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Abstract

This paper proposes a vibration model for modal analysis and vibration control of a vertical cable-dual tuned mass damper (C-DTMD) coupled system. The proposed model introduces several non-dimensional parameters to describe the in-plane vibration of the coupled system. Moreover, the phenomenon of modal split is comprehensively studied to present the complex vibration modes. Parameter analysis is conducted to show the influence of length ratio, mass ratio, damping ratio, and frequency ratio on vibration reduction. In addition, the theoretical model is verified using finite element methods. A practical vertical cable in Jiangyin Bridge with designed wind loads is chosen as a numerical example. The results show that DTMD devices can effectively reduce the modal vibration and have great potential for multi-mode vibration control of vertical cables in suspension bridges. The proposed model can accurately describe the in-plane vibration modes of C-DTMD systems and provides a basis for the optimal design of damper parameters.

Keywords

vibration control dual tuned mass dampers cable modal damping modal split

Introduction

Vertical cables have been widely used in suspension bridges as the main load bearing structures.^1–3 The operating conditions of vertical cables are very important for the overall safety of bridges. With the development of advanced production technology and design theory, the length of vertical cables in suspension bridges has been remarkably enhanced in recent decades. However, vertical cables are usually made of materials that have high-strength steel wires but relatively light mass. The cables may become more sensitive to wind or rain-induced vibration due to their higher structural flexibility. Cables may encounter undesired vibrations, including vortex-induced vibration, buffeting, and galloping.^4–7 The vibration-induced impact force and fatigue loads may cause damage to structures, jeopardizing the integral safety of suspension bridges.^8–10 Therefore, it is necessary to reduce the vibration of cable structures in bridge engineering to ensure its safety.

Currently, various vibration control methods, including active control, passive control, semi-active control, and hybrid control, have been developed to reduce the vibration of cable structures. The vibration control strategy can be divided into the following categories: shape modification,^11,12 cross-ties methods,^13–15 viscous dampers,^16,17 tuned liquid dampers,¹⁸ inertial mass dampers,¹⁹ magnetorheological dampers,^20,21 and tuned mass dampers (TMD).^22–25 Compared with the other methods, the TMD method has been demonstrated to be an effective method for reducing vibrations in bridges due to its low costs, flexible installation position, and reliable damping effect. The vibration energy can be transferred from host structures to additional TMD devices.^25,26 It can realize the passive or semi-active control of structures with many successful cases. For example, Di et al.²⁷ designed an additional device to suppress the vibrations of high-order modes using TMD method. Pacheco et al.²⁸ proposed a universal estimation curve that related the modal damping ratio of the cable with several parameters, including attached dampers, mode numbers, damper sizes, the damper locations, and cable parameters. The mass of the damper was ignored in this study. Wu and Cai²⁹ designed a magnetorheological TMD device. The parameters were analyzed to show the effect of the positions, mass and damping on the dynamic characteristics of the system in terms of modal damping. Liu et al.²³ designed a pounding TMD to reduce the modal vibration of an experimental stay cable with a prominent damping effect. In short, the limitations in using TMD to control cable vibration are mainly two points. Firstly, vibration control by TMD is only effective within a specific frequency region. The regulation capability is limited to single modal vibration, which may not be suitable for multi-modal vibration control. Secondly, the optimal deturning and damping effect of C-TMD coupled structures is difficult to realize in real engineering based on the single-degree-of-freedom model. It is of great importance to know the effect of each factor to design the optimal parameter of the TMD device.

To address this problem, dual tuned mass dampers (DTMD) are used in many civil engineering due to their superb robustness in frequency tuning.^30–32 In this paper, a vibration model of a vertical cable-dual tuned mass dampers (C-DTMD) system is established. Several non-dimensional parameters are introduced to describe the vibration characteristics of coupled systems. In addition, the parameters were analyzed based on the influence of mass ratio, damping ratio, and frequency ratio. Moreover, a numerical model using the finite element method is presented to verify the proposed model. The results first show the vibration characteristic of C-DTMD devices, which is useful for multi-modal vibration control of cable structures.

The rest of this paper is organized as follows. The Vibration model section formulas a vibration model of C-DTMD systems. The Modal damping analysis of a C-DTMD system section discusses the phenomenon of modal split and related influencing parameters, including length ratio, mass ratio, frequency ratio, and damping ratio. The Numerical verification section establishes a numerical example under wind loads based on the finite element model to verify the proposed model. Finally, the conclusions of this study are drawn in the Conclusions section.

Vibration model

Dynamic modal of a vertical cable

The tension of a vertical cable follows a uniform linear distribution under the influence of gravity, as shown in Figure 1. The vibration model of cables is expressed in the equation below³³

m \frac{\partial^{2} v}{\partial t^{2}} = \frac{\partial}{\partial x} (T \frac{\partial v}{\partial x}),

(1)

T = (L - x) m g + T_{B},

(2)

Figure 1.

In-plane vibration of a vertical cable.

where $m$ , $L$ , $T$ , $T_{B}$ are the mass per unit length, cable length, cable tension, and tension of lower anchor end, respectively. According to the method of separation of variables, the transverse displacement can be supposed as $v (x, t) = A (x) e^{j ω t}$ , equation (1) can be rewritten as

δ^{2} \frac{d^{2} A}{d δ^{2}} + δ \frac{d A}{d δ} + δ^{2} A = 0,

(3)

A (δ) = C_{1} J_{0} (δ) + C_{2} N_{0} (δ),

(4)

where

δ = 2 ω / g \sqrt{((L - x) m g + T_{B}) / m}

is a non-dimensional parameter.

J_{0} (δ)

and

N_{0} (δ)

are the first and second kinds of Bessel functions, respectively.

C_{1}

and

C_{2}

are constants that need to be determined by the boundary conditions. If

α = 2 ω / g \sqrt{T_{B} / m}

β = m g L / T_{B}

and

h = \sqrt{β + 1}

, then, the boundary conditions of cables are

{δ |}_{x = 0} = h α,

(5)

{δ |}_{x = L} = α .

(6)

The non-dimensional parameter $α$ is the function of angular frequency $ω$ . The non-dimensional parameter $β$ is defined as the ratio of gravity to tension, which is the function of material parameters and cable tension. Substituting equations (5) and (6) to equation (4), the solution can be solved as

J_{0} (α) N_{0} (h α) - J_{0} (h α) N_{0} (α) = 0 .

(7)

The roots of equation (7) are the natural frequencies of the vertical cable systems. It can be numerically calculated by Maple or Mathematica. Suppose that the n-th root of equation (7) is defined as $α_{n}$ , then the n-th order natural frequency can be calculated by

ω_{n} = \frac{β α_{n}}{2 L} \sqrt{\frac{T_{B}}{m}} .

(8)

Combining equation (8) and the definitions of non-dimensional parameter $β$ , the first-order frequency can be rewritten as

ω_{0 n} = \frac{2 n π}{2 L} \sqrt{\frac{T_{B}}{m}} = n π \sqrt{\frac{g}{β L}},

(9)

Hence, the relative error is calculated as follows

E_{r} = \frac{ω_{n} - ω_{0 n}}{ω_{n}} = 1 - \frac{2 n π}{β α_{n}},

(10)

It is shown in equation (10) that the error is controlled by the non-dimensional parameter $β$ . If the tension is far greater than the gravity, the frequency change can be ignored, and vice versa. For a cable in bridge engineering, the frequency variation caused by non-uniform tension is generally within 2.00% and can be ignored in applications. On the other hand, the influence of non-uniform tension should be considered under low tension conditions.

C-DTMD systems

The vibration model of C-DTMD systems is shown in Figure 2. Without loss of generality, it is assumed that $l_{1} < l_{2}$ , then the vibration equations and continuity conditions are

f_{y} + T \frac{\partial^{2} v_{k} (x_{k}, t)}{\partial x_{k}^{2}} = m \frac{\partial^{2} v_{k} (x_{k}, t)}{\partial t^{2}}, k = 1,2,

(11)

T (- {\frac{\partial v_{1}}{\partial x_{1}} |}_{x_{1} = l_{1}} - {\frac{\partial v_{2}}{\partial x_{2}} |}_{x_{2} = l_{2}}) = \sum_{j = 1}^{2} [K_{j} ({v_{1} |}_{x_{1} = l_{1}} - v_{d_{j}}) + C_{j} ({\frac{\partial v_{1}}{\partial t} |}_{x_{1} = l_{1}} - \frac{\partial v_{d_{j}}}{\partial t})],

(12)

K_{j} ({v_{1} |}_{x_{1} = l_{1}} - v_{d_{j}}) + C_{j} ({\frac{\partial v_{1}}{\partial t} |}_{x_{1} = l_{1}} - \frac{\partial v_{d_{j}}}{\partial t}) = M_{j} \frac{\partial^{2} v_{d_{j}}}{\partial t^{2}}, j = 1,2,

(13)

Figure 2.

In-plane vibration of a vertical cable with dual tuned mass dampers.

where $f_{y}$ is the outside force along with the axis. $k$ represents the two sections of the cable divided by the DTMD devices. $M_{j}$ , $K_{j}$ and $C_{j}$ are the mass, stiffness, and damping of j-th TMD. The above equations can be solved by the method of separation of variables. It is assumed that $f_{y} = f_{y}^{*} (x) e^{s t}$ , $v = v^{*} (x) e^{s t}$ and $v_{d} = v_{d}^{*} {(x) e}^{s t}$ , then, the solution of equation (11) is

v_{k}^{*} = v_{c}^{*} \frac{\sinh (γ x_{k})}{\sinh (γ l_{k})}, k = 1,2,

(14)

where

γ = \sqrt{m s^{2} / T}

is a non-dimensional frequency parameter. The complex variable

s

is a non-dimensional eigenvalue. Then, the fundamental natural frequency

ω_{0}

can be expressed as the function of

s

and

γ

ω_{0} = π / L \sqrt{T / m} = π s / L γ,

(15)

Similarly, substituting equation (14) for equations (12) and (13) yields

v_{d_{j}}^{*} = \frac{K_{j} + C_{j} s}{K_{j} + C_{j} s + M_{j} s^{2}} v_{c}^{*}, j = 1,2,

(16)

K_{j} = M_{j} ω_{d_{j}}^{2}, C_{j} = 2 M_{j} ω_{d_{j}} ξ_{j}, j = 1,2,

(17)

where

ω_{d_{j}}

and

ξ_{j}

are the natural frequency and damping ratio. Besides, by introducing two non-dimensional parameters

ρ^{'} = s / ω_{0}

and

ρ_{j} = ω_{d_{j}} / ω_{0}

, the transverse motions of cable and DTMD are rewritten as

v_{k}^{*} = v_{c}^{*} \frac{\sinh (π ρ^{'} \frac{x_{k}}{L})}{\sinh (π ρ^{'} \frac{l_{k}}{L})}, k = 1,2,

(18)

v_{d_{j}}^{*} = A_{j} v_{c}^{*}, j = 1,2,

(19)

where the amplitude ratio

A_{j}

A_{j} = \frac{ρ_{j}^{2} + 2 ξ_{j} ρ_{j} ρ^{'}}{ρ_{j}^{2} + 2 ξ_{j} ρ_{j} ρ^{'} + {ρ^{'}}^{2}}, j = 1,2,

(20)

Substituting equations (18) and (19) for equation (12) yields

\coth (\frac{π l_{1}}{L} ρ^{'}) + \coth (\frac{π l_{2}}{L} ρ^{'}) + (\frac{π M_{1}}{m L} A_{1} + \frac{π M_{2}}{m L} A_{2}) ρ^{'} = 0,

(21)

The non-zero solution of equation (21) is the natural frequency of C-DTMD systems, which can be taken as an eigenvalue problem. The i-th eigenvalue $ρ_{i}^{'}$ is a complex that can be written as

ρ_{i}^{'} = σ_{i} + j φ_{i},

(22)

where

σ_{i} = ω_{i} / ω_{0} (- ζ_{i})

φ_{i} = ω_{i} / ω_{0} (\sqrt{1 - ζ_{i}^{2}})

are the real part and imaginary part of the complex

ρ_{i}^{'}

j

is the imaginary unit.

ω_{i}

and

ζ_{i}

are the modal frequency and damping of i-th modal, which can be calculated by

ω_{i} = \sqrt{σ^{2} + φ^{2}} ω_{0}, ζ_{i} = {(φ_{i}^{2} / σ_{i}^{2} + 1)}^{- \frac{1}{2}} .

(23)

It can be seen from equation (23) that the system becomes over-damped, that is, modal damping $ζ_{i} \geq 1$ , when $φ_{i}$ is a pure imaginary number or equal to 0. The displacement of each point on cables is a pure hyperbolic function of time. The vibration decreases gradually without oscillation between equilibrium positions. The system is undamped, that is, modal damping $ζ_{i} = 0$ , when $σ_{i}$ is equal to 0. The displacement of the cable becomes a pure trigonometric function without any damping effect. Generally, the eigenvalue $ρ_{i}^{'}$ remains a complex value. Thus, the vibration amplitude decreases gradually with time. If the modal frequency of DTMD $ω_{d_{1}}$ is equal to $ω_{d_{2}}$ , then equation (21) is simplified to the model of TMD systems

\coth (\frac{π l_{1}}{L} ρ^{'}) + \coth (\frac{π l_{2}}{L} ρ^{'}) + \frac{π (M_{1} + M_{2})}{m L} A ρ^{'} = 0 .

(24)

Modal damping analysis of a C-DTMD system

Modal Split

The vibration modes can be solved by equation (18) after the eigenvalue $ρ_{i}^{'}$ is determined. For the C-TMD system, Wu et al.²⁹ proposed that the i-th modal shape may be divided into two similar modes when the natural frequency of the TMD devices approaches the cable, as shown in Figure 3. Similarly, the modal split can also be found in C-DTMD system with more complex cases. The two natural frequencies of the DTMD are generally adjusted to be slightly lesser than and slightly greater than the i-th natural frequency of the cable. The original vibration mode is divided into three homologous modes, as shown in Figure 4(a). The split modals also have an obvious turning point at the installation position. As shown in Figure 4(b), the modal displacements $v_{d_{1}}^{*}$ and $v_{d_{2}}^{*}$ have three vibration modes: (i) $v_{d_{1}}^{*}$ and $v_{d_{2}}^{*}$ all greater than the modal displacement of cable $v_{c}^{*}$ ; (ii) $v_{d_{1}}^{*}$ and $v_{d_{2}}^{*}$ all smaller than $v_{c}^{*}$ ; (iii) one modal displacement of a damper is greater while the other is smaller than $v_{c}^{*}$ .

Figure 3.

Modal split of C-TMD systems. (a) Overall view and (b) regional enlarged view.

Figure 4.

Modal split of C-DTMD systems. (a) Overall view and (b) regional enlarged view.

Parameters analysis

The influence of several parameters is further studied in this section. The basic ranges of the relevant parameters are shown in Table 1. For convenience, the symbols

l_{r}

m_{r j}

ρ_{j}

are defined as the ratio of length, mass, and natural frequency of DTMD to cables, that is,

l_{r} = l_{1} / L

m_{r j} = M_{j} / m l

ρ_{j} = f_{d} / f_{c}

j = 1, 2

, respectively. Afterward, the eigenvalue

ρ_{i}^{'}

can be solved by equation (21).

Table 1.

Basic parameters and value range of DTMD devices.

Option	Basic parameters	Value range
Length ratio $l_{r}$	0.25	0.05∼0.5
Mass ratio-1 $m_{r 1}$	0.005	0.001∼0.05
Mass ratio-2 $m_{r 2}$	0.005	0.001∼0.05
Frequency ratio-1 $ρ_{1}$	1	0.1∼5
Frequency ratio-2 $ρ_{2}$	1	0.1∼5
Damping ratio-1 $ξ_{1}$	0.1	0.01∼1
Damping ratio-2 $ξ_{2}$	0.1	0.01∼1

Length ratio

The variation of the modal damping is shown in Figure 5 with different length ratios. The C-DTMD system is tuned to the first-order modal frequency. Thus, the modal damping of the higher-order is much lesser than that of the first-order. It is obvious from Figure 5(a) that the modal damping is very small when the installation location is close to the anchor region. The modal damping increases significantly to 0.039 when the location moves to about 1/4 span. Afterward, the modal damping remains relatively stable between 1/4 span and mid-span. In this region, the modal damping is slightly larger than 0.040. The length ratios of viscous dampers are generally 0.02 to 0.05 near the anchor region. The modal damping of viscous dampers is about 0.025.³⁴

Figure 5.

Influence of length ratio on 1^st order modal damping. (a) Overall view and (b) regional enlarged view.

Figure 5(b) shows the regional enlarged view to reflect the high-order modal damping. It can be found from the figure that the peak value of the n-th modal damping is located at $l_{r} = j / 2 n (n \geq 2, j = 1, 3, 5, \dots)$ , corresponding to the peak of each modal shape. Meanwhile, the modal damping decreases to 0 at the length ratio $l_{r} = j / n$ due to it being the node point.

The C-DTMD system is then tuned to the second-order modal frequency. The second-order modal damping is much greater than the other modes, as shown in Figure 6. Meanwhile, the second-order modal damping near the 1/4 span is much greater than other situations.

Figure 6.

Influence of length ratio on 2^nd order modal damping. (a) Overall view and (b) regional enlarged view.

Mass ratio

Generally, the mass ratio is within 5% because DTMD devices cannot be designed to be too heavy. For the overall balance, it is assumed that the two masses are equal, that is, $M_{1} = M_{2}$ and vary from 0.1% to 5%. In addition, the stiffness and damping also change accordingly to keep the frequency ratio and damping ratio unchanged.

The change of the first five-order modal damping is shown in Figure 7. The four-order modal damping is identically equal to zero because the 1/4 span is the node. It can be seen from the figure that the first-order modal damping of the system increases greatly from zero to the peak value of 0.044 as the mass ratio increases from 0.10% to 0.65%. Then, the first-order modal damping starts to decrease slowly with the increase in the mass ratio. This is because the frequency ratio is assumed to be a constant instead of decreasing accordingly. Therefore, it does not meet the optimal design principle of dampers. The modal damping may decrease under these conditions.

Figure 7.

Effect of mass ratio on modal damping of C-DTMD systems.

Damping ratio

The damping ratio can control the energy dissipation characteristics of the C-DTMD system. The two damping ratio of DTMD is also assumed to be the same to reduce the complexity. The damping ratios that vary from 0 to 1 are numerically studied in order to provide a comprehensive understanding of the C-DTMD system, although such a large damping ratio may be impractical. The results of the first five modes are shown in Figure 8. The maximum value of the first modal damping is about 0.040 when the damping ratio equals to 0.095. The second-order modal damping increases to 0.006, while the other high-order modes are much smaller than that of the first two modes.

Figure 8.

Effect of damping ratio on modal damping of C-DTMD systems.

On one hand, the damper cannot effectively dissipate vibration energy if the damping ratio of the damping device is too small. The overall damping effect of the C-DTMD system is not obvious under this condition. On the other hand, excessive damping ratio will weaken the movement of the spring, which is equivalent to the damper being rigidly connected to the sling.

Frequency ratio

The frequency ratio has an essential effect on the modal damping of systems. Compared to TMD, DTMD systems have two independent spring devices. This can make them generate two different frequency ratios, which is an outstanding advantage in vibration control. Therefore, the influence of frequency ratio on modal damping is discussed in this section.

Case 1: Same frequency ratios $ρ_{1} = ρ_{2}$

Firstly, the two frequency ratios of DTMD are assumed to be the same. The corresponding modal damping is shown in Figures 9 and 10, respectively. The four-order modal damping is equal to zero due to the 1/4 span being the node. Each order modal damping increases to the peak value when the frequency ratio is equal to the corresponding order.

Figure 9.

Effect of frequency ratio on modal damping of C-DTMD systems ( $ρ_{1} = ρ_{2}$ ).

Figure 10.

Regional enlarged view of the (a) first- and (b) second-order modal damping.

According to the optimal damping theory,³⁵ the natural vibration frequency of the damper generally needs to be adjusted to be slightly lower than the natural vibration frequency of the cable. Therefore, it is shown in Figure 10(a) that the first-order modal damping increases to 0.04,907 as the frequency ratio increases to 0.995. Similarly, the second-order modal damping increases to 0.04,884 when the frequency ratio is equal to 1.964, as shown in Figure 10(b).

Case 2: Different frequency ratios $ρ_{1} \neq ρ_{2}$

The maximum modal damping is not the same as the natural frequency in case $ρ_{1} = ρ_{2}$ . Therefore, the different frequency ratios are necessary to achieve the optimal tuning effect. Firstly, the variation of modal frequency with frequency ratio is shown in Figure 11. In particular, the frequency ratio $ρ_{2}$ is fixed as 1.00, and $ρ_{1}$ is varied from 0.96 to 1.05. The optimal modal damping is 0.04,959 when $ρ_{1}$ is equal to 0.99. Then, the $ρ_{1}$ is fixed as 1.00 and $ρ_{2}$ is varied from 0.96 to 1.05. The frequency ratio of the system increases to 1.00 in this case. The results show that the different frequency ratios give the system the ability to achieve the optimal tuning effect.

Figure 11.

Regional enlarged view of the first-order modal damping. (a) $ρ_{2} = 1.00$ , $ρ_{1} = 0.96 to 1.05$ and (b) $ρ_{1} = 0.99$ , $ρ_{2} = 0.96 to 1.05$ .

In addition, the frequency ratios $ρ_{1}$ and $ρ_{2}$ can control different order modal vibrations. Figure 12 shows the results when $ρ_{1} = 1$ and $ρ_{2} = 2$ . The DTMD can simultaneously control the first two-order modal vibration. In other words, DTMD systems can be used for the multi-modal control of cables.

Figure 12.

Multi-modal vibration control using different frequency ratios ( $ρ_{1} = 1.00$ , $ρ_{2} = 2.00$ ).

Numerical verification

A numerical modal is established in this section to verify the theoretical results of C-DTMD systems under wind loading.

Numerical model of a C-DTMD system

The finite element method is used to solve the vibration equations of C-DTMD systems. For cable vibration, the local mass and stiffness matrix of a cable can be expressed as

M_{e i} = \frac{m_{i}}{6} [\begin{array}{c} 2 & 1 \\ 1 & 2 \end{array}], K_{e i} = \frac{T}{l_{e i}} [\begin{array}{c} 1 & - 1 \\ - 1 & 1 \end{array}],

(25)

where

l_{e i}

and

m_{i}

are the length and mass of the i-th element. Each element length is the same in this study. Considering the boundary conditions of cables, the vibration equation of the system can be written as³⁶

M V^{″} (t) + C V^{'} (t) + K V (t) = F (t),

(26)

where

M

C

K

are the mass, damping, and stiffness matrix of C-DTMD system.

V (t)

F (t)

are the displacement and external force vectors, respectively. The detailed expression can be found in Appendix A.³⁷ The calculated time step is chosen as 0.001 s.

A cable of Jiangyin Bridge is chosen as the numerical example, as shown in Figure 13. The cable is the No. 1 cable of the bridge and is close to the north tower. The cable length is 136.561 m, which is the longest cable of the bridge. The mass per unit length of the cable is about 18.49 kg/m. One accelerometer is installed on the 1/4 span of the cable. The tension of the cable can be calculated by

T = 4 m L^{2} {f_{w}}^{2} / w^{2},

(27)

where

f_{w}

is the

w

-th natural frequency of the cable. The first-order frequency of the cable is 0.633 Hz. Therefore, the tension of the cable is about 553.18 kN.

Figure 13.

Numerical example of a vertical cable in Jiangyin Bridge.

Modal analysis

The modal analysis is firstly conducted to study the phenomenon of modal split. The number of the element is chosen as 80. The calculated modal frequency is compared with the theoretical and experimental values. The results of the first five-order frequencies are shown in Table 2. It can be seen from Table 2 that the error obtained by the numerical method is very small compared to the measurement results in the bridge and theoretical calculation.

Table 2.

Modal frequencies of the cable.

Modal order	Measurement results (Hz)	Theoretical results (Hz)	Numerical results (Hz)	Error with measurement results (%)	Error with theoretical results (%)
1	0.6333	0.6333	0.6333	6.32 × 10⁻⁵	6.32 × 10⁻⁵
2	1.2890	1.2666	1.2669	1.70 × 10⁻²	2.37 × 10⁻⁴
3	2.0430	1.8999	1.9010	6.95 × 10⁻²	5.79 × 10⁻⁴
4	2.7190	2.5332	2.5358	6.74 × 10⁻²	1.03 × 10⁻³
5	3.5050	3.1665	3.1716	9.51 × 10⁻²	1.61 × 10⁻³

The first four-order modal shapes of the cable are shown in Figure 14 as a benchmark. Moreover, the overall modal shapes are calculated by the finite element methods. DTMD is introduced with different frequency ratios, as discussed in Section 3.2.4. Similarly, three cases are separately verified using numerical examples. (i) Case 1: Frequency ratio $ρ_{1} = ρ_{2} = 0.995$ . In this condition, the first-order modal shape is divided into two homologous modes, as shown in Figure 15. The modal shape is not as smooth as the pure cable in Figure 14 at the position of DTMD. (ii) Case 2: Frequency ratio $ρ_{1} = 0.99$ , $ρ_{2} = 1.00$ . The first-order modal shape splits into three similar modes, as shown in Figure 16. (iii) Case 3: Frequency ratio $ρ_{1} = 1.00$ , $ρ_{2} = 2.00$ . The frequency ratios correspond to the first two frequencies of the cable, respectively. Therefore, the first two modal shapes are all divided into two modal shapes, as shown in Figure 17. The modal split results are the same as the theoretical prediction in Section 3.1.

Figure 14.

First four-order modal shapes of the cable.

Figure 15.

First four-order modal shapes of the C-DTMD systems ( $ρ_{1} = ρ_{2} = 0.995$ ).

Figure 16.

First four-order modal shapes of the C-DTMD systems ( $ρ_{1} = 0.99$ , $ρ_{2} = 1.00$ ).

Figure 17.

First four-order modal shapes of the C-DTMD systems ( $ρ_{1} = 1.00$ , $ρ_{2} = 2.00$ ).

Transient dynamic analysis

The transient dynamic analysis is conducted to verify the vibration control effect under wind loads.

Wind loads

Wind-induced vibration is the most common form of cable vibration. The Reynolds number $R_{e}$ of the fluid changes is shown as follows when air flows through cables

R_{e} = \frac{ρ_{a} v_{w} D}{μ_{a}},

(28)

where

ρ_{a}

μ_{a}

are the density and dynamic viscosity coefficient of air.

v_{w}

is the wind speed. The parameter

D

is the diameter of the cable. Suppose that the cable can be seen as a cylinder, the wind loads on the cable can be expressed as

F_{w} = \frac{1}{2} ρ_{a} v_{w}^{2} C_{d} D L,

(29)

C_{d} = \frac{24}{R_{e}} (1 + 0.1806 R_{e}^{0.6459}) + \frac{0.4251}{1 + \frac{6880.95}{R_{e}}},

(30)

where

C_{d}

is the drag coefficient which can be calculated by the Reynolds number

R_{e}

.³⁸ It is assumed that the density

ρ_{a}

and dynamic viscosity coefficient

μ_{a}

of air are 1.205 kg/m³ and 17.9 × 10⁻⁶ Pa·s, respectively. The diameter

D

of the cable is 0.1044 m. The 10 min average wind speed on the height of the bottom of the cable is assumed to be 12 m/s. The wind load can be calculated based on equation (29). To simulate the distribution in the real environment, the wind load is supposed to be a linear function along with the height. The minimum and maximum wind loads of each element are 7.29 N and 14.58 N, respectively. The accumulated wind load on the cable is about 0.8365 kN. The loading time of the wind loads is assumed to be 10 s (Figure 18).

Figure 18.

Designed wind loads of each element in numerical model.

Parameters analysis

The parameters analysis is conducted to validate the accuracy of the proposed theoretical model in this section. The numerical results are compared with the modal damping analysis in Section 3. Without loss of generality, the displacement responses of 3/8 span are chosen as the reference point.

Length ratio

The transient dynamic analysis is conducted to verify the vibration control effect. The frequency ratio is designed as 1 to reduce the first-order modal vibration. The displacement of the reference point is shown in Figure 19. It is shown in Figure 19(b) that the vibration amplitude is not weakened if the DTMD is not installed on the cable. The damping effect is enhanced in turn as the length ratio of the DTMD moves to 1/2.

Figure 19.

Displacement of the reference point under different installation positions of DTMD devices. (a) Overall view and (b) 10–40 s regional enlarged view.

The frequency spectrum of the displacement response is shown in Figure 20. The amplitude of the first-order frequency decreases as the length ratio moves to 1/2. The improved effect of the damper is not obvious when the length ratio is larger than 1/4, corresponding to the results shown in Figure 5.

Figure 20.

Frequency amplitude of the measured displacement. (a) Overall view and (b) regional enlarged view.

Mass ratio

The mass ratios are designed as 0.002, 0.01, and 0.10 to assess the damper effect of DTMD devices, as shown in Figure 21. Firstly, the control effect is poor when the mass ratio is equal to 0.002. The energy transfer is not obvious under a small mass ratio. Then, the damper effect improves as the mass ratio increases to 0.01. However, as the mass ratio increases to 0.10, the damping effect instead decreases, as shown in Figure 21(a). The results are consistent with the theoretical results shown in Figure 7.

Figure 21.

C-DTMD systems with different mass ratios. (a) Time domain and (b) frequency domain.

The above results also can be observed in the frequency domain. For the first case, the frequency amplitude of the first-order frequency hardly changes compared with the results without DTMD when the mass ratio is equal to 0.002. Besides, the frequency amplitude further decreases as the mass ratio increases to 0.01. In addition, the modal split is found when the mass ratio is equal to 0.10. The first-order modal shape is divided into two contiguous vibration modals, as shown in Figure 21(b).

Damping ratio

Three cases of damping ratio are studied to verify the results in Section 3.2.3. Based on the theoretical analysis, the excessive or less damping ratios all reduce the vibration reduction effect of the system. Therefore, the damping ratio is designed as 0.02, 0.10, and 0.20. It can be seen in Figure 22(a) that the damping ratio of 0.10 has a better damping effect than in other cases, which is in line with the theoretical results.

Figure 22.

C-DTMD systems with different damping ratios. (a) Time domain and (b) frequency domain.

On the other hand, the vibration energy of the first-order modal of the cable first decreases and then increases with the increase of the DTMD damping ratio, as shown in Figure 22(b). Meanwhile, the vibration energy of the second-order modal gradually decreases. The law is basically consistent with that shown in Figure 8.

Frequency ratio

The influence of the frequency ratio is extraordinarily complex. Therefore, only two special cases are selected to discuss their influence on vibration control in this section. Firstly, for Case 1, the frequency ratios of two dampers of DTMD are assumed to be the same or very close, that is, $ρ_{1} = ρ_{2} = 0.995$ and $ρ_{1} = 0.99$ , $ρ_{2} = 1.00$ . It is shown in Figure 23 that the damping effect is the same under these two conditions. The frequency amplitude of the first order decreases obviously, as shown in Figure 23(b). On the other hand, for Case 2, the two frequency ratios are assumed to be different and equal to the first and second-order modal frequencies of cables, respectively, that is, $ρ_{1} = 1.00$ , $ρ_{2} = 2.00$ . Compared with Case 1, the displacement response in the time domain and vibration energy in the frequency domain of DTMD (blue, red or black lines) is obviously smaller than that of TMD (green line), respectively. The frequency energy is weakened in both the first two-order modals, especially for the second-order modal vibration. The above conclusions are also in accordance with the theoretical model results.

Figure 23.

C-DTMD systems with different frequency ratios. (a) Time domain and (b) frequency domain.

Conclusions

In this paper, a mechanical model of a C-DTMD system is developed by introducing several non-dimensional parameters. The theoretical analysis of the model is conducted to show the modal split and influence factors. The following conclusions are drawn:

1. The frequency variation caused by non-uniform tension is within 2.00% and can be ignored in applications for vertical cables in suspension bridges. The influence of non-uniform tension must be considered for vertical cables with small tension.

2. The modal shapes of C-DTMD systems may split into three homologous modals. The modal split also has an obvious turning point at the installation position of DTMD devices.

3. Several key parameters have an impact on systematic modal damping, including length ratio, mass ratio, damping ratio, and frequency ratio. The recommended installation position is near the middle of the cable. The mass ratio and damping ratio have an optimal value for each order of modal damping. The frequency ratios can be adjusted to control the different modal vibrations of cables.

The multi-parameters optimization design and experimental validation of DTMD devices in practical engineering need to be comprehensively studied in the future.

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 authors are grateful for the Anhui International Joint Research Center of Data Diagnosis and Smart Maintenance on Bridge Structures [No. 2021AHGHYB04], the University Natural Science Research Project of Anhui Province [No. KJ2021A1098], the Nanjing International Joint Research and Development Program [No. 202112003], the Jiangsu International Joint Research and Development Program [No. BZ2022010], and the Fundamental Research Funds for the Central Universities [No. B220204002].

ORCID iD

Xin Zhang

Appendix

(A1)

M = \frac{m_{1}}{6} {[\begin{array}{c} 4 & 1 \\ 1 & 4 & 1 & 0 \\ ⋱ & ⋱ & ⋱ \\ 1 & 4 & 1 \\ 1 & 4 \\ 0 & m_{d 1} \\ m_{d 2} \end{array}]}_{(n + 1) \times (n + 1)},

where

m_{d j} = 6 M_{j} / m_{1} j = 1, 2

. (A2)

K = \frac{T}{l_{e 1}} {[\begin{array}{c} 2 & - 1 \\ - 1 & 2 & - 1 & 0 \\ ⋱ & ⋱ & ⋱ \\ - 1 & 2 & - 1 \\ - 1 & k_{d 12} & - 1 & 0 & L & 0 & - k_{d 1} & - k_{d 2} \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & ⋱ & ⋱ & ⋱ \\ 0 & ⋮ & - 1 & 2 & - 1 & ⋮ & ⋮ \\ 0 & - 1 & 2 & 0 \\ - k_{d 1} & 0 & \dots & 0 & k_{d 1} & 0 \\ - k_{d 2} & 0 & \dots & 0 & k_{d 2} \end{array}]}_{(n + 1) \times (n + 1)},

where

k_{d j} = \frac{l_{e 1} M_{j} ω_{d j}^{2}}{T}, j = 1, 2

k_{d 12} = 2 + k_{d 1} + k_{d 2}

. (A3)

C = {[\begin{array}{c} 0 \\ ⋱ & 0 \\ 0 \\ C_{1} + C_{2} & 0 & \dots & 0 & - C_{1} & - C_{2} \\ 0 & 0 & 0 & 0 \\ 0 & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 \\ - C_{1} & 0 & \dots & 0 & C_{1} & 0 \\ - C_{2} & 0 & \dots & 0 & C_{2} \end{array}]}_{(n + 1) \times (n + 1)},

where

C_{j} = 2 M_{j} ω_{d j} ξ_{j}, j = 1, 2

. (A4)

V (t) = {[\begin{array}{c} v_{2} (t) & v_{3} (t) & L & v_{n} (t) & v_{d 1} (t) & v_{d 2} (t) \end{array}]}_{(n + 1) \times 1}^{T} .

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Modal analysis and vibration control of a vertical cable with dual tuned mass dampers

Abstract

Keywords

Introduction

Vibration model

Dynamic modal of a vertical cable

C-DTMD systems

Modal damping analysis of a C-DTMD system

Modal Split

Parameters analysis

Length ratio

Mass ratio

Damping ratio

Frequency ratio

Case 1: Same frequency ratios ρ 1 = ρ 2

Case 2: Different frequency ratios ρ 1 ≠ ρ 2

Numerical verification

Numerical model of a C-DTMD system

Modal analysis

Transient dynamic analysis

Wind loads

Parameters analysis

Length ratio

Mass ratio

Damping ratio

Frequency ratio

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Appendix

References

Case 1: Same frequency ratios $ρ_{1} = ρ_{2}$

Case 2: Different frequency ratios $ρ_{1} \neq ρ_{2}$