{-}}) = f_{j} (t), j = 1, 2 \end{array}

(3)

For free vibration of the cable, the cable transverse displacement, the force of the PVMD, and the force of the cross-tie can be expressed as

u (x, t) = U (x) e^{i ω_{n} t}, f_{1} (t) = F_{1} e^{i ω_{n} t}, f_{2} (t) = F_{2} e^{i ω_{n} t}

(4)

where U(x) is the corresponding complex mode shape; and ω_n is the n^th complex natural frequency of the cable; there is

ω_{n} = | ω_{n} | (\sqrt{1 - ξ_{n}^{2}} + i ξ_{n})

(5)

where ξ_n is the n^th supplemental modal damping ratio of the cable.

Substituting equation (4) into equation (1), U(x) in each span needs to satisfy a homogeneous equation^45,46

\frac{d^{2} U (x)}{d x^{2}} + β_{n}^{2} U (x) = 0 {\begin{matrix} 0 \leq x \leq x_{1} \\ x_{1} \leq x \leq x_{2} \\ 0 \leq x^{*} \leq x_{2}^{*} \end{matrix}

(6)

where

β_{n} = ω_{n} \sqrt{m / T}

refers to the wave number.

The general solution of equation (6) can be written in the form

U (x) = {\begin{matrix} U (x_{1}) \frac{\sin β_{n} x}{\sin β_{n} x_{1}} & 0 \leq x \leq x_{1} \\ U (x_{1}) \frac{\sin β_{n} (x_{2} - x)}{\sin β_{n} (x_{2} - x_{1})} + U (x_{2}) \frac{\sin β_{n} (x - x_{1})}{\sin β_{n} (x_{2} - x_{1})} & x_{1} \leq x \leq x_{2} \\ U (x_{2}) \frac{\sin β_{n} x^{*}}{\sin β_{n} x_{2}^{*}} & 0 \leq x^{*} \leq x_{2}^{*} \end{matrix}

(7)

Substituting equation (7) into equation (3) yields

{\begin{matrix} \cot β_{n} x_{1} + \cot β_{n} (x_{2} - x_{1}) - \frac{U_{2}}{U_{1}} \frac{1}{\sin β_{n} (x_{2} - x_{1})} = - \frac{F_{1} / T}{β_{n} U_{1}} \\ - \frac{U_{1}}{U_{2}} \frac{1}{\sin β_{n} (x_{2} - x_{1})} + \cot β_{n} (x_{2} - x_{1}) + \cot β_{n} x_{2}^{*} = - \frac{F_{2} / T}{β_{n} U_{2}} \end{matrix}

(8)

Substituting U₁/U₂ from the second equation of equation (8) into the first equation of equation (8) and rearranging, the characteristic equation of the wave number β_n can be derived as

\begin{array}{l} (\cot β_{n} x_{1} + \frac{F_{1} / T}{β_{n} U_{1}}) (\cot β_{n} x_{2}^{*} + \frac{F_{2} / T}{β_{n} U_{2}}) \\ + (\cot β_{n} x_{1} + \frac{F_{1} / T}{β_{n} U_{1}} + \cot β_{n} x_{2}^{*} + \frac{F_{2} / T}{β_{n} U_{2}}) \cot β_{n} (x_{2} - x_{1}) = 1 \end{array}

(9)

The forces of the PVMD and the cross-tie can be expressed as

f_{1} (t) = c_{d} \frac{\partial u (x_{1}, t)}{\partial t} + b_{d} \frac{\partial^{2} u (x_{1}, t)}{\partial t^{2}}, f_{2} (t) = k_{d} u (x_{2}, t)

(10)

When the PVMD and the grounded cross-tie are installed at the same cable end, that is, x₁≤x₂<l/2, substituting equation (10) into equation (9), equation (9) can be rearranged to the form relating to x₁ and x₂ as

\tan β_{n} l = \frac{A + i B}{C + i D}

(11)

A = - χ \sin^{2} β_{n} x_{1} + κ \sin^{2} β_{n} x_{2} - χ κ \sin β_{n} x_{1} \sin β_{n} x_{2} \sin β_{n} (x_{2} - x_{1})

(11a)

B = η \sin^{2} β_{n} x_{1} + κ η \sin β_{n} x_{1} \sin β_{n} x_{2} \sin β_{n} (x_{2} - x_{1})

(11b)

C = 1 - χ \sin β_{n} x_{1} \cos β_{n} x_{1} + κ \sin β_{n} x_{2} \cos β_{n} x_{2} - κ χ \sin β_{n} x_{1} \cos β_{n} x_{2} \sin β_{n} (x_{2} - x_{1})

(11c)

D = η \sin β_{n} x_{1} \cos β_{n} x_{1} + κ η \sin β_{n} x_{1} \cos β_{n} x_{2} \sin β_{n} (x_{2} - x_{1})

(11d)

where

κ = k_{d} / β_{n} T

denotes the dimensionless stiffness coefficient of the cross-tie and

η = c_{d} / \sqrt{m T}

and

χ = b_{d} ω / \sqrt{m T}

denote the dimensionless damping coefficient and the dimensionless inertial mass of the PVMD, respectively.

When the PVMD and the grounded cross-tie are installed at opposite cable ends, that is, $0 < x_{2}^{*} < l / 2$ , equation (9) can be rearranged to the form relating to x₁ and $x_{2}^{*}$ as

\tan β l = \frac{E + i F}{G + i H}

(12)

E = - χ \sin^{2} β_{n} x_{1} + κ \sin^{2} β_{n} x_{2}^{*} - χ κ \sin β_{n} x_{1} \sin β_{n} x_{2} \sin β_{n} (x_{2}^{*} + x_{1})

(12a)

F = η \sin^{2} β_{n} x_{1} + κ η \sin β_{n} x_{1} \sin β_{n} x_{2}^{*} \sin β_{n} (x_{2}^{*} + x_{1})

(12b)

G = 1 - χ \sin β_{n} x_{1} \cos β_{n} x_{1} + κ \sin β_{n} x_{2}^{*} \cos β_{n} x_{2}^{*} - κ χ \sin β_{n} x_{1} \sin β_{n} x_{2}^{*} \cos β_{n} (x_{2}^{*} + x_{1})

(12c)

H = η \sin β_{n} x_{1} \cos β_{n} x_{1} + κ η \sin β_{n} x_{1} \sin β_{n} x_{2}^{*} \cos β_{n} (x_{2}^{*} + x_{1})

(12d)

The PVMD and the grounded cross-tie at the same cable end

Asymptotic solution and iterative solution

The iterative solution of the complex wave number $β_{n}$ is obtained by the fixed-point iteration method.⁴⁹ The initial value of the iterative equation is the undamped complex wave number $β_{n}^{0}$ . Substituting $β_{n}^{0}$ into the right side of equation (11), the n^th modal resulting value of $β_{n}^{1}$ is obtained. Similarly, with current estimate $β_{n}^{j}$ (j=1, 2, 3, …), a new estimate $β_{n}^{j + 1}$ will be derived. The iterative equation can be written in the form

\begin{array}{l} β_{n}^{j + 1} l = n π + \arctan \frac{A (β_{n}^{j}) + i B (β_{n}^{j})}{C (β_{n}^{j}) + i D (β_{n}^{j})} \\ β_{n}^{0} = \frac{n π}{l}, n = 1, 2, 3, \dots \end{array}

(13)

The n^th supplemental modal damping ratio of the cable can be calculated by

ξ_{n} = \frac{Im (ω_{n})}{| ω_{n} |} = \frac{Im (β_{n})}{| β_{n} |} \approx \frac{Im (Δ β_{n})}{| β_{n}^{0} |}

(14)

Finally, the supplemental modal damping ratio of the cable with a PVMD and a grounded cross-tie can be calculated by equation (14) after solving iteratively the wave number.

The following assumptions are introduced⁴⁵: (1) the PVMD and the cross-tie are installed near the left cable end, that is, $x_{1} = l$ , $x_{2} = l$ , $x_{1} < x_{2}$ and (2) the complex wave number β_n of the cable has a small perturbation $Δ β_{n} = β_{n} - β_{n}^{0} = β_{n}^{0}$ from the undamped value $β_{n}^{0} = n π / l$ . These assumptions yield the following approximations

\tan β_{n} l ≃ β_{n} l - β_{n}^{0} l, \sin β_{n} x_{1} ≃ β_{n}^{0} x_{1}, \sin β_{n} x_{2} ≃ β_{n}^{0} x_{2}, \cos (β_{n} x_{1}) ≃ \cos (β_{n} x_{2}) ≃ 1

(15)

The asymptotic solution for the complex wave number β_n can be expressed as

\frac{Δ β l}{β^{0} l} = \frac{\bar{κ} \frac{x_{2}^{2}}{x_{1}^{2}} - \bar{b} - \bar{κ} \bar{b} \frac{x_{2}}{x_{1}} (\frac{x_{2}}{x_{1}} - 1) + i \bar{c} [1 + \bar{κ} \frac{x_{2}}{x_{1}} (\frac{x_{2}}{x_{1}} - 1)]}{1 + \bar{κ} \frac{x_{2}}{x_{1}} - \bar{b} - \bar{κ} \bar{b} (\frac{x_{2}}{x_{1}} - 1) + i \bar{c} [1 + \bar{κ} (\frac{x_{2}}{x_{1}} - 1)]} \frac{x_{1}}{l}

(16)

where

\bar{κ} = κ β x_{1} = k_{d} x_{1} / T

\bar{c} = η β x_{1} = c_{d} x_{1} ω_{n}^{0} / T

, and

\bar{b} = χ β x_{1} = b_{d} x_{1} {(ω_{n}^{0})}^{2} / T

Substituting equation (16) into equation (14), the asymptotic solution of the n^th supplemental modal damping ratio of the cable can be derived as

ζ_{n} = \frac{\bar{c}}{{(1 - \bar{b} + \bar{κ} \frac{x_{2}}{x_{1}} - \bar{κ} \bar{b} (\frac{x_{2}}{x_{1}} - 1))}^{2} + {\bar{c}}^{2} {[1 + \bar{κ} (\frac{x_{2}}{x_{1}} - 1)]}^{2}} \frac{x_{1}}{l}

(17)

Comparison of asymptotic and iterative solutions

The PVMD and the grounded cross-tie are installed at the positions of x₁=1%l and x₂=2%l from the left cable end, respectively. Figure 2 compares the asymptotic and iterative complex wave numbers of a cable with a PVMD and a cross-tie when the dimensionless stiffness coefficient of the cross-tie κ=0.01/(nπx₁/l). When the dimensionless inertial mass remains constant and the dimensionless damping coefficient of PVMD increases from zero to infinity, the complex wave number traces a semicircle with a positive imaginary part. The semicircle trace starts from an undamped frequency and eventually terminates at a clamped frequency on the real axis. The top point of the semicircle trace suggests the maximum supplemental modal damping ratio of the cable. It can be seen from Figure 2 that the maximum supplemental modal damping ratio of the cable gradually increases with the increase of the inertial mass of the PVMD. Moreover, the asymptotic complex wave number is in good agreement with the iterative complex wave number when the dimensionless inertial mass χ=0/(nπx₁/l), χ=0.3/(nπx₁/l), and χ=0.6/(nπx₁/l). However, the asymptotic and iterative complex wave numbers significantly deviate from each other for the large inertial mass χ=0.9/(nπx₁/l), and the basic assumption of the asymptotic solution is no longer valid under this circumstance. Hence, the iterative solutions are used to accurately predict the maximum supplemental modal damping ratio of the cable as well as the corresponding parameters of the PVMD and the cross-tie in the following discussions.

Figure 2.

Comparisons of the asymptotic and iterative complex wave numbers of a cable with a PVMD and a grounded cross-tie (x₁=1%l, x₂=2%l, κ=0.01/(nπx₁/l)).

Parametric studies

Figure 3 shows the variations of the first supplemental modal damping ratio of a cable with a VD (PVMD) and a grounded cross-tie with the damping coefficient of the VD (PVMD) under the various stiffness coefficients of the cross-tie. It can be observed from Figure 3 that the maximum supplemental damping ratio of a cable with a PVMD and a grounded cross-tie is significantly increased compared with that of a cable with a VD and a grounded cross-tie, which indicates the coupled damping effect of a PVMD and a grounded cross-tie on a stay cable is superior to that of a VD and a grounded cross-tie on a stay cable. However, increasing the dimensionless stiffness coefficient of the cross-tie has negative effects on improving the maximum supplemental damping ratio of the cable.

Figure 3.

Variations of the first modal damping ratio of the cable with the damping coefficient under various stiffness coefficients of the cross-tie (x₁=1%l, x₂=2%l).

Figure 4 shows the variations of the first supplemental modal damping ratio of a cable with a VD (PVMD) and a grounded cross-tie with the damping coefficient of the VD (PVMD) under various installation positions of the cross-tie. It can be seen from Figure 4 that the maximum supplemental modal damping ratio of the cable decreases with the cross-tie moving away from the VD or the PVMD. Thus, when the PVMD and the grounded cross-tie are installed at the same cable end, the increase of installation length of the cross-tie is unfavorable to the damping improvement of the cable.

Figure 4.

Variations of the first modal damping ratio of the cable with the damping coefficient under various installation positions of the cross-tie (x₁=1%l, κ=0.05/(nπx₁/l)).

The PVMD and the grounded cross-tie at opposite cable ends

Asymptotic solution and iterative solution

Similar to the case of a PVMD and a ground cross-tie at the same cable end, when the PVMD and the grounded cross-tie are installed at opposite cable ends, the iterative solution of the complex wave number β_n is also obtained by the fixed-point iteration method,⁴⁹ in which the iterative equation is given by

\begin{array}{l} β_{n}^{j + 1} l = n π + \arctan \frac{E (β_{n}^{j}) + i F (β_{n}^{j})}{G (β_{n}^{j}) + i H (β_{n}^{j})} \\ β_{n}^{0} = \frac{n π}{l}, n = 1, 2, 3, \dots \end{array}

(18)

After solving the wave number, the supplemental modal damping ratio of a cable with a PVMD and a cross-tie at opposite cable ends can also be calculated by equation (14). It is assumed that the PVMD is installed near the left cable end, while the cross-tie is installed near the right cable end, that is,

x_{1} = l

x_{2}^{*} = l

. The approximate solution for the complex wave number β_n can be expressed as

\frac{Δ β l}{β^{0} l} = \frac{\bar{κ} {(\frac{x_{2}^{*}}{x_{1}})}^{2} - \bar{b} - \bar{κ} \bar{b} \frac{x_{2}^{*}}{x_{1}} (1 + \frac{x_{2}^{*}}{x_{1}}) + i \bar{c} [1 + \bar{κ} \frac{x_{2}^{*}}{x_{1}} (\frac{x_{2}^{*}}{x_{1}} + 1)]}{1 + \bar{κ} \frac{x_{2}^{*}}{x_{1}} - \bar{b} - \bar{κ} \bar{b} \frac{x_{2}^{*}}{x_{1}} + i \bar{c} (1 + \bar{κ} \frac{x_{2}^{*}}{x_{1}})} \frac{x_{1}}{l}

(19)

Substituting equation (19) into equation (14), the asymptotic solution of the n^th supplemental modal damping ratio of the cable can be expressed as

ξ_{n} = \frac{\bar{c} {(1 + \bar{κ} \frac{x_{2}}{x_{1}})}^{2}}{{(1 + \bar{κ} \frac{x_{2}}{x_{1}} - \bar{b} - \bar{k} \bar{b} \frac{x_{2}}{x_{1}})}^{2} + {(\bar{c} (1 + \bar{κ} \frac{x_{2}}{x_{1}})}^{2}} \frac{x_{1}}{l}

(20)

Comparison of asymptotic and iterative solutions

The PVMD and the grounded cross-tie at opposite cable ends are installed at the positions of x₁=1%l from the left cable end and $x_{2}^{*} = 1 % l$ from the right cable end. Figure 5 compares the asymptotic and iterative complex wave numbers of a cable with a PVMD and a grounded cross-tie when the dimensionless stiffness coefficient of the cross-tie κ=0.01/(nπx₁/l). It can be seen from Figure 5 that the maximum supplemental modal damping ratio of the cable gradually increases with the increase of the inertial mass of the PVMD. When the dimensionless inertial mass χ=0/(nπx₁/l), χ=0.3/(nπx₁/l), and χ=0.6/(nπx₁/l), the asymptotic complex wave number coincides well with the iterative complex wave number, while the asymptotic and iterative complex wave numbers significant deviate from each other for the large inertial mass χ=0.9/(nπx₁/l). Similarly, the basic assumption of the asymptotic solution is no longer valid when the PVMD has a large inertial mass, which further verified that the applicability of the iterative solution is superior to the asymptotic solution. In addition, compared with installing the PVMD and the cross-tie at the same cable end, the asymptotic and iterative complex wave numbers of the cable with a PVMD and a cross-tie at opposite cable ends are more consistent.

Figure 5.

Comparisons of the asymptotic and iterative complex wave numbers of a cable with a PVMD and a grounded cross-tie ( $x = x_{1} = x_{2}^{*} = 1 % l$ , κ=0.01/(nπx₁/l)).

Parametric studies

Figure 6 shows the variations of the first supplemental modal damping ratio of the cable with a VD (PVMD) and a grounded cross-tie with the damping coefficient of the VD (PVMD) under various stiffness coefficients of the cross-tie. As illustrated in Figure 6, the maximum supplemental damping ratio of a cable with a PVMD and a grounded cross-tie is significantly increased compared with that of a cable with a VD and a grounded cross-tie, which indicates the coupled damping effect of a PVMD and a grounded cross-tie on a stay cable is superior to that of a VD and a grounded cross-tie on a stay cable. Moreover, increasing the dimensionless stiffness coefficient of the cross-tie is conducive to improving the maximum supplemental modal damping ratio of the cable.

Figure 6.

Variations of the first supplemental modal damping ratio of the cable with the damping coefficient under different stiffness coefficients of the cross-tie $(x = x_{1} = x_{2}^{*} = 1 % l)$ .

Figure 7 shows the variations of the first supplemental modal damping ratio of the cable with a VD (PVMD) and a grounded cross-tie with the damping coefficient of the VD (PVMD) under various installation positions of the cross-tie. It can be seen from Figure 7 that the maximum supplemental modal damping ratio of the cable slightly increases with the cross-tie moving away from the right cable end. Consequently, when the PVMD and the grounded cross-tie are installed at opposite cable ends, the increase of installation length of the cross-tie helps to achieve the damping improvement of the cable.

Figure 7.

Variations of the first modal damping ratio of the cable with the damping coefficient under various installation positions of the cross-tie (x₁=1%l, κ=0.05/(nπx₁/l)).

Optimization for the installation positions of the cross-tie

Optimal installation positions of the cross-tie

To analyze the effects of the installation positions of the cross-tie on the damping of a cable with a PVMD and a grounded cross-tie, the optimization for the installation positions of the cross-tie is carried out in this section. Figure 8 shows the variations of the first supplemental modal damping ratio of the cable with the installation positions of the cross-tie and the damping coefficients of the PVMD. It can be observed from Figure 8 that the first modal damping ratio of the cable reaches its maximum value when the installation position of the cross-tie is x₂/l=0.51 and the damping coefficient of the PVMD is η=0.26. The variations of the first supplemental modal damping ratio of the cable with the installation positions and the stiffness coefficients of the cross-tie are further illustrated in Figure 9. It can be seen from Figure 9 that the first supplemental modal damping ratio of the cable reaches the maximum when the cross-tie is installed at the cable mid-span, in which the first supplemental modal damping ratio of the cable increases significantly with the increase of the stiffness coefficient of the cross-tie. Thus, the cable mid-span is the optimal installation position for the cross-tie to improve the vibration control performance of the cable for the first mode.

Figure 8.

Variations of the first supplemental modal damping ratio of the cable with the installation positions of the cross-tie and the damping coefficients of the PVMD (χ=0.9/(nπx₁/l), κ=0.01/(nπx₁/l)).

Figure 9.

Variations of the first supplemental modal damping ratio of the cable with the installation positions and the stiffness coefficients of the cross-tie (χ=0.9/(nπx₁/l), η=0.26/(nπx₁/l)).

The cross-tie at the cable mid-span

To further analyze the effects of the stiffness coefficients of the cross-tie on the damping and frequency of a cable with a PVMD and a grounded cross-tie, the PVMD and the cross-tie are installed at the positions of x₁=2%l and x₂=50%l from the left cable end. Figure 10 shows the first modal complex wave numbers of a cable with a PVMD and a grounded cross-tie under various stiffness coefficients of the cross-tie. It is noteworthy that the frequency and the damping ratio of the cable increase significantly with the increase of the stiffness coefficient of the cross-tie.

Figure 10.

The first modal complex wave numbers of a cable with a PVMD and a grounded cross-tie under various stiffness coefficients of the cross-tie (x₁=2%l, x₂=50%l, χ=0.9/(nπx₁/l)).

Figure 11 shows the influence of the damping coefficients of PVMD and the stiffness coefficients of the cross-tie on the first supplemental modal damping ratio of the cable. It can be seen from Figure 11 that when the stiffness coefficient of the cross-tie is determined, the first supplemental modal damping ratio of the cable increases with the increase of the damping coefficient of the PVMD, reaching the maximum value at the optimum damping coefficient of the PVMD, and then decreases with the further increase of the optimum damping coefficient of PVMD. Furthermore, the optimum damping coefficient of the PVMD decreases with the increase of the stiffness coefficient of the cross-tie, which helps to reduce the damping cost to achieve optimal vibration control performance of the cable.

Figure 11.

Variations of the first supplemental modal damping ratio of the cable with the damping coefficients of PVMD and stiffness coefficients of the cross-tie (x₁=2%l, x₂=50%l, χ=0.9/(nπx₁/l)).

Figure 12 compares the maximum first supplemental modal damping ratio of a cable with a PVMD and a cross-tie, a single PVMD, as well as a VD and a cross-tie. It is found in Figure 12 that when the cross-tie is installed at the cable mid-span, the maximum first supplemental modal damping ratio of a cable with PVMD and cross-tie is greater than that of a cable with VD and cross-tie or single PVMD. In addition, the maximum first supplemental modal damping ratio of the cable increases with the increase of the stiffness coefficient of the cross-tie.

Figure 12.

The maximum first supplemental modal damping ratio of the cable (x₁=1%l, x₂=50%l).

Conclusions

In this paper, the coupled damping effect of a cable with a parallel-connected viscous mass damper (PVMD) and a grounded cross-tie is investigated. The coupled damping effect of a PVMD and a grounded cross-tie on a stay cable is compared with that of a viscous damper (VD) and a grounded cross-tie on a stay cable. Moreover, the effects of the stiffness coefficients and installation positions of the cross-tie on the damping of the cable are emphatically analyzed. The main conclusions are summarized as follows:

(1) The supplemental modal damping ratio of a cable with a PVMD and a grounded cross-tie is significantly improved compared with that of a cable with a VD and a grounded cross-tie, which indicates the coupled damping effect of a PVMD and a grounded cross-tie on a stay cable is superior to that of a VD and a grounded cross-tie on a stay cable.

(2) When a PVMD and a ground cross-tie are installed at the same cable end, increasing the dimensionless stiffness coefficient of the cross-tie has negative effects on improving the maximum supplemental damping ratio of the cable, and the maximum supplemental modal damping ratio of the cable decreases with the cross-tie moving away from the PVMD.

(3) When a PVMD and a ground cross-tie are installed at opposite cable ends, increasing the dimensionless stiffness coefficient of the cross-tie is beneficial to enhance the maximum supplemental damping ratio of the cable, and the maximum supplemental modal damping ratio of the cable increases with the cross-tie moving close to the cable mid-span.

(4) The cable mid-span is determined as the optimal installation position of the cross-tie to improve the vibration control performance of the cable for the first mode. When the cross-tie is installed at the cable mid-span, the first supplemental modal damping ratio of the cable reaches the maximum.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The National Natural Science Foundation of China (Grant No. 51878274).

ORCID iD

Zhi-hao Wang

References

Qie

Hou

. The fastest insight into the large amplitude vibration of a string. Rep Mech Eng 2021; 2: 1–5.

Yang

, et al. A simple frequency formulation for the tangent oscillator. Axioms 2021; 10: 320.

Chen

Sun

, et al. A comparative study of multi-mode cable vibration control using viscous and viscoelastic dampers through field tests on the Sutong Bridge. Eng Struct 2020; 224: 111226.

Zhou

. Wind-rain-induced vibration and control of stay cables in a cable-stayed bridge. Struct Control Health Monit 2007; 14: 1013–1033.

Wang

Tao

Gao

, et al. Measurement of wind effect on a kilometer-level cable-stayed bridge during typhoon Haikui. J Struct Eng 2018; 144: 04018142.

Liu

Zheng

Zhou

, et al. Stability and dynamics analysis of in-plane parametric vibration of stay cables in a cable-stayed bridge with superlong spans subjected to axial excitation. J Aerospace Eng 2020; 33: 04019106.

Kleissl

Georgakis

. Comparison of the aerodynamics of bridge cables with helical fillets and a pattern-indented surface. J Wind Eng Ind Aerod 2012; 104-106: 166–175.

Amer

Elnaggar

, et al. Periodic property and instability of a rotating pendulum system. Axioms 2021; 10: 1–15.

Cai

Wang

, et al. Experimental verification of the effectiveness of elastic cross-ties in suppressing wake wake-induced vibrations of staggered stay cables. Eng Struct 2018; 167: 151–165.

10.

Pacheco

Fujino

Sulekh

. Estimation curve for modal damping in stay cables with viscous damper. J Struct Eng 1993; 119: 1961–1979.

11.

Mekki

Auricchio

. Performance evaluation of shape-memory-alloy superelastic behavior to control a stay cable in cable-stayed bridges. Int J Nonlin Mech 2011; 46: 470–477.

12.

Soltane

Montassar

Ben Mekki

, et al. A hysteretic Bingham model for MR dampers to control cable vibrations. J Mech Mater Struct 2015; 10: 195–206.

13.

Chen

, et al. On the rain-wind induced in-plane and out-of-plane vibrations of stay cables. J Eng Mech 2013; 139: 1688–1698.

14.

Weber

Distl

. Amplitude and frequency independent cable damping of Sutong Bridge and Russky Bridge by magnetorheological dampers. Struct Control Health Monit 2015; 22: 237–254.

15.

Liu

Shen

, et al. Experimental study on high-mode vortex-induced vibration of stay cable and its aerodynamic countermeasures. J Fluid Struct 2021; 100: 103195.

16.

Johnson

Christenson

Spencer

. Semi-active damping of cables with sag. Comput-Aided Civ Inf 2003; 18: 132–146.

17.

Duan

. State-derivative feedback control of cable vibration using semiactive magnetorheological dampers. Comput-Aided Civ Inf 2005; 20: 431–449.

18.

Christenson

Spencer

Johnson

. Experimental verification of smart cable damping. J Eng Mech-ASCE 2006; 132: 268–278.

19.

Chen

Sun

Nagarajaiah

. Cable with discrete negative stiffness device and viscous damper: passive realization and general characteristics. Smart Struct Syst 2015; 15: 627–643.

20.

Zhou

. Modeling and control performance of a negative stiffness damper for suppressing stay cable vibrations. Struct Control Health Monit 2016; 23: 764–782.

21.

Shi

Zhu

. Magnetic negative stiffness dampers. Smart Mater Struct 2015; 24: 072002.

22.

Shi

Zhu

. et al. Dynamic behavior of stay cables with passive negative stiffness dampers. Smart Mater Struct 2016; 25: 75044.

23.

Shi

Zhu

Nagarajaiah

. Performance comparison between passive negative stiffness damper and active control in cable vibration mitigation. J Bridge Eng 2017; 22: 04017054.

24.

Shi

Zhu

Spencer

. Experimental study on passive negative stiffness damper for cable vibration mitigation. J Eng Mech 2017; 143: 04017070.

25.

Javanbakht

Cheng

Ghrib

. Refined damper design formula for a cable equipped with a positive or negative stiffness damper. Struct Control Health Monit 2018; 25: e2236.

26.

Lazar

Neild

Wagg

. Using an inerter-based device for structural vibration suppression. Earthq Eng Strut Dyn 2014; 43: 1129–1147.

27.

Sun

Zuo

Wang

, et al. Exact H₂ optimal solutions to inerter-based isolation systems for building structures. Struct Control Health Monit 2019; 26: e2357.

28.

Huang

Hua

Chen

, et al. Performance evaluation of inerter-based damping devices for structural vibration control of stay cables. Smart Struct Syst 2019; 23: 615–626.

29.

Hao

. Inerter-based structural vibration control: A state-of-the-art review. Eng Struct 2021; 243: 112655.

30.

Luo

Jiang

John

. Cable vibration suppression with inerter-based absorbers. J Eng Mech 2019; 145: 04018134.

31.

Duan

Spencer

, et al. Inertial mass damper for mitigating cable vibration. Struct Control Health Monit 2017; 24: e1986.

32.

Wang

Yue

Gao

, et al. Refined study on free vibration of a cable with an inertial mass damper. Appl Sci 2019; 9: 2271.

33.

Gao

Wang

, et al. Optimum design of viscous inerter damper targeting multi-mode vibration mitigation of stay cables. Eng Struct 2021; 226: 111375.

34.

Wang

Gao

, et al. Vibration control of a stay cable with a rotary electromagnetic inertial mass damper. Smart Struct Syst 2019; 23: 627–639.

35.

Shen

Zhu

. Vibration mitigation of stay cables using electromagnetic inertial mass dampers: full-scale experiment and analysis. Eng Struct 2019; 200: 109693.

36.

Sun

Chen

. In-plane dynamic behaviors of two-cable networks with a pretensioned cross-tie. Struct Control Health Monit 2021; 28: e2755.

37.

Zhou

Yang

, et al. Free vibration of taut cable with a spring. J Vib Eng Technol 2012; 25: 522–526.

38.

Ahmad

Cheng

Ghrib

. Effect of the number of cross-tie lines on the in-plane stiffness and modal behavior classification of orthogonal cable networks with multiple lines of transverse flexible cross-ties. J Eng Mech 2016; 142: 04015106.

39.

Ahmad

Cheng

. Impact of key system parameters on the in-plane dynamic response of a cable network. J Struct Eng 2013; 140: 04013079.

40.

Ahmad

Cheng

Ghrib

. Efficiency of an external damper in two-cable hybrid systems. J Bridge Eng 2018. 23: 04017138.

41.

Zhou

Liu

Xiao

, et al. Feasibility of using a negative stiffness damper to two interconnected stay cables for damping enhancement. Int J Struct Stab Dy 2019; 19: 1950058.

42.

Zhou

Yang

Sun

, et al. Free vibrations of a two-cable network with near-support dampers and a cross-link. Struct Control Health Monit 2015; 22: 1173–1192.

43.

Zhou

Sun

Xing

. Free vibration of taut cable with a damper and a spring. Struct Control Health Monit 2014; 21: 996–1014.

44.

Chen

Sun

. Free vibrations of hybrid cable networks with external dampers and pretensioned cross-ties. J Mech Syst Signal Pr 2021; 156: 107627.

45.

Hoang

Fujino

. Combined damping effect of two dampers on a stay cable. J Bridge Eng 2008; 13: 299–303.

46.

Wang

Yue

Gao

. Free vibration of a taut cable with two discrete inertial mass dampers. J Appl Sci 2019; 9: 3919.

47.

Sun

Chen

. Cable vibration control with internal and external dampers: theoretical analysis and field test validation. Smart Struct Syst 2020; 26: 575–589.

48.

Yang

Chen

Wang

, et al. Optimal design of two viscous dampers for multi-mode control of a cable covering broad frequency range. Eng Struct 2021; 245: 112830.

49.

Krenk

. Vibration of a taut cable with an external damper. J Appl Mech 2000; 67: 772–776.