## 5 Discussion
### Calculation of the Modal Responses  
Figure 31 shows the calculation flow for the first and second modal responses from the nonlinear time-history analysis results. This procedure is based on the procedure originally proposed by [PERSON] [62] for analyzing the nonlinear modal response of a non-isolated planer frame structure extended to analyze that of three-dimensional base-isolated structures. Next, the momentary input energy of the first modal response per unit mass was calculated as follows. From the time-history of the equivalent displacement of the first modal response (\(D_{1U}{}^{*}(t)\)), the momentary input energy of the first modal response per unit mass was calculated as:  
\[\frac{\Delta E_{1U}{}^{*}}{M_{1U}{}^{*}}=-\int\limits_{t}^{t+\Delta t}\dot{D }_{1U}{}^{*}(t)a_{gU}(t)dt. \tag{25}\]  
Figure 30: Peak displacement at isolators for various angles of incidence of seismic input (Model-Tf44): (**a**) UTO0414; (**b**) UTO0416; (**c**) TCU; (**d**) YPT. In Equation (25), \(\dot{D}_{1U^{*}}(t)\) is the derivative of \(D_{1U^{*}}(t)\) with respect to \(t\), \(a_{gU}(t)\) is the ground acceleration component of the U-axis (the principal axis of the first modal response corresponds to \(D_{1U^{*}}\)max = max{\(|D_{1U^{*}}(t)|\)}), and \(\Delta t\) is the duration of a half cycle of the first modal response. The maximum momentary input energy of the first modal response per unit mass (\(\Delta E_{1U^{*}}\)max/\(M_{1U^{*}}\)) is the maximum value of \(\Delta E_{1U^{*}}/M_{1U^{*}}\) calculated by Equation (25) over the course of the seismic event. Figure 32 shows the definition of the maximum momentary input energy of the first modal response per unit mass. As shown in this figure, the half cycle when the maximum equivalent displacement (\(D_{1U^{*}}\)max) occurs corresponds to the half cycle when the maximum momentary energy input occurs. The equivalent velocity of the maximum momentary input energy of the first modal response (\(V_{\Delta E1U^{*}}\)) and the response period of the first modal response (\(T_{1}\)') were calculated such that:  
\[V_{\Delta E1U^{*}}=\sqrt{2\Delta E_{1U^{*}}\mathrm{max}/M_{1U^{*}}}, \tag{26}\]  
\[T_{1}\prime=2\Delta t. \tag{27}\]  
Relationship between the Peak Equivalent Displacement and the Maximum Momentary Input Energy of the First Modal Response  
Figures 33 and 34 show comparisons between the \(V_{\Delta E\mu 1U^{*}}\)-\(D_{1U^{*}}\) curve and the \(V_{\Delta E1U^{*}}\)-\(D_{1U^{*}}\)max relationship obtained from the nonlinear time-history analysis, with  
Figure 31: Calculation flow for the first and second modal responses. the results of Models Tf34 and Tf44 shown in Figures 33 and 34, respectively. These figures confirm that the \(V_{\Delta E\mu 1U}{}^{*}-D_{1U}{}^{*}\) curve fits the plots obtained from the time-history analysis results very well. This is because the contribution of the first modal response to the whole response may be large: The effective modal mass ratio of the first mode with respect to the U-axis is large (more than 0.7) in both Models Tf34 and Tf44. Therefore, the accuracy of the predicted \(D_{1U}{}^{*}\)max relies on the accuracy of the predicted \(V_{\Delta E\mu 1U}{}^{*}\), which is discussed in the next subsection. Figure 32: Definition of the maximum momentary input energy of the first modal response per unit mass (structural model: Model-Tf44, ground motion: TCU, angle of incidence of seismic input: \(\psi=-30^{\circ}\)): (**a**) Hysteresis of the first modal response and (**b**) time-history of the momentary energy input. Comparison of the Maximum Momentary Input Energy and the Bidirectional Momentary Input Energy Spectrum  
Figure 35 shows the prediction of \(V_{\Delta E\Pi U}\)\({}^{*}\) from the bidirectional \(V_{\Delta E}\) spectrum for Model-Tf34. The plots shown in this figure indicate the nonlinear time-history analysis results. In most cases, the bidirectional \(V_{\Delta E}\) spectrum with complex damping \(\beta\) = 0.10 approximated the upper bound of the plot of the time-history analysis results, as shown in Figure 35a,b,e.f.