## 4 Energy Spectra for Damped and Inerter-Based Structures
### Optimal Design and Design Cases  
Inspired by the revealed energy working basis of the inerter system, an optimal energy dissipation enhancement-based design principle is developed herein. As explained for Figure 17, from the energy perspective, the advantageous feature of the TVMD is the enhanced efficiency and ability of the dashpot to conduct energy dissipation. The pursuit of enhanced energy dissipation can potentially utilize the co-working mechanism of the inerter, the tuning spring, and the dashpot. For a structure with a predetermined damping ratio and oscillating period, the ratio of \(f_{EI}/\xi_{d}\) to \(f_{Es}/\xi_{s}\), denoting the contribution of the dashpot \(c_{d}\) and the \(c\) value of the primary structure for energy dissipation, respectively, is selected as the optimization objective for maximization. The performance demand design principle is widely accepted as the orientation to guide structural design, which involves achieving stated performance objectives for structures when they are subjected to a stated intensity level of excitation [20; 57; 58]. Considering structural displacement as a key measurement, a target displacement control effect \(\alpha_{t}\) is considered as a constraint condition in the design problem. In summary, the proposed optimal energy dissipation enhancement-based design problem can be expressed in mathematical form:  
\[\left\{ \begin{array}{ll}\underset{\mu,\xi_{d},\kappa}{\mathrm{Maximize}}& \frac{f_{EI}/\xi_{d}}{f_{Es}/\xi_{s}}\\ \mathrm{subject}&\mathrm{to}\,\alpha(\mu,\xi_{d},\kappa)=\alpha_{t}\end{array}  \right., \tag{20}\]  
where \(\alpha\) is the ratio of the root mean square of the displacement of the SDOF-TVMD system to that of the uncontrolled SDOF element. Considering that the maximum displacement is also a very important design parameter of the controlled system, this ratio can also be used in Equation (20) for optimal design. This optimization problem can be solved by the constrained nonlinear multivariable function fmincon compiled in MATLAB (MathWorks, Natick, MA, USA) [59]. Note that during the calculation of \(f_{Ed}\) and \(f_{Es}\) for the optimal design, the computed ground surface acceleration time histories are suggested as the excitation, especially the ground motions, whose \(f_{EI}\) values closely match the fitted input spectrum in Figure 16. An illustrative case was used to introduce the implementation of the proposed optimal design strategy. An SDOF element with an undamped oscillating period of \(1.0\) s was considered, where \(\alpha_{t}\) was prespecified as \(0.50\). Takingthe ground motion of TAP077 as an example, which matches Figure 16, substituting \(\alpha_{t}=0.50\) into Equation (20), the TVMD parameters can be optimized numerically and summarized as \(\mu\)= 0.136, \(\xi_{d}\)= 0.051, and \(\kappa\)= 0.395. To illustrate the advantages of the TVMD against the single dashpot and the uncontrolled SDOF element intuitively, the displacement and acceleration response results under the excitation of TAP077 and CHY102 are given in Figure 18, where the SDOF element controlled by a single dashpot with the same damping ratio of the TVMD is also analyzed. Inspecting Figure 18, including the root mean square and peak responses, it can be seen that the designed TVMD can provide the target displacement to the controlled system under different seismic excitations. The energy response results under different excitations are given in Figure 19. Comparing the single dashpot and TVMD, the designed TVMD shows increased energy dissipation ability (indicated by the black arrow), while the primary structure has a decreased energy dissipation burden (indicated by the red arrow). Quantitatively, the ratio of \(f_{Ed}\) to \(f_{Es}\) is maximized as 2.09 and 2.10 for TAP077 and CHY102, respectively, which is consistent with the results of the optimization design. In addition, the same operation is conducted for the 40 studied signals, of which the average results are summarized.