## 5 In-Structure Response Spectra
### Effect of Equipment Mass  
The decoupled and coupled mean ISRS at each of the four equipment locations are compared in Figures 12-15, in which the equipment mass is 1%, 0.1%, and 0.01% of the total building mass, and equipment damping is 3%. The ISRS in the H1 and H2 directions at equipment locations 1 and 2 represent in-plane vibration, and as the mass ratio increases, the coupled ISRS decreases below the decoupled ISRS owing to the equipment-structure interaction in the proximity of the dominant frequency. In contrast, the ISRS in the V direction at equipment location 1 represents the out-of-plane vibration, and the acceleration response of the coupled ISRS tends to increase more than that of the decoupled ISRS over frequencies higher than 4 Hz as the mass ratio increases. The V-direction coupled ISRS at equipment location 2, which also represents an out-of-plane vibration, is lower than the decoupled ISRS near the dominant frequency of 15 Hz, but the influence of the mass ratio is unclear. The V-direction coupled ISRS at equipment location 3 showed a significant decrease as the mass ratio increased around the dominant frequency. In the case of the H2-direction coupled ISRS at equipment location 4, representing out-of-plane vibration as well, the equipment-structure interaction effect is noticeable only for 1% mass ratio at a dominant frequency of 5 Hz, while 0.1 and 0.01% mass ratios are more effective at the dominant frequency of 10 Hz. The H1- and V-direction coupled ISRS, representing in-plane vibration, also shows that the equipment-structure interaction becomes more significant as the mass ratio increases. Figure 11: Primary structure displacement at equipment location 3: (**a**) Full FE model; (**b**) Two-DOF model. Figure 14: ISRS for 3% equipment damping at the equipment location 3: (**a**) H1; (**b**) H2; (**c**) V.  
Figure 12: ISRS for 3% equipment damping at the equipment location 1: (**a**) H1; (**b**) H2; (**c**) V.  
Figure 13: ISRS for 3% equipment damping at the equipment location 2: (**a**) H1; (**b**) H2; (**c**) V.  
Figure 16 shows mode shapes corresponding to the dominant frequencies in the V-direction decoupled ISRS for equipment locations 1 to 3. As illustrated in Figure 16a, the dominant mode shape for vertical vibration at equipment location 1 is not as prominent as other parts, and major vertical deformation occurs on the roof and other slabs. In the case of equipment location 2, the local vertical deformation at the location is not prominent compared to other parts too, and the vertical deformation appears to be dispersed in most slabs. Figure 16: _Cont._  
Figure 15: ISRS for 3% equipment damping at the equipment location 4: (a) H1; (b) H2; (c) V.  
Table 4 summarizes the modal mass ratio, defined as the ratio of the equipment mass to the modal mass for the dominant frequency in the decoupled ISRS. Three equipment masses corresponding to 1, 0.1, and 0.01% of the total building mass were considered, and these values are denoted by the total mass ratio below: The modal mass ratio of the equipment is expressed by the following equation [5]:  
\[\Lambda_{i}=M_{s}/M_{pi} \tag{25}\]  
where  
\[M_{pi}=\phi_{ci}^{-2} \tag{26}\]  
and \(M_{s}\) is the equipment mass; \(\phi_{ci}\) is the value of the i-th mode eigenvector normalized to the mass matrix of the primary structure. It is observed that the modal mass ratio is significantly higher than the total mass ratio for 14.4, and 15.3 Hz, both of which are related to out-of-plane vibration. In the case of vertical vibration, the modal mass of 15.3 Hz  
Figure 16: Mode shape for the natural frequency dominant in the vertical direction: (**a**) equipment location 1 (16.4 Hz); (**b**) equipment location 2 (14.4 Hz); (**c**) equipment location 3 (15.3 Hz). vibration mode, of which the deformation is localized to equipment location 3 as shown in Figure 16c, has the smallest value, and the corresponding mass ratio is the highest.