## 2 Fast Frequency Domain Method  
### Theoretical Background  
The governing equation of motion for a single-degree-of-freedom (SDOF) system with complex damping in forced vibration can be expressed as [20]:  
\[m\ddot{u}+(1+i\eta)ku=f(t), \tag{2}\]  
in which \(m\) and \(k\) are the mass and stiffness, respectively, \(\ddot{u}\) and \(u\) are the complex acceleration and displacement of the SDOF system, respectively, and \(f(t)\) is the real-valued force whose duration is \(T_{p}\).  
To solve Equation (2) in the frequency domain, the duration, \(T_{p}\), is divided into \(N\) equal intervals, \(\Delta t\), where \(N\) equals to \(2^{\
u}\) and \(\gamma\) is an integer from fast Fourier transform (FFT). The discrete time, \(t_{n}\), can then be calculated as \(t_{n}=n\Delta t\) (\(n=0,1,2,\,\ldots,\,N\)). The response of displacement at a discrete time instant \(t_{n}\) is \(u_{n}\). The applied force \(f(t)\) at a discrete time instant \(t_{n}\) is \(f_{n}\). Thus, the discrete Fourier transform (DFT) of force \(f_{n}\), \(F_{j}=F(\theta_{j})\), can be given as:  
\[F_{j}=\frac{1}{N}\sum_{n=0}^{N-1}f_{n}e^{-i\theta_{j}t_{n}},(j=0,1,2,\ldots,N /2), \tag{3}\]  
in which, \(\theta_{j}\) is the discrete frequency which can be obtained as \(\theta_{j}=j\Delta\theta\) with \(\Delta\theta=\frac{2\pi}{N\Delta t}\). In the DFT, the highest effective frequency is the Nyquist frequency \(\theta_{N/2}\). The amplitude \(F_{j}\) of \(j>N/2\) is the conjugate of that of \(j<N/2\) on the Nyquist frequency, that is,  
\[F_{N/2+j}=\overline{F}_{N/2-j},(j=1,2,\ldots,N/2-1), \tag{4}\]  
in which, the superscript \"-\" denotes complex conjugation.  
Under harmonic excitation of \(F_{j}e^{i\theta_{j}t_{n}}(j=0,1,2,\ldots,N/2)\), the displacement amplitude of steady-state response can be expressed as:  
\[U_{j}=H_{j}F_{j},(j=0,1,2,\ldots,N/2), \tag{5}\]  
in which \(H_{j}\) is the discrete form of the complex frequency response function \(H(\theta_{j})\). For a single degree-of-freedom (SDOF) system with complex damping in forced harmonic vibration as shown in Equation (2), the frequency response function \(H_{j}\) can be given as [24]:  
\[H_{j}=\frac{1}{-\theta_{j}^{2}m+(1+i\eta)k}, \tag{6}\]  
With an excitation frequency of \(\theta_{N/2+j}(j=1,2,\ldots,N/2-1)\), the amplitude of the steady-state response can be directly calculated by the conjugate method according to the Nyquist criterion, that is,  
\[U_{N/2+j}=\overline{U}_{N/2-j},\,\,j=1,2,\ldots,N/2-1, \tag{7}\]  
Then based on the conventional frequency domain method, the inverse Fourier transform should be used to evaluate the dynamic response in the discrete form, which can be expressed as:  
\[u_{n}=\sum_{j=0}^{N-1}U_{j}e^{i\frac{2\pi ij}{N}}, \tag{8}\]Considering the conjugate of \(U_{j}\) and \(e^{i\frac{2\pi ni}{N}}\) on the Nyquist frequency, Equation (8) can be rewritten as:  
\[u_{n}=\frac{F_{0}}{(1+i\eta)k}+2\sum_{j=1}^{N/2-1}\text{Re}\left(H_{j}F_{j}e^{ \frac{2\pi ni}{N}}\right)+(-1)^{n}H_{N/2}F_{N/2}, \tag{9}\]  
in which \(F_{0}=\frac{1}{N}\sum\limits_{n=0}^{N-1}f_{n}\), Re denotes the real part of the dynamic displacement response. Specifically, the real and imaginary parts (\(u_{Rn}\) and \(u_{In}\)) of dynamic displacement response can be expressed as:  
\[u_{Rn}=\frac{F_{0}}{(1+\eta^{2})k}+2\sum_{j=1}^{N/2-1}\left(\frac{F_{j}\left[ \left(k-\theta_{j}^{2}m\right)\cos\left(\frac{2\pi ni}{N}\right)+\eta k\sin \left(\frac{2\pi ni}{N}\right)\right]}{\left(k-\theta_{j}^{2}m\right)^{2}+ \eta^{2}k^{2}}\right)+\frac{(-1)^{n}F_{N/2}\left(k-\theta_{N/2}^{2}m\right)}{ \left(k-\theta_{N/2}^{2}m\right)^{2}+\eta^{2}k^{2}}, \tag{10}\]  
\[u_{In}=-\frac{F_{0}\eta}{(1+\eta^{2})k}-\frac{(-1)^{n}F_{N/2}\eta k}{\left(k -\theta_{N/2}^{2}m\right)^{2}+\eta^{2}k^{2}}, \tag{11}\]  
It can be seen from Equations (9)-(11) that the complex dynamic displacement response from Equation (11) is very small. Thus, for the conventional frequency domain method (CFDM), only the real part of the dynamic displacement response is considered using Equation (8).