## 3 Numerical Analysis
### Seismic Excitation  
The third example uses seismic excitation with ground acceleration \(\ddot{u}_{g}(t)\) on a SDOF with \(m=1\) kg, \(k=4\pi^{2}/1.28^{2}\) N/m, and \(\eta=0.1\). The equivalent real-valued force \(f(t)\) is:  
\[f(t)=-m\ddot{u}_{g}(t), \tag{34}\]  
In this example, the acceleration \(\ddot{u}_{g}(t)\) was the north-south component of the ground motion recorded at a site in El Centro, California during Imperial Valley earthquake of 18 May 1940. Figure 6 shows the ground motion and its Fourier spectrum. The time interval of seismic wave sampling is 0.02 s and the end of the force is filled with zero after the duration becomes 40.96 s.  
Figure 7 shows the comparison of the dynamic displacement responses from the developed FEDM with the DTDM and the CFDM under El Centro ground motion excitation. Assuming that the exact displacements obtained by the DTDM, Table 3 lists the relative errors of the amplitude of displacements of various methods. Since \(\left|D_{N/2}\right|=2.0819\), \(\mathrm{max}\left|D_{j}\right|=245.7599\), that is \(\left|D_{N/2}\right|<<\mathrm{max}\left|D_{j}\right|\), ignoring the effect of highest effective frequency in Fourier series of force in approximate FEDM would induce very little error. The real and imaginary parts obtained by the accurate FEDM and the approximate FEDM are perfectly in agreement with those by using the DTDM. Therefore, it is reasonable to use the approximate FEDM to analyze the seismic response of system with complex damping. Since the El Centro wave time history applied is a zero-mean value, the real part obtained by the CFDM coincides  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **CFDM** & **DTDM** & **Accurate FEDM** & **Approximate FEDM** \\ \hline \(u_{R}\) & 0.09 & 0 & 0 & 0 \\ \(u_{l}\) & 99.01 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The relative errors of the amplitude of displacements under a non-zero mean value force (%).  
Figure 5: Displacement response time history of different methods under non-zero mean cosine force (\(\theta/\omega=1\)).  
with those by the DTDM, but the imaginary part is almost equal to zero. This finding further confirms that there exists dual imaginary part, although it has not physical meaning, it has theoretical meaning to make the solution complete.  
Table 4 compares the computation time of all the four different numerical methods analyzed in all three examples. It can be seen that the time consumed by the developed FFDM is significantly shorter than the CFDM and the DTDM. The DTDM is the slowest due to the fact that a straightforward evaluation of the summation of Equations (18) and (19) would require lots of multiplications. The main computation time for the CFDM contains two parts, one for solving the FFT of force and the other for solving the IFFT of displacement. The procedure of the approximate FFDM is the same as the CFDM, but the IFFT series in the approximate FFDM is only half of those in the CFDM. In the accurate FFDM, only one extra term adding calculation is demanded on the last term of Equation (23) compared with approximate FFDM, which leads to the time consumed by the accurate FFDM is more than that by  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **CFDM** & **Accurate FFDM** & **Approximate FFDM** \\ \hline \(u_{R}\) & 0 & 0 & \(4.59\times 10^{-5}\) \\ \(u_{I}\) & 100 & 0 & \(4.59\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The relative errors of the amplitude of displacements under El Centro excitation (%).  
Figure 6: The acceleration time history and Fourier spectrum of El Centro wave.  
Figure 7: Displacement time history under El Centro excitation.  
approximate FEDM, but less than by the CFDM. Therefore, the approximate FEDM outperforms the others when considering both the efficiency and accuracy.