## 3 Measured Results Analysis
### Cross Spectral Density Function (CSD), Phase Spectrum (PS), and Coherence Function (CF)  
In the frequency response function, frequencies corresponding with the peak value of the amplitude-frequency curve can be determined as the natural frequency, and the phase spectrum can reflect the specific phase and the phase-difference values of different floors in the resonance region. Taking frequency response as an example:  
\[H_{d}(\omega)=\left|H_{d}(\omega)\right|\exp(j\varphi(\omega)) \tag{6}\]  
\[\left|H_{d}(\omega)\right|=\frac{1}{m\sqrt{\left(\omega_{0}^{2}-\omega^{2} \right)^{2}+\left(2\xi\omega_{0}\omega\right)^{2}}} \tag{7}\]  
\[\varphi(\omega)=\arctan\frac{-2\xi\omega_{0}\omega}{\omega_{0}^{2}-\omega^{2}} \tag{8}\]  
where \(\omega_{0}\) is frequency response function; \(\varphi(\omega)\) is the phase-frequency function; \(\omega\) is the circular frequency, rad/s; \(m\) is mass, kg; and \(\omega_{0}\) is the natural circular frequency of the structure. Coherence function is the index of correlation degree for the two-dimensional random vibration signal in the frequency domain. The coherence function is the quotient obtained by dividing the square of the mode of the cross power spectral density function by the self spectral product of the excitation and response. Equation (9) is the calculation equation of the coherence function:  
\[C_{xy}(k)=\frac{\left|S_{xy}(k)\right|}{S_{xx}(k)S_{yy}(k)} \tag{9}\]  
Figure 10: Acceleration auto-power spectral density functions: (**a**) auto-power spectral density along the X axis; (**b**) auto-power spectral density along the Y axis; (**c**) auto-power spectral density along the torsional direction. where \(S_{xy}(k)\) and \(S_{yy}(k)\) and are the estimations of the auto-power spectrum density function of random vibration excitation and response signals processed by the average periodic diagrams; and \(S_{xy}(k)\) is the estimations of the cross-spectral density function of excitation and response signals. The coherence function (\(C_{xy}(k)\)) was adopted in this paper to evaluate the frequency response function. If \(C_{xy}(k)\) became closer to 1, this meant that the noise disturbance was slight and the estimation results of frequency response function were good. The measured environment was complicated, because several factors, such as typhoon, normal wind, earth pulsation, mechanical vibration, and ambient noise, would affect the accuracy of the excitation function. Therefore, the cross-spectral density curves between the acceleration data on the 6 th, 12 th, 18 th, 24 th, and 30 th floors along the X and Y axes and those of the 32 th floors were presented, and the cross-spectral density curves between the torsional angular acceleration data of the 8 th, 12 th, 16 th, 20 th, 24 th, and 28 th floors and those of the 32 th floor were also given, by which the corresponding phase spectrum and the coherence functions were sketched, and the cross-spectral density function, phase spectrum, and coherence functions along X axis and Y axis and the torsional direction were all obtained. More details are presented in Figures 11-13. Figure 11 shows that wave crests of the cross-spectral density function between the 6 th, 12 th, 18 th, 24 th, and 30 th floors and the 32 th floors along the X axis are clearly visible, and the phase angles of all cross-spectral density function are 180\({}^{\circ}\) when the first-order frequency of wave crest is 0.66 Hz, which explains that the acceleration vibration directions of the 6 th, 12 th, 18 th, 24 th, and 30 th floors are the same, and that their coherence coefficients all closely approximate to 1. This phenomenon also proves that there is strong coherence between different floors, and that the vibration modes of different floors are in good agreement with the first-order vibration modes.