## 5 In-Structure Response Spectra
### Decoupling Criteria  
In general, coupled analysis considering the equipment-structure interaction causes a difference in the dynamic response of both equipment and structure compared to the decoupled analysis. However, the difference between coupled and decoupled analyses is not always significant enough to warrant the effort and time required for the coupled analysis. ASCE/SEI 4-16 provides decoupling criteria to determine whether the decoupled analysis of a coupled system is acceptable and which model is appropriate among the three types illustrated in Figure 17. Models A and B are models for decoupled analysis, in which the response acceleration of the primary system is used as the input to the secondary system. The only difference between these two models is that in model B, the mass of the secondary system is lumped into the mass of the primary system, as shown in Figure 17b. Model C is a coupled model. The decoupling criteria for selecting an appropriate model are plotted in Figure 18 [5]. Region U allows a decoupled analysis so that all three model types are applicable. In Region C, a coupled analysis should be applied, and only model C is applicable. Region A is an area where the response error of Model A increases, and only Models B and C are applicable. Region B is an area where the response error of Model B increases, with only Models A and C being acceptable. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Natural Frequency (Hz)**} & \multirow{2}{*}{**Equipment Location**} & \multirow{2}{*}{**Modal Mass (10\% kg)**} & \multicolumn{3}{c}{**Mass Ratio of the Equipment to the Dominant Mode (\%)**} \\ \cline{5-6}  & & & & **1 (1)** & **0.1 (1)** & **0.01 (1)** \\ \hline \multirow{5}{*}{H1} & 4.65 & 3 & 213,000 & \(994\times 10^{-6}\) & \(99.4\times 10^{-6}\) & \(9.94\times 10^{-6}\) \\ \cline{2-6}  & & 1 & 155 & 1.37 & 0.137 & 0.0137 \\ \cline{2-6}  & 5.33 & 2 & 56.6 & 3.75 & 0.375 & 0.0375 \\ \cline{2-6}  & & 4 & 129 & 1.65 & 0.165 & 0.0165 \\ \hline \multirow{5}{*}{H2} & 4.65 & 3 & 474,000 & \(447\times 10^{-6}\) & \(44.7\times 10^{-6}\) & \(4.47\times 10^{-6}\) \\ \cline{2-6}  & & 1 & 64.0 & 3.31 & 0.331 & 0.0331 \\ \cline{2-6}  & & 2 & 124 & 1.71 & 0.171 & 0.0171 \\ \cline{2-6}  & & 4 & 129 & 1.65 & 0.165 & 0.0165 \\ \hline \multirow{5}{*}{V} & 14.4 & 2 & 29.9 & 7.09 & 0.709 & 0.0709 \\ \cline{2-6}  & 15.3 & 3 & 8.08 & 26.2 & 2.62 & 0.262 \\ \cline{1-1} \cline{2-6}  & 16.1 & 4 & 7830 & 0.0271 & 0.00271 & 0.000271 \\ \cline{1-1} \cline{2-6}  & 16.4 & 1 & 773 & 0.274 & 0.0274 & 0.00274 \\ \hline \end{tabular} \({}^{(\rm I)}\) Total mass ratio, the equipment mass divided by the total building mass, \(212\times 10^{6}\) kg.  
\end{table} 
Table 4: Mass ratio of the equipment to the dominant mode for different equipment locations and directions. The solid boundaries between regions U and A and between regions B and C in Figure 18 are expressed by the following equation based on [PERSON] and [PERSON] (1986) [37]:  
\[\frac{\omega_{e}}{\omega_{s}}=\left|\sqrt{\frac{a^{2}-2a}{2a(1-\mu_{A})-4}}\right| \tag{27}\]  
where  
\[a=\frac{2}{\left(1-e_{A}\right)^{2}} \tag{28}\]  
and \(\omega_{e}\) is the natural frequencies of the equipment; \(\omega_{s}\) is the natural frequency of the decoupled structure in Model A; \(\mu_{A}\) is the modal mass ratio calculated for the modal mass of Model A; \(e_{A}\) is the acceptable error in the natural frequency predicted by Model A compared to that calculated by the coupled Model C. The dashed boundaries between regions U and B and between regions A and C in Figure 18 are expressed by the following equation [37]:  
\[\frac{\omega_{e}}{\omega_{l}}=\left|\sqrt{\frac{\left(1+\mu_{B}\right)^{-1}-b }{\left(b-b^{2}\right)}}\right| \tag{29}\]  
Figure 17: Different models of a coupled system: (**a**) Model A; (**b**) Model B; (**c**) Model C.  
Figure 18: Criteria for modeling a coupled system.