## 3 Measured Results Analysis
### Cross Spectral Density Function (CSD), Phase Spectrum (PS), and Coherence Function (CF)
All the phase angles of cross-spectral density function on the 6 th, 12 th, and 18 th floors are 0\({}^{\circ}\) when the frequency of wave crest is 2.3 Hz, and the coherence coefficient is 1, which shows that the vibration directions of the 6 th, 12 th, and 18 th floors are the same, and have high cooperativity. While the phase angle of the cross-spectral density function on the 30 th floor is 180\({}^{\circ}\), which shows that the vibration direction of the 30 th floor at the second order frequency is opposite to that at the 6 th, 12 th, and 18 th floors. The phase angle of the cross-spectral density function on the 24 th floor is within the range of 0-180\({}^{\circ}\), and is nearly 0\({}^{\circ}\), which indicates that the vibration directions of the 24 th floor have a higher vibration cooperativity with the 6 th, 12 th, and 18 th floors than with the 30 th floor, and that the vibration modes of different floors are in good agreement with the second-order vibration modes. The phase angles of the cross-spectral density function of the 6 th, 12 th, and 30 th floors are 180\({}^{\circ}\) when the frequency of wave crest is 4.2 Hz, and those of the 18 th and 24 th floors are 0\({}^{\circ}\), which demonstrates that there are consistent vibrations among the 6 th, 12 th, and 30 th floors and between the 18 th and 24 th floors. While the vibrations existing among the 6 th, 12 th, and 30 th floors are opposite to those existing between the 18 th and 24 th floors, and the coherence coefficients of the 18 th and 24 th floors all approach 1, which proves that the vibration cooperativity at those two points is extremely high and that vibration modes of different floors are consistent with the third-order vibration modes. The same principle as above is adopted to analyze the measured cross-spectral density, phase spectrum, and coherence functions along the Y axis and the torsional direction. The measured CSD, PS, and CF of the first three modes along the X and Y axes axis are shown in Table 1. The measured CSD, PS, and CF of the first three modes along the torsional direction are shown in Table 2.  
Figure 13: The measured CSD, PS, and CF along the torsional direction. Vibration mode, frequency, and damping ratio are key modal parameters to commonly reflect dynamic characteristics of structures. According to the synthetic evaluation of the auto-power, cross-spectral, phase spectrum, and coherence functions of different floors along the X axis, Y axis, and torsion direction, modal frequencies corresponding to the first three vibration modes can be clearly diagnosed. The first three order frequencies along the X axis were 0.66 Hz, 2.30 Hz, and 4.20 Hz in sequence, and those along the Y axis were 0.56 Hz, 2.15 Hz, and 4.30 Hz, while those along the torsional direction were 0.56 Hz, 2.15 Hz, and 4.35 Hz, respectively. The first three vibration modes along the X axis, Y axis, and the torsional direction of the measured building were calculated based on the parameters of acceleration APSD, CSD, PS, and CF. Besides, the floors from below along the X axis were 0.56 Hz, 2.15 Hz, and 4.30 Hz, while those along the Y axis were 0.56 Hz, 2.15 Hz, and 4.30 Hz, respectively. The first three vibration modes along the Y axis were 0.56 Hz, 2.15 Hz, and 4.35 Hz, respectively. The first three vibration modes along the Y axis were 0.56 Hz, 2.15 Hz, and 4.30  
the X and Y axes were the 6 th, 12 th, 18 th, 24 th, 30 th, and 32 th floors, while those along the torsional direction were the 8 th, 12 th, 16 th, 20 th, 24 th, 28 th, and 30 th floors, and the torsional vibration of the 4 th floor approximated half of that in the 8 th floor. The measured vibration modes at the first three orders along the X and Y axes and the torsional direction are shown in Figure 14.