## 1 Introduction  
Damping is the natural property of a vibration system, which dissipates vibration energy. The widely used models to represent damping in a system are the viscous model and the hysteretic model [1; 2; 3]. Viscous damping, whose force is proportional to the velocity, is the most popular model due to the mathematical convenience. However, it implies a dissipated energy per a vibration cycle depending on the frequency. Hysteretic damping, whose force is proportional to the displacement and in phase with the velocity, leads to a dissipated energy per cycle which is independent of the frequency [4; 5]. Experiments on structural materials indicate that the energy dissipation is independent of the cyclic frequency [6; 7; 8]. In those cases, the hysteretic damping is more precise. However, there are some drawbacks in the solution of the equation of motion with hysteretic damping for both the frequency domain analysis and the direct integration method in the time domain, which hinders the wider application of hysteretic damping models. The frequency domain approach solves the hysteretic damping system by using the Fourier transform for the equation of motion [9; 10; 11], but it has two main drawbacks [12; 13]. (1) Part of transient response may be neglected which may underestimate the response at the initial time and (2) in theory, it is only applicable for the linear elastic system due to the application of the superposition principle, so it cannot be used to analyze the true nonlinear response. For the approximate solution of a nonlinear system, [PERSON] and [PERSON] [14] proposed an equivalent linearization method, which was widely used in the nonlinear seismic response of soil site and soil-structure interaction [15; 16; 17; 18]. In a time-domain analysis, the numerical method is widely used because of its high efficiency, and because it can analyze the nonlinear problems as well [19; 20; 21; 22; 23]. However, the result from the numerical method is divergent when the direct integration method is used to solve the hysteretic damping systems, even if the integration procedure is unconditionally stable [24]. To develop the hysteretic damping in the time domain, a common method is to transfer the hysteretic damping into the approximate viscous damping. [PERSON] [25] used the forcing frequency to replace the natural frequency. Thus, the hysteretic damping matrix can be transferred to a viscous damping matrix, which is independent of frequency. [PERSON] [26] proposed an integral-differential equation (IDE) in the time domain for the free vibration of a single degree-of-freedom (SDOF) system by using viscous damping depending on the frequency to equivalent hysteretic damping. [PERSON] [27] combined the standard linear constitutive models to approximate the hysteretic damping constitutive model in the time domain, and then applied the time-step integral calculation to hysteretic damping model. However, until now, the improvement in the direct integration of a hysteretic damping system is still very slow, and the divergence of the results of the direct integral method has not been solved. In theory, there are two kinds of unstable solutions for the direct integration method. One is caused by the instability of the numerical method, and the other is caused by the divergent solution of the equation itself. The instability of the direct integration method of the hysteretic damping model is caused by the latter one since the pair of eigenvalues of the SDOF hysteretic damping are complex numbers and opposite to each other. The complementary solution of the equation is a linear combination of the two eigenvalue exponential functions. Therefore, the eigenvalue with a positive real part will cause a divergent solution, which is contrary to the natural and physical phenomenon. In the theoretical solution, the divergent term can be arbitrarily omitted, but it is kept in the direct integration procedure.