## 2 Fast Frequency Domain Method
### Development of Fast Frequency Domain Method  
To overcome the limitation of the CFDM for ignoring the complex response of the system and obtain both the real and imaginary parts of the dynamic response, the dual imaginary part of applied force can be brought back to Equation (2). In this case, Equation (2) with the governing equation of motion under complex force excitations can be rewritten as:  
\[m\ddot{u}+(1+i\eta)ku=f(t)+ig(t), \tag{12}\]  
in which \(g(t)\) is dual force which is constructed to match the measured real-valued force \(f(t)\). The force \(g(t)\  
Thus, the complex coefficient Equation (12) can be decomposed into a set of two differential equations with real coefficients:  
\[ \begin{array}{l}\dot{m}_{R}+k(u_{R}-\eta u_{I})=f(t)\\ \dot{m}_{I}+k(\eta u_{R}+u_{I})=g(t)\end{array} \Biggr{\}}, \tag{17}\]  
where, \(u_{R}=\text{Re}(u)\) and \(u_{I}=\text{Im}(u)\) represent the real and imaginary parts of the displacement \(u\), respectively. In the DTDM, the real part and the imaginary part of response satisfy dual relationship and are the exact solution. In the DTDM, the steady-state solution of Equation (17) can be given as:  
\[u_{Rn}=\frac{F_{0}}{k}+2\sum_{j=1}^{N/2-1}\left(\frac{F_{j}\left[\left(k-\theta _{j}^{2}m\right)\cos\left(\theta_{j}t_{n}\right)+\eta k\sin\left(\theta_{j}t_{ n}\right)\right]}{\left(k-\theta_{j}^{2}m\right)^{2}+\eta^{2}k^{2}}\right)+(-1)^{n} \frac{F_{N/2}\left(k-\theta_{N/2}^{2}m\right)}{\left(k-\theta_{N/2}^{2}m \right)^{2}+\eta^{2}k^{2}}, \tag{18}\]  
\[u_{Int}=-2\sum_{j=1}^{N/2-1}\frac{F_{j}\left[\eta k\cos\left(\theta_{j}t_{n} \right)-\left(k-\theta_{j}^{2}m\right)\sin\left(\theta_{j}t_{n}\right)\right] }{\left(k-\theta_{j}^{2}m\right)^{2}+\eta^{2}k^{2}}-(-1)^{n}\frac{F_{N/2}\eta k }{\left(k-\theta_{N/2}^{2}m\right)^{2}+\eta^{2}k^{2}}, \tag{19}\]  
Comparing the results of Equations (10) and (11) with Equations (18) and (19), it can be seen that the real part of the displacement obtained by the CFDM and the DTDM are same except the first term. The CFDM only obtain the real part solution of the \(j=1,2,\ldots,N/2-1\) term, resulting in two issues in the CFDM. The first issue of the CFDM is that the real part of the calculation result is reasonable, but the imaginary part solution of the response cannot be obtained. The second problem is that when the force is non-zero mean value, the CFDM would have calculation error because of the first term of Equation (10). For the DTDM, it has two problems as well, including a need to construct dual force in advance and the extremely intensive computation efforts needed to obtain the solution by direct use of Equations (18) and (19). To address the problems of the two above mentioned algorithms, both the CFDM and the DTDM, a fast frequency domain method (FFDM) can be used to make the solution accurate and meanwhile reduce the computation efforts. Comparing Equation (3) with Equations (14) and (15), the \(F_{j}\), \(a_{j}\), and \(b_{j}\) would satisfy the following relationships:  
\[F_{0}=\frac{a_{0}}{2},F_{j}=\frac{1}{2}(a_{j}-ib_{j}),\ (j=1,2,\ldots N/2-1), \ F_{N/2}=\frac{a_{N/2}}{2}, \tag{20}\]  
Therefore, the complex load can be expressed as:  
\[\widetilde{f_{n}}=f_{n}+ig_{n}=F_{0}(1+i\eta)+2\sum_{j=1}^{N/2-1}F_{j}e^{i \theta_{j}t_{n}}+F_{N/2}e^{i\theta_{N/2}t_{n}}, \tag{21}\]  
In the FFDM, a new Fourier series \(D_{j}\) is defined by the force Fourier series \(F_{j}\) in Equation (3) as follows:  
\[D_{j}= \begin{cases}F_{0}(1+i\eta)&j=0\\ 2F_{j}&j=1,\ldots,N/2-1,\\ F_{N/2}&j=N/2\end{cases}  \tag{22}\]  
Then the dynamic displacement response at time \(t_{n}\) can be rewritten as:  
\[u_{n}=\sum_{j=0}^{N/2-1}H_{j}D_{j}e^{i\frac{2\pi inj}{N}}+(-1)^{n}\frac{D_{N/2 }}{-\theta_{N/2}^{2}m+(1+i\eta)k}, \tag{23}\]Comparing Equation (23) to Equations (18) and (19), it can be seen that they are the same. In fact, the second term on the right of Equation (23) is usually a small quantity because the amplitude \(F_{N/2}\) of the highest effective frequency in the Fourier spectrum is far less than that of the prominent frequency.