*4.2. Asymptotic Solution*

Similarly to the case of two opposite IMDs, assuming two IMDs locations *x*1, *x*2 *L* and the wave number β*n* of the damped cable to be small perturbations from β<sup>0</sup>*n*, Equation (16) can be simplified as:

$$
\beta\_{\rm n} L \cong n\pi + \beta\_n^0 \frac{E\_2 + iF\_2}{G\_2 + iH\_2},
\tag{17}
$$

$$E\_2 = -\overline{b\_1}\mathbf{x}\_1 - \overline{b\_2}\mathbf{x}\_2 + (\overline{b\_1 b\_2} - \overline{c\_1 c\_2})(\mathbf{x}\_2 - \mathbf{x}\_1),\tag{17a}$$

$$F\_2 = \overline{c\_1}\mathbf{x}\_1 + \overline{c\_2}\mathbf{x}\_2 - (\overline{b\_1}\overline{c\_2} + \overline{b\_2}\overline{c\_1})(\mathbf{x}\_2 - \mathbf{x}\_1),\tag{17b}$$

$$G\_2 = 1 - \overline{b\_1} - \overline{b\_2} + (\overline{b\_1 b\_2} - \overline{c\_1 c\_2})(1 - x\_1 / x\_2),\tag{17c}$$

$$H\_2 = \overline{c\_1} + \overline{c\_2} - (\overline{b\_1}\overline{c\_2} + \overline{b\_2}\overline{c\_1})(1 - \mathbf{x}\_1/\mathbf{x}\_2). \tag{17d}$$

From Equations (12) and (17), the asymptotic modal damping ratio of a cable with two IMDs at the same end can be obtained as:

$$
\xi\_{\rm II}^{\rm \zeta} \cong \frac{\text{Im} [\Delta \beta\_n]}{\| \beta\_n^0 \|} = \frac{F\_2 G\_2 - E\_2 H\_2}{(G\_2)^2 + (H\_2)^2}. \tag{18}
$$

## *4.3. Numerical Solution*

Similarly to the case of two opposite IMDs, Equation (16) can be solved for the wavenumber using the fixed-point iteration. Starting from the undamped wavenumber β<sup>0</sup>*n*, the iterative scheme is given by:

$$
\beta\_n^{k+1} L = n\pi + \arctan \frac{\overline{A}\_2 + i\overline{B}\_2}{\overline{C}\_2 + i\overline{D}\_2},
\tag{19}
$$

$$\overline{A}\_2 = -\chi\_1 \sin^2 \beta\_n^k \mathbf{x}\_1 - \chi\_2 \sin^2 \beta\_n^k \mathbf{x}\_2 + (\chi\_1 \chi\_2 - \eta\_1 \eta\_2) \sin \beta\_n^k \mathbf{x}\_1 \sin \beta\_n^k \mathbf{x}\_2 \sin \beta\_n^k (\mathbf{x}\_2 - \mathbf{x}\_1), \tag{19a}$$

$$\overline{B}\_2 = \eta\_1 \sin^2 \beta\_n^k \mathbf{x}\_1 + \eta\_2 \sin^2 \beta\_n^k \mathbf{x}\_2 - (\chi\_1 \eta\_2 + \chi\_2 \eta\_1) \sin \beta\_n^k \mathbf{x}\_1 \sin \beta\_n^k \mathbf{x}\_2 \sin \beta\_n^k (\mathbf{x}\_2 - \mathbf{x}\_1), \tag{19b}$$

$$\begin{cases} \overline{\mathbf{C}}\_{2} = 1 - \chi\_{1} \sin \beta\_{n}^{k} \mathbf{x}\_{1} \cos \beta\_{n}^{k} \mathbf{x}\_{1} - \chi\_{2} \sin \beta\_{n}^{k} \mathbf{x}\_{2} \cos \beta\_{n}^{k} \mathbf{x}\_{2} + \\ (\chi\_{1} \chi\_{2} - \eta\_{1} \eta\_{2}) \sin \beta\_{n}^{k} \mathbf{x}\_{1} \cos \beta\_{n}^{k} \mathbf{x}\_{2} \sin \beta\_{n}^{k} (\mathbf{x}\_{2} - \mathbf{x}\_{1}) \end{cases} \tag{19c}$$

$$\begin{aligned} \overline{D}\_2 &= \eta\_1 \sin \beta\_n^k \mathbf{x}\_1 \cos \beta\_n^k \mathbf{x}\_1 + \eta\_2 \sin \beta\_n^k \mathbf{x}\_2 \cos \beta\_n^k \mathbf{x}\_2 - \\ \left(\chi\_1 \eta\_2 + \chi\_2 \eta\_1\right) \sin \beta\_n^k \mathbf{x}\_1 \cos \beta\_n^k \mathbf{x}\_2 \sin \beta\_n^k (\mathbf{x}\_2 - \mathbf{x}\_1) \end{aligned} \tag{19d}$$

After solving numerically for the wavenumber, the supplemental modal damping ratio of a cable with two IMDs at the same end can be determined from Equation (12).

#### *4.4. Comparison of Asymptotic and Numerical Solutions*

Figure 6 shows the comparison of asymptotic and numerical complex wavenumbers of a cable with two IMDs at the same end for various inertial masses, where two IMDs are installed at distances *x*1 of 1%*L* and *x*2 of 2%*L* from the left end of the cable, i.e., *x*1 = 1%*L*, *x*2 = 2%*L*. Seeing that two IMDs are usually identical, some simplifications in the notation are introduced, i.e., *b*1 = *b*2 = *b*, χ1 = χ2 = χ, *c*1 = *c*2 = *c*, and η1 = η2 = η. Similar to the case of two opposite IMDs, the loci start from the undamped wavenumber along a semicircular contour and finally attach to the real axis when damping coefficients of the IMDs increase from zero to infinity, and the effect of two IMDs installed at the same end of the cable on the cable frequency is also not significant. For a cable with two IMDs at the same end, although the asymptotic complex wavenumber agrees well with the numerical solution when the small inertial mass (χ ≤ 0.3/(*n*π*x*2/*L*)) is used, it will lose accuracy when moderate or large inertial mass (0.6/(*n*π*x*2/*L*) ≤ χ ≤ 0.9/(*n*π*x*2/*L*)) is adopted. Compared to the case of two opposite IMDs, prediction accuracies of the asymptotic solution are found to be relatively poor when two IMDs are installed at the same end of the cable. Hence, numerical results are used for the following parametric study.

**Figure 6.** Comparison of asymptotic and numerical complex wavenumbers of a cable with two identical IMDs at the same end (*<sup>x</sup>*1 = 1%*L*, *x*2 = 2%*L*).

## *4.5. Parametric Studies*

Figure 7 presents the supplemental modal damping ratio of a cable with two identical VDs at the same end or a single VD versus damping coefficients. It is observed that attaching two VDs at the same ends of the cable may help to reduce the damper size but cannot increase the maximum supplemental modal damping ratio. Moreover, its maximum modal damping ratio is slightly smaller than that provided by a single VD at the further distance. These observations are in agreemen<sup>t</sup> with previous findings [76,77].

**Figure 7.** The modal damping ratios curves of a cable equipped with a single VD or two identical VDs at the same end (*<sup>x</sup>*1 = 1%*L*, *x*2 = 2%*L*).

Figure 8 presents the supplemental modal damping ratio of a cable with two identical IMDs at the same end or a single IMD versus damping coefficients of the IMD. If two IMDs with relatively small or big inertial masses (χ ≤ 0.1/(*n*π*x*2/*L*) or χ = 0.9/(*n*π*x*2/*L*)) are installed at the same end of the cable, similarly to the case of two VDs at the same end of a cable, there is no advantage of increasing the maximum modal damping ratio over that of a single IMD. However, if moderate inertial mass (0.4/(*n*π*x*2/*L*) ≤ χ ≤ 0.7/(*n*π*x*2/*L*)) of the IMD is used, it is interesting to observe that two IMDs at the same end can lead to smaller optimum damping coefficients and larger maximum supplemental modal damping ratios than that of a single IMD at a bigger distance.

**Figure 8.** The modal damping ratios curves of a cable equipped with a single IMD or two identical IMDs at the same end (*<sup>x</sup>*1 = 1%*L*, *x*2 = 2%*L*).

The maximum achievable supplemental modal damping ratios of a cable equipped with a single IMD and two IMDs at the same end are directly compared in Figure 9. It is worth noting that the maximum supplemental modal damping ratio provided by two IMDs is higher than the sum of contributions from each IMD when inertial mass χ = 0.7/(*n*π*x*2/*L*) is used. Though the strategy of two opposite IMDs has demonstrated that it can provide superior control performance, installing a damper at cable-tower anchorage is difficult and inconvenient. Thus, attaching two IMDs with appropriate inertial mass installed at the same end of the cable seems to be more promising for practical application.

**Figure 9.** The maximum achievable supplemental modal damping ratio of a cable equipped with a single IMD or two identical IMDs at the same end (*<sup>x</sup>*1 = 1%*L*, *x*2 = 2%*L*).
