## 4 Energy Spectra for Damped and Inerter-Based Structures
### Energy Spectrum for Inerter-Based Structure  
In terms of the energy spectrum analysis for the inerter-based structure, a typical TVMD in Figure 4 was considered to suppress the oscillating vibration of the SDOF element. The key parameters of the TVMD were considered, including the inertance-to-mass ratio \(\mu=0.05\), \(0.10\), and \(0.25\); the nominal damping ratio \(\xi_{d}=0.02\); and the stiffness ratio \(\kappa=0.50\). For a fair comparison, the damping coefficient of the TVMD was deliberately set to be the same as that of the single dashpot and primary SDOF element. Applying this TVMD to the primary SDOF elements considered in Section 4.1, normalized energy spectra were obtained for the input energy (\(f_{EI}\)), primary SDOF element (\(f_{Es}\)), and dashpot of the TVMD (\(f_{Ed}\)) under excitation of 40 computed ground surface acceleration time history responses, as shown in Figure 17.  
Figure 16: Representative and fitted curves for seismic-induced input energy spectrum of SDOF element connected to a single dashpot.  
Figure 17: Seismic-induced energy spectra of SDOF element connected to the TVMD.  
After that, the mean curves of \(f_{EI}\), \(f_{Es}\), and \(f_{Ed}\) were obtained and are given as solid blue, black, and red lines, respectively, in Figure 17. Comparing the input energy spectra \(f_{EI}\) in Figure 17a-c to the SDOF element connected to a dashpot in Figure 15, the input energies of the SDOF element connected to the TVMD and the single dashpot are the same, basically representing the same burden of energy dissipation. The fitted input energy in Figure 16 can be directly used for the SDOF-TVMD system. Faced with this, as evident from Figure 17g-i, with an increased inertance-to-mass ratio \(\mu\), the energy dissipated by the dashpot of the TVMD (characterized by \(f_{Ed}\)) increases rapidly, and it dissipates via the primary SDOF element (characterized by \(f_{Es}\)), as shown in Figure 17d-f, where the decrease is dramatic.  
Quantitatively, we let \(f_{E\mathrm{Imax}}\), \(f_{E\mathrm{smax}}\), and \(f_{Ed\mathrm{max}}\) denote the maximum values of \(f_{EI}\), \(f_{Es}\), and \(f_{Ed}\), respectively. It was shown that when \(\mu=0.05\), \(0.10\), and \(0.25\), the \(f_{E\mathrm{smax}}\) percentages are \(43.02\%\), \(36.16\%\), and \(23.27\%\) of \(f_{E\mathrm{Imax}}\) respectively, whereas for \(f_{Ed\mathrm{max}}\) they are \(56.98\%\), \(63.84\%\), and \(76.73\%\) of \(f_{E\mathrm{Imax}}\), respectively. To make this clear, the ratio of \(f_{E\mathrm{Imax}}/f_{E\mathrm{smax}}\) was calculated to be \(1.33\), \(1.78\), and \(3.30\) for \(\mu=0.05\), \(0.10\), and \(0.25\), respectively. Compared with the single dashpot with the same damping coefficient (\(f_{E\mathrm{d0 max}}/f_{E\mathrm{d0 max}}=1\)), the TVMD exhibits improved ability and efficiency for energy dissipation (corresponding to \(f_{E\mathrm{dmax}}/f_{E\mathrm{Imax}}>50\%\)), which is called the energy dissipation enhancement effect in this study. More importantly, a larger inertance-to-mass ratio is beneficial for this. According to the present study, the inerter of the TVMD amplifies the dashpot deformation and makes it larger than the entire deformation of the TVMD, which lays the foundation for improved energy dissipation. The energy spectrum analysis in this study explicitly reveals the energy mechanism and characteristics of a TVMD for the first time.