## The Formulation for the Virtual Initial Condition
### The Virtual Initial Conditions for Harmonic Force  
If \(f(t)\) is a harmonic force with forcing frequency \(\theta\), written as \(Ae^{i\theta t}\), the particular solution of Equation (1) is  
\[u=Xe^{i\theta t}, \tag{22}\]where \(X=\frac{A}{-\theta^{2}m+(1+i\eta)k}\) The complementary solution is the free vibration response given by Equation (12). The complete solution is the sum of the complementary and particular solutions.  
\[u=De^{st}+Ee^{-st}+Xe^{i\theta t}. \tag{23}\]  
The constants \(D\) and \(E\) are determined by the initial conditions. To illustrate conveniently, the virtual initial conditions associated with the real zero initial condition are deduced first. On applying the zero initial condition with \(u(0)=0\) and \(\dot{u}(0)=0\) to Equation (23), the constants \(D\) and \(E\) are deduced.  
\[D=\frac{-s-i\theta}{2s}X,\ E=\frac{i\theta X-sX}{2s}. \tag{24}\]  
Similar to the free vibration, the right-hand first term \(De^{st}\) is the divergent term and contrary to the physical rule, which should be omitted. The complete solution that is satisfied with physical meaning is  
\[u=C_{f}e^{-st}+Xe^{i\theta t}. \tag{25}\]  
On making use of \(u(0)=0\) and \(\dot{u}(0)=0\) to Equation (25), the constant \(C_{f}\) is deduced as  
\[C_{f}=-\text{Re}(X)-i\frac{\theta\text{Im}(X)-a\omega\text{Re}(X)}{b\omega}, \tag{26}\]  
where \(\text{Re}(X)\) is the real part of \(X\), and \(\text{Im}(X)\) is the imaginary part of \(X\).  
The direct integration solution converges to Equation (23), therefore, let Equation (23) be equal to Equation (25) by making use of the virtual initial condition \(u_{0}^{z2}\) and \(v_{0}^{z2}\). Subjected to initial displacement \(u_{0}^{z2}\) and initial velocity \(v_{0}^{z2}\), the constants \(D\) and \(E\) in Equation (23) are determined from  
\[D+E+X=u_{0}^{z2}, \tag{27}\]  
\[sD-sE+i\theta X=v_{0}^{z2}. \tag{28}\]  
Solving the two algebraic Equations (27) and (28) leads to  
\[D=\frac{su_{0}^{z2}+v_{0}^{z2}-sX-i\theta X}{2s},\ E=\frac{su_{0}^{z2}-v_{0}^{ z2}-sX+i\theta X}{2s}. \tag{29}\]  
On making use of the identity between Equations (23) and (25), one obtains  
\[\frac{su_{0}^{z2}+v_{0}^{z2}-sX-i\theta X}{2s}=0,\ \frac{su_{0}^{z2}-v_{0}^{z2}- sX+i\theta X}{2s}=C_{f}. \tag{30}\]  
Then, it leads to  
\[u_{0}^{z2}=C_{f}+X=i\frac{(b\omega-\theta)\text{Im}(X)+a\omega\text{Re}(X)}{b \omega}, \tag{31}\]  
\[v_{0}^{z2}=-sC_{f}+i\theta X=i\frac{a\theta\text{Im}(X)+(b\theta-a^{2}\omega- b^{2}\omega)\text{Re}(X)}{b}. \tag{32}\]  
The virtual initial conditions \(u_{0}^{v2}\) and \(v_{0}^{z2}\) are also purely imaginary. The member \(C_{f}e^{-st}\) in Equation (25) is free adjoint vibration which is independent of the real initial conditions but depends on the load. \(u_{0}^{v2}\) and \(v_{0}^{v2}\) are artificial load-dependent initial conditions which are developed to make the direct integration solution converges to the exact solution.  
Equations (31) and (32) are deduced by the assumption that the system is initially at rest. For a system with non-zero initial conditions, the virtual initial conditions defined by Equations (31)and (32) should be augmented by the addition of the free vibrational solution defined by Equation (19) and (20). Then the total initial conditions of harmonic responses are  
\[u_{0}^{t}=u_{0}+u_{0}^{v1}+u_{0}^{v2},\;v_{0}^{t}=v_{0}+v_{0}^{v1}+v_{0}^{v2}. \tag{33}\]