3.3.2. Earthquake Excitation

Figures 10 and 11 show the SDOF structures' acceleration and displacement responses. The peak value and RMS value of the displacement and acceleration responses were chosen to evaluate the vibration mitigation performance of CBIS. These values are both important controlling indices in structural vibration control. The peak value reflects the dynamic response at a certain instant, whereas the RMS value indicates the vibration energy and reflects the responses over an entire period.

**Figure 10.** *Cont*.

**Figure 10.** Acceleration response time histories for two test frame roofs during (**a**) the El Centro wave, (**b**) the Tohoku wave recorded by Tohoku University, (**c**) the Kobe record and (**d**) the Chi–chi record.

**Figure 11.** Displacement response time histories for two test frame roofs during (**a**) the El Centro wave, (**b**) the Tohoku wave recorded by Tohoku University, (**c**) the Kobe record and (**d**) the Chi–chi record.

The vibration reduction effect is defined as:

$$\text{Reduction effect} = \frac{\text{Response of uncountable structure} - \text{Response of controlled structure}}{\text{Response of uncountable structure}} \times 100\%. \tag{31}$$

The results are listed in Tables 2 and 3. The reduction effects were favorable under the El Centro wave, Tohoku wave, Kobe record and Chi–chi record. The best vibration control effects for the peak and RMS values of the acceleration responses were 52.08% and 45.71% (marked in bold), respectively. The values for the displacement responses were 55.56% and 52.50%, respectively.


**Table 2.** Acceleration responses at the roof of the test frame (m/s2).

**Table 3.** Displacement responses at the roof of the test frame (m).


*3.4. Ignoring the Flexibility of the Cable*

> To simplify the CBIS analytical model, the flexibility of the cable is neglected as shown in Figure 12.

**Figure 12.** Simplified analytical model of the CBIS-equipped SDOF structure.

Consider this simplified CBIS-equipped SDOF structure as a model with no connection element flexibility. The relationship between the axial deformation of the cable and the rotational angle of the shaft is expressed as:

$$\varphi(t) = \frac{u(t)\cos\theta}{r\_0}.\tag{32}$$

The equation of motion for an SDOF model with a CBIS is given by:

$$\left(m+\frac{f\cos^2\theta}{r\_0^2}\right)\ddot{u}(t)+\left(c\_0+c\_d\cos^2\theta\right)\dot{u}(t)+ku(t)=-ma\_\mathcal{S}(t).\tag{33}$$

In Equation (33), *J* cos<sup>2</sup> θ *<sup>r</sup>*0<sup>2</sup> is *m*d, which is namely the inertance of the inerter. From Equation (33), it can be understood that the utilization of CBIS induces the elongation of the natural period and increases the damping effect. The mass of the primary structure is affected, while the stiffness of the overall structure remains unaffected. By using the Laplace transformations, the transfer function can be easily obtained as:

$$H\iota(\mathbf{s}) = \frac{\mathcal{U}(\mathbf{s})}{A\_{\mathbf{g}}(\mathbf{s})} = \frac{-1}{(1+\mu)s^2 + (2\zeta\omega\_{\mathbf{s}} + 2\zeta\omega\_{\mathbf{s}}\cos^2\theta)\mathbf{s} + \omega\_{\mathbf{s}}^2}.\tag{34}$$

We assume a single-floor frame structure, which can usually be treated as an SDOF structure. The key parameters of the structure and the inerter system are as follows: inherent damping ratio is ζ = 0.02, the tilt angle of the cable is θ = π/4, the inertance–mass ratio is μ = 0.1, 0.2, and the damping ratio is ξ = 0.01, 0.05, 0.1. By using these parameters, the displacement amplification factor can be plotted as shown in Figure 13.

**Figure 13.** Displacement amplification factors of the SDOF structure with different inerter systems.

Figure 13 shows that inerter systems can significantly suppress the resonant response in a narrow band near the natural frequency of the primary structure but does not impact the other range of frequencies.
