A Comparative Study of Inertial Mass Dampers and Negative Stiffness Dampers for the Multi-Mode Vibration Control of Stay Cables

Abstract

Previous studies have demonstrated that two representative passive control devices, including inertial mass dampers (IMDs) and negative stiffness dampers (NSDs), exhibit superior control performance in single-mode vibration control of stay cables. However, observations in recent years have increasingly reported rain–wind-induced multi-mode vibrations of stay cables on actual bridges. Therefore, it is of considerable significance to investigate the control effectiveness of the two representative passive dampers in mitigating multi-mode cable vibrations. For this reason, this study presents a comparative study of the IMD and NSD for the multi-mode vibration control of stay cables. The mechanical models of typical IMDs and NSDs are first introduced, followed by the numerical modeling of the two cable-damper systems using the finite difference method. Subsequently, the effectiveness of three multi-mode optimization strategies is comprehensively assessed, and the most effective strategy is selected for the optimal design of the IMD and NSD. Finally, the effectiveness of the control of the IMD and NSD in suppressing harmonic, white noise and wind-induced multi-mode vibrations of a 493.72 (m) long ultra-long cable is systematically evaluated. The numerical results indicate that the NSD significantly improves the cable damping ratios for multiple vibration modes as its negative stiffness coefficient increases, while IMD performs well only within a small inertia coefficient. Moreover, the NSD outperforms the IMD in suppressing multi-mode cable vibrations induced by harmonic, white noise and wind excitations.

1. Introduction

Stay cables, as essential load-carrying components in cable-stayed bridges, are highly susceptible to excessive vibrations caused by environmental factors such as wind or combined wind–rain effects. This vulnerability stems from their inherent characteristics, including low damping and high flexibility [1,2,3]. Such adverse vibrations not only pose a risk of immediate structural damage but can also lead to fatigue failure at the cable anchorages over time. These issues significantly compromise the durability and lifespan of the cables, while also undermining public confidence in the safety and reliability of the bridge structure [4,5,6]. In response to these challenges, a wide range of vibration control measures have been developed and implemented. According to the working mechanism, these control measures can be roughly grouped into the following categories: active control systems, which rely on external energy input to counteract vibrations [7,8]; semi-active control systems, which combine the high efficiency of active control with the cost-effectiveness of passive methods [9,10]; and passive control systems, which utilize energy-dissipating devices without external power, such as magnetorheological damper [11], dynamic vibration absorber [12], and tuned mass damper (TMD) [13,14,15]. Each approach has its unique advantages and limitations, making the selection of an appropriate control strategy critical for ensuring the long-term performance and safety of cable-stayed bridges.

Among these control measures, transversely attached passive viscous dampers (VDs) are one of the most commonly used. In this regard, the effectiveness of the control of VDs in suppressing cable vibrations has been extensively researched. Some previous studies have analyzed the available damping ratio of a cable with length L, provided by a VD at a specific installation position $x_{\mathrm{d}}$, which is approximately $x_{\mathrm{d}}$/2L [16,17,18]. In addition, the effects of various non-ideal conditions on the dynamic characteristics of the cable–VD system have been investigated, such as internal stiffness and support stiffness [19,20,21], damper nonlinearity [22,23,24], cable inclination [25,26], and sag and flexural rigidity [27,28]. To design VDs for multi-mode cable vibration mitigation, Wang et al. [29] developed a novel approach for optimizing VD parameters to mitigate these adverse vibrations. Weber et al. [30] proposed a practical multi-mode design procedure for the optimal design of VDs. Yang et al. [31] presented a novel control strategy to improve the multi-mode cable vibration mitigation performance of two optimized VDs installed at the lower cable anchorage. The literature review indicates that adopting appropriate control strategies is conducive to improving the cable vibration control performance of VDs. However, the effectiveness of VDs is constrained by installation height, which results in a limited damping ratio provided to the cable, making it difficult to satisfy the multi-mode vibration mitigation requirements of ultra-long cables with lengths over 450 (m) [32,33,34].

Previous investigations have indicated that the enhanced control performance of active and semi-active systems is attributed to the negative stiffness behavior of their control dampers, which allows these control devices to undergo larger displacements and dissipate more energy [35,36,37]. However, the successful on-site implementation of active and semi-active systems relies heavily on a complex control scheme and a dependable power source. Compared to active and semi-active systems, passive dampers are favored in practical applications for their simplicity and reliability [38,39,40]. In this aspect, lots of efforts have been made to develop practical passive control devices incorporating negative stiffness mechanisms. In particular, two representative passive dampers, namely inertial mass dampers (IMDs) and negative stiffness dampers (NSDs), offer a promising solution for addressing the challenges of vibration control in stay cables subjected to wind or wind–rain excitations. The viscous inertial mass dampers (VIMDs) [41,42,43] and electromagnetic inertial mass dampers (EIMDs) [44,45,46] are common types of IMDs, whose hysteresis curves of inertial force and displacement exhibit equivalent negative stiffness characteristics. Theoretical and experimental studies have indicated that they provide superior additional damping to stay cables compared to conventional VDs. The pre-stressed spring NSDs [47,48] and magnetic NSDs [49,50,51] are typical NSDs capable of generating true negative stiffness, which has been demonstrated theoretically and experimentally to offer larger damping to stay cables than conventional VDs. Although some papers have investigated the cable vibration control performance of IMD and NSD, most studies are limited to single-mode cable vibration mitigation, with limited attention paid to multi-mode cable vibration mitigation. Specifically, Chen et al. [52] conducted a unified analysis of NSD and inerter-based dampers for multi-mode cable vibration control. Nevertheless, the IMD is the most typical inerter-based damper; no comparison is made between the IMD and NSD.

Given the above facts, this study conducts a comprehensive comparative analysis of IMDs and NSDs for multi-mode vibration control of stay cables. The primary objective is to evaluate their control performance in mitigating vibrations across multiple modes, which is critical for ensuring the safety and longevity of cable-stayed bridges. The remaining paper is organized as follows: in Section 2, the mechanical models of typical IMD and NSD are first introduced, and the numerical models of the two cable-damper systems are established utilizing the finite difference method; the effectiveness of three multi-mode optimization strategies is comprehensively assessed in Section 3, and the most effective strategy is selected for the optimal design of the IMD and NSD; Section 4 evaluates the performance of the IMD and NSD in suppressing harmonic, white noise and wind-induced multi-mode vibrations of a 493.72 (m) long ultra-long cable; and finally, key conclusions are drawn in Section 5.

2. Dynamic Formulation of a Stay Cable Equipped with an IMD or a NSD

2.1. Mechanical Models of Typical IMD and NSD

Figure 1 shows the mechanical models of two representative passive dampers, i.e., the IMD and NSD. As shown in Figure 1a, the IMD comprises an inerter element and a damping element in parallel, and its axial force $F_{\mathrm{d}1}$ can be expressed as follows:

$$F_{\mathrm{d}1}=b_{\mathrm{d}}\ddot{u}+c_{\mathrm{d}}\dot{u}$$

(1)

where $b_{\mathrm{d}}$ and $c_{\mathrm{d}}$ denote the inertia coefficient and damping coefficient of the IMD, respectively.

It can be found from Figure 1b that the NSD consists of a negative stiffness element and a damping element in parallel, with its axial force $F_{\mathrm{d}2}$ mathematically expressed as follows:

$$F_{\mathrm{d}2}=k_{\mathrm{d}}u+c_{\mathrm{d}}\dot{u}$$

(2)

in which $k_{\mathrm{d}}$ represents the negative stiffness coefficient of the NSD.

Notably, the IMD contains an inerter element that generates an acceleration-dependent force, while the NSD incorporates a negative stiffness element capable of generating a displacement-dependent force. These differences result in distinct mechanical behaviors of the IMD and NSD, namely equivalent negative stiffness for the IMD with a frequency of $f$ (i.e., $k_{\mathrm{e}\mathrm{q}}=-b_{\mathrm{d}}{\left(2\pi f\right)}^2$) and real negative stiffness for the NSD.

In recent years, various practical prototypes of typical IMDs [7,42,53] and NSDs [47,50,51] have been developed for cable vibration control. Given the differences in mechanical performance between the IMD and NSD and their potential applications in multi-mode cable vibration mitigation, there is an urgent need for a comparative study on the control effectiveness of these two representative passive dampers in suppressing multi-mode cable vibrations.

2.2. Numerical Modeling of Two Cable-Damper Systems

As illustrated in Figure 2, an external damper is transversely equipped to an inclined cable at a distance $x_{\mathrm{d}}$ from the lower cable anchorage. The assumptions are as follows [28]: (1) the cable static profile follows a second-order parabola; (2) the cable vibrations are confined to the $x$-$u$ plane, with negligible motion along the $x$-direction; (3) the sag-to-span ratio is small enough; (4) the cable maintains a consistent cross-sectional area throughout its length; and (5) the nonlinear characteristics of the cable, including large deformations, material nonlinearities, and temperature-induced nonlinearity, are neglected. On this basis, the equation governing the motion of the cable-damper system can be formulated as [42,50]

$$EI{\displaystyle \frac{{\partial }^{4}u\left(x,t\right)}{\partial x^4}}-T{\displaystyle \frac{{\partial }^{2}u\left(x,t\right)}{\partial x^2}}+{\displaystyle \frac{{\lambda }^{2}T}{l^3}}{\int }_0^{l}u\left(x,t\right)dx+c{\displaystyle \frac{\partial u\left(x,t\right)}{\partial t}}+m{\displaystyle \frac{{\partial }^{2}u\left(x,t\right)}{\partial t^2}}+F_{\mathrm{d}}\delta \left(x-x_d\right)=f_{\mathrm{e}\mathrm{x}\mathrm{t}}(x,t)$$

(3)

where $EI$, $T$, and $l$ denote the flexural rigidity, the chord tension force, and the length of the cable, respectively; $u(x,t)$ and $f_{\mathrm{e}\mathrm{x}\mathrm{t}}(x,t)$ represent the transverse cable displacement and external load applied on the cable, respectively; $c$ is the inherent damping of the cable; $m$ represents the mass per unit length of the cable; $F_{\mathrm{d}}$ denotes the axial force of the external damper, which can be expressed as Equation (1) for the IMD and Equation (2) for the NSD, respectively; $\delta \left(·\right)$ is the Dirac delta function.

In Equation (3), the sag parameter ${\lambda }^2$ is represented as [50]:

$${\lambda }^2={\left({\displaystyle \frac{mgl\cos \theta }{T}}\right)}^2{\displaystyle \frac{EAl}{T{L}_{\mathrm{e}}}}$$

(4)

where $g$ denotes the gravitational acceleration, which is 9.8 m/s²; $\theta$ is the cable inclination angle; and $L_{\mathrm{e}}=l\left[1+{\left(mgl\cos \theta \right)}^2/\left(8T^2\right)\right]$ represents the static length of the cable under gravity.

Equation (3) can be discretized utilizing the finite difference method provided by Mehrabi and Tabatabai [54], whose effectiveness has been validated in Refs. [11,41,42,50]. Notably, the boundary condition for fixed ends is taken into account, which can be mathematically expressed as

$$\left\{\begin{array}{c}u\left(0,t\right)=u\left(l,t\right)=0\\ {\displaystyle \frac{\partial u\left(0,t\right)}{\partial x}}={\displaystyle \frac{\partial u\left(l,t\right)}{\partial x}}=0\end{array}\right.$$

(5)

It is assumed that the cable is discretized into N + 1 equal segments with N internal nodes ($N$ = 199 in the present study). Accordingly, Equation (3) can be formulated in matrix form:

$$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{f}-\mathsf{\gamma }{F}_{\mathrm{d}}$$

(6)

in which $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ represent the mass matrix, damping matrix, and stiffness matrix of the cable, respectively, with the corresponding mathematical expressions given as follows:

$$\mathbf{M}=ma{\mathbf{I}}_N$$

(7)

$$\mathbf{C}=c{\mathbf{I}}_N$$

(8)

$$\mathbf{K}={\displaystyle \frac{{\lambda }^{2}Ta}{l^3}}{\mathbf{B}}_{N\times N}+{\displaystyle \frac{T}{a}}\epsilon {\chi }_{N\times N}$$

(9)

where ${\mathbf{I}}_N$ is a N-dimensional identity matrix; ${\mathit{B}}_{N\times N}$ represents a full unit matrix; $\epsilon$ = $EI{T}^{-1}{l}^{-2}$ denotes the dimensional flexural rigidity; $a$ = $l$/$N$ is the segment length; and

$${\mathit{\chi }}_{N\times N}=\left[\begin{array}{cccccc}Q& U& W& & & 0\\ & S& U& W& & \\ & & S& U& \ddots & \\ & & & \ddots & \ddots & W\\ & \mathrm{s}\mathrm{y}\mathrm{m}& & & S& U\\ & & & & & Q\end{array}\right]$$

(10)

where

$$\mathrm{Q}=7+{\displaystyle \frac{2}{\epsilon }}$$

(11)

$$S=6+{\displaystyle \frac{2}{\epsilon }}$$

(12)

$$U=-4-{\displaystyle \frac{1}{\epsilon }}$$

(13)

$$W=1$$

(14)

in which $\mathbf{u}$, $\mathbf{f}$, and $\mathsf{\gamma }$ represent the displacement, external load, and damper location vector, respectively, with the corresponding mathematical expressions given as follows:

$$\mathbf{u}=\left[\begin{array}{ccccc}{u}_1& u_2& u_3& \cdots & u_N\end{array}\right]$$

(15)

$$\mathbf{f}=\left[\begin{array}{ccccc}{f}_1& f_2& f_3& \cdots & f_N\end{array}\right]$$

(16)

$$\mathsf{\gamma }=\left[\begin{array}{ccccc}{\gamma }_1& {\gamma }_2& {\gamma }_3& \cdots & {\gamma }_N\end{array}\right]$$

(17)

where ${\gamma }_i$ is determined by the damper location, with its installation at the jth node given by

$${\gamma }_i=\left\{\begin{array}{c}\begin{array}{cc}0& i\ne j\end{array}\\ \begin{array}{cc}1& i=j\end{array}\end{array}\right.$$

(18)

Considering the damper as an intrinsic element of the cable-damper system, Equation (6) can be rewritten as follows:

For a cable–IMD system,

$${\mathbf{M}}_{\mathrm{d}}\ddot{\mathbf{u}}+{\mathbf{C}}_{\mathrm{d}}\dot{\mathbf{u}}+\mathbf{K}\mathit{u}=\mathbf{f}$$

(19)

in which ${\mathbf{M}}_{\mathrm{d}}$ = M + $\mathit{\gamma }{\mathit{b}}_{\mathrm{d}}{\mathit{\gamma }}^{\mathrm{T}}$ and ${\mathbf{C}}_{\mathrm{d}}$ = C + $\mathit{\gamma }{\mathit{c}}_{\mathrm{d}}{\mathit{\gamma }}^{\mathrm{T}}$, respectively.

For a cable–NSD system,

$$\mathbf{M}\ddot{\mathbf{u}}+{\mathbf{C}}_{\mathrm{d}}\dot{\mathbf{u}}+{\mathbf{K}}_{\mathrm{d}}\mathit{u}=\mathbf{f}$$

(20)

where ${\mathbf{C}}_{\mathrm{d}}$ = C + $\mathit{\gamma }{\mathit{c}}_{\mathrm{d}}{\mathit{\gamma }}^{\mathrm{T}}$ and ${\mathbf{K}}_{\mathrm{d}}$ = K + $\mathit{\gamma }{\mathit{k}}_{\mathrm{d}}{\mathit{\gamma }}^{\mathrm{T}}$.

An equivalent state-space form of Equations (19) and (20) can be represented as

$$\dot{\mathbf{Z}}=\mathbf{A}\mathbf{Z}+\mathbf{B}\mathbf{f}$$

(21)

in which Z, A, and B denote state vector, system matrix, and input matrix, respectively, with their mathematical expressions provided as follows

$$\mathbf{Z}=\left[\begin{array}{c}\mathbf{u}\\ \dot{\mathbf{u}}\end{array}\right]$$

(22)

For a cable–IMD system,

$${\mathbf{A}}_1=\left[\begin{array}{cc}{\mathbf{0}}_{N\times N}& {\mathbf{I}}_N\\ -{\mathbf{M}}_{\mathrm{d}}^{-1}\mathbf{K}& -{\mathbf{M}}_{\mathrm{d}}^{-1}{\mathbf{C}}_{\mathrm{d}}\end{array}\right]$$

(23)

$${\mathbf{B}}_1=\left[\begin{array}{c}{\mathbf{0}}_{N\times 1}\\ {\mathbf{M}}_{\mathrm{d}}^{-1}\end{array}\right]$$

(24)

For a cable–NSD system,

$${\mathbf{A}}_2=\left[\begin{array}{cc}{\mathbf{0}}_{N\times N}& {\mathbf{I}}_N\\ -{\mathbf{M}}^{-1}{\mathbf{K}}_{\mathrm{d}}& -{\mathbf{M}}^{-1}{\mathbf{C}}_{\mathrm{d}}\end{array}\right]$$

(25)

$${\mathbf{B}}_2=\left[\begin{array}{c}{\mathbf{0}}_{N\times 1}\\ {\mathbf{M}}^{-1}\end{array}\right]$$

(26)

Based on Equation (21), the dynamic characteristics of the cable-damper system, including frequencies, damping ratios, and mode shapes, can be obtained using complex eigenvalue analysis [42]. Notably, the numerical model employed in the present study was developed using the finite difference method. However, the finite difference method is not the only viable approach, and alternative methods such as the Galerkin method [55,56] and the finite element method [57] may also be applicable.

3. Design IMD and NSD for Controlling Multi-Mode Cable Vibrations

In Section 2, the mechanical models of typical IMD and NSD are introduced in detail, and the numerical models of the two cable-dampers systems are established to obtain their dynamic characteristics. In this section, the effectiveness of three multi-mode optimization strategies is comprehensively assessed, and the most effective strategy is selected for the optimal design of the NSD and IMD. On this basis, the design parameters of IMD and NSD are then determined to suppress the multi-mode vibrations of a 493.72 (m) long ultra-long cable.

3.1. Effectiveness of Multi-Mode Optimization Strategies

In this subsection, the effectiveness of three multi-mode optimization strategies is first assessed. Prior to this, a detailed description of the optimization process is first presented in Figure 3.

Notably, the optimization problem can be regarded as determining the optimal value $c_{\mathrm{d}}$ (namely $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$) under given constraints of $x_{\mathrm{d}}$, $b_{\mathrm{d}}$ for IMD or $k_{\mathrm{d}}$ for NSD, and ${\xi }_n\ge {\xi }_{n,t}$ to maximize the performance index $J_i$, with the corresponding mathematical formulation provided as follows:

$$\underset{c_{\mathrm{d}}}{\mathrm{maximize}}{J}_i$$

(27)

subjected to given values of $x_{\mathrm{d}}$, $b_{\mathrm{d}}$ for IMD or $k_{\mathrm{d}}$ for NSD, and ${\xi }_n$

$$0.005l{\le x}_{\mathrm{d}}\le 0.05l,-{\displaystyle \frac{T}{x_{\mathrm{d}}}}<k_{\mathrm{d}}\le 0 \mathrm{or} b_{\mathrm{d}}\ge 0,{\xi }_n\ge {\xi }_{n,t}$$

(28)

in which $J_i$ is the performance index obtained using the ith optimization strategy; ${\xi }_{n,t}$ denotes the target modal damping ratio.

To address the aforementioned optimization problem, the particle swarm optimization (PSO) algorithm in MATLAB (version R2022a) is utilized in the present study.

Subsequently, to achieve the optimal design of IMD or NSD for mitigating cable multi-mode vibrations, the effectiveness of three multi-mode optimization strategies is comprehensively assessed. The three optimization strategies are described in detail below.

Strategy 1: The optimization of the minimum cable damping ratio in the first mode, with the constraint that the cable damping ratios are provided by the conventional VD in the designed cable modes, is considered [57]. By employing this strategy, the performance index $J_1$ in the designed cable modes can be defined as

$$J_1=\min {\xi }_1$$

(29)

$${\xi }_1>\min {\xi }_n^{\mathrm{V}\mathrm{D}}$$

(30)

in which $n$ is the designed cable modes; ${\xi }_n^{\mathrm{V}\mathrm{D}}$ denotes the minimum cable damping ratio provided by VD in the designed cable modes.

Strategy 2: The optimization of the minimum cable damping ratio provided by the equipped damper in the designed cable modes [58] and the mathematical expression of the performance index $J_2$ is provided as follows:

$$J_2=\min \left(\begin{array}{ccccc}{\xi }_1& {\xi }_2& {\xi }_3& \cdots & {\xi }_n\end{array}\right)$$

(31)

Strategy 3: The average value $\mu \left(c_{\mathrm{d}}\right)$ of cable damping ratios provided by an equipped damper in the designed cable modes is maximized, and in the meantime, the standard deviation $\sigma \left(c_{\mathrm{d}}\right)$ is minimized. $\mu \left(c_{\mathrm{d}}\right)-\sigma \left(c_{\mathrm{d}}\right)$ is optimized to assess the control effectiveness of the equipped damper, with the performance index $J_3$ mathematically expressed as follows [50,59]:

$$J_3=\left(\mu \left(c_{\mathrm{d}}\right)-\sigma \left(c_{\mathrm{d}}\right)\right)$$

(32)

where

$$\mu \left(c_{\mathrm{d}}\right)={\displaystyle \frac{1}{\kappa }}\sum _{n=1}^{\kappa }{\xi }_n\left(c_{\mathrm{d}}\right)$$

(33)

$$\mathsf{\sigma }\left(c_{\mathrm{d}}\right)=\sqrt{{\displaystyle \frac{\sum _{n=1}^{\kappa }{\left({\xi }_n\left(c_{\mathrm{d}}\right)-\mu \left(c_{\mathrm{d}}\right)\right)}^2}{\kappa }}}$$

(34)

The frequencies at which stay cables experienced rain-wind-induced vibrations (RWIVs) are generally below 3 Hz [60], covering multiple vibration modes. A case study is performed on the No. A30U stay cable of the Sutong Bridge (see Figure 4), with its main properties shown in

Table 1. Primary properties of the No. A30U stay cable of the Sutong Bridge.

Parameter	Value
Cable length ($l$)	493.72 (m)
Mass per unit length ($m$)	78.50 (kg/m)
Diameter ($D$)	0.142 (m)
Tension force ($T$)	5347 (kN)
Flexural rigidity ($EI$)	3.460 × 105 (N·m2)
Inclination angle ($\theta$)	25.84 (°)
Sag parameter (${\lambda }^2$)	1.30

. Numerical simulations of the ultra-long stay cable indicate that the first eleven vibration modes must be considered to effectively suppress the cable RWIVs. In the numerical simulations, it is assumed that the IMD or NSD is installed at the location $x_{\mathrm{d}}$ = 2%$l$ from the lower anchorage of the ultra-long cable.

To facilitate designing the IMD and NSD to mitigate the multi-mode vibrations of the ultra-long cable, defining the following dimensionless parameters:

$${\overline{b}}_{\mathrm{d}}={\displaystyle \frac{b_d}{ml}}$$

(35)

$${\overline{k}}_{\mathrm{d}}={\displaystyle \frac{k_{\mathrm{d}}{x}_{\mathrm{d}}}{T}}$$

(36)

where ${\overline{b}}_{\mathrm{d}}$ and ${\overline{k}}_{\mathrm{d}}$ are the dimensionless inertia and negative stiffness coefficients, respectively.

Notably, the dimensionless inertia coefficient ${\overline{b}}_{\mathrm{d}}$ of the IMD is assumed to be in the range of 0 to 0.25. This is because the cable possesses a length of 493.72 (m) and a mass per unit length of 78.50 (kg), with the corresponding ${\overline{b}}_{\mathrm{d}}$ of the IMD ranging from 0 to 9.69 (tons), a value that is relatively easy to achieve in practice. In addition, the dimensionless negative stiffness coefficient ${\overline{k}}_{\mathrm{d}}$ of the NSD ranges from −0.9 to 0, preventing stability issues in the cable–NSD system as ${\overline{k}}_{\mathrm{d}}$ approaches −1.

Figure 5 compares the performance index $J$ and optimal damping coefficient $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$ of the IMD obtained using the three optimization strategies with respect to its dimensionless inertia coefficient ${\overline{b}}_{\mathrm{d}}$. As shown in Figure 5a, $J$ initially increases and then decreases as ${\overline{b}}_{\mathrm{d}}$ increases for the three optimization strategies. It can be observed from Figure 5b that $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$ initially decreases and then increases as ${\overline{b}}_{\mathrm{d}}$ increases for Strategies 2 and 3. In contrast, Strategy 1 leads to a continuously decreasing value of $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$.

Table 2. Multi-mode optimization parameters of the IMD obtained using the three optimization strategies.

Strategy	Optimal Inertia Coefficient ${\overline{\mathit{b}}}_{\mathbf{d}}$	Optimal Damping Coefficient ${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$ (kN·s/m)	Average Value of Cable Damping Ratios (%)	Average Value of Cable Damping Ratios (%)
1	0.19	290.42	0.48	0.19
2	0.04	116.26	0.80	0.54
3	0.06	60.22	1.14	0.30

shows the multi-mode optimization parameters of the IMD obtained using the three optimization strategies. Notably, Strategy 3 achieves the maximum average cable damping ratio, followed by Strategy 2, while Strategy 1 performs the worst. Although the minimum cable damping ratio achieved by Strategy 3 is lower than that achieved by Strategy 2, Strategy 3 requires a smaller damping coefficient for the IMD than Strategy 1. Thus, Strategy 3 is preferred for designing IMD to suppress multi-mode cable vibrations.

Figure 6 compares the performance index $J$ and optimal damping coefficient $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$ of the NSD obtained using the three optimization strategies with respect to its dimensionless negative stiffness coefficient ${\overline{k}}_{\mathrm{d}}$. As illustrated in Figure 6a, $J$ steadily increases as the increasing ${\overline{k}}_{\mathrm{d}}$ for the three optimization strategies. It can be found from Figure 6b that $c_{\mathrm{d}\_\mathrm{o}\mathrm{p}\mathrm{t}}$ gradually decreases with the increasing ${\overline{k}}_{\mathrm{d}}$ for the three optimization strategies.

Table 3. Multi-mode optimization parameters of the NSD obtained using the three optimization strategies.

Strategy	Negative Stiffness Coefficient ${\overline{\mathit{k}}}_{\mathbf{d}}$	Optimal Damping Coefficient ${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$ (kN·s/m)	Average Value of Cable Damping Ratios (%)	Average Value of Cable Damping Ratios (%)
1	−0.4	210.45	0.67	0.29
	−0.6	150.00	0.95	0.41
	−0.8	90.10	1.63	0.70
2	−0.4	73.23	1.20	0.84
	−0.6	53.22	1.71	1.20
	−0.8	34.04	2.96	2.13
3	−0.4	47.13	1.36	0.57
	−0.6	35.20	1.94	0.84
	−0.8	25.30	3.34	1.66

summarizes the multi-mode optimization parameters of the NSD obtained using the three optimization strategies. Similarly to the IMD, a comprehensive comparison of the average and minimum cable damping ratios demonstrates that Strategy 3 outperforms Strategies 1 and 2, as it requires the least damping coefficient for the NSD to achieve relatively optimal performance in mitigating multi-mode cable vibrations. Therefore, Strategy 3 is recommended for the IMD or NSD to achieve better control effectiveness in suppressing multi-mode cable vibrations compared to Strategies 1 and 2. Notably, the NSD significantly improves the cable damping ratios for multiple vibration modes as its negative stiffness coefficient increases, while the IMD performs well only within a small inertia coefficient.

3.2. Design Parameters of IMD and NSD

The multi-mode optimization parameters of the IMD and NSD obtained using the most effective strategy, Strategy 3, are adopted as the final design parameters, as summarized in

Table 4. Design parameters of the IMD and NSD for suppressing multi-mode cable vibrations.

Mode Order $\mathit{n}$	Damper Type	${\mathit{x}}_{\mathbf{d}}$	${\overline{\mathit{b}}}_{\mathbf{d}}$	${\overline{\mathit{k}}}_{\mathbf{d}}$	${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$ (kN·s/m)	$\mathit{J}$ (%)
11	IMD	2%l	0.06	N/A	60.22	0.767
	NSD		N/A	−0.4	47.13	1.070
			N/A	−0.6	35.20	1.545
			N/A	−0.8	25.30	2.772

N/A: not applicable.

. The table shows that the NSD requires a smaller damping coefficient than the IMD, and its required damping coefficient decreases as the negative stiffness increases. Figure 7 compares the cable damping ratios provided by the designed IMD and NSD in the first eleven modes. As shown, the NSD offers higher modal damping ratios to the cable than the IMD, with the enhancement becoming more pronounced as the negative stiffness increases.

Taking into account the cost-effectiveness and manufacturability of the NSD prototype, the configuration with ${\overline{k}}_{\mathrm{d}}$ = $-$0.6 is chosen for the subsequent performance comparison between the optimized IMD and the NSD. To assess their control performance from the perspective of robustness, a sensitivity analysis of the performance index $J$ with respect to the damping coefficients of the IMD (${\overline{b}}_{\mathrm{d}}$ = 0.06) and NSD (${\overline{k}}_{\mathrm{d}}$ = $-$0.6) was performed, and the corresponding results are listed in

Table 5. Sensitivity analysis of the performance index $J$ with respect to the damping coefficients of the IMD and NSD in the first eleven modes.

Mode Order $\mathit{n}$	Damper Type	${\mathit{x}}_{\mathbf{d}}$	${\overline{\mathit{b}}}_{\mathbf{d}}$	${\overline{\mathit{k}}}_{\mathbf{d}}$	${\mathit{c}}_{\mathbf{d}}$ (kN·s/m)	$\mathit{J}$ (%)
11	IMD	2%l	0.06	N/A	1.2${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	0.753
					1.1${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	0.762
					1.0${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	0.767
					0.9${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	0.761
					0.8${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	0.745
	NSD		N/A	−0.6	1.2${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	1.496
					1.1${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	1.531
					1.0${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	1.545
					0.9${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	1.528
					0.8${\mathit{c}}_{\mathbf{d}\_\mathbf{o}\mathbf{p}\mathbf{t}}$	1.473

N/A: not applicable.

. As can be seen, the performance index $J$ progressively decreases as the damping coefficient deviates from its optimal value. When the damping coefficient deviates from the optimal value by +20% and $-$20%, the performance index of the IMD decreases by 1.83% and 2.87%, respectively, relative to its maximum value, whereas that of the NSD decreases by 3.17% and 4.66%, respectively. In comparison, the NSD exhibits greater differences than the IMD, while the overall discrepancy between the two remains relatively small.

Figure 8 compares the sensitivity of the cable damping ratios with respect to the damping coefficients of the IMD and NSD in the first eleven modes. As shown, for modes 1 to 4, deviations in the damping coefficient from its optimal value have little effect on the cable damping ratio. By contrast, for modes 5 to 11, the impact is considerably greater. This indicates that the cable damping ratios of low-order modes are less sensitive to deviations of the IMD or NSD damping coefficient from its optimal value, whereas those of high-order modes exhibit greater sensitivity. Overall, the sensitivities of the multi-mode cable damping with respect to the damping coefficients of the IMD and NSD are comparable, with no significant differences observed.

4. Performance Comparison Between IMD and NSD

In Section 3, the optimal design parameters for the IMD and NSD are determined to mitigate multi-mode vibrations of a 493.72 (m) long ultra-long cable. On this basis, this section systematically evaluates the control effectiveness of the designed IMD and NSD in mitigating harmonic, white noise and wind-induced multi-mode vibrations of the ultra-long cable.

4.1. Harmonic Excitation

To evaluate the control effectiveness of the designed IMD and NSD, the ultra-long cable is first subjected to sinusoidal excitation, and Figure 9 shows the corresponding results. As shown, the mid-span cable vibrates predominantly in modes 1, 3, 5, 7, 9, and 11, while the one-tenth-span cable vibrates predominantly in modes 1 to 9 and 11. Overall, both the IMD and NSD can effectively reduce the cable dynamic responses. However, the NSD outperforms the IMD in reducing cable dynamic responses, highlighting its superior control effectiveness in suppressing multi-mode vibrations of stay cables subjected to harmonic excitation.

4.2. White Noise Excitation

Subsequently, white noise excitation is applied to the ultra-long cable. Figure 10 shows the displacement responses of the ultra-long stay cable subjected to the white noise excitation. As shown in Figure 10a, the displacement response of the cable equipped with the IMD or NSD is significantly less than that of the cable without control. The displacement response of the cable equipped with the NSD is dramatically reduced compared to that with the IMD. To quantitatively compare the cable displacement responses at multiple positions, the root-mean-square (RMS) of the cable displacements is calculated and compared in Figure 10b. For the cable equipped with the IMD, the RMS displacements of the cable at 0.02l (i.e., damper location) 0.25l, 0.5l, and 0.75l are 0.28 mm, 3.20 mm, 4.56 mm, and 3.20 mm, respectively, while in the case of cable equipped with the NSD, they are 0.48 mm, 2.49 mm, 3.34 mm, and 2.30 mm, respectively. Compared to the cable equipped with the IMD, the RMS displacements of the cable equipped with the NSD at 0.25l, 0.5l, and 0.75l are reduced by 22.19%, 26.75%, and 28.13%, respectively. In addition, the NSD exhibits greater displacement than the IMD under white noise excitation.

4.3. Wind Excitation

Finally, wind excitation is applied to the ultra-long cable to further investigate the control effectiveness of the IMD and NSD. Figure 11 depicts the Fourier transform (FFT) spectrum of the wind-induced vibration responses of the uncontrolled cable with a one-tenth-span. As depicted, the wind-induced cable vibrations predominantly occur in modes 1 to 8. Figure 12 compares the displacement responses of the ultra-long cable under wind excitation. As illustrated in Figure 12a, both the IMD and NSD can reduce the displacement response of the ultra-long stay cable. Notably, the NSD shows better performance than the IMD in reducing the cable displacement response. Nevertheless, the reduction ratio varies in different time intervals, which can be attributed to the inherent randomness and complexity of wind loads. To further analyze the underlying reasons, the corresponding FFT spectrums during time intervals Ⅰ (750 s~1050 s) and Ⅱ (1150 s~1450 s) are compared in Figure 12b,c. During both time intervals, the IMD and NSD can effectively mitigate the first eight modal vibrations of the ultra-long cable. However, the IMD shows insufficient vibration mitigation performance in the first mode, whereas the NSD performs better. In comparison, the NSD is more applicable to mitigate wind-induced multi-mode cable vibrations.

To elucidate the underlying reasons for the superior control effectiveness of the NSD over the IMD in multi-mode cable vibration mitigation, Figure 13 compares the hysteresis loops of the IMD and NSD in suppressing wind-induced multi-mode cable vibrations. As shown, the output force and displacement of the NSD are significantly greater than those of the IMD. Consequently, the negative stiffness of the NSD significantly amplifies the displacement of its damping element compared with the inertia of the IMD, resulting in a considerably greater energy dissipation capacity for the NSD in comparison to the IMD when applied to suppress multi-mode cable vibrations. Notably, the integration of artificial intelligence (AI)- and deep learning (DL)-based algorithms can facilitate more effective data analysis [61,62,63], which represents a key direction for our future work.

5. Conclusions

In this paper, a comparative study is conducted to evaluate the control effectiveness of the IMDs and NSDs in suppressing multi-mode cable vibrations. The mechanical models of typical IMD and NSD are introduced in detail, and the numerical models of the two cable-damper systems are established utilizing the finite difference method. The effectiveness of three multi-mode optimization strategies is systematically evaluated, and the most effective strategy is chosen for the optimal design of the IMD and NSD. A performance comparison between the IMD and NSD is performed to assess their control effectiveness in suppressing harmonic, white noise, and wind-induced multi-mode vibrations of a 493.72 (m)-long ultra-long cable. According to the results, the key conclusions can be drawn:

(1)

Strategy 3 is recommended for the IMD or NSD to achieve better control effectiveness in suppressing multi-mode cable vibrations compared to strategies 1 and 2.

(2)

The NSD significantly improves the cable damping ratios for multiple vibration modes as its negative stiffness coefficient increases, while the IMD performs well only within a small inertia coefficient.

(3)

The NSD requires a smaller damping coefficient than the IMD to achieve superior control performance, and its required damping coefficient decreases as the negative stiffness increases.

(4)

The sensitivities of the multi-mode cable damping with respect to the damping coefficients of the IMD and NSD are comparable, with no significant differences observed.

(5)

The NSD outperforms the IMD in suppressing multi-mode cable vibrations induced by harmonic, white noise and wind excitations. The dynamic responses of the ultra-long cable equipped with the NSD are significantly reduced compared to those equipped with the IMD under the three types of external excitations.

(6)

The negative stiffness of the NSD significantly amplifies the displacement of its damping element compared to the inertia of the IMD, resulting in a considerably greater energy dissipation capacity for the NSD in comparison to the IMD when applied to suppress multi-mode cable vibrations.

In the present study, a relatively simple linear model is adopted to investigate the control effectiveness of IMD and NSD. In future studies, the influences of structural nonlinearities will be further investigated.