# Vibration Isolation of a Surveillance System Equipped in a Drone with Mode Decoupling
## 3 Experimental Vibration Characterization
### Frequency Response Function  
A frequency response function is useful to identify the characteristics of a multi-DOF system by analyzing the relationship between an input signal, \(x(t)\), and an output signal, \(y(t)\), in the frequency domain. Moreover, this response can be easily measured with modal testing, which suggests that the frequency response function must be measured to identify the vibration characteristics between a base and a payload. After that, the response at various operational conditions can be predicted to propose the enhanced vibration performance for the system of interest. Therefore, this study focuses on calculating the frequency response function from experiments. Note that simply measuring vibration during operation in the time domain cannot quantitatively compare the VIP because the amplitude of vibration depends on the magnitude of a vibration source. Hence, the frequency response function is the more elaborate metric to identify vibration characteristics from a system dynamic point of view [15]. The frequency response function \(H(f)\)[18] is calculated as  
\[H(f)=\frac{S_{xy}(f)}{S_{xx}(f)}, \tag{19}\]  
where \(S_{xx}\) and \(S_{xy}\) denote the auto spectral density function of an input signal and the cross-spectral density function between the input and output signals, respectively. The auto spectral density function is calculated by multiplying the Fourier transform of input, \(X(f)\), and a complex conjugate form of \(X(f)\), as follows:  
\[S_{xx}(f)=X(f)\times X^{*}(f), \tag{20}\]  
where the Fourier transform of \(x(t)\) is denoted as  
\[X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt. \tag{21}\]  
The cross-spectral density function is calculated by multiplying the Fourier transform of the input signal, \(X(f)\), and output signal, \(Y^{*}(f)\), as follows:  
\[S_{xy}(f)=X(f)\times Y^{*}(f), \tag{22}\]  
In real-world applications, the effects of environmental noises, such as random noise, interfere with vibration measurements. Therefore, averages are considered by repeating the experiments to mitigate the effects of noises, as follows:  
\[\hat{S}_{**}(f)=\frac{1}{N}\sum_{n=1}^{N}(S_{**})_{n}, \tag{23}\]  
where \(N\) denotes the number of averages. Hence, the auto spectral density function from Equation (20) and the cross-spectral density function from Equation (21) was averaged as \(\hat{S}_{xx}(f)\) and \(\hat{S}_{xy}(f)\) to calculate the frequency response function in Equation (19). Note that the system should be excited to measure the vibration signals of input, \(x(t)\), and output, \(y(t)\). There exist various modal testing methods to excite systems and conduct measurements with an impact hammer and vibration exciter [19]. Modal testing using an impact hammer is widely conducted for a small system because it is easy to execute at a laboratory scale. However, it is hard to excite all frequency ranges of interest for a massive system; therefore, a vibration exciter is generally used for such a system. Furthermore, the coherence function can be calculated to validate the relationship between the input and output signals, as follows:  
\[\gamma_{xy}^{2}=\frac{\left|S_{xy}(f)\right|^{2}}{S_{xx}(f)\times S_{yy}(f)}, \tag{24}\]  
where \(S_{yy}\) denotes the auto spectral density function of the output signal, \(y(t)\), indicating the acceleration of a payload. A value of unity in the coherence function shows that the output signal is completely affected by the input signal, \(x(t)\), whereas a value of zero indicates no correlation between the input and output signals.