*3.1. Analysis Index*

The response of a structure at the resonant frequency is much greater than that at another frequency, so the peak value of the system's responses deserves much attention. To find out the maximum displacement response of the structure, the displacement amplification factor is referred to as the *H*∞ norm and can be defined as:

$$H\_{\infty} = \max\{ \left| H\_{\text{II}}(i\beta) \right| \omega\_{\text{s}}^2 \},\tag{25}$$

where β = <sup>ω</sup>/<sup>ω</sup>s, and it can be interpreted as the normalized frequency. This index is independent of the natural circular frequency ωs of the original structure, and can be considered as a function, depending on the original structure's inherent damping ratio ζ and the CBIS parameters, μ, ξ, and κ. Henceforth, the intention of vibration control is to minimize the displacement amplification factor in terms of the *H*∞ norm for a set of optimal parameters of the CBIS.

#### *3.2. Parametric Analysis Results*

Based on the analysis indexes mentioned above, a series of numerical cases were considered. In these cases, the inherent damping ratio for the main frame ζ=0.02 was assumed. Three inertance-mass ratios, namely, μ = 0.01, 0.1 and 1.0, were used to study the vibration control effects by continuously varying κ and ξ within specified ranges. To describe the controlling index of the SDOF with a CBIS, contour plots were illustrated with ξ on the *x*-axis, κ on the *y*-axis, and max*HU*(*<sup>i</sup>*β)ω2s as the height, as shown in Figure 5.

(**c**) *μ* = 1 

**Figure 5.** Displacement amplification factor of a CBIS-equipped SDOF structure with changes in κ ∈ [0.01, 100] and ξ ∈ [0.01, 1].

The displacement amplification factor of any point in the <sup>κ</sup>*-*ξ space is determined by the frequency response function of Equation (25). The magnitude is represented by the color intensity. The lowest displacement amplification factor is in the corner of the <sup>κ</sup>*-*ξ space, where the two parameters reach their upper bounds. As shown in Figure 5a–c, the desirable solution of the dynamic response ratio is determined by the feasible upper bounds of the CBIS parameters in the given ranges. Mathematically, the optimal configuration for vibration control requires the sti ffness ratio and additional damping to be as large as possible. However, both are impossible to realize in actual engineering. It is necessary to introduce appropriate boundary conditions or constraints for practical optimization processes.

When the additional damping ratios are fixed, for example ξ = 0.05, 0.1, closed contour lines can be obtained (as shown in Figure 6). A very low point implies that the parameter set for optimal control can always be found in the inner part of every contour. This means that optimal solutions lie within the inner part of the parametric space. When the sti ffness ratio closes to 1, and the inertance–mass ratio closes to 0.1, the displacement response reaches its lowest point. In this process, the optimal additional damping ratio ξ remains unknown. Therefore, the selection of a rational parameter set for the design of CBIS based on only the displacement response is di fficult. However, for many situations, the optimization will involve a recursive process in which the optimal configuration keeps updating with a prescribed additional damping ratio ξ until the mitigation e ffect satisfies the objective.

**Figure 6.** Displacement amplification factor of a CBIS-equipped SDOF structure with changes in κ ∈ [0.01, 100] and μ ∈ [0.01, 1].

In Figure 6, the displacement responses reach their optimal points when the sti ffness ratio closes to 1, and the inertance–mass ratio closes to [0.1, 0.5],where the force response of the inerter system is relatively large (as shown in Figure 7). In the optimization design, it is unreasonable to consider only the structural displacement, or the force response provided by the inerter element. Therefore, both the displacement responses and the inerter element's force should be considered in the design of CBIS. The demand-oriented multi-objective optimum design method will be introduced in the next section, and the control force response of the CBIS will be brought into the optimization process as the secondary objective.

**Figure 7.** Force amplification factor of the inerter system.

#### *3.3. Multi-Objective H2 Norm Optimum Design*

Three unknown parameters can be designed optimally according to performance demands. An optimization method is proposed to achieve the desired performance levels (structure's displacement *u*) with low control output force (CBIS's force *f* d). In other words, the goal is to suppress the displacement of the structure as thoroughly as possible while minimizing the output force of the inerter system. Therefore, both the displacement response and the CBIS force should be considered in the design of this inerter system. Two dimensionless response variation ratios [32] are defined as: namely, the displacement reduction ratio, γ*U* and the force ratio, γ*F*d , which can be expressed as:

$$\gamma\_{\rm II}(\zeta,\mu,\xi,\kappa) = \frac{\sigma\_{\rm II}}{\sigma\_{\rm II\_0}} = \frac{\sqrt{\int\_{-\infty}^{\infty} \left| H\mu(i\omega) \right|^2 d\omega}}{\sqrt{\int\_{-\infty}^{\infty} \left| H\_{\rm IL\_0}(i\omega) \right|^2 d\omega}}\,\tag{26}$$

$$\gamma\_{F\_{\rm d}}(\zeta,\mu,\xi,\kappa) = \frac{\sigma\_{F\_{\rm d}}}{\sigma\_{F\_{\rm d0}}} = \frac{\sqrt{\int\_{-\infty}^{\infty} \left| H\_{F\_{\rm d}}(i\omega) \right|^{2} d\omega}}{\sqrt{\int\_{-\infty}^{\infty} \left| H\_{F\_{\rm d0}}(i\omega) \right|^{2} d\omega}}. \tag{27}$$

In these expressions, γ*U* is the ratio of the CBIS-equipped structure's displacement, compared with the primary structure, and γ*F*d is the dimensionless force of the CBIS. σ*U* is the root mean square (RMS) of the output displacement response of the structure equipped with CBIS, and <sup>σ</sup>*U*0 is the displacement response of the primary structure. <sup>σ</sup>*F*d is the force of the RMS response of the structure equipped with CBIS, and <sup>σ</sup>*F*d0 is the force RMS response of the original structure. *HU*0 (*<sup>i</sup>*ω) and *HF*d0 (*<sup>i</sup>*ω) are the displacement and damping force (caused by the inherent damping) transfer function moduli of the original structure, respectively. The optimization of CBIS can be expressed mathematically as:

$$\begin{aligned} \underset{\boldsymbol{\mu}, \boldsymbol{\xi}, \boldsymbol{\kappa}}{\text{minimize}} & \left[ \boldsymbol{\gamma}\_{\mathcal{U}} (\boldsymbol{\mu}, \boldsymbol{\xi}, \boldsymbol{\kappa}), \boldsymbol{\gamma}\_{\mathcal{F}\_{\rm d}} (\boldsymbol{\mu}, \boldsymbol{\xi}, \boldsymbol{\kappa}) \right], \\ \text{subject to} & \left\{ \begin{array}{l} \boldsymbol{\mu}\_{\text{min}} \le \boldsymbol{\mu} \le \boldsymbol{\mu}\_{\text{max}} \\ \boldsymbol{\xi}\_{\text{min}} \le \boldsymbol{\xi} \le \boldsymbol{\xi}\_{\text{max}} \\ \boldsymbol{\kappa}\_{\text{min}} \le \boldsymbol{\kappa} \le \boldsymbol{\kappa}\_{\text{max}} \end{array} \right. \end{aligned} \tag{28}$$

where μ, ξ and κ are decision variables, and μmin, ξmin, and κmin are the lower bounds of μ, ξ and κ, while μmax, ξmax, and κmax are the upper bounds, respectively. Multi-objective optimization (MOO) is used to find the boundary of the feasible criterion space where all optimal points lie: namely, Pareto Front (shown in Figure 8). On this boundary, there are many points. Based on the demanding

performance, a reasonable optimization parameter for γ*U* ≤ 40% and the corresponding γ*F*d can be found. Furthermore, a set of parameters was selected for time domain analysis, as shown in Table 1.

**Figure 8.** Pareto optimal front.

**Table 1.** Results of performance-based design optimization.


One of the advantages of a CBIS over a viscous element is that it can enhance the energy dissipation efficiency. The energy dissipation enhancement mechanism is described by the factor ψ [32] (shown in Table 1), which equals the ratio of the response mitigation of an SDOF structure with CBIS to that of an SDOF structure with a viscous element having the same additional damping coefficient as the CBIS, that is:

$$\psi = \frac{\sigma\_{lL\_0}(\zeta) - \sigma\_{lI}(\mu, \underline{\xi}, \kappa)}{\sigma\_{lL\_0}(\zeta) - \sigma\_{lL\_0}(\zeta + \underline{\xi})}. \tag{29}$$

The degree of the energy dissipation enhancement of the CBIS can be adjusted by adding the following supplementary constraint condition to the optimization problem by:

$$
\psi \cong \psi\_{0\prime} \tag{30}
$$

where ψ0 is a constant during and the recommended range 1< ψ0 ≤2, according to the many numerical case studies [32]. Here, ψ is 8.7517, which means that the inerter element has fully played its role, and, in the case of the same additional damping ratio, the energy dissipation efficiency of CBIS is 8.7517 times that of a purely viscous element. To illustrate the effects of CBIS on the seismic performance in the time domain, dynamic time–history analyses were conducted to further verify the design results under harmonic excitation and seismic excitations. Four seismic waves are used as external excitations—the El Centro record (1940, NS), the ground motion recorded at Tohoku University during the 2011 Tohoku earthquake (M = 9.0, PGA = 3.33 m/s2), Kobe record (1995) and Chi–chi record (1999). The Tohoku wave occurred on 11 March 2011 and was part of the most powerful known earthquake in Japan. For the SDOF structure, the inherent damping ratio is ζ = 0.02. The natural period on the rigid base is 1.00 s.

## 3.3.1. Harmonic Excitation

Using the performance-based optimization results in Table 1, the controlled and uncontrolled responses of the structure are compared under harmonic excitations. The natural frequency of the structure is 1 Hz and the frequency range of harmonic excitations is 0–3 Hz. Figure 9 shows displacement amplification factors of the uncontrolled and a CBIS-equipped SDOF structure. The peak value reduction effect is 71.75%. The response of a structure at the resonant frequency is significantly reduced by the CBIS.

**Figure 9.** Displacement amplification factors.
