*2.1. CBIS Concept*

Cable bracing is the proposed mechanism of translation-to-rotation conversion for an inerter connected to a structure. Figure 1 shows an SDOF structure with a CBIS, which consists of a pair of bracing cables, a pair of conductor plates (flywheels), and a shaft. A pair of cables is pre-tensioned connecting the structural frame and the shaft diagonally. Both ends of the shaft are supported by a pair of shaft bearings mounted on the side plates fixed on the ground floor, making sure that only the shaft rotates. When inter-story drift occurs in structures, one of the cables will shorten and drive the shaft into rotation. The low-speed translational movement of the structure can be converted into a high-speed rotational motion of the conductor plates by cable bracing.

**Figure 1.** A single-degree-of-freedom (SDOF) structure with a CBIS: (**a**) structure with a CBIS and (**b**) detail of a CBIS. Note that the conductor plates serve as flywheels.

Conductor plates are fixed on the shaft and rotate together; meanwhile, several magnets with alternating magnetic polarization are allocated on the fixed side plate to generate the electromagnetic field. The rotational conductor plates and shaft function as an inerter. The fixed side plate and conductor plate form one eddy current damper when the rotation conductor plate is cutting through the electromagnetic field, thereby dissipating the vibration energy in the form of heat. As a result, the novel cable-bracing inerter system presented herein can obtain its inertance and enhance the energy dissipation capacity via the additional damping provided by the eddy current damping. Compared to the classic ball-screw mechanism of an inerter, CBIS is cost-effective and very simple.

## *2.2. Inerter Element*

An inerter element is a two-terminal element. The output force is proportional to the relative acceleration between two terminals and can be expressed as:

$$p = m\_{\rm d}(a\_2 - a\_1),\tag{1}$$

where *p* is the output force of the inerter element, and *m*d is the inertance; moreover, *a*1 and *a*2 are the accelerations at the two terminals of the inerter element, as depicted in Figure 2.

 **Figure 2.** Mechanical model of an inerter element.

#### *2.3. Layout of SDOF System with CBIS*

The mechanical system of a CBIS consists of an additional damping element, an inerter element and a spring element. The damping element is set in parallel with the inerter element. The spring element is then connected with the paralleled inerter and damping element in a series. Figure 3 shows the layout and mechanical model of an SDOF structure equipped with a CBIS when the structure is deformed in a horizontal direction.

**Figure 3.** Layout of an SDOF system with a CBIS.

In Figure 3, *m*, *c*, and *k* are the mass, damping coefficient, and stiffness of the SDOF system, the primary structure, respectively. θ is the inclined angle of the diagonal cable; *m*d is the inertance of the CBIS; *c*e = *c*d cos<sup>2</sup> θ is the equivalent damping coefficient considering the inclined angle of the cables, where *c*d is the damping coefficient of the damping element, and *k*b is the stiffness of the supporting spring element. The output force of this inerter system is the resultant force of the inerter element and the eddy current damping element.

#### *2.4. Motion Governing Equation of SDOF System with CBIS*

When the structure is in a balanced state, the prestressed tension forces in both cables are *T*0. If the structure starts to leave the balance position by moving to the right as illustrated in Figure 4, the diagonal cable on the right side drives the inerter to rotate clockwise. At this time, the force increment in the right cable is Δ*T*, and the force becomes *T*2 (*T*2 = *T*0 + Δ*T*). The tension in the left cable decreases Δ*T* and becomes *T*1 (*T*1 = *T*0 − Δ*T*).

**Figure 4.** Transmission mechanism of cable-bracing inerter system.

When the structure has a positive deformation, it moves to the right with a relative displacement *u*(*t*), and the shaft and the conductor plates rotate correspondingly. We assume there is no relative slippage between the cable and the shaft. Considering the axial stiffness of the one-sided cable kb0, the forces in the right cable *T*2 and the left cable *T*1 are as follows:

$$\begin{aligned} T\_2 &= T\_0 + k\_\mathsf{b}^0(\mathfrak{u}(t)\cos\theta - \varphi(t)r\_0) \\ T\_1 &= T\_0 - k\_\mathsf{b}^0(\mathfrak{u}(t)\cos\theta - \varphi(t)r\_0), \end{aligned} \tag{2}$$

where the rotational angle of the conductor plate is ϕ(*t*). As shown in Figure 5, when the conductor plates rotate, the cable moves in its own axial direction. Thus, the angle difference Δθ from its balance position is trivial and can be ignored. *u*(*t*) cos θ − ϕ(*t*)*<sup>r</sup>*0 is the axial elongation of the cable. During the operation of a CBIS, the tension force difference between two cables drives the shaft to rotate and is given as:

$$T\_2 - T\_1 = 2k\_b \, ^0(\mu(t) \cos \theta \, -\, \varphi(t) r\_0). \tag{3}$$

The eddy currents cause a damping force that is proportional to the velocity of the conductive metal, which makes the eddy currents function like a viscous damper. According to the force equilibrium conditions, compatibility condition, and the layout of the system (as shown in Figure 3), the motion equation for this SDOF structure with a CBIS under earthquake excitations can be written as:

$$-m\ddot{u}(t) + c\dot{u}(t) + ku(t) + k\_b(u(t)\cos\theta - \varphi(t)r\_0)\cos\theta = -ma\_\mathcal{J}(t),\tag{4}$$

where *u*(*t*) is the relative displacement of the SDOF system, and the dots represent the derivative with respect to time *t*. *a*g(*t*) is the acceleration of the ground motion, *k*b is the equivalent stiffness of two cables, and it is used to replace <sup>2</sup>*k*b<sup>0</sup> in Equation (3). The motion equation for the CBIS is written as:

$$\left|f\bar{\boldsymbol{\varphi}}(t) + \mathbf{c}\_d \dot{\boldsymbol{\varphi}}(t)\boldsymbol{r}\_0\right|^2 = k\_\mathsf{b} (\boldsymbol{u}(t)\cos\theta - \boldsymbol{\varphi}(t)\boldsymbol{r}\_0)\boldsymbol{r}\_{0\star} \tag{5}$$

where *J* is the moment of inertia for the inerter, and *r*0 is the radius of the shaft. The conductor plate serves as a flywheel whose moment of inertia can be calculated as:

$$J = m\_{\rm l} \mathbb{R}^2 / 2,\tag{6}$$

where *m*I is the physical mass of two conductor plates and the shaft, and *R* is the radius of gyration. The governing motion equation of an SDOF structure can be described as:

**¨**

$$
\begin{bmatrix} m & 0 \\ 0 & J \end{bmatrix} \begin{Bmatrix} \ddot{u} \\ \ddot{\varphi} \end{Bmatrix} + \begin{bmatrix} c & 0 \\ 0 & c\_b r\_0^2 \end{bmatrix} \begin{Bmatrix} \dot{u} \\ \dot{\varphi} \end{Bmatrix} + \begin{bmatrix} k + k\_\mathsf{b} \cos^2 \theta & -k\_\mathsf{b} r\_0 \cos \theta \\ -k\_\mathsf{b} r\_0 \cos \theta & k\_\mathsf{b} r\_0^2 \end{bmatrix} \begin{Bmatrix} u \\ \varphi \end{Bmatrix} = \begin{bmatrix} -ma\_\mathcal{S} \\ 0 \end{bmatrix}. \tag{7}
$$

Therefore, Equation (7) can be expressed in matrix form as:

$$\mathbf{M}\mathbf{X} + \mathbf{C}\mathbf{X} + \mathbf{K}\mathbf{X} = \mathbf{F}\_{\prime} \tag{8}$$

where **M**, **C**, **K** and **F** respectively represent the mass matrix, damping matrix, stiffness matrix and external excitation vector of the SDOF system with a CBIS. Equation (8) is converted into the state space form:

·

$$\mathbf{A}\overline{\mathbf{X}} + \mathbf{B}\overline{\mathbf{X}} = \begin{pmatrix} \mathbf{F} \\ \mathbf{0} \end{pmatrix},\tag{9}$$

where **0**={**0**, **<sup>0</sup>**}*<sup>T</sup>*, **A**, **B**, **X** and **X** are determined as:

·

$$\mathbf{A} = \begin{bmatrix} c & 0 & m & 0 \\ 0 & c\_{\mathrm{d}}r\_{0}^{2} & 0 & I \\ m & 0 & 0 & 0 \\ 0 & I & 0 & 0 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} k + k\_{\mathrm{b}}\cos^{2}\theta & -k\_{\mathrm{b}}r\_{0}\cos\theta & 0 & 0 \\ -k\_{\mathrm{b}}r\_{0}\cos\theta & k\_{\mathrm{b}}r\_{0}^{2} & 0 & 0 \\ 0 & 0 & -m & 0 \\ 0 & 0 & 0 & -I \end{bmatrix}, \overline{\mathbf{X}} = \begin{pmatrix} u \\ \dot{\boldsymbol{q}} \\ \dot{\boldsymbol{u}} \\ \dot{\boldsymbol{q}} \end{pmatrix}, \overline{\mathbf{X}} = \begin{pmatrix} \dot{u} \\ \dot{\boldsymbol{q}} \\ \ddot{\boldsymbol{u}} \\ \ddot{\boldsymbol{q}} \end{pmatrix}. \tag{10}$$

·

Assume that the solution of Equation (9) has the form of:

**X** = ψ**e**λ*t*, **X** = ψλ**e**λ*t*, (11)

where ψ is the eigenvector, substituting ψ= [ψ1 ψ2]<sup>T</sup> into Equation (9), whereby we obtain:

$$(\mathbf{A}\boldsymbol{\lambda} + \mathbf{B}) \begin{pmatrix} \Psi \\ \Psi \boldsymbol{\lambda} \end{pmatrix} = 0. \tag{12}$$

The characteristic equation can be expressed as:

$$\det[\mathbf{A}\lambda + \mathbf{B}] = 0.\tag{13}$$

The *j* th pair of eigenvalues are <sup>λ</sup>2*j*−<sup>1</sup> and <sup>λ</sup>2*j*, and the *j* th fundamental angular frequency <sup>ω</sup>*j* can be obtained as:

$$
\lambda \omega\_j = \left| \lambda\_{2j-1} \right| = \left| \lambda\_{2j} \right|. \tag{14}
$$

#### *2.5. Frequency Response Function*

In this section, the frequency response function of an SDOF structure with a CBIS under seismic excitation is obtained. To normalize the dynamic equation, the parameters can be defined as:

$$
\omega\_{\sf S} = \sqrt{\frac{k}{m'}}\tag{15}
$$

$$\zeta = \mathfrak{c} / 2m\omega\_{\mathfrak{s}\mathfrak{s}} \tag{16}$$

$$\pounds = \mathfrak{c}\_{\mathrm{d}} / 2m\omega\_{\mathrm{s}}.\tag{17}$$

where ωs and ζ are the circular frequency and the inherent damping ratio of the original SDOF structure, respectively. ξ is the additional damping ratio provided by the CBIS. Through dimensionless processing, the following parameters can be defined for designing CBIS:

$$
\mu = m\_{\rm d} / m,\tag{18}
$$

$$\kappa = k\_{\rm b} \cos^2 \theta / k\_{\rm \prime} \tag{19}$$

.

.

where μ is inertance–mass ratio, a ratio of the inertance of CBIS to the mass of primary system. κ is the ratio of supporting spring stiffness in the horizontal direction to the primary stiffness *k*. Substituting the parameters in Equations (15)–(19) into Equation (9), the Laplace transformation of Equation (7) can be written as:

$$\begin{bmatrix} sI \\ s^2I \\ s\Phi \\ s^2\Phi \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -\omega\_s^2 - \kappa\omega\_s^2 & -2\zeta\omega\_s & \kappa\omega\_s^2r\_0/\cos\theta & 0 \\ 0 & 0 & 0 & 1 \\ \kappa\omega\_s^2 mr\_0/I\cos\theta & 0 & -\kappa\omega\_s^2 mr\_0^2/I\cos^2\theta & -2\xi\omega\_s r\_0^2 m/I \end{bmatrix} \begin{bmatrix} U \\ sI \\ \Phi \\ s\Phi \end{bmatrix} - \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} A\_\ddagger(s), \tag{20}$$

where *s* = *i*<sup>ω</sup>, and *<sup>A</sup>*g(*s*) is the Laplace transformation of *a*g(*t*). *U*, *U*, Φ and Φ are the Laplace transformations of *u*, .*u*, ϕ and .ϕ, respectively, and they can be solved from Equation (8):

$$\begin{bmatrix} \boldsymbol{\varPi}(\boldsymbol{s}) \\ \boldsymbol{\varPi}(\boldsymbol{s}) \\ \boldsymbol{\varPhi}(\boldsymbol{s}) \\ \boldsymbol{\Phi}(\boldsymbol{s}) \\ \boldsymbol{\Phi}(\boldsymbol{s}) \end{bmatrix} = \begin{bmatrix} \frac{-(s^{2}\boldsymbol{\varmu} + 2\boldsymbol{\upzeta}\boldsymbol{\upmu}\_{s}\boldsymbol{\upkappa}\cos^{2}\boldsymbol{\upbeta} + \boldsymbol{\upkappa}\boldsymbol{\upkappa}\boldsymbol{\uprho}\_{s}^{2})\boldsymbol{\varLambda}\_{\mathcal{E}}(\boldsymbol{s})}{-s(s^{2}\boldsymbol{\upmu} + 2\boldsymbol{\upzeta}\boldsymbol{\upkappa}\boldsymbol{\uprho}\cos^{2}\boldsymbol{\upbeta} + \boldsymbol{\upkappa}\boldsymbol{\uprho}\_{s}^{2})\boldsymbol{\upbeta}\_{\mathcal{E}}(\boldsymbol{s})} \\ & \frac{\boldsymbol{\upkappa}(\boldsymbol{s},\boldsymbol{\uprho},\boldsymbol{\uprho},\boldsymbol{\uprho},\boldsymbol{\uprho}\_{0})}{-\boldsymbol{\upkappa}\boldsymbol{\uprho}\_{s}^{2}\cos\boldsymbol{\upalpha}\boldsymbol{\uprho}\_{s}(\boldsymbol{s})} \\ & \frac{-\boldsymbol{\upkappa}\boldsymbol{\uprho}\_{s}^{2}\cos\boldsymbol{\uprho}\_{s}\boldsymbol{\uprho}\_{s}\boldsymbol{\uprho}\_{0}(\boldsymbol{s})}{\boldsymbol{\upkappa}\boldsymbol{\uprho}\_{s}(\boldsymbol{s},\boldsymbol{\uprho},\boldsymbol{\uprho},\boldsymbol{\uprho}\_{0},\boldsymbol{\uprho}\_{0})} \end{bmatrix} \tag{21}$$

where

$$\begin{array}{l} \mathbb{C}(\kappa,\xi,\zeta,\mu,\kappa,\mu\_{\sf s}) = s^{4}\mu + 2s^{3}(\xi\cos^{2}\theta + \zeta\mu)\omega\_{\sf s} + s^{2}(\kappa + 4\xi\zeta,\cos^{2}\theta + \mu + \kappa\mu)a\_{\sf s}^{\sf z} \\ + 2s(\zeta\kappa + \xi\cos^{2}\theta + \kappa\zeta\cos^{2}\theta)\omega\_{\sf s}^{\sf z} + \kappa\omega\_{\sf s}^{\sf z} \\\ D(\mathbf{s},\xi,\zeta,\mu,\kappa,\mu\_{\sf s},r\_{0}) = s^{4}\mu r\_{0} + 2s^{3}(r\_{0}\xi\cos^{2}\theta + r\_{0}\zeta\mu)a\_{\sf s} + s^{2}(\kappa\mu r\_{0} + 4\xi\zeta\cos^{2}\theta r\_{0} \\ + \mu r\_{0} + \kappa r\_{0})a\_{\sf s}^{\sf z} + 2s(\xi\kappa r\_{0} + \zeta\kappa r\_{0} + \xi\cos^{2}\theta r\_{0})a\_{\sf s}^{\sf z} + \kappa r\_{0}a\_{\sf s}^{\sf z}. \end{array} \tag{22}$$

The frequency–domain transfer function between *u*(*t*) and input excitations can be easily obtained as:

$$H\_{ll}(\mathbf{s}) = \frac{\mathcal{U}(\mathbf{s})}{A\_{\mathcal{S}}(\mathbf{s})} = \frac{-(s^2\mu + 2\xi\mu\_{\mathbf{s}}\mathbf{s}\cos^2\theta + \kappa\omega\_{\mathbf{s}}^2)}{\mathcal{C}(\mathbf{s}, \xi, \zeta, \mu, \kappa, \omega\_{\mathbf{s}})}. \tag{23}$$

The normalized force of the CBIS is defined as *F*(*t*) = *k*b(*u*(*t*) cos θ − ϕ(*t*)*<sup>r</sup>*0)/*<sup>m</sup>*, which is provided by the inertial mass element and the eddy current damping element. The frequency–domain transfer function between *F*(*t*) and the input excitation is given as:

$$H\_{\rm F}(s) = \frac{F(s)}{A\_{\rm \xi}(s)} = \frac{-(2\kappa \xi \omega\_{\rm s}{}^3 s \cos^2 \theta + \kappa \omega\_{\rm s}{}^2 \mu s^2)}{A(s, \xi, \zeta, \mu, \kappa, \omega\_{\rm s}) \cos \theta}. \tag{24}$$
