## The Formulation for the Virtual Initial Condition
### The Virtual Initial Conditions for Arbitrary Force  
Assuming the SDOF system with hysteretic damping is subjected to real-valued force \(g_{1}(t)\), which varies arbitrarily with time, it is necessary to establish a dual force in the time domain analysis [30]. Then, the governing equation of motion can be written as  
\[m\ddot{u}+(1+i\eta)ku=f(t)=g_{1}(t)+ig_{2}(t), \tag{34}\]  
in which \(g_{1}(t)\) is the observable force, and \(g_{2}(t)\) is the corresponding dual force. By using discrete Fourier transforms (DFT), \(g_{1}(t)\) can be expanded to the sum of a series harmonic function, that is  
\[g_{1}(t)=\frac{A_{0}}{2}+\sum_{j=1}^{N/2-1}\Big{(}A_{j}\cos\theta_{j}t+B_{j} \sin\theta_{j}t\Big{)}+\frac{A_{N/2}}{2}\cos\theta_{N/2}t, \tag{35}\]  
where \(\theta_{j}=\frac{2\pi j}{N\Delta t}\) (\(j=0,1,2,\ldots,N/2-1\),\(N/2\)), \(\Delta t\) is time intervals of discrete, Fourier coefficients \(A_{0}\), \(A_{1}\), \(A_{2}\), \(\ldots,A_{N/2}\) and \(B_{1}\), \(B_{2}\), \(\ldots,B_{N/2-1}\) can be determined from  
\[A_{j}=\frac{2}{N}\sum_{l=0}^{N-1}g_{1}(t_{l})\cos(\theta_{j}t_{l}),j=0,1,2, \ldots,N/2-1,N/2, \tag{36}\]  
\[B_{j}=\frac{2}{N}\sum_{l=0}^{N-1}g_{1}(t_{l})\sin(\theta_{j}t_{l}),j=1,2, \ldots,N/2-1, \tag{37}\]  
where \(t_{i}=l\Delta t\) (\(l=0,1,2,\ldots,N-1\)) is the discrete value of \(t,g_{1}(t_{l})\) is the discrete value of \(g_{1}(t)\). Then dual loading \(g_{2}(t)\) can be expressed as  
\[g_{2}(t)=\eta\frac{A_{0}}{2}+\sum_{j=1}^{N/2-1}\Big{(}A_{j}\sin\theta_{j}t-B_ {j}\cos\theta_{j}t\Big{)}+\frac{A_{N/2}}{2}\sin\theta_{N/2}t. \tag{38}\]  
So,  
\[f(t)=\frac{A_{0}}{2}(1+i\eta)+\sum_{j=1}^{N/2-1}\Big{(}A_{j}-iB_{j}\Big{)}e^{i \theta_{j}t}+\frac{A_{N/2}}{2}e^{i\theta_{N/2}t}. \tag{39}\]  
Equation (39) denotes that arbitrary force can be expressed as the sum of a series of harmonic forces, so the complete solution is the sum of a series of harmonic forces. \[u=\sum_{j=0}^{N/2}\Big{(}D_{j}e^{st}+E_{j}e^{-st}+X_{j}e^{i\theta_{j}t}\Big{)}. \tag{40}\]  
The right-hand first term \(D_{j}e^{st}\) is a divergent term and contrary to the physical rule and should be omitted, so the complete solution which is satisfied with physical meaning is  
\[u=\sum_{j=0}^{N/2}\Big{(}C_{f}e^{-st}+X_{j}e^{i\theta_{j}t}\Big{)}, \tag{41}\]where \(X_{0}=\frac{A_{0}}{2k}\), \(X_{j}=\frac{A_{j}-iB_{j}}{-\theta_{j}^{2}m+(1+i\eta)k}(j=1,\,2,\,\ldots,\,N/2-1)\), and \(X_{N/2}=\frac{A_{N/2}}{2(-\theta_{N/2}^{2}m+(1+i\eta)k)}\). On making use of \(u(0)=0\) and \(i(0)=0\) to Equation (41), the constant \(C_{\bar{f}j}\) is deduced. \[C_{\bar{f}j}=-\text{Re}(X_{j})-i\frac{\theta_{j}\text{Im}(X_{j})-a\omega\text{ Re}(X_{j})}{b\omega}\ (j=\ 0,\ 1,\ 2,\ \ldots,\ N/2). \tag{42}\]  
Similar to the harmonic force, the virtual initial condition associated with each term can be determined from Equations (31) and (32) for the zero initial condition. Therefore, the virtual initial condition for arbitrary force can be expressed as  
\[u_{0}^{v3}=\sum_{j=0}^{N/2}\Bigl{(}C_{\bar{f}j}+X_{j}\Bigr{)}, \tag{43}\]  
\[g_{1}(t)=\frac{A_{0}}{2}+\sum_{j=1}^{N/2-1}\Big{(}A_{j}\cos\theta_{j}t+B_{j} \sin\theta_{j}t\Big{)}+\frac{A_{N/2}}{2}\cos\theta_{N/2}t, \tag{35}\]0  
Then the total initial conditions for a system with non-zero initial conditions are  
\[g_{1}(t)=\frac{A_{0}}{2}+\sum_{j=1}^{N/2-1}\Big{(}A_{j}\cos\theta_{j}t+B_{j} \sin\theta_{j}t\Big{)}+\frac{A_{N/2}}{2}\cos\theta_{N/2}t, \tag{35}\]1  
Algorithm 1 summarizes the above-described procedure, which can be implanted into the computer. The procedure can also be used to calculate the free vibration and harmonic force with slight modification. Let \(f_{n}=u_{n}^{v3}=v_{n}^{v3}=0\) for free vibration. The harmonic force is the particular example of the arbitrary force, which only keeps one term with the with forcing frequency \(\theta\)to calculate \(u_{n}^{v3}\) and \(v_{n}^{v3}\). It is noted that step 2.4 is used to calculate the virtual initial condition for each time step \(n\) to eliminate the effect of the round-off and improve the accuracy.