## 5 In-Structure Response Spectra
### Procedure to Calculate ISRS  
The overall procedure to calculate ISRS through coupled analysis using a two-DOF system in the frequency domain proposed in this study can be summarized as follows.  
Step (1) Uncoupled response analysis of primary structure: Time history analysis of the primary structure is performed for a given ground motion set comprised of two horizontal and one vertical accelerogram and the relative displacement and absolute acceleration responses at the equipment location and in the direction of interest are calculated.  
Step (2) Calculation of mechanical impedance: Harmonic analysis of the primary structure is performed for a unit harmonic force acting at the equipment location and in the direction for which the ISRS is calculated. The displacement response is calculated for the same DOF as the input force. The mechanical impedance of the primary structure is calculated as the ratio of the input force to the output displacement at each frequency.  
Step (3) Calculation of input to coupled analysis: Fourier transformation of the relative displacement response at the equipment location obtained in Step 1 and the ground acceleration in the direction of interest is performed. Then, the input force to the two-DOF model, \(F_{i}^{*}\), is calculated using Equation (19) in the frequency domain.  
Step (4) Coupled analysis using the two-DOF model: Coupled analysis of the two-DOF system is performed using Equation (18) in the frequency domain. The absolute acceleration of the equipment is calculated using Equation (20). Then, the absolute acceleration of the equipment is transformed into the time domain by inverse Fourier transform and the peak absolute acceleration is determined.  
Step (5) Calculation of ISRS: Step 4 is repeated for different equipment frequencies where the ISRS is defined. The calculated peak accelerations are plotted with respect to the equipment frequency.  
In addition, the following procedure was used to calculate ISRSs using decoupled analysis for comparison in this study.  
Step (A) Decoupled analysis using SDOF system: Time history analysis of an SDOF system representing equipment is conducted in the time domain for the base excitation defined by the absolute acceleration response at the equipment location and in the direction of interest calculated in Step 1 for the coupled analysis procedure.  
Step (B) Calculation of ISRS: Step A is repeated for different equipment frequencies where the ISRS is defined. The relationship between the peak acceleration of the equipment and the equipment frequency is plotted.  
Using a computer with a 32-core CPU and 128-GB RAM, the two-DOF system analysis in the frequency domain spent 65 sec to compute an ISRS corresponding to 200 natural frequencies for a 36-sec-long ground motion set. However, one execution of the modal  
Figure 8: Mechanical impedance by harmonic analysis at equipment location 1 in H1 direction.  
time history analysis using the full FE model of the auxiliary building with an SDOF oscillator spent 6060 sec. Moreover, the modal time history analysis shall be repeated for 200 natural frequencies of the equipment, which result in 200 \(\times\) 6060 sec = 337 h. Therefore, the two-DOF system analysis is much more efficient than the conventional coupled analysis using a full FE model.