## The Formulation for the Virtual Initial Condition  
### The Virtual Initial Condition for Free Vibration  
For a system in free vibration, the right-hand member of Equation (1) vanishes and gives  
\[m\ddot{u}+(1+i\eta)ku=0. \tag{7}\]  
The solution for the homogeneous equation is  
\[u=Ce^{\lambda t}, \tag{8}\]  
in which \(C\) is a constant and \(\lambda\) is eigenvalue. Substituting Equation (8) into Equation (7) leads to the algebraic equation:  
\[\lambda^{2}m+(1+i\eta)k=0. \tag{9}\]  
The solutions of Equation (9) are  
\[\lambda_{1}=s,\;\lambda_{2}=-s, \tag{10}\]  
\[s=\omega(a-ib), \tag{11}\]  
where \(\omega=\sqrt{k/m}\) is the undamped natural frequency, \(a=\sqrt{\frac{-1+\sqrt{1+\eta^{2}}}{2}}\), \(b=\sqrt{\frac{1+\sqrt{1+\eta^{2}}}{2}}\). The two eigenvalues are opposite numbers to each other. The general solution of Equation (7) is  
\[u=C_{1}e^{st}+C_{2}e^{-st}. \tag{12}\]The constant \(C_{1}\) and \(C_{2}\) in terms of the real-value initial conditions of Equation (2) may be determined from  
\[C_{1}=\frac{su_{0}+v_{0}}{2s},\;C_{2}=\frac{su_{0}-v_{0}}{2s}. \tag{13}\]  
Since the real part of \(s\) is a positive quantity, there would be a growing response along the time for the right-hand first member \(C_{1}e^{st}\), which is a divergent term which is contrary to the physical rule and should be omitted. Therefore, by introducing \(C_{1}=0\) the correct general solution can be expressed as  
\[u=Ce^{-st}, \tag{14}\]  
where \(C\) is a complex constant. Applying the initial condition of Equation (2) gives  
\[C=u_{0}-i\frac{v_{0}+\alpha au_{0}}{\omega b}. \tag{15}\]  
In the analytical solution, the divergent member can be omitted to obtain the exact solution of Equation (14). However, the direct integration solution converges to Equation (12). Therefore, it is crucial to make Equation (12) equal to Equation (14). In this paper, the virtual initial conditions \(u_{0}^{v1}\) and \(v_{0}^{v1}\) are proposed to achieve this goal. The total initial conditions are the sum of the real initial conditions and corresponding virtual initial conditions, that is  
\[u_{0}^{t}=u_{0}+u_{0}^{v1},\;v_{0}^{t}=v_{0}+v_{0}^{v1}. \tag{16}\]  
Substituting Equation (16) into Equation (13), and making Equation (12) equal to Equation (14), gives  
\[\frac{s(u_{0}+u_{0}^{v1})+(v_{0}+v_{0}^{v1})}{2s}=0, \tag{17}\]  
\[\frac{s(u_{0}+u_{0}^{v1})-(v_{0}+v_{0}^{v1})}{2s}=u_{0}-i\frac{v_{0}+\alpha au _{0}}{\omega b}. \tag{18}\]  
Solving the two algebraic Equations (17) and (18) leads to  
\[u_{0}^{v1}=-i\frac{v_{0}+\alpha au_{0}}{\omega b}. \tag{19}\]  
\[v_{0}^{v1}=i[\frac{a}{b}(v_{0}+\alpha au_{0})+\omega bu_{0}]. \tag{20}\]  
It can be seen that the virtual initial conditions are purely imaginary. The usual real initial conditions are  
\[u_{0}=\text{Re}(u_{0}^{t}),\;v_{0}=\text{Re}(v_{0}^{t}). \tag{21}\]  
It can be observed that the complete initial conditions are complex, but only the real parts can be observed and measured. The corresponding unobserved imaginary part should be the indivisible member of the initial condition. After introducing the virtual initial conditions, the general solution of Equation (7) is naturally exact, the real part of which is the observable solution and free decay.