Vibration Suppression of Two Adjacent Cables Using an Interconnected Tuned Mass Damper/Nonlinear Energy Sink

Abstract

Due to their high flexibility, low damping, and small mass, stay cables are prone to large-amplitude vibrations. Various mechanical measures, typically installed near the cable anchorage to the deck, have been developed to suppress cable vibration. These dampers, however, may not be effective for ultralong cables since the damper is close to the cable anchorage, the cable node. In this paper, a tuned mass damper (TMD)/nonlinear energy sink (NES) are considered for installation between two adjacent stay cables for vibration mitigation. Firstly, the static equilibrium equation of the stay cable–damper system is established, and the influence of the self-weight of the damper on cable shape is investigated. The governing equations describing the motion of the two adjacent cables with a damper are then established using the Hamilton principle, which are then solved by the method of separation of variables. For cases of swept-sine excitation and harmonic excitation, the optimal designs of TMD and NES are achieved with the purpose of suppressing the first- and third-mode-dominated vibrations, respectively. Both optimal TMD and NES may substantially suppress cable vibrations, with each having advantages under certain situations. Finally, the dynamic response characteristics of two adjacent cables with an optimal damper are analyzed. Interesting dynamic behaviors, such as energy input suppression, phase shift, cable frequency shift, and phase diagram boundary rotation, are identified, and their mechanisms are explained.

1. Introduction

Due to their high flexibility, low damping, and small mass, stay cables are prone to large-amplitude vibrations induced by wind, wind and rain excitation, or cable support displacement excitation. These vibrations may accelerate fatigue, cause damage to the cable protection system, and subsequently reduce the life span of both the stay cables and bridges [1]. Notorious examples related to large-amplitude cable vibrations include estimated 2.5 m peak-to-peak vibrations on the Dubrovnik Bridge [1], 0.7 m peak-to-peak vibrations on the Dongting Lake Bridge [2], and high-order mode vortex-induced vibrations on the Sutong Bridge [3]. Therefore, many scholars have studied the vibration response of cable-stayed bridges under various conditions. Tang et al. [4] systematically investigated the seismic performance of a case cable-stayed bridge, considering the loss of one cable or multiple cables during earthquake excitations. It was revealed that the cable loss would lead to larger seismic demands of the cable-stayed bridge whereas the detrimental effect due to cable loss is highly sensitive to the cable loss location and loss stress. Nicoletti et al.’s [5] study concerned the experimental and numerical dynamic characterization of a newly built steel and wooden cable-stayed footbridge. The footbridge was dynamically tested in situ under ambient vibration. The dynamic response under pedestrian dynamic loads was also investigated. A numerical model of the footbridge was also developed and updated based on the experimental outcomes. Wen et al. [6] performed a probability analysis of RWIV of stay cables under a TC-induced wind and rainfall environment using both a wind tunnel test and numerical simulation techniques. Surrogate models including polynomial and Kriging methods were introduced to fit the experiment results, allowing for the construction of a three-dimensional surface representing RWIV amplitude or the capacity of the stay cable. The occurrence probabilities of various RWIV amplitudes during TC process were finally calculated to conduct the probability analysis.

Various mechanical dampers have been developed for cable vibration mitigation. They can be categorized as passive, semi-active, and active dampers. A viscous damper is one of the most widely used passive dampers. Pacheco et al. [7] proposed a universal curve for designing viscous dampers which relates the damping ratio, number of modes, damper size, damper location, cable span, mass per unit length, and fundamental frequency. Krenk [8] derived an analytical formula from the asymptotic results of complex modal analysis. This formula permits explicit determination of the optimal location of the viscous damper, depending on its damping parameter. Tabatabai and Hawileh [9] showed that bending stiffness is a key parameter that affects the modal damping of cables, and derived a numerical formula to calculate the vibration frequency and damping ratio of a cable with a viscous damper. Other passive dampers have been investigated recently. Shi and Zhu [10] investigated the dynamic characteristics of stay cables with an inerter damper, installed close to one end of a cable. It was found that inerter dampers can offer better damping performance than conventional viscous dampers for the target mode of a stay cable. Javanbakht et al. [11] explored the superior performance of negative stiffness dampers in mitigating cable vibrations. They presented a design approach to optimize negative-stiffness dampers for multimode cable vibration control. Generally, passive dampers are mounted near the cable-deck anchorage, usually at 2 to 5% of the cable length to the anchorage, which limits its efficiency. By contrast, semi-active and active control schemes can provide supplemental damping and enhance control performance. Fujino and Susumpow [12] proposed active stiffness control and active sage-induced force control schemes using actuator motion in the cable axial direction. The controller suppresses free vibration of a cable efficiently and the multimodal response of a cable under random excitation. Johnson et al. [13] studied the potential for improved damping using semi-active devices. A semiactive damper was found to dramatically reduce the cable response compared to the optimal passive linear viscous damper for typical damper configurations. Weber and Distl [14] presented two control approaches, i.e., cycle energy control and controlled viscous damping, for magnetorheological (MR) dampers on cables based on collocated control without state estimation. Cycle energy control and controlled viscous damping were experimentally validated by hybrid simulations and free decay tests on stay cables of the Sutong Bridge, China, and the Russky Bridge, Russia, respectively. Zhao et al. [15] discussed an optimal equivalent control algorithm for vibration mitigation of stay cables based on the linear quadratic regulator for MR damper optimal design. The proposed MR semi-active control algorithm was found to perform better than that of optimal passive control, almost achieving the level of LQR control. However, semi-active and active control requires power demand and complicated design. Active control also suffers stability issues.

For ultralong stay cables, the conventional mechanical cable dampers may have limited effects since they are generally mounted near the cable-deck anchorage. To overcome this issue, Sun et al. [16] proposed to mount a tuned inerter damper between two adjacent flat cables. In this study, tuned mass dampers (TMDs) and nonlinear energy sinks (NESs) are considered for cable vibration mitigation since they can be hung at the anti-node of a target mode of a stay cable. The dynamic characteristics of two adjacent sagged stay cables interconnected with a TMD/NES are investigated. The TMD and NES are comprehensively optimized under various conditions. Both TMDs and NESs can substantially reduce the vibration of two adjacent stay cables under harmonic excitation or frequency-swept excitation at the first-mode frequency as well as under harmonic excitation at the third-mode frequency. In particular, the NES performs slightly better under harmonic excitation at the first-mode frequency, and the TMD performs slightly better under frequency-swept excitation for the first mode and under harmonic excitation at the third-mode frequency. Finally, the vibration reduction mechanisms are analyzed, which provides a basis for the subsequent damper design. The difference between this work and that in [16] is that the effect of sage is considered. In addition, nonlinear energy sinks are also considered for the first time to suppress two adjacent sagged cables.

2. Geometric Shape of the Stay Cable–Damper System under Self-Weight

The presence of a TMD or an NES, whose mass is generally 5–10% of the cable modal mass, will have a non-negligible impact on the cable shape according to Irvine’s classic cable theory [17]. Therefore, this section investigates the geometric shape of a stay cable–damper system under its self-weight.

2.1. System Description

A schematic diagram of two adjacent stay cables with a damper installed between their midspan points is shown in Figure 1. Two types of dampers are considered, i.e., a TMD and an NES. In this paper, the first and third modes are considered as target modes for control. Similar control design and analysis can be straightforwardly extended for other modes of the cables if the damper is installed at an appropriate location, which is not included in this paper for the sake of simplicity. The following four cases are considered:

Case 1: A TMD is employed to suppress the first-mode-dominated vibration.

Case 2: An NES is employed to suppress the first-mode-dominated vibration.

Case 3: A TMD is employed to suppress the third-mode-dominated vibration.

Case 4: An NES is employed to suppress the third-mode-dominated vibration.

As shown in Figure 1, a local coordinate system is established for each cable with its upper end as the origin. The chord direction is the x-axis, and the in-plane transverse vibration direction is the y-axis. The distances between the end points of the outermost cable (Cable 1) and second outermost cable (Cable 2) are $l_1$ and $l_2$, respectively. $K_1$ and $C_1$ are the stiffness and damping coefficient of the spring and viscous damper near Cable 1, respectively. $K_2$ and $C_2$ are the stiffness and damping coefficient of the spring and viscous damper near Cable 2, respectively. $M_d$ is the mass of the damper. The two cables are placed in parallel, with an inclination angle of $\theta$ to the ground. The external load is a distributed dynamic force, with $F(x)$ being the external load distribution function and $q_{pi}(t)$ being the external load time history function.

2.2. Derivation of the Cable Shape

The curve shape equations are derived in the global coordinate system, as shown in Figure 2. The classic cable model proposed by Irvine [17] is adopted. The mass of the damper is taken as 5% of the mass of Cable 1.

The vertical coordinate corresponding to a point on the cable with the horizontal coordinate $x_0$ can be expressed as

$$y_0\left({\mathrm{x}}_0\right)=x_0\tan \left(\theta \right)+z\left(x_0\right)+w\left(x_0\right)$$

(1)

where $z\left(x_0\right)$ is the deformation caused by the self-weight of the cable and $w\left(x_0\right)$ is the additional deformation caused by the mass block of the damper, which is a piecewise function.

Let p be the force generated by the mass block of the damper on the cable, m the mass per unit length of the cable, g the gravitational acceleration, $x_p$ the coordinate of the suspension position of the mass block, H₀ the horizontal component of the initial cable force, A the cross-sectional area of the cable, E the elastic modulus of the cable, and l₀ the horizontal distance between the end points of the cable. It is not difficult to show that

$$z\left({\mathrm{x}}_0\right)=\frac{x_0\left(1-\frac{x_0}{l_0}\right)\left({g{l}_0}^{2}m\sec \left(\theta \right)\right)\left[\frac{\left(1-\frac{2x_0}{l_0}\right)\left({\mathrm{gl}}_0\mathrm{msin}\left(\theta \right)\right)}{6H_0}+1\right]}{2H_0{l}_0}$$

(2)

The addition deformation $w\left(x_0\right)$ is a piecewise function which can be expressed as

$$w\left(x_0\right)=\frac{\left(l_{0}p\right)\left[\frac{x_0\left(1-\frac{x_p}{l_0}\right)}{l_0}-\frac{h\mathrm{z}\left(x_0\right)}{p{l}_0}\right]}{H_0},x_0\le x_p$$

(3)

and

$$w\left(x_0\right)=\frac{\left(l_{0}p\right)\left[\frac{x_p\left(1-\frac{x_0}{l_0}\right)}{l_0}-\frac{h\mathrm{z}\left(x_0\right)}{p{l}_0}\right]}{H_0},x_0>x_p$$

(4)

where

$$h=\frac{\left[6p{x}_p\left(1-\frac{x_p}{l_0}\right)\right]\left[\frac{\left(1-\frac{2x_p}{l_0}\right)\left(g{l}_{0}m\sin \left(\theta \right)\right)}{6H_0}+1\right]}{\left(\frac{12}{{\lambda }^2}+1\right){g{l}_0}^{2}m\sec \left(\theta \right)}$$

(5)

$${\mathrm{L}}_e=l_0{\sec }^3\left(\theta \right)\left(\frac{g^2{l_0}^2{m}^2}{8{H_0}^2}+1\right)$$

(6)

and

$$\lambda =\sqrt{\frac{g^2{l_0}^3{m}^{2}se{c}^2\left(\theta \right)E}{A{H_0}^3{L}_e}}$$

(7)

Substituting Equations (2)–(8) into the expression of $y_0(x_0)$ yields the expression of the cable shape. After the arrangement of terms, the cable shape can be analytically expressed in the following compact form.

$$y_0(x_0)=\left\{\begin{array}{l}{a}_1{x_0}^3+b_1{x_0}^2+c_1{x}_0+d_1,x_0\le x_p\\ a_2{x_0}^3+b_2{x_0}^2+c_2{x}_0+d_2,x_0>x_p\end{array}\right.$$

(8)

with the coefficients listed in Appendix A.

To facilitate dynamic calculations, the shape of the cable in the global coordinate system is transformed to that in a local coordinate system with the chord direction as the x-axis and the in-plane transverse vibration direction as the y-axis, as shown in Figure 3.

A coordinate transformation relating the global and local coordinate systems can be found as

$$\begin{gathered} x_0=x\cos \left(\theta \right)-y\sin \left(\theta \right) \\ {y}_0=x\sin \left(\theta \right)+y\cos \left(\theta \right) \end{gathered}$$

(9)

By substituting Equation (9) into Equation (8) and rearranging the terms, we obtain a cubic equation with respect to y.

$$q{y}^3+w{y}^2+ry+j=0$$

(10)

whose solution is always a real number

$$\begin{array}{l}\mathrm{y}=\frac{\sqrt[3]{\sqrt{{\left(-27q^{2}j+9qrw-2w^3\right)}^2+4{\left(3qr-w^2\right)}^3}-27q^{2}j+9qrw-2w^3}}{3\sqrt[3]{2}q}\\ -\frac{\sqrt[3]{2}\left(3qr-w^2\right)}{3q\sqrt[3]{\sqrt{{\left(-27q^{2}j+9qrw-2w^3\right)}^2+4{\left(3qr-w^2\right)}^3}-27q^{2}j+9qrw-2w^3}}-\frac{w}{3q}\end{array}$$

(11)

The cable shape equations $y_i=f\left(x_i\right)$ for Cable 1 and Cable 2 can be obtained in the local coordinate system, correspondingly. Due to space limitations, these equations are not presented here.

2.3. Numerical Example One

Consider two adjacent parallel cables with the following usual parameters: $m=100 \mathrm{kg}/\mathrm{m}$, $H=5.03608\times {10}^6 \mathrm{N}$, $g=9.8 {\mathrm{m}/\mathrm{s}}^2$, $\theta =\pi /4$, $A=0.012 {\mathrm{m}}^2$, $E=2.1\times {10}^{11} \mathrm{Pa}$, $l_{01}=400 \mathrm{m}$, and $l_{02}=380 \mathrm{m}$. A damper is hinged to the midpoints of the two cables. The weight of the damper mass block is set to 5% of the weight of Cable 1. Then, the vertical force of the whole device on a single cable is

$$p=\frac{G}{2}=0.5\times 5\%\times 400\sqrt{2} \mathrm{m}\times 100 \mathrm{kg}/\mathrm{m}\times 9.8 \mathrm{m}/{\mathrm{s}}^2=\mathrm{13,859.258} \mathrm{N}$$

Since the longitudinal deformation of the cable is negligible, any point on the cable after the suspension of the damper can be considered to undergo only a displacement in the direction perpendicular to the cable chord direction. When the inclination angle of the cable is 45°, the midpoint of the line segment formed by the line connecting the two end points of the cable is designated A for the Cable 1 and B for the Cable 2; then, AB must be perpendicular to the cable chords. Since the maximum amplitude positions of the first and third modes are both at the midpoint, the damper is suspended along the line segment AB.

In Figure 2, perpendicular lines to the x₀ axis are drawn and the line connecting the end points of the cable through the suspension point, and the following equation can be derived from the geometrical relationship.

$$\sqrt{2}{x}_p+[y_0(x_p)-x_p]\cdot \sqrt{2}/2=l/2$$

where l is the chord length. By substituting the parameters of each cable into the equation, a sixth-degree equation is obtained, which is solved to yield the following roots after discarding the extraneous roots.

$$x_{p1}=197.111 \mathrm{m}, x_{p2}=187.386 \mathrm{m}$$

The midpoints of the cable chords have coordinates of 200 m and 190 m, respectively, indicating a noticeable offset of the suspension point under the combined action of the suspended damper and the cable self-weight. By substituting all the parameters into the piecewise function of the line shape, the line shape of each cable can be obtained. Keeping all other parameters unchanged and setting the damper load p to zero, the line shape of the cable in the uncontrolled case, i.e., without a control device, can be obtained.

The analytical cable shapes in the local coordinate system are graphically shown in Figure 4.

Table 1. Geometric characteristics of the cables.

	Cable 1 with a Damper	Cable 1 without a Damper	Cable 2 with a Damper	Cable 2 without a Damper
Sag	4.08617 m	3.89200 m	3.69704 m	3.51252 m
Sag-to-span ratio	0.722340%	0.688014%	0.687948%	0.653612%
Distance from the maximum value position to the centre (offset)	0 m	2.59474 m	0 m	2.34177 m
The percentage of increase in sag compared to the no damper case	4.98895%	/	5.25321%	/

summarizes the geometric characteristics of Cables 1 and 2 with /without a damper. The sag-to-span ratio of the cable is approximately 0.7%. The smoothness and maximum value (sag) at the top of the line shape are related to the existence of a damper. When there is a damper, the sag of the cable is larger, and the curve at the position of the suspended damper is not smooth or differentiable. In addition, when there is no damper, the maximum sag position of the cable in the local coordinate system is not at the midpoint of the cable chard but deviates towards the lower end due to the influence of gravity. In contrast, when there is a damper, the maximum sag position strictly appears at the midpoint (which is also the damper suspension position), which is consistent with the assumption of cable deformation, namely, ignoring the longitudinal deformation of the cable. The cable shapes of the two cables have similar geometric characteristics, and Cable 1 has a slightly larger sag-to-span ratio in each case.

3. Dynamic Characteristics of the Cable–Damper System

Equations of Motion for Two Adjacent Cables Interconnected with an NES or a TMD

Consider the two adjacent cables interconnected with an NES/TMD shown in Figure 1. The axial vibration as well as the torsional and shear stiffnesses of the cables are ignored. The equation of motion (EOM) of the two adjacent cables interconnected with an NES is derived using the extended Hamilton principle. The kinetic energy of the system is

$$E_k=\frac{1}{2}{\displaystyle \underset{0}{\overset{l_1}{\int }}{\rho }_1{A}_1}{\stackrel{·}{v}}_1^{2}d{x}_1+\frac{1}{2}{\displaystyle \underset{0}{\overset{l_2}{\int }}{\rho }_2{A}_2}{\stackrel{·}{v}}_2^{2}d{x}_2+\frac{1}{2}{M}_d{{\stackrel{·}{v}}_d}^2$$

(12)

where ${\rho }_i$, $A_i$, and $v_i$$(i=1,2)$ are the density, cross-sectional area, and in-plane transverse displacement of Cable 1 and Cable 2, respectively. $M_d$ is the mass of the damper. $v_d$ is the displacement of the damper mass along the y-axis. The dot represents the differential with respect to the time t. The elastic potential energy of the cable can be expressed as

$$E_p={\displaystyle \sum _{i=1}^2\{{\displaystyle \underset{0}{\overset{l_i}{\int }}[H_i{e}_i(x_i,t)+\frac{1}{2}}}{E}_i{A}_i{e_i}^2(x_i,t)]d{x}_i+\frac{1}{4}{K}_i{[v_i(x{s}_i,t)-v_d]}^4\}$$

(13)

where $H_i$ and $E_i$ are the initial force and elastic modulus of the cable, respectively, $x{s}_i$ is the position coordinate of the damper, and $e_i(x_i,t)$ is the tensile strain of the cable in motion, defined as

$$e_i(x_i,t)={y_i}^{\prime }{v_i}^{\prime }+\frac{1}{2}{v_i}^{\prime 2}(i=1,2)$$

(14)

where the prime represents the differential with respect to the coordinate $x_i$. The virtual work is related to the gravity, external force, and damping force, that is,

$$\delta W={\displaystyle \sum _{i=1}^2\{{\displaystyle \underset{0}{\overset{l_i}{\int }}[{\rho }_i}{A}_{i}g+p_i(x_i,t)-{\mu }_i\stackrel{·}{v_i}]\delta v_{i}d{x}_i+}{C}_i[\stackrel{·}{v_i}(x{s}_i,t)-\stackrel{·}{v_d}]\delta v_d\}$$

(15)

where g is the gravitational acceleration, $p_i(x_i,t)$ is the total load acting on the cable, and ${\mu }_i$ is the damping coefficient per unit length of the cable. $C_i$ is the damping coefficient of the viscous damper near the Cable i. Substituting the kinetic energy (12), potential energy (13), and virtual work (15) into the extended Hamilton principle yields the EOM of the cable and NES system.

$$\delta {\displaystyle \underset{t_1}{\overset{t_2}{\int }}(E_k}-E_p)dt+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}\delta W}dt=0$$

(16)

Combined with the static cable shape derived in Section 1, the EOM of the cable is

$${\rho }_i{A}_i\stackrel{··}{v_i}+{\mu }_i\stackrel{·}{v_i}-[H_i{v_i}^{\prime }+E_i{A}_i({v_i}^{\prime }+{y_i}^{\prime })e_i(x_i,t){]}^{\prime }=p_i(x_i,t)(i=1,2)$$

(17)

The EOM of the NES is

$${\displaystyle \sum _{i=1}^2\{K_i}{[v_i(x{s}_i,t)-v_d]}^3+C_i[\stackrel{·}{v_i}(x{s}_i,t)-\stackrel{·}{v_d}]\}-M_d\stackrel{··}{v_d}+M_{d}g\cos \theta =0$$

(18)

Integrating Equation (14) along the length of the cable and applying the boundary condition of the cable gives the uniform dynamic elongation of the cable as

$$e_i(t)=\frac{1}{l_i}{\displaystyle \underset{0}{\overset{l_i}{\int }}({y_i}^{\prime }{v_i}^{\prime }+\frac{1}{2}{v_i}^{\prime 2}})d{x}_i(i=1,2)$$

(19)

Using Equation (19), the EOM of the cable Equation (17) can be rewritten as

$${\rho }_i{A}_i\stackrel{··}{v_i}+{\mu }_i\stackrel{·}{v_i}-[H_i{v_i}^{″}+E_i{A}_i({v_i}^{″}+{y_i}^{″})e_i(t)]=p(x_i,t)(i=1,2)$$

(20)

where the term $p(x_i,t)$ considers the effect of the NES, that is,

$$p_i(x_i,t)=-f_i(t)\delta (x_i-x{s}_i)+F_i(x_i)q_{pi}(t)(i=1,2)$$

(21)

in which $\delta ()$ is the Dirac delta function, $f_i(t)$ is the force exerted by the NES on the cable, and $f_{ie}(t)$ is exciting load, i.e.,

$$f_i(t)=K_i{[v_i(x{s}_i,t)-v_d]}^3+C_i[\stackrel{·}{v_i}(x{s}_i,t)-\stackrel{·}{v_d}](i=1,2)$$

(22)

$$f_{ei}(t)=F_i(x_i)q_{pi}(t)$$

(23)

Based on the method of separation of variables to solve the partial differential equation, the cable displacement function may be expressed as the product of the mode shape function and the generalized coordinates, i.e.,

$$v_i(x_i,t)={\varphi }_i(x_i)q_i(t)(i=1,2)$$

(24)

Applying the Galerkin method to the cable yields

$${\displaystyle \underset{0}{\overset{l_i}{\int }}{\varphi }_i}(x_i)\{{\rho }_i{A}_i\stackrel{··}{v_i}+{\mu }_i\stackrel{·}{v_i}-[H_i{v_i}^{″}+E_i{A}_i({v_i}^{″}+{y_i}^{″})e_i(t)]-p(x_i,t)\}d{x}_i=0$$

(25)

A three-DOF system can be obtained as follows. The EOM of Cable 1 is

$$\begin{array}{l}{r}_1{q}_1(t)+r_2{q}_1^2(t)+r_3{q}_1^3(t)+r_4{\stackrel{·}{q}}_1(t)+r_5{\stackrel{··}{q}}_1(t)+r_6{v_d}^3(t)+r_7{\stackrel{·}{v}}_d(t)+r_8{q}_1(t){v_d}^2(t)\\ +r_9{q_1}^2(t)v_d(t)+r_{10}{q}_{p1}(t)=0\end{array}$$

(26)

The EOM of Cable 2 is

$$\begin{array}{l}{s}_1{q}_2(t)+s_2{q}_2^2(t)+s_3{q}_2^3(t)+s_4{\stackrel{·}{q}}_2(t)+s_5{\stackrel{··}{q}}_2(t)+s_6{v_d}^3(t)+s_7{\stackrel{·}{v}}_d(t)+s_8{q}_2(t){v_d}^2(t)\\ +s_9{q_2}^2(t)v_d(t)+s_{10}{q}_{p2}(t)=0\end{array}$$

(27)

The coefficients in the formulas are provided in Appendix B.

The EOM of the NES is

$$\begin{array}{l}-K_1{v_d}^3(t)-K_2{v_d}^3(t)+3K_1{q}_1(t){v_d}^2(t){\varphi }_1(x{s}_1)-3K_1{q_1}^2(t)v_d(t){{\varphi }_1}^2(x{s}_1)+K_1{q_1}^3(t){{\varphi }_1}^3(x{s}_1)\\ +3K_2{q}_2(t){v_d}^2(t){\varphi }_2(x{s}_2)-3K_2{q_2}^2(t)v_d(t){{\varphi }_2}^2(x{s}_2)+K_2{q_2}^3(t){{\varphi }_2}^3(x{s}_2)+C_1{\varphi }_1(x{s}_1){\stackrel{·}{q}}_1(t)\\ +C_2{\varphi }_2(x{s}_2){\stackrel{·}{q}}_2(t)-C_1{\stackrel{·}{v}}_d(t)-C_2{\stackrel{·}{v}}_d(t)+M_{d}g\cos (\theta )-M_d{\stackrel{··}{v}}_d(t)=0\end{array}$$

(28)

It can be seen from Equations (26)–(28) that there are terms related to the motion of the NES, e.g., $r_6{v_d}^3(t)$ and $r_8{q}_1(t){v_d}^2(t)$ due to the presence of the NES damper, indicating the generation of a large amount of non-ignorable coupling effects between the two cables and the damper. In the meanwhile, many terms related to cable vibrations appear in the EOM of the NES, indicating that the motion of the NES mass block is also strongly constrained by the motion of the two cables. The interaction between the two cables and the NES provides the possibility for energy transfer between them.

When the two adjacent cables are interconnected with a TMD, the EOMs of the system can be obtained similarly and given as follows.

The EOM of Cable 1 is

$$c_{01}{q}_1(t)+c_{02}{q}_1^2(t)+c_3{q}_1^3(t)+c_4{\stackrel{·}{q}}_1(t)+c_5{\stackrel{··}{q}}_1(t)+c_6{v}_d(t)+c_7{\stackrel{·}{v}}_d(t)+c_8{q}_{p1}(t)=0$$

(29)

The EOM of Cable 2 is

$$d_{01}{q}_2(t)+d_{02}{q}_2^2(t)+d_3{q}_2^3(t)+d_4{\stackrel{·}{q}}_2(t)+d_5{\stackrel{··}{q}}_2(t)+d_6{v}_d(t)+d_7{\stackrel{·}{v}}_d(t)+d_8{q}_{p2}(t)=0$$

(30)

The EOM of the TMD is

$$\begin{array}{l}{M}_{d}g\cos (\theta )-K_1{v}_d(t)-K_2{v}_d(t)+K_1{q}_1(t){\varphi }_1(x{s}_1)+K_2{q}_2(t){\varphi }_2(x{s}_2)+C_1{\varphi }_1(x{s}_1){q_1}^{\prime }(t)\\ +C_2{\varphi }_2(x{s}_2){q_2}^{\prime }(t)-C_1{v_d}^{\prime }(t)-C_2{v_d}^{\prime }(t)-M_d{v_d}^{″}(t)=0\end{array}$$

(31)

The coefficients in the equations are provided in Appendix C.

Compared with the EOMs of the cable–NES system, the EOMs of the cable–TMD system have far fewer nonlinear coupling terms, e.g., $a_8{q}_1(t){v_d}^2(t)$ mentioned earlier. This indicates that the coupling effect between the two cables and the damper is much less in the case of a TMD than in the case of an NES. In particular, the direct coupling term has disappeared. In the case of a TMD, the remaining nonlinearity mostly originates from the geometric nonlinearity of the cables themselves.

4. Optimal Design of the Damper

4.1. Optimal Problem Description

The stiffness and damping coefficients of the damper are optimized such that the maximum generalized amplitude of the two adjacent cables under external excitation is minimized. The generalized amplitude is defined as the half of the difference between the maximum and minimum displacements of the midpoint of the cables over a certain period. The considered external excitations include a harmonic excitation of the frequency of the first or third mode and a swept-sine excitation with the frequency range covering the frequency of the first or third mode. Each set of differential equations of motion are solved. The maximum number of integrating steps is ${10}^{10}$, which is same in all subsequent integral calculations. Once the steady state response is obtained, the maximum value of the generalized amplitude of the two cables is estimated. In this paper, the considered period for generalized amplitude estimation is over a time range of $5000 \mathrm{s}-4T_n~5000 \mathrm{s}$, where $T_n$ is the vibration period of the structure when the n-th modal vibration is excited, that is, $T_n=1/\left(f_{2,n}-f_{1,n}\right)$. A brute-force method is employed for searching optimal stiffness and damping coefficients, corresponding to the minimum generalized amplitude, over their prescribed feasible ranges with step size of 1% of the whole range.

For the sake of simplicity, the springs and dampers on the two sides of the mass block are identical, that is, $K_1=K_2=K,C_1=C_2=C$. The optimization problem can be then formulated as

$$\begin{array}{l}\arg \min A_m(K,C)\\ A_m(K,C)=\max \{A_{m1},A_{m2}\}\\ A_{mi}(K,C)=\frac{v_i{(t)}_{\max }-v_i{(t)}_{\min }}{2},5000\mathrm{s}-4T_n\le t\le 5000\mathrm{s}\end{array}$$

(32)

4.2. External Excitation Description

External excitations considered may be a single-frequency harmonic excitation or a swept-sine excitation, acting on the cable in the bridge longitudinal direction and uniformly distributed on the two cables along the elevation direction. In the later analysis, a 5000 s long harmonic external excitation is considered and can be expressed as

$$f_{ei,n}(t)=F_i(x_i)\cos ({\Omega }_{i,n}t),0<t<t_{\max }$$

(33)

where $F_i(x_i)$ is the external excitation amplitude and ${\Omega }_{i,n}$ is selected as the circular frequency of the n-th mode of Cable i.

The considered swept-since excitation may be expressed as

$$f_{ei}(t)=F_i(x_i)\sin (a_{ei}{t}^2+b_{ei}t),0<t<t_{\max }$$

(34)

where $a_{ei}$ and $b_{ei}$ are the frequency coefficients, that is,

$$a_{ei}=\frac{{\omega }_{i\max }-{\omega }_{i\min }}{2t_{\max }},b_{ei}={\omega }_{i\min }$$

$${\omega }_{i\min ,1}=0,{\omega }_{i\max ,1}=2{\omega }_{i,1},t_{\max }=5000\mathrm{s}$$

The magnitude of the external excitation is assumed to be that of the static wind load, which is estimated according to reference [18] as,

$$F_i(x_i)=\frac{1}{2}\rho U^2{\sin }^2\theta C_{D0}D$$

(35)

where $\rho$ is the air density, $U$ is the horizontal wind speed along the longitudinal bridge direction, $C_{D0}$ is the drag coefficient of the cable when it is perpendicular to the incoming wind, and D is the cable diameter. In this study, $\rho =1.29 \mathrm{kg}/{\mathrm{m}}^3,U=38.9 \mathrm{m}/\mathrm{s}, \theta =\frac{\pi }{4},C_{D0}=1.2,D=\sqrt{\frac{4A}{\pi }},A=0.012 {\mathrm{m}}^2$. Therefore, the external excitation amplitude is found to be $F_i(x_i)=72.3862 \mathrm{N}/\mathrm{m}$.

4.3. Estimation of Fundamental Frequency of the Two Cables

The fundamental frequency of a sagged cable will deviate from that of the corresponding taut string $f_{i,1}=\frac{1}{2l_i}\sqrt{\frac{H_i}{m_i}}$ [19]. In this paper, an accurate estimate of fundamental frequency is obtained by identifying the frequency corresponding to the peak of the frequency response curve of the cable. The frequency response curve is obtained by calculating the time-domain response of the cable (without damper) under a swept-sine excitation with the frequency varying from 0 Hz to $4\pi f_{i,1}$. However, the frequencies of higher modes are still estimated as $f_{i,n}=\frac{n}{2l_i}\sqrt{\frac{H_i}{m_i}}$.

Consider two cables having $l_1=400\sqrt{2} \mathrm{m}$, $l_2=380\sqrt{2} \mathrm{m}$, $H_i=7.1221\times {10}^6 \mathrm{N}$, and $m_i=100 \mathrm{kg}/\mathrm{m}$.

$F_i(x_i)=72.3862 \mathrm{N}/\mathrm{m}$, $x{s}_i=\frac{l_i}{2}$, ${\varphi }_i(x_i)=\sin (\frac{n\pi x_i}{l_i})$, ${\mu }_i=0.5 \mathrm{kg}/(\mathrm{m}\cdot \mathrm{s})$, ${\rho }_i=8333.33 {\mathrm{kg}/\mathrm{m}}^3$, $A_i=0.012 {\mathrm{m}}^2,E_i=2.1\times {10}^{11} \mathrm{Pa},M_d=G/g=2828.42 \mathrm{kg}$. The approximate fundamental frequencies of the two cables, indicated by the taut string theory, are $\widetilde{{\omega }_{1,1}}=1.48 \mathrm{rad}/\mathrm{s}$ and $\widetilde{{\omega }_{2,1}}=1.56 \mathrm{rad}/\mathrm{s}$. The frequency response curves are calculated and shown in Figure 5, which indicates the peak frequency as ${\omega }_{1,1}=1.64 \mathrm{rad}/\mathrm{s}, {\omega }_{2,1}=1.72 \mathrm{rad}/\mathrm{s}$. It can be seen that under external excitations the fundamental frequency of the cable may experience a variation up to 10%.

4.4. Initial Guess of Damping and Stiffness Coefficients

In order to improve the optimization efficiency, a reasonable initial guess of the damper parameters is necessary. The two adjacent cables interconnected with a damper can be considered to be the combination of two substructures. Each substructure consists of a cable and a half of the damper having stiffness $K$, damping $C$, and mass $M_d/2$.

For the TMD case and the n-th mode as the target mode, the initial guess of the stiffness $K$ and damping $C$ can be estimated by designing an optimal half-damper attached to Cable 1 using the Den Hartog formula. That is,

$$K=K_{1n}=\frac{M_d{\lambda }^2{{\omega }_{1,n}}^2}{2},C=C_{1n}=\xi \lambda {\omega }_{1,n}{M}_d$$

(36)

where

$$\lambda =\frac{1}{1+\mu },\xi =\sqrt{\frac{3\mu }{8(1+\mu )}},\mu =\frac{M_d}{2m{l}_1}$$

From the two cables considered in Section 4.3, the initial values for the stiffness and damping coefficients of the damper aiming at suppressing the first and third mode can be calculated as

$$K_{11}=3639.81 \mathrm{N}/\mathrm{m},C_{11}=433.960 \mathrm{Ns}/\mathrm{m};K_{13}=26611.3 \mathrm{N}/\mathrm{m},C_{13}=1173.39 \mathrm{Ns}/\mathrm{m}$$

For the case of an NES, the initial damping and cubic stiffness coefficients take the values of the initial damping and linear stiffness coefficients for the case of a TMD.

4.5. Optimization Result Analysis

The generalized amplitude (Am) of the two cables described in Section 4.3 interconnected with a TMD or NES under the external excitation described in Section 4.2 can be calculated against a different stiffness $K$ and damping coefficient $C$. The results for the case where the first mode is the target mode are presented in Figure 6, while those for the case where the third mode is the target mode are presented in Figure 7. The purple plane represents the generalized amplitude (Am) of the two cables without a damper.

The comparison between Figure 6a,b shows that under a harmonic excitation at the first-mode frequency, an NES produces better control performance than a TMD. NESs with the damping and stiffness coefficient in the considered feasible ranges, except for those with extremely small damping and stiffness coefficients, can always significantly reduce the generalized amplitude of the two cables. It also indicates that the control performance is insensitive to the variation of the damping and stiffness parameters. In contrast, many TMDs with the damping and stiffness coefficient in the considered feasible ranges amplify the generalized amplitude of the two cables. In addition, there are chaotic regions for structural vibration for both damper cases, with fluctuating surfaces in these regions. That is, the generalized amplitude of vibration is very prone to small changes with the damper parameters, indicating very poor robustness. However, the parameter region for the occurrence of chaotic response is different for both dampers. Under the control of an NES, this region appears at low stiffness and low damping, especially when both are low, with a pronounced increase in amplitude at low stiffness. In contrast, under TMD control, this region experiences low stiffness and damping levels, with a clear rising platform. There are obvious boundaries between these different regions, resulting from the transformation of the dynamic behavior of the system beyond the critical range. These chaotic regions with an obvious increase in amplitude (at locations with low stiffness for the NES and at locations with high stiffness and low damping for the TMD) are parametric regions that should be avoided in design.

Under swept-since excitations targeting at the first mode, the two cable continuously absorb the vibration energy from external loads in the absence of vibration control, resulting in a very high generalized amplitude. However, in the presence of a TMD or NES control device, the vibration can be effectively controlled. The figure clearly shows that, except for the region with extremely small damping and stiffness coefficients, the vibration amplitude in other regions (the maximum value of the generalized amplitude of the two cables) is reduced significantly. Under NES control, there are more nearly completely flat regions. Outside the flat regions are sharp chaotic regions and slow vibration reduction regions with no stiffness. These regions are unstable and have an amplitude control effect that is not as good as that in flat regions and should be avoided. Under TMD control, the surface is mainly divided into flat regions, folded regions, and sharp regions. The sharp regions appear in undamped and lightly damped regions; the folded regions mainly start at the stiffness–damping coordinates (5000, 0) and extend towards large stiffness–damping regions, forming an area. The cause of this area is also the repeated transformation of the dynamic behavior of the system beyond the critical range, and the dynamic behavior of the system is the same on some approximately straight lines on the stiffness–damping plane (at the folds and gullies). It can also be seen from the overall properties of the first mode (single-point/frequency-swept) optimization graph that a TMD with low damping and an NES with low stiffness result in dangerous regions and hence should be avoided in design.

Under harmonic excitation at the third-mode frequency, except for the sharp-angled regions under TMD control in Figure 7a and the undamped chaotic sharp regions under NES control in Figure 7b, the other regions are relatively flat and approximately cylindrical, indicating that damping in most of the other flat regions has little impact on structural vibration and provides excellent control performance. In general, an NES results in larger and flatter regions, but the maximum amplitude it generates in chaotic regions is much higher than that of a TMD. Therefore, these undamped chaotic sharp regions should also be avoided as much as possible in the design. In addition, the peak pattern in the case of a TMD is similar to the shape of the frequency response function, which might be caused by the multiple natural frequencies generated in the structure after adding stiffness. The folding of the surface at low stiffness also indicates that a sufficiently high stiffness level is required for a more effective control of vibration.

After organizing the data, the optimal controller parameters under different conditions are shown in

Table 2. Optimal controller parameters under different conditions.

	TMD (Stiffness; Damping)	NES (Stiffness; Damping)
Harmonic excitation at the first-mode frequency	0; 538.111 Ns/m	7971.18 N/m3; 0
Swept-sine excitation (excitation of the first mode)	3457.82 N/m; 542.45 Ns/m	400.379 N/m3; 1254.14 Ns/m
Harmonic excitation at the third-mode frequency	52,424.30 N/m; 3074.28 Ns/m	67,858.80 N/m3; 3426.30 Ns/m

and

Table 3. Control effects of different devices (maximum value of the generalized amplitude of the two cables $\psi$).

	Uncontrolled Group (No Control Device)	Best TMD Controller	Best NES Controller
Harmonic excitation at the first-mode frequency	10.9992 m	3.57421 m	2.55049 m
Swept-sine excitation (excitation of the first mode)	44.8049 m	7.5476 m	8.81287 m
Harmonic excitation at the third-mode frequency	0.925372 m	0.309824 m	0.315037 m

.

Based on the data in Table 2 and Table 3, we obtain a relatively peculiar result. Under harmonic excitation for the first mode, the optimal TMD controller does not require spring stiffness, and the optimal NES controller does not require damping. That is, the optimal controller is either a single viscous damping device or a single cubic nonlinear spring. Under swept-sine excitation, the optimal TMD device exhibits high stiffness and low damping, while the optimal NES device exhibits low stiffness and high damping. Under harmonic excitation at the third-mode frequency, the optimal TMD and NES both exhibit high damping and stiffness.

A comparison of the diagrams of the global optimization processes shows the reasons for these phenomena. Under harmonic excitation for the first mode, an increase in stiffness is detrimental to the structural performance of the TMD and may even lead to chaos. However, in the case of the NES, there is a significant slight decrease in the surface at the regions of low damping or no damping, and conversely, increasing damping slightly will cause the system to enter different mechanical behavior regions, thereby reducing the control effectiveness. Under frequency-swept excitation for the first mode, for the TMD, the lowest point of the surface is located at the position of relatively high stiffness and low damping, which is also the point of minimum value. At this time, for the NES, the surface is flat except for some peak regions, but there are some differences in the heights of the flat regions beyond different peaks, indicating that as long as chaos does not occur, the system is always in a stable state under other conditions, but it may have entered different stable branches. At the same time, under frequency-swept excitation, the NES needs more damping for energy dissipation, while the TMD requires greater stiffness. This may be because the TMD needs a higher stiffness level to achieve vibration frequency capture, while the nonlinear spring of the NES captures the vibration frequency more easily due to the broadband effect and instead requires more damping to limit some uncontrollable nonlinear factors.

Under harmonic excitation at the third-mode frequency, the curved surface is similar to a cylindrical surface, which means that the damping coefficient has little impact on vibration control. This may be because the vibration velocity of the structure in the third mode is not as rapid as that in the first mode, and hence the role of viscous damping is limited. At this time, under TMD control, the surface has distinct peaks similar to those of the frequency response function. This indicates that the natural frequencies of the structure may have changed or that other natural frequencies may have been generated after installing the TMD device. After the NES device is installed, the surface exhibits chaos in the undamped region, indicating that a certain amount of damping is needed to suppress the chaotic response of the structure and that increasing the damping has almost no effect. At this point, the surface tends to be flat, indicating that there may be a stiffening effect, i.e., due to the high stiffness and damping levels, it is almost equivalent to a rigid connection, and the control effect is generally good in this case.

5. Performance Analysis of the Optimal Damper

After the optimal damper parameters are obtained, because different dampers have different vibration reduction mechanisms and control effects, the performances of different dampers are analyzed from two aspects: amplitude and frequency robustness and dynamic response characteristics (time–history curves, phase diagrams, and spectra).

5.1. Robustness Analysis

After obtaining the optimal damper parameters under different conditions, the robustness of the main structure–damper system is tested by adjusting the frequency and amplitude of the external load. Since the frequency-swept test has already been performed for the first mode, its frequency robustness is no longer studied.

5.1.1. Amplitude Robustness

Amplitude Robustness under Harmonic Excitation at the First-Mode Frequency

The optimal parameters of each damper under harmonic excitation at the first-mode frequency are adopted, and the amplitude of the external load is changed (the horizontal axis $\eta$ is the percentage ratio of the external load amplitude to the determined load amplitude used in the previous section) to generate generalized amplitude (Am) response curves, which are compared with those of the uncontrolled group. The amplitude varies from 50% to 150%.

Figure 8 shows that under harmonic excitation at the first-mode frequency, when the amplitude changes, with the NES damper installed, although the generalized amplitude fluctuates, the vibration energy is generally suppressed so that the vibration amplitude remains at a low level and is consistently near a straight line; at this point, the graph exhibits obvious nonlinear characteristics. In contrast, when the TMD damper is used or in the case of the uncontrolled group, the generalized amplitude of the structure jumps significantly near the resonance point, possibly due to entering different stable branches. It is evident that the vibration of the structure is linear. Overall, the system installed with the NES damper has the highest amplitude robustness in this case.

Amplitude Robustness under Swept-Sine Excitation (Excitation of the First Mode)

Figure 9 shows that under swept-sine excitation (excitation of the first mode), the absence of a damper will lead to the continuous input of vibration energy and an extremely high amplitude, which is dangerous. Installing a damper can effectively control the structural vibration. However, due to its nonlinear nature, the NES damper may lead to unstable situations, where the amplitude of the structure increases abruptly. In this case, the TMD has the best control performance, with the generalized amplitude of the structure maintained at a low and very stable level.

Amplitude Robustness under Harmonic Excitation at the Third-Mode Frequency

Figure 10 shows that under harmonic excitation at the third-mode frequency, all three curves are smooth. A structure without control has a high vibration amplitude, while a structure with control exhibits good results. The TMD has a slightly better control effect than the NES, and its curve is at the bottom. In this case, the lack of nonlinearity in the structure may be due to the high stiffness and damping coefficient of the damper, resulting in a stiffening effect.

5.1.2. Frequency Robustness

In this study, dampers are mostly designed at fixed frequencies, but in reality, we hope that they can achieve a control effect within a certain range, especially considering the complex excitation frequencies of stay cables under various conditions. Therefore, a frequency robustness analysis is carried out on the optimal NES and TMD dampers under third-mode frequency excitation without performing a frequency-swept analysis. The frequency range $\varsigma$ is taken to be 200% to 400% of the fundamental frequency, and the load remains unchanged. The analysis results are shown in Figure 11, where the horizontal axis $\varsigma$ is the percentage ratio of the external load frequency to the structural fundamental frequency and Am is generalized amplitude.

The figure clearly shows that in the nearby frequency range, both the TMD and NES can effectively control the vibration of the structure, with the NES exhibiting the best control effectiveness. Moreover, the figure shows that the vibration reduction mechanisms of the dampers are almost the same at this time, as both decompose the main frequency peak into four peaks, and the positions of the four peaks for the TMD and NES are identical. Therefore, the vibration reduction mechanism at this time is that after the installation of a damper, the main resonant frequency of the system is dispersed to form several different frequencies, thus enlarging the energy absorption range and significantly reducing the overall vibration.

5.2. Dynamic Response Characteristics

Studying the dynamic response characteristics of the vibrating system helps to analyze the vibration reduction principle. Therefore, the time–history curves, phase diagrams, and frequency spectra for the uncontrolled group, the structure installed with the optimal NES device, and the structure installed with the optimal TMD device are plotted and studied.

5.2.1. Time–History Analysis

The time–history curves for the uncontrolled group, the TMD group, and the NES group are plotted as follows.

Time–History Analysis under Harmonic Excitation at the First-Mode Frequency

The amplitude when assembled with the NES is the lowest, indicating the best vibration control effect. Figure 12 shows that cables installed with the NES and with the TMD ultimately exhibit beat vibrations. This is because the two cables have very close chord lengths and natural frequencies, resulting in long-period beat vibrations with a frequency difference from the vibration frequency. The beat vibrations occur because the amplitude is variable, and the generalized amplitude used in this study is half of the difference between the maximum and the minimum displacements. In reality, after the NES is installed, the average amplitude decreases after the occurrence of beat vibrations, the overall vibration energy of the system decreases, and the drag effect between the two cables increases, resulting in a better control effect. Moreover, the phenomenon of beat vibrations indicates the occurrence of energy transfer between the two cables as well as between the cables and the damper.

Time–History Analysis under Swept-Sine Excitation (Excitation of the First Mode)

Figure 13 shows that after installing the damper, both the NES and TMD can effectively suppress the vibration, and the significant vibration phenomenon only occurs when sweeping to the fundamental frequency of the structure. For the uncontrolled group, a large amount of energy is suddenly absorbed near the fundamental frequency, and the vibration is not effectively suppressed afterwards, resulting in a sustained increase in amplitude (which eventually decays, while the length of time selected here is finite). Due to the stiffness of the vibration reduction device, the displacement of the structure under frequency-swept excitation is limited, and its vibration amplitude is suppressed, while the damping device continuously dissipates the vibration energy, thus achieving vibration reduction.

Time–History Analysis under Harmonic Excitation at the Third-Mode Frequency

Figure 14 shows that after the damper is installed, the vibration is effectively controlled, and the degree of control is almost consistent, demonstrating the effectiveness of the device. In addition, since the stiffness and damping of the device at this time are already high, a stiffening effect occurs, resulting in almost the same vibration curves. The time from the beginning to entering a stable state is also very short, indicating better stability of the system at this time. However, due to the lower energy of the third mode, its amplitude and nonlinearity are significantly weaker than those under excitation of the first mode.

5.2.2. Phase Diagram Analysis

Phase diagram analysis helps to explain the resonant characteristics of a structure, including properties such as energy transfer, the resonant frequency ratio, and phase difference. Standard Lissajous figures(Figure 15) are used for comparative analysis with structural vibration phase diagrams.

Phase Diagram Analysis under Harmonic Excitation at the First-Mode Frequency

The phase diagrams of the uncontrolled group, the TMD group, and the NES group are plotted as Figure 16 (the time range chosen is four beat vibration periods, i.e., 4677.808 s to 5000 s).

These phase diagrams are normalized such that the aspect ratio of the images is set to one. First, comparison to the standard Lissajous figures shows that the phase diagrams of the motion of the midpoints of the cables of the uncontrolled group are all close to a perfect circle, indicating that they all undergo harmonic motion. For the TMD group, the q₁ phase diagram is also a slightly thickened circle, while the q₂ phase diagram is a significantly thickened circle, indicating that their motion is still close to harmonic motion. However, according to the physical meaning of the phase diagrams, circles with different radii represent different total values of mechanical energy. Therefore, it can be seen that their vibration energy is constantly changing and converted, which is also observed from the beat vibration of the time–history curve. In comparison, the NES group exhibits more pronounced nonlinear properties, with more chaotic patterns in the diagrams. However, first, its maximum range envelope is circular, indicating that there is an upper limit to the energy. Second, observing the q₁ and q₂ phase diagrams at this time, it can be found that the interior is also composed of some fine circles, which indicates that it also undergoes irregular harmonic motion, suggesting that there is more than one frequency component. The difference is that the total value of the motion mechanical energy of q₁ motion can even be zero (passing through the origin), but the mechanical energy of q₂ motion is always above a certain value.

According to the relationship curve between q₁ and q₂, since the two cables in the uncontrolled group are not connected, the diagram is very similar to the Lissajous figure, with a small difference in frequency but an unstable phase difference. In the TMD group, this diagram is similar to that of the uncontrolled group but with a tilt, and a 3D structure similar to a cylinder even appears. This occurrence may be caused by the structure having another longer operating period, similar to a precession phenomenon. In the NES group, the relationship curve of q₁ and q₂ clearly forms a parallelogram region, indicating that there is some form of linear coupling in the energy boundary of the structure. The original energy boundary with a fixed value (the straight line on the vertical axis) becomes a linear combination of the two (general position straight line equation). Moreover, it can be seen from the tilt direction that q₁ and q₂ have a phase difference close to 45° at this time, indicating that there is already a strong coupling effect within the system.

Phase Diagram Analysis under Swept-Sine Excitation (Excitation of the First Mode)

The phase diagrams of the uncontrolled group, the TMD group, and the NES group are generated as Figure 17. (The time range selected is the period when the frequency of swept-sine excitation is near the fundamental frequency, i.e., 2350 s to 2650 s).

First, a comparison with the standard Lissajous figures shows that the phase diagrams of the motion of the midpoint of the cables in the uncontrolled group are close to circular, but traces of circles can be observed. Based on the time–history curves and physical meaning, harmonic motion occurs with increasing amplitude, and external energy is continuously input into the system. Observation of the phase diagrams of the motion of the midpoint of the cables in the TMD group and the NES group reveals that they also undergo harmonic motion with continuously changing amplitudes, but the circles overlap at this time, which indicates that the vibration energy does not always increase, suggesting that the damper has a significant suppressing effect on the vibration energy absorption and possibly that different branch responses are excited in the process of increasing energy.

Observing the relationship curve between q₁ and q₂, in the uncontrolled group, there are many similar squares of different sizes in the relationship curve, indicating that the vibration energy gradually increases in stages, and the development direction of the circles is basically along the diagonal direction of the square, from which it is seen that q₁ and q₂ have two phase differences: 45° and 135°. When the TMD is installed, the phase diagram of the structure exhibits a clear envelope. A comparison with the Lissajous figure shows that this envelope is the Lissajous figure in the case of 1:1 resonance and a phase difference of 135°. Therefore, the TMD device changes the internal resonance mode of the structure and generates a more pronounced drag effect. In comparison, when the NES is installed, the development of curves is clearly divided into several directions, indicating that the phase difference of the structure has experienced several significant changes. This is also related to the more prominent nonlinearity of the structure and the occurrence of significant energy exchange at this time.

Phase Diagram Analysis under Harmonic Excitation at the Third-Mode Frequency

The phase diagrams of the uncontrolled group, the TMD group, and the NES group are generated as Figure 18 (the time range chosen is four beat vibration periods, i.e., 4892.603 s to 5000 s).

Under the third-mode excitation, the phase diagram of the uncontrolled group is closer to the standard perfect circle, indicating that the vibration has fully approached harmonic motion and shows almost no nonlinear characteristics. At the same time, with increasing modal order, the vibration energy weakens and the amplitude decreases. At this time, the response characteristics of the structure installed with the TMD and NES are very similar, indicating that the damper has undergone a stiffening effect. More scrutiny reveals that, especially in the q₂ phase diagram, there are clear tangential relationships between the small circles in the middle, with the tangent points being exactly one diameter apart. This is a beat vibration phenomenon. At the same time, energy absorption and release occur at each antiphase. This is due to the drag effect between the two cables and the stable phase difference caused by the damping of the damper, which can also be seen from the relationship curve between q₁ and q₂.

The relationship curves of q₁ and q₂ in the damper group are very similar, indicating that the vibration reduction mechanisms of the dampers are the same in this case. A comparison to the uncontrolled group shows that the rectangle has clearly rotated by an angle. This is because there are certain linear coupling forms in the energy boundary of the structure. The original energy boundary with its own fixed value (straight line perpendicular to the coordinate axis) has changed into a linear combination of the two (the equation of a straight line at a general position). Moreover, the inclination direction shows that q₁ and q₂ have a phase difference close to 45° or 135° at this time, indicating a strong coupling effect within the system. In addition, the vibration reduction mechanism is also related to the energy transfer caused by this coupling effect and the damping loss due to the energy transfer.

5.2.3. Spectral Characteristics

The spectra of the uncontrolled group, the TMD group, and the NES group are obtained as follows. The calculation time range is completely consistent with that of the phase diagram.

Spectral Analysis under Harmonic Excitation at the First-Mode Frequency

According to the spectra in Figure 19, the first-mode vibration is very effectively controlled after the installation of the damper, with the NES showing the best control effect. Nevertheless, under NES control, the main peak of the vibration frequency of the system shifts, and there are even two separate main peaks in the vibration of Cable 1, indicating that due to nonlinearity, the vibration frequency of the structure changes greatly, even leading to multibranch responses and causing different frequency components, resulting in multiple-frequency resonance. At the same time, a very interesting phenomenon occurs: when the NES is installed, the right resonance peak of Cable 1 coincides with the resonance peak of the uncontrolled group of Cable 2, while the resonance peak of Cable 2 is completely assimilated by Cable 1 and is equal to the main peak frequency of the uncontrolled group of Cable 1, indicating a strong coupling and drag effect between the two cables due to the presence of the NES damper and the simultaneous occurrence of energy exchange. These phenomena are obviously not observed in the structure with an installed TMD or in the uncontrolled group.

Spectral Analysis under Swept-Sine Excitation (Excitation of the First Mode)

Figure 20 shows that under swept-sine excitation, the normalized amplitude of the vibration of Cable 1 decreases significantly when the damper is installed; the greatest decrease occurs when the NES is installed, while the normalized amplitude of Cable 2 increases. However, as the normalized amplitude increases, the vibration bandwidth decreases, and the spectral integral decreases significantly. Since frequency-swept excitation is a continuous process of energy absorption and release, the actual amplitude is still effectively suppressed (as seen in the time–history curve). The observation of the side lobes of the spectrum shows that the side lobes of Cable 2 are the most symmetrical after NES installation, indicating that the vibration is the most stable, while the vibration of Cable 1 is the most stable when the TMD is installed. A comparison of the first-mode frequency of the structure reveals that the maximum normalized amplitude of Cable 1 occurs at the fundamental frequency only when the TMD is installed, while the maximum normalized amplitude of Cable 2 occurs at the fundamental frequency after installing each damper.

Spectral Analysis under Harmonic Excitation at the Third-Mode Frequency

Figure 21 shows that under harmonic excitation at the third-mode frequency, it can be clearly seen that the curves of the optimal TMD and NES almost overlap, which further confirms the results observed from the time–history curves and phase diagrams, indicating that at this time, the vibration reduction mechanisms of the best TMD and the best NES are almost identical.

In addition, at this time, the subpeak frequency of each cable is the main peak frequency of the other cables. This phenomenon is also observed under the first-mode excitation, suggesting a strong coupling and drag effect between the two cables due to the presence of a damper (both TMD and NES), along with energy exchange. Energy exchange eventually leads to a substantial reduction in the total energy of vibration.

6. Conclusions

In this study, the control performance and mechanical behavior of a TMD/NES interconnected between two sagged stay cables are analyzed. First, the influence of the damper mass on the cable shape is investigated. The governing equations of motions for the two adjacent cables are established using the Hamilton principle and solved by the method of separation of variables. The TMD and NES are optimized when two cables are excited by a harmonic or swept-sine load targeting at the first or third mode. Finally, the control performance (robustness and dynamic response characteristics) of the optimal dampers is analyzed. The following conclusions are drawn:

(1)

The damper mass has a non-negligible impact on the static cable shape.

(2)

It is shown that TMDs and NESs with appropriate damping and stiffness parameters can achieve satisfactory control performance by substantially reducing the vibration of two adjacent cables under a harmonic or swept-sine excitation targeting at the first or third mod. In particular, the NES performs slightly better under harmonic excitation of the first-mode frequency, and the TMD performs slightly better under swept-sine excitation targeting at the first mode, and under harmonic excitation of the third-mode frequency.

(3)

An NES exhibits better robustness to the variations in the amplitude of the external harmonic excitation of the first-mode frequency, while a TMD exhibits better robustness to the variations in the amplitude of swept-sine excitation targeting at the first mode. Both dampers exhibit good robustness to variations in the amplitude of the external harmonic excitation of the third-mode frequency. An NES shows better frequency robustness.

(4)

The steady-state time histories of the two adjacent cables with an optimal TMD are typically periodic responses of constant amplitude under a harmonic excitation of the first mode frequency. The phase diagram of the cables interconnected with an NES is more complicated than the cases of TMD and no damper. Additionally, the spectra show that the NES has an obvious frequency coupling capability and can shift the vibration frequency from one cable to another. Under swept-sine excitations, both the TMD and NES can effectively suppress energy input, and the TMD can form a stable phase difference region. It can be seen from the frequency domain results that the area enclosed by the spectral curve is substantially reduced, indicating a decrease in total vibration energy. Under harmonic excitations of the third-mode frequency, the two dampers have similar effects in that both can effectively suppress vibration, produce almost the same drag effect on the phase difference, and generate a vibration frequency shift between the two cables.

(5)

From dynamic characteristic analyses, some nonlinear dynamic behaviors of the system, such as multiple-frequency resonance, multiple-branch response, and different frequency components, can be identified. Coupling effects and drag motion phenomena, such as energy transfer (especially when the phase is opposite), phase diagram boundary rotation, frequency shift, and phase difference, can also be observed.