## 1 Introduction  
In structural dynamics, the damping force of a complex damping system [1; 2; 3] which is also known as hysteretic damping, rate-independent damping, structural damping, \(f_{\mathrm{d}}\), can be calculated as:  
\[f_{\mathrm{d}}=-i\eta ku, \tag{1}\]  
in which \(i=\sqrt{-1}\) is a pure imaginary number, and \(k\) is the stiffness of the structure, \(\eta\) is the complex damping factor, and \(u\) is the displacement. From Equation (1), it can be seen that the energy dissipation of the complex damping is independent of the excitation frequency, which is satisfied with the energy dissipation characteristics of many materials and has been widely used. Taking soil as an example, overserved from numerous dynamic triaxial tests, the dissipated energy per cycle of many kinds of soil is almost independent of the loading frequency [4; 5]. Laboratory shaking table test results of a single-layer reticular dome [6] showed that comparing with the identified damping ratios of the first ten vibration modes, complex damping model has smaller errors than Rayleigh damping model. The generalized complex damping is used to uncouple the coupled damping matrix caused by the offshore and added hydrodynamic damping [7]. In addition, the damping of sandwich-like plate and beam with viscoelastic cores is usually also expressed as complex damping [8; 9]. For the forced vibration of system with complex damping, two methods are commonly used to obtain the steady-state responses of such a system, including the conventional frequency domain method (CFDM) and the dual-force-based time domain method (DTDM). The seismic response of soil layer and soil-structure interaction [10; 11; 12] are often calculated by the CFDM due to the complex damping of soil deposit. For example, the CFDM has been used to compute the nonlinear seismic response of one-dimensional soil layers [13; 14], two-dimensional irregular-shaped basins with horizontal soil layers [15], and a hill in a layered elastic half space [16]. The CFDM also was used to analyze complex damping system under forced vibrations of axial and torsional forces [17]. In theory, the differential equation with complex coefficients can be equivalent to two differential equations with real coefficients which use the real part and imaginary part of solution as variable. The real and imaginary parts of the solution not only affect each other but also have a dual relationship. The real parts of displacement response and force are observed and measured. Although the imaginary part is not the actual process, it is the complete condition for the equation with the complex constitutive. In the process of establishing and solving the equation, the influence of the imaginary part should be considered, although in most of the current engineering applications, the imaginary part may not be necessary to be applied due to complexity in solving the imaginary part. Therefore, methods using frequency domain which can only obtain the real part of the solution satisfy the engineering demand. To obtain both the real and imaginary parts of the solution, the DTDM can be applied. Previous research demonstrated that the solution of differential equation with complex coefficients in the time domain analysis under the real-valued force is inaccurate, and the imaginary part of the complex force corresponding to real-valued force should be established according to the dual principle [18]. Thus, the calculation method of dual force was proposed to be improved [19]. However, since the eigenvalues of the complex damping system are complex eigenpairs that are opposite numbers to each other [20], it is inevitable that the real part of one eigenvalue is positive. Since the real part of eigenvalue is a positive quantity, the complementary solution would be divergent, resulting in contrary to the physical rule. For an analytical solution, the divergent term can be manually deleted to make the calculation result stable.