## 2 The Basic Theory of Shock Response Prediction  
A new prediction method of shock response based on the hybrid FE-SEA method is presented in this paper. The flow chart for the shock response predictions process in this article is illustrated in Figure 1. It is known that the amplitude of the response is mainly determined by the frequency response function, and the typical spacecraft structure is a small damping vibration system. Therefore, we start with the analysis of the loading signal. The quasi-steady state loading method is adopted. Firstly, based on short-time Fourier transform (\(STFT\)) technology, the time-frequency domain transform of the shock load signal is studied. The amplitude-frequency value of the shock load and the acceleration phase-frequency characteristics of the shock loading in a single degree of freedom system are obtained. Then, the hybrid FE-SEA model of the structure is established. By using the force load input method, the model is loaded. The acceleration frequency response of each frequency band is calculated separately. Finally, the time domain response results are reconstructed based on the inverse short-time Fourier transform (ISTFT) technique by using a series of acceleration frequency response and the corrected phase information. At the same time, the \(SRS\) are calculated based on the shock response spectrum theory. The shock load force function acting on the structure is defined as \(x(t)\). \(STFT\) is used for the time-frequency analysis of the shock time domain signal. It has superior performance in non-linear and non-stationary signal processing. The basic idea of \(STFT\) is to divide the original signal into a number of stationary or appro  
Figure 1: Flow chart for the shock response predictions process. by a window function, which is shifted with time. Then, the frequency spectrum of each time period is calculated and determined. For the convenience of numerical calculation, _STFT_ is discretized in the time domain and frequency domain. After deduction and analysis, the basic form of discrete _STFT_ is written as follows:  
\[S(n,\omega_{r})=STFT_{x}(n,r)=\sum_{m=-\infty}^{\infty}x(t)w(n-t)e^{-i\frac{2 \pi}{N}rt} \tag{1}\]  
in which \(S(n,\omega_{r})\) contains the results of _STFT_ related to frequency and time, \(n\) is the discrete time, \(r\) is the angular frequency, \(\omega_{r}=\frac{2\pi}{N}r\), \(x(t)\) symbolizes the original signal, and \(w(n-t)\) symbolizes a window function. According to the _STFT_ theory, the result of the time-frequency analysis is calculated by using MATLAB tools for programming. From the above Equation (1), we can obtain the following:  
\[F(\omega)_{n}=abs(S(n,\omega_{r})) \tag{2}\]  
\[Ph(\omega)_{n}=angle(S(n,\omega_{r})) \tag{3}\]  
where \(F(\omega)_{n}\) symbolizes the amplitude-frequency value of the shock load, and \(P(\omega)_{n}\) symbolizes the phase-frequency characteristics. The phase information of the response of the \(i^{th}\) subsystem corrected on the basis of original signal could be written as follows:  
\[Ph(\omega)_{j}=F(Ph(\omega))+K_{i} \tag{4}\]  
where \(K_{i}\) is the phase delay factor of the shock load passing through the \(i^{th}\) subsystem. \(K_{i}\) is related to the damping of the \(i^{th}\) subsystem and the distance between the \(i^{th}\) subsystem and the shock source. The sine shock load signal is loaded into the single degree of freedom system with a different damping ratio. The acceleration response analysis of the system is shown in Figure 2. Figure 2a,b shows the acceleration responses of the system with a large damping ratio (0.5) and small damping ratio (0.02), respectively. It is found that the phase of the acceleration response is almost the same as that of the loading signal in the case of small damping. The above study reveals that when the shock load acts on the structure piecewise and the response time is very short, the phase change is very small in the whole loading time domain. The damping coefficient of the system is generally less than 0.1.