## 2 Equation of Motion and the Direct Integration Method  
The governing equation of motion of a single-degree-of-freedom (SDOF) system with hysteretic damping in forced vibration can be expressed as [29]  
\[m\ddot{u}+(1+i\eta)ku=f(t), \tag{1}\]  
where \(m\) and \(k\) are the mass and stiffness, respectively, \(\eta\) is the hysteretic damping factor, \(i=\sqrt{-1}\), \(\ddot{u}\) and \(u\) are the acceleration and displacement of the SDOF, respectively, and \(f(t)\) is the applied force. This equation is to be solved for the displacement, \(u(t)\), subjected to the initial conditions as  
\[u(0)=u_{0}\text{ and }\dot{u}(0)=v_{0}, \tag{2}\]  
where \(u_{0}\) and \(v_{0}\) are the displacement and velocity at the time \(t=0\).  
For the direct integration method, let the time axis be divided into small intervals of equal length \(\Delta t\), and let \(t_{n}=n\Delta t(n=0,\,1,\,2,\,\dots\,\,N)\) be the division times. The response of displacement, velocity, and acceleration at the discrete time instants \(t_{n}\) are \(u_{n}\), \(\dot{u}_{n}\), and \(\ddot{u}_{n}\). The applied force \(f(t)\) at \(t_{n}\) is \(f_{n}\).  
So far, several direct integration methods have been proposed to obtain the solution at time \(t_{n+1}\) assuming that the solutions at times \(t_{i}\) (\(i=0,\,1,\,2,\,\dots\,\,n\)) are known. The Newmark method is used to investigate the stability of the solution of the hysteretic damped equation in the following illustration. In the Newmark method, the velocity and acceleration at time \(t_{n+1}\) can be expressed as  
\[\dot{u}_{n+1}=\frac{\gamma}{\beta\Delta t}(u_{n+1}-u_{n})+(1-\frac{\gamma}{\beta })\dot{u}_{n}+(1-\frac{\gamma}{2\beta})\ddot{u}_{n}\Delta t, \tag{3}\]  
\[\ddot{u}_{n+1}=\frac{1}{\beta\Delta t^{2}}(u_{n+1}-u_{n})-\frac{1}{\beta \Delta t}\dot{u}_{n}-(\frac{1}{2\beta}-1)\ddot{u}_{n}, \tag{4}\]  
in which parameter \(\gamma\) and \(\beta\) are parameters that can be determined to obtain integration accuracy and stability. Substituting Equation (4) into Equation (1) gives:  
\[\dot{k}u_{n+1}=\hat{f}_{n+1}, \tag{5}\]  
where \(\dot{k}=(1+i\eta)k+\frac{1}{\beta\Delta t^{2}}m\) and \(\hat{f}_{n+1}=f_{n+1}+[\frac{1}{\beta\Delta t^{2}}u_{n}+\frac{1}{\beta\Delta t }\dot{u}_{n}+(\frac{1}{2\beta}-1)\ddot{u}_{n}]m\). Solve the displacement, \(u_{n+1}\), by Equation (5), then \(\dot{u}_{n+1}\) may be determined from Equation (3) and the acceleration may be determined from the following expression deduced from Equation (1):  
\[\ddot{u}_{n+1}=\frac{f_{n+1}-(1+i\eta)ku_{n+1}}{m}. \tag{6}\]  
It is well known that the Newmark method is unconditionally stable for \(\gamma=\frac{1}{2}\) and \(\beta=\frac{1}{4}\), which is called the average acceleration method. However, the results obtained by the average acceleration method will blow up for the hysteretic damping system, giving meaningless results. The reason is that the solution will converge to the divergent solution which is caused by the divergent member of the complementary solution of Equation (1) if the real-valued initial condition is used directly in the Newmark method. Therefore, in this paper, a modified initial condition in each time step is proposed to address the unstable phenomenon for direct integration.