**1. Introduction**

With the flourishing development of materials and construction technologies, civil engineering structures are becoming larger, lighter, and more flexible, especially for long-span bridges. Cable-stayed is a common option for bridges in the medium to long-span ranges due to its unique structural formation, economic advantage, and esthetic value [1]. However, as important load-bearing components of cable-stayed bridges, stay cables are highly susceptible to dynamic excitations due to their high flexibility and low intrinsic damping [2,3]. Frequent and excessive amplitude cable vibrations may lead to fatigue failure of cables. These problems may inevitably shorten the service life and cause the risk of losing public confidence in cable-stayed bridges. To guarantee structural safety, several solutions have been proposed to dampen cable vibrations, which include modifying aerodynamic surface of cables [4], connecting cables together via cross tie [5], and attaching external dampers on cables [6–9].

Though these practical measures have been well applied in the field, each has its own shortcomings. Changing the surface of the cable is difficult to implement for retrofit and may increase drag forces at high wind velocities [10]. Cross-ties are incapable of direct energy dissipation and make the aesthetics

of cable-stayed bridges deteriorate [11]. Compared to the two methods above, attaching external dampers on the cable seems to be more promising. Nevertheless, the installation location of a passive viscous damper is typically restricted to within a few percentage points of the cable length from the cable anchorage [12]. As expected, passive viscous dampers cannot provide su fficient damping to eliminate vibrations for super-long cables, such as the Sutong Bridge, with cables nearly 600 m long. Moreover, the results based on both theoretical and experimental studies indicated that the existence of the cable sag [13,14], the cable flexural rigidity [15,16], the damper sti ffness [17], and the damper support sti ffness [18] or their coexistence [19–23] would have adverse impacts on the e fficiency of passive viscous dampers.

An active damper can produce a force-deformation relationship with the negative-sti ffness behavior that benefits damper e fficiency when the linear quadratic regulator (LQR) algorithm is employed [24,25]. However, active dampers often require high power sources beyond practical limits and are thus rarely used for cable vibration mitigation in real bridges. Alternatively, semiactive dampers, which can produce similar hysteresis and achieve control performance comparable to that of active dampers, were proposed [26–29]. For instance, the semiactive control based on magnetorheological dampers has been successfully applied on the Dongting Lake Bridge [30], Binzhou Bridge [31], and Sutong Bridge [32]. Compared to active dampers, semiactive dampers require less power. Nevertheless, possible implementations of semiactive dampers on site still require an external stable power supply, a sensing system, and a controller, which seems to be complicated and costly. This fact has inspired researchers to introduce a negative sti ffness mechanism into passive dampers to mitigate cable vibrations.

Recently, several representative passive dampers with negative sti ffness mechanisms, including pre-spring negative sti ffness dampers (pre-spring NSDs) [33,34] and magnetic negative sti ffness dampers (magnetic NSDs) [35,36], have been successfully developed. Negative sti ffness dampers have well demonstrated to be capable of providing superior damping over that of traditional passive viscous dampers [37–39]. However, extremely large passive negative sti ffness may make the NSD lose its stability. Alternatively, an inerter has the potential to provide similar negative sti ffness without a stability problem [40]. Many inerter-based absorber layouts have been proposed, and their control performance advantages have been proven for civil engineering structures [41–59]. As for the vibration suppression of cables, typical inertial mass dampers (IMDs) [60–65] and tuned inerter dampers [66,67] were well developed, and their significant improvement on the achievable modal damping ratio of the cable was verified via both theoretical and experimental investigations.

With the increasing cable length, it may be di fficult to attain a desired level of supplemental modal damping with a single damper or a pair of dampers installed near the deck anchorage. Hence, some hybrid techniques have been further proposed. The idea of combining external dampers with cross-ties for cable vibration control was considered, which not only addresses the deficiencies of these two commonly used countermeasures but also still retains their respective merits [68–73]. A hybrid damper system, combining a viscous damper and a tuned mass damper, can overcome the shortcomings of single type of dampers and improve e ffectiveness and robustness in suppressing cable vibration [74]. In addition, application of two viscous dampers or two high-damping rubber dampers at di fferent locations of a cable was proposed [75–77]. The results have shown that when two viscous dampers are installed at opposite ends of a cable, their damping e ffects are approximately the sum of the contributions from each damper [77]. However, when they are at the same cable end, the maximum modal damping ratio of the cable is determined by a single damper at the further distance, indicating no benefits over a single damper configuration [77].

Inspired by the potential advantages of attaching two external discrete viscous dampers (VDs) on a cable, this study aimed to evaluate the feasibility of a cable with two discrete IMDs, either on the opposite end or on the same end of the cable, to improve the vibration mitigation performance of the cable in each mode. Complex modal analysis based on the taut-string model was employed and extended to allow for the existence of two external discrete IMDs. The formulation for free vibration of a taut cable with two discrete IMDs was established, and corresponding complex wavenumber equations of free damped vibration were derived. The asymptotic and numerical solutions of the wavenumber equation were obtained, and the applicability of asymptotic solutions was then evaluated. Finally, parametric studies were performed to investigate the e ffects of damper positions and damper properties on the control performance of the cable with two discrete IMDs.

#### **2. Formulation of the Cable–IMD System**

A taut cable with two transversely attached inertial mass dampers is shown in Figure 1. The length, the mass per unit length, and the tension of the cable are *L*, *m*, and *T*, respectively. The coordinate system defines that the *x*-axis and the *v*-axis are along the cable chord and the transverse direction, respectively. *x*<sup>∗</sup> = *L* − *x* represents the coordinate from the right end of the cable. Two discrete IMDs are respectively installed at distances *x*1 and *x*2 from the left end of the cable (*<sup>x</sup>*2 ≥ *x*1). The distance between the right IMD and the right end is denoted as *x*<sup>∗</sup> 2 = *L* − *x*2. The damping coe fficient and inertial mass of the *jth* IMD are denoted as *cj* and *bj* (*j* = 1, 2), respectively. The equation of motion of the cable–IMD system is given by:

$$T\frac{\partial^2 \upsilon}{\partial \mathbf{x}^2} - m\frac{\partial^2 \upsilon}{\partial t^2} = \sum\_{j=1}^2 F\_{\text{IMD}j}(t)\delta(\mathbf{x} - \mathbf{x}\_j),\tag{1}$$

where *<sup>v</sup>*(*<sup>x</sup>*,*<sup>t</sup>*) is the cable transverse displacement and <sup>δ</sup>(·) is delta function to specify the location of the damping force *<sup>F</sup>*IMD*j* at *x* = *xj*.

**Figure 1.** The taut cable with two discrete inertial mass dampers.

Under free vibration, applying separation of variables, the cable transverse displacement and the IMD force can be respectively expressed as:

$$w(\mathbf{x},t) = \overline{v}(\mathbf{x})e^{i\omega t}, \mathbf{F}\_{\text{IMD}j}(t) = \overline{F}\_{\text{IMD}j}e^{i\omega t} \tag{2}$$

where *i* 2 = −1, ω is a complex natural frequency of the cable, and *v*(*x*) is the corresponding complex mode shape. To find *<sup>v</sup>*(*x*), the cable can be considered as a multispan structure connected at the damper locations [77]. Substituting Equation (2) into Equation (1), *v*(*x*) in each span needs to satisfy a homogeneous equation [77]:

$$\frac{d^2\overline{\overline{v}}}{dx^2} + \beta^2 \overline{\overline{v}} = 0\\ \begin{cases} 0 \le \mathbf{x} \le \mathbf{x}\_1\\ \mathbf{x}\_1 \le \mathbf{x} \le \mathbf{x}2\\ 0 \le \mathbf{x}^\* \le \mathbf{x}\_2^\* \end{cases},\tag{3}$$

where β = ω √ *m*/*T* refers to the wavenumber. *Appl. Sci.* **2019**, *9*, 3919

Applying boundary conditions at cable ends, i.e.,*v*(0) = *v*(*L*) = 0, and the transverse displacement compatibility conditions at damper locations, i.e., *<sup>v</sup>*(*<sup>x</sup>*1) = *<sup>v</sup>*1, *<sup>v</sup>*(*x*<sup>∗</sup>2) = *<sup>v</sup>*2, the general solution of Equation (3) can be further written in the form [77]:

$$
\widetilde{\boldsymbol{w}}(\mathbf{x}) = \begin{cases}
\overline{\boldsymbol{\upsilon}}\_1 \frac{\sin \beta \mathbf{x}}{\sin \beta \mathbf{x}\_1} & 0 \le \mathbf{x} \le \mathbf{x}\_1 \\
\overline{\boldsymbol{\upsilon}}\_1 \frac{\sin \beta (\mathbf{x}\_2 - \mathbf{x})}{\sin \beta (\mathbf{x}\_2 - \mathbf{x}\_1)} + \overline{\boldsymbol{\upsilon}}\_2 \frac{\sin \beta (\mathbf{x} - \mathbf{x}\_1)}{\sin \beta (\mathbf{x}\_2 - \mathbf{x}\_1)} & \mathbf{x}\_1 \le \mathbf{x} \le \mathbf{x} \mathbf{2}, \\
\overline{\boldsymbol{\upsilon}}\_2 \frac{\sin \beta \mathbf{x}^\*}{\sin \beta \mathbf{x}\_2^\*} & 0 \le \mathbf{x}^\* \le \mathbf{x}\_2^\*
\end{cases} \tag{4}
$$

where *vj* is the mode shape amplitude at the *jth* damper location. 

At damper locations, there is:

$$T(\frac{\mathrm{d}v}{\mathrm{d}x}\Big|\_{\overline{\gamma}}^{\cdot} - \frac{\mathrm{d}v}{\mathrm{d}x}\Big|\_{\overline{\gamma}}^{\cdot}) = \overline{F}\_{\mathrm{IMD}/}(t). \tag{5}$$

Substituting Equation (4) into Equation (5), it yields:

$$\begin{cases} \cot \beta \mathbf{x}\_1 + \cot (\mathbf{x}\_2 - \mathbf{x}\_1) - \frac{\widetilde{\mathbf{v}}\_2}{\widetilde{\mathbf{v}}\_1} \frac{1}{\sin \beta (\mathbf{x}\_2 - \mathbf{x}\_1)} = -\frac{\overline{F}\_{\text{IMD1}}/T}{\overline{\beta} \widetilde{\mathbf{v}}\_1} \\\ -\frac{\widetilde{\mathbf{v}}\_2}{\widetilde{\mathbf{v}}\_1} \frac{1}{\sin \beta (\mathbf{x}\_2 - \mathbf{x}\_1)} + \cot \beta (\mathbf{x}\_2 - \mathbf{x}\_1) + \cot \beta \mathbf{x}\_2^\* = -\frac{\widetilde{F}\_{\text{IMD2}}/T}{\beta \widetilde{\mathbf{v}}\_1} \end{cases} \tag{6}$$

Substituting *<sup>v</sup>*1/*v*2 from the second one into the first one of Equation (6) and rearranging, the characteristic equation of the wavenumber β is derived as:

$$\begin{array}{l} \left(\cot \beta \mathbf{x}\_1 + \frac{\overline{F}\_{\text{DMD}}/T}{\beta \overline{v}\_1}\right) \left(\cot \beta \mathbf{x}\_2^\* + \frac{\overline{F}\_{\text{DMD}}/T}{\beta \overline{v}\_2}\right) +\\ \left(\cot \beta \mathbf{x}\_1 + \frac{\overline{F}\_{\text{DMD}}/T}{\beta \overline{v}\_1} + \cot \beta \mathbf{x}\_2^\* + \frac{\overline{F}\_{\text{DMD}}/T}{\beta \overline{v}\_2}\right) \cot \beta (\mathbf{x}\_2 - \mathbf{x}\_1) = 1 \end{array} \tag{7}$$

#### **3. Two Opposite IMDs**

#### *3.1. The Wavenumber Equation*

The damper force of the *jth* IMD can be expressed as [45]:

$$F\_{\rm IMDj}(t) = b\_{\dot{\cdot}} \frac{\partial^2 v(\mathbf{x}\_{\dot{\prime}}, t)}{\partial t^2} + c\_{\dot{\cdot}} \frac{\partial v(\mathbf{x}\_{\dot{\prime}}, t)}{\partial t} \text{ or } \widetilde{F}\_{\rm IMDj} = -b\_{\dot{\cdot}} \omega^2 \widetilde{\boldsymbol{\nu}}\_{\dot{\cdot}} + c\_{\dot{\cdot}} \omega \widetilde{\boldsymbol{\nu}}\_{\dot{\cdot}}.\tag{8}$$

When two IMDs are installed at different ends of the cable, substituting Equation (8) into Equation (7) and using trigonometric relations, Equation (7) can be rearranged to the form relating to *x*1 and *x*∗2as:

$$\tan \beta L = \frac{A\_1 + iB\_1}{C\_1 + iD\_1} \,\text{'}\tag{9}$$

$$A\_1 = -\chi\_1 \sin^2 \beta \mathbf{x}\_1 - \chi\_2 \sin^2 \beta \mathbf{x}\_2^\* + (\chi\_1 \chi\_2 - \eta\_1 \eta\_2) \sin \beta \mathbf{x}\_1 \sin \beta \mathbf{x}\_2^\* \sin \beta (\mathbf{x}\_1 + \mathbf{x}\_2^\*),\tag{9a}$$

$$B\_1 = \eta\_1 \sin^2 \beta \mathbf{x}\_1 + \eta\_2 \sin^2 \beta \mathbf{x}\_2^\* - (\chi\_1 \eta\_2 + \chi\_2 \eta\_1) \sin \beta \mathbf{x}\_1 \sin \beta \mathbf{x}\_2^\* \sin \beta (\mathbf{x}\_1 + \mathbf{x}\_2^\*),\tag{9b}$$

$$\begin{cases} \mathbf{C}\_1 = 1 - \chi\_1 \sin \beta \mathbf{x}\_1 \cos \beta \mathbf{x}\_1 - \chi\_2 \sin \beta \mathbf{x}\_2^\* \cos \beta \mathbf{x}\_2^\* + \\ \left(\chi\_1 \chi\_2 - \eta\_1 \eta\_2\right) \sin \beta \mathbf{x}\_1 \sin \beta \mathbf{x}\_2^\* \cos \beta (\mathbf{x}\_1 + \mathbf{x}\_2^\*) \end{cases} \tag{9c}$$

$$\begin{aligned} D\_1 &= \eta\_1 \sin \beta \mathbf{x}\_1 \cos \beta \mathbf{x}\_1 + \eta\_2 \sin \beta \mathbf{x}\_2^\* \cos \beta \mathbf{x}\_2^\* - \\ &\quad \left(\chi\_1 \eta\_2 + \chi\_2 \eta\_1\right) \sin \beta \mathbf{x}\_1 \sin \beta \mathbf{x}\_2^\* \cos \beta (\mathbf{x}\_1 + \mathbf{x}\_2^\*) \end{aligned} \tag{9d}$$

where η*j* = *cj* √*mT* and χ*j* = *bj*ω √*mT* represent the dimensionless damping coefficient and the dimensionless inertial mass of the *jth* IMD, respectively.

The form of Equation (9) is suitable for solutions, either for asymptotic solutions or numerical solutions by iteration.
