## 4 Energy Spectra for Damped and Inerter-Based Structures
### Energy Spectrum for Damped SDOF Element  
Considering a series of SDOF elements with an inherent damping ratio where \(\xi_{s}=\ c/2\sqrt{km}=0.02\), a single dashpot where \(\xi_{d}=0.02\) and the damping coefficient of the primary SDOF element is the same, and an undamped period varying from 0.02 s to 10 s, the normalized energy spectra can be calculated by Equations (13) and (17) for input energy (\(f_{EO}\)) dissipated energy of the primary SDOF element (\(f_{E\otimes}\)), and dissipated energy of the single dashpot (\(f_{E\otimes}\)), shown in Figure 14, where the results of TAP077 are shown as an example by a thicker line. The subscript 0 refers to the condition of an SDOF element with a single dashpot. As seen in Figure 14, the normalized curves of the energy spectra, including \(f_{EI0}\), \(f_{E\otimes}\), and \(f_{E\otimes}\), can be divided into three parts: ascending in the short period, where the spectral values increase with the period; plateau in the moderate period, where the spectral values fluctuate at some value; and descending in the long period, where the spectral values decline with the period. In addition, the normalized dissipated energy spectra of the primary SDOF element \(f_{E\otimes}\) and single dashpot \(f_{E\otimes}\) are approximately half of the input energy spectra \(f_{E\otimes}\). Note that the contributions of energy dissipation are the same for the SDOF element and the dashpot when their damping coefficients are the same. Inspired by the results obtained in Figure 14, a model was adopted to approximate the energy spectra consisting of three regions: linear variation in the short periods, a constant region in the moderate periods, and a decreasing curve in the long periods. The three-region model can summarize  
Figure 14: Normalized seismic-induced energy spectra of SDOF element connected to a single dashpot. (**a**) Input energy spectra of SDOF element connected to a single dashpot. (**b**) Dissipated energy spectra of primary SDOF element. (**c**) Dissipated energy spectra of single dashpot. the statistical characteristics of the seismic-induced energy spectrum. To provide an average result, the mean and mean plus one standard deviation (SD) curves of the normalized seismic-induced input energy spectra were calculated, shown in Figure 15. The mean plus one SD curve was used as a representative curve and the following equations were introduced to characterize the fitted three-region seismic input energy spectrum [23; 56]:  
\[f_{EI0}=\left\{ \begin{array}{ll}a+\left(f_{EI0\max}-a\right)\ \frac{T-T_{0}}{T_{1}-T_{0}}&T_{0}\leq T<T_{1}\\ f_{EI0\max}&T_{1}\leq T<T_{g}\\ \left(\frac{T_{g}}{T}\right)^{\prime}f_{EI0\max}&T\geq T_{g}\end{array} \right., \tag{18}\]  
where \(a\) is the normalized spectral ordinate, corresponding to the lower considered period, 0.02 s (\(T_{0}\)); \(f_{EI0\max}\) is the spectral value of the plateau of the energy spectrum; \(T_{1}\) and \(T_{g}\) denote the start and end periods of the plateau of the energy spectrum, respectively; and \(r\) is the parameter governing the velocity of decay in the descending part of the energy spectrum. The least squares method was applied to the energy spectrum representative curve [56], thus:  
\[\Delta_{\min}=\sum_{i=1}^{n}\left(f_{EI0}-\overline{f_{EI0}}\right)^{2}, \tag{19}\]  
where \(\overline{f_{EI0}}\) is the representative value and \(f_{EI0}\) is the fitted value of the seismic-induced input energy spectrum of the SDOF element with a single dashpot. The fitted input spectrum is shown in Figure 16 and the corresponding parameters are given in Table 5. As illustrated in Table 5, the energy responses of structures with a long period (\(T_{g}=3.1\) s) were large and located in the plateau. Consistent with the results from Section 3.3, the deep cover soft deposit causes a larger response for long-period structures, and this should be quantitatively considered.