This study proposes the MPR, which is an index for the damage detection of buildings under ambient excitation. An analysis of the relationship between the MPR index and the story stiffness of a damaged building was conducted. Without using modal parameters such as the natural frequency, mode shape, and damping ratio, which were widely adopted in the existing studies of damage detection, this study extracts a modal response from dynamic responses measured for a building 124 H.S. Park, B.K. Oh / Journal of Sound and Vibration 418 (2018) 122e143 structure with no SI process. The presented MPR is a representative value of the modal response and is obtained in the manner described in this section. Since each mode has a different contribution to the behavior and response of a building structure, the impact of each mode needs to be considered to reflect the vibration characteristics of the structure for damage detection. In this regard, since as many modal responses of a measurement location or a DOF, which are extracted from measured dynamic responses such as acceleration, velocity, and displacement responses, as the number of modes to be considered are extracted, the influences of all modes of interest are reflected in the MPR [15]. 

In this study, bandpass filtering [45] is used to extract modal responses. The filtering of dynamic responses is conducted based on the value of the natural frequency shown in the fast Fourier transform (FFT) of the measured dynamic responses. When bandpass filtering is applied, a filtering range needs to be set up. A simple rule for setting up such a range is presented below. In the frequency domain of a measured response, ranges between peaks for natural frequencies, as shown in Fig. 1, and twice the value of the natural frequency for the first mode are compared. The smallest one is set as a bandpass range, which can be expressed as in Eq. (1). Rb ¼ g Minf2 f1; Rig i ¼ 1 to nm (1) where Rb is the selected bandpass range, Ri is the interval between each natural frequency, f1 is the natural frequency of the first mode, nm is the number of modes of interest, and g is a range adjustment coefficient, which can be selected by the user. As shown in Fig. 1, in the case in which the number of modes under consideration is four, the values in the range of 1e3 are calculated. Among those values and the value of twice the natural frequency for the first mode, the smallest one is set as the bandpass range. 

This method enables a set range that sufficiently reflects the response of each mode in the frequency domain, and it also prevents a modal response for a specific mode from being combined with vibration characteristics for other modes. In addition, the filtering can be conducted with the same range for each mode, which results in the consistent extraction of each modal response. The dynamic responses, which are measured from a building structure, are converted to modal responses with a set bandpass range, as expressed in Eqs. (2)e(4