The transient responses include exponential growth terms and the time-domain responses are divergent based on the complex damping constitutive model. This study aims to propose a convergent timedomain direct integration method. Based on the relationship of the displacement and velocity, an improved time-domain complex damping motion equation is obtained. Combined with Newmark-β method, an improved time-domain direct integration method of complex damping constitutive model is proposed for single-degree-of-freedom (SDOF) systems and multi-degree-of-freedom (MDOF) systems. The harmonic vibration responses of SDOF systems show that the proposed method can eliminate the divergent components of the responses within each time step, and the time-domain results are convergent. The random excitation responses of SDOF and MDOF systems show that the calculation results with the proposed method are approximately equal to those with the frequencydomain method for the systems with large natural frequencies. The relative error of peak displacements with the frequency method may be greater than 10% for the systems with small natural frequencies. The improved time-domain direct integration method can overcome the divergent limitation and obtain the stable time-domain responses. Besides, compared with the frequency-domain method, the proposed method can further consider the influence of the transient responses. The calculation accuracy is improved.
Shan et al. (Thu,) studied this question.